A study on interaction control for seismic response of parallel structures

A study on interaction control for seismic response of parallelstructures

Hongping Zhu a,*, Yinping Wen a, Hirokazu Iemura b

a School of Civil Engineering, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of Chinab Graduate School of Civil Engineering, Kyoto University, Sakyo-ku, 606-01, Japan

Received 12 October 1998; accepted 12 January 2000

Abstract

A structure response control approach which uses controlled interactions between two parallel structures (a primary

structure and an auxiliary structure) to reduce the seismic response of the primary structure (P-structure) during

earthquake excitation, is proposed. Three strategies of the control approach, including optimal passive control, semi-

active control, and active control are examined. The optimum passive-coupling elements between two parallel structures

under di�erent circumstances are ®rstly investigated. Then, emphasis has been placed on the derivation of a simple

e�ective algorithm for semi-active control which guarantees the power absorption of the required control force from the

P-structure at every time step. Finally, numerical results in the frequency and time domains are presented to demon-

strate the e�ectiveness of these proposed control strategies. The comparison of the structure response time histories

when subjected to El Centro 1940 ground excitation with di�erent control methods also illustrates that the semi-active

control based on optimum passive coupling element can achieve improvement in reducing the response of P-structure

compared to the passive control and is almost comparable to the active control system with the same degree of

magnitude of control force, and that the proposed instantaneous power absorption semi-active control algorithm is

always superior to conventional linear control law. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Interaction control; Parallel structures; Power absorption; Semi-active control; Earthquake excitation

1. Introduction

An earthquake directly a�ects a structure by in-

creasing the energy within the structural systems. A

signi®cant portion of this energy can be dissipated

through the introduction of a supplemental energy dis-

sipation system placed as structural elements within a

conventional construction. Although a variety of sup-

plemental energy dissipation systems has been proposed

for the purpose of mitigating the harmful e�ects of

earthquakes, all such systems may be categorized under

three basic headings: passive control, active control, and

semi-active control systems [1].

Both active [2±4] and passive [5] control systems have

been studied extensively and used to protect structures

against wind excitation and earthquake. Active control

can be extremely e�ective; however, it has several dis-

advantages [6]. In addition to the requirements of large

control forces and detailed state information, active

control may also su�er from instability [7]. Passive

control does not have these drawbacks, but it is not as

e�ective [1].

A compromise between active and passive control

systems is available in the form of semi-active control

systems which have been developed to take advantage of

the best features of the two. As in an active control

system, the mechanical properties of semi-active control

Computers and Structures 79 (2001) 231±242

www.elsevier.com/locate/compstruc

* Corresponding author. Address: School of Civil Engineer-

ing, Asian Institute of Technology, P.O. Box 4, Klongluang

Pathumthani 12120, Thailand. Tel.: +66-2-5245787; fax: +66-2-

5246059.

E-mail address: [email protected] (H. Zhu).

0045-7949/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0 04 5 -7 94 9 (00 )0 0 11 9 -X

are typically adjusted based on feedback from the

structural system to which they are attached. As in a

passive control system, semi-active control systems uti-

lize the motion of the structure to develop the control

force [8].

In this study, an innovative control approach is

proposed which can be realized by means of installing an

interaction element between two parallel substructures.

This takes advantage of the interaction between the

parallel systems to supply a larger control force. The

strategy of the control approach is to remove energy

associated with vibration only from the primary struc-

ture (P-structure), which is done by transferring energy

to the auxiliary structure (A-structure) or directly dissi-

pating energy by means of controllers (passive, semi-

active or active) between the two systems.

Since Klein studied the possibility of using dissipative

links and semi-active devices to control the response of

adjacent buildings to wind excitation [9], there have been

many advances in this ®eld. Gurley et al. [10] modeled a

system of two adjacent buildings by a couple of uniform

shear beams coupled by a single ¯exible and damped

link and obtained the optimal sti�ness and damping in

the link when the P-structure was subjected to wind

excitation. The possibility of using active or passive

control devices interconnecting adjacent buildings each

modeled as a discrete multi-degree-of-freedom damped

shear beam to reduce the seismic or wind response of

P-structure has been theoretically investigated by Luco

and coworkers [11,12]. The optimal control forces for

the active control case and optimal values for the

distribution of passive dampers interconnecting two

adjacent buildings of di�erent heights subjected to a

horizontal ground harmonic motion are determined

[11,12].

