Stiffness controllable isolation system for near-fault seismic isolation

Engineering Structures 30 (2008) 747–765 www.elsevier.com/locate/engstruct

Stiffness controllable isolation system for near-fault seismic isolation Lyan-Ywan Lu a,∗ , Ging-Long Lin b , Tzu-Ching Kuo a a Department of Construction Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan b Graduate Institute of Engineering and Technology, National Kaohsiung First University of Science and Technology, Kaohsiung 824, Taiwan

Received 18 September 2006; received in revised form 17 May 2007; accepted 23 May 2007 Available online 2 July 2007

Abstract A seismic structure isolated by a conventional passive isolation system is usually a long-period structural system with a fixed fundamental vibration frequency. Even though conventional isolation systems may effectively mitigate the dynamic responses of structures in a regular earthquake, they may also encounter a low-frequency resonance problem when subjected to a near-fault earthquake that usually has a long-period pulse-like waveform. This long-period wave component may result in an enlargement of the base displacement as well as decrease isolation efficiency. To overcome this problem, a sliding base isolation system with controllable stiffness is proposed in this study. By varying the stiffness of the isolation system, the restoring force provided by the system can be controlled by a proposed semi-active control method that is developed based on active feedback control. The result of numerical simulation in this paper has shown that the proposed system is able to effectively mitigate the effect of low-frequency resonance induced by a near-fault earthquake. As a result, the base displacement and super-structure acceleration of the isolated structure can be reduced simultaneously, which is a major improvement over the conventional passive isolation system. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Controllable stiffness; Semi-active control; Seismic isolation; Near-fault earthquake; Sliding isolation

1. Introduction Seismic isolation was developed decades ago, and has been proven to be a practical and effective technology for seismic protection of structural systems or equipment [1–3]. A typical seismic isolation system usually consists of a flexible mount (or a sliding interface) to reduce the transmission of the ground motion to the superstructure. It also has a resilient mechanism to reduce the residual base displacement. Because of the resilient mechanism, an isolation system usually possesses a fixed isolation frequency. Recent studies have shown that when a conventional isolation system with a constant frequency is subjected to a near-fault earthquake, which usually possesses a long-period pulse-like waveform, the system may suffer from low-frequency resonance that can cause considerable amplification of the isolator displacement thereby endangering the isolated structure [4,5]. To overcome the resonance problem associated with earthquakes with near-fault characteristics, some researchers ∗ Corresponding author. Tel.: +886 7 6011000x2127; fax: +886 7 6011017.

E-mail address: [email protected] (L.-Y. Lu). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.05.022

proposed sliding isolators with variable curvature [6,7], where the isolation frequency is a function of the isolator displacement rather than a constant, so that the resonant behaviour can be attenuated. Other researchers suggested adding supplementary passive damping in the isolation system [8,9], in order to suppress the large isolator displacement induced by a nearfault earthquake. These above mentioned research results all have one thing in common, i.e. they all suggested using a passive device to solve the problem. Once designed, the parameters of a passive device cannot be adjusted online in response to seismic excitations whose characteristics may not be predictable. Consequently, the passive system may not perform well when it is subjected to a seismic load that is significantly different from the design load. In order to improve the seismic performance of isolation systems, some researchers investigated the possibility of using an active isolation system (also called hybrid control system), which combines a passive isolation system with an active device [10–12]. The active device provides an active control force to the isolation system. Due to its adaptive nature, an active isolation system is generally able to considerably improve the seismic performance of an isolation system

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subjected to earthquakes with a wide range of characteristics, provided that a proper control law is employed. However, an active control system usually requires a larger amount of control energy or control force. This is especially the case when one considers that the structure to be isolated may have a very large mass. This hinders and limits the application of active isolation systems. In addition, an active control system may also have a control stability problem if an improper control method is adopted or if a malfunction of the control system occurs. In order to control a seismic isolation system using less control energy; yet at the same preserving the adaptability and controllability of an active system, some researchers proposed the concept of a semiactive isolation system. A semiactive isolation system generally consists of a set of passive isolators and a certain type of semiactive control device, such as the MR damper [13,14], variable friction device [15,16], tuned interaction damper [17], variable stiffness device [18–21], etc. The function of the semiactive device, which is usually installed under the base of the structure, is to attenuate the seismic motion of the isolation system according to the current system response or excitation, so that the response of the superstructure can be mitigated. A semiactive isolation system may also be called a smart isolation system. The Technical Committee on Structural Control of ASCE (American Society of Civil Engineering) has even developed a benchmark problem for the study of smart base-isolated buildings [14,22] in order to facilitate the comparison between different isolation systems and control strategies. The internal parameters of a semiactive device can be adjusted online by a controller. Therefore, like an active control system, a semiactive system usually requires sensor measurements and a proper control law [23]. Nevertheless, a semiactive device is basically a “variable” passive one, and it has the following features that are very different from an active system. (1) The control force of a semiactive device, which is exerted by the relative motion between the device and the structure, is a passive (resistance) force. In other words, the direction and the magnitude of the semiactive control force depend on those of the relative motion of the system, which makes that the semiactive force is not fully controllable. The direction of the control force is always opposite to the direction of the relative motion of the system. (2) Controlling the internal parameters of a semiactive device generally requires much less control energy, as compared to that of an active device. (3) A semiactive device basically generates a passive force, so it will not pump energy into the controlled structures. As a result, the effect of control spillover and control instability can be avoided. Based on the above reasons, an isolation system involving a semiactive device may be a promising protection system. The purpose of this study was to investigate the possibility of using a semiactive system, referred to as an “stiffness controllable isolation system”, (SCIS) in order to improve the effectiveness of seismic isolation systems subjected to earthquakes with near-fault characteristics. By controlling the stiffness of the SCIS, the restoring force (a semiactive control force) provided by the system can be varied. Nevertheless, since the restoring force of the SCIS is equal to the product

Fig. 1. Schematic diagram of a structure isolated by the SCIS system.

