Viscous Dampers for Optimal Reuction in Seismic Response Navin Prakash Verma (ABSRA) o moel issipation of energy in vibrating civil structures, existence of viscous amping is commonly assume primarily for mathematical convenience. In such a classical amper, the amping force is assume to epen only on the velocity of eformation. Flui viscous ampers that provie this type of amping have been manufacture to provie supplementary amping in civil an mechanical systems to enhance their performance. Some flui ampers, however, exhibit stiffening characteristics at higher frequencies of eformation. he force eformation relationship of such ampers can be better represente by the Maxwell moel of visco-elasticity. his moel consists of a viscous ashpot in series with a spring, the latter element proviing the stiffening characteristics. his stuy is concerne with the optimal utilization of such Maxwell ampers for seismic performance improvement of civil structures. he force eformation relationship of Maxwell ampers is escribe by a first orer ifferential equation. Earlier stuies ealing with these ampers, use an unsymmetric set of equations for combine structure an amper system. he solution of such equations for response analysis or for optimization calculation by a moal analysis approach woul require the pair of the left an right eigenvectors. In this stuy, an auxiliary variable is introuce in the representation of a Maxwell amper to obtain symmetric equations of motion for combine structure an amper system. his eliminates the nee for working with two sets of eigenvectors an their erivatives, require for optimal analysis. Since the main obective of installing these ampers is to reuce the structural response in an optimal manner, the optimization problem is efine in terms of the minimization of some response-base performance inices. o calculate the optimal parameters of ampers place at ifferent location in the structure, Rosen s graient proection metho is employe. For numerical illustration, a 24-story shear builing is consiere. Numerical results are obtaine for seismic input efine by a spectral ensity
function; however, the formulation permits irect utilization of response spectrum-base escription of esign earthquake. hree ifferent performance inices -- inter story riftbase, floor acceleration-base, an base shear-base performance inices-- have been consiere to calculate the numerical results. A computational scheme is presente to calculate the amount of total amping require to achieve a esire level of response reuction. he effect of ignoring the stiffening effect at higher frequencies in the Maxwell moel on the optimal performance is evaluate by parametric variation of relaxation time coefficient. It is observe that the moels with higher relaxation time parameter show a ecrease response reucing amping effect. hus ignoring the stiffening effect when it is, inee, present woul provie an unconservative estimation of the amping effect. he effect of brace flexibilities on ifferent performance inices is also investigate. It is observe that flexibility in a brace reuces the effectiveness of the amper. iii
o My Dear Parents iv
Acknowlegments It was first time that I move out of my town in Inia for so long an that too 0,000 miles away from home to pursue grauate eucation at Virginia ech. But, my two years on the campus have been extremely satisfying an enoyable. his is a goo place to express few wors for the people who have mae my stay memorable. I express my eep sense of gratitue to my avisor, Professor Mahenra P. Singh, for all the help not only uring the course of my thesis an acaemics but my stay at Virginia ech. His help, in the form of moral an technical support, affection towars me an the spirite scientific temperament towars the work was key for the completion of this work. I woul also like to express my appreciation to Dr R.. Batra an Dr. S. L. Henricks for kinly serving in my committee an giving their valuable time to rea this manuscript. I take this moment to thank all the people an friens I have met in Blacksburg uring past two years. I have great respect for Pramo who has acte as my frien cum mentor from time to time. I express my gratitue to Sarbeet, Prahala, Prateep, Raan, Prashanth, Srinivas, Jiten, Nirmala, Kona an Ravi. I will always remember the goo time spent with Nishant, Deepak, Surya, Bipul, Naren, Shailesh an many others. he wonerful moments share with them in the trips outsie Blacksburg an in ay-to-ay life will always be cherishe. I woul also like to thank Luis Moreschi, who at time to time gave me his expert avice on various things relate to thesis. I eicate my work to my ear parents who have struggle throughout life to give their chilren the very best. Both my parents have been supportive of my ecisions an I am thankful for all their faith in me. his research was financially supporte by National Science Founation Grant No. MS-9987469. his support is gratefully acknowlege. Navin Prakash Verma v
ontents. Introuction.... Overview of Passive Structural ontrol....2 Viscoelastic Devices... 2.3 hesis Organization... 3 2. Analytical Formulation... 5 2. Introuction... 5 2.2 Equations of Motion... 5 2.2. lassical Viscous Dampers... 6 2.2.2 Kelvin Damper Moel... 8 2.2.3 Maxwell Damper Moel... 9 2.3 Equations of Motion with Maxwell Dampers... 2 Symmetric Equations of Motion with Maxwell Dampers... 3 2.4 Response Analysis... 6 2.5 hapter Summary... 2 3. Optimal Damper Parameters... 27 3. Introuction... 27 3.2 Performance Functions... 28 3.3 Formulation of Optimization Problem... 29 3.4 Graient alculations... 34 3.5 Numerical Results... 36 Builing Structure Moel... 36 Supplementary Dampers... 37 Performance Inices... 37 Seismic Input... 37 A omputational Issue... 38 vi
3.5. Damping Distributions for Different Inices... 38 3.5.2 Effect of Using the Maxwell Moel... 40 3.5.3 ross-effectiveness of the Drift-Base an Acceleration-Base Designs... 4 3.6 hapter Summary... 4 4. Effect of Bracing Stiffness... 55 4. Introuction... 55 4.2 Equivalent Bracing Stiffness... 55 4.3 Numerical Results... 58 4.4 hapter Summary... 59 5. Summary an oncluing Remarks... 67 5. Summary... 67 5.2 onclusions... 68 Appenix... 70 A. Partial Fraction oefficients... 70 B. Derivatives of Eigenproperties... 7. Graients alculations Formulas... 74 References... 77 vii
