Seismic Response of Torsionally Coupled System with Magnetorheological Dampers

Abstract

The seismic response of linearly elastic, idealized single-storey, one-way asymmetric building with semiactive magnetorheological (MR) dampers with clipped-optimal algorithm is investigated. The response is obtained by numerically solving the governing equations of motion. The effects of eccentricity ratio, uncoupled time period, and ratio of uncoupled torsional to lateral frequency are investigated on peak responses which include lateral, torsional and edge displacements and their acceleration counter parts, base shear, and control forces. To study the effectiveness of control system, the controlled response of asymmetric system is compared with the corresponding uncontrolled response. Further, controlled response of asymmetric system is compared with corresponding symmetric system to study the effects of torsional coupling. It is shown that the implementation of semiactive dampers reduces the deformations significantly. Also, the effects of torsional coupling on effectiveness of semiactive system are found to be more sensitive to the variation of eccentricity and torsional to lateral frequency ratio.

1. Introduction

Many times, most of the real structures are prone to the severe response and damage during a seismic event due to their asymmetric nature. This asymmetry in the buildings arises primarily due to uneven distribution of mass and/or stiffness of the structural components for elastic range. Due to the asymmetric nature of buildings causing the torsional deformations, they are more vulnerable to the earthquake induced damage. The prime focus of the structural engineer is to reduce the torsional response mainly by avoiding the eccentricity which is produced due to uneven mass and stiffness distribution. However, there are many limitations for avoiding the eccentricity between mass and stiffness due to stringent architectural and functional requirements and hence in such cases, implementation of supplemental energy dissipation devices proves to be an effective solution to minimize the lateral-torsional response of the buildings.

In past, many researchers have investigated the performance of seismic control aspects of lateral-torsional deformations using various techniques such as passive control namely base isolation and supplemental dampers and active control. Hejal and Chopra [1] investigated the effects of lateral-torsional couplings and demonstrated that the building response significantly depends on structural eccentricity and frequency ratio. Jangid and Datta [2] investigated that the effectiveness of base isolation for an asymmetric system reduces for higher eccentricity. Jangid and Datta [3] found that the effectiveness of multiple tuned mass dampers is overestimated by ignoring the system asymmetry. Goel [4] investigated that by implementing proper supplemental damping, edge deformations in asymmetric-plan systems can be reduced than those in the corresponding symmetric systems. Date and Jangid [5] investigated the effectiveness of active control system for an asymmetric system in controlling torsional and corner displacements and shown that effectiveness is overestimated by ignoring the effects of torsional coupling. Lin and Chopra [6] studied the effects of plan-wise distribution of nonlinear viscous and visco-elastic dampers and found that the asymmetric distribution of damping reduces the response more effectively as compared to symmetric distribution. De La Llera et al. [7] proposed the weak torsional balance condition for system installed with friction dampers such as to minimize the correlation between translation and rotation. Petti and De Iuliis [8] proposed a method to optimally locate the viscous dampers for torsional response control in asymmetric plan systems by using modal analysis techniques.

The semiactive control systems combine the attractive features of passive and active control systems and hence drawn the attention of many researchers, in recent past. Chi et al. [9] evaluated an asymmetric building with base isolation augmented with semiactive magnetorheological (MR) damper and found improvement in torsional behavior. Yoshida et al. [10] and Yoshida and Dyke [11] investigated the torsional response of asymmetric building using semiactive MR damper and noticed an increase of torsional response and decrease of translational response due to asymmetry. Moreover, the base torque increases and base shear decreases with an increase in eccentricity for strongly coupled system. Shook et al. [12] investigated the effectiveness of semiactive MR damper and observed that fuzzy logic controller is effective in decoupling lateral and torsional response with reduction in displacement and acceleration responses. H. N. Li and X. L. Li [13] developed an MR damper based on semi-geometric model for asymmetric building and found a greater reduction in displacement and acceleration responses compared to passive control case. Although, the above studies reflect the effectiveness of some of the semiactive systems in controlling the lateral-torsional responses, however, no specific study has been carried out to investigate the effects of torsional coupling especially by considering important system parameters on edge deformations as well as on control forces using semiactive MR dampers.

In this paper, the seismic response of idealized single storey, one-way asymmetric building, is investigated under different real earthquake ground motions. The specific objectives of the study are summarized as (i) to study the effects of torsional coupling on the effectiveness of the semiactive MR damper control system, (ii) to investigate the effectiveness of semiactive MR dampers with clipped-optimal control algorithm in controlling lateral, torsional and specifically the edge deformations, and (iii) to investigate the influence of important parameters on the effectiveness of semiactive MR dampers for asymmetric systems. The important parameters considered are eccentricity ratio of superstructure and uncoupled time period and ratio of uncoupled torsional to lateral frequency.

2. Model of Magnetorheological Damper

Magnetorheological (MR) dampers are the semiactive devices which use MR fluids to provide control forces and they are quite promising for civil engineering applications. They offer highly reliable operation at a modest cost and can be considered as fail-safe in the case of hardware malfunction in that they become passive dampers. MR fluids typically consist of micron-sized magnetically polarizable particles dispersed in a carrier medium such as silicon oil. When a magnetic field is applied to the fluids, particle chains form, and the fluid becomes a semisolid, exhibiting plastic behavior. MR fluids have high yield strength, low viscosity, and stable hysteretic behavior over a wide temperature range. Transition to rheological equilibrium can be achieved in a few milliseconds, providing devices with high bandwidth. MR devices are capable of generating large forces [15].

In this study, the Modified Bouc-Wen model proposed by Spencer et al. [14] as shown in Figure 1 is used. The model has shown to accurately predict the behavior of the prototype MR damper over a broad range of inputs. The equations governing the force predicted by this model is [14]𝑓_𝑖=𝛼_𝑖𝑧_𝑖+𝑐_0𝑖𝑥̇_𝑑𝑖−𝑦̇_𝑑𝑖+𝑘₀𝑥_𝑑𝑖−𝑦_𝑑𝑖+𝑘₁𝑥_𝑑𝑖−𝑥₀=𝑐_1𝑖𝑦̇_𝑑𝑖+𝑘₁𝑥_𝑑𝑖−𝑥₀, (1) where the evolutionary variable 𝑧_𝑖 is governed by 𝑧_𝑖̇=−𝛾_𝑚𝑥̇_𝑑𝑖−𝑦̇_𝑑𝑖𝑧_𝑖𝑧_𝑖^𝑛−1−𝛽_𝑚𝑥̇_𝑑𝑖−𝑦̇_𝑑𝑖𝑧_𝑖^𝑛+𝐴_𝑚𝑥̇_𝑑𝑖−𝑦̇_𝑑𝑖,𝑦̇_𝑑𝑖=1𝑐_0𝑖+𝑐_1𝑖𝛼_𝑖𝑧_𝑖+𝑐_0𝑖𝑥̇_𝑑𝑖+𝑘₀𝑥_𝑑𝑖−𝑦_𝑑𝑖. (2)

Figure 1 

Mechanical model of the MR damper (Modified Bouc-Wen model) [14].

