Guidelines for Structural Control

Transcription

1 SAMCO Final Reort 6 Guidelines for Structural Control by Felix Weber, Glauco Feltrin, and Olaf Huth Structural Engineering Research Laboratory, Swiss Federal Laboratories for Materials Testing and Research Dübendorf, Switzerland Dübendorf, March 6 Page of 55

2 SAMCO Final Reort 6 CONTENTS LIST OF FIGURES...4 LIST OF TABLES...9 INTRODUCTION.... GOAL.... DEFINITION OF THE TERM STRUCTURAL CONTROL....3 STRUCTURE OF GUIDELINES... INCREASING DAMPING.... DAMPING DEVICES..... Passive damers Linear viscous damer Nonlinear viscous damer Viscoelastic damer Coulomb friction damer Structural friction damer Hysteretic damer Shae memory alloy damer Passive tuned mass damer Tuned liquid damer Semi-active devices Defintion of term semi-active Magnetorheological/electrorheological fluid damer Controlled shae memory alloy Actuators Actuators for increase of daming Hydraulic aggregate Piezo actuator CONTROL ALGORITHMS Control strategies Feedback control Feed forward control Passive control References Notations Active daming with collocated airs Linear comensators Active daming with collocated actuator sensor airs References Notations Otimal control State sace reresentation Linear quadratic regulator Linear Quadratic Gaussian noise control References Notations Other control algorithms Page of 55

3 SAMCO Final Reort Fuzzy control Neural network control References Notations VIBRATION ISOLATION PASSIVE VIBRATION ISOLATION Theoretical background Design issues Low tuning High tuning Base isolation Testing and validation Imlementation References Notations ACTIVE VIBRATION ISOLATION Scoe Passive isolator Active isolator Velocity feedback Force feedback Flexible clean body References Notations Page 3 of 55

4 SAMCO Final Reort 6 LIST OF FIGURES Figure : Linear viscous daming: a) force dislacement and b) force velocity trajectories for eriodic excitation... Figure : Nonlinear viscous daming at small velocities due to seal friction.... Figure 3: Viscous daming device (Connor ())...4 Figure 4: Viscous damer, 45 kn (htt:// Figure 5: Schematic test set-u...5 Figure 6: Measurement of damer behaviour at Ema...5 Figure 7: Mean ower equivalent nonlinear viscous damer...8 Figure 8: Behaviour of: a) ure elastic materials, b) and c) of viscoelastic materials.... Figure 9: Examle of a viscoelastic daming device according to Connor ().... Figure : Single bearing test machine (Aiken et al. (989))....3 Figure : Cyclic hysteresis loo of a bolted rubber bearing (Aiken et al. (989))....3 Figure : Bearing system design roosed by Aiken et al. (989)...4 Figure 3: Coulomb friction: a) force dislacement trajectory, b) force velocity trajectory; c) force velocity trajectory of real Coulomb friction damers (e.g. MR damers at constant current)....6 Figure 4: Limited Sli Bolted Joints (Pall et al. (98))....8 Figure 5: X-braced Friction Damer (Pall and Marsh (98))....8 Figure 6: Sumitomo Friction Damer (Aiken and Kelly (99))...9 Figure 7: Energy Dissiating Restraint (EDR) (Nims et al. (993))...9 Figure 8: Structural daming: a) force dislacement trajectory and b) force velocity trajectory...3 Figure 9: Hysteretic daming: a) stress strain relation, b) real hysteretic damer behaviour, c) idealized hysteretic damer behaviour...33 Figure : Ideal elastic-lastic daming device according to Jones ()...34 Figure : X-shaed Plate Damer...34 Figure : Stress strain curves and transformation temerature rofiles for actuator like change of strain and stiffness at constant stress (Janke et al. (5))...39 Figure 3: Stress strain curves and transformation temerature rofiles for shae memory effect in the case of free strain recovery and constraint strain recovery (Janke et al. (5))....4 Figure 4: Stress strain curves and transformation temerature rofiles for the so-called suerelastic behaviour (Janke et al. (5))....4 Figure 5: Stress strain curves and transformation temerature rofiles for martensitic hysteretic daming (Janke et al. (5))...4 Figure 6: Hysteresis loos for: a) suerelastic SMA behaviour and b) martensitic hysteretic daming (Janke et al. (5))....4 Figure 7: Torsion bar design (Witting and Cozzarelli (99))...4 Figure 8: Stress strain diagram used for bilinear damer model (Witting and Cozzarelli (99)) Figure 9: Force dislacement loos (Witting and Cozzarelli (99)) Figure 3: Tyical imlementation of a TMD for vertical vibration mitigation of bridges...48 Figure 3: Two degree of freedom model of a TMD attached to a rimary structure. Left: Excitation force f ( t ) acting on the rimary mass. Right: Excitation through base acceleration u&& g ( t)...49 Figure 3: Methods for H norm otimization of a TMD alying the fixed oints method...5 Figure 33: Methods for H norm otimization of a TMD alying the real roots method....5 Figure 34: Effects of oor frequency tuning of a TMD on the dynamic magnification factor of the dislacement of the rimary system Page 4 of 55

5 SAMCO Final Reort 6 Figure 35: Effects of oor daming tuning of a TMD on the dynamic magnification factor of the dislacement of the rimary system...5 Figure 36: Otimal TMD frequency for minimizing the dislacement of the rimary system with resect to the H norm Figure 37: Otimal TMD daming ratios for minimizing the dislacement of the rimary system with resect to the H norm Figure 38: Maximum of the non-dimensional transfer function of the dislacement of the rimary system for arameters minimizing the dislacement of the rimary system with resect to the H norm Figure 39: Maximum of the non-dimensional transfer function of the relative dislacement for arameters minimizing the dislacement of the rimary system with resect to the H norm Figure 4: Mitigation factor of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm Figure 4: Amlification due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm ( δ = ηt / ηt,, µ =. )...53 Figure 4: Normalized variance of the dislacement of the rimary system...55 Figure 43: Otimal TMD frequency for minimizing the variance of the dislacement of the rimary system with resect to the H norm...56 Figure 44: Otimal TMD daming ratios for minimizing the variance of the dislacement of the rimary system with resect to the H norm...56 Figure 45: Minimum of the non-dimensional variance of the dislacement of the rimary system for arameters minimizing the variance of the dislacement of the rimary system with resect to the H norm Figure 46: Non-dimensional variance of the relative dislacement of the mass of the TMD for arameters minimizing the variance of the dislacement of the rimary system with resect to the H norm...56 Figure 47: Mitigation factor of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm...56 Figure 48: Amlification due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm ( δ = ηt / ηt,, µ =. )...56 Figure 49: Schematic illustration of a sloshing (a) and a column tuned liquid damer (b) couled to a rimary structure...67 Figure 5: Princial arameters of sloshing and column TLD...67 Figure 5: Definitions and characteristics of daming devices often used in structural control.7 Figure 5: Rotational MR damer, force range aroximately [, 3] N...73 Figure 53: Cylindrical MR damer with gas ocket (accumulator) for comensation of iston volume; force range aroximately [7, 8] N Figure 54: Schematic steady-state damer behaviour of MR damers...74 Figure 55: Measured MR damer behaviour in re- and ostyield regions (MR damer of Maurer Söhne)...74 Figure 56: Force dislacement trajectories including transient measurement data from reyield to ostyield regions (MR damer of Maurer Söhne) Figure 57: Simle Bouc-Wen modelling aroach for MR-damers...76 Figure 58: Fitted inverse MR damer function Figure 59: Testing daming devices...77 Figure 6: Transient measurement data due to MR damer behaviour in reyield and ostyield regions with force overshoot, fairly small MR fluid viscosity...77 Figure 6: Transient measurement data due to MR damer behaviour in reyield and ostyield regions, no force overshoot, fairly large MR fluid viscosity Page 5 of 55

6 SAMCO Final Reort 6 Figure 6: MR damer force resonse on aroximate velocity ste at A Figure 63: MR damer force resonse on aroximate velocity ste at 4 A Figure 64: Cable-stayed Dongting Lake Bridge in China, equied with MR damers of LORD Cororation...79 Figure 65: Cable-stayed Eiland Bridge nearby Kamen, The Netherlands, equied with one MR damer of Maurer Söhne for long-term field tests and daming measurements...79 Figure 66: Cable-stayed Dubrovnik Bridge nearby Dubrovnik, Croatia....8 Figure 67: Flow chart of damer design rocedure...8 Figure 68: Cable/damer system with distributed disturbance force according to the shae of the excited mode (here shown for the first vibration mode)....8 Figure 69: Characteristics of linear viscous and friction damers...83 Figure 7: Force dislacement trajectories of MR damer of the Eiland Bridge Figure 7: Force current relation of MR damer of the Eiland Bridge Figure 7: Instrumentation of MR damer on the Eiland Bridge nearby Kamen, The Netherlands Figure 73: Taking decay measurements on the Eiland Bridge nearby Kamen, The Netherlands Figure 74: Sketch of decay measurements on the Eiland Bridge nearby Kamen, The Netherlands Figure 75: Claming of cables...86 Figure 76: Measured vibration decay at % cable length and damer osition with MR damer at A...86 Figure 77: Measured vibration decay at % cable length and damer osition with MR damer at.4 A...86 Figure 78: Measured vibration decay at % cable length and damer osition with MR damer at. A...87 Figure 79: Measured vibration decay at % cable length and damer osition of free cable.87 Figure 8: Exonential aroximation of linear decay within the amlitude range from 95% to 5% Figure 8: Evaluated mean logarithmic decrement...89 Figure 8: Schematic controller structure...89 Figure 83: Dislacement sensor and MR damer on the Eiland Bridge nearby Kamen, The Netherlands....9 Figure 84: Solar anel and controller box mounted on the lightning ylon on the Eiland Bridge nearby Kamen, The Netherlands...9 Figure 85: Emulation of desired daming characteristics using feedback controlled actuators.9 Figure 86: Characteristics of real linear viscous damers....9 Figure 87: Stress-strain curves for the suerelastic behaviour of SMAs at different temeratures Figure 88: Different moduli of elasticity for martensite and austenite Figure 89: Hydraulic aggregate, mounted on a steel frame.... Figure 9: Schematic of a iezo-electric single layer element (htt:// Figure 9: Schematic of an axially acting multilayer iezo stack (htt:// Figure 9: Schematic of a stacked ring actuator (htt:// Figure 93: Schematic of an actuator with restress (htt:// Figure 94: Schematic of a iezo actuated system (htt:// Figure 95: Schematic voltage/stroke diagram of a stack actuator (htt:// Figure 96: Schematic voltage force relation of a iezo stack (htt:// Page 6 of 55

7 SAMCO Final Reort 6 Figure 97: Maximum iezo actuator stroke versus maximum (blocking) iezo actuator force (htt:// Figure 98: Aroximately linear relation between stroke and charge content of iezo actuators (htt:// Figure 99: Cree of iezo actuators (htt:// Figure : Closed-loo structure of feedback control...3 Figure : Closed-loo structure of feedback control including actuator and sensor dynamics....3 Figure : Structure of feed forward control...4 Figure 3: Oen-loo structure of assive control....4 Figure 4: Structure of arallel PID controller with inverse lant model NL Figure 5: Bode diagram of P, I, D elements, and of low ass filter of nd order...7 Figure 6: Bode diagram of a arallel PID controller with low ass filter of nd order for a good roll-off....8 Figure 7: Characteristics of Lead and Lag elements shown in the Bode Diagram....8 Figure 8: Plant of 3rd order with roortional gain only... Figure 9: Root Locus of 3rd order lant with feedback roortional gain K P only.... Figure : Closed-loo structure of direct velocity feedback for collocated actuator sensor airs... Figure : Four mass system... Figure : Transfer function of four mass system with force inut and velocity outut at mass no... Figure 3: Placement of closed-loo ole for maximum additional daming of mode (direct velocity feedback)... Figure 4: Comarison between very low damed lant and lant with otimally damed mode...3 Figure 5: Location of closed-loo oles of all four modes for maximum daming of mode...3 Figure 6: State sace reresentation of the lant....9 Figure 7: Plant in SSR, all state variables measured, full state feedback with stochastic LQ Regulator...9 Figure 8: Real lant with full state observer and regulator...35 Figure 9: Observer and regulator design...35 Figure : Schematic oerating range of magnetorheological fluid damers...36 Figure : Inverted endulum on a cart...4 Figure : Structure of a fuzzy controller...4 Figure 3: The stes of fuzzification, inference, and defuzzification...4 Figure 4: Black box identification by training of neural networks Figure 5: Neural network consisting of neurons (weights and transfer function) and layers Figure 6: Schematic reresentation of vibration isolation systems. Left: Isolation reducing the effects of the excitation force acting on the structure. Right: Isolation reducing the effects of a forced excitation of the suort Figure 7: Force transmissibility for an isolator consisting of a linear sring and dashot acting in arallel for different daming ratios Figure 8: Rubber bearing alied for base isolation Figure 9: Shear dislacement test of a rubber bearing for base isolation Figure 3: i) Oerating system () generating disturbance forces. ii) Sensitive device () excited by vibrating suort structure (). iii) Active vibration isolation with velocity feedback and force actuator ( skyhook damer ) Figure 3: Magnitude of transmissibility for different daming ratios Page 7 of 55

8 SAMCO Final Reort 6 Figure 3: i) Velocity feedback for flexible clean body. ii) Force feedback for flexible clean body Page 8 of 55

9 SAMCO Final Reort 6 LIST OF TABLES Table : Non-dimensional transfer functions for various excitations and resonse arameters....6 Table : Aroximate solutions of the H otimization roblem for rimary systems with vanishing structural daming ( ζ = )...63 Table 3: Closed form exressions of the non-dimensional H norm for various excitations and resonse arameters Table 4: Exact solutions of the H otimization roblem for rimary systems with vanishing structural daming ( ζ = )...65 Table 5: Aroximate solutions of the H otimization roblem for rimary systems with structural daming ( ζ > )...66 Table 6: Proerties of longest stay cable of Eiland Bridge Table 7: Possible fuzzy rules for inverted endulum...4 Table 8: Fuzzy logic oerations...4 Table 9: Proerties of critically damed active isolator Page 9 of 55

10 SAMCO Final Reort 6 INTRODUCTION. Goal The goal of this guideline is to give an overview of daming devices and control algorithms that are often used in the field of structural control of civil structures. From this oint of view, the guideline may reresent a helful document and a source of ideas and references for engineers and scientists working in this research area.. Definition of the term Structural Control This guideline focuses on Structural Control of civil and large scale mechanical structures but not on macro or micro sized structures. Prestressing of structures using strain actuators in order to roduce a desired shae or restress of the structure will not be treated within this document although shae control belongs to the wide field of structural control. The resented guideline describes the common daming devices and control methods alied for suression of undesired structural vibrations that occur after the erection of the structure. Hence, this guideline describes the tools for vibration mitigation of civil structures..3 Structure of guidelines In the second chater, the guideline describes control devices and control methods that are imlemented in order to increase the daming of structures. This is the case when additional damers or actuators, resectively, are attached to vibrating structures and are controlled by a control law. The control algorithms may be art of an oen loo system or may work in a closed loo, which allows for controlling the device force according to the actual vibration state. In contrast to the second chater, the third chater gives an overview of vibration isolation systems. Here, the target is not to enhance the structural daming using external daming devices but to isolate sensitive structures from disturbances. This chater is divided into two arts. First, assive isolators are described, then, active vibration isolation is resented. Page of 55

11 SAMCO Final Reort 6 INCREASING DAMPING Within the first subchater, an overview of existing daming devices is given. These daming devices include assive damers, controllable damers, and actuators. Passive damers are characterized by constant daming roerties given by their materials and construction. Controllable damers are often called semi-active devices since their daming force may be adjusted or controlled, resectively. Actuators comrise such devices that are able not only to dissiate but also to generate energy. Thus, actuators are able to ut energy into the vibrating structure. The second subchater describes the control algorithms that are usually alied when controllable damers and actuators are used for vibration control. First, the main control strategies such as feedback, feed forward and assive control are shortly introduced. Then, active daming with collocated actuator sensor airs is described, followed by the otimal control aroach within the third subchater. Finally, the most common, other linear control aroaches used for structural control are resented.. Daming devices In the following, the hysics of the most common daming devices are described. The daming devices are slit u in the sections assive daming devices, semiactive daming devices, and actuators... Passive damers... Linear viscous damer Theoretical background The equation of motion of a single degree of freedom system (SDOFS) with external damer force f d and external disturbance force f is w m & x + c x& + k x = f d + f w () where m denotes the SDOFS mass, c the SDOFS eigendaming and k the SDOFS stiffness. The viscous damer is a hyothetical device that creates a daming force which is directly roortional to the SDOFS velocity but with oosite sign (Fig. ) f d = c x& () The roortionality factor c d is the daming coefficient of the external viscous damer. The reason why ure linear viscous damers may be a referable daming device demonstrates the following equation, which results from the substitution of Eq. () in Eq. () d ( c + cd ) x& + k x = f w m & x + (3) 3 This equation is again linear and therefore analytically solvable. Moreover, the main benefit of external viscous damers is that they seem to enhance the material daming of the structure, in this case of the SDOFS, which is in good aroximation ure viscous daming. The external viscous damer cannot comensate for the distur- ctot Page of 55

12 SAMCO Final Reort 6 bance force f since the disturbance force w f w reresenting excitation forces caused by wind, earthquakes and other imacts is unknown. Viscous daming devices are based on material daming which may be assumed to behave linear viscous. However, it must be mentioned that most damers classified as viscous damers do not behave comletely linear over the entire velocity range due to sealing friction and nonlinear material behaviour which ends u in a nonlinear viscous behaviour at small velocities (Fig. ). The work W dissiated within the time interval [ t,t ] can be formulated as t = W f d x& dt (4) For eriodic excitation, the energy dissiated er cycle becomes (Bachmann et al. (995), Weber ()) max t W = π f x = π c x (5) max d max Fig. : Linear viscous daming: a) force dislacement and b) force velocity trajectories for eriodic excitation. Fig. : Nonlinear viscous daming at small velocities due to seal friction. Design issues In ractice, in order to roduce daming devices with almost linear viscous characteristics, damers on the base of fluids are roduced. Basic rinciles of balance laws and exlanations about inviscid fluids, Newtonian fluids, and temerature deendent Newtonian fluids are given in the books of Soong and Dargush (997) and Constantinou et al. (998). A ossible design for a viscous daming device is shown in Fig. 3 (Connor ()). The ga between lunger and external lates is filled with a linear viscous fluid which can be characterized by τ G & γ (6) = v Page of 55

13 SAMCO Final Reort 6 The variables τ and γ& are the shear stress and strain rate, resectively, and G v is the viscosity coefficient. In case that sli does not exist between fluid and lunger, the shear strain is related to the lunger motion by x γ = (7) where t d reresents the thickness of the viscous layer. With L as the wetted length and w as the width of the lunger, the daming force may be written as Substituting τ by Eq. (6) and γ& by Eq. (7) in Eq. (8) yields t d f d = wlτ (8) f d wlg = td v x& (9) According to Eq. (), the viscous daming coefficient of the daming device becomes c wlg v d = () td Hence, the design arameters of such linear viscous damers based on fluids are: thickness t d of the viscous layer, the wetted length L, the width w of the lunger, and the fluid viscosity coefficient G v. The design of viscous damers (Fig. 3, Fig. 4) in order to enhance the overall viscous daming of, e.g., civil structures, comrises the following stes and may need iterations to meet the requirements:. Determination of structural roerties of the civil structure and analysis of the structure.. Determination of the desired daming ratio. 3. Satial lacement of the daming devices. 4. Design of the viscous fluid damer. 5. Estimation of structural daming ratio. 6. Analysis of the overall structural behaviour with the enhanced daming ratio. Page 3 of 55

14 SAMCO Final Reort 6 Fig. 3: Viscous daming device (Connor ()). Fig. 4: Viscous damer, 45 kn (htt:// m). If the damers are well distributed over the structure, the design rocess can be reduced rimarily to the selection of the desired daming ratio. The otimal locations are found if the results of structural analysis show that the targeted mode/modes are reduced maximally in amlitude. However, to find the otimal damer locations is a challenging and nontrivial issue. Since the imlementation of damers requires bracings, the structural stiffness and therefore its resonance frequencies are also affected. The design methodology has to be modified in resence of nonlinearities. The hysical roerties of thick viscoelastic fluids are often quite sensitive to temerature changes. Therefore, temerature comensation is often included in the form of a bi-metallic orifice working similarly to a thermostat (Fig. 4). The thermal stability is combined with the steel construction that has internally threaded joints and no welded or bonded arts. A critical issue in the design of orifice damers concerns the durability of the high strength seals. Additionally, these of the high strength seals may be resonsible for nonlinear viscous daming at small iston velocities since the sealing friction behaves like the force offset of friction damers (Fig. ). Usually, tyical buildings have internal critical structural daming of % to 3%. The critical daming of buildings should be increased by external viscous damers to aroximately % to 5% in order to guarantee that the building is not suscetible to vibrations of large amlitudes anymore. As a comarison, most conventional assenger cars use damers with % to 3% ercent of critical daming. Testing and validation The damer force characteristics are obtained with a simle testing rocedure. The daming device is clamed between an actuator and a fixed anel, with a force transducer in series connected (Fig. 5, Fig. 6). Due to the sinusoidal dislacement control of the actuator, the daming device is tested from zero velocity to maximum velocity. The first derivation of the measured dislacement delivers the velocities. In conjunction with the measured force, the daming characteristics of the tested device are known. In case that the force transducer is moving together with the daming device, the measured force value has to be comensated for the inertia term due to the sensor s mass. Page 4 of 55

15 SAMCO Final Reort 6 Fig. 5: Schematic test set-u. Fig. 6: Measurement of damer behaviour at Ema. There exist damers like, e.g., the orifice fluid damer (Constantinou and Symans (993)), which behave as linear viscous damers if oerating at low frequencies. At high frequencies, many damers based on fluids show also elastic force values caused by the fluid stiffness. These damers are then called viscoelastic damers (Jones (), see chater...3). Imlementation Viscous fluid damers have been successfully used for structural control in the fields of: Automotive industries. Bridge design, foundation of suerstructures. High buildings for the alleviation of wind and seismic vibrations (Martinez-Romero and Romero (3)). References Bachmann, H., et al. (995), Vibration Problems in Structures: Practical Guidelines, Birkhäuser Verlag Basel, ISBN Connor, J. J. (), Introduction to Structural Motion Control, htt:// Constantinou, M. C., and Symans, M. D. (993), Exerimental Study of Seismic Resonse of Buildings with Sulemental Fluid Damers, Structural design Tall Buildings,. Constantinou, M. C., Soong, T. T., and Dargush, G. F. (998), Passive Energy Dissiation Systems for Structural Design and Retrofit, Multidiscilinary Center for Earthquake Engineering Research. Jones, D. I. G. (), Handbook of Viscoelastic Vibration Daming, John Wiley & Sons LTD. Page 5 of 55

16 SAMCO Final Reort 6 Martinez-Romero, M., and Romero, M. L. (3), An otimum retrofit strategy for moment resisting frames with nonlinear viscous damers for seismic alications, Engineering Structures, 5, Soong, T. T., and Dargush, G. F. (997), Passive Energy Dissiation Systems in Structural Engineering, John Wiley & Sons, New York. Weber, B. (), Daming of vibrating footbridges, Proceedings of the International Conference on Footbridge, Paris, France, - November, on CD, AFGC OTUA (eds.). htt:// Notations Symbol Descrition Unit G v viscosity coefficient Ns/m L length m W work, energy J c viscous daming coefficient kg/s f force N k stiffness kg/s m mass kg t thickness m w width m x dislacement m γ shear strain - τ shear stress N/m Subscrits d max tot w damer maximum total disturbance Page 6 of 55

17 SAMCO Final Reort 6... Nonlinear viscous damer Theoretical background The force follows f d of a damer with nonlinear viscous daming behaviour may be exressed as f d = cd nonlin η u& sgn(x) & () where x& is the relative velocity between the two damer ends, c is the viscosity of the nonlinear damer and η is a frictional ower law exonent (Fig. 7, Tsai and Poov (998), Weber ()). The case of η = describes the linear viscous damer behaviour. Velocity exonents smaller than one are often chosen because of the fast increase of the daming force at low velocities (Fig. ). The main drawback of this kind of damer is that it may clam the structure at small vibration velocities due to the relatively large damer force in the case of η <. If the structure is clamed at damer osition, damer relative dislacement and therefore energy dissiation become zero. Then, the structure is undamed and the structure at damer osition becomes a nodal oint. Consequently, the entire excitation energy excites the rest of the structure. d nonlin Design issues In the case of linear viscous damers, the otimum viscosity may be determined using the Matlab tool root locus (Preumont ()), which determines the otimum viscosity deending on: the structural roerties, damer osition, and the target mode that shall be damed maximally. However, this tool fails for nonlinear viscous damers. In this case, two ossible design strategies are resented. Martinez-Rodrigo and Romero (3) roosed a trial and error method, where the maximum damer force of the nonlinear viscous damer is chosen to be smaller than its counterart of the linear viscous damer. The relation is exressed by the so-called force index FI. f = d nonlin f d lin max max * FI [%] () In a first ste, the maximum required daming force is estimated assuming linear viscous daming. Then, the maximum force of the nonlinear viscous damer is determined using Eq. (). As a next ste, the structure is damed with the nonlinear viscous damer with a chosen value of the nonlinear exonent η. Both damer arameters, FI and η (note: the arameter c lin is already given by the otimum linear viscous damer, Krenk ()), are varied within an otimization rocedure until the otimization criterion is fulfilled. Possible otimization criteria the may be: maximum daming of one target mode, maximum daming of several target modes, or minimum ti dislacement of a high rise building for a given excitation time history (earthquake). Page 7 of 55

18 SAMCO Final Reort 6 Fig. 7: Mean ower equivalent nonlinear viscous damer. Another design rocedure is roosed by Pekcan et al. (999). The idea is to determine a mean ower equivalent nonlinear viscous damer based on the otimum linear viscous damer. The damer energy er cycle is W T d = f d x& dt (3) The mean ower over one cycle is the first time derivative, hence d Td Td dw d 4 P = = f d x dt = f d dx = f d dx = dt dt & & 4 & 4 T ( grey area) (4) The mean ower over one cycle of the otimal linear viscous damer is P lin c = 4 ot d lin x& lin max The mean ower over one cycle of the nonlinear viscous damer is P nonlin c = 4 d nonlin η + x& η + nonlin max (5) (6) Equating both mean ower values yields the tuning of mean ower equivalent nonlinear viscous damers c d nonlin η+ x& η + nonlin max c = d lin x& lin max (7) Page 8 of 55

19 SAMCO Final Reort 6 The formula shown in Eq. (7) indicates that the mean ower equivalent nonlinear viscous damer still needs the choice of one of its design arameters and the assumtion of equal maximum damer velocities. References Krenk, S. (), Vibrations of a Taut Cable With an External Damer, Journal of Alied Mechanics, 67, Martinez-Romero, M., and Romero, M. L. (3), An otimum retrofit strategy for moment resisting frames with nonlinear viscous damers for seismic alications, Engineering Structures, 5, Pekcan, G., Mander, J. B., and Chen, S. S. (999), Fundamental considerations for the design of nonlinear viscous damers, Earthquake Engineering and Structural Dynamics, 8(), Preumont, A. (), Vibration Control of Active Structures, Kluwer Academic Publishers, Dordrecht. Tsai, K. C., and Poov, E. P. (998), Steel beam-column joints in seismic moment resisting frames, Berkeley, CA, Earthquake Engineering Research Center, University of California, 998, reort No. UCB/EERC-88/9. Weber, B. (), Daming of vibrating footbridges, Proceedings of the International Conference on Footbridge, Paris, France, - November, on CD, AFGC OTUA (eds.). Notations Symbol Descrition Unit FI force index % P ower W W work, energy J c viscous daming coefficient kg/s f force N x dislacement m η frictional ower law exonent - Suerscrits ot a otimum mean value of a Subscrits d lin nonlin max damer linear nonlinear maximum Page 9 of 55

20 SAMCO Final Reort Viscoelastic damer Theoretical background The total force of a viscoelastic damer is the some of its viscous and stiffness arts (Jones (), Weber ()) f d = k x c x& () The stiffness art causes that the ellise curve in the force dislacement ma has a mean sloe of k d (Fig. 8). The force velocity trajectory is also an ellise curve. Its mean sloe still indicates the damer viscosity c d. The dissiated energy over one cycle is d T d d E = f d x& dt () In contrast to ure linear viscous damers, the force velocity ma does not only show dissiative but also active force values due to the damer s stiffness (within quadrants I and III in Fig. 8). Viscoelastic behaviour is tyical for material damers, not fluid damers, since materials can withstand constant deformations. Fig. 8: Behaviour of: a) ure elastic materials, b) and c) of viscoelastic materials. The stress strain relationshi of a linear elastic material is τ = G e γ (3) Page of 55

21 SAMCO Final Reort 6 where τ and γ are the shear stress and strain, resectively, and G e is the elastic shear modulus. The stress strain curve is deicted in Fig. 8. For the elastic behaviour, time lag between stress and strain and force and dislacement, resectively, does not exist. The roortionality factor of course is G. In contrast, stress is π/ radians out of hase with strain for linear viscous daming only. The linear combination of both roduces the behaviour shown by the viscoelastic trajectories in Fig. 8 with a hase between stress and strain between and π/ according to the linear combination. The hysteresis loos deliver an initial base for analytical modelling the viscoelastic material behaviour as a function of strain, frequency and temerature. For the eriodic excitation, the basic strain stress relations may be written as follows max e γ = γ sin( ω t) (4) max [ G sin( ω t) + G cos( t) ] τ = γ ω (5) where G s is the storage modulus and G l is the loss modulus. The ratio of loss modulus and storage modulus is called loss factor s G η = G l s = tan The angle δ signs the hase shift between stress and strain and ranges from for elastic behaviour to π/ for ure viscous behaviour. Equation Eq. (5) written in an alternative form yields l ( δ ) (6) τ = γ G sin( ω t + δ ) (7) max max G = G s + Gl = + (8) max η Chang et al. (993) erformed several tests of three tyes of viscoelastic damers. They concluded that the storage modulus and the loss modulus are functions of ambient temerature, material temerature, excitation frequency and shear strain. During damer oeration, the energy is dissiated in form of heat roducing temerature rise in the viscoelastic material. Therefore, the damer roerties deend - u to a certain degree - on the number of loading cycles and the range of deformation, esecially under large strain due to temerature increase of the damer material. Bagley and Torvik (983) characterized the frequency and temerature deendent shear storage modulus and shear loss modulus for most viscoelastic materials. Ferry (98) roosed the method of reduced variables to determine the deendence of G and G on ambient temerature. This method is based on a simlification in l s searating the two variables, frequency and temerature, on which the viscoelastic roerties deend. These roerties are exressed in terms of a single function of each. Page of 55

