Studying the Behavior of Active Mass Drivers during an Earthquake Using Discrete Instantaneous Optimal Control Method

Transcription

1 Stuying the Behavior of Active Mass Drivers uring an Earthquae Using Discrete Instantaneous Optimal Control Metho O. Bahar 1, M. R. Banan 2, an M. Mahzoon 3 1. Structural Engineering Research Center, International Institute of Earthquae Engineering an Seismology (IIEES), Iran, omibahar@iiees.ac.ir 2. Department of Civil Engineering, School of Engineering, Shiraz University, Shiraz, Iran 3. Department of Mech. Engineering, School of Engineering, Shiraz University, Shiraz,Iran ABSRAC: In orer to control the responses of a builing, ifferent control systems may be employe. o recognize an select a proper control system, a esigner has to analyze many cases. his paper investigates the behavior of some control systems with respect to changes in ifferent parameters of an AMD, an various combinations of masses an control forces of two or three AMDs, an also ifferent locations of an AMD along the height of a builing. In this stuy we use a recently propose control algorithm, name iscrete instantaneous optimal control metho. A new iscrete stable weighting matrix strengthens this metho. Keywors: Active control; Discrete instantaneous optimal control; Digital state-space equation; Discrete stable weighting matrix; Active Mass Driver (AMD) 1. Introuction In orer to control a builing, ifferent control systems may be employe. In a builing, there are many parameters that may be controlle (e.g. responses of the floors, maximum an average require control force, an responses of Active Mass Drivers, AMDs). herefore, a esigner has to recognize the main features of an efficient control system an contemplate the effect of any change in the main parameters of a control system on its final performance. When an AMD is use for controlling the excessive responses of a builing, the values of mass, amping, frequency, an its installation location have to be specifie base on the specifications of the builing an the characteristics of a esign earthquae groun motion that may occur in the future. If the control system consists of two or more AMDs, selection of the main parameters will be more ifficult, because the mass ratio an the control force ratio of the two or three AMDs will also be ae to the parameters. Many researchers have been woring on the subject of the active structural control an have alreay presente many recommenations in this fiel [1-6]. hey investigate ifferent control systems an compare their performances. In general, in their wors, the total mass of a control system an/or the total require control force was not fixe. hese parameters strongly affect the behavior an efficiency of a control system an their changes cause the primary control system an the change one to behave completely ifferent. In this paper in orer to compare efficiencies of ifferent cases, the total mass an the average require control force (as a criterion for the value of the require external energy) are given prescribe values. During this research, the frequencies of the AMDs are assume to be close to the funamental frequency of the moel builing, an they are not change. In this paper by using a recently propose JSEE: Spring 2005, Vol. 7, No. 1 / 1

2 O. Bahar, et al algorithm, name Discrete Instantaneous Optimal Control (DIOC) metho [7, 8], it is trie to investigate the behavior of a controlle builing equippe by Active Mass Drivers (AMDs) uring an earthquae groun motion. In the following sections, a brief escription of the Discrete Instantaneous Optimal Control an the proceure base on which a esigner can fin a stable weighting matrix, is presente. Using this algorithm, the behavior of an AMD with ifferent mass an amping values an also its various locations along the height of a moel builing are investigate. he behavior of two or three AMDs with ifferent mass ratios an/or control force ratios is also examine an iscusse in etail. 