In this model, the accumulator stiffness is represented by 𝑘₁ and the viscous damping observed at large velocities by 𝑐_0𝑖. A dashpot, represented by 𝑐_1𝑖, is included in the model to introduce the nonlinear roll-off in the force-velocity loops at low velocities, 𝑘₀ is present to control the stiffness at large velocities, and 𝑥₀ is the initial displacement of spring 𝑘₁ associated with the nominal damper force due to the accumulator.

To account for the dependence of the force on the voltage applied to the current driver and the resulting magnetic current, the suggested parameters are 𝛼_𝑖=𝛼_𝑎+𝛼_𝑏𝑢_𝑑𝑖,𝑐_0𝑖=𝑐_0𝑎+𝑐_0𝑏𝑢_𝑑𝑖,𝑐_1𝑖=𝑐_1𝑎+𝑐_1𝑏𝑢_𝑑𝑖. (3)

In addition, the dynamics involved in the MR fluid reaching rheological equilibrium are accounted for through the first-order filter 𝑢̇_𝑑𝑖=−𝜂𝑢_𝑑𝑖−𝑣_𝑖, (4) where 𝑣_𝑖 is the commanded voltage sent to the current driver.

3. Structural Model and Solution of Equations of Motion

The system considered is an idealized one-storey building which consists of a rigid deck supported on columns as shown in Figure 2. Following assumptions are made for the structural system under consideration: (i) floor of the superstructure is assumed as rigid, (ii) force-deformation behaviour of the superstructure is considered as linear within elastic range, (iii) the structure is excited by uni-directional horizontal component of earthquake ground motion and the vertical component of earthquake motion is neglected, and (iv) the effect of time delay is neglected for control algorithm. The mass of deck is assumed to be uniformly distributed and hence centre of mass (CM) coincides with the geometrical centre of the deck. The columns are arranged in a way such that it produces the stiffness asymmetry with respect to the CM in one direction and hence, the centre of rigidity (CR) is located at an eccentric distance, 𝑒_𝑥 from CM in 𝑥-direction. The system is symmetric in 𝑥-direction and therefore, two degrees-of-freedom are considered for model, namely, the lateral displacement in 𝑦-direction, 𝑢_𝑦 and torsional displacement, 𝑢_𝜃 as represented in Figure 2. The governing equations of motion of the building model with coupled lateral and torsional degrees-of-freedom are obtained by assuming that the control forces provided by the MR dampers are adequate to keep the response of the structure in the linear range. The equations of motion of the system in the matrix form are expressed as 𝐌𝐮̈+𝐂𝐮̇+𝐊𝐮=−𝐌𝚪𝐮̈_𝐠+𝚲𝐟, (5) where 𝐌, 𝐂, and 𝐊 are mass, damping, and stiffness matrices of the system, respectively; 𝐮={𝑢_𝑦𝑢_𝜃}^T is the displacement vector; Γ is the influence coefficient vector; 𝐮̈_𝐠={𝑢̈_𝑔𝑦0}^T is the ground acceleration vector; 𝑢̈_𝑔𝑦 is the ground acceleration in 𝑦-direction, Λ is the matrix that defines the location of control devices, and 𝐟 is the vector of control forces.

(a)

(b)

Figure 2 

Plan and isometric view of one-way asymmetric system showing arrangement of dampers and accelerometers.

The mass matrix can be expressed as 𝐌=𝑚00𝑚𝑟², (6) where 𝑚 represents the lumped mass of the deck; 𝑟 is the mass radius of gyration about the vertical axis through CM which is given by, 𝑟=(𝑎²+𝑏²)/12; where 𝑎 and 𝑏 are the plan dimensions of the building.

The stiffness matrix of the system is obtained as follows, after doing some algebraic manipulations [4] 𝐊=𝐾_𝑦1𝑒_𝑥𝑒_𝑥𝑒²_𝑥+𝑟²Ω²_𝜃,𝑒_𝑥=1𝐾_𝑦𝑖𝐾_𝑦𝑖𝑥_𝑖,Ω_𝜃=𝜔_𝜃𝜔_𝑦,𝜔_𝜃=𝐾_𝜃𝑟𝑚𝑟²,𝜔_𝑦=𝐾_𝑦𝑚,𝐾_𝜃𝑟=𝐾_𝜃𝜃−𝑒²_𝑥𝐾_𝑦,𝐾_𝜃𝜃=_𝑖𝐾_𝑥𝑖𝑦²_𝑖+_𝑖𝐾_𝑦𝑖𝑥²_𝑖, (7) where 𝐾_𝑦 denotes the total lateral stiffness of the building system in 𝑦-direction; 𝑒_𝑥 is the structural eccentricity between CM and CR of the system; Ω_𝜃 is the ratio of uncoupled torsional to lateral frequency of the system; 𝐾_𝑦𝑖 indicates the lateral stiffness of 𝑖th column in 𝑦-direction; 𝑥_𝑖 is the 𝑥-coordinate distance of 𝑖th element with respect to CM; 𝜔_𝑦 is uncoupled lateral frequency of the system; 𝜔_𝜃 is uncoupled torsional frequency of the system; 𝐾_𝜃𝑟 is torsional stiffness of the system about a vertical axis at the CR; 𝐾_𝜃𝜃 is torsional stiffness of the system about a vertical axis at the CM; 𝐾_𝑥𝑖 indicates the lateral stiffness of 𝑖th column in 𝑥-direction; 𝑦_𝑖 is the 𝑦-coordinate distance of 𝑖th element with respect to CM.

The damping matrix of the system is not known explicitly and it is constructed from the Rayleigh’s damping considering mass and stiffness proportional as 𝐂=𝑎₀𝐌+𝑎₁𝐊 (8) in which 𝑎₀ and 𝑎₁ are the coefficients depends on damping ratio of two vibration modes. For the present study 5% damping is considered for both modes of vibration of system.

The governing equations of motion are solved using the state space method [16, 17] and rewritten as 𝐳̇=𝐀𝐳+𝐁𝐟+𝐄𝐮̈_𝐠, (9) where 𝐳={𝐮𝐮̇}^T is a state vector; 𝐀 is the system matrix; 𝐁 is the distribution matrix of control forces; 𝐄 is the distribution matrix of excitations. These matrices are expressed as, 𝐀=𝟎𝐈−𝐌^−𝟏𝐊−𝐌^−𝟏𝐂𝟎𝐌,𝐁=_−𝟏𝚲,𝟎𝚪𝐄=−, (10) in which 𝐈 is the identity matrix.

