Wind and Seismic Response Control of Dynamically Similar Adjacent Buildings Connected Using Magneto-Rheological Dampers

Abstract

Wind and/or earthquake-imposed loadings on two dynamically similar adjacent buildings cause vigorous shaking that can be mitigated using energy dissipating devices. Here, the vibration response control in such adjacent structures interconnected with semi-active magneto-rheological (MR) dampers is studied, which could also be used as a retrofitting measure in existing structures apart from employing them in new constructions. The semi-active nature of the MR damper is modeled using the popular Lyapunov control algorithm owing to its least computational efforts among the other considered control algorithms. The semi-active performance of the MR damper is compared with its two passive states, e.g., passive-off and passive-on, in which voltage applied to the damper is kept constant throughout the occurrence of a hazard, to establish its effectiveness even during the probable electric power failure during the wind or seismic hazards. The performance of the MR damper, in terms of structural response reduction, is compared with other popular energy dissipating devices, such as viscous and friction dampers. Four damper arrangements have been considered to arrive at the most effective configuration for interconnecting the two adjoining structures. Structural responses are recorded in terms of storey displacement, storey acceleration, and storey shear forces. Coupling the two adjacent dynamically similar buildings results in over a 50% reduction in the structural vibration against both wind and earthquake hazards, and this is achieved by not necessarily connecting all the floors of the structures with dampers. The comparative analysis indicates that the semi-active MR damper is more effective for response control than the other passive dampers.

1. Introduction

Extreme natural events, such as strong winds and earthquakes, are some of the greatest threats to socioeconomic life, causing severe damage worldwide, mainly to civil infrastructures [1]. The response of a structure to such extreme events depends upon its energy dissipating characteristics, i.e., altering the kinetic (structural vibrations) and potential or strain (inelastic deformations or damping) energies suitably. Employing external energy dissipating devices in the structure helps decrease structural vibration and damage incurred by dissipating a significant portion of the input energy in the so-designed specialized devices. These energy dissipating devices can be broadly classified into three categories, viz., passive devices, active devices, and semi-active devices. Passive devices such as viscous dampers, friction dampers, etc., control the structural responses by their natural virtue or inherent properties, whereas active systems control the structural responses based on the feedback from the structure and the load incident on it. Semi-active devices combine the advantages of both passive and active devices in the sense that they can generate large resistive forces based on the feedback from the structure at a comparatively lower energy demand, which a battery source can meet, and in the absence of a feedback system (in the case of probable power failure during the hazard), they are still able to control the dynamic response through their passive characteristics. One example of such a device is a magneto-rheological (MR) damper [2,3].

Vibration control by connecting two adjacent structures was first proposed by Klein et al. [4]. Westermo [5] suggested using hinged links to connect two neighbouring floors if the floors of the adjacent buildings are in alignment. This method not only helps to decrease the structural response but also decreases the risk of pounding between the structures, which was observed in some past earthquakes [6,7,8,9,10]. The connection between the two adjacent similar or dissimilar structures can be established with the use of viscous dampers [11,12,13,14,15,16,17], viscoelastic dampers [18,19], friction dampers [20,21,22,23], mass dampers [24,25,26,27,28,29], liquid dampers [30], hysteretic dampers [31,32,33], MR dampers [34,35,36,37,38,39], hybrid devices [40,41,42], etc. These techniques can even be used for existing structures as a retrofitting measure [43]. However, investigating dynamically similar adjacent structures connected with semi-active MR dampers and subjected to multiple hazards, e.g., wind and an earthquake, has not yet been considered. The present study aims to bridge this gap. Thus, the objectives of the present study are: (i) to study wind- or earthquake-induced vibration response reduction in the two adjacent buildings having similar dynamic properties connected at different floors by the MR dampers; (ii) to determine effective placement (configuration/arrangement) of the MR dampers connecting the adjacent buildings of similar dynamic properties; and (iii) to compare the effectiveness in response reduction achieved by introducing the MR damper and other passive dampers, such as viscous dampers and friction dampers when acted upon by wind or seismic loadings.

This study uses MR dampers as coupling devices between dynamically similar adjacent structures. Dampers are connected in cross alignments, as shown in Figure 1, to induce relative displacement and velocity between both ends. This structural system is subjected to the individual action of site-specific wind excitations and selected historical earthquakes to evaluate the structural response. The advantage of using dampers between the structures is that it reduces the possibility of pounding between the adjacent structures when subjected to a non-uniform action such as wind loading. In addition, it uses the free space between the structures that otherwise is not fully utilized. Therefore, this is a viable technology for structural vibration control in dynamically similar buildings. The effectiveness of the MR damper is obtained for its semi-active state and both passive states, passive-off (V = 0 V) and passive-on (V = V_max). Four damper arrangements are considered to check the possibility of reducing the number of dampers while not compromising the degree of response reduction. Furthermore, the dynamic response reduction obtained using the MR dampers as coupling devices is compared with that obtained using other popular passive devices—viscous and friction dampers.