List of Figures Figure.: Passive response control systems, (a) seismic isolation, (b) energy issipation evices, (c) ynamic vibration absorbers (Moreschi, 2000).... 4 Figure.2: A aylor Device Damper (onstantinou et. al., 993).... 4 Figure 2.: Different Moels for Viscous Dampers.... 22 Figure 2.2: Maxwell Moel of a Damping Device Representing the Deformations in the Spring an the Damping Element... 23 Figure 2.3: Force-Deformation Responses for Different Values of Parameter a (0.2, 3.0, 0.0).... 24 Figure 2.4: Frequency Depenency of the Stiffness an Damping Parameters.... 25 Figure 2.5: A Bay with hevron Bracing Depicting Auxiliary DOF an Floor DOF for Floor i... 26 Figure 3.: Flowchart Showing Steps of Rosen s Graient Proection echnique.... 50 Figure 3.2: Power spectral ensity function of the Kanai-aimi form (ω g = 8.85 ra/s, β g = 0.65 an S = 0.069 m 2 /s 3 /ra)... 5 Figure 3.3: Evolution of Optimal Solution in Different Iterations for the Drift-base Performance inex.... 52 Figure 3.4: omparision of Drift Response Reuctions by Viscous an Maxwell Moels for τ = 0.04 an τ = 0.4.... 52 Figure 3.5: omparision of Acceleration Response Reuctions by Viscous an Maxwell Moels for τ = 0.4 an τ = 0.4... 53 Figure 3.6: Effect of τ on Drift an Acceleration Performance Inices for Uniform Distribution of otal Damping... 53 Figure 3.7: ross-effectiveness of Acceleration-base an Drift-base Designs... 54 Figure 4.: ypical configurations of amping evices an bracings, (a) chevron brace, (b) iagonal bracing, (c) toggle brace-amper system... 63 Figure 4.2: A Bay with Flexible hevron Bracing for Floor i... 64 viii
Figure 4.3: Variation of Drift Base Performance Inex with Stiffness Ratio... 64 Figure 4.4: omparision of Reuction in Drift Responses for Rigi an Flexible Bracings.... 65 Figure 4.5: omparision of Reuction in Base Shear Responses for Rigi an Flexible Bracings.... 66 ix
List of ables able 3.: Properties of 24-Story Builing... 43 able 3.2: Numerical Results Showing Incremental Refinement of the otal Damping oefficient an its Distribution to Obtain a Given Value of Drift Base Performance Inex... 44 able 3.3: Optimal Designs Obtaine for Different Initial Guesses for =.32 0 N-s/m... 45 able 3.4: Numerical Results Showing Incremental Refinement of the otal Damping oefficient an its Distribution to Obtain a Given Value of Acceleration Base Performance Inex... 46 able 3.5: Numerical Results Showing Incremental Refinement of the otal Damping oefficient an its Distribution to Obtain a Given Value of Base Shear Base Performance Inex... 47 able 3.6: omparision of Results for Maxwell an lassical Moels for τ = 0.04... 48 able 3.7: omparision of Results for Maxwell an lassical Moels for τ = 0.4... 49 able 4.: Numerical Results for Drift Base Response for total amping 9 =.32 0... 6 8 able 4.2: Numerical Results for Base Shear for total amping = 3.88 0... 62 9 x
hapter Introuction. Overview of Passive Structural ontrol Design of structures to reuce vibrations ue to external environmental forces such as wins, earthquakes etc, has been a maor concern of engineers for many years. Earthquake inuce groun motions often give a large amount of energy to structures an thus make them more susceptible to suen amage. he current esign practices allow the structure to yiel so as to absorb this huge amount of external energy without getting collapse. his is achieve by allowing inelastic cyclic eformations in specially etaile regions in a structure. his strategy is effective but it may rener a structure irreparable. he consieration of actual ynamic nature of environmental forces has given rise to newer an innovative techniques of structural protection. he focus of research in past few years has been shifte in reucing the response of the structures ue to external forces by employing special protective systems. Besies reucing amage, these methos have been successful in increasing safety of the structure. hese protective systems work on the following philosophy. he earthquake motions impart potential an kinetic energy to the structure, which makes them vibrate. A part of this energy is also absorbe by inherent amping characteristics of the structure. he protective systems either prevent the energy from reaching the structure or enhance the energy issipation capacity. Base isolation systems are esigne to prevent the energy reaching the structure by filtering it out. he energy issipation systems on the other han absorb the energy that reaches the structure in special evices calle ampers. Several ifferent types of energy issipation systems have been evelope. hey are viscous ampers, visco-elastic ampers, friction ampers, an yieling metallic ampers. here are also tune mass or tune liqui ampers, which primarily store some of the structural
hapter. Introuction 2 energy in their eformation an motion, an rely much less on energy issipation. More complete etails on the mechanics an working principles of these evices can be foun in the excellent treatise by Soong an Dargush (997) an onstantinou, Soong an Dargush (998). he basic philosophy of passive energy control system is shown in Figure.. his stuy focus will focus on flui visco-elastic ampers, an especially on those that nee to be moele by Maxwell viscoelastic amper moels..2 Viscoelastic Devices he visco-elastic evices are of two types: (a) soli viscoelastic evices an, (b) flui visco-elastic evices. he soli visco-elastic ampers consist layers of polymeric material that are allowe to eform in shear between two or more steel plates. When mounte in a structure, the structural vibration inuces relative motion between the steel plates, causing shear eformation of the visco-elastic material an hence energy issipation. For convenience of analysis, the force eformation characteristics of these viscoelastic evices can be moele by the Kelvin moel. he flui visco-elastic evice on the other han operates on the principle of resistance of a viscous flui flow through a constraine opening. here are also flui ampers that eform highly viscous flui in shear to issipate energy an to provie stiffness. he focus of this stuy is, however, only on the flui orifice ampers. he issipation of energy occurs via conversion of mechanical energy to thermal energy as a piston forces a viscous flui substance through a restricte opening (Figure.2). he force-eformation characteristics of such evices can be esigne to behave as linear viscous ampers in which forces are only proportional to the velocity of eformation (Makris et. al., 993, 995). his is the classical form of a viscous amper that is so commonly use in analytical stuies. Such ampers give rise to amping forces that are out-of-phase with eformation an eformation epenent forces. So, they o not a to maximum forces evelope in structural elements. hey can be very effective in reucing rifts. However, some of these flui orifice ampers cannot be accurately moele by the classical velocity-epenent moel as they also offer resistance an provie stiffness to the system at higher frequencies of eformation. Different mathematical moels have been propose to capture their behavior in analysis.