Recently, a control strategy for reducing wind or

seismic response of a mega-structure utilizing the mega-

substructure con®guration of tall buildings has been

considered by Feng and Mita [13]. The optimal values of

the damping ratio and sti�ness of the substructure are

determined on the basis of single-degree-of-freedom

models for both the mega- and sub-structures [13]. Luco

and coworkers have considered possible optimal con-

®gurations for this type of system, modeled by a couple

of equivalent shear beams connected at several locations

along the height by rigid or ¯exible links [14].

Here, the ®rst study uses two damped single-degree-

of-freedom systems coupled by a viscous damper or by

an active or semi-active device to present two parallel

structures with an interaction element. The system is

subjected to the horizontal ground white noise excita-

tion, and for simplicity, the e�ects of soil-structure in-

teraction and the possible lateral variation of the ground

motion are ignored. A formulation for time-averaged

energy of the P-structure in terms of the structural pa-

rameters of the P-structure and A-structure has been

developed so that the optimum values of the passive

coupling element can be obtained simply based on

minimizing the time-averaged energy of the P-structure.

The in¯uence of mass ratio, natural frequency ratio and

damping ratio of the P-structure to A-structure on the

optimal values of the passive coupling element is dis-

cussed in detail. Also, the e�ective semi-active control

design methodology based on the optimum passive

coupling element has been presented. Especially, an in-

stantaneous power absorption algorithm is proposed to

determine the o�±on switch for the semi-active control

system. The root mean square (RMS) and peaks of

relative displacements and absolute acceleration of the

P-structure with di�erent types of control strategy sub-

jected to El Centro 1940 are compared.

2. The proposed interaction control system

The subject of structural control o�ers opportunities

to design new structures and to retro®t existing struc-

tures by application of counter-forces, smart materials,

frictional devices, etc., instead of just increasing the

strength of the structure at a greater cost. As tall

buildings become even higher, the mass becomes larger

and the natural frequency lower, it will be di�cult to

realize a vibration device with su�cient control force.

Even though there are now more than 20 active or hy-

brid control devices installed in large civil structures,

and even though these devices have been performing

¯awlessly in providing comfort control under wind and

minor earthquakes, none of these devices worked during

the Kobe earthquake [15]. The level of shaking in the

Kobe earthquake caused their motion to exceed design

speci®cations; consequently, the control devices were

shut down in an orderly manner as a precaution to

prevent damage to the active control system. It does

dramatically illustrate that there is a quantum jump

needed in research and development on the mitigation of

the dynamic response of large civil structures under

strong earthquake loads.

The proposed interaction control strategy takes ad-

vantage of the con®guration and interaction of two

di�erent structural systems (primary±auxiliary structure)

to reduce the vibration response of the P-structure or the

total vibration of primary±auxiliary structure (P±A

structure) during an external excitation. This paper fo-

cuses on the simplest form which reduces only the re-

sponse of the P-structure. The large mass and more than

one vibration mode of the A-structure implies that an

extremely high level of response reduction in P-structure

can be achieved. Thus, it will be very e�ective in re-

ducing the vibration of the P-structure during large

earthquakes.

The application of the interaction control method

may be categorized into several aspects: one is that the

232 H. Zhu et al. / Computers and Structures 79 (2001) 231±242

systems represent two adjacent multi-storey buildings

[9±12] or a tall building and its skirt structures; another

is that the P-structure represents a complete structure

while the A-structure represents some other substructure

element, such as the mega-substructure con®guration of

a tall building [13,14]. In the ®rst case, the control de-

vices are added between parallel structures; and in the

other, the control actions are added to the ¯oors be-

tween the P-structure and substructures.

In this paper, the study is limited to the ®rst case

under earthquake excitation. Fig. 1 outlines two ¯exible

structures arranged in parallel connected by means of

controllable elements to control the vibration of the P-

structure. The mechanism of the interaction control is to

remove vibration energy from the P-structure by means

of transferring energy to the A-structures or dissipating

energy directly in interaction elements. The control de-

vices may be either active actuators or passive energy

dissipated elements or semi-active controllers. The

physical properties of the interaction elements can be

justi®ed according to the control signals. For the passive

control strategy, the purpose of the study is to compute

the optimum structural parameters of interaction ele-

ments; for semi-active or active control, the objective is

to ®nd a suitable control algorithm to actively change

the interaction.