Equation (9) is discretized in time domain, and the excitation and control forces are assumed to be constant within any time interval, the solution may be written in an incremental form [16, 17] 𝐳[𝑘+1]=𝐀_𝐝𝐳[𝑘]+𝐁_𝐝𝐟[𝑘]+𝐄_𝐝𝐮̈_𝐠[𝑘], (11) where 𝑘 denotes the time step; and 𝐀_𝐝=𝑒^𝐀Δ𝑡 represents the discrete-time system matrix with Δ𝑡 as the time interval. The constant coefficient matrices 𝐁_𝐝 and 𝐄_𝐝 are the discrete-time counterparts of the matrices 𝐁 and 𝐄 and can be written as 𝐁_𝐝=𝐀⁻¹𝐀_𝐝−𝐈𝐁,𝐄_𝐝=𝐀^−𝟏𝐀_𝐝−𝐈𝐄. (12)

4. Semiactive Clipped-Optimal Control Algorithm

The clipped-optimal control algorithm based on acceleration feedback is used herein as shown in Figure 3(a). This algorithm has been found to be among the best performing of several non linear semiactive controllers for MR devices [15, 18].

(a)

(b)

Figure 3 

Graphical representation of clipped-optimal control algorithm [15] and semiactive control system.

The measurement equation is given by 𝐲_𝑚=𝐂_𝑚𝐳+𝐃_𝑚𝐟+𝐯, (13) where 𝐲_𝑚 is the vector of measured outputs, and 𝐯 is the measurement noise vector. In this study the measurements considered for control force determination include the accelerations at flexible and stiff edge of the structure.

The matrices used in the measurement equation have the following form:𝐂_𝑚=−𝐏_𝑚𝐌^−𝟏𝐊−𝐏_𝑚𝐌⁻¹𝐂,𝐃_𝑚=𝐏_𝑚𝐌^−𝟏𝚲,𝐏_𝑚=1𝑙_𝑎11𝑙_𝑎2, (14) where 𝑙_𝑎1 and 𝑙_𝑎2 are the coordinates of the acceleration measurements of flexible and stiff edges, respectively. For the study carried out herein, the accelerometers are assumed to be fixed at the edges of building as shown in Figure 2(b).

The approach considered is to append a force feedback loop to induce the MR damper to produce approximately a desired control force 𝑓_𝑐𝑖. The force generated by the MR damper cannot be commanded; only the voltage 𝑣_𝑖 applied to the current driver for the MR damper can be directly changed as follows. When the MR damper is providing the desired optimal control force (i.e., 𝑓_𝑖=𝑓_𝑐𝑖), the voltage applied to the damper should remain at the present level. If the magnitude of the force produced by the damper is smaller than the magnitude of the desired optimal control force and the two forces have the same sign, the voltage applied to the current driver increases to the maximum level so as to increase the force produced by the damper to match the desired control force. Otherwise, the commanded voltage is set to zero. The algorithm for selecting the command signal is graphically represented in Figure 3(a) and can be stated as [15, 18] 𝑣_𝑖=𝑉_max𝐻𝑓_𝑐𝑖−𝑓_𝑖𝑓_𝑖, (15) where 𝑉_max is the voltage sent to the current driver, associated with saturation of the magnetic field in MR damper, and 𝐻(⋅) is the Heaviside step function. In this study, H₂/LQG (Linear Quadratic Gaussian) strategy is employed as a nominal controller to have desired control forces. Figure 3(b) represents the block diagram of the semiactive control system.

For the design of H₂/LQG controller, the absolute acceleration of the ground, 𝑢̈_𝑔𝑦, is taken to be a stationary white noise, and an infinite horizon performance index is chosen as 𝐽=lim_𝜏→∞1𝜏𝐸^𝜏₀𝐲^T_𝑚𝐐𝐲_𝑚+𝐟^T𝐑𝐟dt, (16) where 𝐐 and 𝐑 are weighting matrices for the vector of regulated/measured responses 𝐲_𝑚={𝑢̈_𝑚𝑓𝑢̈_𝑚𝑠}^T and of control forces 𝐟={𝐹_𝑑𝑓𝐹_𝑑𝑠}^T, respectively. Where, 𝑢̈_𝑚𝑓 and 𝑢̈_𝑚𝑠 are the flexible and stiff edge accelerations measured by the accelerometers which are placed at the respective edges on the floor, as shown in Figure 2(b). For design purposes, the measurement noise vector 𝐯 is assumed to contain identically distributed, statistically independent Gaussian white noise processes, with 𝑆_{̈𝑢𝑔𝑦̈𝑢𝑔𝑦}/𝑆_{𝑣𝑖𝑣𝑖}=𝛾_𝑔=50.

The Nominal controller is represented as [10] 𝐳̇̂=𝐀−𝐋_𝑔𝐂_𝑚𝐳̂+𝐋_𝑔𝐲_𝑚+𝐁−𝐋_𝑔𝐃_𝑚𝐟𝐟,_𝑐=−𝐊_𝑔𝐳̂, (17) where 𝐋_𝑔 is the gain matrix for the state estimator; 𝐳̂ is the estimated state vector; 𝐊_𝑔 is the gain matrix for the linear quadratic regulator obtained as follows: 𝐊_𝑔=𝐁^′𝐑⁻¹𝐏, (18) where 𝐏 is the solution of the algebraic Ricatti equation given by 𝟎=𝐏𝐀+𝐀^′𝐏−𝐏𝐁𝐑^−𝟏𝐁^′𝐏+𝐂^′_𝑚𝐐𝐂_𝑚,𝐋_𝑔=𝐂_𝑚𝐒^′, (19) where 𝐒 is the solution of the algebraic Ricatti equation given by 𝟎=𝐒𝐀^′+𝐀𝐒−𝐒𝐂^′_𝑚𝐂_𝑚𝐒+𝛾_𝑔𝐄𝐄^′. (20)

The response weighting matrix 𝐐 corresponding to the regulated output vector 𝐲_𝑚 is considered as follows: 𝐐=𝑞_𝑤⋅1001, (21) where 𝐪_𝑤 is the coefficient for weighting matrix 𝐐.

The weighting matrix 𝐑 which is corresponding to the control force vector is considered as follows: 𝐑=𝐫_𝑤⋅1001, (22) where 𝑟_𝑤 is the coefficient for weighting matrix 𝐑.