hapter. Introuction 3 Such ampers have been moele by classical Maxwell moel or its moifie version with fractional erivatives (Makris an onstantinou, 99). In this stuy, the ampers characterize by the classical Maxwell moel will be use as the evices of choice for energy issipation of structures expose to earthquake inuce groun motion. he formulation to analyze structures installe with such ampers is evelope. his analytical proceure is use to select optimal esign parameters of these ampers to achieve pre-selecte performance obectives such as reuction of certain response quantities or minimize certain performance inices. In the following section, we provie the organization of this stuy..3 hesis Organization he thesis is organize in five chapters incluing hapter, which gives a backgroun an a brief overview of passive energy issipation system. hapter 2 escribes the moels that are commonly use to characterize the viscous ampers. Equations of motion of a structure installe with viscous ampers moele as Maxwell ampers are evelope in analytically convenient form. he methoology to calculate response of structures with such equations of motions is then presente. hapter 3 escribes the formulation of the constraine optimization problem to calculate the optimal parameters of the evices to achieve certain performance obectives in an optimal manner. he etails of a graient-base approach suitable to solve the optimization problem are provie. Numerical results are obtaine to emonstrate the application of the optimization scheme. he amping evices are attache to the builing structure through braces installe in each story. hese braces were consiere rigi for the formulation evelope in hapter 3. In hapter 4, the effect of the flexibility of braces on the optimal esign is examine. A new optimization problem which now treats the brace stiffness as esign variables in aition to the amping coefficients is formulate an solve. he effect of bracing flexibility is then evaluate through several numerical examples, presente in this chapter. hapter 5 presents the summary an concluing remarks about this stuy.
hapter. Introuction 4 (a) (b) (c) Figure.: Passive response control systems, (a) seismic isolation, (b) energy issipation evices, (c) ynamic vibration absorbers (Moreschi, 2000). Damper Flui Piston Ro Figure.2: A aylor Device Damper (onstantinou et. al., 993).
(2.) hapter 2 Analytical Formulation 2. Introuction In this chapter we evelop the equations of motion of structure installe with supplementary viscous amping evices. Different methos use for moeling viscous ampers are iscusse in brief. Since the main focus of this work is to stuy the optimal esign of structures with ampers escribe by Maxwell moel, more attention is pai to evelop the equations of motion with this type of moel. In earlier works by onstantinou an Symans, (993); Soong an Dargush, (997); Moreschi (2000), nonsymmetric equations were use to investigate the effect of these ampers in structures. It was, however, realize that one coul also evelop symmetric equations of motion, which will simplify the response analysis as well as optimal esign investigation. herefore, in Section 2.3, the symmetric equations of motion have been evelope. In Section 2.4, the steps of the approach use to solve these equations of motion for earthquake inputs are escribe, an the necessary expressions use to calculate the response are provie. hese expressions are use in the subsequent chapters an the numerical stuy. 2.2 Equations of Motion onsier an n-egree of freeom builing structure with supplementary energy issipation evices installe at ifferent locations. For a structure subecte to groun excitations at the base, the equations of motion can be written as: Mx!!() t + x! () t + Kx() t + rp () t = ME X!! () t () s s s s g = n l 5
hapter 2. Analytical Formulation 6 he system matrices involve in the equations M s, K s an s represent, respectively, the n n mass, structural stiffness an inherent structural amping matrices; X!! ( t) is the g groun acceleration time history, E is the groun motion influence coefficients; x ( t) is the n-imensional relative isplacement response vector measure with respect to the base. A ot over a quantity inicates time erivative of the quantity. P (t) represents the force in the amper at the th location, an there are n l number of possible locations where the evices can be installe on the structure. he influence of the amper force on the structure is consiere through the n-imensional influence vector r. In general form, the force in a amper can be escribe by the expression of the following form: P [,,, h ( t), ( t),! ( t), t] = 0 (2.2) where n i represents the mechanical parameters relate with the behavior of the evices, h (t) is an internal variable of the evice require in some moels, an (t) is the local eformation of the th evice. he local eformation of the evice can be relate to that of the main structure by the simple expressions as: () t =r () t x () t (2.3) 2.2. lassical Viscous Dampers Depening upon the characteristics of the ampers an the analytical moels use, Eq. (2.2) can take ifferent forms. he most common moel often use in classical vibration stuies is the linear ashpot moel (Figure 2.). In this moel the amper force is assume to be irectly proportional to the amper velocity as follows: P () t = c! () t (2.4) In elementary vibration stuies, this moel is use primarily for mathematical simplification of the energy amping mechanism that is inherent in mechanical an structural systems. In Eq. (2.), the term associate with the amping matrix is precisely base on this moel. Actual ampers have also been manufacture to provie such linear velocity epenent forces.
hapter 2. Analytical Formulation 7 Using the relationship Eq. (2.3) between the local eformation at a amper an the isplacement vector x ( t) into Eq. (2.4), the amper force can be expresse in terms of the structural response as follows: P () t = c r () t x! () t (2.5) Substituting Eq. (2.5) into Eq. (2.), the combine equations of motion for the case of pure viscous amper can be written as n l Mx!! s () t + s + rcr () t x! () t + Kx s () t = MD s X!! g() t (2.6) = he amping matrix of the system is thus moifie by the installation of the ampers. his combine amping matrix of the system can be written as n l s = x! () t r c r () t (2.7) = + It is note that even if the initial structure is classically ampe, the combine amping matrix, Eq. (2.7), of the system can become non-classically ampe because of the supplementary amping terms. his means that one cannot use the unampe moe shapes to uncouple the equations of motion, but must use the state vector formulation if moal analysis is to be performe. he use of the state vector-base moal analysis approach also becomes necessary if a response spectrum analysis is to be performe. In the state vector formulation, Eq. (2.6) can be written in the following form: 0 As z! () t + Bs z() t = Ds X!! g() t (2.8) E where 0 Ms Ms 0 0 0 As = ; B s = ; D s = Ms 0 Ks 0 Ms (2.9) In this case, it is note that the state matrices are symmetric. o solve these equations by a moal analysis approach, one first calculates the eigenproperties of the first orer system, an then uses them to uncouple the system of equations. Since the matrices are symmetric, the left an right eigenvectors remain the same. he approach is escribe in several publications (Singh, 980, Igusa et al., 984).