2. Numerical Modeling of Coupled Buildings

Certain assumptions are made to meet the objectives of the present study, and they are stated below:

The building frames are idealized as multi degree-of-freedom (MDOF) shear buildings with lateral degrees of freedom at floor levels. The mass of both structural and non-structural elements is lumped at the joint nodes. Therefore, columns are treated as massless elements. This two-dimensional model, as shown in Figure 1, is subjected to unidirectional wind or earthquake excitation.

Soil-structure interaction (SSI) is not considered in the current study; hence, the base of the columns is rigid. Furthermore, the beam-slab assembly at each floor level provides a rigid diaphragm action to columns. As a result, both ends of each column are rigid, i.e., in fixed-end conditions.

The present study focuses on the effectiveness of dampers as coupling devices between structures. Therefore, it is assumed that the lateral resistance of the buildings is sufficiently large that it does not adversely affect the dampers’ performance. This assumption is in line with past studies on a similar topic wherein the damper–structure connection is assumed to be infinitely rigid. However, it is recommended that the nature of the damper–structure connection should be considered while designing this type of structural system in the field.

There is no time lag between the feedback system and the resultant damper force for the semi-active MR dampers.

Although the nonlinear force-deformation characteristics are considered for the external dampers, the structures remain within the linear elastic range during wind and earthquake excitation. This assumption is valid for the specialized design of the energy dissipation devices to ensure undamaged structures.

2.1. Formulation of the Equation of Motion of Coupled Buildings

Let the two structures with n floors have mass, damping, and stiffness values for each storey be $m_i$, $c_i$, and $k_i$, respectively. Thus, the coupled building has 2n degrees of freedom. The governing equation of motion under the independent action of wind and earthquake excitation can be written in matrix form, as shown in Equations (1) and (2), respectively.

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\}=\left\{P\right\}+\left[\Gamma \right]\left\{f\right\}$$

(1)

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\}=-\left[M\right]\left\{\Lambda \right\}{\ddot{x}}_g+\left[\Gamma \right]\left\{f\right\}$$

(2)

where, [M], [C], and [K] are mass, damping, and stiffness matrices of the coupled structures, respectively; {x} is the relative displacement vector with respect to the ground, which contains storey displacement values of Structure 1 followed by Structure 2; {P} is the wind force vector; [Γ] is the damper location matrix, of which each column contains +1 and −1 values at structural degrees of freedom to which a particular damper is connected and zeros at all other degrees of freedoms; {f} is the damper force vector; {Λ} is an influence coefficient vector which contains all its matrix elements as unity since the spatial variation in the ground motion is neglected because the plan area of the structures is too small to observe effects of the variability in the excitation with the propagation of the ground motion; and ${\ddot{x}}_g$ is the earthquake ground acceleration. Mass, damping, and stiffness matrices of connected structures, as shown in Figure 1, can be obtained using Equations (3)–(5), respectively.

$$\underset{\left(2n,2n\right)}{\left[M\right]}=\left[\begin{array}{cc}\underset{\left(n,n\right)}{\left[M_1\right]}& \underset{\left(n,n\right)}{\left[O\right]}\\ \underset{\left(n,n\right)}{\left[O\right]}& \underset{\left(n,n\right)}{\left[M_2\right]}\end{array}\right]$$

(3)

$$\underset{\left(2n,2n\right)}{\left[C\right]}=\left[\begin{array}{cc}\underset{\left(n,n\right)}{\left[C_1\right]}& \underset{\left(n,n\right)}{\left[O\right]}\\ \underset{\left(n,n\right)}{\left[O\right]}& \underset{\left(n,n\right)}{\left[C_2\right]}\end{array}\right]$$

(4)

$$\underset{\left(2n,2n\right)}{\left[K\right]}=\left[\begin{array}{cc}\underset{\left(n,n\right)}{\left[K_1\right]}& \underset{\left(n,n\right)}{\left[O\right]}\\ \underset{\left(n,n\right)}{\left[O\right]}& \underset{\left(n,n\right)}{\left[K_2\right]}\end{array}\right]$$

(5)

where, [M₁] and [M₂] are individual mass matrices of Structure 1 and Structure 2, respectively, which are defined in Equation (6); [K₁] and [K₂] are individual stiffness matrices of Structure 1 and Structure 2, respectively, which are defined in Equation (7); [C₁] and [C₂] are individual damping matrices of Structure 1 and Structure 2, respectively; and [O] is the null matrix.