5. Numerical Study

The seismic response of linearly elastic, idealized single-storey, one-way asymmetric system installed with semiactive MR dampers with clipped-optimal control algorithm is investigated by numerical simulation study. The response quantities of interest are lateral and torsional displacements of the floor mass obtained at the CM (𝑢_𝑦 and 𝑢_𝜃), displacements at stiff and flexible edges of the system (𝑢_𝑦𝑠 and 𝑢_𝑦𝑓=𝑢_𝑦±𝑏𝑢_𝜃/2), lateral and torsional accelerations of the floor mass obtained at the CM (𝑢̈_𝑦 and 𝑢̈_𝜃), accelerations at stiff and flexible edges of the system (𝑢̈_𝑦𝑠 and 𝑢̈_𝑦𝑓=𝑢̈_𝑦±𝑏𝑢̈_𝜃/2), base shear (𝑉_𝑦), as well as damper control forces at stiff edge (𝐹_𝑑𝑠) and at flexible edge (𝐹_𝑑𝑓) of the building. The response of the system is investigated under following parametric variations: structural eccentricity ratio (𝑒_𝑥/𝑟), uncoupled lateral time period of system (𝑇_𝑦=2𝜋/𝜔_𝑦) and ratio of uncoupled torsional to lateral frequency of the system (Ω_𝜃=𝜔_𝜃/𝜔_𝑦). The peak responses are obtained corresponding to the important parameters which are listed above and variations are plotted for the eight considered earthquake ground motions namely, Imperial Valley (1940), Loma Prieta (1989), Northridge (Sylmar, 1994), Kobe (1995), Erzincan (1992), Northridge (Newhall, 1994), Northridge (Rinaldi, 1994), and Chi-Chi (1999) for the present study with corresponding peak ground acceleration (PGA) values of 0.31 g, 0.96 g, 0.89 g, 0.82 g, 0.51 g, 0.59 g, 0.82 g, and 0.57 g as per the details summarized in Table 1. The input parameters considered for the present study are; Total weight of building, 𝑊=10,000 kN and aspect ratio of plan dimension is kept as unity. For the numerical simulation study carried out herein, two MR dampers, each having 200 kN capacity [20] are assumed to be installed at the edges of the building as shown in Figure 2. The parameters of mechanical model of a damper are presented in Table 2 [19].

Table 1 

Details of earthquake motions considered for the numerical study.

Table 2 

Parameters for the MR damper model (200 kN capacity) [19].

In order to study the effectiveness of implemented semiactive control system, the response is expressed in terms of indices, 𝑅_𝑒 and 𝑅_𝑡 are defined as follows:𝑅_𝑒=Peak(orRMS)responseofcontrolledasymmetricsystem(),𝑅PeakorRMSresponseofcorrespondinguncontrolledsystem_𝑡=Peak(orRMS)responseofcontrolledasymmetricsystem().PeakorRMSresponseofcorrespondingsymmetricsystem (23)

The value of 𝑅_𝑒 less than one indicates that the implemented semiactive control system is effective in controlling the responses. On the other hand, the value of 𝑅_𝑡 reflects the effects of torsional coupling on the effectiveness of semiactive control system and seismic behavior of system. The value of 𝑅_𝑡 greater than one indicates that the response of controlled asymmetric system increases due to torsional coupling as compared to the corresponding symmetric system. The effects of various levels of excitations for particular earthquake motion are not studied by considering the fact that the main objective is to investigate the effects of torsional coupling on the effectiveness of semiactive device for asymmetric systems as compared to symmetric systems and the responses are obtained in terms of response ratios. Moreover, in past, some work have been reported with the use of semiactive MR damper with the assumption that the building behaves in linear range and also without considering the effects of different levels of excitations [15, 18, 21].

In order to determine the optimum coefficients of weighting matrices 𝐐 and 𝐑, the response ratio, 𝑅_𝑒 for peak and root mean square (RMS) values of displacements as well as accelerations of the floor mass are obtained for the symmetric system with 𝑇_𝑦=1 s. Initially, the coefficient, 𝑟_𝑤 of the matrix, 𝐑 is considered as unity and the coefficient, 𝑞_𝑤 of matrix, 𝐐 is varied by placing higher weightage on the response quantities. The response are obtained under Imperial Valley, 1940 earthquake and shown in Figure 4(a). From the figure, it can be noticed that initially with the increase in 𝑞_𝑤, the ratio, 𝑅_𝑒 decreases up to certain limit and then increases with further increase in value of 𝑞_𝑤. This means there exists an optimum value of coefficient, 𝑞_𝑤 for which 𝑅_𝑒 is minimum, which shows the maximum reduction in various responses. Thus, the optimum value for 𝑞_𝑤 is found to be 10¹¹. Further, trials have been done by keeping 𝑞_𝑤 as constant (i.e., 𝑞_𝑤=10¹¹) and by varying the coefficient, 𝑟_𝑤 and the results are shown in Figure 4(b). From the figure, it can be noticed that the ratio, 𝑅_𝑒 is minimum corresponding to the value of 𝑟_𝑤=1. Moreover, the above results are also verified for the systems with different time period, 𝑇_𝑦 (i.e., varied from 0.1 s to 3 s), and it is found that the optimum values of 𝑞_𝑤 and 𝑟_𝑤 remains almost in the similar range such as to have optimum compromise between the reduction in peak and RMS displacement and acceleration responses. Thus, for the study carried out herein, the optimum coefficients for the weighting matrices 𝐐 and 𝐑 are considered as 𝑞_𝑤=10¹¹ and 𝑟_𝑤=1.

(a)

(b)

Figure 4 

Effect of weighting parameters on response ratio, 𝑅𝑒 for peak and RMS responses under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, 𝑒𝑥/𝑟=0).

The present study is carried out by considering two MR dampers (one on each side of building), each of 200 kN capacity with semiactive clipped-optimal control algorithm with maximum applied voltage of 10 V. The responses are also obtained with passive-off and passive-on cases, with constant zero voltage and with constant maximum voltage, respectively, and compared with the semiactive control case. However, the main focus is to study the behavior of the system equipped with semiactive dampers. Figure 5 shows the time histories of various displacement and acceleration response quantities of the uncontrolled building compared with the corresponding building controlled with semiactive MR dampers with clipped-optimal control algorithm.

Figure 5 

Time histories for various uncontrolled and controlled displacements and accelerations under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, Ω𝜃=1, 𝑒𝑥/𝑟=0.3).

The responses are obtained for the system with 𝑇_𝑦=1 s, Ω_𝜃=1, and 𝑒_𝑥/𝑟=0.3 (intermediate eccentricity) under Imperial Valley, 1940 earthquake. The significant reduction in displacement and acceleration responses at CM, at flexible and stiff edges as well as torsional response can be noticed from the figure. Thus, the semiactive system is quite effective in reducing lateral-torsional responses.

Furthermore, Table 3 depicts the comparison between the three control strategies, namely, passive-off, passive-on, and semiactive clipped-optimal control for the system with 𝑇_𝑦=1 s, Ω_𝜃=1, 𝑒_𝑥/𝑟=0.3 under Imperial Valley, 1940 earthquake. The ratio, 𝑅_𝑒 for various responses based on peak and RMS values are summarized herein. In general, ratio 𝑅_𝑒 comes out be less than unity for all control strategies implying the effectiveness of control systems. It is further observed that passive-on strategy gives higher reduction in peak and RMS displacements and accelerations as compared to passive-off case. Also, it is observed that the reduction in peak displacements and accelerations obtained at CM (𝑢_𝑦 and 𝑢̈_𝑦), at edges (𝑢_𝑦𝑠, 𝑢_𝑦𝑓, 𝑢̈_𝑦𝑠, and 𝑢̈_𝑦𝑓) as well as its torsional components (𝑢_𝜃 and 𝑢̈_𝜃) for the semiactive case is found as little higher or comparable with those obtained with passive-on case. Moreover, the reduction in RMS of various displacement quantities is slightly higher for passive-on case as compared to corresponding semiactive case, whereas higher reduction in RMS of various acceleration quantities is found for semiactive case as compared to passive-on case. The table also shows the normalized control forces of dampers which are placed at the edges of the building. It is important to note that the RMS control forces required for passive-on case is much more higher than the passive-off case while for the semiactive case it is only little higher than passive-off case. This reflects that, for the semiactive clipped-optimal control algorithm, with little increase in RMS damper forces as compared to passive-off case, the higher or comparable reduction in peak and RMS values of lateral, edge, and torsional responses is obtained than those which is obtained by passive-on case.