22 SAMCO Final Reort 6 Fig. 9: Examle of a viscoelastic daming device according to Connor (). A serious roblem with viscoelastic devices is an increase in force at low temeratures couled with an accomanying overloading of the bonding agent used to glue the viscoelastic material to its steel attachments. Design issues There exists a large variety of different damer design rocedures, which are collected in Soong and Dargush (997). The damer design rocedure for alication in structures follows aroximately the stes described for ure viscous damers. Aiken et al. (989) reorted that the general design methodology for elastomeric bearings for seismic isolation is substantial different from the currently used for bridge bearings, in articular in asects of slenderness, rated loads and lateral dislacements. A ossible viscoelastic damer design with bonding steel sheets of a viscoelastic material to steel lates is illustrated in Fig. 9 (Connor ()). The equation of shear strain rewritten becomes x γ = (9) For a given shear strain γ, one determines τ with the stress strain relation. Then, the daming force may be estimated as t d f d = w L τ () For eriodic excitation and substituting τ from Eq. (4), one obtains Then, the dissiated energy er cycle becomes f d w L = Gs xmax [ sin( ω t) + η cos( ω t) ] () t d w L W = π η Gs xmax () t d Page of 55

23 SAMCO Final Reort 6 Fig. : Single bearing test machine (Aiken et al. (989)). Fig. : Cyclic hysteresis loo of a bolted rubber bearing (Aiken et al. (989)). Testing and validation The testing rocedure of a viscoelastic damer needs one ste more comared to the testing of ure viscous damers because the elasticity has to be identified too. The daming device is clamed between actuator and fixed anel with a force transducer connected in series (Fig. ). The sring constant k d can be easily identified by alying a constant load and measuring the resulting dislacement. With the known sring constant, the testing rocedure follows the one of viscous damers (Fig. ). If the actuator is dislacement controlled, then the resulting force reresents the measured system outut. The first derivation of the dislacement delivers the velocity. Together with the measured force, the daming coefficient c d may be identified as the mean sloe of the force velocity trajectory (Fig. 8). Please note that the measured total damer force has to be reduced by the inertia force of the sensor if the force transducer is moving together with the daming device. Imlementation Basically, most material damers behave viscoelastic. Even some fluid damers show viscoelastic behaviour due to the fluid stiffness although the force due to their stiffness is small comared to the force resulting from their viscosity. Therefore, numbers of viscoelastic damers have been imlemented on real civil structures for structural control. They have been used in the fields of: vibration control and vibration mitigation of sensitive machines (e.g. nuclear industry and aerosace technology), base isolation systems, also foundation of suerstructures (Huffmann (985)), daming of vibrations of high rise buildings, and daming of bridge deck and cable vibrations (elastomeric bearings). Elastomeric bearings The bearing systems which have been imlemented in buildings are either systems which include additional devices to enhance the overall daming of the isolation system or do not have extrinsic devices which increase the system daming. The latter tye can be of the lead-rubber tye, in which elastomeric bearings contain a lead- Page 3 of 55

24 SAMCO Final Reort 6 lug insert to sulement daming. Another realization is the filled tye, in which a filler material is added to the rubber to enhance the daming and stiffness roerties of the comound. A bearing design, roosed by Aiken et al. (989), is shown in Fig.. The bearing consists of three rubber layers searated by two steel shims and two end lates. An external rubber cover layer is used for rotection. The material used for the bearings is a filled, natural rubber with high daming. Kelly and Edgardo (99) reorted that if an isolation system uses high strength elastomers, such as olychlororene rubber, connected with high quality bonding techniques, the ultimate caacity can be accurately redicted and very substantial safety margins can be established. Fig. : Bearing system design roosed by Aiken et al. (989). References Aiken, I. D., Kelly, J. M., and Tajirian, F. F. (989), Mechanics of Low Shae Factor Elastomeric Seismic Isolation Bearings, Reort No. UCB/EERC-89/3, Earthquake Engineering Research Center, College of Engineering, University of California at Berkeley. Bagley, R. L., and Torvik, P. J. (983), Fractional Calculus A Different Aroach of the Analysis of Viscoelastic Damed Structures, AIAA Journal, 983. Connor, J. J. (), Introduction to Structural Motion Control, htt:// Chang, K. C., Lai, M. L., Soong, T. T., Hao, D. S., and Yeh, Y. C. (993), Seismic Behaviour and Design Guidelines for Steel Frame Structures with Added Viscoelastic Damers, NCEER 93-9, National Center for Earthquake Engineering Research, Buffalo. Ferry, J. D. (98), Viscoelastic Proerties of Polymers, John Wiley, New York. Page 4 of 55

25 SAMCO Final Reort 6 Huffmann, G. K. (985), Full Base Isolation for Earthquake Protection by Helical Srings and Viscodamers, Nuclear Engineering Design, 84(). Jones, D. I. G. (), Handbook of Viscoelastic Vibration Daming, John Wiley & Sons LTD. Kelly, J. M., and Edgardo, Q. (99), Mechanical Characteristics of Neorene Isolation Bearings, Reort No. UCB/EERC-9/, Earthquake Engineering Research Center, College of Engineering, University of California at Berkeley. Soong, T. T., and Dargush, G. F. (997), Passive Energy Dissiation Systems in Structural Engineering, John Wiley & Sons, New York. Weber, B. (), Daming of vibrating footbridges, Proceedings of the International Conference on Footbridge, Paris, France, - November, on CD, AFGC OTUA (eds.). Notations Symbol Descrition Unit G e elastic shear modulus N/m G s storage modulus N/m G l loss modulus N/m L length m W work, energy J c viscous daming coefficient kg/s f force N k stiffness kg/s t thickness m w width m x dislacement m δ hase angle rad γ shear strain - η loss factor - τ shear stress N/m ω radial frequency rad/s Subscrits d max damer maximum Page 5 of 55

26 SAMCO Final Reort Coulomb friction damer Theoretical background In friction damers, irrecoverable work is done by tangential force required to slide one surface of solid body across to another. Contacting surfaces are intended to remain dry during oeration. No additional hydrodynamic lubricating layer is needed. More detailed information in conjunction with numerous references can be found in Larssen-Basse (99) and Taylor (98). Basic theory of solid friction is inferred from hysical exeriments involving lanar sliding of rectangular blocks: The total frictional force is indeendent of the aarent surface area of contact The total frictional force is roortional to the total normal force acting across the interface. In case of sliding with low relative velocities, the total frictional force is indeendent of velocity. Hence, the behaviour of a Coulomb friction damer may be described as f d ( sign(x) & ) = µ N () where f d reresents the damer force, N the normal force and µ the friction coefficient for the two layers. In some cases, the friction coefficient is somewhat higher when sliage is imminent than during sliding. This is indicated by the two searate friction coefficients of static µ sta and kinetic µ kin friction coefficients. The Coulomb friction force acts always against the direction of movement and therefore behaves dissiative like any ure daming force (Fig. 3). The Coulomb friction force is zero in the case of zero velocity since relative dislacement between the two surfaces vanishes (Fig. 3). Therefore, the force of an ideal Coulomb friction damer jums from its ositive value to zero and then to its negative value when the velocity changes its sign. In contrast, real damers that are seen as Coulomb friction damers (e.g. MR damers at constant current) show a force velocity trajectory as deicted in Fig. 3 c). Assuming eriodic excitation, the work er one full cycle is reresented by the rectangular area deicted in Fig. 3 (Weber ()) W = 4 f d x () max max Fig. 3: Coulomb friction: a) force dislacement trajectory, b) force velocity trajectory; c) force velocity trajectory of real Coulomb friction damers (e.g. MR damers at constant current). Page 6 of 55

27 SAMCO Final Reort 6 In order to generalize and extend the theory, involving non-uniform distributions or non-lanar surfaces, the basic assumtions mentioned above are often abstracted to the infinitesimal limit. The generalization of Eq. () becomes t n ( sign(x) & ) τ = µ τ (3) where τ t reresents the tangential traction and τ n the normal traction. This equation can be also used for determining nominal contact stresses. Design issues In ractice, the force trajectory of Coulomb friction damers will follow qualitatively the trajectory deicted Fig. 3 c) due to sealing friction and viscous material behaviour. However, the main and only design variable of Coulomb friction damers is their friction force level f. This force level has to be chosen the way that the damer d does not clam the main structure due to the constant damer force. However, the friction force level of the originally designed and manufactured Coulomb friction damer may change during lifetime for several reasons: The damer force level may change due to varying conditions of the sliding interfaces. Bimetal interfaces are suscetible to this behaviour because ongoing hysical and chemical rocesses change the friction coefficient considerably. At microscoic level one finds that natural and engineered surfaces are not smooth. These irregularities are often categorized as waviness and roughness and they are tyically resent over a wide range of scales. It should be mentioned that true contact does occur directly between metals, adhesive bonds form across the interface often roducing a friction coefficient larger than one ( µ > ). Aging and corrosion of the surfaces may also affect the originally roduced friction damer. The investigation of friction rocesses under such circumstances is a challenging issue since the mechanical characteristics of oxide films are not well understood. Finally, local deformational rocesses which occur in the vicinity of surfaces may change the friction force behaviour (Soong and Dargush (997)). During the sliage, the dissiated energy will cause local heating along the interface of the constituent materials. The thermal effects can result in an aging rocess of surfaces caused by material softening or romoting oxidation. However, it may be assumed that the system resonse is not sensitive to relatively small variations of ambient temerature. The attention has to be more directed on hysiochemical rocesses, often triggered by atmosheric moisture or containments. These rocesses may change the hysical and chemical character of the surfaces, caused by formation of oxide layers. In more aggressive environments the interfaces tend to corrode. The formation of the geometry of the comonent is of rime imortance to attenuate corrosion. For instance, exosed surfaces become a surlus of oxygen and other inaccessible regions have much less dissolved oxygen. Therefore, a mass transfer does occur between anode and cathode (Soong and Dargush (997)). Imlementation In the following, several tyes of friction daming devices are resented. Page 7 of 55

28 SAMCO Final Reort 6 a) Limited Sli Bolted Joints Pall et al. (98) started with the realization of friction damers as Limited Sli Bolted Joints varying simle sliding elements which have different surface treatments. Several surfaces like mill scale, sand blasted, inorganic metallic zinc-rich aint, brake lining ads and olyethylene coating were investigated by static and dynamic tests evaluating load dislacement resonse under constant cyclic loading (Fig. 4). The maximum static sli is obtained for metallic surfaces. The cyclic resonse is quite erratic, with considerable stick-sli associated with the transition from static to kinetic frictional resonse. Anagnostides and Hargreaves (99) reorted exerimental results from a number of other frictional materials. Brake lining materials would erform well in dynamic tests. The characterization of their simle brake lining frictional system in terms of an elastic erfectly lastic model is quite aroriate. A mathematical model is given in Soong and Dargush (997). Fig. 4: Limited Sli Bolted Joints (Pall et al. (98)). Fig. 5: X-braced Friction Damer (Pall and Marsh (98)). b) X-braced Friction Damer A much more effective oeration can be reached secial damer mechanism, roosed by Pall and Marsh (98) (Fig. 5). These devices utilize the brake lining ads. One rincile of a tyical X-braced system is based on braces that are designed to buckle at relatively low comressive loads, so the braces contribute only when subjected to tension. Installing uni-axial friction elements within each brace, sliage would only occur in the tensile direction and little energy dissiation would result during cyclic loading. The mechanism tends to straighten buckled braces and also enforces sliage in both tensile and comressive directions. Pall and Marsh (98) used a simle elastolastic model to describe the behaviour of this X-braced friction damer. Filiatrault and Cherry (988) roosed a more refined macroscoic model for this device, since the Pall-Marsh model overestimates the energy dissiation and the simle elastolastic model is only valid if the device slis every cycle and if that the sliage is always sufficient to straighten comletely the buckled braces. Wu et al. (5) ublished an imroved variant of the Pall-tyed frictional damers. Page 8 of 55

29 SAMCO Final Reort 6 c) Sumitomo Friction Damer The uni-axial friction damer made by Sumitomo Metal Industries Ltd. utilizes a more sohisticated design (Fig. 6). A recomressed internal sring exerts a force that is converted trough the action of inner and outer wedges into a normal force on the friction ads. The use of grahite lug inserts ensures dry lubrication in order to maintain a consistent friction coefficient between the ads and the inner surface of the steel casing. Aiken and Kelly (99) reorted that the resonse of this damer is extremely regular and reeatable with rectangular hysteresis loos. Effects of loading frequency, number of cycles, and ambient temerature are indicated as not sensitive. Fig. 6: Sumitomo Friction Damer (Aiken and Kelly (99)). Fig. 7: Energy Dissiating Restraint (EDR) (Nims et al. (993)). d) Energy Dissiating Restraint The Energy Dissiating Restraint (EDR), designed by Fluor Daniel, Inc., is similar to the Sumitomo concet, since this device also includes an internal sring and wedges encased in a steel cylinder (Fig. 7). A detailed descrition of the damer design is rovided in Nims et al. (993). The EDR uses steel comression wedges and bronze friction wedges in order to transform the axial sring force into normal ressure acting outward on the cylinder wall. The frictional surface is formed by the interface between the bronze edges and the steel cylinder. Internal stos are ensured within the cylinder in order to create the tension and comression gas. It should be mentioned that the length of the internal sring can be altered during oeration, roviding a variable frictional sli force. Other realized friction damers can be found in the book of Soong and Dargush (997). References Aiken, I. D., and Kelly, J. M. (99), Earthquake Simulator Testing and Analytical Studies of Two Energy-Absorbing Systems for Multistory Structures, Reort No. UCB/EERC-9/3, University of California, Berkeley, CA. Anagnostides, G., and Hargreaves, A. C. (99), Shake Table Testing On an Energy Absortion Device for Steel Braced Frames, Soil Dynamics Earthquake Engineering, 9(3). Filiatrault, A., and Cherry, S. (988), Comarative Performance and Friction Damed Systems and Base Isolation Systems for Earthquake Retrofit and Aseismic Design, Earthquake Engineering Structural Dynamics, 6. Page 9 of 55

30 SAMCO Final Reort 6 Larsen-Basse, J. (99), Basic Theory of Solid Friction, in Friction, Lubrication and Wear Technology, ASM International Handbook Committee, American Society for Metals, Materials Park, Ohio. Pall, A. S., Marsh, C., and Fazio, P. (98), Friction Joints for Seismic Control of Large Panel Structures, Journal Prestressed Concrete Inst., 6(6). Pall, A. S., and Marsh, C. (98), Resonse of Friction Damed Braced Frames, Journal Structural Division, ASCE, 8(ST6). Nims, D. K., Richter, P. J., and Bachmann, R. E. (993), The Use of the Energy Dissiating Restraint for Seismic Hazard Mitigation, Earthquake Sectra, 9(3). Soong, T. T., and Dargush, G. F. (997), Passive Energy Dissiation Systems in Structural Engineering, John Wiley & Sons, New York. Taylor, D. (98), Friction The Present State of Our Understanding, Journal Lubr. Technics, ASME, 3. Weber, B. (), Daming of vibrating footbridges, Proceedings of the International Conference on Footbridge, Paris, France, - November, on CD, AFGC OTUA (eds.). Wu, B, Zhang, J., Williams, M. S., and Ou, J. (5), Hysteretic behaviour of imroved Pall-tyed Frictional damers, Engineering Structures, 7(3), Notations Symbol Descrition Unit W work, energy J N normal force N f force N x dislacement m µ friction coefficient - τ traction N/m Subscrits d kin max n sta t damer kinetic maximum normal static tangent Page 3 of 55

31 SAMCO Final Reort Structural friction damer Theoretical background Structural daming is the use of internal friction in a material to transform structural vibration energy into heat roduced within the damer material. This reduces vibration amlitudes of the main structure. The definition equation of friction damers is (Weber ()) f d ( sgn(x) & ) = k x () where k s is called seudo-stiffness factor. For eriodic excitation, Fig. 8 shows the force dislacement and force velocity trajectories of structural friction damers. Please notice that the damer force jums just before reaching its maximum value to zero (see the ath direction in Fig. 8 indicated by arrows) because of the damer velocity that is zero at t = π ( ω) and at t = 3 π ( ω). The dissiated energy er full cycle is equal to d k d x max W = 4 = kd xmax () Fig. 8: Structural daming: a) force dislacement trajectory and b) force velocity trajectory. Testing and validation The goal of testing structural friction damers is to identify the single damer arameter d k. Therefore, the testing rocedure of structural friction damers follows the one of linear viscous damers excet that not the damer s viscosity but the damer s stiffness is identified. The identification can be realized by simly measuring airs of force deformation values. Several airs have to be measured in order to be able to identify nonlinear stiffness behaviour. Page 3 of 55

32 SAMCO Final Reort 6 Imlementation Structural friction damers are often imlemented as vibration mitigation devices directly within ylons and struts in order to dissiate vibration energy and therefore to mitigate the structure. Esecially, they are used in: base isolation systems, daming of belfries, gear boxes as daming materials, hollow arts or rofiles as daming material, and automotive alications. References Weber, B. (), Daming of vibrating footbridges, Proceedings of the International Conference on Footbridge, Paris, France, - November, on CD, AFGC OTUA (eds.). Notations Symbol Descrition Unit W work, energy J f force N k stiffness kg/s x dislacement m Subscrits d max damer maximum Page 3 of 55

33 SAMCO Final Reort Hysteretic damer Theoretical background The inelastic deformation caability of metallic substances reresents an effective energy dissiation mechanism for daming of engineering structures. The stress strain curve deicted in Fig. 9 a) is tyical for most metals. For strain values less or equal than ε Y, the stress strain relation is linear and the initial state O is fully recoverable if the alied load is removed. For strain values larger than ε Y, the metal yields and deforms irreversible. The lastic deformation occurs directly with energy dissiation. If the load in oint B is removed, static deformation remains due to the inelastic deformation from Y to B, therefore denoted as ε ine. Only the elastic art ε ela is recovered. At oint M, the maximum load is reached. The ultimate failure occurs at oint Ω (Soong and Dargush (997)). Fig. 9: Hysteretic daming: a) stress strain relation, b) real hysteretic damer behaviour, c) idealized hysteretic damer behaviour. The shae of the force dislacement trajectory deends on the stress strain relationshi of the material and the device design (Fig. 9 b)). For the case of an ideal elastic-lastic material behaviour, the relation between daming force and dislacement is as shown in Fig. 9 c). The dissiated energy er cycle of an ideal elastic-lastic material becomes y ( x x ) = 4 ( k x ) ( x x ) = 4 k x ( ) W = 4 f µ max y where f y is the yield force, x y the dislacement at which the material starts to yield, x max the maximum dislacement, k y the elastic damer stiffness and the ductility ratio µ denotes the ratio between maximum dislacement and yield dislacement (Jones ()) y y x max x y max y y y () µ = () Page 33 of 55

34 SAMCO Final Reort 6 Design issues An examle Jones () resents an examle how to design a hysteretic daming device. The daming device consists of a cylindrical rod of length L and cross sectional area A (Fig. ). Provided that the damer material behaves ideal elastic-lastic, the yield force may be exressed as follows Considering the relationshi between strain and deformation f y = A σ = k x (3) x y y y y y = L ε (4) the linear stiffness describing the elastic behaviour becomes k y = A L σ y ε y = A E L (5) Equation Eq. (5) describes the known relation between stiffness and elasticity modulus E. Hence, by aroriate choice of material ( E ) and geometry ( A, L ), the desired damer with maximum exected deformation and maximum tolerable damer force may be designed. Fig. : Ideal elastic-lastic daming device according to Jones (). Fig. : X-shaed Plate Damer. The idea of utilizing searate metallic hysteretic damers in order to imrove the resistance of structures in the case of earthquakes was formulated firstly by the concetual work of Kelly et al. (97) and Skinner et al. (975). In recent years, many new designs of metallic damers have been roduced. Two examles are shown in Fig.. An X-shaed late damer or ADAS (Added Daming And Stiffness) device has been studied via exeriments by Bergman and Goel (987), Whittaker et al. (989), Whittaker et al. (993), and subsequently emloyed in the seismic retrofit rojects discussed by Martinez-Romero (993) and Perry et al. (993). The hourglass shae roduces nearly uniform curvature throughout the late during infinitesimal deformation. Similar reasoning has also led to the develoment of triangular late systems by Tsai et al. (993). Modelling aroaches In order to characterize the metallic damer resonse, a coule of different suitable force dislacement models can be taken. The first aroach involves the direct Page 34 of 55

35 SAMCO Final Reort 6 use of exerimental data from damer tests. The whole damer behaviour is described by the static damer ma, which consists of the fitted force dislacement trajectories. Another way of modelling is that the damer model is constructed from an aroriate constitutive relationshi for metals by alying the rinciles of mechanics. From a constitutive model of a metal, the force dislacement relationshi can be develoed introducing a geometric descrition of the device and emloying rinciles of mechanics. The geometric descrition may require a finite element discretization or can be adequately modelled, for instances by simle strength of materials reresentation. For the examle of a triangular late damer, Tsai et al. (993) introduced a set of equations to describe the following force dislacement model f d N E w 6 L = 3 where f d reresents the damer force, N the number of identical triangular structural steel lates, which are ositioned in arallel, x the damer dislacement, h the thickness of the cantilevered late, L the length and w the base width. This model agrees reasonably well with erformed exeriments. However, the model is valid only for the elastic resonse of the damer. Thus, the dissiated energy of the device cannot be estimated using this model. Soong and Dargush (997) recommended a finite element analysis methodology. For the examle of triangular late damer model, develoment and subsequent calculations were described. Generally, in the same way, a wide range of metallic damers, including the X-shaed late damers, can be develoed. A more detailed descrition of a mechanic based modelling aroach for triangular late damers is rovided in Dargush and Soong (99) and in Tsai and Tsai (995). In the analysis of single-degree-of-freedom systems (SDOFS), the addition of a ure viscous device increases the system daming indeendently of the excitation forces. Although the insertion of metallic damers into SDOFS generally reduces the resonse by the ortion of the dissiated energy, the SDOFS resonse may increase for some secific seismic inuts due to the added stiffness of the hysteretic damer. Soong and Dargush (997) have illustrated a articular case in which an elastolastic damer is either ineffective or even detrimental under secial ground motion. This exerience suggests that a detailed analysis of the nonlinear transient dynamics is required to evaluate the effectiveness of metallic damers for seismic rotection of civil structures. On the basis of the finite element method, the Newton-Rahson time domain aroach is directly alicable for structures that include metallic damers. Alications of nonlinear structural analysis to building frames using X-shaed damers are rovided by Xia et al. (99) and for the case of using triangular late damers by Tsai et al. (993). Another aroach is rovided by Soong (99), which allows for rewriting the governing equations in state sace reresentation and solving in an efficient and accurate way by alication of first order differential equation solvers. h 3 x (6) Effect of viscolasticity For steel at aroximately room temerature and in the case of moderate strain rates, it can be assumed that the lastic flow occurs instantaneously comared to the time variation of the alied load. In cases of lead damers or steel at high temera- Page 35 of 55

36 SAMCO Final Reort 6 ture or under very high strain rates, the cree and relaxation henomenon must be considered. Cree signifies increasing strain with time under constant stress. Relaxation is defined as continual reduction of stress with time for a material under constant strain. In order to model the motion of dislocations, it is necessary to incororate the hysics of time indeendent lastic strains and time deendent cree strains to a unified theory. Özdemir (976) roosed the following uni-axial model n σ σ d σ σ b & (7) ε = + E τ E where ε& is the strain decrement, τ and σ b reresent the relaxation time and back stress, resectively. The drag stress σ d is a material constant. The quantity σ σ b is denoted as overstress. This allows the modelling of the kinematical hardening effect. A detailed discussion can be found in the book of Soong and Dargush (997). σ d Effect of temerature The mechanical roerties of steel in structures at aroximately room temerature are both consistent and stable. Therefore, steel is often used as a building material. In case of imacts by major earthquakes, the structural steel within metallic damers will cycle into the inelastic range. A significant ortion of dissiated energy will be converted into heat. The surrounding metal will be heated. The quantity of temerature increase can be estimated by considering the energy balance. The dissiated energy reresents a heat source, while conduction and convection rocesses lead to a redistribution of energy. For reasonable large steel damer design, it may be assumed that temerature increase does not significantly alter the mechanical roerties of the device. The behaviour of lead damers is much more sensitive to moderate increases beyond room temeratures. Thus, in this case, thermal effects must be considered. Consideration of failure The theories of lasticity and viscolasticity describe the behaviour of metals in the inelastic range during cycling loading but not the fail due to fatigue. The henomenon of the low-cycle fatigue is of main interest. Low cycle fatigue results from a limited number of excursions into the inelastic range. Growth and interconnection of micro sized cracks lead eventually to failure at macroscoic level. In ractice, a more henomenological aroach is used based uon the concet of material damage. It should be recognized that these late damers reresent critical elements in the overall seismic rotective system and therefore must be engineered to a high level of reliability. Dargush and Soong (995) used a simle modelling aroach for the mechanics of materials in order to study the behaviour of triangular metallic late damers under cyclic load. Testing and validation Soong and Dargush (995) reorted that, from a review of literature, much effort has been exended on exerimental testing of metallic damers and test structures. For determination of the mechanical characteristics of the daming devices, an in-lane Page 36 of 55

37 SAMCO Final Reort 6 testing frame is emloyed and the horizontal dislacement and the vertical force as well are controlled. The investigations should include: the force resonse at several dislacement levels, fatigue history, and temerature rise of the secimen during cycling loading. Imlementation Hysteretic damers have been successfully alied for structural control in the fields of: design of seismic assive energy dissiation systems (Constantinou et al. (998), Paulay and Priesley (99)), and base isolation systems and foundation of suerstructures (Bhatti et al. (978)). References Bergman, D. M., and Goel, S. C. (987), Evaluation of Cyclic Testing of Steel Plate Devices for Added Daming and Stiffness, Reort No. UMCE 87-, University of Michigan, Ann Arbor, MI. Bhatti, M. A., Pister, K. S., and Polek, E. (978), Otimal design of an Earthquake Isolation System, Reort No. UCB/EERC-78/, University of California. Constantinou, M. C., Soong, T. T., and Dargush, G. F. (998), Passive Energy Dissiation Systems for Structural Design and Retrofit, MCEER Monograh Series, Multidiscilinary Center for Earthquake Engineering Research, Buffalo, NY. Dargush, G. F., and Soong, T. T. (995), Behaviour of Metallic Plate Damers in Seismic Passive Energy Dissiation Systems, Earthquake Sectra, 9(3). Jones, D. I. G. (), Handbook of Viscoelastic Vibration Daming, John Wiley & Sons LTD. Kelly, J. M., Skinner, R. I., and Heine, A. J. (97), Mechanisms of Energy Absortion in Secial devices for Use in Earthquake Resistant Structures, Bull. N.Z. Society Earthquake Engineering, 5(3), Martinez-Romero, E. (993), Exeriences on the Use of Sulemental Energy Dissiators on Building Structures, Earthquake Sectra, 9(3). Özdemir, H. (976), Nonlinear Transient Dynamic Analysis of Yielding Structures, Ph.D. Dissertation, University of California, Berkeley, CA. Paulay, T., and Priesley, M. J. N. (99), Seismic Design of Reinforced Concrete and Masonry Buildings, John Wiley & Sons, New York. Perry, C. L., Fierro, E. A., Sedarat, H., and Scholl, R. E. (993), Seismic Ugrade in San Francisco Using Energy Dissiation Devices, Earthquake Sectra, 9(3). Skinner, R. I., Kelly, J. M., and Heine, A. J. (975), Hysteresis Damers for Earthquake-Resistant Structures, Earthquake Engineering and Structural Dynamics, 3, Soong, T. T. (99), Active Structural Control: Theory and Practise, Wiley, New York. Page 37 of 55

38 SAMCO Final Reort 6 Soong, T. T., and Dargush, G. F. (997), Passive Energy Dissiation Systems in Structural Engineering, John Wiley & Sons, New York. Tsai, K. C., Chen, H. W., Hong, C. P., and Su, Y. F. (993), Design of Steel Triangular Plate Energy Absorbers for Seismic-Resistant Construction, Earthquake Sectra, 9(3). Tsai, C. S., and Tsai, K. C. (995), TPEA Device as Seismic Damer for High- Rise Buildings, Journal Engineering Mechanics, ASCE, (). Whittaker, A., Bertero, V., Alonso, J., and Thomson, C. (989), Earthquake Simulator Testing of Steel Plate Added Daming and Stiffness Elements, Reort No. UCB/EERC-89/, College of Engineering, University of California at Berkeley. Whittaker, A., Aiken, I., Bergman, D., Clark, P., Cohen, J., Kelly, J., and Scholl, R. (993), Code Requirements for the Design and Imlementation of Passive Energy Dissiation Systems, Proc. ATC 7- on Seismic Isolation, Energy Dissiation, and Active Control,, Xia, C., Hanson, R. D., and Wight, J. K. (99), A Study of ADAS Element Parameters and Their Influence on Earthquake Resonse of Building Structures, Reort No. UMCE 87-, The University of Michigan, Ann Arbor, MI. Notations Symbol Descrition Unit A cross sectional area m E elasticity modulus N/m L length m W work, energy J f force N k stiffness kg/s x dislacement m ε strain m µ ductility ratio - σ stress N/m τ relaxation time s Subscrits d max y damer maximum yield Page 38 of 55

39 SAMCO Final Reort Shae memory alloy damer Theoretical background Shae memory alloys (SMAs) are metallic materials that mainly exist in two different crystal structure tyes, namely martensite and austenite or a combination of both. Martensite is characterized by a body centred tetragonal crystal structure and austenite by a body centred cubic crystal structure. Since the roerties of SMAs change according to the two inut variables temerature and strain, SMAs are often called smart materials. The hase transformation of SMAs is reversible. Profound information about material behaviour of SMAs and their alications may be found in Duerig (99), Funakubo (987), Humbeeck (), Janke et al. (5), Otsuka and Wayman (999), and in Otsuka and Kakeshita (). Due to the fairly high daming characteristics of SMAs, these materials are also used as damers. If the material roerties are changed according to the actual material temerature and material strain, resectively, SMAs reresent controllable damers or semi-active daming devices, resectively. Within this chater, SMAs used as assive damers are introduced, whereas the use of controllable SMAs is described in chater... According to Janke et al. (5), rimarily, one may distinguish between four different behaviours of SMAs:. SMAs may behave actuator like (Fig. ),. SMAs show the so-called shae memory effect (Fig. 3), 3. SMAs behave suer elastically (Fig. 4), and 4. SMAs may act as hysteretic material damers in ure martensitic state (Fig. 5). curve of the austenite heating to Af cooling below Mf z % ambient temeratures for outdoor civil alications A Mf As loadingthe martensite % Ms Af A - 6 T [ C] Fig. : Stress strain curves and transformation temerature rofiles for actuator like change of strain and stiffness at constant stress (Janke et al. (5)). If SMAs behave actuator like, ambient temerature is below the temerature martensite finish (Mf). After ure martensite has been deformed, the crystal structure of the SMA may be changed from martensite to austenite and back by heating and cooling (horizontal dashed line in Fig. ). Since austenite has a larger elasticity modulus than martensite, a device with two different elasticity moduli may be roduced. If a sring is roduced with such an SMA and connected with another steel Page 39 of 55