hapter 2. Analytical Formulation 8 he classical ashpot moel provies a simple approach to inclue the energy issipation mechanisms inherent in the systems an evices if it is appropriate. However, previous research has shown that the classical ashpot moeling of viscoelastic evices is effective only at low frequency range of operation, an that the viscous amping evices exhibit some stiffening characteristics at high frequencies of cyclic loaing. o incorporate this observe stiffening characteristics at high frequencies, a linear spring is ae to the ashpot in the moel. he aition of spring may be one in two ways. he spring may be attache in parallel or in series to a ashpot. he moel in which the spring is ae in parallel is referre to as the Kelvin moel whereas the other one in which the spring is ae in series is known as the Maxwell moel. he analytical escription an characteristics of these two moels are escribe in the following paragraph. 2.2.2 Kelvin Damper Moel In this moel, a spring is attache in parallel to the ashpot as shown in Figure 2.. he force-isplacement relations for the resulting moel is given as (Zhang an Soong, 989): where k an P ( t) = k ( ω) ( t) + c ( ω)! ( t) (2.0) c enote, respectively, the stiffness an amping coefficients of the moel. o inclue the frequency epenence in this moel, the coefficients have been expresse as functions of ω to rener them frequency epenent. his moel has been use to characterize the soli visco-elastic (VE) material in a amper. Soli VE ampers utilize copolymers or glossy substances, which issipate energy when subecte to shear eformation. A typical VE amper consists of viscoelastic layers bone with steel plates, which are subecte to shear eformations uring vibrations. In that case, simplifie expressions for the stiffness an amping coefficients in Eq. (2.3) can be written in terms of the storage an loss moulus of the material an the imensions of the viscoelastic layers as (Zhang an Soong, 992; Soong an Dargush, 997) AG ( ω) AG ( ω) k ( ω ) =, c ( ω ) =, (2.) h ωh
hapter 2. Analytical Formulation 9 where G (ω) an G (ω) are efine, respectively, as the shear storage moulus an shear loss moulus of the viscoelastic material, an ω is the frequency at which the eformation occurs, an A an h are the shear area an thickness, respectively, of the material uner shear. For practical applications, the mechanical properties k an c, though epenent on the eformation frequency ω, are consiere nearly constants within a narrow frequency ban an operating temperature. For the analysis of structures installe with these ampers, the values of these parameters corresponing to the ominant frequency of the structure can be use to obtain useful results. 2.2.3 Maxwell Damper Moel In the Maxwell moel, the spring an ashpot are place in series. his moel is schematically shown in Figure 2.2. he figure also shows the eformations of the spring an amping elements uner the applie force. Let these eformations of the spring element an amper element be an 2, respectively. he figure 2.2 also shows the free boy iagrams of each element. For a given force in this amping system, the force balance equation gives, k = c! = P (2.2) 2 he eformations of the two elements are relate to the total amper eformation follows Differentiating Eq. (2.3), we obtain, as + 2 = (2.3) +!! 2 =! (2.4) Using the force relationships of Eq. (2.2) in Eq. (2.4), we obtain the following relating the force in the amper with the amper eformation P! P! (2.5) k + = c We re-write this first orer equation in the following stanar form: P () t + τ P! () t = c! () t (2.6)
hapter 2. Analytical Formulation 0 where, τ = c / k is referre to as the relaxation time constant. It is of interest to examine the force eformation characteristics of this moel. For this, consier the solution of this equation for a harmonic eformation as follows: = osin ω t (2.7) he solution for this harmonic eformation with the zero-start initial conition can be written as: where a ωt/ a P () t = P 0 e + cos( ω t) + asin( ωt) 2 + a 0 0 (2.8) a =ωτ ; P = k (2.9) he parametric Eqs. (2.7) an (2.8) can be use to create a force-eformation plot. Figure 2.3 shows these plots for ifferent values of the parameter a =ωτ. he Figure clearly shows that the moel shows increasing stiffness as the frequency of the eformation is increase for a fixe value of the relaxation time parameter. In the steay state situation, when the first term becomes insignificant the solution in Eq. (2.8) can be written as follows: ka P t t a t + a [ ] 0 () = cos( ω ) + sin( ω ) 2 (2.20) Using Eq. (2.7) into Eq. (2.20), we obtain the following equation relating P ( t ) with () t : 2 ka c P () t = () t +! () t 2 2 + a + a = k ( a) ( t) + c ( a)! ( t) e e (2.2) where ke( a ) an ce ( a) are frequency epenent stiffness an amping coefficients of the moel efine as: k ( a) e ka a 2 = c ( ) + 2 e a 2 c = (2.22) + a
hapter 2. Analytical Formulation Figure 2.3 shows the variation of these equivalent coefficients with parameter a, which is the prouct of the frequency of eformation ω an relaxation time parameter τ. he equivalent stiffness coefficient increases with the frequency whereas the amping coefficient ecreases with the frequency. One can also obtain the following nonparametric equation relating force with eformation by eliminating (2.7) an (2.20): where 2 2 2 2 P + a P 2 2 2 ( + a ) + ( + a ) = P a P 0 0 0 0 P = k 0 0 ω t between Eqs. his equation escribes the ellipses, which were obtaine in Figure 2.3 uring the steay state situation. A viscous amping evice can also be moele by a more complex combination of the springs an ashpots in parallel an series. One such moel is calle Wiechert moel as shown in Figure 2.(c). It fins applications in the evices exhibiting stiffness at very low frequencies. Bituminous flui amper is one of its examples. he force in the evice is obtaine as: P () t +τ P! () t =τ k! () t + k () t (2.23) g e where k g an k e are, respectively, the stiffness coefficients of the glossy an rubbery materials. his stuy will focus only on the use of the Maxwell moel for flui viscous ampers. As inicate by onstantinou an Symans (993), some commercially available ampers nee to be characterize by this moel. he following analytical evelopment an subsequent analysis an numerical results are thus obtaine only for ampers using this particular moel. In the following we first evelop the equations of motion an then their solution approach for the structures installe with such a ampers.
