Active Multi Degree-of-Freedom Pendulum Tuned Mass Damper

Transcription

1 Proceeding of the 3 rd World Congre on Civil, Structural, and Environmental Engineering (CSEE 18) Budapet, Hungary April 8-10, 018 Paper No. ICSENM 107 DOI: /icenm Active Multi Degree-of-Freedom Pendulum Tuned Ma Damper M. Eltaeb 1, R. Kahani 1 Miurata Univerity Miurata, Libya Univerity of Dayton Dayton, OH, USA reza.kahani@udayton.edu Abtract - A novel active pendulum tuned ma damper (APTMD) configuration along with a multi-layer control trategy for mitigating the vibration of tall tructure, are preented. The control cheme a) increae the damping effectivene of the APTMD and b) enable the APTMD to imultaneouly tune itelf to multiple frequencie adding tuned damping to the correponding mode of thoe frequencie. The enhanced damping effectivene component of the propoed control cheme and the multi-frequency tuning attribute of the propoed control trategy, eliminating the need for uing multiple TMD tuned to different frequencie, reult in the control of tructural vibration uing a relatively mall ma. The uperior damping effectivene and multi-frequency tuning capacity of the propoed APTMD are numerically demontrated by incorporating it into the model of a multi-degree of freedom aymmetrical tall building. The APTMD introduced a much damping to the target mode a a paive PTMD twice a large, while tuned to three mode of the building. Keyword: Active Tuned Ma Damper, Vibration Control, High-Rie Building, Multi-Frequency Tuning. 1. Introduction Tuned ma damper have been extenively ued for adding tuned damping to variou tructure. Conidering that the firt vibrational mode of mot tructure play the dominant role in their dynamic repone, TMD are normally tuned (and mut remain tuned) to the firt natural frequency of thee tructure. The damping effectivene of a TMD depend on a) the ize of it ma compared to the modal ma of it target mode, i.e., it ma ratio, b) it internal damping, and c) the accuracy of it tuning. Due to the tight retriction on their weight and pace requirement, the ma ratio of a TMD ued in a large tructure uch a a tall building, i very mall. The mall ma ratio neceitate the ue of low internal damping in the TMD, itelf. The highly underdamped nature of uch TMD reult in o a very narrow operating frequency range (bandwidth) making the effectivene of the TMD highly enitive to it tuning accuracy (i.e., a light change in the dynamic of the target tructure would render uch TMD detuned and hence le effective), o a rather long delay in reaching teady tate motion (and thu becoming fully effective) when a diturbance tart perturbing the tructure and alo a long delay to top when the diturbance ha ceaed, o exceive excurion, and o low damping effectivene even when the TMD with mall ma ratio i well-tuned. The impact of ma ratio on the damping effectivene of a TMD i evident from Figure 1 depicting the frequency repone function of a tructure without and with two different TMD, one with a mall (0.15%) and the other with a large (0.5%) ma ratio. A enumerated above and alo clear from Figure 1, the TMD with maller ma ratio ha 1) lower bandwidth, and ) le damping effectivene than the TMD with larger ma ratio. The above-lited iue could have been addreed if the ma ratio of the TMD wa large. But a large ma ratio would make the TMD conidered for a high-rie building very maive, exceively large, and unacceptable in mot application. ICSENM 107-1

2 Fig. 1: Frequency repone function of an underdamped tructure without and a mall (red trace) and large (blue trace) ma ratio. Clear from Figure 1, increae in ma ratio and the correponding damping ratio of a TMD increae the energy diipation (damping) effectivene a well a the bandwidth of that TMD. A tated earlier, the problem i that in many application there i neither enough pace to accommodate large a large TMD nor the target tructure can take the weight of a TMD with large enough ma ratio. The effectivene of a tuned ma damper can be increaed by introducing a controllable element into it make-up, reulting in an active tuned ma damper (ATMD) [1], [], [3], [4]. The ue of an active tuned ma damper make it poible to overcome the drawback of a paive TMD with mall ma ratio. A properly controlled active TMD tuned to a ingle mode a) ha a wider frequency range and thu i more robut (than a paive TMD of the ame ize) to tuning inaccuracie, b) can elf-tune itelf, and c) poee higher effectivene in adding damping to it target mode. Moreover, an ATMD can be ued to add damping to more than one mode of vibration when multiple mode are in need of being damped. Thi eliminate the need for uing multiple paive TMD with different dynamic characteritic, each tuned to one of the target mode. Conidering that a properly controlled active TMD i a) by far more effective than an equally ized paive TMD and b) can be tuned (add damping) to more than one mode of the tructure, it could be an economically attractive olution in certain application. With cot comparion baed on effectivene, an ATMD would be no more expenive than either a paive TMD of many time larger in ize or multiple paive TMD tuned to multiple frequencie. A to the running cot, with proper deign and control trategy an ATMD can witch to a paive ytem under typical loading condition, conuming no energy and revert back to active only when the tructure reache a high enough vibration level which require more effectivene. Alo in the unlikely event of loing power, uch active TMD revert to a paive TMD, automatically. In thi work, a novel APTMD i preented which 1. with a maller ma poee the bandwidth and energy diipation effectivene of a more maive paive PTMD.. It ha multiple, adjutable tuning frequencie (and the correponding internal damping) in multiple direction. With it mall ize and it multi-frequency tuning capacity, the propoed APTMD i a effective a a paive PTMD many time more maive. 3. Moreover, it can add damping to more than one mode in more than one direction, replacing multiple more maive PTMD. A tated earlier, the above-lited attribute lower the cot, weight, and pace requirement aociated with dampening multiple mode uing multiple PTMD. Following a hort review of exiting active tuned damping technique, the paper introduce the propoed active pendulum tuned ma damper along with it correponding control trategy. In ubequent ection, the paper continue ICSENM 107-