2. Discrete Instantaneous Optimal Control Metho By employing the igital state-space equation an a new efinition of the time-epenant performance inex, DIOC metho presents a powerful closeopen loop control rule similar to that of the classical optimal control metho. A new Discrete Stable Weighting Matrix (DSWM ) warrants the stability of the DIOC metho. A brief escription of this metho is presente hereafter [7, 8]. he matrix equations of motion of a structure subjecte to a groun acceleration x& 0 (t) that is controlle by AMDs can be iealize by an n-egree of freeom linear system as follows: M x&& ( t ) + C x& ( t ) + K x ( t ) = Du ( t ) + Mex && 0( t ) (1) in which M, C, an K are n x n mass, amping, an stiffness matrices, respectively, x ( t), x& ( t) an x &&(t) are n-imensional isplacement, velocity, an acceleration vectors, respectively, D is an n x r matrix that specifies the locations of active controllers, u(t) is an r-imensional control force vector, e = [ ] is an n-imensional vector which efines the groun acceleration influence on masses of the whole builing. he first orer igital state-space equation of motion of such structural system is efine as follows: z 1 = A z + B u + w1 x& + 0, z ( t0) = & z (2) where t 0 is the initial time an subscript refers to time instant t, such that t = t, an t is the time increment. Vector z is the state vector at time instant t, [ x x& ], matrices A, B, an w 1 are the transition matrices corresponing to A, B, an w 1, respectively an are efine as follows: 0 A 1 1 t 0 = ex p( A t), F = ex p( Aη) η, B = F B, w 0 A = M w = 1 [ 0 e] n 1 n n = Fw I n n K M, B = C [ 0 M D] n r, (3) (4) A new efinition for quaratic time-epenent performance inex J(t) in iscrete form is presente by authors [7, 8] as follows: 1 J ( t) = ( z+ 1 Qz + 1+ u Ru ) (5) 2 in which 2 n x 2 n positive semi-efinite Q matrix an r x r positive efinite R matrix are weighting matrices relate to the state variables an the control force, respectively. Minimizing J(t) subject to the constraint of Eq. (2) at each time instant, the close-open loop control force vector is obtaine as follows: u = [ R + B QB ] B Q( A z + w x& ) (6) his equation is equivalent to the first two terms of the control force vector in the close-open loop metho of the classical optimal control [9]. he only ifference is the appearance of the weighting matrix Q in Eq. (6) instea of the Riccati matrix in close-open loop classical optimal metho. he thir term in classical metho, which is relate to the external loa an obtaine from a bacwar solution of a matrix ifferential equation in time, is not appeare in Eq. (6). 3. Stable Weighting Matrix A proceure base on the Lyapunov irect metho [7-9] in iscrete form is propose. Consier a positive semi-efinite matrix Q, such that V ( z ) = z Qz 0 (7) which is a possible Lyapunov function. If the first ifference of Eq. (7) with respect to the state-space vector results in a negative semi-efinite matrix, the Q matrix is a iscrete stable weighting matrix (DSWM). he above-mentione first ifference of Eq. (7) is as follows: V ( z) = z B A Q A A QB ( R + B Q A Q ( A B G) Q B G Q B ) z (8) 2 / JSEE: Spring 2005, Vol. 7, No. 1

3 Stuying the Behavior of AMDs uring an Earthquae Using DIOC Metho As a sufficient conition, it can be assume that the sum of the first three terms of the bracet in Eq. (8) is equal to a negative semi-efinite matrix, -I 0 in which I 0 is an arbitrary positive semi-efinite matrix. By this efinition we get A B QA A QA Q+ I QB ( R+ B 0 = 0 QB ) (9) his is the iscrete Riccati matrix equation. By selecting a positive semi-efinite matrix I 0, Eq. (9) is solve an the weighting matrix Q is obtaine. Now, if as a necessary conition, the consiere values of Eq. (7) in all time steps are non-negative, an as a sufficient conition, the bracet in Eq. (8) or its simpler form (i.e.,[-i 0 -(A -B G) QB G]) is a negative semi-efinite matrix, the compute weighting matrix Q is a DSWM for the ifferential equation of motion in Eq. (1). 4. Specifications of the Builing an the Earthquae Recor 4.1. Moel Builing An eight-story planar shear-type builing frame with similar story properties is selecte as the moel builing. he structural properties of each story are as follows: floor mass is 345.6tons, elastic stiffness is 3.404e5N/m, an internal amping coefficient is 2937tons/sec that correspons to a 2% viscous amping of the first moe of the builing without control system [1] Control System Active Mass Drivers (AMDs) are use as an active control system. he properties of AMDs are as follows: the frequency of each river mass is 98% of the funamental frequency of the builing without control [1, 7, 10], the amping of each river mass is 25tons/sec, such that their amping ratio are approximately 7.3%. In orer to compare per - formances of ifferent control systems, their average require control forces an the total mass of the AMDs are fixe to constant values, 72.68N, an 29.63tons, respectively. he average require control force is etermine as follows: AC F = 1 t f tf 0 u ( τ ) τ where t f is the terminating time. (10) 4.3. Discrete Stable Weighting Matrix First, the esigner has to assign a proper I 0 matrix. hen by solving Eq. (9) the Q matrix will