Table 3 

Response ratio, 𝑅_𝑒 for Peak and RMS Responses for various control strategies under Imperial Valley, 1940 earthquake (𝑇_𝑦=1, Ω_𝜃=1, 𝑒_𝑥/𝑟=0.3).

Moreover, Figures 6, 7, and 8 represents the hysteresis loops for normalized damper force with displacement and velocity for the considered control strategies, namely, passive-off, passive-on, and semiactive clipped-optimal control case, respectively, for the system with 𝑇_𝑦=1 s, Ω_𝜃=1, 𝑒_𝑥/𝑟=0.3 under Imperial Valley, 1940 earthquake. Further, Figure 9 shows the time variation of applied voltage for the semiactive clipped-optimal control case for the dampers located at stiff and flexible edges.

Figure 6 

Normalized damper force-displacement and velocity loops for dampers located at stiff and flexible edges for passive-off algorithm under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, Ω𝜃=1, 𝑒𝑥/𝑟=0.3).

Figure 7 

Normalized damper force-displacement and velocity loops for dampers located at stiff and flexible edges for passive-on algorithm under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, Ω𝜃=1, 𝑒𝑥/𝑟=0.3).

Figure 8 

Normalized damper force-displacement and velocity loops for dampers located at stiff and flexible edges for semiactive clipped-optimal control algorithm under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, Ω𝜃=1, 𝑒𝑥/𝑟=0.3).

Figure 9 

Time variation of applied voltage for dampers located at stiff and flexible edges for semiactive clipped-optimal control under Imperial Valley, 1940 earthquake (𝑇𝑦=1 s, Ω𝜃=1, 𝑒𝑥/𝑟=0.3).

One of the key parameters affecting the lateral-torsional response of buildings is the structural eccentricity. In order to study the effects of eccentricity and torsional coupling on the seismic response of asymmetric systems, the variations of ratio, 𝑅_𝑡 for lateral and edge displacements are plotted against eccentricity ratio, 𝑒_𝑥/𝑟 (which is varied from 0 to 1) for eight selected earthquakes. The variations are shown for torsionally flexible (Ω_𝜃=0.5), strongly coupled (Ω_𝜃=1) and torsionally stiff (Ω_𝜃=2) systems with 𝑇_𝑦=1 s in Figures 10, 11, and 12, respectively. It is observed that for the systems with Ω_𝜃=0.5, with increase in 𝑒_𝑥/𝑟, the ratio, 𝑅_𝑡 for lateral displacement, 𝑢_𝑦 decreases and remains less than unity, in general, whereas, the ratio, 𝑅_𝑡 for 𝑢_𝑦𝑠 increases first, up to an intermediate eccentricity (i.e., 𝑒_𝑥/𝑟<0.5) and then decreases with further increase in eccentricity ratio. For flexible edge displacement, 𝑢_𝑦𝑓, the ratio, 𝑅_𝑡 first decreases up to an intermediate eccentricity and then increases for higher eccentricity. This shows that for torsionally flexible system with an intermediate eccentricity, 𝑢_𝑦 and 𝑢_𝑦𝑓 reduces and 𝑢_𝑦𝑠 increases due to torsional coupling as compared to corresponding symmetric system and for the system with higher eccentricity, 𝑢_𝑦𝑓 increases, and 𝑢_𝑦 and 𝑢_𝑦𝑠 decrease due to torsional coupling except for Erzincan (1992) and Chi-Chi (1999) earthquakes. Hence, for torsionally flexible systems with an intermediate eccentricity, the effectiveness of control system is more for asymmetric system for reducing 𝑢_𝑦 and 𝑢_𝑦𝑓 as compared to corresponding symmetric systems and hence it will be underestimated by ignoring the eccentricity and modeling the building as 2D. Whereas, the effectiveness of control system is less for asymmetric system for reducing 𝑢_𝑦𝑠 as compared to corresponding symmetric systems and hence it will be overestimated for 𝑢_𝑦𝑠. Whereas, for torsionally flexible systems with higher eccentricities, effectiveness of control system will be underestimated for reducing 𝑢_𝑦 and 𝑢_𝑦𝑠 and it will be overestimated for 𝑢_𝑦𝑓.

Figure 10 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak displacements (𝑇𝑦=1 s, Ω𝜃=0.5).

Figure 11 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak displacements (𝑇𝑦=1 s, Ω𝜃=1).

Figure 12 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak displacements (𝑇𝑦=1 s, Ω𝜃=2).

Moreover, as shown in Figures 11 and 12 for the systems with Ω_𝜃=1 and 2, the ratio, 𝑅_𝑡 for 𝑢_𝑦𝑠 decreases and for response, 𝑢_𝑦𝑓, it increases with increase in eccentricity ratio. This implies that the effectiveness of control system is more for asymmetric system as compared to corresponding symmetric system in reducing 𝑢_𝑦𝑠 and hence it will be underestimated and the effectiveness is less for asymmetric system in reducing 𝑢_𝑦𝑓 and hence it will be underestimated by ignoring the effect of torsional coupling. Also, for the systems with Ω_𝜃=1, the ratio, 𝑅_𝑡 for 𝑢_𝑦 remains less that unity except for the systems with higher eccentricities. However, for torsionally stiff systems, the ratio, 𝑅_𝑡 for lateral and edge displacements, remains very less sensitive to the variation of 𝑒_𝑥/𝑟. Thus, the effects of torsional coupling can be ignored for such systems while estimating the effectiveness of semiactive control system for asymmetric system as compared to symmetric system. Further, the difference between edge displacements of asymmetric and corresponding symmetric system is higher for strongly coupled and torsionally flexible systems as compared to torsionally stiff systems and the difference increases with the increase in structural eccentricity. Thus, for torsionally flexible and strongly coupled systems, the effects of torsional coupling can not be neglected while estimating the effectiveness of semiactive control system for asymmetric systems as compared to corresponding symmetric systems in controlling edge displacements.