of the base displacement and the present value of the isolation stiffness that is always positive, the direction of the restoring force must be the same as that of the base displacement. In order to release this constraint (in the control derivation of this paper) the restoring stiffness of the SCIS is first divided into two parts, the static and the dynamic parts, so that the dynamic stiffness which represents the variation of the stiffness can be either positive or negative. The restoring force provided by the dynamic stiffness is then controlled according to a proposed semiactive control law that is modified from an active control algorithm. A numerical method able to simulate the dynamic response of the SCIS while considering the sliding friction, will also be presented in this paper. Through numerical simulation, both the near-fault and far-field seismic performance of the SCIS will be compared with their passive and active isolation counterparts. 2. Formulation for a structure isolated by SCIS A structural system, isolated by the SCIS system, is shown schematically in Fig. 1. To simplify the problem, the superstructure in the figure is modelled as a single-degreeof–freedom (SDOF) system, so that the study can focus on the performance of the SCIS system itself. As shown in Fig. 1, a SCIS consists of a set of pure sliding isolators and a stiffness controllable device. The stiffness device provides the isolation system with a restoring force and gives it the ability to recentre. For the convenience of deriving the dynamic equation, Fig. 1 is further transformed into a mathematical model, as shown in Fig. 2, where the sliding isolators and the stiffness control device are, respectively, modelled by a friction element with a friction coefficient of µ and a nonlinear spring with a stiffness of kr (t). In Fig. 2, symbols xs and xb denote the relativeto-ground displacements of the superstructure and the base, respectively; m s and m b are their mass; ks and cs represent the stiffness and damping of the superstructure. In Fig. 2, kr (t) is a controllable parameter, and therefore a time-variant quantity. It is this quantity that makes the behaviour and the analysis method of a SCIS different from those of a conventional sliding isolation system. From Fig. 2, the equation of dynamics for a SCIS isolated structure subjected to a seismic load may be expressed as M¨x(t) + C˙x(t) + Kx(t) + B0 (u r (t) + u f (t)) = −ME0 w(t)

(1)

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Fig. 2. Mathematical model for a structure with SCIS.

where x(t) = {xs (t), xb (t)}T denotes the vector of the system displacements relative to the ground; M, C and K represent the mass, damping and stiffness matrices of the isolated system; u r (t) and u f (t) denote the restoring force of the nonlinear spring and the resistant force provided by the friction element, respectively; w(t) is a vector of ground accelerations; B0 and E0 denote the placement matrices for the isolation system and the excitation, respectively. Note that the matrix K does not include the isolator stiffness, therefore it is a singular stiffness matrix. Moreover, since the controllable restoring force u r (t) is produced by the stiffness kr (t), it may be written as u r (t) = kr (t)xb (t).

(2)

In the last equation, since physically the stiffness kr (t) can only be positive, the direction of the force u r (t) must be in the same direction of xb (t) and can not be freely determined. In order to release this restraint, let us rewrite kr (t) as kr (t) = kr 0 + 1kr (t) = kr 0 (1 + α(t))

(3)

where α(t) = 1kr (t)/kr 0 .

(4)

As illustrated in Fig. 3, Eq. (3) implies that the stiffness of the SCIS can be divided into two components, the constant component kr 0 and the time-variant component 1kr (t). The constant component represents a static stiffness, while the timevariant component can be treated as a dynamic stiffness. Unlike kr (t), 1kr (t) can be either positive or negative. The dynamic stiffness 1kr (t) will be the quantity to be controlled. Moreover, in this paper the parameter α(t) defined in Eq. (4), whose physical meaning is the ratio of the dynamic and static stiffness, is referred to as the stiffness ratio. Accordingly, by substituting kr (t) from Eq. (3) into Eq. (2), the restoring force u r (t) can also be divided into two parts, i.e. u r (t) = u r 0 (t) + 1u r (t)

(5)

where u r 0 (t) = kr 0 xb (t)

(6a)

1u r (t) = 1kr (t)xb (t) = α(t)kr 0 xb (t).

(6b)

In the above equations, u r 0 (t) and 1u r (t) denote the restoring forces provided by the static and dynamic stiffness components, respectively. Note that, unlike u r 0 (t), 1u r (t) is a controllable nonlinear force, since it is not linearly proportional to the base displacement xb (t). Moreover, 1u r (t) is denoted by an incremental symbol 1, since it is also an incremental force due to the incremental stiffness 1kr (t) (see Eq. (6b)).

Fig. 3. Static and dynamic restoring stiffness of SCIS.

Next, if we further substitute u r (t) from Eq. (5) in Eq. (1), and rearrange the equation, we obtain M¨x(t) + C˙x(t) + Kr 0 x(t) = −B0 (1u r (t) + u f (t)) − ME0 w(t)

(7)

where  Kr 0 =

ks −ks

 −ks . k s + kr 0

(8)

Unlike K in Eq. (1), Kr 0 contains kr 0 of SCIS and is a nonsingular constant matrix. Note that in Eq. (7) both nonlinear forces, 1u r (t) and u f (t), have been moved to the right-hand side of the dynamic equation for the convenience of deriving a numerical solution, which will be discussed in a later section. The dynamic equation shown in Eq. (7) can be expressed as a first order ordinary differential equation, i.e. z˙ (t) = Az(t) + B(1u r (t) + u f (t)) + Ew(t)

(9)

where    x˙ (t) −M−1 C −M−1 Kr 0 z(t) = , A= x(t) I 0     −E0 −M−1 B0 B= , E= . 0 0 

(10) (11)

Eq. (9) is similar to the standard state-space equation for an actively controlled structure [24], except that the control force term is replaced by the nonlinear forces 1u r (t) and u f (t). As per Eq. (6b), it is evident that 1u r (t) is actually a controllable force that can be varied by controlling the dynamic stiffness component 1kr (t) (or α(t)) of the SCIS. However, even though 1u r (t) has controllability, in the end it is a passive force induced by the base displacement xb (t). For this reason, in this paper, 1u r (t) is referred to as the semiactive control force of SCIS, to be distinguished from an active control force. In the next section, we will propose a semiactive control method for deciding 1u r (t). On the other hand, the friction force u f (t) in Eq. (9) is a purely passive and uncontrollable force. Because of the friction effect, the motion of a SCIS isolated structure has two possible states, stick and sliding. By Coulomb’s model of friction, the

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quantities u f (t) and x˙b (t) must satisfy the following kinetic and kinematic conditions, respectively, for different motion states: (1) for the stick state: u f (t) ≤ u f,max = µMg (12a) x˙b (t) = 0.