$$\underset{\left(n,n\right)}{M_i}=\left[\begin{array}{ccccc}{m}_{1, i}& 0& 0& \cdots & 0\\ 0& m_{2, i}& 0& \cdots & 0\\ 0& 0& m_{3, i}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & m_{n, i}\end{array}\right]\hspace{1em}\forall \hspace{1em}i=1,2$$

(6)

$$\underset{\left(n,n\right)}{K_i}=\left[\begin{array}{ccccc}{k}_{1, i}+k_{2, i}& -k_{2, i}& 0& \cdots & 0\\ -k_{2, i}& k_{2, i}+k_{3, i}& -k_{3, i}& \cdots & 0\\ 0& -k_{3, i}& k_{3, i}+k_{4, i}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & k_{n, i}\end{array}\right]\hspace{1em}\forall \hspace{1em}i=1,2$$

(7)

where, m_*,i and k_*,i, respectively, denote the lumped mass and stiffness of the beam elements in structure i, used in modeling coupled buildings. The structural damping matrix is evaluated using the classical Rayleigh damping relation [44] for mass and stiffness proportional damping as given in Equation (8), as:

$$C_i=a_0{M}_i+a_1{K}_i\hspace{1em}\forall \hspace{1em}i=1,2$$

(8)

in which, coefficients for mass (a₀) and stiffness (a₁) are derived using relations given in Equations (9) and (10), respectively.

$$a_0=\frac{2\xi {\omega }_1{\omega }_c}{{\omega }_1+{\omega }_c}$$

(9)

$$a_1=\frac{2\xi }{{\omega }_1+{\omega }_c}$$

(10)

where, ξ is the damping ratio, assumed to be 5% for the reinforced concrete (RC) structure; ω₁ and ω_c correspond to natural frequencies of the structure in mode 1 and cut-off mode, respectively. The third mode is assumed to be the cut-off mode for calculating these coefficients.

The dynamic similarity between two adjacent buildings can be represented mathematically as [M₁] = [M₂] and [K₁] = [K₂]. Applying these conditions in Equation (8) results in [C₁] = [C₂]. In other words, dynamically similar buildings have the same natural frequencies and mode shapes.

2.2. Mathematical Models for Dampers

Force exerted by a linear viscous damper is directly proportional to the relative velocity between its ends, v, and the coefficient of proportionality is its damping coefficient, $c_d$. Mathematically, it can be described as in Equation (11) [14].

$$f=c_{d}v$$

(11)

The force exerted by a friction damper can be calculated using the hysteretic model proposed by Constantinou et al. [45]. The frictional forces mobilized in the damper are calculated using the relation given in Equation (12) as,

$$f=f_s{Z}_f$$

(12)

where, f_s is a slip force, and Z_f is a non-dimensional hysteretic component satisfying the nonlinear first-order differential equation, given in Equation (13) as,

$$q_f{\dot{Z}}_f=A_{f}v-{\beta }_f\left|v\right|Z_f{\left|Z_f\right|}^{n_f-1}-{\tau }_{f}v{\left|Z_f\right|}^{n_f}$$

(13)

where, q_f is the yield displacement of the friction damper, and A_f, β_f, n_f, and τ_f are the non-dimensional parameters of the hysteretic loop, which are responsible for the shape of the force deformation curve of the damper. Values of these parameters used in the current study are adopted as q_f = 0.1 mm; A_f = 1; β_f = 0.5; τ_f = 0.5; and n_f = 2 [46].

In this study, the popular 14-parameter modified Bouc–Wen model proposed by Spencer et al. [47] is used to model the nonlinear behavior of the magneto-rheological (MR) damper. Thus, the force exerted by the MR damper is calculated using Equation (14) as,

$$f=\alpha Z_m+c_0\left(v-{\dot{y}}_m\right)+k_0\left(x_d-y_m\right)+k_1\left(x_d-x_0\right)$$

(14)

where, Z_m is an evolutionary variable that is calculated using Equations (15) and (16) as,

$${\dot{Z}}_m=-{\gamma }_m\left|v-{\dot{y}}_m\right|Z_m{\left|Z_m\right|}^{n_m-1}-{\beta }_m\left(v-{\dot{y}}_m\right){\left|Z_m\right|}^{n_m}+A_m\left(v-{\dot{y}}_m\right)$$

(15)

$${\dot{y}}_m=\frac{1}{\left(c_0+c_1\right)}\left\{\alpha Z_m+c_{0}v+k_0\left(x_d-y_m\right)\right\}$$