40 SAMCO Final Reort 6 sring, the connecting oint will move when heating and cooling. This is the reason why this SMA behaviour is called actuator like. If ambient temerature lies between martensite finish and austenite start (As), the so-called shae memory effect of SMA occurs. First, ure martensite is deformed lastically. Then, the load is removed and lastic deformation results (Fig. 3). From this oint, two scenarios are feasible. If the SMA is heated without geometrical constraints, martensite will change to austenite which ends u in a full shae recovery of the initial shae in oint A. If the SMA is heated at constant strain, stress increase results due to the hase change to austenite, which has a larger elasticity modulus. heating to Af z % Mf A As A loading the martensite heating to Af free unloading % Ms - 6 Af T [ C] Fig. 3: Stress strain curves and transformation temerature rofiles for shae memory effect in the case of free strain recovery and constraint strain recovery (Janke et al. (5)). If ambient temerature lies above austenite finish (Af), the SMA behaves suer elastically (Fig. 4). Pure austenite is deformed until it reaches lastic state. Then, austenite will transform to martensite with increasing strain (see lateau in Fig. 4). When the state of ure martensite is reached, the lateau ends and the stress strain curve goes u with smaller sloe due to smaller elasticity modulus of martensite relatively to austenite. During unloading, martensite will change to austenite on a lower stress lateau. Finally, the SMA consists again of ure austenite. The area enclosed by the stress strain trajectory is equivalent to the dissiated energy er cycle. Therefore, SMA working in suerelastic mode may be used as damers. stress induced martensite loading z % Mf As A unloading austenite % Ms Af - 6 T [ C] Fig. 4: Stress strain curves and transformation temerature rofiles for the so-called suerelastic behaviour (Janke et al. (5)). A Page 4 of 55

41 SAMCO Final Reort 6 If ambient temerature lies between martensite finish and austenite start, SMAs may work as ure martensitic, hysteretic damers if temerature is constant (Fig. 5). Basically, also ure austenitic SMAs may work as hysteretic damers. However, the main advantages of martensitic hysteretic damers are: A The elasticity modulus of martensite is smaller than of austenite. Therefore, the stress strain trajectory encloses a larger area for the same stress which results in larger daming for the target structure. Due to the reorientation of martensite variants, martensite can withstand many more deformation cycles than austenite without failure for the same dissiated energy. z loadingthe martensite unloading % % - Mf Ms As 6 Af T [ C] Fig. 5: Stress strain curves and transformation temerature rofiles for martensitic hysteretic daming (Janke et al. (5)). A Due to the larger hysteresis loo of SMAs working as ure martensitic hysteretic damers comared to the hysteresis area of SMAs oerating in suerelastic mode (Fig. 6), SMA assive damers are usually based on the former effect. Fig. 6: Hysteresis loos for: a) suerelastic SMA behaviour and b) martensitic hysteretic daming (Janke et al. (5)). Fig. 7: Torsion bar design (Witting and Cozzarelli (99)). Page 4 of 55

42 SAMCO Final Reort 6 Design issues Witting and Cozzarelli (99) investigated four rincial design mechanisms for roducing an SMA based daming device. They used a Cu-Zn-Al alloy. It was a bar in torsion and beam in bending, which was axially loaded, and the clamed late was loaded in the centre (Fig. 7). They found that the annular clamed late was much too stiff and resulted in very small strains. The axially loaded beam was also found unsuitable because the constraints of stiffness and strain would cause the beam to buckle. A direct comarison between torsion and bending beam showed that more energy was ut into higher strain regions in the torsion bar design than in the bending beam design. The larger strain resulted in a bigger amount of energy absorbed. Therefore, the torsion bar design aarently resulted in a more effective damer. For the first assessment of the damer caabilities, a linear constitutive mode was used. The basic dimensions of the damer are torsion arm length D, radius of torsion bar R, and length of torsion bar L (Fig. 7). The angle of twist Θ of a solid round bar of radius R, length L, with torque T alied at the end of the bar and shear modulus G is T L Θ = () 4 G R The angle of twist Θ due to the dislacement x is exected to be small, which allows the following simlification x = sin( Θ) Θ D The shear strain ε xy may be exressed in terms of L, D, x and the radial distance r from the centre of the bar as follows The damer force is given by () ε Θ r x r xy = = L L D (3) f d 4 G R x = π (4) L D Considering that the damer force is the roduct of damer stiffness and damer dislacement, the damer stiffness can be exressed as k d 4 G R = π (5) L D The maximum shear strain occurs at r = R if the dislacement x reaches its maxi- x. This yields for the maximum strain mum max xmax R ε max = (6) L D Solving Eqs. (5) and (6) for D yields Page 4 of 55

43 SAMCO Final Reort 6 D = 3 π G R ε x max k d max (7) which leads to the length L as follows L = k d x max 4 π G R ε max (8) Uon substitution of Eq. (7) into Eq. (3) leads to the shear strain as follows ε r ε xy = max (9) R The equation of strain energy density is U = G ε () xy This equation must be integrated over the high strain region. Since the strain is indeendent of y, the integration limits for Θ become and π. The maximum of r is R. The minimum value of r can be calculated by substituting ε min for ε xy in Eq. (9) which leads to = R ε min r min () ε max Using Eq. (7) in order to eliminate L, the total strain energy may be estimated by W π a R G ε min 3 = r dr dy dθ R b () xmax kd with: a = 4 π G R ε max (3) = R ε ε min b (4) For the final design, a more accurate nonlinear model is develoed. The constitutive law used in the analysis is bilinear τ max xy = G xy + { ( G G ) ( ε ε )} U ( ε ε ) ε (5) The two shear moduli G and G reresent the elastic and inelastic shear modulus. The variable ε t is the strain value at which the stress strain trajectory changes its sloe. The term U ε ε ) denotes the ste function as follows ( xy t t xy : x U ( x) = (6) : x > xy t Page 43 of 55

44 SAMCO Final Reort 6 The stress strain curve described by Eq. (5) is deicted in Fig. 8. The torque roduced from the two torsion bars within the damer (Fig. 7) may be calculated as T π R = f D = ( τ r) r dr dθ d (7) xy Substituting τ xy from the constitutive law in Eq. (7) and using Eq. (3) for the shear strain, the damer force becomes f Assuming d 4 D π R G x L D x L D = + ( G G ) ε r W ( ε ε ) r dr dθ t (8) R > L D ε t / x, integration of Eq. (8) simlifies to f d xy t ( L D ε ) 4 π G R 8 π ε t 3 = x + ( G G ) R 3 L D 3 D x The two resented design aroaches delivered two estimates of the damer force. By choosing the geometrical design arameters L, D, and R and selecting the aroriate material, which defines G, G, and ε t, the damer force may be estimated using the simlified aroach of Eq. (4) or the more accurate aroach given by Eq. (9). t 3 (9) Fig. 8: Stress strain diagram used for bilinear damer model (Witting and Cozzarelli (99)). Fig. 9: Force dislacement loos (Witting and Cozzarelli (99)). The alication of the SMA damers is limited due to dimensions of civil structures which makes relatively high daming forces necessary. Therefore, in case of daming civil structures using SMAs, large amount of material is needed comared to other alications as in the case of, e.g., automotive engineering or medical micro technologies. Probably one of the most imortant reasons why SMA damers are not so often used in civil engineering for vibration mitigation is that the increase of daming is low comared to the financial investment. Another reason might be that a value Page 44 of 55

45 SAMCO Final Reort 6 of 4% material daming is quite high but only a art of all mechanical energy can be transmitted to the daming art in devices (Humbeeck and Kustov (5)). Testing and validation SMA damers are tested similarly to other damer tyes; see the test set-u configuration deicted in Fig. 5 and Fig. 6. An examle of measured force dislacement trajectories of SMA damers is deicted in Fig. 9 (Witting and Cozzarelli (99)). The tested SMA is a CU-Zn-Al alloy. Imlementation Due to the fairly large material costs of SMAs, such damers are rarely imlemented in civil structures in order to enhance structural daming. If SMA based assive damers shall be used for vibration reduction, the SMA oerating as a martensitic hysteretic damer should be used because this oerating mode rovides maximum additional daming to the structure. Basically, the main motivation to use SMA as daming element is to have a semiactive daming device available, where, e.g., the force may be adated to the structure by heating and cooling the SMA device (Li et al. (4), Rustighi et al. (5), Williams et al. ()). Such kind of alications is shortly described in chater...3. References Duerig, T. W. (99), Engineering Asects of Shae Memory Alloys, Butterworth- Heinemann, London. Funakubo, H. (987), Precision Machinery and Robotics, Vol. Shae Memory Alloys, Gordon and Breach. Humbeeck, J. V. (), Shae memory alloys: A material and a technology, Adv. Eng. Mater, 3(), Humbeeck, J. V., and Kustov, S. (5), Active and assive daming of noise and vibrations through shae memory alloays: alications and mechanisms, Smart materials Structures, 4, Janke, L., Czaderski, C., Motavalli, M., and Ruth, J. (5), Alications of shae memory alloys in civil engineering structures Overview, limits and new ideas, Journal of Materials and Structures, 38(79), Li, H., Liu, M., and Oh, J. (4), Vibration mitigation of a stay cable with one shae memory alloy damer, Journal of Structural Control and Health Monitoring, (), -36. Otsuka, K., and Wayman, C. M. (999), Shae Memory Materials, Cambridge University Press. Otsuka, K., and Kakeshita, T. (), Science and technology of shae-memory alloys. New develoments", Mrs Bulletin 7(), 9-. Rustighi, E., Brennan, M. J., and Mace, B. R. (5), A shae memory alloy adative tuned vibration absorber: design and imlementation, Journal of Smart Materials and Structures, 4, Page 45 of 55

46 SAMCO Final Reort 6 Williams, K., Chiu, G., and Berhard, R. (), Adative-assive absorbers using shae-memory alloys, Journal of Sound and Vibration, 49(5), Witting, P.R., and Cozzarelli, F.A. (99), Shae Memory Structural Damers: Material Proerties, Design and Seismic Testing, Reort No. NCEER-9-3, State University of New York at Buffalo. Notations Symbol Descrition Unit D torsion arm length m G shear modulus N/m L length m R radius m T temerature; torque K; Nm W work, energy J f force N k stiffness kg/s r radial distance m x dislacement m Θ angle of twist rad ε shear strain - τ shear stress N/m Subscrits d max damer maximum Page 46 of 55

47 SAMCO Final Reort Passive tuned mass damer Introduction A assive tuned mass damer (TMD) or tuned vibration absorber is basically an energy dissiation device that in its simlest form consists of a mass that is attached to a structure (rimary system) with sring and damer elements (Fig. 3). Due to the damer, energy dissiation occurs whenever the mass of the TMD oscillates with non-vanishing dislacement or velocity relative to the rimary system. This is achieved by transferring as much energy as ossible from the rimary system to the TMD by a careful tuning of the natural frequency and daming ratio of the TMD. Since the mass of the TMD is significantly smaller that that of the rimary system, transferring energy from the rimary system to the TMD generates a great relative oscillation of the mass of the TMD. Contrary to a standard damer which generally rovides additional energy dissiation in a wide frequency band, a TMD oerates efficiently only in a narrow frequency band. This behaviour is closely related to the mechanism of energy transfer from the rimary system to the TMD. High energy transfer arises whenever the natural frequency of the TMD is tuned to the natural frequency of the rimary structure. Therefore, if attached to a continuous structure, the TMD mitigates only one secific vibration mode. The concet of a TMD without integrated daming device was invented by Frahm in 99 (Frahm (99)) to reduce the rolling motion of shis. A first theory of the TMD with integrated daming device was resented years later by Ormondroyd and Den Hartog (Ormondroyd and Den Hartog (98)). In Den Hartog s monograh (Den Hartog (947)), a detailed analysis of otimal arameters of a TMD is resented. There, the model of a rimary structure with an attached TMD is reresented by the two degree of freedom system shown in Fig. 3. Den Hartog used the fixed-oints method for obtaining an accurate aroximate solution of the otimal arameters, natural frequency f t and daming ratio ζ t, of a TMD that minimizes the dislacement of the rimary structure where the latter has vanishing structural daming. A list of otimal arameters for different minimization objectives obtained by the fixed-oints method is given in Warburton (98) and in the textbook of Korenev and Reznikov (993). Recently, a method for comuting exact closed form solutions of otimal arameters that minimizes the maximum of transfer functions was resented by Nishihara and Asami (). For structures with vanishing structural daming, exact solutions with resect to different minimization objectives are given in Asami and Nishihara (3). If the rimary system is subjected to random white noise excitation, that is, an excitation that contains any frequency with exactly the same amlitude, the design of a TMD is based on the minimization of the integral of the square of the absolute value of the transfer function (Crandall and Mark (963)). Exact closed form solutions of the otimal arameters of a TMD for different minimization objectives and undamed rimary system are given by Warburton (98) and Korenev and Reznikov (993). In ractical alications, the rimary system has non-vanishing structural daming and therefore many attemts have been made to find exact, closed form solutions of the otimal arameters. Unfortunately, so far, no exact algebraic solutions to this roblem are known. Emirical formulas for several minimization objectives, that are based on numerical otimization results, were given by Ioi and Ikeda (978). Recently, analytical aroximations obtained by erturbations techniques were constructed by Asami et al. (). Page 47 of 55

48 SAMCO Final Reort 6 Fig. 3: Tyical imlementation of a TMD for vertical vibration mitigation of bridges. Basic equations of the assive TMD The standard model of a linear elastic structure with a linear TMD is the two degree of freedom model dislayed in Fig. 3. The equations of motion of this model are given by mu&& ( t) + ( c + ct ) u& ( t) + ( k + kt ) u ( t) ctu& t ( t) ktut ( t) = f ( t) m% u&& g ( t),, m u&& ( t) + c u& ( t) + k u ( t) c u& ( t) k u ( t) = m u&& ( t), t t t t t t t t t g where u ( t ) is the dislacement of the rimary system ut ( t ) the dislacement of the mass of the TMD (see Fig. 3). Both dislacements are measured relative to the base. m, c, k are the modal mass, daming constant and stiffness of the rimary system, and mt, ct, k t are the mass, daming constant and stiffness of the TMD. f ( t ) is the force acting the rimary system, u&& g ( t) is the base acceleration and m% is the articiating mass of the rimary system (see the end of this section for further details). When formulated with resect to u ( t ) and ur ( t) = ut ( t) u ( t), the relative dislacement of the TMD mass with resect to the dislacement of the rimary system, the equations of motion are m u&& ( t) + c u& ( t) + k u ( t) c u& ( t) k u ( t) = f ( t) m% u&& ( t), t r t r g m u&& ( t) + m u&& ( t) + c u& ( t) + k u ( t) = m u&& ( t), t t r t r t r t g The standard model of a structure with a TMD is obtained by modelling the structure as a one-degree-of-freedom system using modal dislacements. This dramatic simlification is justified by the articularity of roerly tuned TMDs to oerate effectively only in a narrow frequency band. TMDs are therefore designed to mitigate the oscillations of a secific vibration mode. A linear elastic structure with TMD can be mod- Page 48 of 55 () ()

49 SAMCO Final Reort 6 elled by multi-degree-of-freedom system with N degree-of-freedoms where the TMD is connected with the k th degree-of-freedom of the structure. The N + equations of motion are given by Fig. 3: Two degree of freedom model of a TMD attached to a rimary structure. Left: Excitation force f ( t ) acting on the rimary mass. Right: Excitation through base acceleration u&& ( t). g M U&& ( t) + C U& ( t) + K U ( t) ( c u& ( t) + k u ( t)) E = F( t) ( M E) u&& ( t), t t t t k g m u&& ( t) + m u&& ( t) + c u& ( t) + k u ( t) = m u&& ( t), t, k t r t r t r t g where M, C and K are the mass, daming and stiffness matrices of the multidegree-of-freedom model of the structure, E is the column vector with a one in each row, E is the column vector with a one in the k th row and a zero in all other rows k and u, k is the dislacement of the k th degree-of-freedom of the structure. We assume that the TMD shall be designed to mitigate the vibrations associated to the mode shae ψ, where ψ is a solution of the eigenvalue roblem k k k k k M ψ ω K ψ = and is normalized in such a way to dislay a one in its k th row. Using the mode shae ψ k, the dislacement of the rimary structure associated to this mode shae can be described by U, k = ψ ku, k. The two-degree-of-freedom model is obtained by multilying the first equation in Eq. (3) from the left with the transose of the mode shae ψ k. By this oeration, the relationshi between the arameters describing the structure of the two-degree-of-freedom system and the N- degree-of-freedom system are given by m = ψ T M ψ, c = ψ T C ψ, k = ψ T K ψ, f ( t) = ψ T F( t), m% = ψ T M E. k k k k k k k k T Observe that ψ k Ek = because of the secific normalization of the mode shae ψ k. In general, the articiating mass is not equal to the modal mass of the rimary system: m% m. By introducing the articiation factor (3) (4) Page 49 of 55

50 SAMCO Final Reort 6 T T ψ k M E ψ k M ( E ψ k ) Γ m = = + (5) T T ψ M ψ ψ M ψ k k k k m% can be exressed as = Γm m% m. In general, because TMDs are connected to the degree-of-freedom with maximum modal amlitude, the articiation factor Γ m is greater than unity. Γ m has an effect on otimal tuning arameters of the TMD in the cases of base excited structures (e.g. earthquake loadings). Since the ratio of the mass of the TMD and the modal mass m is small, in general, this effect is also small. H otimization Until now, many otimization criteria for the TMD have been roosed. The most common criteria are the H and H norm otimization criteria. The H norm of a scalar transfer function G( z ) is defined by G( z) = max G( z), (6) that is, the H norm reresents the maximum amlitude of the absolute value of the transfer function. The absolute value of the comlex transfer function G( z ) is also known as the dynamic magnification factor. The H norm rovides a bound of the outut gain, where the gain is defined in terms of the H norm: z y ( t ) G ( z ) f ( t ). (7) That is, given a square integrable excitation f ( t ), the H norm rovides a bound of the integral of the square resonse. Near equality is achieved for a harmonic excitation f ( t) = F sin( ωt) rovided that the integral is taken over a finite time interval K t max, since any harmonic function is not square integrable over an infinite time interval. Therefore, Eq. (7) rovides a shar bound for harmonic excitations acting over a long but finite eriod of time. It is imortant to note that Eq. (7) does not rovide any information about the maximum amlitude of the outut y( t ) within a time interval for an arbitrary excitation f ( t ). Dimensionless comlex transfer functions G( z ) of various resonse arameters of the rimary system u ( t ) and the relative dislacement ur ( t ) of the TMD mass with resect to the rimary system are given in Table. The objective of the H norm otimization is to find the frequency ratio η t and daming ratio ζ t of the TMD that minimize G( z) : min G( z) = min max G( z). (8) ηt, ζ t ηt, ζ t z With resect to the grah of the absolute value of the transfer function G( z ), the minimum of G( z) is achieved if and only if there exist two local maxima and both have exactly the same amlitude (see Fig. 3 and Fig. 33). Den Hartog s fixed-oints method (Den Hartog (947)) for comuting aroximations of the otimal arameters η and ζ t, is based on the observation that for rimary Page 5 of 55

51 SAMCO Final Reort 6 systems with vanishing daming ( ζ ), there exist at least two frequencies z and z where G( z ) is invariant with resect to a variation of ζ t ( η t is fixed), that is, G( z ) G( z ) = =. (9) ζ ζ t t h > h 5 (z,g(z )) (z,g(z )) 5 h = h (z,h ) (z,h ) h < h G(z) G(z) 5 ζ = ζ t t, ζ t =.5 ζ t, 5 ζ =.5 ζ t t, z Fig. 3: Methods for H norm otimization of a TMD alying the fixed oints method z Fig. 33: Methods for H norm otimization of a TMD alying the real roots method. Hence, all grahs of G( z ) with fixed η t and variable ζ t cross the oints ( z, G( z ) ) and ( z, G( z ) ). The aroximations of the otimal arameters ηt, and ζ t, are obtained by maximizing G( z ) and G( z ) using the condition G( z ) = G( z ). The fixed-oints method rovides simle closed form aroximations of the otimal arameters ηt, and ζ t, that are accurate to a few ercent within the range of mass ratios that are tyical for civil engineering alications of a TMD (. µ.) (Fig. 3). Unfortunately, the fixed-oints method is limited to rimary systems with vanishing daming since the resence of structural daming destroys the fixed oints. A method that rovides exact closed form solutions of the H norm otimization roblem was recently roosed by Nishihara and Asami (). Their method is based on the observation that the function H ( z) = h G( z), where h is an arbitrary real constant, has exactly two ositive real roots z and z of multilicity two if and only if h = h = min G( z). This concet is sketched in Fig. 33. For h = h and ηt, and ζ t, η, ζ t the grah of G( z ) touches the horizontal line h at the two oints ( z, G( z ) ) and ( z, G( z ) ). The algebraic exressions of the exact closed form solutions are more comlicated than those obtained by the fixed-oints method. Table lists aroximate solutions of the H norm otimization roblem for a rimary structure with vanishing daming. The formulas are obtained by simlifying the exact solutions by using a rational functions aroximation technique. The error of the formulas with resect to the exact solutions is smaller than % for µ.. The formulas regarding h = min G( z) show that the effectiveness of the TMD increases with η, ζ t increasing mass ratio µ. The effectiveness is very sensitive to an inaccurate frequency tuning. A tuning error of ±5% with resect to the otimal frequency increases significantly the eak value of the transfer function of the rimary structure (see Fig. Page 5 of 55

52 SAMCO Final Reort 6 34). Much less critical is the correct tuning of the daming ratio of the TMD. A tuning error of ±5% has little influence on the eak value of the transfer function of the rimary structure (see Fig. 35). 5 4 no TMD η t = η t, η t =.95 η t, µ =. ζ t = ζ t, 5 4 no TMD ζ t = ζ t, ζ t =.95 ζ t, µ =. η t = η t, η t =.5 η t, ζ t =.5 ζ t, G(z) 3 G(z) z Fig. 34: Effects of oor frequency tuning of a TMD on the dynamic magnification factor of the dislacement of the rimary system z Fig. 35: Effects of oor daming tuning of a TMD on the dynamic magnification factor of the dislacement of the rimary system. In most ractical alications, the rimary structure has non-vanishing daming so that the equations given in Table reresent an aroximation of the otimal arameters. Unfortunately, no simle equations exist for non-vanishing daming. However, if the daming ratio of the rimary system ζ is less than %, the otimal arameters given in Table are still a reasonable good aroximation for engineering alications. With increasing daming of the rimary system, the equations of Table become increasingly inaccurate. Fig. 36 and Fig. 37 dislay the otimal arameters ηt, and ζ t, that achieves the minimum of the dislacement of the rimary system u ( ) z for several daming ratios ζ of the structure. The H norm of the nondimensional transfer function of the dislacement of the rimary system u ( z ) and of the relative dislacement of the TMD mass ur ( z ) is dislayed in Fig. 38 and Fig. 39. The figures show that the maximum dislacement of the rimary system as well as the maximum relative dislacement decreases with increasing mass ratio. It is imortant to note that the maximum amlitude of the relative dislacement ur ( z ) is large for small mass ratios µ. The effectiveness of a TMD can be described by using the mitigation factor that is defined as the ratio of the non-dimensional H norm of the rimary system without TMD G ( z), to the non-dimensional H norm of the rimary system with otimally tuned TMD G( z) : Page 5 of 55 G ( z) λ =. () G( z) ζ G( z) Fig. 4 shows that that the mitigation factor of an otimally tuned TMD increases with increasing mass ratio and decreases with increasing daming ratio of the structure. The increase with increasing mass ratio is articularly strong for small structural daming. In the range of ractically realizable mass ratios, a TMD rovides good erformance only for lightly damed structures ( ζ.). The smaller the daming

53 SAMCO Final Reort 6 ratio the better is the erformance of a TMD. In structures with a structural daming ratio of 5%, the dynamic resonse of the rimary system can only be reduced by aroximately a factor of two η t, ζ =.9 ζ =. ζ =. ζ = µ Fig. 36: Otimal TMD frequency for minimizing the dislacement of the rimary system with resect to the H norm. ζ t,..5 ζ = ζ =. ζ =. ζ = µ Fig. 37: Otimal TMD daming ratios for minimizing the dislacement of the rimary system with resect to the H norm. 5 ζ = ζ =. ζ =. ζ =.5 5 ζ = ζ =. ζ =. ζ =.5 G ot (z) 5 G ot (z) µ Fig. 38: Maximum of the non-dimensional transfer function of the dislacement of the rimary system for arameters minimizing the dislacement of the rimary system with resect to the H norm µ Fig. 39: Maximum of the non-dimensional transfer function of the relative dislacement for arameters minimizing the dislacement of the rimary system with resect to the H norm. λ 5 5 ζ =.5 ζ =. ζ =. ζ =.5 G ε / G ot ζ = ζ =. ζ =. ζ = µ Fig. 4: Mitigation factor of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm δ Fig. 4: Amlification due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm ( = /,, µ =. ). δ ηt ηt Page 53 of 55

54 SAMCO Final Reort 6 The erformance of a TMD with resect to H norm is very sensitive to incorrect frequency tuning. The frequency detuning amlification with resect to an otimally tuned damer can be defined as the ratio of the non-dimensional H norm of the resonse of the detuned rimary system Gε ( z), to the non-dimensional H norm of the rimary system with otimally tuned TMD Got ( z) : λ ε, Gε ( z) =. () G ( z) ot λε, increases with increasing frequency detuning and decreasing mass ratio, and decreases with increasing daming of the structure. Fig. 4 dislays the frequency detuning amlification due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm. The amlification is considerable and must be accounted for in the design since live loads or environmental effects induce changes of the natural frequencies of a structure. Frequency detuning increases also the H norm of the relative dislacement. However, the detuning amlification is much smaller than that of the resonse of the rimary system. H otimization The H norm of a transfer function G( z ) is defined by t= z= π () t= z= G( z) = G( t) dt = G( z) dz and is essentially a measure of the area between the square of the absolute value of the transfer function G( z ) and the z -axis. If the system is subjected to a random, white noise excitation, the H norm of the transfer function G( z ) is directly related to the variance of the outut associated to the transfer function. That is y [( ( ) [ ( )]) ] ( ) σ = E y t E y t = G z S, (3) where y( t ) is the resonse of the structure defined by y( t) = G( t) f ( t), is symbolizing the convolution of the transfer function G( t ) with the excitation f ( t ), E[ y( t )] is the average of the outut that is usually zero, and S f is the sectral density of the excitation f ( t ). For ergodic rocesses, the variance can be estimated via temoral averages f σ y = lim ( ( ) [ ( )]) = ( ) T T t= T y t E y t dt G z S f. (4) t= T The objective of the H norm otimization is to find arameters η t, and ζ t, that minimize G( z ) : z= min G( z) = min G( z) dz η, t ζ t ηt, ζ t π. (5) z= Page 54 of 55

55 SAMCO Final Reort 6 Fig. 4: Normalized variance of the dislacement of the rimary system. G( z ) is a function of η t, ζ t, ζ and µ. Generally, closed form exressions of G( z ) can be obtained by evaluating the integral in Eq. () (Crandall and Mark (963)) or by comuting the observability grammian of the linear system described by the transfer function G( z ). Table 3 lists closed form exressions of G( z ) for the most common minimization objectives. Fig. 4 dislays the normalized variance of the dislacement of the rimary system. Minimization of G( z ) is achieved if G( z) η t = G( z) and ζ t = (6) hold. These conditions are usually sufficient and yields two nonlinear equations defining the otimal arameters η t, and ζ t,. Exact closed form solutions have been obtained for systems with vanishing structural daming ( ζ = ). Table 4 lists exact solutions for various excitations and resonse arameters. For rimary systems with structural daming, accurate closed form exressions can be obtained in form of a ower series with resect to the daming ratio ζ. A list of first and second order terms for various excitations and resonse arameters can be found in Table 5. In general, the otimal arameters with resect to the H norm are less sensitive to structural daming ζ than the otimal arameters with resect to the H norm (see Fig. 43 and Fig. 44). For small mass ratios, the otimal daming ratio ζ t, which minimizes the variance of the dislacement of the rimary system with resect to the H norm is virtually indeendent of the structural daming ζ. Fig. 45 and Fig. 46 dislay the non-dimensional variance the dislace- ment of the rimary system u ( z ) and of the relative dislacement of the TMD mass ur ( z ) of a TMD otimally tuned with resect to minimization of dislacement of the rimary system. Page 55 of 55

56 SAMCO Final Reort 6 Fig. 43: Otimal TMD frequency for minimizing the variance of the dislacement of the rimary system with resect to the H norm. Fig. 44: Otimal TMD daming ratios for minimizing the variance of the dislacement of the rimary system with resect to the H norm. Fig. 45: Minimum of the non-dimensional variance of the dislacement of the rimary system for arameters minimizing the variance of the dislacement of the rimary system with resect to the H norm. Fig. 46: Non-dimensional variance of the relative dislacement of the mass of the TMD for arameters minimizing the variance of the dislacement of the rimary system with resect to the H norm. Fig. 47: Mitigation factor of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm. Fig. 48: Amlification due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm ( δ = ηt / ηt,,. µ = ). Similar to the case of H norm otimization, the mitigation factor is defined as the ratio of the non-dimensional H norm of the rimary system without TMD G ( ) z to Page 56 of 55

57 SAMCO Final Reort 6 the non-dimensional H norm of the rimary system with otimally tuned TMD G( z ) : λ G ( z) =. (7) G( z) 4 ζ ( ) G z The effects of mass ratio and daming ratio of the structure on the mitigation factor λ are equal to those observed for λ (see Fig. 47). For small structural daming, the overall erformance is aroximately % smaller than that of a TMD otimized with resect to H norm. Similar to the H norm, the H norm is sensitive to changes of the frequency ratio η t and much less sensitive to changes of the daming ratio ζ t. This is evident by observing the shae of the H norm shown in Fig. 4. Defining the frequency detuning amlification factor with resect to an otimally tuned damer as G ( z) ε λ ε, =, (8) Got ( z) where Gε ( z) is the non-dimensional H norm of the resonse of the detuned rimary system and Got ( z ) is the non-dimensional H norm of the rimary system with otimally tuned TMD. λ ε, increases with increasing frequency detuning and decreasing mass ratio, and decreases with increasing daming of the structure. Fig. 48 dislays the frequency detuning amlification factor due to a erturbation of frequency tuning of a TMD otimized for minimizing the dislacement of the rimary system with resect to the H norm. Since the H norm is defined via an integral, the amlification is significantly smaller than the amlification observed for TMDs otimized with resect to the H norm. Frequency detuning increases also the H norm of the relative dislacement. The amlification factor is of similar magnitude of the amlification of the resonse of the rimary system. Design consideration Based on the theory develoed in the last sections, a design rocedure can be develoed. The very first ste in designing a TMD is to take a decision according to which norm the TMD should to be otimized. An otimization with resect to H norm is recommended for structures that are excited by loads exhibiting mainly eriodical time comonents (e.g. loads generated by human activities like walking, running, juming, dancing etc. or machines). For loads having mainly a wide band stochastic character (e.g. wind loads, earthquake loads), an otimization with resect to H norm is more aroriate. The second ste is to identify the vibration mode of the structure that shall be damed. Using the shae of this mode, an equivalent single-degree-of-freedom model of the structure is generated by comuting the associated modal mass and stiffness. For obtaining a scaling of the modal mass that is comatible with Eq. () or (), the amlitude of the mode shae at the osition where the TMD is attached to the rimary structure has to be chosen equal to unity in the direction of action of the TMD. The modal stiffness is estimated using the modal mass and the natural frequency of the vibration mode. The natural frequency should be determined with field tests because the natural frequencies comuted from analytical or numerical models Page 57 of 55