be foun. If this matrix satisfies the necessary an sufficient conitions of the Lyapunov stability metho [7, 8], presente in the previous section, the Q matrix will be a DSWM. After extensive analysis, the following matrix is selecte for the I 0 matrix: α Kγ O I 0=, where O M β K1,1 K1,2 K = γ (11) 1) K2,1 γ in which M is the mass matrix of the whole builing (builing with control system), sub-matrix K i,j in matrix K γ is a partition of the stiffness matrix of the controlle builing, an factors α, β, an γ are three arbitrary scalar factors. hese factors are assigne such that the Lyapunov stability conitions are satisfie an the average require control force remains equal to the prescribe value Control Force Relate Weighting Matrix his matrix, R, is a iagonal matrix with a imension equal to the number of the AMDs, multiplie by a constant factor equal to For cases with one river mass this matrix is reuce to 1, an for more AMDs with force ratios equal to 1, this matrix is a unit matrix Earthquae Recor he N-S component of the 1940 El Centro earthquae recor is use as the input excitation. he time increment is 0.02sec. 5. Behavior of the Builing with Respect to Changes in Different Parameters of One AMD Achieving high efficiency of a control system is mainly relate to the employe control metho, selecte weighting matrices an the parameters of the applie control mechanism. he consiere parameters of a control system with one AMD are the frequency, amping, mass, an the installation location of the river mass. In this section, the behavior of the sample builing controlle by an AMD with respect to changes in its ifferent parameters is investigate. he investigate parameters inclue the value of mass an amping of the river mass, an its location along the height of the builing. JSEE: Spring 2005, Vol. 7, No. 1 / 3

4 O. Bahar, et al 5.1. Effects of Different Values of the Mass of One AMD In orer to investigate the performance of the control system with respect to the ifferent values of the mass of an AMD, seven cases are consiere. he values of the masses of these cases are as follows: 1% of the effective moal mass relate to the funamental moe, (i.e., 85.63%), 0.5%, 1%, 1.5%, 2%, 12.5%, an finally 25% of the total mass of the builing. he last two cases refer to the weight of one an two floors of the builing, which are use as a river mass, respectively. After extensive analysis, the proper values for the three factors in I 0 matrix, Eq. (11), are specifie as follows. he values of α for these cases are equal to 2.31, 7, 7, 7, 7, 8, an 4.19, respectively. he values of β are equal to 0.1, 0.21, 0.35, 0.54, 0.68, 1.0, an 0.5, respectively an finally the values of γ are equal to 21, 21, 21, 41, 71, 5001, an As mentione earlier, these factors are specifie such that the conitions of the Lyapunov stability metho are satisfie as well as the average require control forces are fixe to 72.68KN. In able (1), comparing the first row with the other rows shows that, using AMD remarably ecreases the responses of the builing, but increasing the mass value of the AMD oes not consierably mae more ecrease in the responses of the floors. he heavier river masses prouce more reuction in the responses of the AMD an the maximum require control force. he ecrease in the acceleration responses of the floors in the last two rows may be referre to the use of a larger value for the factor γ [7]. herefore in orer to control tall builings, using heavy equipments of the builing as AMDs is strongly recommene. For instance for the moel builing, using 12.5% of the total mass of able 1. he resulting responses for ifferent values of the mass of AMD. Mass Value (%) Displ. (cm) 8 th floor Driver Mass Responses Accel. Displ. * Velocity * Accel. (m/sec 2 ) (m) (m/sec) (m/sec 2 ) * Relative to the 8 th floor responses. Max. Control Max. Base Shear Force Reuction (KN) (%) the builing as a river mass means that the 9 th floor acts as a river mass for the eight floors below. In such a case, the inter-story eformation of the 8 th floor with respect to the 7 th floor without an with control system are 0.80 an 0.57cm, respectively, while the relative isplacement of the river mass (i.e., the 9 th floor), with respect to the 8 th floor is about 15cm. he last row of able (1) is only presente to clarify the following point. Many researchers believe that when the mass value of a control system increases to a value about 5% of the total mass of the builing, the control system may fail an the controlle builing may be unstable. In general, this iea may be correct. But, the results of our