(16)

where, x_d is the relative displacement between the damper ends; y_m is the pseudo-displacement of the damper; k₁ is the stiffness of the accumulator; c₀ is the viscous damping observed at large velocities; c₁ is the viscous damping for force roll-off at low velocities; k₀ is the controlling stiffness at large velocities; x₀ is the initial displacement in spring k₁ associated with the nominal damper force due to accumulator; and γ_m, n_m, β_m, and A_m are the shape control parameters [48]. Voltage dependency of the force exerted by the MR damper is expressed in terms of model parameters c₀, c₁, and α as given in Equations (17)–(19), respectively.

$$c_0=c_{0a}+c_{0b}{u}_m$$

(17)

$$c_1=c_{1a}+c_{1b}{u}_m$$

(18)

$$\alpha ={\alpha }_a+{\alpha }_b{u}_m$$

(19)

where, u_m is the output of the first-order filter, which is determined by solving the first-order differential equation given in Equation (20) as:

$${\dot{u}}_m=-{\eta }_m\left(u_m-V\right)$$

(20)

where, V is the voltage supplied to the MR damper. The corresponding MR damper parameters adopted in the current study are: α_a = 140 N.cm⁻¹; α_b = 695 N.cm⁻¹.V⁻¹; c_0a = 21.0 N.s.cm⁻¹; c_0b = 3.50 N.s.cm⁻¹.V⁻¹; c_1a = 283 N.s.cm⁻¹; c_1b = 2.95 N.s.cm⁻¹.V⁻¹; η_m = 190 s⁻¹; k₀ = 46.9 N.cm⁻¹; k₁ = 5.0 N.cm⁻¹; x₀ = 14.3 cm; γ_m = 363 cm⁻²; β_m = 363 cm⁻²; A_m = 301; n_m = 2 [47].

2.3. Semi-Active Control Algorithm for MR Dampers

As discussed earlier, the MR dampers are semi-active devices, i.e., their dynamic properties can be varied with an application of voltage through a battery source. One challenge in using the semi-active nature of the MR dampers is the choice of an appropriate nonlinear control algorithm, which depends on the available feedback measurements from the structures. The performance of three popular control algorithms, viz., Lyapunov control algorithm (LCA) [49], decentralized bang-bang control algorithm (DBBCA) [50], and clipped optimal control algorithm (COCA) [48], is compared in terms of the computation efforts. The flowcharts to model the LCA, DBBCA, and COCA are presented in Figure 2, Figure 3 and Figure 4, respectively. All the algorithms are tested on the three-storey structure [48]. The structural response obtained by applying different control algorithms and using the two passive states of the MR dampers is presented in

Table 1. Peak responses in a 3-storey structure [48] when subjected to a time-scaled 1940 Imperial Valley earthquake under different control strategies.

Response Quantity	Storey	Uncontrolled	LCA	DBBCA	COCA	Passive-Off	Passive-On
Storey displacement (cm)	1	0.55	0.16	0.14	0.19	0.23	0.08
	2	0.84	0.25	0.25	0.31	0.37	0.18
	3	0.97	0.30	0.31	0.39	0.47	0.29
Inter-storey drift (cm)	1	0.55	0.16	0.14	0.19	0.23	0.08
	2	0.32	0.11	0.11	0.13	0.15	0.16
	3	0.20	0.09	0.07	0.10	0.10	0.11
Storey acceleration (cm/s2)	1	880.42	398.57	504.23	685.10	419.96	283.93
	2	1069.38	580.24	470.77	604.60	489.10	501.11
	3	1411.84	608.59	662.15	669.65	725.26	771.57
f (N)	-	-	859.42	944.86	615.78	330.77	1054.00

. The results of the semi-active control strategies are comparable in terms of peak storey displacement, peak storey acceleration, and peak inter-storey drift. The time taken by the Lyapunov control algorithm is the least among the three algorithms owing to the absence of complex numerical functions such as Laplace and inverse Laplace transformations which are otherwise present in other control algorithms. Therefore, this algorithm is adopted to model the semi-active nature of the MR damper in this study.

2.4. Solution Procedure

The voltage supplied to the MR damper is governed by the Lyapunov control algorithm, assuming that Q_p is a unit matrix [34]. The resultant damper force is obtained using relations given in Equations (14)–(20) using a finite difference scheme. The finite difference scheme required a pretty small time step (dt) value for attaining the MR damper’s output stability and thereby increased the computation efforts in the structure’s dynamic wind and earthquake analysis. The solution to the governing equation of motion (Equations (1) and (2)) is obtained using the state-space formulation [51].