58 SAMCO Final Reort 6 are too unreliable to be used for design uroses. As already ointed out, an inaccurate frequency tuning reduces significantly the effectiveness of a TMD. Although much less critical as the natural frequency, the structural modal daming should also be determined by field tests or estimated conservatively based on exerience. The third ste regards the choice of the mass of the TMD. For a given daming ratio ζ of the structure, the mass ratio µ determines the otimal arameters η t, ot and ζ t, ot as well as the resonse of the structure. The mass ratio is chosen to satisfy the maximum accetable resonse level of the structure. In rincile, the latter can always be achieved since the resonse of the rimary structure decreases monotonically with increasing mass ratio. The maximum accetable resonse level may be given by codes, guidelines, rovisions or generally acceted rules. In the design of a TMD, a number of issues have to be considered:. A TMD is effective if a sufficiently large TMD mass (mass ratio µ. ) can be accommodated and the vibration mode exhibit small structural daming. ζ. Beyond a mass ratio of µ =.5 a further increase of TMD mass results only in a modest further reduction of the resonse of the rimary system.. A TMD is only effective when roerly tuned to a secific natural frequency or, equivalently, to a secific vibration mode. Structures with several closely saced natural frequencies which are all inside the frequency band of the excitation require a TMD for each natural frequency for achieving an effective vibration mitigation. Since each natural frequency is associated to a vibration mode and the couling between the vibration modes is usually negligible, the design rocedure for single modes can be alied. 3. The relative dislacement, the dislacement of the mass of the TMD with resect to the structure, can be considerable and requires sace that must be accommodated within the structure. If the relative dislacement of an otimally tuned TMD exceeds existing sace constraints, the relative dislacement can be reduced by increasing the mass of the TMD or, if the latter is not feasible, by increasing the daming ratio ζ of the TMD. For structures with structural daming ( ζ > ) a limitation of relative dislacement can always be achieved. However, any restriction of the relative dislacement that can not be accommodated by an increase of the mass of the TMD reduces the effectiveness of the TMD. 4. The mass of the TMD with resect to the modal mass of the structure determines the effectiveness of the TMD. In existing structures, the mass of the TMD may be restricted by the load bearing caacity of the structure and the installation of a single TMD may require a strengthening of the structure. A design with multile TMDs having smaller masses may avoid a strengthening. However, the effectiveness decreases if not all TMDs can be installed at the locations with maximum amlitude of the mode shae. Furthermore, because of the smaller mass, the TMDs will exhibit a larger relative dislacement and hence require more sace than a single TMD and, finally, the costs of multile TMDs is generally larger than the costs of a single TMD. 5. Natural frequency and daming ratio of structures may change because of the effect of live loads (e.g. edestrians on footbridges) or environmental arameters (e.g. temerature). Since the effectiveness and resonse of Page 58 of 55 t

59 SAMCO Final Reort 6 TMDs are very sensitive to an incorrect frequency tuning, changes of natural frequencies have to be considered in the design. Bad frequency tuning increases the resonse of the structure as well as the relative dislacement of the TMD. 6. Since the relative dislacement can be considerable, the fatigue behaviour of the TMD s srings has to be considered. Testing and validation For determining reliably the relevant absorber arameters, the natural frequencies and the daming ratios of the structure s relevant modes have to be established exerimentally. This can be accomlished by ambient or forced vibration tests. A reliable estimation of the daming ratios can usually only be established by forced vibration tests. For lightweight structures, a simle and effective method is an imact test (e.g. by droing a sandbag). A TMD should be tuned in the laboratory before roceeding to it s installation on a structure. Natural frequency and daming ratio of a TMD are established by free vibration tests. Notice that the daming devices may contribute to the stiffness of the TMD and therefore affect its natural frequency. This has to be considered during the tuning of a TMD because of its high sensitivity with resect to frequency tuning. The final tuning of the TMD s natural frequency is erformed after installation. The easiest method for frequency tuning is to increase or reduce TMD s mass to fit the correct natural frequency. For lightweight structures, the validation can be erformed by imact test (e.g. by droing a sandbag). The imulsive force covers a wide frequency band so that the resonse of the structure is similar to the dynamic magnification curve shown in Fig. 34. Structures with an otimally tuned TMD show an amlification curve with two hums of equal height. Another useful test is to comare the resonse of the structure with locked and unlocked TMD. The effectiveness of the TMD can be obtained by comuting the ratio of the maximum resonse with locked and unlocked TMD. Obviously, a reliable estimation requires a test with controllable excitation forces. Imlementation examles The most common imlementation of a TMD consists of a mass either hanging or being suorted by several srings as shown in Fig. 3. This design is often used for controlling bridge and floor motions. However, there exist a large variety of different designs. Pendulum TMD are very often used for motion control of steel smoke stacks (Petersen ()). Solutions for sace-limited alications include the tuned roller endulum damer and the multi stage endulum damer (Soong and Dargush (997)). TMDs have been successfully installed for vibration resonse control of: Pedestrian bridges, stairs, sectator stands excited by walking or juming eole (Bachmann and Weber (995)). Lightweight factory floors excited in one of their natural frequencies by machines. Tall free-standing structures (bridges, ylons of bridges, smoke stacks, towers) excited by wind-induced loads (Kwok (984), Kwok and Samali (995), Ohtake et al. (99), Petersen (), Petersen (98), Ueda et al. (99)). Page 59 of 55

60 SAMCO Final Reort 6 References Asami, T., et al. (), "Analytical solutions to H-infinity and H- otimization of dynamic vibration absorbers attached to damed linear systems", Journal of Vibration and Acoustics-Transactions of the Asme, 4(), Asami, T., and Nishihara, O. (3), "Closed-form exact solution to H-infinity otimization of dynamic vibration absorbers (Alication to different transfer functions and daming systems)", Journal of Vibration and Acoustics-Transactions of the Asme, 5(3), Bachmann, H., and Weber, B. (995), "Tuned Vibration Absorbers for "Lively" Structures", Structural Engineering International, 5(), Crandall, S. H., and Mark, W. D. (963), Random vibration in mechanical systems, New York: Academic Press. Den Hartog, J. P. (947), Mechanical Vibrations, Third ed, New York and London: McGraw-Hill Book Comany. Frahm, H. (99), Device for Damed Vibrations of Bodies. U. S. Patent No Ioi, T., and Ikeda, K. (978), "On the Dynamic Vibration Damed Absorber of the Vibration System", Bulletin of the Jaanese Society of Mechanical Engineering,, (5), Korenev, B., and Reznikov, L. M. (993), Dynamic Vibration Absorbers: Theory and Technical Alications, Chichester, Eng-land: John Wiley & Sons Ltd.. Kwok, K. C. S. (984), "Daming Increase in Building with Tuned Mass Damer", Journal of Engineering Mechanics-Asce, (), Kwok, K. C. S., and Samali, B. (995), "Performance of Tuned Mass Damers under Wind Loads", Engineering Structures, 7(9), Nishihara, O., and Asami, T. (), "Closed-form solutions to the exact otimizations of dynamic vibration absorbers (mini-mizations of the maximum amlitude magnification factors)", Journal of Vibration and Acoustics-Transactions of the Asme, 4(4), Ohtake, K., et al. (99), "Full-Scale Measurements of Wind Actions on Chiba Port Tower", Journal of Wind Engineering and Industrial Aerodynamics, 43(-3), Ormondroyd, J., and Den Hartog, J. P. (98), "The Theory of the Dynamic Vibration Absorber", Transactions of the ASME, APM-5-7, 9-. Petersen, C. (), Schwingungsdämfer im Ingenieurbau, München: Fa. Maurer und Söhne, GmbH & Co. KG. Petersen, N. R. (98), "Design of Large Scale Tuned Mass Damers", Structural control, June 4-7, 979. Ontario, Soong, T. T., and Dargush, G. F. (997), Passive Energy Dissiation Systems in Structural Engineering, Chichester: John Wiley & Sons. Ueda, T., et al. (99), "Suression of Wind-Induced Vibration by Dynamic Damers in Tower-Like Structures", Journal of Wind Engineering and Industrial Aerodynamics, 43(-3), Page 6 of 55

61 SAMCO Final Reort 6 Warburton, G. B. (98), "Otimum Absorber Parameters for Various Combinations of Resonse and Excitation Parameters", Earthquake Engineering & Structural Dynamics, (3), Often used notations Symbol Descrition Unit G ( z) non-dimensional comlex transfer function - G ( z) H norm of non-dimensional comlex transfer function - G ( z) H norm of non-dimensional comlex transfer function - c dashot constant kg/s f frequency Hz k stiffness constant kg/s m mass kg η frequency ratio - µ mass ratio - ω circular frequency rad/s ζ daming ratio - Subscrits r t Descrition main structure relative tuned mass damer Page 6 of 55

62 SAMCO Final Reort 6 Aendix Table : Non-dimensional transfer functions for various excitations and resonse arameters. Excitation F F F F u&& g u&& g u&& g u&& g Outut arameter u u u& ω u && u ω u u u r ω u && u g ω u& && u g && u + && u && u g ω u r && u g A(z) B(z) η z ηtζ tz t η ζ z t t z ( η z ) t z( η z ) t η ζ z z ( + µ ) η z ( + µ ) ηtζ tz t ( + µ ) η ζ z z (( + µ ) η z ) t t t t z(( + µ ) η z ) t 3 t ( + µ ) η ζ z t t 3 Legend A(z) + ib(z) G(z) = : non-dimensional transfer function with coefficients A(z), B(z),C(z) and C(z) + id(z) D(z), where D(z) = z((( + µ ) ζ η + ζ )z ( ζ + ζ η ) η ) C(z) = z (( + µ ) η + 4 ζ ζ ) η + )z + η and t t t t t 4 t t t t F : u&& g : excitation (force) alied to the rimary system excitation (acceleration) alied to the base u : reference dislacement ( u F / k u = S w / k for = for harmonic excitation with white noise excitation w(t) with autocorrelation R ww (t) = S wδ (t) ) i t Fe ω and z : non-dimensional frequency z = ω/ ω Page 6 of 55

63 SAMCO Final Reort 6 Table : Aroximate solutions of the H otimization roblem for rimary systems with vanishing structural daming ( ζ = ). Excitation Minimized arameter Transfer function η u r (z) ζ t, min G(z) u i t Fe ω u u u + µ 3µ 3 + µ 8( + µ ) µ µ µ 5 µ i t Fe ω u& u& ω u µ + + µ 3µ 5 + µ 8( + µ ) µ / µ + 3 µ / µ µ 49 ( + )( + µ µ ) 5 µ 5 i t Fe ω u&& && u ω u + µ 3µ 7 + µ 8( + µ ) 3 µ + 3 µ /3 4 + µ + µ / 5 µ 4 µ / 5 + µ / i t && uge ω u ω u && u g µ / + µ 3µ 3 + µ 8( + µ ) µ 3 4 ( + µ + µ ) ( ) 4 + µ µ + µ 5 µ 5 75 i t && uge ω u& ω u& && u g + µ 3µ 8( + µ ) + µ µ ( + µ ) µ + µ 5 µ 6 i t && uge ω && u + && ug && u + && u && u g + µ 3µ 3 + µ 8( + µ ) µ µ 3 ( + µ ) µ + µ 5 µ 5 Legend i t Fe ω : harmonic excitation alied to the rimary system i t && uge ω : harmonic excitation alied at base Pa

64 SAMCO Final Reort 6 Table 3: Closed form exressions of the non-dimensional H norm for various excitations and resonse arameters. Excitation F F F F u&& g u&& g u&& g u&& g Resonse arameter E[u ]m ω 3 πs f G(z) ( + µ ) ζ η + ( µ + 4( + µ ) ζ ) ζ η ( + µ 4( + µ ) ζ 4 ζ ) η + 4ζ ζ η + ζ 4 ( + µ ) ζ ζ η + ( µ + 4( + µ ) ζ ) ζ η ( ( + µ ) ζ ζ ) ζ ζ η + ( µ + 4 ζ ) ζ η + ζ ζ 4 3 t t t t t t t t t 4 3 t t t t t t t t t t 4 3 & ω ( + µ ) ζ tη t + ( µ + 4 ζt ) ζηt ( ζ t ζ) η t + 4ζ t ζη t + ζt 4 3 πs 4 ( + µ ) ζ f tζη t + ( µ + 4( + µ ) ζ t ) ζηt ( ( + µ ) ζt ζ) ζtζη t + ( µ + 4 ζ) ζt η t + ζtζ E[u ]m E[u && ]m πs ω f E[u ]m ω 3 r πs f E[u ] ω 3 πs f N( µ, η, ζ, ζ ) 4 ( + µ ) ζ ζ η + ( µ + 4( + µ ) ζ ) ζ η ( ( + µ ) ζ ζ ) ζ ζ η + ( µ + 4 ζ ) ζ η + ζ ζ t t 4 3 t t t t t t t t t t 4( µηζ + ζ ) N( µ, η, ζ, ζ ) = ( + 4( + µ ) ζ ) µ ζ η + ( + 4( + 3µ + µ ( + ζ )) ζ ) ζ η + ( µ 4( + µ ) ζ + 6µ ζ + 4( µ + 4( + µ ) ζ ) ζ ) ζ η t t t t t t t t t t t ( µ 4( + µ µ ) ζ + 4( 4( + µ ) ζ ) ζ 6 ζ ) η + 4 ζ ζ ( + µ + 4 ζ ) η + ζ ( + 4 ζ ) 4 t t t t t t ζ η + ζ 4 ( + µ ) ζ ζ η + ( µ + 4( + µ ) ζ ) ζ η ( ( + µ ) ζ ζ ) ζ ζ η + ( µ + 4 ζ ) ζ η + ζ ζ t t 4 3 t t t t t t t t t t ( + µ ) ζ η + ( + µ ) ( µ + 4( + µ ) ζ ) ζ η ( + µ ) ( + µ 4( + µ ) ζ 4 ζ ) η + (4( + µ ) ζ + µ ) ζ η + ζ 4 ( ) ( 4( ) ) ( ( ) ) ( 4 ) t t t t t t t t t µ ζtζη t + µ + + µ ζt ζηt + µ ζ t ζ ζ tζη t + µ + ζ ζt η t + ζ tζ & ω ( + µ ) ζ tη t + ( + µ )( µ + 4( + µ ) ζt ) ζηt ( + µ ( + µ ) ζ t ζ) η t + 4ζ t ζη t + ζ t 4 3 π 4 ( + µ ) ζ f tζη t + ( µ + 4( + µ ) ζ t ) ζηt ( ( + µ ) ζt ζ) ζtζη t + ( µ + 4 ζ) ζt η t + ζtζ E[u ] S 4 3 && + && ( + µ )( + µ + 4 ζ) ζtη t + ( µ + 4( +µ + 4 ζ) ζ t + 4 µζ ) ζηt ( + µ + 4( 4 ζ) ζ 4( +µ + 4 ζ) ζ t ) ζ tη t + 4( + 4 ζ) ζ t ζη t + ( + 4 ζ) ζt 4 3 πs ω 4 ( + µ ) ζ tζη t + ( µ + 4( + µ ) ζ t ) ζηt ( ( + µ ) ζt ζ) ζ tζη t + ( µ + 4 ζ) ζ t η t + ζtζ E[(u u g ) ] f E[u ]m ω 3 r πs f ( + µ )( + µ + 4 ζ ) ζ η + ( µ + 4( + µ ) ζ + 4 ζ ) ζ η + (4 ζ + µ ) ζ η + ζ 4 ( + µ ) ζ ζ η + ( µ + 4( + µ ) ζ ) ζ η ( ( + µ ) ζ ζ ) ζ ζ η + ( µ + 4 ζ ) ζ η + ζ ζ 3 t t t t t t 4 3 t t t t t t t t t t t Legend F : u&& g : E[u ], E[u r ] : E[u & ], random excitation alied to the rimary system with sectral density S f random excitation at base with sectral density S f && : mean of the square of the dislacement E[u ] u, velocity mean of the square of the relative dislacement u r u& and acceleration u&& Pa

65 SAMCO Final Reort 6 Table 4: Exact solutions of the H otimization roblem for rimary systems with vanishing structural daming ( ζ = ). Excitation F F F E[u ]m ω 3 πs E[u & ]m ω f πs f E[u && ]m πs ω f Minimized arameter Parameter otimized u u& η ζ t, + µ / + µ µ + 3 µ / 4 4 ( + µ ) + µ / min G(z) + 3 µ / 4 µ + µ µ + µ 4 µ +µ µ ( µ / 4)( + µ ) u&& µ / 4 ( + µ ) ( µ / )( µ + µ ) ( µ + µ )( µ / 4) µ E(u )m ω 3 r πs 3/ + µ µ + 3 µ / 4 f + µ µ 3/ 3/ ( µ + µ ) µ µ / 4 u&& g u&& g u&& g E[u ] ω 3 πs f E[u & ] ω πs E[(u u g ) ] f f && + && πs ω && u u r u& r r + && u g µ / + µ µ µ / 4 4 ( + µ ) µ / µ + µ 4 ( + µ ) + µ / + µ µ + 3 µ / 4 4 ( + µ ) + µ / µ / 4 ( + µ ) µ + µ ( + µ ) µ + 3 µ / 4 µ + µ 3 / + µ + µ / ( + µ ) µ µ / µ / 4 + µ µ 3 / 3/ ( + µ ) + µ + 3 µ / ( + µ ) µ + µ / + 3 µ / 4 3 Legend F : random excitation alied to the rimary system with sectral density S f u&& g : random excitation at base with sectral density S f E[u ], E[u r ] : E[u & ], && : mean of the square of the dislacement E[u ] u, velocity mean of the square of the relative dislacement u r u& and acceleration u&& Pa

66 SAMCO Final Reort 6 Table 5: Aroximate solutions of the H otimization roblem for rimary systems with structural daming ( ζ > ). Excita tion Minimized arameter η ζ t, F F F u&& g u&& g u&& g E[u ]m ω 3 πs f / 3 ( + µ / 4) µ µ (3 + 9 µ / µ /6 + 3 µ / 64) 3/ / / 3 3/ ζ + ζ ( + µ ) ( + µ / ) ( + 3 µ / 4) ( + µ ) ( + µ / ) ( + 3 µ / 4) & / / ω µ µ ( + 7 µ /8) 7 µ ( + 5 µ / 8) 3 ζ + ζ 3/ + ζ πs ( + µ ) ( + µ ) 4 ( + µ ) f E[u ]m && / µ µ + µ µ µ ( µ / 4) ( µ / )( µ 3 µ ) ( µ / 8) f E[u ]m πs ω E[u ] ω 3 πs f 3 ( 5 / 3 / 4) ( / 4) ( 435 /) ζ + µ / / 3 ( + µ / 4) µ µ ( + 83 µ / 4 + µ / µ / 64) 3/ / / 3/ ζ + ζ ( + µ ) ( + µ / ) ( + 3 µ / 4) 8 ( + µ ) ( + 3 µ / 4) ( + µ / ) & / ω 3 µ µ ( 7 µ / 8) 5 ( + µ ) 3 ζ 3/ ζ + ζ 5/ π ( + µ ) ( + µ ) 6 ( + µ ) f E[u ] S && + && / 3 µ ( + µ / 4) µ ( + 83 µ / 4 + µ / µ / 6 ζ 3/ / / + 3/ πs ω ( + µ ) ( + µ / ) ( + 3 µ / 4) 4 ( + µ ) ( + 3 µ / 4) ( + µ / ) E[(u u g ) ] f 3 5/ µ ( + 5 µ / 4 5 µ / 64) µ ζ 3/ ζ 3/ 5/ 3( + µ )( + 3 µ / 4)( + µ / ) 64( + µ ) ( + µ / ) ( + 3 µ / 4) / 3 µ 3 µ ( + 9 µ /) µ ( + 63 µ / 64) 3 ζ / + ζ + ζ 3/ 4 ( + µ ) 4 ( + µ ) ( + µ ) ( 79 µ /59) µ ( µ / 4) ( 459 µ / 59) ( µ / )( µ 3 µ µ / ζ + ( + 9 µ / µ / 4) 3 3/ 3 µ µ ( + 5 µ / µ / µ /8) ζ 3/ + 3/ 5/ 3( + µ )( + 3 µ / 4)( + µ / ) ( + µ ) ( + µ / ) ( + 3 µ / 4) µ µ µ µ + µ 4 ( + µ ) 4 ( + µ ) ( + µ ) / ( 3 / 4) 3 ( / 48) 3 ζ ζ 3/ + ζ ( 5 / / /8) µ ζ 3/ + µ + µ + µ µ 3/ 5/ 5/ 64( + µ )( + 3 µ / 4)( + µ / ) ( + µ ) ( + µ / ) ( + 3 µ / 4) 3 3 3/ Legend F : u&& g : E[u ], E[u r ] : E[u & ], random excitation alied to the rimary system with sectral density S f random excitation at base with sectral density S f && : mean of the square of the dislacement E[u ] u, velocity mean of the square of the relative dislacement u r u& and acceleration u&& Pa

67 SAMCO Final Reort Tuned liquid damer Theoretical background The basic rincile of a tuned liquid damer (TLD) is similar to that of a tuned mass damer (TMD). In articular, a moving secondary mass which is realized by the fluid mass is introduced into the rimary structural system. The gravity acts as restoring force and energy dissiation is rovided through viscous action mainly in the boundary layers of the fluid. A simle realization of TLD consists of rectangular or circular container with water inside (Fig. 49). A horizontal motion of the container roduces a sloshing motion of the fluid. A variation of TLDs is the tuned liquid column damer (TLCD) which consists of a U-shaed container artially filled with fluid (Fig. 5). The fluid column starts oscillating as soon as the container is subjected to horizontal motion. The dynamic vibration absorber for shi alications roosed by Frahm was actually a forerunner of modern tuned liquid column damer (Den Hartog (997), Frahm (99)). For civil engineering alications, the first investigations started in the mid-98s by the work of Bauer (984). mass TLD m T mass TLD m T Bauteil force f(t) Bauteil force f(t) dashot c stiffness k mass m dashot c stiffness k mass m u(t) u(t) Fig. 49: Schematic illustration of a sloshing (a) and a column tuned liquid damer (b) couled to a rimary structure. Unlike a TMD, generally the resonse of a TLD is highly nonlinear. Liquid sloshing or fluid flow through orifices is tyically a highly nonlinear henomenon. The resonse of the rimary structure with a TLD is therefore amlitude deendent, even h L m f A f m f Fig. 5: Princial arameters of sloshing and column TLD. for rimary structures oerating within the linear elastic regime. Considerable research effort has been focused in understanding and quantifying the dynamic behaviour of nonlinear sloshing TLD. The most romising macroscoic models are based on extensions of classical theory of shallow water gravity waves with finite amlitude (Leelletier and Raichlen (988), Shimizu and Hayama (987), Sun et al. (99)). These models contain secial terms to account for daming and fit reasonably well with test results. The steady state resonse curves of nonlinear sloshing TLD show some similarity with the steady state amlitude resonse curves of oscillating systems with nonlinear hardening roerties (Fujino et al. (99)). The fundamental frequency of sloshing TLD can be estimated from linearized small amlitude theory and is given by Page 67 of 55

68 SAMCO Final Reort 6 f = π πg πh tanh L L () The energy dissiation er cycle of a TLD deends strongly from the excitation frequency. For frequencies in the neighbourhood of the natural frequency the energy dissiation was found to be significantly higher than elsewhere. The dynamic behaviour of tuned liquid column damers (TLCD) is more suited for a descrition with simle one degree of freedom systems. Nevertheless, one degree of freedom systems are still nonlinear for significant vibration amlitudes of the fluid column (Hochrainer (), Saoka et al. (988)). The nonlinearity mainly involves the daming term because of the energy dissiation due to turbulent flow. This term is usually modelled to be roortional to the square of the velocity of the oscillating fluid column. The natural frequency of a CTLD can be estimated using g f = () π L Design issues Because of the lack of reasonably simle models for describing the dynamic behaviour of a TLD, no generally acceted design rocedure exists so far. In rincile, because of the close similarity in the basic rincile of oeration between TMD and TLD, the same aroach as described for TMDs can be alied for TLDs. In fact, a tuned mass damer analogy for the sloshing TLD was roosed for design issues with frequency deendent virtual mass and dashot (Sun et al. (995)). However, because the virtual mass and dashot are amlitude-deendent, these had to be determined by exeriments and may therefore change due to imlementation details. Because of the much lower mass density of water with resect to steel, TLD needs much more sace than TMD for installation. In addition, sloshing TLDs are usually imlemented as a rack of tanks therefore requiring even more sace. The fundamental frequency of sloshing TLDs can be estimated using the linearized theory of shallow water waves. Much more difficult is redicting the energy dissiation of a TLD. Therefore, laboratory tests are mandatory to otimize the dynamic behaviour of TLD with resect to the rimary structure. Because of the nonlinear hardening behaviour of sloshing TLD, its effectiveness is less sensitive to a correct frequency tuning of the damer. Rectangular tanks are alied for structures with different natural frequencies in the two rincial directions. For a given water deth, the frequencies are tuned by an adequate selection of the lan dimensions of the tank. However, because the existing theories are only tested for unidirectional excitation, secial care should be taken in designing sloshing TLD acting in two directions. For structures with nearly equal natural frequencies in the two rincial directions, a circular tank may be used. Testing and validation The testing and validation of TLDs follows the same rocedures as described for TMD. Page 68 of 55

69 SAMCO Final Reort 6 Imlementation Tuned liquid damers have been rimarily used for mitigating wind-induced vibrations of tall structures with very small natural frequencies. The first alications of nonlinear sloshing TLD were imlemented in Jaan and include the Nagasaki Airort Tower, the Yokohama Marine Tower and the Shin-Yokohama Prince Hotel (Tamura et al. (995), Tamura et al. (996), Wakahara et al. (99)). The hardware imlementation of sloshing TLDs is much simler as that for TMD. Each damer, generally, consists of a rack of fluid containers. The installation requirements are minimal. Sloshing TLD may be easily added in existing buildings and roduce very little maintenance costs. References Bauer, H. F. (984), Oscillations of Immiscible Liquids in a Rectangular Container - a New Damer for Excited Structures, Journal of Sound and Vibration, 93(), Den Hartog, J. P. (947), Mechanical Vibrations. Third ed. 947, New York and London: McGraw-Hill Book Comany. Frahm, H. (99), Device for Damed Vibrations of Bodies. U. S. Patent No , 99. Fujino, Y., Sun, L., Pacheco, B. M., and Chaiseri, P. (99), Tuned Liquid Damer (Tld) for Suressing Hori-zontal Motion of Structures, Journal of Engineering Mechanics-Asce, 8(), 7-3. Hochrainer, M. (), Control of Vibrations of Civil Engineering Structures with Secial Emhasis on Tall Building, Institut für Allgemeine Mechanik, Vienna University of Technology, Wien. Leelletier, T. G., and Raichlen, F. (988), Nonlinear Oscillations in Rectangular Tanks, Journal of Engineering Mechanics-Asce, 4(), -3. Saoka, Y., Sakai, F., Takaeda, S., and Tamaki, T. (988), On the Suression of Vibrations by Tuned Liquid Column Damers, in Annual Meeting of JSCE, Tokyo: JSCE. Shimizu, T., and Hayama, S. (987), Nonlinear Resonses of Sloshing Based on the Shallow-Water Wave Theory, Jsme International Journal, 3(63), Sun, L. M., Fujino, Y., Pacheco, B. M., and Chaiseri, P. (99), Modelling of Tuned Liquid Damer (Tld), Journal of Wind Engineering and Industrial Aerodynamics, 43(-3), Sun, L. M., Fujino, Y., Chaiseri, P., and Pacheco, B. M. (995), The Proerties of Tuned Liquid Damers Using a Tmd Analogy, Earthquake Engineering & Structural Dynamics, 4(7), Tamura, Y., Fujii, K., Ohtsuki, T., Wakahara, T., and Kohsaka, R. (995), Effectiveness of Tuned Liquid Damers under Wind Excitation, Engineering Structures, 7(9), Tamura, Y., Kohsaka, R., Nakamura, O., Miyashita, K., and Modi, V. J. (996), Wind-induced resonses of an airort tower - efficiency of tuned liquid damer, Journal of Wind Engineering and Industrial Aerodynamics, 65(-3), Page 69 of 55

70 SAMCO Final Reort 6 Wakahara, T., Ohyama, T., and Fujii, K. (99), Suression of Wind-Induced Vibration of a Tall Building Using Tuned Liquid Damer, Journal of Wind Engineering and Industrial Aerodynamics, 43(-3), Notations Symbol Descrition Unit A cross sectional area m L container length in the sloshing direction m g gravity constant m/s h deth m m mass kg ρ mass density kg/m 3 µ mass ratio - Subscrits f Descrition fluid Page 7 of 55

71 SAMCO Final Reort 6.. Semi-active devices... Defintion of term semi-active Fig. 5: Definitions and characteristics of daming devices often used in structural control. Semi-active devices are actuators that cannot roduce ower and therefore, e.g., excite structures (Gavin and Alhan (5), Sencer Jr. et al. (997), Sencer Jr. and Nagarajaiah (3), Xu et al. (3)). However, they are able to dissiate structural vibration and - in contrast to assive damers - the amount of energy dissiation within the semi-active device may adjusted by control of the device force. Thus, the oerating range of semi-active devices may be defined by the force, which is desired by the controller, and the device velocity, which is the velocity of the iston, rod, shaft etc deending on the device design, as follows P sa device = P des : : ( f des x& sa device ) ( f x& ) des sa device > () Page 7 of 55

72 SAMCO Final Reort 6 The different oerating ranges of assive damers, semi-active devices such as controllable damers and fully active devices, usually called actuators, are illustrated in Fig. 5. The oerating range of many semi-active devices is not only constraint to quadrants II and IV, which include dissiative force values only (in view of the device), but further constraint by a maximum and minimum device force (Fig. 5, bottom right). The force limitation by a maximum value is nothing secial since this roerty is common for any semi-active and fully active control device. However, the additional restriction by a minimum device force that is relatively large comared to the minimum force of actuators given by friction of seals, bearings, and gears, lays an imortant issue in control oint of view. This fact imlies that the desired force trajectory is not only bounded by the maximum device force but also constraint by a considerable minimum force value. Esecially in the field of vibration mitigation, the minimum device force leads to claming effects at osition of the semi-active device if the desired daming force is far smaller than the minimum force of the semi-active device because then minimum force is alied to mitigate the structure. Claming the structure at device osition is equivalent with generating a nodal oint of the structure at that osition. Then, vibration velocity at device osition is zero. As a result, structural vibration energy may not be dissiated anymore within the semi-active device. Hence, structural vibrations decay as without external semi-active daming device. Consequently, not only the maximum force of semi-active devices but also the minimum device force must be taken into account for the design of semi-active devices. In accordance with customers, engineers have to assume the maximum magnitude of exected structural vibrations (assumtion of worst case scenario) in order to determine the maximum device force which is mandatory so that the desired maximum control force may be tracked. In order to determine the minimum force value of the semi-active device, engineers and customers together have to define this vibration magnitude threshold below which the semi-active device may clam the structure and therefore cannot damen vibrations. According to the values of maximum and minimum daming forces, the semi-active device may be evaluated or designed and manufactured, resectively. References Gavin, H. P., and Alhan, C. (5), Guidelines for low-transmissibility semi-active vibration isolation, Journal of Smart Materials and Structures, 4, Sencer Jr., B. F., Dyke, S. J., Sain, M. K., and Charlson, J. D. (997), Phenomenological Model of a Magnetorheological Damer, Journal of Engineering Mechanics, ASCE, 3(3), Sencer Jr., B. F., and Nagarajaiah, S. (3), State of the Art of Structural Control, Journal of Structural Engineering, 9(8), Xu, Z.-D., Shen, Y.-P., and Guo, Y.-Q. (3), Semi-active control of structures incororated with magnetorheological damers using neural networks, Journal of Smart Materials and Structures, (), Page 7 of 55