(12b)

(2) for the sliding state: u f (t) = u f,max

(13a)

x˙b (t) 6= 0

(13b)

where µ denotes the friction coefficient of the sliding bearings, M = m b +m s the total mass of the isolated structure, and u f,max denotes the maximum friction force.

Fig. 4. An actively controlled sliding isolation system.

3. Semiactive control law for determining the dynamic stiffness of SCIS Let us consider an actively controlled sliding isolation system as shown in Fig. 4. Assume that the control force delivered by the active device shown in Fig. 4 can be determined by an active control law with direct output feedback, where the control forces or control commands are obtained by directly multiplying the sensor outputs with a feedback gain, i.e. ˆ ˆ u(t) ˆ = Gy(t) = GDz(t)

(14)

ˆ represent the control force and feedback where u(t) ˆ and G gain obtained according to an active control law, respectively. Hereafter, a notation with an accent circumflex “ ˆ ” represents a target value decided by an active control. Also, in Eq. (14) y(t) and D denote the sensor measurement vector and sensor placement matrix, respectively. Next, equating the dynamic restoring force 1u r (t) of SCIS to the control force decided by the active control law equation (14), i.e. 1u r (t) = u(t). ˆ

(15)

Moreover, substituting 1u r (t) from Eq. (6b) into Eq. (15), one can solve for the “target” dynamic stiffness of the SCIS at certain time instant, i.e. 1kˆr (t) = u(t)/x ˆ b (t),

for xb (t) 6= 0.

(16)

By using Eqs. (16) and (14) in Eq. (4), one may further obtain a formula for determining the “target” stiffness ratio α(t) ˆ = G(t)y(t)

(17)

where G(t) represents a time-variant adaptive gain that is determined by ˆ G(t) = (kr 0 xb (t))−1 G,

for xb (t) 6= 0.

(18)

Note that the target stiffness ratio α(t) ˆ computed by Eq. (17) may not be achievable, since physically the variation of the stiffness 1kr (t) must be limited within a certain range. Accordingly, the actual stiffness ratio α(t) defined by Eq. (4)

Fig. 5. Block diagram for semiactive control of SCIS system.

must also be bounded by a given range. Let us assume that αmax and αmin are the upper and lower limits of the achievable stiffness ratio, and as a result the actual value of α(t) shall be determined by the following equation:  ˆ for αmin ≤ α(t) ˆ ≤ αmax α(t) ˆ α(t) = αmax for αmax < α(t) (19)  αmin for α(t) ˆ < αmin . Eqs. (17)–(19) form a semi-active control law for the SCIS system. After α(t) is decided by the above control law, it can be substituted back to Eq. (3), in order to obtain the actual stiffness kr (t). From the above derivation, it is known that α(t) is a very important control parameter for the proposed semiactive control law. Fig. 5 shows the control flowchart of the above semiactive control law. Note that, when xb (t) in Eq. (18) is very close to zero, α(t) ˆ in Eq. (17) may become very large and exceed αmax . In this case, since α(t) ˆ is bounded by Eq. (19), α(t) ˆ will be automatically set to αmax . 4. Method of numerical simulation Since a structure isolated by a SCIS system and controlled by the above proposed semiactive control law becomes a highly nonlinear system, a numerical method is generally required to analyse it. In this section, a numerical approach based on the discrete-time solution of the state space formulation will be proposed. First, assuming that the time interval (sampling period) 1t for the numerical analysis is small, the forcing term on the right-hand side of Eq. (9) can be treated as a constant within each time step, i.e. 1u r [τ ] ≈ 1u r [tk ] = constant u f [τ ] ≈ u f [tk ] = constant w[τ ] ≈ w[tk ] = constant,

for tk ≤ τ < tk+1

(20)

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where a quantity associated with the symbol [tk ] denotes that the quantity is evaluated at the k-th time step of the time instant tk = k1t. The assumption made in Eq. (20) is commonly adopted in the numerical solution of a nonlinear system. With the above assumption, and by the numerical method of step-bystep integration, the discrete-time solution of Eq. (9) may be written in an incremental form [24], i.e. z[tk+1 ] = Ad z[tk ] + Bd (1u r [tk ] + u f [tk ]) + Ed w[tk ]

(21)

where Ad = eA1t Bd = A−1 (Ad − I)B Ed = A

−1

(22)

(Ad − I)E.

Eq. (21) represents the discrete-time solution of the statespace equation of Eq. (9), and the constant coefficient matrices Ad , Bd and Ed can be treated as the discrete-time counterparts of the matrices A, B and E in Eq. (9). Eq. (21) states that response z[tk+1 ] of the (k + 1)-th time step can be computed based on the previous state z[tk ] of the system and the forcing terms w[tk ], 1u r [tk ], u f [tk ] which are all evaluated at the previous time step, i.e. the k-th step. Among the three forcing terms, 1u r [tk ] and u f [tk ] are interactively determined by the system state z[tk ], so their solutions can not be readily obtained and require a special treatment, which is illustrated by Fig. 6 and is explained below. (1) Determine the value of 1u r [tk ]: From Eq. (6b), we have 1u r [tk ] = α[tk ]kr 0 xb [tk ] = α[tk ]kr 0 Dd z[tk ]