3 with numerical demontration of the enhanced damping effectivene a well a multi-mode damping capacity of the propoed APTMD by incorporating it into the numerical model of a high-rie building perturbed in multiple direction.. Review of ATMD Control Scheme Over the lat two decade, a number of reearcher have uggeted different control trategie, motly for one degree of freedom tuned ma damper. Chang and Soong [5], and Iao [5] calculated the required active force uing an optimization proce and howed ignificant improvement in the damping effectivene of the ATMD without increaing it excurion, compared to a paive TMD of equal ize. Abdel-Rahman [7] ued pole aignment method to evaluate the active force. In eparate tudie, Chang and Yang [8] and Sehaayee and Yang (1996) computed the active force by feeding back the acceleration, velocity, and diplacement of the tructure. In addition to the ue of claical feedback control (feeding back diplacement, velocity, and acceleration of the tructure), modern control technique including linear quadratic regulator (LQR) Alavinaab and Moharrami [5], linear quadratic Gauian (LQG), H and H inf control a well a nonlinear control technique uch a liding-mode control Adhikari and Yamaguchi [9], fuzzy logic Samali et. al. [10], and even neural network control Ghaboui and Joghataie [11] have been uggeted to obtain the mot ideal control force for mitigating the vibration of the tructure. Depite the ucceful application of all of thee control cheme to a one degree-of-freedom tructure, mot of thee controller are a) either model-baed and require a reaonably accurate model of the tructure plu TMD which i rarely available epecially in tuned damping application of very large tructure, b) are in need to feeding back all the tate of tructure and TMD, and/or c) are omewhat complex to implement. Thee obervation and control claification have been made by Nerve and Krihnan [1] baed on the reult obtained from the imulation tudy of applying variou active control cheme to large tructure. A will be decribed in the ubequent ection, the author have expanded the non-model-baed claical feedback control propoed by Nihimura et al. [5] to a multi-degree-of-freedom APTMD and cacaded it with a novel multifrequency tuning trategy. 3. The Propoed Active Pendulum Tuned Ma Damper (APTMD) Figure depict the propoed APTMD with the ma upended by teel wire rope and actuated by 6 hydraulic cylinder (actuator). The hydraulic cylinder along with the moving ma form a 6-legged cloed-chain mechanim with 6 degree of freedom. Thi mechanim, commonly known a Stewart platform, i capable of moving in any direction and orientation, generating controllable dynamic motion [6] [7] [8]. The hydraulic cylinder (leg) are the active element of the APTMD realizing the patial motion of the pendulum ma, decided by the control algorithm. Except for the ue of hydraulic cylinder in place of paive vicou damper, the propoed APTMD reemble a paive PTMD commonly ued in many modern high-rie building and tower. In it default tate, the hydraulic cylinder are configured to act a paive vicou damper turning the device to a paive PTMD. The length of the upenion teel wire rope are elected uch that the default paive PTMD i tuned to the lowet target frequency. When the need arie (depending on the extent and frequency content of the tructural vibration) the hydraulic cylinder witch to active actuator and turn the device into an active PTMD with enhanced damping effectivene a well a multi degree-offreedom tuning capacity (capability). The hydraulic actuation ytem of the propoed paive/on-demand active PTMD conit of ix double acting ingle ended cylinder. Each hydraulic cylinder i equipped with a temperature-compenated flow control valve and a 4-way electrohydraulic ervo-valve; the former i ued when the device i in it paive tate and the latter along with the hydraulic power upply are ued when the device i in active tate. Refer to Eltaeb [4] for more on the hydraulic circuit of the APTMD. ICSENM 107-3