investigations show that this problem is only raise from the selecte weighting matrices. In other wors, a esigner can always fin a proper stable weighting matrix for a heavy river mass such that it maes the control system very efficient without inucing instability in the controlle builing Effects of Different Values of the Damping of One AMD In orer to investigate performance of the control system with respect to the ifferent values of the amping ratio of an AMD, five cases are consiere. he selecte values of the amping ratios inclue zero an the values of the 1 st to the 4 th amping ratios of the consiere builing without control, i.e., 2.5%, 7.41%, 12.07%, an 16.32%. For these analyses, other parameters are fixe to the beforementione parameters of one AMD. he coefficients of the I 0 matrix in Eq. (11) are specifie as follows: the values of α an γ for all cases are equal to 7, an 21, respectively, an the values of β for the consiere cases are equal to 0.478, 0.417, 0.429, 0.496, an he resulte responses of these consiere cases are summarize in Figures (1) an (2). he values of the maximum require control forces of these cases are 592.6, 608.8, 631.5, 661.8, an 669.9N, respectively. he results show that, by increasing the value of the amping of an AMD, the efficiency of the control system ecreases, i.e., the responses of the floors smoothly increase, an a larger maximum control force is neee. herefore, choosing amping ratio of the AMD between the 1 st an the 2 n moe amping ratio of the builing causes the most significant reuction in the responses of the builing while the responses of the river mass are maximize. 4 / JSEE: Spring 2005, Vol. 7, No. 1

5 Stuying the Behavior of AMDs uring an Earthquae Using DIOC Metho able 2. Comparison of the resulting responses ue to varying location of an AMD along the height of the controlle builing. Max. Max. Base Driver Mass Location of Control Shear Driver Mass Displ. * Velocity * Accel. Force Reuction (m) (m/sec) (m/sec 2 ) (KN) (%) 8 th floor th floor th floor th floor * Relative to the responses of the floor that is equippe by AMD. Figure 1. he isplacements of the floors an river mass for ifferent amping of the AMD. Figure 3. Comparison of the responses ue to ifferent locations of one AMD. Figure 2. he accelerations of the floors an river mass for ifferent amping of the AMD Proper Location of One AMD Analyses show that to achieve a better performance for a control system, when there is not any limitation for installation of AMD along the height of a builing the best location is the top of the builing. Four cases are presente such that in each case the AMD is installe on one floor among 8 th, 7 th, 6 th, an 5 th floors of the builing, respectively. he coefficients of the I 0 matrix in Eq. (11) are specifie as follows. he values of α an γ for all these cases are equal to 7, an 21, respectively, an the values of β are equal to 0.417, 0.389, 0.337, an 0.296, respectively. he resulte responses of these cases are compare in able (2) an Figure (3). Base on the results set forth in able (2), by moving the AMD to the lower floors, the responses of the control system an the maximum require control force remarably increases, an maximum reuction of the base shear of the builing effectively ecreases. On the other han, the results in Figure (3) show that, installing the AMD at the lower floors may slightly affect the isplacement an acceleration responses of the floors. he velocity responses of the floors are not change. herefore, moving the location of an AMD system towar the higher floors causes the control system to be more efficient. 6. Behavior of the Builing with respect to Changes in Different Parameters of Few AMDs In this section, the behavior of the moel builing controlle by two or three AMDs with respect to changes in ifferent parameters of AMDs is investigate. he investigate parameters inclue; various mass ratio an/or force ratio of two AMDs installe at the top floor. It is note that, the average require control force is fixe to KN, an the total mass of the two or three AMDs are also fixe to ton. By using two AMDs the imensions of I 0, Q an R matrices are equal to 20, 20, an 2, respectively. So, the factor γ in Eq. (11) an also, the force relate weighting matrix, i.e., R matrix, are change as follows: γ (in Eq. (11) ) 1 γ 0 0 p an 1 0 R = (12) 0 m JSEE: Spring 2005, Vol. 7, No. 1 / 5