(23)

where Dd = [0, 0, 0, 1] is a row matrix. In Eq. (23), since z[tk ] is a known vector at the current step, once the control parameter α[tk ] is computed, the term 1u r [tk ] is also determined. On the other hand, parameter α[tk ] has to be decided by the proposed control law as described by Eq. (19), in which the target value α[t ˆ k ] has to be determined first by Eqs. (17) and (18), i.e. ˆ k ]. α[t ˆ k ] = G[tk ]y[tk ] = (kr 0 xb [tk ])−1 Gy[t

(24)

After α[t ˆ k ] is obtained by Eq. (24), it is substituted into Eq. (19) to determine the actual α[tk ]. (2) Determine the value of u f [tk ]: As mentioned earlier, at any given time instant, the friction behavior of a SCIS system may be either in its sliding state or stick state. The friction force u f [tk ] may not be determined unless the motion state is known. In this study, a numerical approach modified from the one proposed by Lu et al. [25], which can efficiently simulate the response of structures containing friction elements, is adopted to determine the motion state and the friction force u f [tk ]. This approach is explained as follows: First of all, at the (k + 1)-th time step, assume that the SCIS is in its stick state, so it must satisfy the kinematic condition described in Eq. (12b), i.e. x˙b [tk+1 ] = Dv z[tk+1 ] = 0

(25)

Fig. 6. Computational flowchart for the numerical method.

where Dv = [0, 1, 0, 0] is a row matrix, which extracts x˙b [tk+1 ] from the state vector. Next, substituting z[tk+1 ] from Eq. (21) into Eq. (25) and further solving for the friction force, we get u˜ f [tk ] = −(Dv Bd )−1 Dv (Ad z[tk ] + Bd 1u r [tk ] + Ed w[tk ]). (26) Note that in Eq. (26), u f [tk ] is replaced with the symbol u˜ f [tk ] to signify that the obtained friction force is under the assumption of a stick state. Although u˜ f [tk ] in Eq. (26) may or may not be the actual friction force, u˜ f [tk ] will help determine the actual value. Moreover, the sign of u˜ f [tk ] physically represents the direction of the actual friction force. If the absolute value of u˜ f [tk ] is less than the maximum friction force u f,max , then according to Eq. (12a) this implies that the SCIS is in its stick state, so u˜ f [tk ] becomes the actual friction force. On the other hand, if the absolute value of u˜ f [tk ] is greater than u f,max , it implies that the SCIS is in its sliding state, and then according to Eq. (13a) the magnitude of u f [tk ] must be equal to u f,max . From the above discussion, regardless of the motion state of the SCIS, the actual friction force can be computed by the following single equation u f [tk ] = min( u˜ f [tk ] , u f, max )sgn(u˜ f [tk ]) (27)

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where a function min(a, b) means taking the minimum value of a and b, and a function sgn(x) represents taking the sign of x. After 1u r [tk ] and u f [tk ] have been computed by Eqs. (23) and (27), respectively, they can be substituted into Eq. (21), in order to compute the complete (k + 1)-th step response of the SCIS isolated structure. This numerical procedure will be conducted step by step until the complete time history of the structural response is obtained. Fig. 6 shows the computational flowchart for the numerical method discussed above. 5. Computation of feedback gain

where the notations ∈ C r ×q and ∈ R r ×q indicate the dimensions of a complex-numbered and a real-numbered matrix with r rows and q columns, respectively. When the feedback gain is designed according to Eq. (28), the controlled system has a very important feature, i.e. the number of controllable eigenvalues is equal to the number of sensors used for feedback control, provided that the number of sensors is larger than, or equal to, that of the actuators (q ≥ r ). Here, a controllable eigenvalue means that its value can be arbitrarily assigned. Furthermore, since the system matrix A is usually a realnumber matrix, its eigenvalues must be either real numbers or conjugated complex numbers. Each pair of conjugated eigenvalues corresponds to one structural vibration mode in the modal domain. Let (λ2i−1 , λ2i ) be the i-th pair of conjugated eigenvalues, then their relation with the modal parameters can be expressed as q λ2i−1 , λ2i = −ζi ωi ± jωi 1 − ζi2 (30)

As shown in Eq. (18), the proposed semiactive control law ˆ which can for the SCIS requires a constant feedback gain, G, be determined by any widely used active control law, such as the optimal control or modal control, etc. In this paper, for the simulation of the seismic response of the SCIS, two modal control methods will be employed to determine the gain, ˆ The modal control, in which the motion of a structure is G. reshaped by simply controlling some selected vibration modes, is one of several well-known techniques of active structural control. By properly selecting the target modal values, it has been experimentally proven that modal control can be an effective method for suppressing the seismic response of a structure [26]. The two modal control methods employed in this paper are referred to as the “modal control with direct output feedback” [26] and the “modal control with multiplestep feedback” [27], respectively. The latter requires less sensor measurements than the former. As a result, the effect of different sensor placement can be studied. Both methods will be briefly reviewed.

where ωi and ζi denote the target modal frequency and damping ratio, respectively. Once the desired ωi and ζi for a controlled mode are selected, they can be converted into one pair of the conjugated target eigenvalues (λ2i−1 , λ2i ) by Eq. (30). After all target eigenvalues are computed they can be substituted ˆ In the latter into Eq. (28) to compute the feedback gain G. numerical simulation of this paper, the criteria for selecting the target modal parameters was set to increase the modal damping ratios of the structure, and at the same time to keep the modal frequencies unchanged. In other words, we varied ζi but preserved ωi for a selected controlled mode.