To study the effects of torsional coupling on base shear, 𝑉_𝑦, the variations of ratio, 𝑅_𝑡 for 𝑉_𝑦 are plotted against 𝑒_𝑥/𝑟 for three values of frequency ratios (i.e., Ω_𝜃=0.5, 1 and 2) as shown in the Figure 13. From the figure, it can be observed that with increase in 𝑒_𝑥/𝑟, ratio, 𝑅_𝑡 decreases, in general. Further, it is noticed that the ratio, 𝑅_𝑡 for 𝑉_𝑦, is more sensitive to variation of 𝑒_𝑥/𝑟 for systems with Ω_𝜃=1 followed by the systems with Ω_𝜃=0.5 and Ω_𝜃=2. This indicates that the torsional coupling reduces the base shear for controlled asymmetric systems as compared to corresponding symmetric systems and hence, the effectiveness of semiactive control system in reducing base shear is more for asymmetric systems as compared to corresponding symmetric systems and the effectiveness will be underestimated, if torsional coupling is ignored and building is modeled as 2D. This effect is more pronounced for the systems with higher superstructure eccentricities. Thus, the difference between the base shear of asymmetric and corresponding symmetric system increases with the increase in eccentricity ratio and it is significant for strongly coupled systems followed by torsionally flexible and torsionally stiff systems.

Figure 13 

Effect of eccentricity on response ratio, 𝑅𝑡 for peak base shear, 𝑉𝑦 (𝑇𝑦=1 s).

Moreover, the variations of ratio, 𝑅_𝑡 for lateral and edge accelerations are shown for the systems with Ω_𝜃=0.5, 1, and 2 in Figures 14, 15, and 16, respectively. In general, it is found that the trends for edge accelerations are opposite to those obtained for the edge displacements. It is further observed that for the systems with Ω_𝜃=0.5, with increase in 𝑒_𝑥/𝑟, the ratio, 𝑅_𝑡 for lateral acceleration, 𝑢̈_𝑦 and flexible edge acceleration, 𝑢̈_𝑦𝑓 decreases and that of stiff edge acceleration, 𝑢̈_𝑦𝑠 increases and remains more than a unity. Thus, for torsionally flexible systems, by ignoring the eccentricity, the effectiveness of semiactive control system will be underestimated for reducing 𝑢̈_𝑦 and 𝑢̈_𝑦𝑓 and overestimated for 𝑢̈_𝑦𝑠. Similarly, for the system with Ω_𝜃=1, the ratio, 𝑅_𝑡 for 𝑢̈_𝑦 reduces with the increase in 𝑒_𝑥/𝑟 and for 𝑢̈_𝑦𝑠, the ratio remains slightly more that unity, in general. Whereas, for 𝑢̈_𝑦𝑓, the ratio remains less than or close to unity. However, the variation of 𝑅_𝑡 for 𝑢̈_𝑦𝑓 remains less sensitive to 𝑒_𝑥/𝑟 as compared to those of responses, 𝑢̈_𝑦 and 𝑢̈_𝑦𝑠. Also, it is found that the variation of 𝑅_𝑡 with 𝑒_𝑥/𝑟 for edge accelerations are more pronounced for systems with Ω_𝜃=0.5 as compared to Ω_𝜃=1. Moreover, as shown in Figure 16, for systems with Ω_𝜃=2, with increase in 𝑒_𝑥/𝑟, the ratio, 𝑅_𝑡 for 𝑢̈_𝑦 and 𝑢̈_𝑦𝑠 slightly decreases and for 𝑢̈_𝑦𝑓, it increases slightly. However, the rate of change of ratio, 𝑅_𝑡 with 𝑒_𝑥/𝑟 is very less sensitive for such systems as compared to systems with Ω_𝜃=0.5 and 1. Thus, the effects of torsional coupling are more pronounced for torsionally flexible systems than strongly coupled systems while estimating the effectiveness of semiactive control system for asymmetric systems in controlling lateral and edge accelerations as compared to corresponding symmetric systems.

Figure 14 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak accelerations (𝑇𝑦=1 s, Ω𝜃=0.5).

Figure 15 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak accelerations (𝑇𝑦=1 s, Ω𝜃=1).

Figure 16 

Effect of eccentricity on response ratio, 𝑅𝑡 for various peak accelerations (𝑇𝑦=1 s, Ω𝜃=2).

Figures 17 and 18 show the variations of response ratio, 𝑅_𝑡 against 𝑒_𝑥/𝑟 for three values of Ω_𝜃, for normalized RMS control forces developed by the dampers which are installed at stiff edge (𝐹_𝑑𝑠) and at flexible edge (𝐹_𝑑𝑓) of the building. It can be observed that for torsionally flexible systems, the ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 increases with increase in 𝑒_𝑥/𝑟 and remains more than unity and for 𝐹_𝑑𝑓, it decreases and remains less than unity except for the systems with higher eccentricities under Erzincan (1992) and Chi-Chi (1999) earthquakes. Hence, by ignoring the effects of torsional coupling for torsionally flexible systems, control forces at stiff edge, 𝐹_𝑑𝑠, will be underestimated, and at flexible edge, 𝐹_𝑑𝑓, it will be overestimated as compared to corresponding symmetric systems. Further, for strongly coupled systems, the ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 decreases with the increase in 𝑒_𝑥/𝑟 and remains less than or very close to unity, whereas for 𝐹_𝑑𝑓, it increases with the increase 𝑒_𝑥/𝑟 and remains more than or close to unity. This implies that for strongly coupled systems, by ignoring the effects of torsional coupling, the control forces at stiff edge will be overestimated and at flexible edge, those will be underestimated as compared to corresponding symmetric systems. Whereas, for torsionally stiff systems, the ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 decreases with the increase in 𝑒_𝑥/𝑟 and for 𝐹_𝑑𝑓, it increases with the increase 𝑒_𝑥/𝑟. Thus, by ignoring the effects of torsional coupling, control forces at stiff edge will be overestimated and at flexible edge, those will be underestimated as compared to corresponding symmetric systems. However, the rate of change for ratio, 𝑅_𝑡 against 𝑒_𝑥/𝑟 for control forces is more sensitive for systems with Ω_𝜃=0.5 followed by the systems with Ω_𝜃=1 and 2. Thus, the effects of torsional coupling on the estimation of control forces significantly depend on superstructure eccentricity and torsional to lateral frequency ratio.

Figure 17 

Effect of eccentricity on response ratio, 𝑅𝑡 for normalized stiff edge RMS damper force (𝑇𝑦=1 s).

Figure 18 

Effect of eccentricity on response ratio, 𝑅𝑡 for normalized flexible edge RMS damper force (𝑇𝑦=1 s).