4 Fig. : The active PTMD. 4. Active Control Strategy The control cheme propoed for the multi-degree-of-freedom PTMD i made up of a parallel cacade of a) active damping effectivene enhancement controller and b) active multi-frequency tuning controller. In addition, a high-level uperviory controller overee the vibration attribute of the tructure both in term of everity a well a frequency and decide to bring none, one or both controller() online Active Damping Effectivene Enhancement Conidering that the inertia force generated by TMD ma i what add damping to the tructure, increaing uch inertia force by active mean will reult in increae in the effectivene of the TMD. A uggeted by Nihimura et. al. [3], the inertia force of a one degree of freedom ATMD appended to a tructure can be increaed by having the active element actuate the ytem proportional to the acceleration of the tructure. The three degree-of-freedom verion of uch control trategy formulated in the global Carteian coordinate ytem i hown in Equation (1). GX 0 0 X U 0 GY 0 Y 0 0 G (1) where U i the vector of ATMD active force, X, Y, and are the abolute acceleration of the tructure along x and y and around z axe, and G, G, and G are the correponding feedback gain. X Y Since the hydraulic cylinder (leg) are to actuate the APTMD, the control force vector which i formulated in the Carteian coordinate ytem need to be tranformed to the leg pace prior to realization. The control force vector U of Equation (1) i tranformed from the global coordinate ytem to the leg coordinate ytem (leg pace) uing the principle of virtual work, Eltaeb [4]. Auming that the friction force in the joint and the gravitational effect of the leg are negligible and conidering that the main gravitational effect due to the weight of the TMD ma i taken up by the teel wire rope (not the leg of the mechanim), the virtual work contributed by all the active force i u p U P () T T 0 ICSENM 107-4

5 where p and P are the virtual diplacement in the local leg pace and the global Carteian coordinate ytem intalled on the tructure, repectively. Moreover, u i the active force vector (realized by hydraulic cylinder) correponding to U, but defined in the leg pace. Clear from Equation (), the principle of virtual work indicate that the work done by the global force vector U i the ame a the work done by the local force vector u defined in the leg pace. Uing the Jacobian matrix of the cloed-chain mechanim J, relating mall motion of the TMD ma P to the correponding mall motion of the leg p, i.e., and ubtituting p in Equation () from Equation (3) reult in Equation (4) p J P (3) T T 0 u J U P (4) Conidering that the above equation hold for any virtual diplacement, one can conclude that u J U 0 u J U (5) T T T The actuation force vector of Equation (5) could increae the excurion of the APTMD. A uggeted by Nihimura et. al. [3] uch excurion can be lowered by augmenting the actuation with feeding back the excurion velocity, making the actuator to act a an active vicou damper, a well. It hould be noted that the active cheme of Equation (5) with or without velocity feedback can be turned on and off, depending on the frequency and everity of the tructural vibration. 4.. Active Multi-Frequency Tuning To tune the APTMD to different frequencie in different direction and add the deired internal damping to the APTMD correponding to thoe tuning frequencie, the APTMD frequency/direction dependent tiffne and damping coefficient matrice need to be realized actively by the actuator. Thi i achieved by active, imultaneou actuation of the actuator (hydraulic cylinder) ubjecting the APTMD ma a well a the tructure to the combined tiffne plu damping force vector of U hown in Equation (6) K X 0 0 X CX 0 0 X U K P C P 0 KY 0 Y 0 CY 0 Y 0 0 K 0 0 C (6) where Ki and Ci are the additional tiffne and damping coefficient needed in i=x, Y, and direction, over and beyond what the pendulum itelf i providing. And P X Y T and P X Y are the poition and velocity vector of the center of ma of the PTMD ma meaured in the Carteian coordinate ytem intalled on the tructure, i.e. the relative motion of the APTMD ma with repect to the tructure. Note that the formulation of Equation (6) i baed on the aumption of the three target mode of the tructure are nearly decoupled. If the natural mode were not nearly decoupled then the deired tiffne and damping coefficient matrice hown in Equation (6) would not be ymmetric. T ICSENM 107-5