hapter 2. Analytical Formulation 2 2.3 Equations of Motion with Maxwell Dampers Substituting for the following equation for the amper force.! in terms of structural response x! ( t) from Eq. (2.3), we obtain τ P! ( t) + P ( t) c r x! ( t) = 0 ; =,, n (2.24) l Assembling these equations for iniviual ampers in matrix form we obtain: Where, ΓP! () t Dx! () t + P () t = 0 (2.25) cr D = # cn l r n l ; τ $ 0 Γ = # % # (2.26) 0 $ τn l his equation can be combine with the equation of motion (2.) to provie the following state space system of equation of the first orer as: where the system matrices an are efine as: M 0 0 A s = 0 I 0 ; 0 0 Γ E A sz! ( t) + B sz( t) = Ds 0 X!! g ( t) (2.27) 0 A s, B s an s K s L B s = I 0 0 ; D 0 I D are now of imension 2n + n ) (2n + n ), s M 0 0 D s = 0 0 0 ; 0 0 0 ( l l x! ( t) z( t) = x( t) (2.28) P ( t) L r $ r (2.29) = n n n l n l Eq. (2.27) can be solve by a generalize moal analysis approach as has been one by Moreschi (2000). However, it is note that the system matrix B s is not symmetric an the system is not self-aoint. In this case, one has to solve for both the right an left eigenvectors if a moal solution is esire. he graient calculations that are require for the optimization analysis also become more complicate an cumbersome. It is therefore esirable to evelop the symmetric equations of motion. In the following section we,
hapter 2. Analytical Formulation 3 therefore, evelop these symmetric equations of motion for the structures installe with Maxwell ampers using the Lagrange equations. Symmetric Equations of Motion with Maxwell Dampers o obtain the equations of motion in symmetric form, we will use the Lagrange equations. onsier a bay of a shear builing in Figure 2.5 installe with a Maxwell amper. Assume that the amper is installe in a hevron brace as shown in the Figure 2.5. One can also evelop similar equations of motion for these ampers installe on other type of brace. Figure 2.5 shows the egrees of freeom associate with this system. In aition to the egrees of freeom representing the floor isplacement, we have now introuce another egree of freeom to efine the eformations of the spring an ashpot separately. In terms of these egrees of freeom, the kinetic energy, potential energy, an issipation function can be expresse as follows: = m x + x n 2 i(! i! g) ; i= 2 V = k x x + k x x n n 2 2 i( i i ) ( ) 2 2 i a i i i= i= (2.30) where of D= c x! x! + c x! x! ; x0 = 0; x! 0 = 0 (2.3) n n 2 2 i( i i ) ( ) 2 2 i i a i= i= i m i, k i an c i, respectively, enote mass, inherent stiffness an inherent amping th i story; c an i k represent amping an stiffness coefficient of the evice in i th i story; x i an x! i are the relative isplacement an relative velocity of the i th floor; an x a i an freeom. x! a are the isplacement an velocity terms associate with an auxiliary egree of i he following Lagrange s equation is use to obtain the equation of motion for each egree of freeom. t q! k qk V + q k D + q! k = 0 ; k =,..., n (2.32) where represents total kinetic energy of the system, V is the total potential energy, an D, known as Raleigh issipation function which takes into account amping forces which are assume to be proportional to velocities.
hapter 2. Analytical Formulation 4 Using Eq. (2.32) for the egree of freeom x i associate with the i th mass we obtain, or mx!! + c( x! x! ) c ( x! x! ) + c ( x! x! ) + i i i i i i+ i+ i i i ai k ( x x ) k ( x x ) k ( x x ) = mx!! i i i i+ i+ i i+ ai+ i i g mx!! cx! + ( c + c + c ) x! c x! c x! kx i i i i i i+ i i i+ i+ i ai i i + ( k + k + k ) x k x k x = mx!! i i+ i+ i i+ i+ i+ ai+ i g (2.33) (2.34) For the first floor mass an top floor mass, these equations are specialize as follows: For i =, mx!! + ( c + c + c ) x! cx! c x! + ( k + k + k ) x k x k x = mx!! (2.35) 2 2 2 a 2 2 2 2 2 a2 g For i = n, mx!! cx! + ( c + c ) x! c x! kx + kx = mx!! (2.36) n n n n n n n n an n n n n n g Similarly, using the Lagrange equation with the auxiliary egree of freeom associate with the amper gives the following equation or c ( x! x! ) + k ( x x ) = 0 (2.37) i i ai i ai i c x! + c x! k x + k x = (2.38) 0 i i i ai i i i ai Again writing this equation for i = an i = n, respectively, we get c x! + c x! + k x = (2.39) a 0 a c x! + c x! k x + k x = (2.40) 0 n n n an n n n an he equations of motion (2.34) an (2.38) can be combine into the matrix form as follows: Mx!! x! x! K K x K x ME!! (2.4) s () t + ( s + ) () t a() t + ( s + ) () t a() t = s Xg() t x a i x! + x! Kx + Kx = 0 (2.42) () t a() t () t a() t