To study, the effects of torsional coupling on the seismic behavior of laterally stiff and laterally flexible structural systems, the variations of ratio, 𝑅_𝑡 for edge displacements and accelerations as well as for normalized RMS control forces are plotted against uncoupled lateral time period, 𝑇_𝑦 (i.e., varied from 0.1 s to 3 s). The results are obtained for an intermediate eccentricity ratio (𝑒_𝑥/𝑟=0.3) and shown in Figure 19 for the systems with Ω_𝜃=0.5, 1 and 2. The response ratios are obtained for eight selected earthquakes and the average trend from those earthquakes are shown in Figure 19. It can be noticed from the figure that for laterally stiff systems with Ω_𝜃=0.5 and 1, the difference between the edge displacements of asymmetric and corresponding symmetric systems is higher as compared to laterally flexible systems and as the flexibility of superstructure increases the difference tends to decreases. Whereas, for torsionally stiff systems, variation in ratio 𝑅_𝑡 for edge displacement and acceleration response remains insensitive to the change in values of 𝑇_𝑦. Further, for laterally stiff systems with Ω_𝜃=0.5, the difference between the flexible edge displacement of asymmetric and corresponding symmetric systems is significantly higher as compared to laterally flexible systems. This means that for laterally stiff systems, the torsional coupling significantly increases the flexible edge displacement for asymmetric system as compared to symmetric system. In addition, for laterally stiff systems with Ω_𝜃=0.5, the values of ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 and 𝐹_𝑑𝑓 remains more than unity and for laterally flexible systems with Ω_𝜃=0.5, the ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 remains more than unity and for 𝐹_𝑑𝑓, it remains less than unity. Thus, by ignoring the effects of torsional couplings for laterally stiff systems with Ω_𝜃=0.5, the control forces at stiff edge and flexible edge damper will be underestimated for asymmetric systems as compared to corresponding symmetric systems. Whereas, for laterally flexible systems with Ω_𝜃=0.5, the control forces at stiff edge damper will be underestimated and at flexible edge damper, it will be overestimated. Similarly, for laterally stiff systems with Ω_𝜃=1, the values of ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 remains more than unity and for 𝐹_𝑑𝑓, it remains more than unity and for laterally flexible systems with Ω_𝜃=1, the ratio, 𝑅_𝑡 for 𝐹_𝑑𝑠 and 𝐹_𝑑𝑓 remains close to the unity. Thus, by ignoring the effects of torsional couplings for laterally stiff systems with Ω_𝜃=1, the control forces at stiff edge will be overestimated and at flexible edge that will be underestimated. Moreover, for laterally stiff to laterally flexible systems with Ω_𝜃=2, the values of ratio, 𝑅_𝑡, for 𝐹_𝑑𝑠 remains slightly less than unity and for 𝐹_𝑑𝑓, it remains slightly more than unity implying that by ignoring the effects of torsional couplings for such systems, the control forces at stiff edge will be overestimated and at flexible edge that will be underestimated. Further, it is important to note that for the systems with Ω_𝜃=0.5, 1 and 2, the variations in values of ratio, 𝑅_𝑡, for normalized damper control forces, 𝐹_𝑑𝑠 and 𝐹_𝑑𝑓, are much more sensitive and significant for laterally stiff systems as compared to laterally flexible systems. Thus, the difference between the damper control forces for asymmetric systems and corresponding symmetric systems is higher for laterally stiff systems as compared to laterally flexible systems and this difference tends to reduce with the increase in lateral flexibility of the superstructure.

Figure 19 

Effect of time period, 𝑇𝑦 on response ratio, 𝑅𝑡 for peak edge displacements and accelerations and RMS damper forces under average earthquakes (𝑒𝑥/𝑟=0.3).

In order, to study the effectiveness of semiactive MR dampers with clipped-optimal control algorithm in reducing the coupled responses, the variations of ratio, 𝑅_𝑒 against eccentricity ratio, 𝑒_𝑥/𝑟 for the system with 𝑇_𝑦=1 are shown in Figure 20. The response ratios, R_e, for lateral, torsional, and edge displacements as well as accelerations for different values of Ω_𝜃 for eight selected earthquakes are obtained and the average trends from these earthquakes are shown in Figure 20. It may be recalled that, the ratio, 𝑅_𝑒 is between the responses of controlled asymmetric system and corresponding uncontrolled system and its value less than one indicates that the control system is effective in reducing the responses. It can be noticed from the figure that the ratio, 𝑅_𝑒, for lateral, edge, and torsional displacements and accelerations comes out to be less than unity for all considered values of 𝑒_𝑥/𝑟 implying that the semiactive MR dampers using clipped-optimal algorithm is effective in reducing peak lateral-torsional responses. It is further noticed that for strongly coupled system (Ω_𝜃=1), the values of ratio, 𝑅_𝑒, for torsional displacement, 𝑢_𝜃, and torsional acceleration, 𝑢̈_𝜃, increases with increase in 𝑒_𝑥/𝑟. This means the effectiveness of control system for controlling 𝑢_𝜃 and 𝑢̈_𝜃 reduces for higher eccentricities for such systems. Moreover, for torsionally stiff systems, the values of ratio, 𝑅_𝑒 for all responses quantities remains very less sensitive to the change in 𝑒_𝑥/𝑟. However, for such systems with lower eccentricities ratio, the effectiveness of control system is less in reducing torsional acceleration. Furthermore, for strongly coupled system, the values of 𝑅_𝑒 for torsional displacement and acceleration responses are less than those obtained for lateral and edge displacement and acceleration responses for considered range of values of 𝑒_𝑥/𝑟, implying that the implemented control system is more effective for reducing the torsional responses than lateral responses of strongly coupled system. Moreover, the variation in values of ratio, 𝑅_𝑒 for edge displacements and accelerations strongly depends on values of 𝑒_𝑥/𝑟 for the systems with Ω_𝜃=0.5 and 1. Thus, the effectiveness of semiactive control system in reducing the various responses has strong dependence on superstructure eccentricity for torsionally flexible and strongly coupled systems and it is very less dependent for torsionally stiff systems.

Figure 20 

Effect of eccentricity on response ratio, 𝑅𝑒 for various peak response quantities under average earthquakes (𝑇𝑦=1).

The investigations carried out in this research paper are based on the idealized single-storey building model with dampers located only at periphery (edges) of the building as it would develop the higher resisting forces. The dampers are assumed to be symmetrically placed in the building. Moreover, from the practical application point of view also, it shall be convenient to install dampers at the edges of the building to satisfy the functional requirements also. However, it shall be further useful to study the effects of torsional coupling for multistory building with different configuration of dampers in plan and/or in elevation.