6 O. Bahar, et al where p an m are arbitrary values. For three AMDs the matrices given in Eq. (12) are efine as unit matrices with a imension equal to Different Combinations of Mass Values of the AM Drivers Sometimes, ue to some practical limitations, the esigner is force to use few AMDs instea of one AMD. Here, some possible choices are investigate that inclue two AMDs with three ifferent mass ratios equal to 1, 3 an 9, respectively, an three AMDs with equal masses an three AMDs with masses equal to 1/2, 1/3 an 1/6 of the total mass. It is note that, the total mass values of all cases are fixe to 29.63tons an the total average require control forces are ientical. In orer to compare the efficiency of the control systems computations were one an the results of these cases an also one AMD are summarize in able (3). All AMDs are installe on the top of the builing consiere. able 3. he responses of the entire builing using few AMDs installe on the top floor. Case Mass Displacement Acceleration Max. otal No. Ratio 8 th floor 2 n floor DM # 8 th floor 2 n floor DM # Force (cm) (cm) (m) (m/sec 2 ) (m/sec 2 ) (m/sec 2 ) (N) * * * * ** * ** * # Driver Mass * Relative to the 8 th floor responses ** Ratio of the largest value to the smallest value Base on these results, iviing a total mass into two or three parts oes not significantly affect the responses of the builing. In some cases the maximum responses of the control system may slightly increase. But, it is clear that, more AMDs nee a greater pea in the total control force, an greater mass ratio nees smaller maximum of the total control force. his is because when more than one AMD is use the control force for each AMD is smaller than the total value. his certainly affects the performance of the control system. he frequency response of the acceleration of the top floor of the builing controlle by one AMD an two AMDs with ifferent mass ratios are shown in Figure (4). One can observe that the frequency responses for ifferent mass ratios are completely ientical. By using only one AMD the first moe of the builing can be well controlle. But, this control system has small effects on the other moes. Since, the accelerations of the floors highly epen on the higher moes, controlling them with one AMD is very ifficult. On the other han, a control system with two or more AMDs may affect more than one moe of the builing. herefore, in orer to control the longituinal responses of builings lie the moel builing, by using a control system with a specifie total mass an by tuning all frequencies of the AMDs to values close to the funamental frequency of the builing, some practical aspects are recommene which are using a fewer number of Active Mass rivers if possible, either using a greater mass ratio for two AMDs, or using equal mass values for three AM rivers. Figure 4. he acceleration frequency response of the top floor using one AMD an two AMDs; (a) the 1 st ominant frequency, (b) the 2 n ominant frequency, an (c) the 3 r ominant frequency of the controlle builing. 6 / JSEE: Spring 2005, Vol. 7, No. 1

7 Stuying the Behavior of AMDs uring an Earthquae Using DIOC Metho 6.2. Different Combinations of the Control Force Ratios of the wo AMDs he values of the coefficients p an m efine in Eq. (12) were selecte to be equal to 1 before. But, selecting larger values for these coefficients causes the require control force of the greater mass to be larger than the require control force of the smaller one. Extensive analysis shows that, the behavior of a control system using two AMDs with the same frequencies tune to values close to the funamental frequency of the builing follows almost a regular tren with respect to changes in control force ratio of the two mass rivers. In orer to confirm this regularity, three combinations for the mass values of the two AMDs are evaluate. he factors γ an β are fixe to 21 an 0.5, respectively. he two coefficients p an m that will be note hereafter by a couple value (p, m) are change. he coefficient α is selecte such that the total average require control forces of all cases be equal to the specifie value. hree couple values, which are (1,1), (1,9) an (9,9), for each mass ratio are examine. he results of analysis are presente in able (4). he results given in able (4) show that: (a) using ifferent control force ratios oes not consierably affect the responses of the floors, (b) using smaller control force ratios for a fixe mass ratio causes the maximum responses of the control system to ecrease, but maximum of the total require control force to increase, an (c) using greater mass ratios for an almost fixe control force ratio causes the maximum responses of the control system an the maximum of the require total control force to ecrease (e.g., the 3 r, 5 th, an 7 th rows in able (4)). 