3. Basic equations

The analysis of P±A structure is inherently complex

because both P- and A-structures are multi-degree-of-

freedom systems and the number of degrees of freedom

of the combined system can be prohibitively large.

However, important physical insights into complex P±A

structure behavior can be gained by using more simpli-

®ed procedures while demanding less-detailed response

information. Only a simple P±A structure coupled by

two single-degree-of-freedom systems subjected to seis-

mic excitation, with one interaction element, is consid-

ered here. Fig. 2 schematically represents the controlled

structure. The P- and A-structures are respectively

speci®ed by their ®rst modal masses, MP and MA, along

with the horizontal relative displacement, XP and XA; the

system spring constant, KP and KA; damping constant,

CP and CA; and the ground horizontal motion acceler-

ation, �Xg. The control device acts between the P-struc-

ture and the A-structure, which is represented by U.

For the P±A structure of Fig. 2, the equations of

motion are derived as follows:

MP�XP � CP

_XP � KPXP � ÿMP�Xg ÿ U ; �1a�

MA�XP

�� Z

�� CA

_XP

�� _Z

�� KA XP� � Z�

� ÿMA�Xg � U : �1b�

In state space form,

_X � AX � BU � C �Xg �2�

in which X � �XP Z _XP_Z�T; where Z � XA ÿ XP; �3�

A �0 0 1 00 0 0 1ÿx2

P 0 ÿ2nPxP 0x2

P ÿ x2A ÿx2

A 2 nPxP ÿ nAxA� � ÿ2nAxA

26643775;�4a�

B � 0 0

�ÿ 1

MP

1� lMP

�T

and

C � 0 0� ÿ 1 0�T�4b; c�

with the following parameter de®nitions:

Fig. 1. Con®guration of parallel buildings with control device.

Fig. 2. Schematic representation of P±A structure with control

device.

H. Zhu et al. / Computers and Structures 79 (2001) 231±242 233

xP ��KP

MP

r; nP �

CP

2��MPKP

p ; xA ��KA

MA

r;

nA �CA

2��MAKA

p ; l � MP

MA

: �5�

4. Control strategies

For the purpose of demonstrating the interaction

control mechanism between P±A structure, a total of

four cases of interest are presented by considering pas-

sive, active and semi-active control strategies. The con-

trol algorithm and control parameters for each of these

cases are derived next.

4.1. Case 1: Optimum passive control

In this case, the passive energy dissipation coupling

element is installed between the P- and A-structures. The

passive control force, UP, is given by

UP�t� � ÿKCZ ÿ CC _Z: �6�

Substituting Eq. (6) into Eqs. (1a) and (1b) gives the

relative displacement XP of the P-structure as

XP�t� � ÿ�ix�2 � ix�DA � DC� � �x2

A � x2C�

h iD

�Xg

� aP

D�Xg; �7�

where

D � �ix�4 � �ix�3�DP � DA � DC� � �ix�2�x2P � x2

A

� x2C � DPDA � DPDCA � DADCP�

� �ix� DP�x2A

� � x2CA� � DA�x2

P � x2CP�

� DCPx2A � DCAx2

P

�� x2Px

2A � x2

Px2CA

� x2Ax2

CP�� a4�ix�4 � a3�ix�3 � a2�ix�2 � a1�ix� � a0: �8�

The other parameters are de®ned as follows:

DP � gPxP � CP=MP, bandwidth of the P-structure;

DA � gAxA � CA=MA, bandwidth of the A-structure;

DCP � CC=MP;DCA � CC=MA; DC � DCP � DCA, cou-

pling damping parameter;

xCP ��KC=MP

p;xCA �

��KC=MA

p,

xC ��x2

CP � x2CA

p, coupling sti�ness parameters.

The relative vibrational energy of the P-structure, EP,

is de®ned as

EP � 1

2MP

_X 2P �

1

2KPX 2

P : �9�

It can be shown using Fourier transform methods

that the time-averaged total relative energy of the P-

structure is [16]

In order to simplify the computation, the dampings

of the P- and A-structures are assumed to be zero. The

simpli®ed expression of the time-averaged energy of the

P-structure can be obtained by substituting DP � DA � 0

into Eq. (10):

where b1 � xA=xP is the frequency ratio of the A-

structure to the P-structure and, b2 � KC=KP, the sti�-

ness ratio of the passive coupling element to the P-

structure.