3. Numerical Study

3.1. Seismic Hazard Scenario

A seismic hazard is a function of peak ground acceleration (PGA), duration, and frequency content. Each ground motion has unique characteristics in terms of these controlling parameters. Hence, specific cases of well-known historical earthquakes are considered in the present study and are specified in

Table 2. General characteristics of the seismic events considered in the current study.

Sl no.	Earthquake	Event Date	Recording Station	Component	PGA (g)
1.	Imperial Valley	18 May 1940	El Centro	NS	0.35
2.	Loma Prieta	18 October 1989	LGPC	00	0.57
3.	Northridge	17 January 1994	Sylmar	360	0.84
4.	Kobe	17 January 1995	JMA	NS	0.83

. The time history data for each earthquake are obtained from the Pacific Earthquake Engineering Research (PEER) strong motion database. Each earthquake’s acceleration and displacement response spectrum is obtained for the damping ratio of 5% and is presented in Figure 5.

3.2. Wind Hazard

While earthquakes last for a short duration, like a few minutes, the cyclonic wind tends to have a longer duration to the order of 30–60 min [52,53,54]. The wind force acting at a location is a function of wind velocity at that site and the drag coefficient. In dynamic wind analysis, the fluctuant wind speed component is obtained from the peak gust velocity, U_3s-10 (3 s gust velocity at a 10 m height from the ground), defined by the design standards. The site-specific fluctuating wind velocity component is obtained in NatHaz online wind simulator [55] using a discrete frequency function with Cholesky decomposition and fast Fourier transform (FFT). The corresponding wind force acting on the structure can be determined using the expression in Equation (21).

$$P=\frac{1}{2}\rho C_d{A}_b{\left[U\left(t\right)+u\left(t\right)\right]}^2$$

(21)

where, ρ is the air density; C_d is the air drag coefficient; A_b is the bay area perpendicular to the wind velocity profile; U(t) is the static or mean wind speed component, and u(t) is the fluctuating wind velocity component. For the current study, four wind load cases are obtained considering the variation in peak gust velocity [56] and exposure category [57] Details on these load cases are given in

Table 3. General characteristics of site-specific wind load cases considered in the current study.

Sl no.	Wind Load Case	Location	U3s-10 (m/s)	Exposure Category
1.	Case 1	Bengaluru	33	C
2.	Case 2	Mumbai	44	B
3.	Case 3	Delhi	47	C
4.	Case 4	Kolkata	50	B

.

The cut-off frequency is kept equal to the fundamental natural frequency of the structure under consideration to obtain the maximum possible response. The air drag coefficient for an isolated rectangular-shaped building is 1.2 [54]. Previous studies suggest that it needs to be modified for closely spaced structures. The building on the upwind side experiences negative pressure due to the small gap between the adjacent structures. In contrast, the building on the downwind side experiences fluctuating pressure due to the eddies shed. Sakamoto and Haniu [58] and Lam et al. [59] performed wind tunnel tests on closely spaced rectangular structures to study variations in the air drag coefficient with spacing between the structures. Thus, the air drag coefficients for buildings on the upwind and downwind sides are evaluated at 1.2 and −0.2, respectively. The negative sign of the air drag coefficient indicates that the direction of wind pressure is opposite to the wind velocity. This further increases the possibility of pounding between two closely spaced structures. For the current study, values of ρ and A_b are considered 1.225 kg·m⁻³ and 9 m², respectively. Top storey velocity time history of the wind load case 1, along with its fast Fourier transform (FFT) and autoregressive power spectral density (PSD) function obtained from Burg’s method for Building 1 (10 storey structure) and Building 2 (20 storey structure), are presented in Figure 6.

3.3. Characteristics of the Coupled Structures

The mass and stiffness of each storey of both buildings are uniform throughout the structure’s height. They are chosen to yield a fundamental time period of 1 s for a 10-storey structure (stiff building) and 2 s for a 20-storey structure (flexible building). The coupled structural arrangement involves connecting two adjacent buildings that have the same dynamic properties with dampers. In other words, the 10-storey structure is connected to an adjacent 10-storey structure, and the 20-storey structure is connected to an adjacent 20-storey structure. These structural arrangements in this study are called coupled stiff and flexible buildings, respectively. Four damper arrangements, referred to as a damper configuration, are considered in this study. They are presented in Figure 1 (Configuration 1), Figure 7a (Configuration 2), Figure 7b (Configuration 3), and Figure 7c (Configuration 4) [14,60]. Different damper configurations are selected to determine each hazard’s most effective damper placement. The structures interconnected with the dampers under different configurations are subjected to wind and earthquake excitations separately. The responses are measured in storey displacement, storey acceleration, and storey shear.