5.1. Modal control with direct output feedback

5.2. Modal control with multiple-step feedback

Generally speaking, the purpose of a modal control is to ˆ such that the gain will alter the design a feedback gain, G, eigenvalues and eigenvectors of the open-loop system to the values that correspond to preselected target modal parameters, namely, modal frequencies, damping ratios, mode shapes, etc. For modal control with direct output feedback Lu [26] derived a ˆ very general formula that makes the computation of the gain G systematic. This formula is given in a concise matrix form as follows:

From the above discussion it is evident that for the modal control with Eq. (28) the number of the controllable eigenvalues (vibration modes) is restrained by the sensor number q. In order to reduce the number of sensors needed, Lu and Chung [27] further relaxed the restraint by using the method of the augmented state matrix. The design of their control gain was formulated in a discrete-time domain. The dimensions of the discrete-time state matrix and the feedback gain were all augmented by using a multiple-step feedback technique. In other words, in computing the control command at a certain time step, the method makes use of the sensor measurement taken at several previous time steps, in addition to the current time step. For instance, for the control command u[k] ˆ at the k-th step, we have

ˆ = B−1 [Zc1 diag(λi )c − A1 Zc ](DZc )−1 G 1

(28)

where the diagonal matrix diag(λi )c and the rectangular matrix Zc contain the target eigenvalues and eigenvectors, respectively (the subscript “c” denotes the entity is associated with the controlled mode). These target values are selected by the control designer. Also, in Eq. (28) the matrices A1 , B1 and Zc1 are the upper portions of matrices A, B and Zc . If we let r and q be the numbers of actuators and sensors, respectively, then the dimensions of the above matrices can be expressed as (in current case, r = 1): ˆ ∈ C r ×q , G

B1 ∈ R r ×r ,

diag(λi )c ∈ C q×q ,

Zc1 ∈ C r ×q ,

A1 ∈ R r ×2n ,

Zc ∈ C 2n×q

(29)

u[t ˆ k] =

m X

ˆ¯ y[t ] ˆ i y[tk−i ] = G¯ G k

i=0

where   y[tk ]       y[tk−1 ]   y¯ [tk ] = ..  .        y[tk−m ]

(31)

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D 0  0 = . . . 0

0 D 0 .. .

0 0 D .. .

··· ··· ··· .. .

0

0

···

 0  z[tk ]     0     z[tk−1 ]  0 ¯ z[tk ]. = D¯ .. ..  .       .   z[tk−m ] D

Table 1 Values of parameters for numerical simulation

(32)

In the above equations, m represents the total number of the previous time steps of sensor measurements taken for the ˆ i denotes the gain associated with the ifeedback control, G ˆ¯ ∈ C r ×q(m+1) denotes th time-step sensor measurement, G the augmented gain for the discrete-time system, and y¯ [tk ] ∈ R q(m+1) denotes the augmented output vector. The above discussion shows that after the augmentation, the ˆ¯ has increased from (r × q) dimension of the gain matrix G ˆ (see to (r × q(m + 1)), compared with the dimension of G Eq. (29)); therefore, the number of the controllable structural eigenvalues is also increased from q to q(m + 1), without physically increasing the sensor number. The enlarged gain can be expressed in the following form: ˆ¯ = B ¯ ¯ ¯ ¯ −1 ¯ −1 ¯ G d1 [Zc1 diag(γ¯i )c − Ad1 Zc ](DZc )

(33)

where

System

Item

Value

Super-structure

Structural mass (m s ) Structural stiffness (ks ) Damping coefficient (cs ) Damping ratio (ζs ) Frequency ( f s )

3.0 × 105 kg 3.289 × 104 kN/m 3.141 × 102 kN s/m 5% (fixed-base) 1.67 Hz (fixed-base)

Isolation system SCIS

Mass of base mat (m b ) Friction coefficient (µ) Static stiffness (kr 0 ) Dynamic stiffness ratios (αmax , αmin ) Frequency (uncontrolled) ( fr 0 )

1.0 × 105 kg 0.03 2.527 × 103 kN/m (0.5, −1.0)

Table 2 Structural matrices used for simulation Properties Mass matrix Damping matrix Stiffness matrix

diag(γ¯i )c ∈ C ¯ c ∈ C 2n(m+1)×q(m+1) , Z

q(m+1)×q(m+1)

¯ c1 ∈ C r ×q(m+1) Z ¯ d1 ∈ R r ×2n(m+1) , A

, (34)

¯ d1 ∈ R r ×r B

¯ c contain the target values of the discretewhere diag(γ¯i )c and Z time eigenvalues and eigenvectors of the controllable modes. Eq. (33) is similar to Eq. (28) except that matrices A1 , B1 , D, Zc and Zc1 in Eq. (28) are replaced by their augmented discrete¯ d1 , B ¯ d1 , D, ¯ Z ¯ c and Z ¯ c1 . time counterparts A 6. Results of numerical simulation 6.1. Numerical model and parameters In order to demonstrate the effectiveness of seismic isolation of a SCIS system, the structure model shown in Fig. 2 controlled by the proposed semiactive control law will be studied numerically in this section. The performance of the SCIS will be compared with those of its passive and active isolation counterparts (see Fig. 4). Here, the passive isolation system is to be modeled by Eq. (9) with 1u r (t) = 0 (i.e., α(t) = 0 in Eq. (3)); therefore, the passive isolation can also be treated equivalently as the uncontrolled case of SCIS. Table 1 lists the numerical values of the parameters of the superstructure and the SCIS isolation system, and Table 2 shows the structural matrices used for simulation. These values were used throughout the numerical simulation. In all simulation, the time interval of numerical analysis was taken to be 0.005 s. To be practical, the upper and lower limits of the control parameter α for SCIS were chosen to be αmax = 0.5 and αmin = −1.0, respectively. From Eq. (3), due to the limits of α, the range of kr (t) is between 1.5kr 0 and