73 SAMCO Final Reort 6... Magnetorheological/electrorheological fluid damer Main working rinciles Magnetorheological and electrorheological fluid damers (MR damers, ER damers) reresent a class of controllable fluid damers where the shear force of the fluid is controlled by an external magnetic and electric field, resectively (Chen and Yang (3), Gordaninejad et al. (), Han et al. (), Kornbluh et al. (4), Nakamura et al. (), Oh and Onoda (), Sencer Jr. and Nagarajaiah (3)). Basically, MR/ER damers consist of a housing including the MR/ER fluid and the moving art of a disc or iston, deending if the damers are built in rotational or cylindrical form (Fig. 5, Fig. 53, Liao and Lai (), Weber et al. (), Weber et al. (5a), Weber et al. (5c)). The MR/ER fluid is a susension of oil, additives and articles whose olarisation may be influenced by an external magnetic/electric field. Hence, the control inut variable of MR and ER damers is current and voltage, resectively. The amount of olarised articles deends on the magnetic/electric field magnitude and influences the shear stress of the MR/ER fluid. Therefore, according to the alied current/voltage, the shear force of the MR/ER fluid and eventually the damer force acting on the iston may be controlled. Fig. 5: Rotational MR damer, force range aroximately [, 3] N. Fig. 53: Cylindrical MR damer with gas ocket (accumulator) for comensation of iston volume; force range aroximately [7, 8] N. Besides the control inut variable (current/voltage), the main difference between these two semi-active daming devices is that MR damers roduce far larger daming forces relatively to ER damers for the same damer size. That is why MR damers are often used for vibration mitigation in the field of large civil structures whereas ER damers are used for vibration control of smaller structures. In the following, the working rinciles of these two damer tyes shall be exlained on behalf of MR damers. Characteristics of MR damers The MR fluid consists of oil, magnetizable articles and additives. The viscosity of the oil defines the sloe of the force-velocity trajectories at zero current (Fig. 54). The magnetizable articles may be olarized by the alied magnetic field which ends u in a controllable shear force deending on the amount of olarized articles. Since the shear force is controlled, the force trajectory in the force-velocity diagram aroximately describes a Coulomb friction, suerosed by oil viscosity (Guglielmino et Page 73 of 55

74 SAMCO Final Reort 6 al. (5), Zhou and Sun (5)). Thus, MR damers may be seen as controllable Coulomb damers where the sloe of the Coulomb force trajectories deends on the oil viscosity. Usually, the sloe is of minor imortance or even negligible comared to the force value at a certain velocity. At very small velocities, the force trajectories go into the zero-oint tracking a very large viscosity. The reason for that henomenon is that MR fluids behave more like rigid bodies (elastic) below yield stress and therefore at very small velocity values, whereas the fluid only starts to flow above yield stress, thus behaves lastic-viscous (Fig. 55, Fig. 56, Pignon et al. (996), Powell (995), Weiss et al. (994)). This ends u in the fact that force-dislacement trajectories or force-velocity trajectories of sinusoidal measurements include transient measurement data when the fluid changes between these two regions (Jung et al. (3), Gordaninejad et al. (), Yang et al. (), Yang ()). Fig. 54: Schematic steady-state damer behaviour of MR damers. Fig. 55: Measured MR damer behaviour in reand ostyield regions (MR damer of Maurer Söhne). Fig. 56: Force dislacement trajectories including transient measurement data from reyield to ostyield regions (MR damer of Maurer Söhne). Additives within the fluid should hel to avoid agglomeration and sedimentation of the magnetizbale articles. Sedimentation effects are not of high riority if MR damers are used to mitigate, e.g., cable vibrations due to wind loading because such vibrations exist for fairly long time comared to the time of one cycle. In this case, the MR fluid will be mixed u during the first coule of cycles. After that time, the articles are Page 74 of 55

75 SAMCO Final Reort 6 distributed homogeneously again. Then, the MR damer works as exected. However, if damers should otimally dam structural vibrations resulting from earthquakes, they have to react according to the rogrammed control strategy. Hence, damers with sedimentation which changes the damer behaviour instantaneously are not aroriate. Modelling Numbers of henomenological MR damer models have been resented by many researchers (Dominguez et al. (4), Jiménez and Alvarez-Icaza (5), Ramallo et al. (4), Rosenfeld and Wereley (4), Sahasrabudhe and Nagarajaiah (5), Sencer Jr. et al. (997), Wand and Liao (5), Yang et al. (4)). Most models are based the so-called Bouc-Wen model. This aroach tries to model the MR damer behaviour by combining linear sring elements, linear dash-ot elements, nonlinear daming elements, friction elements with the Bouc-Wen element that describes the hysteresis henomenon of MR damers. Exemlarily, the simle Bouc- Wen model roosed by Sencer Jr. et al. (997) is deicted in Fig. 57. The governing equation of the MR damer force including the force offset f describing the friction due to seals or the so-called accumulator is f f = m & x + c { x& x& ( ) + k x + α z () nonlinear daming coefficient where z describes the evolutionary variable which is defined as follows z& = γ x& z z n β x& z n + A x& () The nonlinear daming coefficient c (x& ) describes the lastic, nonlinear daming characteristics within the ost-yield region. According to Yang et al. (4), this daming coefficient may be modelled as follows ( a x& ) c x& = a e ( ) (3) A comletely different modelling aroach is the simle aroach of fitting a twodimensional function using measurement data (Song et al. (5), Weber et al. (), Weber et al. (5c)). The function is two-dimensional since the MR damer force deends on the actual damer iston velocity and actual damer current. The advantage of this maing rocedure is that the function may be simly inverted in order to have an inverse model of the steady-state MR damer behaviour available which comensates for the stationary main nonlinearities of MR damers (Fig. 58). Page 75 of 55

76 SAMCO Final Reort 6 Fig. 57: Simle Bouc-Wen modelling aroach for MR-damers. Fig. 58: Fitted inverse MR damer function. Testing MR/ER damers The common way to determine the MR damer behaviour is to measure the MR damer characteristics at constant current level for sinusoidal controlled damer iston dislacement (Oyadiji and Sarafianos (), Tse and Chang (4), Zhang et al. (4)). The reasons for sinusoidal controlled dislacement are: a) usually, hydraulic and neumatic aggregates are dislacement controlled (Fig. 59). b) sinusoidal dislacement with defined amlitude and frequency roduces the velocity range of [, x& ]. Hence, the deendency of the damer force on max the velocity may be measured over the entire desired velocity range. However, it must be mentioned that this measurement data also includes transient data due to the change of MR damer behaviour between reyield and ostyield regions (Fig. 56 and Fig. 6). That is the reason why force-velocity diagrams do not only show aroximate Coulomb force trajectories at constant damer current but also trajectories including non-dissiative force values and transient force values, see Fig. 6. Deending on the viscosity of the MR fluid, the steady-state arts of the force trajectories describe a Coulomb friction behaviour suerosed by a viscous art (Fig. 6). The following oints for damer measurements with high accuracy have to be resected: The joints must not have lay. Therefore, fitting bolts have to be used (Fig. 59). Otherwise, knocking effects will falsify measurements. If the force sensor is mounted between aggregate and daming device, the sensor moves u and down according to the iston dislacement. Due to the acceleration of the force sensor, the sensor signal has to be comensated by the acceleration term which results from the sensor s mass. It must be ensured that the aggregate iston at high frequencies of the desired dislacement sinus really tracks the desired sinus signal. Deending on the aggregate roerties, the actual iston dislacement tends to triangular shae with increasing frequency of the desired dislacement sinus. Page 76 of 55

77 SAMCO Final Reort 6 The force resonse of MR/ER damers may be measured if the desired dislacement is triangular. Then, the force resonse on velocity ste may be measured (Fig. 6, Fig. 63). Fig. 59: Testing daming devices. Fig. 6: Transient measurement data due to MR damer behaviour in reyield and ostyield regions with force overshoot, fairly small MR fluid viscosity. Fig. 6: Transient measurement data due to MR damer behaviour in reyield and ostyield regions, no force overshoot, fairly large MR fluid viscosity. Page 77 of 55

78 SAMCO Final Reort 6 Fig. 6: MR damer force resonse on aroximate velocity ste at A. Fig. 63: MR damer force resonse on aroximate velocity ste at 4 A. Imlementation Since forces of MR damers are larger than those of ER damers for the same housing dimensions, MR damers have been installed on numbers of civil structures for mitigation of vibrations caused by wind, combined wind-rain, combined wind-snow, edestrian, traffic, and earthquake loading. Wind and earthquake loading may evoke unaccetable large vibration amlitudes of tall buildings, esecially skyscraers. Large cable vibrations of stay cables of long san bridges are mainly due to wind loading or wind loading combined with rain or snow, which change the aerodynamic rofile of cables. Pedestrian loading is mainly reorted in the case of footbridges. In this case vibration amlitudes become too large if the first eigenfrequency of the footbridge is near the frequency of walking, namely around two Hertz (Bachmann et al. (995)). Although e.g. bridge deck vibrations due to traffic loading exist, these vibrations are of minor imortance due to the large deck mass and deck stiffness comared to the exciting forces caused by traffic. MR damers are rimarily manufactured by LORD Cororation and Maurer Söhne, Munich, Germany. MR damers of LORD have been installed on the Dongting Lake Bridge in China on all cables because vibrations due to combined rain-wind effects were unaccetable large (Fig. 64, Ko et al. ()). Each cable is equied with two MR damers in order to damen in-lane and out-of-lane vibrations. In the case of cable amlitudes above a defined level, these MR damers oerate in the so-called assive-on mode which means that MR damer current is hold on a constant value different from zero (Lee and Jeon (), Wan and Gordaninejad (), Weber et al. (5a)). If vibration amlitudes are below this defined threshold, MR damer current is zero. Then, MR damers work in the so-called assive-off mode. Maurer Söhne had the oortunity to install one MR damer on the longest stay cable of the Eiland Bridge nearby Kamen, The Netherlands, manly for long-term field tests (Fig. 65, lease see the following case study, Weber et al. (5b)). It must be mentioned that unaccetable large cable vibrations on this cable-stayed bridge have not been observed so far. In collaboration with Ema, the Swiss Federal Laboratories for Materials Testing and Research, this MR damer has been model-based designed using cable roerties, damer osition and assuming a worst-case scenario which has to be damed. When the MR damer was installed in October 4 on the bridge, decay measurements of the cable without and with MR damer at different Page 78 of 55

79 SAMCO Final Reort 6 current levels have been carried out. The target was to determine exerimentally the otimal current level deending on vibration amlitude for maximum additional daming. Finally, an intelligent on/off control strategy was imlemented and runs since late October 4. Fig. 64: Cable-stayed Dongting Lake Bridge in China, equied with MR damers of LORD Cororation. Fig. 65: Cable-stayed Eiland Bridge nearby Kamen, The Netherlands, equied with one MR damer of Maurer Söhne for long-term field tests and daming measurements. The Dubrovnik Bridge nearby Dubrovnik, Croatia, is an examle of a cable-stayed bridge, where cable vibration amlitudes during combined wind-snow events were that large that severe material damage resulted (Fig. 66). The damage was caused by contact friction between cables and bridge deck lightings. Again in collaboration with Ema, the Swiss Federal Laboratories for Materials Testing and Research, MR damers were model-based designed and manufactured for cables no. -6, no. - Page 79 of 55

80 SAMCO Final Reort 6 3 and for the backstays. These MR damers were installed in March 6. Maurer and Ema will erform field tests with forced cable vibrations in order to measure the additional daming rovided by feedback controlled MR damers. Each MR damer will be equied with one dislacement sensor between damer iston and damer cylinder. Based on the measured relative damer dislacement at Hz samling frequency, two control concets will be tested: velocity feedback and controlled friction force level. It is lanned that all MR damers will be real-time controlled by the friction force control. Fig. 66: Cable-stayed Dubrovnik Bridge nearby Dubrovnik, Croatia. MR damers have been also installed as tuneable daming element in tuned mass damers in footbridges to mitigate vertical bridge deck vibrations due to edestrian s imact from walking (Seiler et al. ()). In the field of earthquake engineering, MR damers have been installed in skyscraers using two different daming concets. One idea is to increase the overall building daming by introducing daming forces that decrease the relative dislacement between adjacent floors. Hence, MR damers are installed on each floor and are connected with the adjacent floor. The other concet is base isolation. Here, damers shall isolate the sensitive system - in this case the tall building from the imact of the earthquake. Then, the MR damer istons are, e.g., connected to the ceiling of the ground floor which is built as a soft story, whereas the MR damer cylinder must be fixed on the ground outside of the building. Hence, the whole soft story including the MR damers reresent the isolating system. Case Studies The case study described here is the imlementation of one feedback controlled MR damer of Maurer Söhne on the Eiland Bridge nearby Kamen, The Netherlands (Weber et al. (5b)). The entire rocedure of imlementation comrises the following stes:. MR damer design based on known cable roerties, intended damer osition on the cable and assumed worst case vibration scenario which still shall be ossible to be damed. Page 8 of 55

81 SAMCO Final Reort 6. MR damer fabrication and measurement of the force-dislacement characteristics at constant damer current. 3. Measurements of the decay rate of the free cable and of the cable with connected MR damer oerating at different settings. 4. Evaluation of measurement data in order to determine the otimum controller tuning that controls the MR damer force. 5. Controller design and imlementation on the bridge. ) Model-based MR damer design The basic design variables of MR damers are: Maximum damer force for given damer iston velocity: f x&, I ) MR max ( MR MR max Minimum damer force at zero current for given damer iston velocity: f x&, I ) MR min ( MR MR min = Maximum damer iston dislacement: x MR max According to these design variables, the MR fluid may be evaluated. Based on that evaluation, the MR damer geometry may be designed. In order to be able to determine the above listed design variables, a worst-case stay cable amlitude of the exected redominant mode has to be assumed for the cable with connected MR damer, thus for the damed cable (Fig. 67). Based on that assumtion, the maximum damer force and maximum damer iston dislacement may be model-based estimated. The basic idea is to simulate the vibrating cable with linear viscous damer that is otimally tuned for the target mode (Fig. 67). Although the linear viscous damer is characterized by a different force trajectory than the one of MR damers oerating at constant current, the simulated maximum damer force and damer dislacement values may be transformed to their corresonding values of a Coulomb friction damer whose behaviour is close to the one of MR damers oerating at constant current. The reason to simulate a fairly simle linear cable model with a linear viscous damer is that the connected system is again linear. Thus, comuting time is small and the correction of the disturbance force amlitude may be done also linearly (Fig. 67). One run of the simulation is comleted when steady-state conditions occur. Then, error ε and correction factor A may be determined using the steady-state results. The whole design rocedure stos when the error of the actual amlitude is smaller than a defined maximum error. Then, the steady-state simulation results deliver the wanted values of maximum damer force and maximum damer dislacement of the linear viscous damer that is otimally tuned for the vibrating mode according to (Fig. 68, Krenk ()) Page 8 of 55

82 SAMCO Final Reort 6 Fig. 67: Flow chart of damer design rocedure. c ot n T = (4) x ω d n The distribution of the disturbance force f w accords to the shae of the mode to be excited (Fig. 68). This guarantees that other modes are not excited. The amlitude A of the external disturbance force is chosen the way that the steady-state amlitude of the cable with linear viscous damer accords to the worst-case amlitude (Fig. 67). Fig. 68: Cable/damer system with distributed disturbance force according to the shae of the excited mode (here shown for the first vibration mode). The maximum force of the linear viscous damer may be transformed to its counterart of the energy equivalent Coulomb friction damer rovided that the maximum dislacements of both damer tyes are equal (Fig. 69) f π 4 fri max = f vis max (5) with: x x (6) fri max vis max Page 8 of 55

83 SAMCO Final Reort 6 Fig. 69: Characteristics of linear viscous and friction damers. Since MR damers working at constant current almost roduce Coulomb force trajectories, the transformed values are the wanted values of maximum MR damer force and maximum MR damer dislacement f f (7) MR max fri max x x (8) MR max fri max The minimum friction force cannot be smaller than the sum of sealing friction and shear force of MR fluids at zero current. If the target is to roduce MR damers with maximum large oerating range, then the friction force at zero current will be chosen to be equal to the minimum friction force mentioned above. If the target is to manufacture MR damers that roduce fairly large friction forces at zero current in order to enhance the fail safe behaviour against vibrations of large amlitudes (Weber and Feltrin (3)), then the friction force at zero current may be increased by additional sealing and different tye of MR fluids. f min = Min( seal friction + MR fluid friction@ A): maximum oerating range f MR min = > f min : enhanced fail safe behaviour (9) Summarizing, the resented design rocedure yields the maximum values of force and dislacement of MR damers oerating at constant current for maximum daming of the target mode with defined, steady-state worst-case amlitude. The minimum MR damer force at zero current is bounded by MR fluid characteristics and damer construction. The minimum MR damer force may be chosen to be larger than that minimum value but not smaller. The test MR damer had to be designed for the longest, 63 meter long stay cable of the Eiland Bridge. The worst-case assumtions for that cable with MR damer were: Page 83 of 55

84 SAMCO Final Reort 6.4 meter amlitude at mid san, and only the first mode vibrates. Using the known cable data listed in Table 6, the first eigenfrequency was estimated using the formula of a linear cable T = = () m L f n Table 6: Proerties of longest stay cable of Eiland Bridge. L m T x d [m] [kg/m] [kn] [m] The estimated first eigenfrequency became.844 Hz. Alying Eq. (4), the otimum viscosity of a linear viscous damer located at 7.88 meter was determined as.6e3 kg/s. Using this otimum linear viscous damer, the results of the modelbased damer design were the following required maximum MR damer values: f MR max = 4 kn, and x MR max =.35 m. Further desired design roerties were: The MR damer force at zero current shall be as small as ossible in order to roduce a maximum large oerating range of the MR damer. The deendency of the MR damer force at constant current on the damer iston velocity shall be as small as ossible. Then, the MR damer force is only a function of current. Hence, the inversion of the steady-state damer function for control uroses becomes simle and the actual velocity does not have to be measured. Fig. 7: Force dislacement trajectories of MR damer of the Eiland Bridge. Fig. 7: Force current relation of MR damer of the Eiland Bridge. Page 84 of 55

85 SAMCO Final Reort 6 ) Measured MR damer characteristics Based on the choice of MR fluid, the geometry of the MR damer was designed by Maurer Söhne GmbH & Co. KG, Munich, Germany, which also manufactured the MR damer. It was tested at the Lehrstuhl für Konstruktiven Ingenieurbau der Universität der Bundeswehr in Neubiberg, nearby Munich. As can be readily seen from Fig. 7 and Fig. 7, the required MR damer characteristics could be fulfilled. 3) Daming measurements on the bridge In order to know the amount of additional daming rovided by the MR damer of Maurer Söhne on the Eiland Bridge (Fig. 7, Fig. 73), the decay rate of both the free cable and the cable with MR damer at different constant current levels were measured. The stay cable was excited by man ower with a erendicularly connected roe at the frequency of the first vibration mode using a metronome (Fig. 74). The decay rate of the cable was evaluated at % relative cable length. The accelerometers on the to and bottom of the MR damer and the dislacement sensor as well detected claming of the cable due to the MR damer force which remained aroximately constant during decay time as a result of constant current. The MR damer clams the cable in its equilibrium osition as soon as the cable force comonent in damer direction is smaller than the damer force (Fig. 75, Weber et al. (5d)) f MR sin( α ) ( I ) < T ( sin( α ) sin( α) ) x& MR P :, f ( I ) T ( sin( α )) x& =, x = P = : claming MR MR MR no claming () Fig. 7: Instrumentation of MR damer on the Eiland Bridge nearby Kamen, The Netherlands. Fig. 73: Taking decay measurements on the Eiland Bridge nearby Kamen, The Netherlands. Page 85 of 55

86 SAMCO Final Reort 6 Fig. 74: Sketch of decay measurements on the Eiland Bridge nearby Kamen, The Netherlands. Fig. 75: Claming of cables. Fig. 76: Measured vibration decay at % cable length and damer osition with MR damer at A. Fig. 77: Measured vibration decay at % cable length and damer osition with MR damer at.4 A. Page 86 of 55

87 SAMCO Final Reort 6 Fig. 78: Measured vibration decay at % cable length and damer osition with MR damer at. A. Fig. 79: Measured vibration decay at % cable length and damer osition of free cable. Due to the aroximate Coulomb friction behaviour of the MR damer under consideration, vibrations at % relative cable length decay aroximately linearly as long as the MR damer works. If the MR damer works, the MR damer relative dislacement is larger than a certain band value which results from out-of-lane vibrations of the damer iston (Fig. 76, Fig. 77). If claming occurs, the relative MR damer dislacement becomes zero or is smaller than the band value, resectively. With the start of claming, the decay enveloe changes to an aroximate exonential function. This indicates that the daming is dominated by the aroximate viscous material daming of the cable only. If the damer current is set to A, the MR damer fully clams the cable at damer osition. Hence, the decay enveloe has an exonential shae and the decay rate is that of the free cable (Fig. 78, Fig. 79). At such high current level, the MR damer behaves like an almost comletely stiff bearing. 4) Evaluation rocedure of measurement data Although the measured acceleration at % relative cable length decays almost with linear enveloe during unclamed conditions (Fig. 76, Fig. 77), the daming was evaluated alying the aroach of the logarithmic decrement that basically assumes linear viscous daming (Bachmann et al. (995)). In that case, the ratio of two following maxima is constant Page 87 of 55

88 SAMCO Final Reort 6 X& ( ti ) X& ( t + T ) i d = constant () where d π T denotes the radial eigenfrequency of the damed structure. The logarithmic decrement becomes X& ( t ) i δ = ln ln( X ( ti )) ln( X ( ti ntd )) n = + X ( ti ntd ) & & (3) & + n which leads to the equivalent daming ratio as follows δ δ ζ = (4) 4π + δ π Fig. 8: Exonential aroximation of linear decay within the amlitude range from 95% to 5%. In order to kee the evaluation error due to the aroximately linear decay enveloe fairly small, the local maxima within the amlitude range of 95% to 5% of the maximum value were fitted by an exonential function in order to determine the logarithmic decrement and the equivalent daming ratio, resectively (Fig. 8). Moreover, this way, the decay rate is evaluated within the amlitude range of large values which could cause material damage. Hence, if the MR damer current is chosen for maximum logarithmic decrement for that amlitude range, the MR damer force is tuned in order to revent from ossible material damage. Due to some small modulation of the decay measurements by higher modes, the evaluation rocedure of the logarithmic decrement is modified as follows (Weber et al. (5d)):. Band ass filtering of the measured signals for the excited mode, in that case mode number.. Determination of the local maxima within the amlitude range of 95% - 5%. 3. Linear fitting of the natural logarithm of the selected local maxima in order to derive an estimate of the logarithmic decrement with small error, Eqs.(6)- (8). Page 88 of 55

89 SAMCO Final Reort 6 ln { X& ( t )}, i i = [, k] linear fitting fit = a + b t (5)-(7) b δ = ( t t ) k k For the cable amlitudes within the range of 95% - 5% of the maximum mid san amlitude of aroximately.75 m,.4 A turned out to be the otimum value of constant damer current for minimum decay time within the chosen amlitude (Fig. 8). The logarithmic decrement of the free stay cable was measured as.96% which is a tyical value for lightly damed steel structures. By attaching the MR damer oerating in the assive-off mode, thus A, the overall daming could be increased by an aroximate factor of four (Fig. 8). If the MR damer current is otimally tuned to the actual vibration amlitude, the overall daming was aroximately eight times larger than the daming of the free cable. If the MR damer clamed the cable at damer osition due to a far too large current, the overall daming was the same as for the free cable (see logarithmic decrement for A in Fig. 8). Fig. 8: Evaluated mean logarithmic decrement. Fig. 8: Schematic controller structure. 5) Controller The basic controller structure consists of analogue/digital converter, the band ass filter, the control law, and the digital/analogue converter (Fig. 8). The inut is the measured relative damer dislacement and the controller outut is a voltage signal [-, ] V, which is the command signal of the ower module that generates the desired MR damer current (Fig. 83). The controller arameters such as samle frequency, cut-off frequencies and control law arameters may be defined using a labto comuter connected to the controller by USB interface. In the case of the Eiland Bridge, a fairly simle control aroach has been imlemented. Basically, it is an on/off strategy that alies.4 A to the MR damer if the actual relative damer dislacement amlitude is above a defined threshold for a certain time and A for relative damer dislacement amlitude below the threshold for a certain time. Page 89 of 55

90 SAMCO Final Reort 6 The memory function is taken into consideration in order to revent from numerous on-off switching. The definition of a dislacement amlitude threshold results in: Low couling of cable to bridge deck in the case of relative dislacement amlitude below the threshold. This guarantees that the MR damer does not oerate in assive-on mode due to small, traffic induced bridge deck vibrations. MR damer energy consumtion is small. The required ower in the assive-off mode counts for.5 W with a current of ma and in the assive-on oerating mode at.4 A is.6 W with a current of ma. The required ower is sulied by a solar anel und a battery 4Ah / V, made lead/gel technology, reresenting the necessary energy buffer (Fig. 84). Fig. 83: Dislacement sensor and MR damer on the Eiland Bridge nearby Kamen, The Netherlands. Fig. 84: Solar anel and controller box mounted on the lightning ylon on the Eiland Bridge nearby Kamen, The Netherlands. The control law may be extended to a gain scheduling aroach where different current levels corresond to different relative damer dislacement amlitude ranges. Due to the fairly small mid san amlitude that could have been excited by the resented man owered tests, only one otimum damer current value for one amlitude range could have been determined. Therefore, the control law consists of only one gain. The hysical reason for the gain scheduling aroach deending on the actual relative damer dislacement amlitude may be exlained in the following. Deending on the MR fluid viscosity, the force trajectory of MR damers describes more or less a Coulomb force trajectory (Fig. 54). Hence, the force of MR damers oerating at constant current is indeendent of the actual collocated cable velocity or dislacement, resectively. As a result, during the decay of cable vibrations, the MR damer force remains constant and finally clams the cable at damer osition as soon as the MR damer force is larger than the cable force in damer direction (Fig. 75). In contrast, linear viscous damers will not clam the cable during a decay hase due Page 9 of 55

91 SAMCO Final Reort 6 to the roortionality between force and velocity. Concluding, if MR damers shall be used for cable vibration mitigation, control of their friction force is mandatory. The control law may be derived from the energy equivalent, otimum linear viscous damer as follows (Fig. 69) vis = f vis max xvis max = W fri = 4 f fri max x fri max W π (8) Assuming sinusoidal dislacement at damer osition, the velocity amlitude may be exressed in terms of dislacement amlitude and frequency W vis = W = π c x& π c x ω (9) fri ot n vis max xvis max = ot n vis max The otimum viscosity given by Eq. (4) may be substituted in Eq. (9). For the same maximum damer dislacements, the force level of the energy equivalent friction damer becomes f equiv fri max = x fri max π 4 Hence, if friction force damers shall dissiate the same amount of vibration energy as an otimally tuned linear viscous damer, then the actual friction damer force has to be controlled roortionally to the actual damer dislacement amlitude. The absolute force value of friction damers remains constant during one cycle of steady state vibrations in contrast to the force of linear viscous damers which changes sinusoidally during one cycle of steady state vibrations. For MR damers with force trajectories at constant current with fairly small sloe, the equations derived for the friction damer may be taken also for MR damers working at constant current. Provided that damer dislacement amlitude may be measured directly, the feedback gain of energy equivalent friction damers becomes g equiv fri π = 4 Linear viscous damers reresent the case of an ideal daming element where the daming coefficient is a constant indeendent variable. The force of an otimally tuned linear viscous damer is the roduct of viscosity and collocated velocity f ot vis ot n T x d vis T x d n () () = c x& () If such an otimum linear viscous damer has to be realized using an ideal actuator (Fig. 5), then the otimum viscosity reresents the otimum feedback gain if the collocated velocity is measured directly (Marathe et al. (4), Preumont ()) g = c (3) ot vis ot n Page 9 of 55

92 SAMCO Final Reort 6 Fig. 85: Emulation of desired daming characteristics using feedback controlled actuators. Fig. 86: Characteristics of real linear viscous damers. Generating the force trajectory of a, e.g., linear viscous damer by feedback control of actuators is called emulating linear viscous damers (Fig. 85, Nakamura and Nakazawa (), Marathe et al. (4), Stelzer et al. (3)). If a feedback controlled actuator emulates a linear viscous damer, the control strategy is called velocity feedback. There are several reasons why emulation of, e.g., linear viscous damers using feedback controlled actuators may be sensible:. the wanted linear viscous damer with the desired otimum viscosity is not available on the market,. the offered linear viscous damers often do not have a ure linear force trajectory, esecially below a certain velocity limit x& lim due to the sealing friction which ends u in a highly nonlinear force trajectory below x& lim (Fig. 86), and 3. the viscosity of a linear viscous damer cannot be tuned on site to the real structural roerties in contrast to feedback controlled actuators, where the feedback gain allows for a fine tuning. References Bachmann, H., et al. (995), Vibration Problems in Structures: Practical Guidelines, Birkhäuser Verlag Basel, ISBN Page 9 of 55

93 SAMCO Final Reort 6 Chen, S. H., and Yang, G. (3), A method for determining locations of electrorheological damers in structures, Journal of Smart Materials and Structures, (), Dominguez, A., Sedaghati, R., and Stiharu, I. (4), Modelling the hysteresis henomenon of magnetorheological damers, Journal of Smart Materials and Structures, 3, Gordaninejad, F., Saiidi, M., Hansen, B. C., Ericksen, E. O., and Chang, F.-K. (), Magneto-Rheological Fluid Damers for Control of Bridges, Journal of Intelligent Material Systems and Structures, 3(/3), Guglielmino, E., Stammers, C. W., Edge, K. A., Sireteanu, T., and Stancioiu, D. (5), Dam-By-Wire: Magnetorheological Vs. Friction Damers, Proceedings of the 6th IFAC World Congress, Prague, Czech Reublic, 4-8 July 5. Han, Y. M., Ham, M. H., Han, S. S., Lee, H. G., and Choi, S. B. (), Vibration Control Evaluation of Commercial Vehicle Featuring MR Seat Damer, Journal of Intelligent Material Systems and Structures, 3(9), Jiménez, R., and Alvarez-Icaza, L. (5), LuGre friction model for a magnetorheological damer, Journal of Structural Control and Health Monitoring,, 9-6. Jung, H.-J., Sencer Jr., B. F., M.ASCE; and Lee, I.-W., M.ASCE (3), Control of Seismically Excited Cable-Stayed Bridge Emloying Magnetorheological Fluid Damers, Journal of Structural Engineering, 9(7), Ko, J. M., Zheng, G., Chen, Z. Q., and Ni, Y. Q. (), Field vibration tests of bridge stay cables incororated with magneto-rheological (MR) damers, Proceedings of the International Conference on Smart Structures and Materials : Smart Systems of Bridges, Structures, and Highways, S.-C. Liu and Darryll J. Pines (eds.), Proceedings of SPIE (ubl.), 4696, 3-4. Kornbluh, R., Prahlad, H., Pelrine, R., Stanford, S., Rosenthal, M., and von Guggenberg, P. (4), Rubber to rigid, clamed to undamed: Toward comosite materials with wide-range controllable stiffness and daming, Proceedings of the International Conference on Smart Structures and Materials 4: Industrial and Commercial Alications of Structures and Technologies, E. H. Anderson (ed.), Proceedings of SPIE (ubl.), Vol. 5388, Krenk, S. (), Vibrations of a Taut Cable With an External Damer, Journal of Alied Mechanics, 67, Lee, Y., and Jeon, D. (), A Study on The Vibration Attenuation of a Driver Seat Using an MR Damer, Journal of Intelligent Material Systems and Structures, 3(7/8), Liao, W. H., and Lai, C. Y. (), Harmonic Analysis of a Magnetorheological Damer for Vibration Control, Journal of Smart Materials and Structures, (), Marathe, S. S., Wang, K. W., and Gandhi, F. (4), Feedback linearization control of magnetorheological fluid damer based systems with model uncertainties, Journal of Smart Materials and Structures, 3(5), 6-6. Nakamura, T., Saga, N., and Nakazawa, M. (), Imedance Control of a Single Shaft-tye Clutch Using Homogeneous Electrorheological Fluid, Journal of Intelligent Material Systems and Structures, 3(7/8), Page 93 of 55