hapter 2. Analytical Formulation 5 he vector x () t contains the relative isplacement values of the floor masses. he vector x () t contains the relative isplacement values associate with the auxiliary egrees of a freeom introuce to efine the eformation of the amper elements. M s, s an are the usual mass, amping an stiffness matrices of the structure without the supplementary amping evices. he matrices,,, K, K an K s K represent the contributions of the installe ampers to the system amping an stiffness matrices. hese matrices as well the other matrices appearing in Eq. (2.4) are efine as follows. c $ 0 = # % # 0 c $ n na na ; c $ 0 = # % # 0 c $ n n n ; c $ 0 = # % # 0 c $ n na n (2.43) k $ 0 K = # % # ; k 0 $ n na na K k 0 0 0 2 # % # # = 0 $ k 0 n 0 $ 0 0 n n ; K 0 0 $ 0 0 k 0 0 0 $ 2 = # % % # # 0 0 % 0 0 0 0 k 0 $ n na n (2.44) Here, an are shown to be same. his will be the case if there is a amper in each story. However, if there is no amper in a story, then in matrix the rows an columns associate with that amper are elete, in only the columns associate with the amper is elete, an in the element associate with the amper is set to zero without eleting the rows an columns. o facilitate writing the equations of motion in the state space form, the following auxiliary equation is use Mx! () t Mx! () t = 0 (2.45) s s
hapter 2. Analytical Formulation 6 ombining Eqs. (2.45), (2.42) an (2.4) in this orer, we obtain the following system of first-orer state equations: 0 A z! ( t) + Bz( t) = D 0 X!! g ( t) (2.46) E where for an n-storie builing, z(t) is the 2n + na - state vector consisting of the relative velocity vector x! (t), the vector of auxiliary variables x a of imension n a, an the relative isplacement vector x (t). his vector an the matrices A, B, D of imension 2 n + n 2n + are efine as: a n a 0 0 Ms A 0 = s s M + ; M s 0 0 B = 0 K K ; 0 K + K s K 0 0 0 D = 0 0 0 (2.47) 0 0 M s x! ( t) z( t) = x a ( t) x( t) (2.48) We note that all state matrices in Eq. (2.46) are symmetric. his will permit us to solve the equations using only one set of eigenvectors, as escribe in the following section. 2.4 Response Analysis In the following chapter we will conuct an optimization stuy to calculate the optimum parameters of the chosen ampers. In this stuy we will nee to calculate the response of Eq. (2.46) for seismic motion. We will also nee to calculate the performance function an its graients to perform the optimal search. In this section, therefore, we escribe the response analysis approach that we have use in this stuy. his solution approach is similar to the one mentione in Section 2.2, an previously evelope by Singh (980). However, since the matrices involve are ifferent in this system, the etails are ifferent. In the following we briefly escribe the steps involve in the solution, an provie necessary equations that are use in this stuy. Since the matrices of the system are symmetric, we only nee to calculate the right
hapter 2. Analytical Formulation 7 eigenvectors to uncouple the system of equations. For this the following eigenvalue problem associate with Eq. (2.46) is solve: where µ is the th eigenvalue an µ A{} φ = B {} φ ; =,, 2n + na (2.49) {φ} is the corresponing n + na 2 -imensional eigenvector. he matrices A an B have been alreay efine in Eqs. (2.47). he eigenvectors an eigenvalues of Eq. (2.49) coul be real or complex. he complex quantities will occur in the pairs of complex an conugate. he real part of eigenvalues shoul be negative for a stable structural system. o uncouple the system of equations, we use the following transformation of coorinates: z() t = Φy () t (2.50) where Φ is the moal matrix containing the eigenvectors of the system. Using Eq. (2.50) an pre-multiplying Eq. (2.49) with the transpose of the moal matrix, we obtain the following 2n+ n uncouple equations for the principal coorinates y ( t ). where F is the L/3 φ is the part of a y! () t µ y() t = FX!! g() t ; =,, 2n + na (2.5) th participation factor an efine as: { } F = φ Mr ; =,, 2n + na (2.52) L/3 g th eigenvector { φ } containing lower n elements associate with x in the state vector an r g is the vector of groun motion influence coefficients. Eq. (2.5) implicitly assumes that { φ } are normalize with respect to matrix A. Although the general approach to obtain uncouple equations (2.6) an (2.5) remain the same, the ifferences in the equations are note. In the former case, { φ } ha 2n elements whereas now it has 2n+ n elements. a he solution of Eq. (2.5) can be foun for any given groun motion escription. his solution y ( t ) can, then, be use to obtain any response quantity Rt () of interest such as story isplacement, story rift, an absolute floor acceleration, base shear, over
hapter 2. Analytical Formulation 8 turning moment an others. A response quantity Rt () is linearly relate to the principal coorinate vector y ( t ) as: 2n+ n a Rt () = ρy() t (2.53) = in which ρ is the moal response of the quantity of interest. his moal response for a quantity of interest is relate to eigenvector { φ } by linear transformation as ρ = { φ } (2.54) where is the associate transformation vector. For various response quantities calculate in this stuy, this transformation vector will be part of the following transformation matrices: For Displacements of All Floors: x n n n n a n n n 2 n + n a = 0 0 I (2.55) For All Story Drifts: 0 0 $ 0 0 % # = ; 0 0 0 n 0 n n n a n n = % % n 2 n + n a # % % % 0 0 $ 0 n n (2.56) For All Story Shears: & s = 0n n 0n n k a si n n ' n n n 2n+ na (2.57)