4. Results and Discussion

4.1. Response of Coupled Similar Structures to Earthquake Excitation

Time history responses of the coupled stiff and flexible structures in terms of top floor displacement, top floor acceleration, and normalized base shear when they are subjected to earthquake ground excitation, as discussed in Section 3.1, are presented in Figure 8, Figure 9 and Figure 10, respectively. The responses obtained from the semi-active control strategy of the MR damper, as discussed in Section 2.4, are compared with the uncontrolled responses (without the presence of any dampers between the structures) of the structures. It can be inferred that the MR dampers, as coupling devices in adjacent similar buildings, are quite effective for dynamic response reduction. Figure 11, Figure 12 and Figure 13 compare response reduction characteristics of the MR damper in its different control states. The passive-on state of the MR damper turned out to be the most effective for response reduction in peak storey displacement and peak storey shear forces for both the cases of coupled structures; however, more damper forces tend to increase storey acceleration in lower floors which is observed from Figure 12 and therefore, it is inferred that the semi-active control strategy provided the most optimum results in the coupled structures. Figure 14, Figure 15 and Figure 16 depict a comparison of response quantities obtained in different damper configurations considered in this study. From the figures, it is quite certain that the MR dampers are effective in any of the configurations considered compared to the uncontrolled structure state. Configuration 1 has shown the maximum response reduction among all the damper arrangements considered in this study. This result is quite understandable as this configuration contained twice the number of MR dampers compared to all other damper configurations. However, it is worth noting that Configuration 3 (dampers connected in the lower half region) has output response quantities quite comparable to those obtained from Configuration 1. Therefore, this configuration can be termed the most effective damper configuration for adjacent similar coupled buildings under earthquakes.

It was observed, in the previous studies, that there exists an optimum value of the control parameter for passive dampers, e.g., damping value for a viscous damper, slip force for a friction damper, etc., for which maximum response reduction could be observed in a structure. Therefore, optimum controller values for viscous and friction dampers are evaluated for both the structures (a stiff building and a flexible building) separately, and the response quantities evaluated at these optimum controller values are compared with those obtained when the MR dampers are used as coupling devices. In all these cases, the damper arrangement, as presented in Configuration 1, is adopted for coupling the adjacent structures. A comparison of the responses is presented in Figure 17, Figure 18 and Figure 19. These plots show that the MR damper’s semi-active control strategy is more effective for response control in the coupled buildings compared to the viscous and friction dampers under the action of an earthquake. Response quantities obtained for the coupled stiff and flexible structures under different control strategies discussed previously are summarized in

Table 4. Comparison of top storey peak displacement of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Displacement (cm)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	16.01	5.00	10.84	3.97	12.17	12.71
Loma Prieta, 1989	32.56	19.97	29.50	15.58	29.72	30.29
Northridge, 1994	27.20	17.77	22.73	15.53	24.15	25.09
Kobe, 1995	49.33	29.72	40.11	25.37	43.05	44.62

,

Table 5. Comparison of top storey peak acceleration of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Acceleration (g)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	0.94	0.22	0.48	0.23	0.70	0.73
Loma Prieta, 1989	1.82	0.74	1.09	0.53	1.46	1.50
Northridge, 1994	2.09	0.89	1.22	0.75	1.46	1.87
Kobe, 1995	2.86	1.56	1.84	1.07	2.24	2.49

,

Table 6. Comparison of peak normalized base shear of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Normalized Base Shear
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	0.46	0.17	0.36	0.16	0.37	0.38
Loma Prieta, 1989	0.92	0.62	0.85	0.52	0.83	0.85
Northridge, 1994	1.02	0.62	0.84	0.54	0.86	0.91
Kobe, 1995	1.18	0.71	0.97	0.62	1.07	1.08

,

Table 7. Comparison of top storey peak displacement of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Displacement (cm)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	22.95	11.63	18.77	11.12	20.92	21.65
Loma Prieta, 1989	83.85	58.32	75.12	49.21	78.07	79.33
Northridge, 1994	79.19	59.46	69.58	52.40	71.54	72.90
Kobe, 1995	59.43	34.61	47.05	28.76	50.82	54.15

,

Table 8. Comparison of top storey peak acceleration of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Acceleration (g)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	0.64	0.15	0.34	0.14	0.49	0.60
Loma Prieta, 1989	1.61	0.48	0.91	0.54	1.27	1.49
Northridge, 1994	1.58	0.61	0.96	0.55	1.17	1.32
Kobe, 1995	2.58	0.60	1.04	0.37	1.65	2.03

and

Table 9. Comparison of peak normalized base shear of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Ground Excitation	Normalized Base Shear
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Imperial Valley, 1940	0.20	0.10	0.16	0.09	0.16	0.17
Loma Prieta, 1989	0.56	0.37	0.45	0.34	0.43	0.48
Northridge, 1994	0.58	0.39	0.47	0.35	0.49	0.51
Kobe, 1995	0.39	0.23	0.33	0.19	0.31	0.35

. A wide range of response reduction (20% to 75%) is observed for the coupled structures with the MR dampers as coupling devices.