94 SAMCO Final Reort 6 Oh, H.-U., and Onoda, J. (), An Exerimental Study of a Semiactive Magneto-Rheological Fluid Variable Damer for Vibration Suression of Truss Structures, Journal of Smart Materials and Structures, (), Oyadiji, S. O., and Sarafianos, P. (), Comaring the Dynamic Proerties Conventional and Electro-Rheological Fluid Shock Absorbers, Proceedings of the 7 th International Conference on Recent Advances in Structural Dynamics, Southamton, England, July, Pignon, F., Magnin, A., and Piau, J.-M. (996), Thixotroic colloidal susensions and flow curves with minimum: Identification of flow regimes and rheometric consequences, Journal of Rheology, 4(4), Powell, J.A. (995), Alication of a nonlinear henomenological model to the oscillatory behavior of ER materials, Journal of Rheology, 39(5), Preumont, A. (), Vibration Control of Active Structures, Kluwer Academic Publishers, Dordrecht. Ramallo, J. C., Yoshioka, H., and Sencer Jr, B. F. (4), A two-ste identification technique for semi-active control systems, Journal of Structural Control and Health Monitoring, (4), Rosenfeld, N. C., and Wereley, M. (4), Volume-constrained otimization of magnetorheological and electrorheological valves and damers, Journal of Smart Materials and Structures, 3, Sahasrabudhe, S., and Nagarajaiah, S. (5), Exerimental Study of Sliding Base-Isolated Buildings with Magnetorheological Damers in Near-Fault Earthquakes, Journal of Structural Engineering, 3(7), Seiler, C., Fischer, O., and Huber, P. (), Semi-active MR damers in TMD s for vibration control of footbridges, Part : Numerical analysis and ractical realisation, Proceedings of the International Conference on Footbridge, Paris, France, - November, AFGC OTUA (eds.), on CD. Song, X., Ahmadian, M., and Southward, S. C. (5), Modeling Magnetorheological Damers with Alication of Nonarametric Aroach, Journal of Intelligent Material Systems and Structures, 6(5), Sencer Jr., B. F., Dyke, S. J., Sain, M. K., and Charlson, J. D. (997), Phenomenological Model of a Magnetorheological Damer, Journal of Engineering Mechanics, ASCE, 3(3), Sencer Jr., B. F., and Nagarajaiah, S. (3), State of the Art of Structural Control, Journal of Structural Engineering, 9(8), Stelzer, G., J., Schulz, M., J., Kim, J., and Allemang, R., J. (3), A Magnetorheological Semi-active Isolator to Reduce Noise and Vibration Transmissibility in Atomobiles, Journal of Intelligent Materials Systems and Structures, 4(), Tse, T., and Chang, C. C. (4), Shear-Mode Rotary Magnetorheological Damer for Small-Scale Structural Control Exeriments, Journal of Structural Engineering, 3(6), Wan, X., and Gordaninejad, F. (), Lyaunov-Based Control of a Bridge Using Magneto-Rehological Fluid Damers, Journal of Intelligent Material Systems and Structures, 3(7/8), Page 94 of 55

95 SAMCO Final Reort 6 Wand, D. H., and Liao, W. H. (5), Modeling and control of magnetorheological fluid damers using neural networks, Journal of Smart Materials and Structures, 4, -6. Weber, F., Feltrin, G., Motavalli, M., and Aalderink, B. J. (), Cable Vibration Mitigation Using Controlled Magnetorheological Fluid Damers: A Theoretical and Exerimental Investigation, Proceedings of the International Conference on Footbridge, Paris, France, November -, AFGC OTUA (eds.), on CD. Weber, F., and Feltrin, G. (3), Influence of the offset friction of rheological fluid damers on the vibration mitigation erformance, Proceedings of Fifth International Symosium on Cable Dynamics, Santa Margherita Ligure (Italy), Setember 5-8 3, Weber, F., Feltrin G., and Motavalli, M. (5a), Passive daming of cables with MR damers, Journal of Materials and Structures, 38(79), Weber, F., Distl, H., and Nützel, O. (5b), Versuchsweiser Einbau eines adativen Seildämfers in eine Schrägseilbrücke, Beton- und Stahlbetonbau (5), Heft 7, Weber, F., Feltrin, G., and Motavalli, M. (5c), Measured LQG Controlled Daming, Journal of Smart Materials and Structures, 4, Weber, F., Distl, H., Feltrin, F., and Motavalli, M. (5d), Evaluation Procedure of Decay Measurements of a Cable with assive-on oerating MR Damer, Proceedings of Sixth International Symosium on Cable Dynamics, Charleston, SC (USA), Setember 9-5, Weiss, K. D., Carlson, J. D., and Nixon, D. A. (994), Viscoelastic roerties of magneto- and electro-rheological fluids, Journal of Intelligent Material Systems and Structures, 5(), Williams, K., Chiu, G., and Berhard, R. (), Adative-assive absorbers using shae-memory alloys, Journal of Sound and Vibration, 49(5), Yang, G., Ramallo, J.C., Sencer, B.F., Jr., Carson, J.D., and Sain, M.K. (), Large-scale MR fluid damers: Dynamic erformance considerations, Proceedings of the International Conference on Advances in Structure Dynamics, Hong Kong, Yang, G. (), Large-scale magnetorheological fluid damer for vibration mitigation: Modeling, testing and control, PhD dissertation, University of Notre Dame, Notre Dame, Indiana. Yang, G., Sencer Jr., B. F., Jung, H.-J., and Carlson, J. D. (4), Dynamic Modeling of Large-Scale Magnetorheological Damer Systems for Civil Engineering Alications, Journal of Engineering Mechanics, 3(9), 7-4. Zhang, T., Jiang, C., Zhang, H., and Xu, H. (4), Giant magnetostrictive actuators for active vibration control, Journal of Smart Materials and Structures, 3(3), Zhou, H. and Sun, L. (5), A full-scale cable vibration mitigation exeriment using MR damer, Proceedings of 6th International Conference on Cable Dynamics, Charleston, SC (USA), Setember 9, 5, Page 95 of 55

96 SAMCO Final Reort 6 Notations Symbol Descrition Unit I current A L cable length m P ower W T cable force N W energy, work J X amlitude of x m c viscosity kg/s f force, frequency N, Hz g gain - k stiffness kg/s m mass; cable mass er unit length kg; kg/m x dislacement m α angle rad δ logarithmic decrement - ε error - ω radial frequency rad/s ζ daming ratio - Suerscrits ot x Subscrits MR act d des fri max min sa u vis w Descrition otimal mean value of x Descrition MR damer actual damer desired friction damer maximum minimum semi-active control inut linear viscous damer disturbance Page 96 of 55

97 SAMCO Final Reort Controlled shae memory alloy Theoretical background Shae memory alloys (SMAs) are mostly known for their so-called shae memory effect, where SMAs revert to their initial shae uon heating after having been deformed at low temeratures (Duerig (99), Funakubo (987), Humbeeck (), Janke et al. (5), Otsuka and Wayman (999), Otsuka and Kakeshita ()). This effect is also called seudolasticity. If the deformation recovery is restrained, mechanical stress results within the SMA. The stress may be used to introduce external forces into structures. Another effect occurs when austenite hase transforms to martensite hase due to external mechanical load at constant temerature. This is called suer- or seudoelasticity, resectively. The alloy retransforms automatically to austenitic state if the SMA is unloaded. The stress-strain curve shows a hysteresis built by the two aths of transformation and retransformation. The two way SMA effect describes the behaviour of some SMAs that may remember two different shaes deending on different temeratures. Besides the transformation rocesses briefly described above, SMAs are characterized by high daming caacity due to internal friction which is based on mainly two different mechanisms. If martensite is the stable hase, martensite variants are reoriented within the SMA uon loading above yield stress. This effect roduces large hysteresis areas if the loading is sinusoidal around zero. If the SMA is heated to a higher temerature, the hysteresis loo changes its form until it reaches the shae similar to the suerelastic behaviour of SMAs. However, due to the comlicated shae of the hysteresis trajectories between ambient temerature and high temeratures (Otsuka and Wayman (999)), this adative material behaviour is not used to roduce controllable daming devices. The second mechanism is based on the suerelasticity of SMAs (Fig. 87). The relative dislacement between austenite and martensite interfaces and between martensite variants, resectively, roduces friction and therefore daming. The hysteresis area during loading and unloading in the suerelastic state is equivalent to the dissiated energy. Since the hysteresis curve aears within the stress-strain ma, this kind of daming is structural daming. If the temerature of the SMA is increased, the hysteresis does change its area, form and location within the stress/strain ma (Fig. 87). For small temerature variations, the hysteresis curve does hardly change its area and form, but moves slightly to higher stress/strain values (comare the thick solid line and thick dashed line in Fig. 87). Adative shae memory alloy In the field of structural control, the suerelastic behaviour may be used to roduce a daming device with variable daming characteristics. The crucial oints of such a controllable daming device may be: Heating u the entire SMA is energy consuming and a constant, homogeneous temerature of the entire SMA is difficult to achieve and maintain. Fast cooling of SMAs is a challenging issue. Both, heating and cooling are rather slow rocesses due to the thermal caacity of the SMA material which results in a very small bandwidth of the semi-active daming device. Page 97 of 55

98 SAMCO Final Reort 6 The working area of the semi-actively working SMA is small comared to the entire area of the stress/strain area. Consequently, in many cases, the desired otimum control force cannot be tracked. The resulting subotimal force leads to mediocre vibration mitigation erformance. Due to the difficulties listed above, the otential of SMAs working in the suerelastic mode at different temerature is not a current issue. However, researchers have investigated numerically the daming otential of SMAs in suerelastic mode at constant temerature (Li et al. (4)). A different way of using SMAs as an adative material is roosed by, e.g., Rustighi et al. (5) and Williams et al. (). Here, the SMA is used as a sring element with two different stiffness values deending on the temerature. At ambient temerature, the SMA is comletely in the martensite hase (Fig. 88). At temerature austenite finish, the SMA is comletely in the austenite hase. The two hases are characterized by two different moduli of elasticity within the elastic stress/strain region. By combining several SMA elements which are either in martensite or austenite state, a tuneable sring can be designed. Such an adative sring may be used in order to tune the frequency of a tuned mass damer to the actual structural target frequency (Rustighi et al. (5), Williams et al. ()). It must be mentioned that the SMA stiffness does not change smoothly with increasing temerature. Therefore, each SMA element of the overall sring is either in full martensite or full austenite state (Williams et al. ()). That is the reason why the number of SMA elements has to be fairly high in order to be able to roduce a sring element with almost continuously tuneable stiffness, bounded by the low stiffness of martensite and the high stiffness of austenite. Fig. 87: Stress-strain curves for the suerelastic behaviour of SMAs at different temeratures. Fig. 88: Different moduli of elasticity for martensite and austenite. References Duerig, T. W. (99), Engineering Asects of Shae Memory Alloys, Butterworth- Heinemann, London. Funakubo, H. (987), Precision Machinery and Robotics, Vol. Shae Memory Alloys, Gordon and Breach. Humbeeck, J. V. (), Shae memory alloys: A material and a technology, Adv. Eng. Mater, 3(), Page 98 of 55

99 SAMCO Final Reort 6 Janke, L., Czaderski, C., Motavalli, M., and Ruth, J. (5), Alications of shae memory alloys in civil engineering structures Overview, limits and new ideas, Journal of Materials and Structures, 38(79), Li, H., Liu, M., and Oh, J. (4), Vibration mitigation of a stay cable with one shae memory alloy damer, Journal of Structural Control and Health Monitoring, (), -36. Otsuka, K., and Wayman, C. M. (999), Shae Memory Materials, Cambridge University Press. Otsuka, K., and Kakeshita, T. (), Science and technology of shae-memory alloys. New develoments", Mrs Bulletin 7(), 9-. Rustighi, E., Brennan, M. J., and Mace, B. R. (5), A shae memory alloy adative tuned vibration absorber: design and imlementation, Journal of Smart Materials and Structures, 4, 9-8. Williams, K., Chiu, G., and Berhard, R. (), Adative-assive absorbers using shae-memory alloys, Journal of Sound and Vibration, 49(5), Notations Symbol Descrition Unit E elasticity modulus N/m T temerature K Page 99 of 55

100 SAMCO Final Reort 6..3 Actuators..3. Actuators for increase of daming In the field of structural control, increasing structural daming is the main target because, in general, both frequency and satial distribution of disturbance forces due to, e.g., wind loading, are unknown. If the structural daming c shall be augmented by the external control force f u, the control force has to be controlled roortionally to the collocated structural velocity, which is shown for the simle case of a single degree of freedom system (SDOFS), see Eqs. () and (). f & + & && & f w () x& u m x cx + kx = f u + f w mx + c x + kx = f u = c x& () u The overall daming of the SDOFS becomes ( c ) c = + (3) tot c u The control concet described by Eq. () is called velocity feedback and will be discussed in detail in Chater.. According to Eq. (), the control forces required are dissiative. Consequently, controllable damers are sufficient to track the desired control force f. However, there are mainly three reasons, why actuators that can u aly active forces to structures may be mandatory in the field of structural control: Besides increasing structural daming, structural control comrises also, e.g., dislacement and shae control of structures. If the control force shall comensate for disturbance forces that influence the structure only at well known ositions of the structure (Achkire et al. (998), Caruso et al. (3), Raja et al. ()). There exist many other control concets used to mitigate structural vibrations which require dissiative and active force values (d Azzo and Houis (995)). In the secial case of sky hook daming requires dissiative and active force values (Lee and Jeon (), Preumont (), Preumont (4), Stelzer et al. (3)). From the wide range of actuator tyes, two of them are described in the following which are fairly often used in the field of structural control, namely hydraulic aggregates and iezo actuators. References Achkire, Y., Bossens, F., and Preumont, A. (998), Active daming and flutter control of cable-stayed bridges, Journal of Wind Engineering and Industrial Aerodynamics, 74-76, d Azzo, J. J., and Houis, C. H. (995), Linear Control System Analyzes and Design, Conventional and Modern, Fourth Edition, McGraw-Hill (eds.), ISBN Page of 55

101 SAMCO Final Reort 6 Caruso, G., Galeani, S., and Menini, L. (3), Active Vibration Control of an Elastic Plate Using Piezoelectric Sensors and Actuators, Journal of Simulation Modelling Practice and Theory, (5-6), Lee, Y., and Jeon, D. (), A Study on The Vibration Attenuation of a Driver Seat Using an MR Damer, Journal of Intelligent Material Systems and Structures, 3(7/8), Preumont, A. (), Vibration Control of Active Structures, Chater, Kluwer Academic Publishers, Dordrecht. Preumont, A. (4), Semi-active sky hook, does it work?, EUROMECH Colloquium 455 on Semi-Active Vibration Suression, M. Valasek and A. Preumont (Eds.), CTU in Prague, Czech Reublic, July 5-7 4, on CD. Raja, S., Pratha, G., and Sinha, P. K. (), Active vibration control of comosite sandwich beams with iezoelectric extension-bending and shear actuators, Journal of Smart Materials and Structures, (6), Stelzer, G., J., Schulz, M., J., Kim, J., and Allemang, R., J. (3), A Magnetorheological Semi-active Isolator to Reduce Noise and Vibration Transmissibility in Atomobiles, Journal of Intelligent Materials Systems and Structures, 4(), Notations Symbol Descrition Unit c viscous daming coefficient kg/s f force N k stiffness kg/s m mass kg x dislacement m Subscrits tot u w total control disturbance Page of 55

102 SAMCO Final Reort Hydraulic aggregate Theoretical background The force of hydraulic aggregates is roduced by the oil ressure within the aggregate cylinder where is the oil ressure and f = A () a sta A the cross sectional area of the iston. The force level is controlled by servo valves (Fig. 89). Usually, hydraulic aggregates are equied with both dislacement and force sensor. This allows for controlling the hydraulic aggregate in dislacement control mode or force control mode. The most often used control mode is dislacement control and the reaction force between tested device and aggregate iston is measured by an additional force sensor or by the force sensor of the aggregate. The maximum available dislacement amlitude deends on the frequency assuming sinusoidal dislacement because the dynamic force of the hydraulic aggregate is limited by a maximum tolerable value in order to avoid damage of the aggregate (Eq. ()). The common way is that data sheets give information about the maximum values of frequency ω and amlitude X of a sinusoidal dislacement. f a dyn max max ( ω X ) & x () Fig. 89: Hydraulic aggregate, mounted on a steel frame. Page of 55

103 SAMCO Final Reort 6 The maximum value of the static aggregate force is usually aroximately % higher than the maximum value of the dynamic aggregate force. However, since aggregates are used in dynamic testing of structures, the maximum dynamic force is relevant. Imlementation Hydraulic aggregates are mainly used to test novel materials in the laboratory. Usually, the test material or test device is dislacement controlled deformed and the reaction force as system outut is measured (Krenk et al. (996)). The results are the well known stress-strain or force-dislacement curves, resectively. Another alication of hydraulic aggregates is the forced excitation of structures such as buildings, bridges, and towers in order to identify structural arameters based on forced vibrations (Irwin and Stoyanoff (5), Occhiuzzi et al. (3), Stoyanoff et al. (5), Weber and Huth (5), Yang et al. (4)). A third variant is simly to control the dislacement or shae of large civil structures installing hydraulic aggregates between stories or between the first floor and ground. As a result of the ability to roduce active forces, the following oints should be included in the dislacement or force control loo of such actuators in order to avoid unintentional material or actuator damage: A maximum force value should be defined above which the controller shuts off. A maximum dislacement value should be defined above which the controller shuts off. The first test should be run using a dummy. Design and testing Since hydraulic aggregates are commercially available, they are designed, manufactured and tested by the corresonding comany. Oerating mas of the aggregate are delivered together with the aggregate. References Irwin, P., and Stoyanoff, S. (5), Use of vibration testing as a diagnostic tool, Tutorial of 6th International Conference on Cable Dynamics, Charleston, SC (USA), Setember 9, 5. Krenk, S., Jönsson, J., and Hansen, L. P. (996), Fatigue Analysis and Testing of Adhesive Joints, Journal of Engineering Fracture Mechanics, 53(6), Occhiuzzi, A., Sizzuoco, M., and Serino, G. (3), Exerimental analysis of magnetorheological damers for structural control, Journal of Smart Materials and Structures, (5), Stoyanoff, S., Theryo, T., and Garcia, P. (5), Full dynamic test of the stay cables of Sunshine Skyway Bridge, Proceedings of 6th International Conference on Cable Dynamics, Charleston, SC (USA), Setember 9, 5, Weber, F., and Huth, O. (5), Charakterisierung der dynamischen Eigenschaften eines Einfamilienhauses in Monthey / Wallis, Ema Bericht Nr. 8473, Swiss Federal Laboratories for Materials Testing and Reserach, Dubendorf, Switzerland. Page 3 of 55

104 SAMCO Final Reort 6 Yang, J., N., Lei, Y., Lin, S., and Huang, N. (4), Identification of Natural Frequencies and Damings of In Situ Tall Buildings Using Ambient Wind Vibration Data, Journal of Engineering Mechanics, 3(5), Notations Symbol Descrition Unit A cross sectional area m X dislacement amlitude m f force N ressure Pa x dislacement m ω radial frequency rad/s Subscrits a dyn max sta actuator dynamic maximum iston static Page 4 of 55

105 SAMCO Final Reort Piezo actuator Besides books, information about iezo actuators may be found in the internet, e.g., on the following ages: htt://de.wikiedia.org/wiki/piezo htt:// htt:// htt:// htt:// (PIEZO SYSTEMS, INC., MA 39 USA) htt:// (Piezo Kinetics, Inc. Bellefonte, PA 683, USA) htt:// (Piezomechanik GmbH, D-8673 Munich, Germany) Theoretical background Piezo actuators like stacks, benders, tubes, and rings make use of the deformation of electroactive PZT-ceramics (PZT: lead (Pb), zirconia (Zr), Titanate (Ti)) when they are exosed to electrical fields. This deformation can be used to roduce motions or forces if the deformation of the iezo element is constraint. The above effect is the comlementary effect to iezo electricity, where electrical charges are roduced uon alication of mechanical stress to the ceramics. As an analogy, the term iezo mechanics was introduced in the early 8 s of the ast century by L. Pickelmann to describe the conversion of electricity into a mechanical action by iezo materials. For iezo mechanical conversion in the simlest case a single PZT layer is used. Such a PZT monolayer structure as shown in Fig. 9 is a caacitive element. It consists of two thin conductive electrode coatings enclosing the iezo ceramic as dielectric. Alying a voltage to this iezo caacitor, the caacitor is charged and deformation results. Hence, PZT actuators may be seen as actuating caacitors. If the maximum deformation of PZT actuators shall be increased, PZT actuators consist of several layers. These iezo stack actuators and stacked iezo rings make use of the increase of the ceramic thickness in direction of the alied electrical field (Fig. 9). This is called d33 effect. Stacking of several layers towards a multilayer structure increases equivalently the total stroke. In ractice, axial strain rates u to of stack s length can be achieved under certain conditions. Page 5 of 55

106 SAMCO Final Reort 6 Fig. 9: Schematic of a iezo-electric single layer element (htt:// Fig. 9: Schematic of an axially acting multilayer iezo stack (htt:// Similar to elastic deformation of solid bodies, the thickness exansion of a PZT layer is accomanied by an in-lane shrinking (Fig. 9). This is called the d3 effect, being comlementary in motion and showing roughly half linear strain comared to the d33 effect. The d3 effect is mainly used for bending structures. Design Piezo actuators were used referentially in the ast for quasistatic recision ositioning tasks, but find now increasing interest in comletely new fields of alication like dynamically actuated mechanics, e.g., valves, fuel injection devices, or adative smart structures such as shae tuning, vibration generation and cancellation, and mode tuning (Caruso et al. (3), Herold et al. (4), Ma (), Raja (), Sadri et al. ()). Of course, such a broad variety of alications cannot be covered by one general actuator tye. The main arameters to adat a iezo actuator to a distinct alication are:. Selection of roer PZT material defining achievable strain, stroke, energy balance, temerature range etc.. Prearation of a highly reliable and efficient stack structure which is, e.g., shock and vibration resistant. 3. Sealing for corrosion resistance. 4. Packaging of the ceramic stack. 5. Actuator system design, e.g., systems able to ush and ull. 6. Performance and reliability of actuator systems deend further on the electrical driving characteristics like voltage / charge / current control oerating strategies and uniolar / semibiolar / biolar oeration. Page 6 of 55

107 SAMCO Final Reort 6 Fig. 9: Schematic of a stacked ring actuator (htt:// Fig. 93: Schematic of an actuator with restress (htt:// Some of the currently used basic stack designs are: Bare stacks: The mounting and motion transfer is always done via end faces. Bare stack must not be hold by sideway claming. Ring stacks with centre hole (Fig. 9): Ring stacks are needed if an accessible system axis is needed for transmissive otical set-us or the feed through of mechanical arts is required. Ring stacks may be also required in order to increase the bending stiffness by diameter enlargement of the stack without the need of increase of the oerated ceramic volume (e.g. for long-stroke elements). Ring actuators rovide further a fairly high cooling erformance due to ossible access of the inner and outer surface by cooling media. Cased stacks with internal reload mechanism (Fig. 93): The incororation of iezo stacks into a metallic casing generally imroves reliability and stability against mechanical imact and deteriorating environmental influences. The imlementation of a reload mechanism comensates for tensile stress. In contrast, ceramics are extremely vulnerable to such imacts. Actuator characterization Like any other electrical actuator, iezo actuators convert electrical energy into motion and force, resectively, deending on the actuator stroke constraints. The generated motion or force is couled to an external device (Fig. 94). In the simlest case, this device shall be shifted from the actual osition to the desired one. More comlex alications may be, e.g., osition control of valves or controlled iezo mechanical elements incororated in smart structures. In all these cases the interaction between the actuator and the driven mechanism must be analyzed. The mechanical interaction is defined by the stiffness values of both systems, namely of the iezo actuator and of the actuated mechanical system. Page 7 of 55

108 SAMCO Final Reort 6 Fig. 94: Schematic of a iezo actuated system (htt:// Fig. 95: Schematic voltage/stroke diagram of a stack actuator (htt:// Two basic exeriments are carried out in order to determine the actuator characteristics: actuator stroke generation as a function of the alied voltage (Fig. 95), and actuator force generation as a function of the alied voltage (Fig. 96). The condition for determination of the stroke voltage relation is that the stiffness of the actuated mechanical system is zero or in other words that the iezo actuator dislacement is not constraint. Then, the iezo actuator generates maximum stroke and minimum force. The necessary condition for determination of the maximum actuator force, also called blocking force, is infinite large stiffness of the actuated mechanical system or zero iezo actuator dislacement, resectively. The force resonse shows remarkably lower hysteresis than the stroke resonse. It must be noted that the hysteresis deends on reload conditions. In ractice, iezo actuators interact always with mechanical systems showing an intermediate stiffness value between the two theoretical limits and. Then, the iezo actuator distributes the electrical energy artially into generation of stroke and artially into generation of force (Fig. 97). The ratio between stroke and force generation deends on the quantitative relation of iezo actuator stiffness and mechanical system stiffness. The achievable force-stroke relations of a real system can be derived in the following way:. Draw a line from maximum stroke to maximum blocking force.. Draw a line from origin with a sloe according the stiffness of the couled mechanical system. 3. The intersection oint A of these two lines describes the achievable stroke and force at maximum voltage of the iezo actuator. Notice that the schematic deicted in Fig. 97 reresents a linearized actuator resonse. It does not take into account stroke enhancement effects. If the stiffness values of the iezo actuator and of the couled mechanical system are equal, the achievable dislacement is 5% of its maximum value and the achievable force variation is 5% of the blocking force. Under this condition, the mechanical energy transfer efficiency from the iezo actuator to the mechanical system is maximized. Page 8 of 55

109 SAMCO Final Reort 6 Fig. 96: Schematic voltage force relation of a iezo stack (htt:// Fig. 97: Maximum iezo actuator stroke versus maximum (blocking) iezo actuator force (htt:// Piezo actuators convert electrical energy into a mechanical energy, often called iezo mechanical resonse. The electrical energy content W of an electrically charged iezo actuator is C U W = () where C is the actuator caacitance and U the alied voltage. The iezo mechanical resonse arameters are: stroke generation l, force generation F, and mechanical energy W = l F. Control of iezo actuators The user of an actuator is mainly interested in the dislacement of the actuator and, erhas, in the ultra fine ositioning caability down to the sub nano meter range. A variation of force or generation of mechanical energy during the action of the iezo actuator is not of high riority. However, iezo actuators then do not oerate energy efficient because only a small art of the actuator energy content is needed to be transferred to the couled mechanics. Otimizing iezo actuators for dynamic ositioning uroses means minimizing the electrical energy inut for given desired dislacement. This is equivalent with minimum actuator caacitance in relation to the driving voltage. Then, the electrical ower requirements for a distinct dynamic resonse, e.g., oscillating arrangements, are minimal. Consequently, a PZT material with low dielectric constant ε together with a high iezoelectric constant d33 is required. Position control of iezo actuators is the classical control target. However, in the case of smart structures of high stiffness, which shall be highly deformed by integrated iezo actuators, large dislacements together with high force generation to achieve high mechanical energy transfer are required. Examles are adative frame struc- Page 9 of 55

110 SAMCO Final Reort 6 tures of machines, car bodies, wings of air lanes for active vibration excitation and cancellation or shae otimization (Caruso et al. (3), Herold et al. (4), Ma (), Raja (), Sadri et al. ()). To rovide this large mechanical energy outut, an increased electrical energy inut is required via a reasonable large electrical caacitance of the actuator to get a high mechanical energy density within the ceramics. Therefore, PZT materials are requested showing an elevated dielectric constant ε combined with a very high strain and force generation rate. Notice that materials with large dielectric constant ε are widely offered but often do not show the wanted large iezo mechanical resonse. Fig. 98: Aroximately linear relation between stroke and charge content of iezo actuators (htt:// Fig. 99: Cree of iezo actuators (htt:// According the simle caacitance equation Q = U C () the voltage U at caacitor s contacts is roortional to the stored electrical charge content Q. For commercially available caacitors, the caacitance C is constant. Piezo actuators reresent also electrical caacitors. In contrast to normal caacitors, the caacitance of iezo actuators is not constant but deends to some extent on driving conditions like voltage, load, temerature etc. Therefore, the osition, velocity, and acceleration of iezo actuator do not deend linearly on the alied voltage U (Fig. 95). However, osition, velocity, and acceleration are roortional to the charge content Q (Fig. 98). Consequently, if the nonlinear relation between voltage and dislacement shall not be comensated, iezo actuators should be charge -controlled (Preumont ()). Summarizing, the controlled variables of the iezo actuator are: osition, which is roortional to the electrical charge content x Q (3) velocity, which is roortional to time derivative of the electrical charge content thus the velocity is roortional to current: dq / dt = I (4) x& I (5) Page of 55