0.4 Hz

Values   300 000 0 M= kg 0 100 000   314.1 −314.1 C= kN/(m/s) −314.1 314.1   328 900 −328 900 Kr 0 = kN/m −328 900 (328 900 + 2527)

zero. Moreover, in order to quantify the isolation effectiveness, two structural responses, i.e. the base displacement (isolator drift) and structural acceleration, are chosen as the performance indices to be compared in the following discussions: In addition, because the location and type of the sensors used may affect the control performance of the SCIS isolation system, three control cases with different sensor placements shown in Fig. 7 will be considered in the later discussion. These three control cases are labelled as D12V12, D1V1 and V2 in Fig. 7. In the name of each control case, the notation D1 or V1 signifies that a sensor is used to measure either the relative-to-the-ground displacement or the velocity of the first floor (the base), respectively. The three control cases have the same control goal and the same target values for the controlled ˆ was set to modes. The control goal for computing gain G, increase the modal damping ratios of both modes to 30%, and at the same time to keep their modal frequencies and the mode shapes unchanged. Table 3 summarizes the features of the three control cases. It should be noted that case D12V12 is a full-state feedback control, while cases D1V1 and V2 are direct-output feedback controls, which require fewer sensor measurements. Therefore, in order to achieve the same control goal of D12V12, cases D1V1 and V2 employed the method of “modal control with multiple step feedback” mentioned in the previous section. As shown in the fourth row of Table 3, the numbers of feedback time steps required for cases D1V1 and V2 are two and four, respectively. 6.2. Comparison of hysteretic properties The mechanical behaviour of a structural control device usually can be characterized by its hysteretic diagram, which

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Table 3 Features of three control cases with different sensor placements Control case

D12V12

D1V1

V2

Control law

Modal control with direct output feedback Full state 1

Modal control with multiple step feedback Direct output 2

Modal control with multiple step feedback

Feedback type No. of time steps feedback (m + 1) No. of sensors Sensor type Sensor locations Gain computation G Control target modes Control goal

4 Displacement and velocity Base and superstructure Eq. (28)

2 Displacement and velocity Base Eq. (33) 1st and 2nd modes (1) Increase damping ratios of both modes to 30%. (2 Keep modal frequencies and mode shapes unchanged.

Direct output 4 1 Velocity Super-structure Eq. (33)

Fig. 7. Three control cases with different sensor placements.

shows the force–displacement relation of the device. In this subsection, the hysteretic property of the SCIS system will be investigated. In order to do so, a harmonic ground acceleration x¨ g (t) of the following form is considered in the simulation: x¨ g (t) = 0.4g sin(2πt).

(35)

Fig. 8 compares the base hysteresis loop of SCIS with those of the passive and active isolation systems. Fig. 8(a) shows that due to the isolator friction the passive isolation system has a bilinear hysteresis loop with a yielding strength equal to the sliding friction force and a postyielding stiffness equal to its constant isolation stiffness kr 0 . Also, note that in Fig. 8(a) the hysteresis loop looks slim, because a lower friction coefficient µ = 0.03 (see Table 1) was adopted in all isolation systems, in order to achieve better isolation performance. On the other hand, Fig. 8(b) depicts that due to the variation of the isolation stiffness of the SCIS, the hysteresis loop of the SCIS system is bounded by two straight lines with slopes equal to the upper stiffness 1.5kr 0 and the lower stiffness zero, respectively. Also, since the area in the hysteresis loop of a control device represents the amount of energy dissipated by the device, Fig. 8(a) and (b) illustrate that the SCIS is able to dissipate

more energy per cycle than the passive isolation system. The energy dissipation ability in an isolation system is crucial, since it helps reduce the maximum isolator displacement due to by an earthquake. As for the active isolation, Fig. 8(c) shows that the hysteresis loop of the active system has no obvious straight lines representing the isolation stiffness. Also shown in Fig. 8(c) is the fact that the active system may dissipate more seismic energy per cycle than the SCIS and the passive system. This is mainly due to the fact that an active control system usually has less restraint on the control force, so it is able to offer more control freedom than a passive or semiactive system. 6.3. Comparison of earthquake responses This subsection aims to investigate the seismic responses of the SCIS system. Four earthquake acceleration records will be used as the input excitations in the simulation. The detailed information about these four earthquake records is given below: (1) El Centro (S00E) Earthquake, 18 May 1940, peak acceleration: 341.0 cm/s2 .

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(2) Northridge (Sylmar) Earthquake, Channel 3 360◦ , 17 January 1994, peak acceleration: 826.8 cm/s2 . (3) Imperial Valley (El Centro Array 6) Earthquake, Channel 1 230◦ , 15 October 1979, peak acceleration: 428.1 cm/s2 . (4) Chi-Chi (TCU068) Earthquake, Channel East-West, 21 September 1999, peak acceleration: 497.4 cm/s2 .

(a) Passive isolation (uncontrolled case).

(b) SCIS (D1V1 sensor placement case).

(c) Active isolation (D1V1 sensor placement case). Fig. 8. Comparison of base hysteretic loops of different isolation systems.

The first record is a famous earthquake that has been widely used in many earthquake engineering studies [21]. The other three earthquakes are recommended by Narasimhan and Nagarajaiah et al. [22] for the study of a benchmark problem on smart base-isolated buildings. The waveforms of these four records are shown in Fig. 9, while Fig. 10 depicts the 5%-damping-ratio acceleration response spectra of these earthquakes with PGA (peak ground acceleration) values normalized to 1g. Based on Figs. 9(a) and 10(a), since El Centro (S00E) earthquake exhibits no long-period characteristics, it is used to represent a far-field earthquake in this study. On the other hand, as shown in Fig. 9(c)–(d), long-period pulse-like waveforms can be clearly observed in Imperial Valley (Array 6) and Chi-Chi (TCU068) earthquakes; as a result, Fig. 10(c) and (d) show that these two earthquakes induce a relatively larger acceleration spectra value up to 1g for long-period structures (structural period larger than 1 s), as compared with Fig. 10(a). Therefore, Imperial Valley (Array 6) and Chi-Chi (TCU068) earthquakes may be classified as earthquakes with strong nearfault characteristics. Finally, from Fig. 10(b), it is evident that the acceleration spectrum of Northridge (Sylmar) earthquake is quickly decreasing below 1g for a structural period larger than 1.5 s, even though the earthquake has a pulse-like waveform shown in Fig. 9(b). Therefore, Northridge (Sylmar) earthquake may only be considered as an earthquake with moderate nearfault characteristics. (1) Time history response: Figs. 11–14 compare the time history responses of the SCIS and the passive isolation system, when both systems are subjected to the four chosen earthquakes. In all figures, the responses of the structural acceleration, base displacement and total base shear (i.e. sum of isolator friction u f (t) and restoring force u r (t)) are depicted. Note that in these figures the PGA of all earthquakes has been scaled to 0.4g for the purpose of comparison. Also, the sensor placement case D1V1 mentioned previously is considered for the SCIS. From Figs. 11 and 12, it is observed that in El Centro and Northridge earthquakes that have fewer near-fault characteristics, the SCIS reduces the maximum base displacement and at the same time maintains a structural acceleration level roughly equal to that of the passive system. On the other hand, Figs. 13 and 14 show that in Imperial Valley and Chi-Chi earthquakes, both of which have strong near-fault characteristics, the SCIS considerably and simultaneously reduces the maximum structural acceleration and base displacement. This implies that the SCIS is able to effectively suppress the large base displacement induced by the long-period pulse-like waveform contained in the near-fault earthquake, without scarifying the response of the structural acceleration. In addition, in Figs. 13 and 14, it is evident that due to the pulse excitation in the near-fault earthquakes, the passive system exhibits an obvious low-frequency oscillation