6 Conidering the inconvenience of meauring X Y T and yet the eae of meauring the diplacement of the actuator (leg of the Stewart platform) the control force U of Equation (6) i tranformed from the Carteian coordinate ytem intalled on the tructure to the control force defined in the local coordinate ytem of the leg. Thi reult in the leg pace control force vector u hown by Equation (7) u k p c p (7) where p and p are the diplacement and the velocity vector of the leg, k and c are the tiffne and damping coefficient matrice of the leg defined by Equation (8) and (9) k J KJ T 1 (8) c J CJ T 1 (9) and upercript -T ignifie the invere of tranpoe ; ee Eltaeb [4]. A tated earlier the uperviory controller continuouly aee the vibration condition of the tructure, both in term of everity a well a frequency, decide to activate none, one (either u or u ), or both of the two control cheme dicued above. The block diagram preentation of the tructure and the paive/on-demand active PTMD i hown in Figure 3. Fig. 3: Block diagram of the tructure + APTMD. 5. Illutrative Example The enhanced effectivene and multi-frequency tuning capability of the propoed APTMD are numerically demontrated by introducing it into the model of a multi-degree of freedom aymmetrical high-rie building. The building ha 41 torie with each floor having three degree of freedom, two tranlational in X and Y direction and one rotational, denoted by, around the Z axi. Due to the uniformity in geometry and ma ditribution in each floor, vibration in thoe three direction i nearly decoupled from each other. The numerical model of the building, in which the firt 15 mode of vibration are included, i developed. The nearly decoupled mode of vibration of the building allow for aociating each mode with the vibration in one direction, only. ICSENM 107-6

7 The propoed APTMD with it ix hydraulic cylinder (actuator) i yntheized to add damping to the firt three nearly-decoupled mode of the aforementioned tall building with the natural frequencie of 0.18, 0.9, 0.56 Hz, imultaneouly. Figure 4 (a, b, and c) how the frequency repone function a well a the reonant time trace of the tructure acceleration along the Y (1 t mode), X ( nd mode), and (3 rd mode) direction meaured at the top floor, without and with the APTMD. Harmonic perturbation with patially varying amplitude along the height of the tructure i ued to perturb all the floor, in three direction, imultaneouly. The effectivene of the APTMD i increaed (beyond that of a paive PTMD of the ame ize) by feeding back the top floor acceleration meaured in the three direction of X, Y, and to the controller of Equation (1). In addition to being able to replace multiple paive PTMD each tuned to a different frequency (in application where more than one mode are in need of damping), the propoed APTMD perform a robutly and a effectively a a paive PTMD twice a maive. 6. Summary A novel paive pendulum tuned ma damper (PTMD) which on-demand can witch to an active PTMD i propoed. In it default tate, the propoed device i paively tuned to the firt mode of the tructure and act a a traditional paive PTMD. In it active mode, the parallel cacade of two control cheme a) increae the damping effectivene of a mall APTMD to the ame level a a larger paive PTMD tuned to the ame frequency and b) tune the APTMD to multiple frequencie. With it mall ize and multi-frequency tuning capacity, the propoed APTMD i a effective a a paive PTMD twice a maive. Moreover, it add damping to more than one mode in more than one direction, replacing multiple more maive PTMD each tuned to a different frequency. The enhanced effectivene and multi-frequency tuning capacity of the propoed APTMD are numerically demontrated by introducing it into the model of a multi degree-of-freedom aymmetrical, 41 tory high-rie building perturbed by wind in multiple direction. (a) (b) (c) Fig. 4: FRF, and the reonant time trace of the tructure linear acceleration along (a) Y, (b) X, and (c) direction meaured at the top floor. Reference [1] J. C. Chang, T. T. Soong, Structural control uing active tuned ma damper, Journal of the Engineering Mechanic Diviion, vol. 106, no. 6, pp , [] K. C Kwok, B. Samali, Performance of tuned ma damper under wind load, Engineering Structure., vol. 17, no. 9, pp , [3] I. Nihimura, T. Kobori, M. Sakamoto, N. Kohika, K. Saaki, S. Ohrui, Active tuned ma damper, Smart Material and Structure, vol. 1, no. 4, p. 306, 199. [4] M. A. Eltaeb, Active Control of Pendulum Tuned Ma Damper for Tall Building Subject to Wind Load, Ph.D. diertation, The Univerity of Dayton, 017. [5] A. Alavinaab, H. Moharrami, A. Khajepour Active Control of Structure Uing Energy Baed LQR Method, Computer Aided Civil and Infratructure Engineering, vol. 1, no. 8, pp , 006. ICSENM 107-7

8 [6] B. Samali, M. Al-Dawod, K. C. Kwok, F. Naghdy, Active control of cro wind repone of 76-tory tall building uing a fuzzy controller, Journal of engineering mechanic., vol. 130, no. 4, pp. 49-8, 004. [7] J. Ghaboui, A. Joghataie, Active control of tructure uing neural network, Journal of Engineering Mechanic., vol. 11, no. 4, pp , [8] R. Adhikari, H. Yamaguchi, Sliding mode control of building with ATMD, Earthquake engineering & tructural dynamic., vol. 6, no. 4, pp. 409-, [9] A. Alavinaab, H. Moharrami, A. Khajepour, Active Control of Structure Uing Energy Baed LQR Method, Computer Aided Civil and Infratructure Engineering., vol. 1, no. 8, pp , 006. ICSENM 107-8