111 SAMCO Final Reort 6 acceleration, which is roortional to the second time derivative of the electrical charge content ( d Q / dt = di / dt ), thus the acceleration is roortional to the change of current (& x I& ). As exlained above, charge -control of iezo actuators within an oen loo system since sensors are not needed results in linear osition control. Besides this very imortant issue, charge control leads also to the following benefits: increase of actuator stiffness, and increasing actuator s reliability under very high dynamic oerating conditions. The slight nonlinear relation between stroke and charge shown in Fig. 98 may be comensated by the inverse function of that relation. The most imortant oint is that hysteresis does almost not exist (aroximately %) as in the case of voltage control (Fig. 95). The main advantage of linearization via the charge hilosohy is the ure mode excitation. A sinusoidal electrical current will be converted into a sinusoidal oscillation of the iezo actuator velocity without any modulations due as it is the case for voltage controlled stroke due to nonlinearities and hysteresis. Piezo actuators show small ositive drift in exansion over a distinct time when a voltage ste is alied (Fig. 99). This is called cree of iezo actuators. It is based on a ferroelectric effect, where the olarization state of the ceramic alters as long as an electrical charge flows that is delivered by the voltage suly. Hence, cree may be immediately stoed, when the charge content of iezo actuators is ket constant. Sensing using iezo actuators is also ossible (Law et al. (3)). Then, the dislacement of the connected mechanical system roduces a charge and therefore a voltage within the iezo actuator. References Caruso, G., Galeani, S., and Menini, L. (3), Active Vibration Control of an Elastic Plate Using Piezoelectric Sensors and Actuators, Journal of Simulation Modelling Practice and Theory, (5-6), Herold, S., Mayer, D., and Hanselka, H. (4), Transient Simulation of Adative Structures, Journal of Intelligent Material Systems and Structures, 5(3), 5-4. Law, W. W., Liao, W.-H., and Huang, J. (3), Vibration control of structures with self-sensing iezoelectric actuators incororating adative mechanisms, Journal of Smart Materials and Structures, (5), Ma, K. (), Vibration control of smart structures with bonded PZT atches: novel adative filtering algorithm and hybrid control scheme, Journal of Smart Materials and Structures, (3), Preumont, A. (), Vibration Control of Active Structures, Chater 3, Kluwer Academic Publishers, Dordrecht. Raja, S., Pratha, G., and Sinha, P. K. (), Active vibration control of comosite sandwich beams with iezoelectric extension-bending and shear actuators, Journal of Smart Materials and Structures, (6), Sadri, A. M., Wright, J. R., and Wynne, R. J. (), LQG control design for anel flutter suression using iezoelectric actuators, Journal of Smart Materials and Structures, (6), htt://de.wikiedia.org/wiki/piezo htt:// Page of 55

112 SAMCO Final Reort 6 htt:// htt:// htt:// (PIEZO SYSTEMS, INC., MA 39 USA) htt:// (Piezo Kinetics, Inc. Bellefonte, PA 683, USA) htt:// (Piezomechanik GmbH, D-8673 Munich, Germany) Notations Symbol Descrition Unit C caacitance F F force N I electrical current A Q charge C U voltage V W work, energy J l stroke, dislacement m t time s x dislacement m difference - ε dielectric constant - Page of 55

113 SAMCO Final Reort 6. Control algorithms The three basic control strategies, such as feedback, feed forward, and assive control are described in the first subchater. The following subchaters resent different control aroaches often used in the field of structural control. Active daming using the Root Locus method is documented in the second subchater. It is extended with a short introduction into PID control. The third subchater describes the aroach of otimal control. This chater also introduces the state sace reresentation. The fourth subchater summarizes other linear control aroaches often used in the field of structural control... Control strategies Basically, there exist three ways how to control actuators connected to dynamic systems: feedback control, feed forward control, and assive control. Within the following subchaters, these three control strategies are exlained. In the case of structural control, controlled actuators may also be controllable daming devices since the main target of structural control in the field of civil engineering is daming increase. Actuators and damers will be called control device, the controlled dynamic systems such as tall buildings, slender bridges, long cables, and other civil structures will be called lant.... Feedback control When the main disturbance w of the lant G P (s) is unknown or not measurable (Fig. ), feedback control may be alied. The basic idea is to measure the system resonse y, which includes measurement noise v, then to comare the variable y to the desired value of y, the so-called reference signal r, and feed back the resulting error e through the controller G C (s) to the lant G P (s) (d Azzo and Houis (995), Geering (99), Preumont (), Soong (99)). The controller is designed in order to minimize the error signal e with the constraint that the closed-loo system must not become unstable. w v w v r + - e G (s) C u G (s) P + + y r + - e G (s) C u c G (s) A u G (s) P G (s) S + + y Fig. : Closed-loo structure of feedback control. Fig. : Closed-loo structure of feedback control including actuator and sensor dynamics. In reality, the closed-loo structure additionally consists of the control device dynamics and of the sensor dynamics, reresented by the corresonding transfer functions G A (s) and G S (s) in Fig. (Dyke and Sencer Jr. (997), Yeo et al. ()). The Page 3 of 55

114 SAMCO Final Reort 6 controller outut u C, e.g., a voltage signal, is the inut signal of the actuator which eventually roduces the desired control signal u, e.g., the desired force. Usually, it is assumed that both the dynamics of the control device and the dynamics of the sensor as well are much faster than the dynamics of the lant and therefore negligible for model-based controller design. However, the dynamics of the control device may limit the controller band width, esecially in the case of vibration control of high frequencies. The sensors have to be chosen the way that the lant state variables of interest may be measured without any aliasing.... Feed forward control Feed forward control can be alied when a reference signal is available, which is correlated to the disturbance acting on the lant (Fig. ). The measured reference signal is assed to an adative filter G F (s) whose outut signal influences the structure as a second disturbance. The criterion for the adatation of the filter coefficients is the minimization of the measured error e, e.g., minimizing the dislacement at a certain location of the structure (Ma ()). Thus, the basic idea of adative filtering is roducing such an additional disturbance that the effect of the rimary disturbance on the lant is comensated. However, the full comensation is only ossible at sensor location. r + w - e G (s) F u G (s) P + + v y uc=const. G (s) A w u G (s) P v + + y - + r e Fig. : Structure of feed forward control. Fig. 3: Oen-loo structure of assive control....3 Passive control In the field of structural control, assive control (Fig. 3) has been imlemented in many cases thanks to its simlicity and robustness (Gordaninejad et al. (), Ko et al. (), Xu and Yu (998a/b): Passive control does not require any PC including control soft- and hardware. Stability is not an issue because the loo is not closed. Usually, the otimal value of the constant inut signal u C of the control device is determined exerimentally. The criterion for the otimal value u C is minimum error e, e.g., the minimum of the measured velocities at certain ositions of the lant....4 References d Azzo, J. J., and Houis, C. H. (995), Linear Control System Analyzes and Design, Conventional and Modern, Fouth Edition, McGraw-Hill (eds.), ISBN Dyke, S. J., and Sencer Jr., B. F. (997), A Comarison of Semi-Active Control Strategies for the MR Damer, Proceedings of the IASTED International Conference, Intelligent Information Systems, The Bahamas, 8- December 997, Page 4 of 55

115 SAMCO Final Reort 6 Geering, H. P. (99), Mess- und Regelungstechnik,. Auflage, Sringer-Verlag Berlin Heidelberg. Gordaninejad, F., Saiidi, M., Hansen, B. C., Ericksen, E. O., and Chang, F.-K. (), Magneto-Rheological Fluid Damers for Control of Bridges, Journal of Intelligent Material Systems and Structures, 3(/3), Ko, J. M., Zheng, G., Chen, Z. Q., and Ni, Y. Q. (), Field vibration tests of bridge stay cables incororated with magneto-rheological (MR) damers, Proceedings of the International Conference on Smart Structures and Materials : Smart Systems of Bridges, Structures, and Highways, S.-C. Liu and Darryll J. Pines (eds.), Proceedings of SPIE (ubl.), 4696, 3-4. Ma, K. (), Vibration control of smart structures with bonded PZT atches: novel adative filtering algorithm and hybrid control scheme, Journal of Smart Materials and Structures, (3), Preumont, A. (), Vibration Control of Active Structures, Kluwer Academic Publishers, Dordrecht. Soong, T. T. (99), Active structural control: theory and ractice, Cheng, W.F. (advisory editor), Longman Scientific & Technical, ISBN Xu, Y. L., and Yu, Z. (998a), Mitigation of three dimensional vibrations of inclined sag cable using discrete oil damers I. formulation, Journal of Sound and Vibration, 4(4), Xu, Y. L., and Yu, Z. (998b), Mitigation of three dimensional vibrations of inclined sag cable using discrete oil damers II. alication, Journal of Sound and Vibration, 4(4), Yeo, M. S., Lee, H. G., and Kim, M. C. (), A Study on the Performance Estimation of Semi-active Susension System Considering the Resonse Time of Electro-rheological Fluid, Journal of Intelligent Material Systems and Structures, 3(7/8), Notations Symbol Descrition Unit G (s) transfer function - e error - r desired value / desired trajectory - s comlex frequency variable - u controlled inut variable of the lant - v measurement noise - w system noise / disturbance inut variable of the lant - y lant outut variable - Subscrits A C Descrition actuator controller Page 5 of 55

116 SAMCO Final Reort 6 F P S filter lant sensor Page 6 of 55

117 SAMCO Final Reort 6.. Active daming with collocated airs The following chater describes active vibration suression using collocated airs of actuators and sensors (Achkire et al. (998), d Azzo and Houis (995), Preumont (), Soong (99)). Actuator and sensor airs are denoted as collocated if they are hysically located at the same lace and energetically conjugated, e.g., force and velocity or torque and angle. One main roerty of such collocated airs is that the closed-loo structure of each air is indeendent from the others. This leads to as many indeendent single inut single outut (SISO) loos as collocated airs. In order to exlain the different control schemes of active daming, first, linear comensators will be introduced. Within this subchater, the terms of loo shaing, Nyquist Diagram, Bode Diagram, and Root Locus will be described (d Azzo and Houis (995), Geering (99), Preumont ()). Then, the different closed-loo structures using different combinations of collocated actuator sensor airs will be resented.... Linear comensators PID control elements The transfer function of a arallel PID controller (Fig. 4, Fig. 5) consists of the roortional gain K P, integral gain K I, and derivative gain K D (d Azzo and Houis (995), Geering (99), Preumont (), Soong (99)) G K I ( s = K + + K D s = K P + + TV s () s TN s PID ) The roortional gain is resonsible for the basic work of the controller. The roortional gain K P amlifies the error e. This forces the measured lant outut y to follow the desired variable r ( y tracks r ). The integral gain makes it ossible to reach a tracking error equal to zero during steady-state conditions. The derivative gain adds daming to the closed-loo structure which makes controlled lant resonse fast to changes of r. PID controller & comensation of non-linearity K P + + r e K + I dt NL - u G (s) y P - + K D d dt Bode Diagram Magnitude (db) Phase (deg) k/tn k/tv /T *log(k) P art I art D art low ass (th order) +db/dec -4dB/dec -db/dec Frequency (rad/sec) Fig. 4: Structure of arallel PID controller with inverse lant model NL -. Fig. 5: Bode diagram of P, I, D elements, and of low ass filter of nd order. However, large values of K D may lead to unstable closed-loo behaviour. Page 7 of 55

118 SAMCO Final Reort 6 Since PID control assumes a linear behaviour of the lant, the main non-linearity (NL) between u and y must be comensated within the control algorithm by an inverse model of the nonlinearity (NL - ). Besides non-linearities of the lant itself, the nonlinear relation between actuator inut (voltage, current) and actuator outut (force, strain, dislacement, ) has to be considered as well. In order to suress measurement noise, a low ass filter of order n is added in series to the PID controller. The low ass filter is resonsible for the so called roll-off behaviour of the controller, which suresses measurement and system noise (Fig. 6). A first order filter roduces - db/dec roll-off, which is not seen as sufficient for measurement and system noise suression. Usually, filters of nd order (- db/dec) or higher are alied. G PID ( s) rf = K P + + T TN s V s ( T s + ) n ( n ) () Magnitude (db) Phase (deg) integral art roortional art Bode Diagram PID PID & low ass (th order) derivative art low ass art: "roll-off" Frequency (rad/sec) Magnitude (db) Phase (deg) (T = / [s]) lead (T = / [s]) lag lead element lag element α lag ( αt) lag Bode Diagram Frequency (rad/sec) ( αt) lead α lead α lag= α lag= α lag= α lead α lead=. α lead=. α lead=.5 α lag Fig. 6: Bode diagram of a arallel PID controller with low ass filter of nd order for a good roll-off. Fig. 7: Characteristics of Lead and Lag elements shown in the Bode Diagram. Lead / Lag element The Lead and Lag elements allow for adjusting the transfer function of the controller in the frequency domain (d Azzo and Houis (995), Preumont ()). Since these elements change the shae of the transfer function in the Nyquist Diagram, where the transfer function is a trajectory in the real-imaginary-lane and therefore aears as a line or loo, this rocedure is called loo shaing. The Lead element may increase the hase margin, whereas the Lag element may increase the gain margin (Fig. 7). T s + G Lead ( s) = : α < (3) α T s + T s + G Lag ( s) = : α > (4) α T s + Page 8 of 55

119 SAMCO Final Reort 6 Controller tuning Besides the trial and error method for tuning the arameters of PID controllers, the most common way of model-based controller arameter design is resented here (Geering (99)):. Given the transfer function of the lant together with a single P controller, the value of K P may be determined where the closed-loo structure just becomes unstable. This value of K is called K Then, the real arts of the P closed-loo oles are zero. The imaginary arts of the closed-loo oles are identical to the critical frequency in radiant er second ( T ). For higher values of K P, the real arts of the closed-loo oles get ositive and the closed-loo structure becomes unstable.. Based on the values of K P crit and T crit be tuned according to Ziegler/Nichols: P controller: P crit K P crit, Tcrit crit K crit = π ω (5), (6), the PID controller arameters may =.5 (7) K P crit PI controller: K =.45 K,T =.85 T (8), (9) P crit N crit PID controller: K =.6 K P criti,t N =.5 T criti,t V =. T crit Please notice that the determination of reason, the controller design may be called model-based. P crit ()-() K requires a model of the lant. For this Root Locus method In the following, the determination of the values of K P crit and crit T shall be demonstrated. It is assumed that the lant has the only real oles -, -, and -3 (Fig. 8, Geering (99)). The transfer function of the closed-loo system with the P controller only and negative unity feedback becomes G cl Y ( s) GC(s) GP(s) ( s) = = = U ( s) + G (s) G (s) s C P 3 + 6s K P + s + (6 + K The oles of G C are the zeros of the denominator of G cl. The oles of the closedloo system are lotted in the real-imaginary-lane (Nyquist Diagram) with arameter K P (Fig. 9). The resulting line or trajectory is called Root Locus of the closedloo. For increasing values of K P, the closed-loo inclines more and more to vibrate. For real arts of the closed-loo oles equal to zero, the closed-loo system does not have any daming. Then, the closed-loo system is on the border of instability. For the examle resented, the critical values are: P ) (3) Page 9 of 55

120 SAMCO Final Reort 6 K P crit = 6 [ ], T crit = π [s] (4), (5) If K P is further increased, the closed-loo system gets unstable or the daming of the closed-loo system becomes negative, resectively. r + - e G (s) C K P u G (s) P (s+)(s+)(s+3) y Im(s) ole 3 K P> Root Locus K: 6 Pole: i Daming: Frequency: (rad/s) ole 3 (K =) P K: Pole: + i Daming: Frequency: (rad/s) K P > Re(s) ole ole (K =) P ole (K =) P K > P stable ole unstable Fig. 8: Plant of 3rd order with roortional gain only. Fig. 9: Root Locus of 3rd order lant with feedback roortional gain K P only.... Active daming with collocated actuator sensor airs Main idea The equation of motion of a system excited by the disturbance force w and controlled by the force f is M & z + C z& + K z = w + f (6) When the control target is increasing the daming of the lant, then the control force f must be roortional to the structural velocity at actuator location in order to increase aarently the lants daming ( C z& f ) + K z = w f z& M & z + ~ (7), (8) This leads to a control force, which acts like an external viscous damer. The collocated sensor may be, e.g., a velocity sensor. The matrix notation of the equations above oints out that the daming of the closed-loo structure may be increased using numbers of collocated actuator sensor airs (Fig. ). Then, a decentralized controller may be realized by a roortional negative feedback T T f = [ g g... g n ] [ z& z&... z& n ] (9) The controller is called decentralized because every sensor builds a closed-loo only with its collocated actuator. Due to the ositive definite values of g (similar to P element), this control law (negative feedback) guarantees ower dissiation. i Page of 55

121 SAMCO Final Reort 6 Page of 55 The locations of the velocity sensors must be chosen the way that the vibration velocity of the mode of interest is not zero. If this is the case, the collocated actuator sensor air is located at the structure where the mode of interest has a nodal oint. In order to guarantee a roer working behaviour of the roortional controller, a low ass filter should be added, which attenuates the measurement noise. The resulting closed-loo structure is called direct velocity feedback (DVF, Nakamura et al. (), Preumont (), Subramanian ()). Tuning the feedback gain for maximum additional daming Given the four mass system deicted in Fig. with identical values for mass, stiffness, and daming elements, the equation of motion with the external control forces i f becomes = f f f f z z z z k k k k k k k k k k z z z z c c c c c c c c c c z z z z m m m m & & & & && && && && () z z z3 z 4 m k c m k c m k c m k c k c Fig. : Four mass system. For the following investigation, one force actuator and one velocity sensor at the first mass are assumed. Then, as seen in the revious chater, negative feedback with roortional gain P K is able to dam structural vibrations. The measurement equation becomes g r e y + - u G (s) P g i g n A A i A n S S i S n G (s) F collocated air Fig. : Closed-loo structure of direct velocity feedback for collocated actuator sensor airs.

122 8 8 SAMCO Final Reort 6 T y = [ ] [ z z z3 z4 z& z& z& 3 z& 4 ] () Due to the low structural daming and the collocation, the transfer function shows the alternating resonance/anti-resonance attern (Fig., Preumont ()). The resonances are the oles of the lant, where the velocity z& (outut) is infinite large for the external force f (inut) near to zero. The anti-resonances are the lant s zeros, where the velocity z& becomes zero for any external force. The alternating ole/zero attern is also visible in the real-imaginary lane (Fig. 3 and Fig. 5). The control target shall be maximum daming of the structural mode with the lowest frequency, because this mode usually roduces the largest dislacement (the modes in Fig. 3 are numbered with increasing frequency). The goal is to find the otimum feedback gain g for maximum additional daming. Obviously, there exist two cases for g that cannot be otimal: g = : The feedback gain is zero. Hence, the closed-loo system behaves like the oen-loo system. Then, the ole of the oen-loo system is identical with the ole of the closed-loo system (Fig. 3). g = : The feedback gain is infinite large. Then, the control force is also infinite large, which results in a nodal oint of the structure at actuator osition. The actuator force behaves like a comletely stiff bearing. The closed-loo ole is identical with the oen-loo zero due to vanishing velocity and dislacement, resectively, at actuator osition (Fig. 3). Phase (deg) Magnitude (db) Bode Diagram lant inut: force f lant outut: velocity x - Frequency (rad/sec) K:.4 Pole: i Daming:.345 Frequency:.85 (rad/s) Im(s).5 closed-loo ole of mode i moving from oen-loo ole (K=) to oen-loo zero (K= ) of mode i location of closed-loo ole for maximum additional daming ζi K otimal Root Locus ζ i ω i ω i Re(s) ζ i arcsin( ) K: Pole: i Daming:.953 Frequency:.5 (rad/s) oen-loo ole of mode i oen-loo zero of mode i K: Pole: i Daming:.58 Frequency:.63 (rad/s) Fig. : Transfer function of four mass system with force inut and velocity outut at mass no.. Fig. 3: Placement of closed-loo ole for maximum additional daming of mode (direct velocity feedback). As exlained in the receding chater, the root locus is the trajectory of all closedloo oles for g increasing from zero to infinity. Considering the following characteristics of the closed-loo ole ( ole cl i ) = ζ i i real ω, olecl i = ωi (), (3) Page of 55

123 8 SAMCO Final Reort 6 the closed-loo structure with maximum additional daming shows its ole on the root locus where the angle between real and imaginary arts reaches its maximum (Preumont ()) maximum daming of mode i : ζ i max real max ωi ( ole ) This fact also oints out that large daming may only be achieved for trajectories going far into the left half lane in the Nyquist Diagram. This is only ossible for oles and zeros being well searated. The transfer function of the lant with otimally damed mode is shown in the Bode Diagram deicted in Fig. 4. The resulting ole locations of all four structural modes are deicted in Fig. 5. This figure clearly demonstrates that single actuator sensor airs may only dam one mode otimally. Phase (deg) Magnitude (db) g = 43: otimal for maximum daming of mode Bode Diagram - Frequency (rad/sec) lant controlled lant roortional gain: g = 43 Im(s) closed-loo oles of modes -4 for g = 43 oen-loo oles: closed-loo oles of modes -4 for g = oen-loo zeros: closed-loo oles of modes -4 for g = Root Locus maximum additional daming for mode cl i (4) Re(s) Fig. 4: Comarison between very low damed lant and lant with otimally damed mode. Fig. 5: Location of closed-loo oles of all four modes for maximum daming of mode. Active daming with force actuators As seen in the examle of direct velocity feedback, the idea is to feed the force actuator with a signal being roortional to the actual vibration velocity at actuator osition. Negative feedback guarantees energy dissiation and thus stability of the control loo. Basically, there are three ways how to measure the actual vibration velocity. These three ossibilities are described in the following. a) Dislacement measurement Notice that measuring dislacements requires the differentiation of these signals in real time in order to get the actual velocities (Gandhi and Munsky ()). Due to the transfer function of derivative elements, measurement noise will be amlified. Therefore, when dislacement sensors are used for active daming, the derivative element must be extended by a low ass filter of n th order ( G F ) for a sufficient good roll-off behaviour of the closed loo system Page 3 of 55

124 SAMCO Final Reort 6 G C n ωrf ( s) = g s (5) { ( s + ω ) active with dislacement daming sensor rf 443 The filter order should be at least three ( n = 3 ) in order to roduce at least a roll-off of -4 db/dec (d Azzo and Houis (995), Preumont ()). The filter corner frequency is the frequency above which the signal arts are cut off. The filter is designed the way that the steady-state amlification is one ( G F ( s = ) = ). b) Velocity measurement Although the signal does not have to be differentiated, again, it is recommended to add a low ass filter to the roortional gain for measurement noise suression. The filter should be of nd order or higher in order to roduce -4dB/dec or more. G C G F n ωrf ( s) = g (6) { ( s + ω ) sensor with active velocity daming n rf 443 G F n c) Acceleration measurement Measuring the acceleration requires the integration of the signal in real time. The integration hels already to suress measurement noise, but only with a roll-off of - db/dec G C ( s) = g (7) { s active with accelerometer daming In order to reach a roll-off behaviour of -4 db/dec, a low ass filter of first order may be added. It may hel not destabilizing modes of high frequencies. The first order low ass filter together with the integration of the acceleration signal gives the form of a nd order filter G C ( s) = g s + ζ Fω F s + ωf The tuning arameters of the second order filter, the filter frequency and the filter daming ratio, allow for tuning the filter recisely to the mode of the lant to be damed. According to Preumont (), the rule of thumb for the filter tuning becomes (index F for filter, index P for lant, index i for mode number): F G F s ( ω ω ) F i close to P i F i > F i (8) ω ω (9) ζ = ζ ( increase of daming ratio) F (3) Page 4 of 55

125 SAMCO Final Reort 6 By designing several second order filters, several structural modes may be damed. Here, the usual rocedure is to determine the gains g by exeriment, starting with the filter design for the mode with highest frequency. These n filters targeted to dam n structural modes do not necessarily need n actuator sensor airs. The number of actuator sensor airs may be smaller than n. i Active daming with strain actuators Strain actuators may work in two rincile ways: i) If iezo elements deform freely, they roduce a voltage signal (outut) that is roortional to the deformation of the iezo element. ii) If iezo elements are, e.g., glued to the structure surface, their deformation is not free. Then, they aly a strain to the structure which is roortional to the inut voltage signal. As a consequence of these two different working rinciles, either the dislacement of the structure or the force generated by the iezo element is measured. a) Dislacement measurement The outut signal of the iezo element which acts as a strain sensor is roortional to dislacements. Then, active vibration mitigation is feasible by roducing a stress being roortional to the dislacement if the entire feedback loo is ositive (Preumont ()) G C ( s) = { g (3) n s + ζ active daming Fω F s + ω F with strain sensor That is why this active control scheme is called Positive Position Feedback (PPF). For a sufficient roll-off, a second order filter should be designed similarly to the filter design for DVF using accelerometers. G F b) Force measurement In the case of force measurements, iezo elements are collocated with force transducers. If the force alied to the structure is measured, the measurement signal must be integrated and fed back in a ositive feedback loo in order to roduce a root locus being stable for all roortional gains. This active control scheme is called Positive Force Feedback (PFF). In order to avoid the roblem of saturation for integral gains, the forgetting factor ε should be introduced, which moves the controller ole from the origin slightly to the negative real axis G C ( s) = g 443 s + ε active daming ( with including force sensor forgetting factor ) G F (3) Page 5 of 55

126 SAMCO Final Reort References Achkire, Y., Bossens, F., and Preumont, A. (998), Active daming and flutter control of cable-stayed bridges, Journal of Wind Engineering and Industrial Aerodynamics, 74-76, d Azzo, J. J., and Houis, C. H. (995), Linear Control System Analyzes and Design, Conventional and Modern, Fouth Edition, McGraw-Hill (eds.), ISBN Gandhi, F., and Munsky, B. (), Comarison of Daming Augmentation Mechanisms with Position and Velocity Feedback in Active Constrained Layer Treatments, Journal of Intelligent Material Systems and Structures, 3(5), Geering, H. P. (99), Mess- und Regelungstechnik,. Auflage, Sringer-Verlag Berlin Heidelberg. Nakamura, T., Saga, N., and Nakazawa, M. (), Imedance Control of a Single Shaft-tye Clutch Using Homogeneous Electrorheological Fluid, Journal of Intelligent Material Systems and Structures, 3(7/8), Preumont, A. (), Vibration Control of Active Structures, Kluwer Academic Publishers, Dordrecht. Soong, T. T. (99), Active structural control: theory and ractice, Cheng, W.F. (advisory editor), Longman Scientific & Technical, ISBN Subramanian, P. (), Vibration suression of symmetric laminated comosite beams, Journal of Smart Materials and Structures, (6), Notations Symbol Descrition Unit G (s) transfer function - K feedback gain - M, C, K mass, daming, stiffness matrices - T time constant s e error - f force N g feedback gain - m, c, k mass, viscosity, stiffness kg, kg/s, kg/s n order - s comlex frequency variable - t time s u controlled inut variable of the lant - v, w measurement noise, disturbance - y lant outut variable - z, z&, & z dislacement, velocity, acceleration m, m/s, m/s Page 6 of 55

127 SAMCO Final Reort 6 α tuning factors for Lead / Lag elements - ε forgetting factor - ω circular frequency rad/s ζ daming ratio - Suerscrits T Descrition transosed Subscrits C D F I P cl crit f i n rf Descrition controller derivative filter integral lant, roortional closed-loo critical filter mode number number of collocated actuator sensor airs roll-off Page 7 of 55

128 SAMCO Final Reort 6..3 Otimal control Otimal controllers are state-feedback controllers which try to equalize the states to zero. The states are the state variables of the lant model that is reresented in the so-called state sace form. Hence, first, the state sace reresentation of linear dynamic systems will be introduced. Then, the otimal control aroaches such as Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian noise control algorithm (LQG) will be described...3. State sace reresentation The reresentation of any dynamic system in state sace reresentation shall be exlained on behalf of a single degree of freedom system (SDFS) subjected to the external control force f and the external disturbance force w (d Azzo and Houis (995), Geering (99)). The differential equation of the SDFS becomes The following substitutions m & z&+ cz& + kz = f + w () k c ω =, ζ =, u = f ()-(4) m mk yield to the differential equation of a SDFS with resonance circular frequency ω of the undamed SDFS and daming ratio ζ u & z + ζω z& + ω z = (5) m By introducing the state variables x x z = z& (6) the second order differential equation may be rewritten in state sace reresentation (SSR) including only first order differential equations x& x& = ω x + u + ζω x m w In order to measure the vibration state of this SDFS, the following ossibilities exist: Measuring the dislacement ( x ). Measuring the velocity ( x ). Measuring the acceleration ( x& ). According to the sensor tye, the measurement equation including measurement noise v looks as follows Page 8 of 55 m (7) y = x = [ ] x + [] u + [] w + v (8)

129 SAMCO Final Reort 6 y = x = [ ] x + [] u + [] w + v (9) y = x& = [ ω ζω ] + [ ] u + [ ] w v () x m m + The SSR of a any dynamic, linear, time-invariant system with several measured outut variables (MO: mixed outut), one inut variable (SI: single inut), the unknown rocess/system w, and the measurement noise v becomes in matrix notation (Fig. 6) x& = A x + B u + G w () y = C x + D u + H w + v () The SSR is called time-invariant if the system matrices A, B, C, D, G, and H do not vary in time. Usually, the measurement noise is assumed to be a zero mean white noise signal. A white noise rocess is a mathematical idealization of a stationary random rocess, whose elements are comletely uncorrelated in time. The matrix G describes the local distribution of the rocess/system noise w. Besides modelling errors, it may include unknown, external forces and noise in the vector of the inut variable. Modelling errors may result from: neglected nonlinearities of the lant, neglected very fast dynamics of the lant, neglected very slow dynamics of the lant, and neglected nonlinearities and dynamics of actuators and sensors...3. Linear quadratic regulator A full state feedback may be designed with an Otimal Linear Quadratic Regulator (LQR, d Azzo and Houis (995), Geering (99), Preumont (), Ribakov et al. (3)). This feedback aroach requires that all ( k ) state variables are known and therefore measurable. The goal in structural control often is to lace the oles of the closed-loo system with LQ Regulator the way that the daming of the closed-loo structure is increased comared to the daming of the uncontrolled lant. dynamics of real lant real measurements dynamics of real lant real measurements H w u x=ax+bu+gw x H D C v y w u x=ax+bu+gw x D C v y -K regulator Fig. 6: State sace reresentation of the lant. Fig. 7: Plant in SSR, all state variables measured, full state feedback with stochastic LQ Regulator. Page 9 of 55

130 SAMCO Final Reort 6 One distinguishes between stochastic and deterministic LQ Regulators, according to the fact if an unknown disturbance w (assumed as white noise) is resent or not. The stochastic LQ Regulator has to be designed for the following lant subjected to the white noise excitation w with intensity W x& = A x + B u + G w with E [ w w T ] = W (3), (4) Then, the full state feedback u = K x (5) minimizes the erformance index (cost functional in the case of deterministic LQR) J T = E[ x T Q x + u R u] (6) q L q M Q =, R = (7), (8) M O L q j j This examle assumes one actuator, so u and R are scalar. The inut weight factor R and the factors q ii are design arameters of the LQ Regulator. For large values of R, the control outut u will be large and consequently force the lant raidly ( ) q ii into its equilibrium osition. Thus, the tracking error will be small for large values of ( q ii R), but at the exense of large control effort. The state feedback, which is based on measurements, is not otimal due to the measurement noise v y = I x + v u = K x K y (9), () Besides the regulator tuning arameters Q and R, the design of the regulator matrix K requires the knowledge of the system matrices A and B. These matrices are based on the hysical model of the real lant. Hence, they include modelling errors. In order to distinguish between the real lant behaviour and the model behaviour, the model matrices used for model-based controller design are indicated in the following with the index c K = f ( A,B, Q, R) () The resulting control system is a Single-Inut-Mixed-Outut System (SIMO, Fig. 7), however, also MIMO (mixed inut, mixed outut), SISO (single inut, single outut), and MISO (mixed inut, single outut) are conceivable. As mentioned above, LQR design makes it absolutely necessary knowing all state variables, but almost in all ractical cases, the state variable vector is not comletely known. In this case, the state variables must be reconstructed using a model of the lant (Zhang and Roschke (999)). The rocedure of full state reconstruction will be exlained in the next subchater. c c Page 3 of 55