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(a) El Centro earthquake (PGA = 341.0 cm/s2 ).

(b) Northridge (Sylmar) earthquake (PGA = 826.8 cm/s2 ).

(c) Imperial Valley (El Centro Array 6) earthquake (PGA = 428.1 cm/s2 ).

(d) Chi-Chi (Tcu068) earthquake (PGA = 497.4 cm/s2 ). Fig. 9. Four ground acceleration records used for simulation.

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(a) El Centro earthquake.

(b) Northridge (Sylmar) earthquake.

(c) Imperial Valley (El Centro Array 6).

(d) Chi-Chi (TCU068) earthquake.

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Fig. 10. Normalized acceleration response spectra of the four earthquakes.

behaviour in both acceleration and displacement responses; whereas, the SCIS effectively suppresses this oscillation motion. As for the comparison of the total base shears, from Fig. 12(c), 13(c) and 14(c), it is observed that the total base shear of the SCIS has been influenced by the controllable isolator stiffness and resulted in a lower base shear than that of the passive system, even though the base shear of the SCIS seems to contain more high-frequency components in the range of low-level ground excitation. Nevertheless, the effect of this low-level high-frequency shear force on the response of the isolated structure is insignificant. Figs. 15–18 compare the time history responses of the SCIS with those of the active isolation system, when both systems are subjected to each of the four chosen earthquakes. In these figures the PGA of all earthquakes has been scaled to 0.4g and the sensor placement case D1V1 is considered for both the SCIS and the active system. From these four figures it is evident that the SCIS is able to closely follow the responses of the active system in all types of earthquakes, even though the maximum

response values of the SCIS are slightly higher than those of the active system. However, it must be remembered that unlike in the active system, the control force 1u r (t) in the SCIS (see Eq. (7)) is a passive force rather than an active force, and may consume more control energy. (2) Maximum responses vs. PGA: When the isolated structure is subjected to each of the four earthquakes, Figs. 19–22 compare the maximum structural acceleration and maximum base displacement as a function of the PGA level for the three isolation systems, namely, passive, SCIS and active systems. To show the isolation effectiveness, the maximum acceleration of a fixed-base structure (structure without isolation) is also depicted in the figures of the maximum acceleration. Figs. 19(a) and 20(a) indicate that the three isolation systems are equally effective in reducing the acceleration responses of El Centro and Northridge earthquakes that have less near-fault characteristics; however, Figs. 19(b) and 20(b) also shows that the base displacement of the SCIS is less than that of the passive system but more than that of

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(a) Structure acceleration.

(b) Base displacement.

(c) Total base shear. Fig. 11. Responses of passive isolation and SCIS due to El Centro earthquake (PGA = 0.4g).

(a) Structure acceleration.

(b) Base displacement.

(c) Total base shear. Fig. 12. Responses of passive isolation and SCIS due to Northridge (Sylmar) earthquake (PGA = 0.4g).

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(a) Structure acceleration.

(a) Structure acceleration.

(b) Base displacement.

(b) Base displacement.

(c) Total base shear. Fig. 13. Responses of passive isolation and SCIS due to Imperial Valley (El Centro Array 6) earthquake (PGA = 0.4g).

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(c) Total base shear. Fig. 14. Responses of passive isolation and SCIS due to Chi-Chi (TCU068) earthquake (PGA = 0.4g).

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(a) Structure acceleration.

(a) Structure acceleration.

(b) Base displacement.

(b) Base displacement.

(c) Total base shear.

(c) Total base shear.

Fig. 15. Responses of active isolation and SCIS due to El Centro earthquake (PGA = 0.4g).

Fig. 16. Responses of active isolation and SCIS due to Northridge (Sylmar) earthquake (PGA = 0.4g).

L.-Y. Lu et al. / Engineering Structures 30 (2008) 747–765

(a) Structure acceleration.

(a) Structure acceleration.

(b) Base displacement.

(b) Base displacement.

(c) Total base shear. Fig. 17. Responses of active isolation and SCIS due to Imperial Valley (El Centro Array 6) earthquake (PGA = 0.4g).

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(c) Total base shear. Fig. 18. Responses of active isolation and SCIS due to Chi-Chi (Tcu068) earthquake (PGA = 0.4g).

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(a) Maximum structure acceleration.

(b) Maximum base displacement.

(a) Maximum structure acceleration.

(b) Maximum base displacement.

Fig. 19. Maximum responses for different control strategies subjected to El Centro earthquake.