hapter 2. Analytical Formulation 9 Base shear: 0 b = 0 k s # 0 2n+ na (2.58) Absolute Acceleration of all Floors: a = M s s + s + K K K K K (2.59) n 2n+ na Eq. (2.53) is use to efine the time history of a response quantity. For earthquake motions efine in stochastic terms such as by a spectral ensity function or by esign response spectra, we procee as follows. Since the earthquake input motions can be consiere as zero mean process, we first obtain the mean square value of the response. Assuming earthquake motions can be efine by a stationary ranom process, it can be shown (Malonao an Singh, 99) that the stationary mean square value of response Rt () can be expresse as follows: he quantity q q = Φ ω ω (2.60) 2n+ na 2n+ na 2 E R () t g ( ) = k= µ iω µ k + iω q is relate to moal quantities as q = ρ F = a + ib; =,, 2n + na (2.6) where a an b are the real an imaginary parts of q. For some eigenproperties, the imaginary part can also be zero as mentione above. he quantity Φ ( ω) in Eq. (2.60) represents spectral ensity function of the groun motion with ω being the frequency parameter. o be able to use the earthquake input efine in terms of esign groun response spectra, we must express the integral of Eq. (2.60) in terms of groun response spectra. his requires that we efine Eq. (2.60) in terms of frequencies an amping ratios of g
hapter 2. Analytical Formulation 20 single egree of freeom oscillators representing the moes of the system. hese parameters can be obtaine from the complex an real eigenvalues of the system as escribe now. For each real eigenvalue an corresponing eigenvector we efine the following quantities: α =µ ; e = q;,, nr = (2.62) where n r is the number of real eigenproperties. o efine the moal frequency an amping ratio associate with a pair of complex an conugate eigenvalues, we express the real an imaginary parts of an eigenvalue as: Re[ µ + n] r ω + n = µ ; r + n β r + n = r ω + nr ; =,, nc (2.63) µ + n = β ; r ω + iω β q+ n = a r + ib Using these forms for the real an the complex an conugate eigenvalues µ an moal quantities q in Eq. (2.63) an after some simplification one can express the mean square value of a response quantity as summation of three terms as follows: where the components of the quantities (Moreschi, 2000): 2 E Ri () c = S i + S2i + S3i (2.64) S i, S 2 i an S 3 i are efine as follows n r n 2 = + 2 r n r eieik S i ei J ( α J + α k J k ) (2.65) = = k = + ( α + α ) n r n r = k= ( A J + B I + I ) k S2 = 2 e 2 (2.66) i i k k k k k S ( I I ) n 3 = 2 c n c k i ik i W k I + Q 2 2 + + 4 4 k k I 4 k = k = + I Ω g Ω g a i a ik I 2k (2.67) Here, Ω is the ratio of th an th k system frequencies; an the quantities e i, a i an g i are efine in terms of the real an imaginary parts of the system eigenproperties.
hapter 2. Analytical Formulation 2 he explicit expressions for W, Q k k,, were evelope for the nonclassically ampe case of Eqs. (2.66) an (2.67) by (Malonao an Singh, 99). Here they were obtaine for the problem at han, an are given in Appenix A. he terms J, I, an I 2 in Eqs. (2.65), (2.66), an (2.67)are mean square values of the response quantities efine in terms of the input motion spectral ensity function as below: J Φ g ( ω) = ω ; 2 2 ( α + ω ) I Φ g ( ω) 2 Φ g ( ω) ω = ω; I 2 2 2 2 2 2 = ω 2 2 2 2 2 ( ω ω ) + 4ω β ω ( ω ω ) + 4ω β ω (2.68) It is note that that I an I 2 are the mean square value of the relative isplacement an relative velocity responses, respectively of a single egree of freeom oscillator of parameters ω, β excite by groun motion component f (t). J represents the mean square response E[ ν 2 ( t)] of the following first orer equation ν! ( t) + α ν( t) = f ( t) (2.69) he quantities I, I 2 an J can also be efine in terms of the esign groun response spectra. his will permit the use of the response spectra in the analysis as well as in the optimization stuy if esire. 2.5 hapter Summary he chapter focuses on the evelopment of equations of motion for structures installe with supplementary amping evices moele by Maxwell moel. hese equations being symmetric are more convenient for analysis than the unsymmetric equations use in earlier stuies. he chapter ens with the escription of the response analysis proceures to be use in the optimization algorithm in next chapter.
hapter 2. Analytical Formulation 22 (t) P (t) (a) Viscous Dashpot (t) k P (t) c (b) Kelvin Moel (t) k = k g - k e c P (t) k e (c) Wiechert Moel Figure 2.: Different Moels for Viscous Dampers.
hapter 2. Analytical Formulation 23 Un eforme State k c P (t) P (t) Deforme State 2 c! 2 c! 2 k k Figure 2.2: Maxwell Moel of a Damping Device Representing the Deformations in the Spring an the Damping Element.
hapter 2. Analytical Formulation 24.50 0.50 P P 0 -.20-0.80-0.40 0.00-0.50 0.40 0.80.20 a = 0.2 -.50 0.50 0.50 P P 0 -.20-0.80-0.40 0.00-0.50 0.40 0.80.20 a = 3.0 -.50 0.50 0.50 P P 0 -.20-0.80-0.40 0.00-0.50 0.40 0.80.20 a = 0.0 -.50 0 Figure 2.3: Force-Deformation Responses for Different Values of Parameter a (0.2, 3.0, 0.0).
hapter 2. Analytical Formulation 25.0 0.9 0.8 c 0.7 e ( a) c 0.6 ke ( a) 0.5 k k ( ) 0.4 e a k 0.3 0.2 0. 0.0 ce ( a) c 0 0.5.5 2 2.5 3 a Figure 2.4: Frequency Depenency of the Stiffness an Damping Parameters.
hapter 2. Analytical Formulation 26 Figure 2.5: A Bay with hevron Bracing Depicting Auxiliary DOF an Floor DOF for Floor i.