4.2. Response of Coupled Similar Structures under Wind Excitation

The MR dampers’ effectiveness as coupling devices between two adjacent dynamically similar buildings in response control under wind excitation is studied on similar lines to that in earthquake excitation. Figure 20, Figure 21 and Figure 22 represent the time history response of top storey displacement, top storey acceleration, and normalized base shear in uncoupled and coupled states of the stiff buildings, respectively. Similarly, Figure 23, Figure 24 and Figure 25 represent the time history response of top storey displacement, top storey acceleration, and normalized base shear in uncoupled and coupled states of the flexible buildings, respectively. Figure 20b and Figure 23b highlight the importance of installing dampers between the adjacent buildings as the direction of vibration is reversed, thereby reducing the possibility of pounding between the structures. From Figure 20b, Figure 21b, Figure 22b, Figure 23b, Figure 24b and Figure 25b, it can be inferred that the response of the building on the downwind side is negligible compared to those of the building on the upwind side. Therefore, it is decided that the responses of the building on the downwind side need not be presented henceforward. From the time history of structural responses of the building on the upwind side, it is evident that the semi-active control strategy of the MR damper is effective against wind excitation. However, due to the small time step in the analysis, as discussed in Section 3.3, structural analysis for long-duration winds poses memory issues in the system, which has a random access memory (RAM) of 32 Gb. The primary objective of the present study is to study the response reduction using dampers, and it is observed from Figure 20a, Figure 21a, Figure 22a, Figure 23a, Figure 24a and Figure 25a that the response reduction characteristics are similar through the entire wind load duration; therefore, to save on the computation efforts and simplifying the problem, long-duration winds are curtailed to short-duration, to the order of 25 s, and the results thus obtained are compared with those obtained from long-duration winds. In addition, the air drag coefficient for the building on the downwind side is taken as 0, and the additional effect of negative wind pressure is incorporated by increasing the air drag coefficient for the building on the upwind side to 1.4. This assumption is justifiable as the presence of dampers between the adjacent buildings makes the structural arrangement a single unit wherein the principle of superposition is applied as the structures are assumed to remain linear elastic. The responses thus obtained for the modified structural and loading arrangement are presented in Figure 26, Figure 27 and Figure 28. From these figures, it is worth noting that the responses of coupled stiff and coupled flexible buildings are within the 10% range to those obtained when structures are subjected to long-duration winds, proving the credibility of the analysis; however, there is an increase in structural responses when structures are not connected to each other which is attributed to the increase in the air drag coefficient, thereby increase in the incident wind force. Thus, the response reduction characteristics are exaggerated by approximately 10%, which is adjusted while writing the conclusions of the present study.

Response quantities obtained through various control states of the MR damper (semi-active, passive-off, and passive-on) against wind excitation were observed to be very close to each other and, therefore, are not presented herein. This behavior resulted from small relative displacement and velocity values generated between both ends of the MR dampers so that the damper forces are in the same range in all three states. Figure 29, Figure 30 and Figure 31 depict a comparison of response quantities obtained in different MR damper configurations considered in this study. It is to be noted here that the MR dampers are effective in controlling the dynamic responses in any of the configurations considered in this study, compared to the uncontrolled state of the structures. Damper Configuration 1 has shown the maximum reduction in responses compared to other configurations in wind excitation due to more dampers being used. However, Configuration 2 (dampers connected in the upper half region) has output response quantities comparable to those obtained from Configuration 1. This observation is because larger values of wind forces are observed at the top of the building, thereby inducing greater values of relative displacement and velocity between the damper. Therefore, this configuration can be termed the most effective damper configuration for adjacent coupled similar buildings against wind excitation. The optimum control parameters are obtained for viscous and friction dampers for both the stiff and flexible structures and are used to study the dampers’ effectiveness against wind excitation. Comparison of the response quantities thus obtained are presented in Figure 32, Figure 33 and Figure 34. The figures show that a viscous damper is more effective than a friction damper in controlling the structural responses under wind excitation. In addition, response quantities obtained by connecting adjacent structures with viscous dampers are comparable to those obtained using the MR dampers as coupling devices. Response quantities obtained for both cases of coupled structures under different control strategies are summarized in