7. Conclusions Behavior of ifferent Active Mass Driver Systems (AMDs) with respect to various changes in their parameters was investigate an their efficiencies were compare, in etail. All of analyses are carrie out by using a recently propose control algorithm name DIOC. A new iscrete stable weighting matrix forme base on the Lyapunov irect metho, strengthens this metho. Different parameters of one AMD incluing mass an amping values an its location along the height of a sample builing are investigate. It is shown that, although an AMD with mass value more than 0.85% of the total mass of the builing, cannot mae more reuction in the responses of the floors, but a heavy AMD, even 25% of the total mass of the builing, can effectively reuce the responses of the control system. It can also lessen the maximum require control force without inucing instability in the controlle builing. his is because of using a proper stable weighting matrix. Installing an AMD above the mi height of the builing can properly reuce the responses of the floors. But, installation in the higher floors prouces smaller responses of the AMD an the maximum require control force. An extensive analysis shows that, iviing a total mass into two or three portions oes not prouce consierable effects on the responses of the builing. But, more AMDs nee greater maximum of the total control force, an greater mass ratio nees smaller maximum of the total control force. he results of the frequency response of the acceleration of the top floor show that, by using only one AMD the first moe of the builing can be well controlle. But, this control system has small effects on the other able 4. he responses of the builing using two AMDs with the frequencies tune close to the funamental frequency of the builing. Couple Mass Displacement Acceleration Control Max. otal Value Ratio 8 th floor 2 n floor DM # 8 th floor 2 n floor DM # Force Control (cm) (cm) (m) (m/sec 2 ) (m/sec 2 ) (m/sec 2 ) Ratio Force (N) (1,1) * (1,9) * (9,9) * (1,1) * (1,9) * (9,9) * (1,1) * (1,9) * (9,9) * * Relative to the 8 th floor responses # Driver Mass JSEE: Spring 2005, Vol. 7, No. 1 / 7

8 O. Bahar, et al moes, while a control system with two or more AMDs can affect more than one moe of the builing. herefore, controlling the longituinal responses of builings lie the moel builing using a control system with a specifie total mass can be unertaen by: 1) using Smaller number of AMDs if possible, 2) tunning their frequencies close to the funamental frequency of the builing, 3) using a greater mass ratio for two AMDs, 4) using equal mass values for three AM rivers. Finally, various control force ratios for two AMDs are investigate. he results show that, using ifferent control force ratios oes not consierably affect the responses of the floors, an using greater mass ratio for an almost fixe control force ratio causes the maximum responses of the control system an the maximum of the total require control force to ecrease. Acnowlegments he first author is grateful to emeritus Prof.. Kobori (Kobori research complex, Kajima Corp., Japan) an Prof. Kitagawa (the first author s supervisor in Hiroshima Univ., Japan) for their support an cooperation for examining some actively controlle builings in oyo, an attaining actual information for checing the computer program. References 1. Yang, J.N. (1975). Application of Optimal Control heory to Civil Engineering Structures, J. Engr. Mech., ASCE, 101, Yao, J..P. an Sae-Ung, S. (1978). Active Control of Builing Structures, J. Engr. Mech., ASCE, 104, Abel-Rohman, M., Quintana, V.H., an Leipholz, H.H.E. (1980). Optimal Control of Civil Engineering Structures, J. Engr. Mech., ASCE, 106, Chang, J.C.H. an Soong,.. (1980). Structural Control Using Active une Mass Dampers, J. Engr. Mech., ASCE, 106, Kobori,., et al (1991). Seismic -Response- Controlle Structures with Active Mass Driver System, Part 1: Design, Earth. Engr. Struc. Dyn., 20, Bahar, O. (2001). Stuying the Behavior of a Builing Controlle by Active Mass Dampers/ Drivers During an Earthquae with Introucing wo New Instantaneous Optimal Control Algorithms, PhD hesis, Shiraz University, Shiraz, Iran. 8. Bahar, O., Mahzoon, M., Banan, M.R., an Kitagawa, Y. (2004). Discrete Instantaneous Optimal Control Metho, Iranian Journal of Science an echnology, 28(B1), Borgan, W. L. (1991). Moern Control heory. Prentice Hall, Englewoo Cliffs. N.J. 10. Bahar, O., Banan, M.R., Mahzoon, M., an Kitagawa, Y. (2003). Instantaneous Optimal Wilson-Θ Control Metho, ASCE, Eng. Mech., 129(11), / JSEE: Spring 2005, Vol. 7, No. 1