hapter 3 Optimal Damper Parameters 3. Introuction he main obective of installing supplementary ampers is to reuce the ynamic response such as the forces an acceleration of a structure. he level of response reuction will, however, epen on the location of amper installation an the amping force that amper can apply. herefore, a question that naturally arises is how one shoul istribute these energy issipation evices on the structure to reuce a response quantity optimally. One may also be intereste in knowing the total amount of amping an its istribution in the structure so as to reuce a response quantity by a pre-ecie amount in an optimal manner. A few ifferent approaches have been suggeste earlier to obtain better performance from the ampers. Hanson et. al, (993) suggeste the installation of evices to maximize amping in the first moe of the structure. Zhang an Soong (992) use a sequential optimal scheme. In absence of any other clear choice, often a uniform istribution is assume. his chapter eals with this issue as an optimization problem. he problem consists of a performance function that measures the esire attribute of the structure. he obective then is to minimize or maximize this function by optimal istribution of the parameters of the amping evices. he evelopment of optimization techniques or more formally calle mathematical programming techniques ates back to the ays of Newton, Lagrange, an auchy. However, the maor evelopments in this area took place aroun the 960s. Since then there has been a large amount of research in this fiel. Several optimization approaches an algorithms are available toay. hey are classifie base on the nature of esign variables, constraint equations an of course the obective function. 27
hapter 3. Optimal Damper Parameters 28 he chapter first escribes the performance functions that have been consiere in this stuy. It then formulates the optimization problem. A graient-base approach calle the Rosen s graient proection metho is selecte to solve the problem at han. A brief escription of this approach, as it is applie to the problem of the optimal placement an parameter selection of the ampers, is then provie. he formulas to calculate the graients of the response quantities an the eigenproperties require in the optimal search proceure are provie. his is followe by the numerical results of an example builing structure, which illustrate the application of the propose optimal approach. 3.2 Performance Functions he performance obective that we want to achieve by installation of amping evices coul be as simple as reuction of a specific response quantity such as a floor acceleration, floor isplacement, base shear, overturning moment, etc. It coul also involve more than one response quantity of similar type such as the sum of the squares of the floor accelerations or sum of the squares of the story rifts. It may also be a weighte average of ifferent response quantities such floor accelerations an story rifts. More complex performance functions can also be chosen such as minimization of the life-cycle cost as it is influence by the installation of the supplementary evices. he performance function use in this work is in the form of normalize inices, calle performance inices. he inices are efine in terms of response quantities of the structure installe with evices normalize by the corresponing response quantity for the structure without any aitional evices. We shall use two forms of performance inices. he first form is where response quantities of interest like floor acceleration, rift or isplacement of a particular story or base shear themselves represents the performance function. his performance function in its normalize form is give as follows: f R( c) c = (3.) [ R( )] where R (c) is the response quantity of interest. he goal is to minimize or reuce the maximum value of this quantity by using aitional amping evices. he performance R o
hapter 3. Optimal Damper Parameters 29 function is normalize by the corresponing response quantity R o of the original unmoifie structure. In this stuy, this inex shall be use for base shear response. Another form of performance function is efine in terms of the secon norm of a vector of response quantities, such as root mean square values of the story rifts, or acceleration of ifferent floors, etc. his secon form of performance function is expresse in normalize form as below: () f 2 [ Rc ()] = Rc (3.2) R where R (c) an R o, respectively, are the vectors of the response quantities of interest of the moifie an unmoifie structures; an = R 2 2 R (c) i = i an R o = i = Roi are the square roots of the secon norm response of the moifie an unmoifie structures, respectively. 3.3 Formulation of Optimization Problem o n n he obective of the optimization stuy is to etermine the parameter values of ampers, an also their istribution in the structure, to achieve the maximum reuction in the esire response measure in terms of a performance inex. he parameter values are boune by practical limits on them. hus, the minimization of the performance function is to be obtaine uner constraine parameter values. Mathematically, this optimization problem can be pose as: subect to Minimize n i= f (c) (3.3) c = 0 (3.4) i u 0 ci ci i =,..., n (3.5) where f (c) is the obective function or performance inex, c is the vector containing esign variables c i, u c i is the upper boun value of the amping coefficient for the i th amper, is the total amount of amping coefficient value to be istribute in the
hapter 3. Optimal Damper Parameters 30 structure an n is the number of stories. Although, there are two parameters in a Maxwell amper - amping coefficient for the amping element an the stiffness coefficient for the spring element - here only the amping coefficient values are chosen as inepenent variables. his assumes that for a given amper, the stiffness coefficient is ust a constant factor times the amping coefficient value. It can be shown that the optimal solution of such a problem must satisfy the Kuhn-ucker conitions (Rao, 996). For the present problem, these conitions can be state as follows: f (c c i ) + λ = 0 i =,..., n A (3.6) n i= c = 0 (3.7) i c = 0 A (3.8) c c u = 0 A (3.9) λ > 0 A (3.0) where A represents the set of active constraints with the associate Lagrange multipliers λ, c is the vector containing esign variables that minimizes the performance function. Having efine the optimization problem, one nees to select an optimization technique to solve Eqs. (3.6) to (3.0). As mentione earlier, Rosen s graient proection metho is a suitable technique for the present case. A brief escription of this technique is now presente. he etails can be foun in literature (Rosen, 960; Haftka an Gural, 992; Rao, 996). As with many other graient-base schemes, the technique is base on a step-bystep iterative proceure. Starting with an initial guess of the esign variables, one successively upates the initial an subsequent estimates using the following recursive scheme: c = c +α s (3.) + k k k k
hapter 3. Optimal Damper Parameters 3 where c k an k + c are the vectors of esign variables at the th k an k + th step, α k is the step length, an s k is the search irection for succeeing estimate with k being the current iteration step. hus to upate the vector of esign variables, we nee two quantities: () the best search irection s k an, (2) a largest possible step size α k that will avoi instability in the search for the optimal solution. he strategy is to approach the minimum of the function in the irection of steepest escent. Mathematically, it implies that the irection s k must be such that it minimizes its scalar prouct with the graient vector of the performance function of Eq. (3.3). Furthermore, the search for a feasible irection must be confine in the subspace efine by the active constraints. For this, the feasible irection must also be orthogonal to the graients of the constraints of Eqs. (3.4) an (3.5). In case of no constraints, this steepest escent irection is the negative of graient vector of the function. Since the present problem has constraints, the problem of fining a feasible irection can be state as follows: subect to where efine as: Fin s k which minimizes f an s f () c (3.2) k ( ) sk = 0 ; =,..., g = = p (3.3) g are the graients of the performance inex an th active constraint, f f f f () c = $ c c2 c n (3.4) g g g g ( c ) = $ ; =,,p c c2 c n (3.5) with p being the number of active constrains. It is note that the number of active constraints can change from one step to another. Eq. (3.3) ensures that the search