131 SAMCO Final Reort Linear Quadratic Gaussian noise control Modelling the lant The rocedure how to design a Linear Quadratic Gaussian noise control algorithm (LQG, d Azzo and Houis (995), Christenson et al. (), Kim et al. (3), Preumont ()) will be exlained on behalf of a beam model. It is subjected to a white noise rocess w and the measured state variables include the measurement noise v. The one-dimensional finite element model (FE model) of a beam with dislacement z erendicular to the beam length and control force f becomes M & z + C z& + K z = Θ w + Λ f () fe The SSR of this differential equation may be written with the following state variable fe = & & & fe T x [ z n zn... zn nn zn zn... zn nn ] (3) This state vector x has twice as many elements as nodal oints ( nn ) of the FE model. For high satial resolution, the order of the roosed SSR is fairly large (dimensions of SSR matrices large). Thus, the simulation of such an SSR would need large comuting time. Moreover, vibrating modes with only a coule of nodal oints er wave length may lead to noisy or scattering simulation results. In order to reduce simulation time and to avoid scattering effects, the original FE model may be: a) simulated with the Newmark integration method, which adds fictitious daming to higher modes and this way hels to damen vibrations or scattering roduced by higher modes with insufficient satial resolution, or b) transformed from Cartesian coordinates into modal coordinates in order to truncate higher modes with insufficient satial resolution. Since the Newmark integration algorithm is commercially available, the second method of coordinate transformation shall be described hereafter. The SSR in modal coordinates includes the dislacements and velocities of the first mm modes of the lant and not of all nn nodal oints of the original FE model. One method of transforming the FE model from Cartesian coordinates into modal coordinates is the Galerkin method (Ni et al. (), Raja et al. ()) mm i i q j j i < j= z ( r, t) The shae functions may be aroximated by sinusoidal functions j i fe fe ( t) ϕ ( r ) ( mm nn) (4) { j ( r L )} The resulting beam model in modal coordinates becomes with the lant s matrices in modal coordinates ϕ ( r ) = sin π (5) i beam M q& + C q& + K q = Θ w + Λ f (6) m T T T T M m = Φ M feφ, Cm = Φ C feφ, Dm = Φ D feφ, Θ m = Φ Θ fe, Λ m = and the transformation matrix m m (7)-(3) m m Φ T Λ fe Page 3 of 55

132 SAMCO Final Reort 6 ϕ( r ) L ϕ mm ( r ) Φ = M O M (3) ϕ ( ) ( ) rnn L ϕ mm rnn The beam model may take the first m modes into account. The according state variable vector becomes x = q & & & ] [ m qm... qm mm qm qm... qm mm Assuming that rocess noise w and measurement noise v are white noise rocesses with intensity matrices W and V >, and for the case that all nodal accelerations are measured, then the state and measurement equations for one actuator (u is scalar) become with T (33) x& = A x + B u + G w (34) y = C x + D u + H w + v (35) = I A, B = M m K m M m C, G = m M m Λ m M m Θ m (36)-(38) T T C [ Φ M C Φ M K ] T T =, D = Φ M Λ ], H = Φ M Θ ] m Full state reconstruction m m m [ m m [ m m (39)-(4) Since it would be too exensive or simly not feasible to measure all state variables because they are not available, the idea is to reconstruct the full state vector by a model of the lant (model matrices indicated by index c ). The state reconstruction of the lant with the rocess and measurement noise described above is realized by the observer of the form ( y C xˆ D u) xˆ& = A xˆ + B u + L (4) c c Here, L is the observer gain matrix, which is chosen the way that the difference between true state x and estimated state xˆ converges to zero c red e = x xˆ (43) If L is chosen the way that the error covariance matrix is minimized, the designed observer is called minimum variance observer or steady-state Kalman-Bucy Filter (KBF, Geering (99)). In order to design the observer gain matrix L, model matrices must be available and the intensities of rocess and measurement noise must be assumed. The observer gain matrix is defined by the following inut data c red Page 3 of 55

133 L SAMCO Final Reort 6 T T T ( A,G,C,H, E[ w w ], E[ v v ], E[ w v ]) = f c c c red c red Since only some of the nn nodal accelerations are measured, the system matrices C, D, and H must be reduced to those columns where the sensors are located ( C ) = ( number of nodes) ( number of modes) dimension c (44) (45) ( C ) = ( number of sensors) ( number of modes) dimension c red ( ) = ( number of nodes) (46) dimension D (47) c ( ) = ( number of ) dimension D sensors (48) c red ( ) = ( number of nodes) dimension H (49) c ( ) = ( number of ) dimension H sensors (5) Process and measurement noise must be assumed. Hence, they may be seen as observer design arameters (Kim et al. (3), Sadri et al. ()). If measurements of, e.g., the disturbance forces and some nodal velocities are available, the intensities of rocess and measurement noise may be estimated from the alternating signal arts (AC) of these measurements. In this examle, the measurement noise is assumed to count for 5% of the maximum value E E c red T [ w w ] ( AC( w )) var (5) mea (. AC( z& )) T [ v v ] var 5 mea (5) Regulator design The full state feedback (single actuator) u = K xˆ (53) minimizes the erformance index J T = E[ x T Q x + u R u] (54) Besides the system matrices A c and B c, the regulator gain matrix requires the state weight factor Q and the inut weight factor R (single actuator) as regulator design arameters K = f ( A,B, Q, R) (55) c c Page 33 of 55

134 SAMCO Final Reort 6 q L q M Q =, M O L qmmmm R = (56), (57) The value of R may be set to because the observer design only deends on the ratio of q ii / R. When designing the regulator, the following rule of thumb should be resected (Preumont ()): For every mode to be damed, the ratio of q ii / R should be chosen the way that the negative real art of the observer ole is aroximately -6 times larger than the negative real art of the corresonding regulator ole. The reason for this rule is that regulator oles are the oles of the closed-loo structure or, in other words, of the controlled lant (Fig. 8). Considering that large negative real arts of observer oles imly that estimation errors decay faster than the relevant dynamics of the lant, then, the reconstructed state variables follow closer the actual true state variables. Thus, small values of q ii / R will roduce a state estimate of high accuracy but with the drawback of increased tracking error. The ratio q ii / R is the regulator tuning variable for the controller designer in order to find the desired comromise between tracking and estimation error. The examle of ole lacement deicted in Fig. 9 is the result of the following control requirements: The control target is to dam the first four structural modes. The observer oles should be at least 4 times faster than the regulator oles. If only the first four modes (control target) were observed, the estimated vibration state would be falsified due to the influence of the higher modes (5- ) of the real lant on the actual vibration state. However, a falsified state estimate leads to a non-otimal or even wrong state feedback and therefore to a deteriorated control erformance. Hence, higher modes than only the controlled ones should be observed. The effect that unobserved modes falsify the state estimation and evoke a false state feedback is called sillover instability. The fact that the state estimation may only be based on a model of the lant is indicated again by the index c of the system matrices (Fig. 8). In the examle shown here, the first eight modes are observed, although only the first four modes are to be controlled (damed). The matrix Q weights the modal state variables (Johnson et al. (999)). Page 34 of 55

135 numbers of residual modes of real lant SAMCO Final Reort 6 dynamics of real lant real measurements observer ole of mode closed-loo ole of mode with increased daming w H D x + u x=ax+bu+gw C x -K x=a x+b u + L(y-C x-d u) c c c c regulator observer v y lant model includes modes -8 in order to avoid sillover instability Im(s) control target: increasing daming of the first 4 modes oles of real lant oles of lant model regulator oles observer oles (time continuous) Re(s) ole of mode of real lant with low structural daming LQG controller Fig. 8: Real lant with full state observer and regulator. Fig. 9: Observer and regulator design. Discrete-time control algorithm For the imlementation of control algorithms, nowadays, it is common to transform the algorithm into the discrete time domain. This enables to run the algorithm a defined clock rate (controller frequency). The algorithm is written in commercially available software which can be rogrammed on the host PC. The use of a PC makes it ossible to adjust controller arameters on site. For real-time control, the software has to be downloaded on either a real-time controller box or on a PC real-time card. This is the normal rocedure as long as the control algorithm is within the develoment rocess. In the case of LQG control, the control algorithm consists of the following three stes: The measurement udate at the time k, the extraolation ste from time k to ( k +), and the ste of state feedback (discrete-time system matrices indicated by index dt ) xˆ [ k k] xˆ[ k k ] + M ( y[ k] C xˆ[ k k ] D u[ k] ) = dt c red dt c red dt The estimated measurements become as follows (58) x ˆ[ k k] = A xˆ[ k k] + B u[ k] (59) c + c dt c dt [ k] K ˆx [ k k] u = dt (6) [ k] C xˆ [ k k] + D u[ k] = c red dt c red dt y (6) The discretization may be realized using the Tustin aroximation. Since the measurements y [ k = ] at time k = already exist, the observation starts with a measurement udate, followed alternating by an extraolation ste and measurement udate (Geering (99)). Page 35 of 55

136 SAMCO Final Reort 6 Semi-active control When controlling a structure using semi-active control devices such as, e.g., magnetorheological and electrorheological fluid damers, the regulator should feed its outut to the daming device only if structural energy shall be dissiated. Otherwise, if the state feedback desires active forces, the regulator outut should be set to zero (Ying et al. ()), because semi-active devices cannot roduce ower (Fig. ). This switching is called cliing of the control force which ends u in high frequency excitation and the fact that the clied control force is not otimal anymore. Fig. : Schematic oerating range of magnetorheological fluid damers. The decision, if the regulator outut is an active or assive force, may be based on the measured or estimated structural velocity at the osition of the semi-active control device u sa K xˆ = : : {( K xˆ z& < } { ˆ u mea ) or ( K x z& u est ) < } {( K xˆ z& ) } or {( K xˆ z& ) } u mea The force of magnetorheological and electrorheological fluid damers, which reresent often used semi-active control devices in the field of structural control, is not only limited by the maximum damer force but also by the residual friction at zero command inut (Fig. ). Hence, the otimum control force is not only constraint by the decision between active and dissiative forces but also by the limited oerating range of magnetorheological and electrorheological fluid damers. Not all desired forces, which are dissiative, may be tracked due to the limited damer s oerating range. This may be seen as a secondary cliing. u est (6)..3.4 References d Azzo, J. J., and Houis, C. H. (995), Linear Control System Analyzes and Design, Conventional and Modern, Fouth Edition, McGraw-Hill (eds.), ISBN Christenson, R. E., Sencer Jr., B. F., and Johnson, E. A. (), Exerimental Verification of Semiactive Daming of Stay Cables, Proceedings of the American Control Conference, IEEE, Piscataway, NJ, USA, 6, Page 36 of 55

137 SAMCO Final Reort 6 Geering, H. P. (99), Mess- und Regelungstechnik,. Auflage, Sringer-Verlag Berlin Heidelberg. Johnson, E. A., Sencer Jr., B. F., and Fujino, Y. (999), Semiactive Daming of Stay Cables: A Preliminary Study, Proceedings of the th International Modal Analysis Conference - IMAC,, Kim, D.-H., Han, J.-H., Yang, S.-M., Kim, D.-H., Lee, I., Kim, C.-G., and Hong, C.- S. (3), Otimal Vibration Control of a Plate Using Otical Fiber Sensor and PZT Actuator, Journal of Smart Materials and Structures, (4), Ni, Y. Q., Cao, D. Q., Ko, J. M., and Chen, Y. (), Neuro-Control of Inclined Sagged Cables Using Semi-Active MR Damers, Proceedings of International Conference on Advances in Structural Dynamics, J. M. Ko and Y. L. Xu (eds.), Elsevier Science Ltd. (ubl.),, Preumont, A. (), Vibration Control of Active Structures, Kluwer Academic Publishers, Dordrecht. Raja, S., Pratha, G., and Sinha, P. K. (), Active vibration control of comosite sandwich beams with iezoelectric extension-bending and shear actuators, Journal of Smart Materials and Structures, (6), Ribakov, Y., and Reinhorn, A. M. (3), Design of Amlified Structural Daming Using Otimal Considerations, Journal of Structural Engineering, 9(), Sadri, A. M., Wright, J. R., and Wynne, R. J. (), LQG control design for anel flutter suression using iezoelectric actuators, Journal of Smart Materials and Structures, (6), Ying, Z. G., Ni, Y. Q., and Ko, J. M. (), Non-cliing otimal control of randomly excited nonlinear systems using semi-active ER/MR damers, Proceedings of SPIE, Smart Structures and Materials : Smart Systems for Bridges, Structures, and Highways, 4696, 9-8. Zhang, J., and Roschke, P. N. (999), Active Control of a Tall Structure Excited by Wind, Journal of Wind Engineering and Industrial Aerodynamics, 83(-3), Notations Symbol Descrition Unit A, B, C, D, G, H matrices of SSR - C daming matrix - I, identity matrix, matrix of zeros - J erformance index / cost functional - K stiffness matrix, regulator gain matrix - L observer gain matrix - L length m M mass matrix, diskrete-time observer gain matrix - Q state weight matrix - Page 37 of 55

138 SAMCO Final Reort 6 R inut weight factor - V intensity matrices of measurement noise - W intensity matrices of rocess noise - Φ transformation matrix - Λ distribution vector of control forces - Θ distribution vector of disturbance forces - c viscosity kg/s e error vector - f force; function N; - k discrete-time variable; stiffness -; kg/s m mass kg mm number of modes - nn number of nodal oints - q only time-deendent vector of modes - r Cartesian coordinate in the direction of the beam m t time s u controlled inut variable of the lant N var variance - v vector of measurement noise - w vector of disturbance - x vector of state variable - xˆ vector of estimated state variable - y vector of lant outut variable - z dislacement m ϕ mode shae - ω fundamental circular frequency of undamed structure rad/s ζ daming ratio - Suerscrits T Descrition transosed Subscrits c dt Descrition controller discrete-time Page 38 of 55

139 SAMCO Final Reort 6 est fe m mea mm n nn red sa u estimated finite element model mode, modal coordinates measured number of modes nodal oint number of nodal oints reduced semi-active controlled inut variable of the lant Page 39 of 55

140 SAMCO Final Reort 6..4 Other control algorithms This chater will give a brief introduction in fuzzy control and neural network control reresenting linear control aroaches that are used fairly often in the field of structural control. Sometimes, they are used in combination with control aroaches described in the revious chaters...4. Fuzzy control Fuzzy control reresents an alternative control aroach to the classical control theory which relies on the mathematical formulation of the lant (Liu et al. ()). The classical control strategies have been widely used for systems that may be aroximated by linear, time-invariant models. However, these control strategies are rarely imlemented for non-linear, time-variant, comlex systems including noisy measurements. Here, fuzzy control may have its advantages. The main idea of fuzzy control is to build a model of an exert oerator who is caable of controlling the lant without thinking in mathematical terms. The erson controls the lant by a set of linguistically exressed rules that result from the a riori know-how of the hysics of the lant. Some characteristics of fuzzy logic may be listed here: Fuzzy logic is concetually easy to understand. Fuzzy logic is tolerant of imrecise data. Fuzzy logic can model nonlinear functions of arbitrary comlexity. Fuzzy logic is able to match any set of inut-outut data. This rocess is made articularly easy by adative techniques. Fuzzy logic can be built on to of the exerience of exerts. In direct contrast to neural networks, which take training data and generate oaque, imenetrable models, fuzzy logic relies on the exerience of eole who already understand the lant. When fuzzy control is called adative fuzzy control, then neural networks are used for the udate of the imlemented rules. Fuzzy logic can be blended with conventional control techniques and it does not necessarily relace conventional control methods. In many cases, fuzzy control systems augment them and simlify their imlementation. Fuzzy logic is based on natural language. Page 4 of 55

141 SAMCO Final Reort 6 α L m Actuator u w Structure Sensors y f m Defuzzifier Fuzzy Controller Fuzzy Inference Fuzzifier x Fuzzy Rule Base Membershi Functions Fig. : Inverted endulum on a cart. Fig. : Structure of a fuzzy controller. Develoing a fuzzy controller for the stabilization of the endulum in the uer osition using a motor-driven cart (Fig. ) shall demonstrate how fuzzy control may work. The linearized equations for small angles and small angular velocities become ( m + m ) gα f m & x = f m gα Lα& =, & & (), () m The fuzzy controller (Fig. ) may be designed based on the measured angle and angular velocity = [ α, & α ] T y (3) Assuming a erson should stabilize the inverted endulum, the rules for ushing and ulling the cart may be similar to the fuzzy rules as following: If the angle is slightly to the left (SL) and the angular velocity is moderate to the right, then do not move the cart, i.e., the force shall be zero (Z). If the angle is moderately to the left (ML) and the angular velocity is zero (Z), then move the cart moderately to the left, i.e., the force shall accelerate the cart moderately to the left (ML). All rules for balancing the endulum in the uer osition are listed in Table 7. Each rule takes the two inut variables, measured angle and measured angular velocity, and roduces the outut variable force, which drives the cart. The rules are IF-THEN rules, connected by logical oerators such as AND, OR, and NOT. The resulting five membershi functions for the examle described here are: moderately to the left (ML), slightly to the left (SL), zero force (Z), slightly to the right (SR), and moderately to the right (MR). Table 7: Possible fuzzy rules for inverted endulum. force α& =ML α& =SL α& =Z α& =SR α& =MR α =ML ML ML ML ML SL α =SL ML ML SL Z Z α =Z SL SL Z SR SR α =SR Z Z SR MR MR α =MR SR MR MR MR MR Page 4 of 55

142 SAMCO Final Reort 6 fuzzification inference defuzzification µ ML SL Z SR MR µ (α) Z =.73 µ (α) SL = α actual µ ML SL Z SR MR µ Z (α) =.55 µ SR(α) = α actual α [ o ] α [ o /s] case : IF α is Z AND α is Z THEN f is Z ( α =.73) ( α=.55) µ min( α)=.55 case : IF α is Z AND α is SR THEN f is SL ( α =.73) ( α=.45) µ SL min( α)=.45 case 3: IF α is SL AND α is Z THEN f is SR ( α =.7) ( α=.55) µ min( α)=.7 Z SR f f (centroid calculation) µ force f case 4: IF α is SL AND α is SR THEN f is Z ( α =.7) ( α=.45) µ min( α)=.7 Z f f result of aggregation (maximum): µ SL Z SR f Fig. 3: The stes of fuzzification, inference, and defuzzification. The fuzzy controller may be divided into the following three stes:. Fuzzification: All measured variables are transformed to the degrees of membershi µ (Fig. 3 left). For this ste, the range of each measured variable must be divided into sections.. Inference: According to the actual fuzzy variables, the corresonding rules are alied using the fuzzy oerations for the logic oerators, such as AND, OR, and NOT (Table 8). The imlication method limits the membershi function according to the degree of truth (minimum method, Fig. 3 middle). The result of the inference ste is the aggregated membershi function of the outut variable (maximum method), in this case, the cart force. 3. Defuzzification: The most common defuzzification method is the centroid calculation, which returns the centre of area under the aggregated curve (Fig. 3 right). The osition of the centre defines the value of the outut variable, in this case the value of the force alied to the cart. Table 8: Fuzzy logic oerations. AND OR NOT min max additive comlement..4. Neural network control Generally, neural networks are models of static and dynamic structures that are not based on hysical relations but consist on mathematical functions, whose coefficients result from learning or training using measurement data (Fig. 4). Besides the varied coefficients, also the functions itself and the structure of the neural network Page 4 of 55

143 SAMCO Final Reort 6 may be varied. Once the structure and the functions are chosen, the neural network is trained by adjusting the coefficients using measured inut data as long as the error between model outut (outut of the neural network) and the target data (measured lant outut) is smaller than the desired error (Fig. 4). target data (measured lant outut) inut data (measured lant inut) neural network (including connections, called weights, bewteen neurons) outut data ('simulated' model outut) comare (error) adjusting weights until error < error max Fig. 4: Black box identification by training of neural networks. Neural networks consist of neurons working in arallel and of several layers working in series (Fig. 5). The coefficients of the neurons are the weight factor w and the bias factor b, which scale the inut i. The result n asses the transfer function f. Tyical transfer functions (TF) may be: Hard limit TF, symmetric hard limit TF. Log sigmoid TF. Positive linear TF, linear TF. Saturation TF, symmetric saturation TF. Hyerbolic tangent sigmoid TF. Triangular basis TF. Several neurons together build one layer (Fig. 5). The number of neurons of each layer deends on the design of the neural network. One layer includes the same tye of transfer function which must be assumed by the neural network designer. The choice of the tye of transfer functions deends on the a riori know-how of the designer about the lant behaviour. Several layers together build one neural network. Due to the black-box characteristics of neural networks, they may be the referable modelling aroach for non-linear, comlex systems where hysically based models lack. Within the field of structural control, neural networks are often used in order to relace a state estimator since state estimators often are based on rather simle finite element modelling aroaches and therefore modelling errors may be too large (Xu et al. (3a), Xu and Shen (3b)). Then, using vibration measurement data of the target civil structure, a neural network may be trained in order to have a state estimator of high accuracy available for structural control. Page 43 of 55

144 SAMCO Final Reort 6 inut i i i R neuron (iw, ), (iw S,R ), (lw, ), first layer second layer k th layer outut (lw, ) 3, (n ) (f) (o ) (n ) (f) (o ) (b ) (b ) (lw, ) k,(k-) (n ) (f) (o ) (n ) k (f) k (o ) k o (n ) (f) (o ) (b ) (b ) k (b ) (n Sk ) k (f) k (o Sk ) k o Sk (n S ) (o S ) (b Sk ) k (f) (b S ) (lw S,S ), (n S ) (f) (o S ) (lw Sk,S(k-) ) k,(k-) (b S ) (lw S3,S ) 3, Fig. 5: Neural network consisting of neurons (weights and transfer function) and layers References Liu, Y., Gordaninejad, F., Evrensel, C. A., Hitchcock, G., and Wang, X. (), Variable Structure System Based Fuzzy Logic Control of Bridge Vibration Using Fail- Safe Magneto-Rheological Fluid Damers, Proceedings of SPIE, S.-C. Liu and D. J. Pines (eds.), 4696, 9-7. Xu, Z.-D., Shen, Y.-P., and Guo, Y.-Q. (3a), Semi-active control of structures incororated with magnetorheological damers using neural networks, Journal of Smart Materials and Structures, (), Xu, Z.-D., and Shen, Y.-P. (3b), Intelligent Bi-state Control of the Structure with Magnetorheological Damers, Journal of Intelligent Material Systems and Structures, 4(), Notations Symbol Descrition Unit L Length m b bias factor - c viscosity kg/s f force; transfer function N; - g gravitational acceleration m/s i layer inut variable, layer outut variable - k stiffness kg/s lw weight factor - m mass kg n neuron outut variable - Page 44 of 55

145 SAMCO Final Reort 6 o layer outut variable - u controlled inut variable of the lant - w disturbance N y vector of lant outut variables - x dislacement m α angle rad µ degree of membershi - Suerscrits T k Descrition transosed number of layers Subscrits R Sj Descrition number of inut variables of the first layer number of neurons of layer j Page 45 of 55

146 SAMCO Final Reort 6 3 VIBRATION ISOLATION The first subchater resents assive isolators, whereas the second subchater describes the concet of active vibration isolation. 3. Passive vibration isolation 3.. Theoretical background The goal of vibration isolation is to mitigate vibrations by reducing the mechanical interaction between the vibration source and the structure, equiment etc. to be rotected. The general aroach is to insert secondary mechanical comonents (antivibration mounts) between the vibration source and the receiver with the objective to avoid any state of significant excitation in the system. The concet of vibration isolation is illustrated in Fig. 6. A single degree of freedom system whose mass reresents a structure, an equiment etc. is connected to a base suort through an isolator. Generally, in vibration isolation, two cases are considered. In the first, the isolator rotects the base suort from the effects of dynamic force F S (t) acting directly on the suorted structure. The goal of vibration isolation is to reduce the force F T (t) transmitted to the suort. In the second case, the isolator rotects the structure from the motion u S (t) of the base suort. The goal is to reduce the absolute or relative dislacement u T (t) resectively u ( t) u ( t) T S of the structure. The erformance of an isolator is usually characterized by the transmissibility. It is defined by the ratio of the magnitude of the item (dislacement, velocity, force etc.) transmitted by the isolator with resect to the magnitude of the exciting item. A mechanical isolator is generally imlemented as a system consisting of masses, srings and damers. A simle model of an isolator consists of a linear sring and dashot acting in arallel. For this isolator model, the transmissibility of the force T F for harmonic force F (t) acting on the structure is given by T F + (ζs) = Tu = () ( s ) + (ζs) Force F(t) u(t) Transmitted Dislacement u(t) Mass m Mass m Isolator Transmitted Force F T (t) Suort Isolator Base Dislacement u S (t) Suort Fig. 6: Schematic reresentation of vibration isolation systems. Left: Isolation reducing the effects of the excitation force acting on the structure. Right: Isolation reducing the effects of a forced excitation of the suort. Page 46 of 55

147 SAMCO Final Reort 6 Transmissibility F S /F ζ S = ζ S = 5% ζ S = % ζ S = % Frequency ratio f T /f S Fig. 7: Force transmissibility for an isolator consisting of a linear sring and dashot acting in arallel for different daming ratios. The absolute dislacement transmissibility T for a harmonic forced dislacement u S (t) at the suort is described by exactly the same function as T F. The force transmissibility T F is illustrated in Fig. 7 for different daming ratios. This Figure shows that a significant reduction of the force as well as dislacement amlitude is achieved for excitation frequencies being greater than natural frequency of the system. The alication of this observation in vibration isolation is called low frequency tuning. It is characterized by selecting the natural frequency of the system to be significantly smaller than the main frequency content of the vibration source. Remarkably, in low frequency tuning, increasing the daming of the system does not imrove the effectiveness of the isolation system. An alternative vibration isolation aroach consists in high tuning: The natural frequency of the system is chosen to be much greater than the main frequency content of the vibration source. In this aroach, the amlitude of the transmitted force or dislacement is similar to the amlitude of the excitation force or dislacement. Therefore, no significant reduction can be achieved. However, high tuning reduces the risk of amlification due to resonance effects. In high tuning, the effect of daming is comletely negligible. Similar transmissibility roerties are obtained for nonviscous tye of daming (Den Hartog (93), Kelly (997)). Vibration isolation of equiment is a well established technology. Its theory and ractice are covered in many books and a vast number of aers (e.g. Beranek and Vér (99), Snowden (979)). The first building to be isolated from ground-born vibration was situated directly above a station of the London Underground (Derham et al. (975)). Now, the use of vibration isolation for entire building from rail traffic is becoming a common ractice (Crockett (98), Derham and Thomas (98), Grootenhuis and Crockett (978), Grootenhuis (98)). u 3.. Design issues 3... Low tuning Low tuning is the standard aroach in vibration isolation because of its high effectiveness. The effectiveness increases by reducing the natural frequency of the isolated system with resect to the dominant frequency content of the excitation force. Therefore, the lowest ossible frequency ratio should be addressed. However, in ractice, the low tuning of natural frequency is mainly limited by dislacement or velocity restrictions. In vibration isolation of manufacturing machines, roduction requirements usually restrict the softness of the isolation to avoid too large dislacements under dead loads. The criteria rovided by machine manufacturers have to be considered in the design rocess. The effect of all degrees of freedom (translational Page 47 of 55

148 SAMCO Final Reort 6 and rotational as well as) has to be considered in the design in order to avoid unaccetable arasitic vibrations induced by the isolation devices. Secial attention has to be aid for start-u and shut-down of machines. During these oerations, the dominant excitation force frequencies may meet for a certain time eriod the natural frequency of the secondary system roducing significant dynamic dislacements. These dislacements have to be controlled either by additional vibration control devices or by a sufficiently fast start-u and shut-down oeration High tuning High tuning is usually alied when low tuning turns out to be imractical. High tuning is mainly alied in isolating machine vibrations when low tuning fails to meet roduction requirements because of a too soft secondary system. The main design issue is to rovide the secondary system with a natural frequency being much larger than the highest relevant excitation force comonents. Detailed information about the frequency sectrum of excitation forces is mandatory for a correct tuning. If this information is artially missing, a safety factor reflecting these uncertainties may be included when secifying the natural frequency of the secondary system. Generally, in high tuning, start-u and shut-down of a machine is of little concern Base isolation Base or seismic isolation of a comlete structure is based on a low tuning concet (Kelly (997)). However, because of the high energy involved during earthquake motion, several secific asects have to be considered in the design rocess. The most imortant requirements for base isolation bearings concern the lateral flexibility and energy dissiation during earthquake loading as well as rigidity under dead loads and low level horizontal loading (e.g. wind). Shear and axial stiffness, daming roerties of rubber bearing are usually rovided by the manufacturer. The eriod of vibration of the based isolated structure is designed to be significantly greater than the eriod of natural frequency of the non-isolated structure as well as the dominant eriod of excitation of earthquake motion. For structures with regular shae, the fundamental frequency of the isolated structure can be roughly estimated by the formula f m = () π k where m is the total mass of the building and k is the shear stiffness of the rubber bearing Testing and validation The testing and validation of vibration isolated structures are usually done under oerational conditions. In vibration isolation for machine induced vibrations, the tests are done by running the machines under the relevant oerational conditions. The vibrations under start-u and shut-down rocedures of the machines should also be recorded. For base isolated structures, ambient vibration tests are the cheaest method to validate the design for low level horizontal loadings. An exerimental validation of high Page 48 of 55

149 SAMCO Final Reort 6 level loading scenarios is in most cases not feasible. A long term continuous monitoring system may be able to rovide information for validation. Fig. 8: Rubber bearing alied for base isolation. Fig. 9: Shear dislacement test of a rubber bearing for base isolation Imlementation The vibration isolation devices used in vibration isolation comrise steel srings, elastomeric and natural rubber ads as well as air susension systems. These devices differ considerably with resect to size, mass, stiffness and oerating frequency. For vibration isolation of machines, the vibration isolation devices may be mounted directly under the machine. Mounting machines on a base is suitable for large dynamic forces, strongly asymmetrical machines or machines with a small mass to avoid a too soft suort. The mass of the base should be very stiff and exceed significantly the mass of the machine (u to a ratio of 8:). A large base mass reduces significantly the dynamic dislacements of the secondary system. For vibration isolation in existing structures, the mass of the base may be limited by the strength of the suorting structural elements. In base isolation of structures, natural rubber bearings are the most widely used devices. Such bearings consist of layers of rubber sheets bonded on steel lates (Fig. 8). The steel lates constrain the lateral deformation of the relative soft rubber under vertical dead loads. The vertical stiffness of rubber bearings are several order of magnitude greater than the horizontal stiffness. The lateral flexibility of bearings is tested under extreme dislacements (u to 5% shear strain, Fig. 9) References Beranek, L. L., and Vér, I. L. (99), Noise and vibration control engineering: rinciles and alications, Beranek, L. L. and Vér, I. L. (eds.), New York, J. Wiley and Sons. Crockett, J. H. A. (98) Early Attemts, Research and Modern Techniques for Insulating Buildings, Proceedings of the International Conference on Natural Rubber for Earthquake Protection of Buildings and Vibration Isolation, Kuala Lumur, Malay- Page 49 of 55