Table 10. Comparison of top storey peak displacement of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Displacement (cm)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	3.21	1.69	1.73	1.61	2.06	2.46
Case 2	3.28	1.79	1.79	1.79	2.26	2.43
Case 3	7.48	4.68	4.68	3.79	4.16	5.33
Case 4	4.58	2.79	2.88	2.55	3.58	3.81

,

Table 11. Comparison of top storey peak acceleration of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Acceleration (g)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	0.20	0.15	0.15	0.15	0.17	0.18
Case 2	0.23	0.20	0.20	0.20	0.22	0.23
Case 3	0.43	0.33	0.33	0.30	0.31	0.40
Case 4	0.31	0.25	0.25	0.24	0.29	0.31

,

Table 12. Comparison of peak normalized base shear of the coupled stiff building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Normalized Base Shear
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	0.10	0.05	0.05	0.05	0.06	0.07
Case 2	0.08	0.05	0.05	0.05	0.07	0.07
Case 3	0.24	0.15	0.15	0.12	0.13	0.16
Case 4	0.14	0.08	0.08	0.07	0.10	0.10

,

Table 13. Comparison of top storey peak displacement of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Displacement (cm)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	10.22	5.37	5.38	5.36	7.35	7.78
Case 2	17.71	9.39	9.39	8.80	9.25	11.85
Case 3	25.68	16.82	16.91	16.60	19.73	22.85
Case 4	20.38	13.20	13.28	13.10	15.88	17.74

,

Table 14. Comparison of top storey peak acceleration of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Acceleration (g)
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	0.17	0.13	0.13	0.13	0.15	0.16
Case 2	0.26	0.17	0.17	0.17	0.19	0.25
Case 3	0.43	0.34	0.34	0.34	0.37	0.41
Case 4	0.35	0.30	0.30	0.30	0.33	0.35

,

Table 15. Comparison of peak normalized base shear of the coupled flexible building under different control strategies when dampers are arranged in Configuration 1.

Wind Load	Normalized Base Shear
	Unconnected	Semi-Active MR	Passive-Off	Passive-On	Viscous	Friction
Case 1	0.07	0.04	0.04	0.04	0.05	0.06
Case 2	0.12	0.06	0.06	0.06	0.06	0.08
Case 3	0.19	0.12	0.12	0.12	0.15	0.16
Case 4	0.14	0.09	0.09	0.09	0.11	0.12

. A wide range of response reductions (15 to 50%) is observed for the coupled structures with the MR dampers as coupling devices.

5. Conclusions

The effectiveness of magneto-rheological (MR) dampers, when used to couple two adjacent dynamically similar structures, is evaluated against multiple independent hazards, i.e., wind and earthquake hazards. It is observed that the technique of coupling two structures by an energy dissipating device, such as the MR damper, is quite effective for controlling structural vibrations under the two distinct dynamic excitations. There is a significant reduction observed in the dynamic response quantities, i.e., storey displacement, acceleration, and shear force, in both the cases of connected structures up to the order of 75% in earthquake excitation and 40% (adjusted value due to the assumptions in structural analysis) in wind excitation. Based on the results obtained from the numerical study, the following conclusions are drawn:

• The coupled building control using the MR damper is an effective response control technique for adjacent dynamically similar stiff and flexible structures. The only requirement of the coupling of both the structures is that the dampers are to be connected in cross alignment rather than to inline floors to develop relative displacement and velocity between both ends of a damper.
• Significant dynamic response reduction in terms of peak storey responses is observed in both stiff and flexible structures. The order of reduction goes up to 75% for earthquake excitation and 40% for wind excitation, which is significant compared to other control strategies studied previously.
• The other two control states of the MR dampers, i.e., passive-off and passive-on, are equally effective for structural response control compared to the semi-active control state. It indicates that the MR damper proves to be effective in structural response control, even in probable power failure during one of the hazards.
• Connecting all the floors with the MR dampers is always effective against wind and earthquake excitations. However, it is also possible to obtain a similar degree of response control after reducing the number of dampers by effectively placing them in an appropriate configuration. In the current study, connecting only the bottom half of floors with the MR dampers provided approximately similar response control against earthquake excitation as what would have been obtained by connecting all the floors with dampers (using twice the number of dampers). Similarly, connecting only the top half of floors by the MR dampers provided approximately similar response control for wind excitation.
• A semi-active MR damper, as a coupling device between two adjacent dynamically similar structures, has exhibited increased effectiveness in terms of structural vibration control when compared with other passive energy dissipating devices such as viscous dampers and friction dampers in earthquake as well as wind excitations.