Fig. 20. Maximum responses for different control strategies subjected to Northridge (Sylmar) earthquake.

the active system. On the other hand, Figs. 21 and 22 compare the maximum responses of the three isolation systems subjected to Imperial Valley and Chi-Chi earthquakes that have stronger near-fault characteristics. Generally speaking, Figs. 21 and 22 show that the performance of the SCIS in the near-fault earthquakes is superior to the passive system and is close to the active system, both in reducing the structural acceleration and base displacement. Especially, in Figs. 21(b) and 22(b), it is evident that the base displacement of the passive system is considerably amplified as the PGA value increases, whereas the SCIS suppresses the maximum base displacement to about one half of that of the passive system. In addition, it should be reminded that the base displacement of a friction type isolation system will approach the peak ground displacement only if both the friction coefficient and the restoring stiffness are zeros. For the above simulated cases, the base displacements shown in Figs. 19–22 cannot be equal to the peak ground displacement,

because the friction coefficient µ and the constant restoring stiffness kr 0 for all isolation systems have been taken to be nonzero values (see Table 1). 6.4. Effect of different sensor placements Tables 4 and 5 compare the maximum response values of the SCIS when the isolated structure with the three different sensor placements shown in Fig. 7 is subjected to El Centro (farfield) earthquake and Imperial Valley (near-fault) earthquake, respectively, with the PGA scaled to 0.4g. The responses of the passive (uncontrolled) isolation and fixed-base case are also listed in these two tables for the purpose of comparison. From the third and fourth columns of Tables 4 and 5, it can be seen that, for all three different sensor placements, the SCIS is almost equally effective on reducing the maximum responses of the isolated system. More specifically, in Table 4 for the El

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(a) Maximum structure acceleration.

(a) Maximum structure acceleration.

(b) Maximum base displacement.

(b) Maximum base displacement.

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Fig. 21. Maximum responses for different control strategies subjected to Imperial Valley (El Centro Array 6) earthquake.

Fig. 22. Maximum responses for different control strategies subjected to ChiChi (TCU068) earthquake.

Centro (far-field) earthquake, the SCIS with all cases of the sensor placements is able to reduce the structural acceleration and isolator displacement to around 90% and 80% of those of the passive isolation. On the other hand, in Table 5 for Imperial Valley (near-fault) earthquake, the SCIS with all three cases of the sensor placements is able to further reduce the structural acceleration and isolator displacement to around 75% and 60% of those of the passive isolation. These observations imply that by employing a proper control law, the proposed SCIS may maintain its effectiveness even when the number of sensors used for feedback control is reduced. Nevertheless, the last columns of Tables 4 and 5 also show that when the number of sensors is substantially reduced (e.g. in case V2), the requirement for the semiactive control force may be increased. It must be remembered that the semiactive control force means the restoring force 1u r (t) provided by the dynamic stiffness of SCIS. This force is eventually a variable passive force.

7. Conclusions When subjected to an earthquake with near-fault long-period characteristics, the seismic response of a conventional sliding isolation system may be considerably amplified. In order to mitigate this problem, a new type isolation system called the “Stiffness Controllable Isolation System” (SCIS) is considered in this study. The SCIS is defined as a system composed of a set of sliding isolators and a stiffness controllable device. By controlling the isolation stiffness of the SCIS the restoring force provided by the SCIS can be varied in response to the system response and seismic excitations. In the formulation of the control design, the isolation stiffness of the SCIS is divided into two parts, static and dynamic. Although physically the total stiffness can only be positive, the dynamic part of the stiffness can be either positive or negative, so the formulation is able to provide better control. This formulation also allows

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Table 4 Comparison of isolation performance for structure subjected to the El Centro (far-field) earthquake (PGA = 0.4g) Isolation type

Control case

Max. structural accelerationa (m/s2 )

Max. isolator displacementa (m)

Max. semiactive control forceb (kN)

Fixed base Passive SCIS

– – D12V12 D1V1 V2

9.67(7.06) 1.37(1.00) 1.25(0.91) 1.25(0.91) 1.28(0.93)

– 0.14(1.00) 0.11(0.79) 0.11(0.79) 0.12(0.86)

– – 156.37(0.039) 151.80(0.039) 216.14(0.055)

a Numbers in parentheses denote the ratios to the corresponding value of passive isolation. b Numbers in parentheses denote the ratio to the structural weight.

Table 5 Comparison of isolation performance for structure subjected to the Imperical Valley (near-fault) earthquake (PGA = 0.4g) Isolation type

Control case

Max. structural accelerationa (m/s2 )

Max. isolator displacementa (m)

Max. semiactive control forceb (kN)

Fixed base Passive SCIS

– – D12V12 D1V1 V2

5.92(1.42) 4.16(1.00) 3.05(0.73) 3.06(0.74) 3.30(0.79)

– 0.61(1.00) 0.35(0.57) 0.35(0.57) 0.37(0.61)

– – 345.32(0.088) 338.15(0.086) 681.10(0.174)

a Numbers in parentheses denote the ratios to the corresponding value of passive isolation. b Numbers in parentheses denote the ratio to the structural weight.

for the restoring force of the dynamic stiffness to be controlled according to an online reference value that may be determined by any well-developed active feedback control law. A numerical procedure that is able to simulate the dynamic response of this stiffness controllable system, with consideration of sliding friction, was also introduced in the study. The isolation performance of the proposed SCIS was compared with those of its passive and active isolation counterparts. The results of the numerical simulation demonstrated that, like the active system, the proposed system is able to effectively suppress the maximum response induced by a near-fault earthquake. Therefore it is concluded that for earthquakes with either far-field or near-fault characteristics, the SCIS is superior to the passive isolation system in reducing both the structural acceleration and the base displacement. Acknowledgement The authors gratefully acknowledge that this research was sponsored in part by the National Science Council of Taiwan, ROC through Grant NSC 94-2625-Z-327-004. References [1] Naeim F, Kelly JM. Design of seismic isolated structures: From theory to practice. John Wiley & Sons; 1999. [2] Lu LY, Yang YB. Dynamic response of equipment in structures with sliding support. Earthquake Engineering and Structural Dynamics 1997; 26:61–76. [3] Yang YB, Lu LY, Yau JD. In: de Silva CW, editor. Structure and equipment isolation vibration and shock handbook. CRC Press, Taylor & Francis Group; 2005 [chapter 22]. [4] Jangid RS, Kelly JM. Base isolation for near-fault motion. Earthquake Engineering and Structural Dynamics 2001;30:691–707.

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