6. Conclusions

The seismic response of linearly elastic, single-storey, one-way asymmetric building installed with semiactive MR dampers subjected to different earthquake ground motions is investigated. The response is evaluated considering clipped-optimal control algorithm with parametric variations to study the effectiveness of semiactive control system for torsionally coupled system and the influence of important parameters on the effectiveness of semiactive control system for asymmetric systems. The important parameters considered are: eccentricity ratio of superstructure, uncoupled lateral time period, and ratio of uncoupled torsional to lateral frequency. From the trend of the results of the present study, the following conclusions can be drawn.(1)For the semiactive clipped-optimal control algorithm, with little increase in RMS damper forces as compared to passive-off case, the higher or comparable reduction in peak and RMS values of lateral, edge and torsional responses is obtained than those which is obtained by passive-on case.(2)For torsionally flexible and strongly coupled systems, the effects of torsional coupling can not be neglected while estimating the effectiveness of semiactive control system for asymmetric systems as compared to corresponding symmetric systems in controlling edge displacements.(3)The difference between the base shear of asymmetric and corresponding symmetric system increases with the increase in eccentricity ratio and it is significant for strongly coupled systems followed by torsionally flexible and torsionally stiff systems.(4)The effects of torsional coupling are more pronounced for torsionally flexible systems than strongly coupled systems while estimating the effectiveness of semiactive control system for asymmetric systems in controlling lateral and edge accelerations as compared to corresponding symmetric systems.(5)The effects of torsional coupling on the estimation of control forces significantly depend on superstructure eccentricity and torsional to lateral frequency ratio.(6)The difference between the damper control forces for asymmetric systems and corresponding symmetric systems is higher for laterally stiff systems as compared to laterally flexible systems and this difference tends to reduce with the increase in lateral flexibility of the superstructure.(7)The effectiveness of semiactive control system in reducing the various responses has strong dependence on superstructure eccentricity for torsionally flexible and strongly coupled systems and it is very less dependent for torsionally stiff systems.

Notations

𝑎:	Plan dimension of building, parallel to the direction of ground motion
𝑎0 and 𝑎1:	Coefficients for Rayleigh’s mass and stiffness proportional damping matrix, respectively
𝐀:	System matrix
𝐀𝐝:	Discrete-time system matrix
𝐴𝑚 and 𝑛:	Parameters for MR damper model
𝑏:	Plan dimension of building, perpendicular to the direction of ground motion
𝐁:	Distribution matrix of control forces
𝐁𝐝:	Discrete-time counterpart of distribution matrix of control forces
𝑐0𝑎 and 𝑐0𝑏:	Parameters for MR damper model
𝑐0𝑖 and 𝑐1𝑖:	Parameters for viscous damping observed at large and low velocities, respectively, for 𝑖th MR damper model
𝑐1𝑎 and 𝑐1𝑏:	Parameters for MR damper model
𝐂:	Structural damping matrix of the system
𝐂𝑚 and 𝐃𝑚:	Matrices used for measurement equation
𝑒𝑥:	Structural (superstructure) eccentricity between centre of mass (CM) and centre of rigidity (CR) of the building system
𝐄:	Distribution matrix of excitation forces
𝐄𝐝:	Discrete-time counterpart of distribution matrix of excitation forces
𝐟:	Damper control force vector
𝑓𝑐𝑖:	Desired control force for 𝑖th damper
𝐟𝑐:	Desired control force vector
𝑓𝑖:	Control force in 𝑖th damper
𝐹𝑑𝑓:	Control force of damper located at flexible edge of building
𝐹𝑑𝑠:	Control force of damper located at stiff edge of building
𝑔:	Acceleration due to gravity
𝐻(⋅):	Heaviside step function
𝐈:	Identity matrix
𝐽:	Infinite horizon performance index for the design of controller
𝑘:	Time step
𝑘0:	Stiffness at large velocities of MR damper model
𝑘1:	Accumulator stiffness of MR damper model
𝐊:	Stiffness matrix of the system
𝐊𝑔:	Gain matrix for the design of controller
𝐾𝑥𝑖:	Lateral stiffness of 𝑖th column in 𝑥-direction
𝐾𝑦:	Total lateral stiffness of the system in 𝑦-direction
𝐾𝑦𝑖:	Lateral stiffness of 𝑖th column in 𝑦-direction
𝐾𝜃𝑟:	Torsional stiffness of the system about a vertical axis at the CR
𝐾𝜃𝜃:	Torsional stiffness of the system about a vertical axis at the CM
𝑙𝑎1:	Coordinate of position of the accelerometer 1, located at flexible edge
𝑙𝑎2:	Coordinate of position of the accelerometer 2, located at stiff edge
𝐋𝑔:	Gain matrix for the state estimator
𝑚:	Lumped mass of the deck
𝐌:	Mass matrix of the system
𝐏:	Matrix of algebraic Ricatti equation
𝑞𝑤:	Coefficient for weighting matrix, 𝐐
𝐐:	Weighting matrix for the vector of regulated/measured responses
𝑟:	Mass radius of gyration about a vertical axis through CM
𝑟𝑤:	Coefficient for weighting matrix, 𝐑
𝐑:	Weighting matrix for the vector of control forces
𝑅𝑒:	Response ratio to study the effectiveness of control system
𝑅𝑡:	Response ratio to study the effects of torsional coupling
𝐒:	Matrix of algebraic Ricatti equation
𝑇𝑦:	Uncoupled lateral time period of system
𝐮 and 𝐮̇:	Displacement and velocity vectors, respectively
𝑢𝑑𝑖:	Filtered voltage of 𝑖th MR damper
𝑢𝑦:	Lateral displacement at CM, in 𝑦-direction
𝑢𝑦𝑓:	Flexible edge displacement
𝑢𝑦𝑠:	Stiff edge displacement
𝑢𝜃:	Torsional displacement, in 𝜃-direction
𝐮̈:	Acceleration vector
𝐮̈𝐠:	Ground acceleration vector
𝑢̈𝑔𝑦:	Ground acceleration in 𝑦-direction
𝑢̈𝑚𝑓:	Flexible edge acceleration measured by the accelerometer
𝑢̈𝑚𝑠:	Stiff edge acceleration measured by the accelerometer
𝐯:	Measurement noise vector
𝑣𝑖:	Applied voltage sent to the current driver of 𝑖th damper
𝑉max:	Maximum applied voltage
𝑉𝑦:	Base shear
𝑊:	Weight of deck
𝑥0:	Initial displacement of spring, 𝑘1
𝑥𝑑𝑖:	Displacement of 𝑖th MR damper
𝑥𝑖 and 𝑦𝑖∶	𝑥-coordinate and 𝑦-coordinate distances of 𝑖th element from CM
𝑥̇𝑑𝑖:	Velocity of 𝑖th MR damper
𝑦𝑑𝑖 and 𝑦̇𝑑𝑖:	Displacement and velocity at middle plate of 𝑖th MR damper model
𝐲𝑚:	Vector measured responses
𝑧𝑖:	Evolutionary variable
𝐳:	State vector
𝐳̂:	Estimated state vector
𝛼𝑎 and 𝛼𝑏:	Parameters for MR damper model
𝛼𝑖:	Parameter for 𝑖th MR damper model
𝛽𝑚 and 𝛾𝑚:	Parameters for MR damper model
𝛾𝑔:	Parameter for statistically independent Gaussian white noise processes
𝜂:	Parameter for MR damper model
Γ:	Influence coefficient vector
Δ𝑡:	Time interval
Λ:	Location matrix for control forces
𝜔𝑦:	Uncoupled lateral frequency of the system
𝜔𝜃:	Uncoupled torsional frequency of the system
Ω𝜃:	Ratio between uncoupled torsional to lateral frequency.