The optimizing conditions for the strategy to mini-

mize the time-averaged vibrational energy of the P-

structure are

EP � MPh _X 2P �t�i �

1

2pMP

Z 1

ÿ1S _XP

_XP�x� dx � 1

2pMPSgg

Z �1

ÿ1

�ixaP��ixaP��DD�

dx

� ÿMPSgg

2

a1a2 ÿ a0a3 � x2A � x2

C

ÿ �2a3a4 ÿ 2 x2

A � x2C

ÿ �ÿ DA � DC� �2h i

a1a4

a4 a4a21 � a0a2

3 ÿ a1a2a3� � : �10�

�EP � MPSgg

2

x2PDCbl�1� l��1ÿ b2

1�2 ÿ 2l�1� l�2�1ÿ b21�b2 � �1� l�4b2

2c � D3C�1� l��l� b2

1�x2

PD2Cl�1ÿ b2

1�2; �11�


oEP

ob2

� 0 andoEP

oDC� 0: �12�

The optimum parameters of the passive coupling el-

ement can be obtained by substituting Eq. (11) into Eq.

(12):

KC-opt � KP

l 1ÿ b21

ÿ ��1� l�2 and

CC-opt � MPxP

1� l

��l 1ÿ b2

1

ÿ �2

�1� l��l� b21�

vuut :

�13a; b�

4.2. Case 2: Active control with classical linear optimal

algorithm

In classical linear optimal control, the desired active

control force, Ua, is assumed to be given by the fol-

lowing:

Ua�t� � GX �t�; �14�where G is a pre-determined constant gain matrix.

The optimal constant gain matrix, G, can be deter-

mined by minimizing a linear quadratic performance

index, J, de®ned as

J �Z 1

0

X TQX� � U T

a RUa

�dt; �15�

where Q and R are weighting matrices whose magni-

tudes are assigned according to the relative importance

attached to the state variables and to the control forces

in the minimization procedure.

The gain matrix can be determined from the solution

of the well known time invariant Ricatti equation [17] as

G � ÿ 1

2Rÿ1BTP �16�

in which the symmetric matrix, P, satis®es the Riccati

matrix equation

PAÿ 1

2PBRÿ1BTP � ATP � 2Q � 0 : �17�

The following are for two semi-active control cases.

The semi-active device can be considered as a passive

damper with time-varying, controllable damping and

sti�ness properties. In its simplest form, the damping

coe�cient CC and sti�ness coe�cient KC can only be

switched to either one of two ®xed values: in the acti-

vated (i.e. on) state, CC � CC-opt and KC � KC-opt; here,

the semi-active device behaves as an energy storage and

energy transfer device for the purpose of extracting

vibrational energy from the P-structure. In the deacti-

vated (i.e. o�) state, CC � 0, and KC � 0; the semi-active

device now yields quickly, rapidly reducing the reaction

force level to zero and dissipating the strain energy ac-

cumulated at the activated state. Thus, the passive

damper provides the desired control force in the acti-

vated state which depends on the semi-active control

algorithm.

4.3. Case 3: Semi-active control with classical linear

optimal algorithm

For this ®rst semi-active control case, the control

force, Usa1, is that exerted by the semi-active device ac-

cording to the classic linear optimal control algorithm. It

is positive when the semi-active damper is in ``tension''

with the sign convention of Fig. 2. If the product of the

passive control force, UP (Eq. (6)) and the required ac-

tive control force, Ua (Eq. (14)) is positive, the semi-

active control device is at the activated state.

Thus, the on±o� semi-active device can produce the

desired force, Usa1, only when feasible, i.e.

o� state when

UaUP < 0;Usa1 � 0; �18a�

on state when

UaUP P 0; Usa1 � ÿKC-optZ ÿ CC-opt_Z: �18b�

In which Ua is the LQ required active control force.

4.4. Case 4: Semi-active control with instantaneous power

absorption algorithm

The instantaneous energy balance for the P-structure

[18] is

PPg � PPf � oEP

ot� P diss

P ; �19�

where PPg � ÿMP�Xg _XP is the input power by the ground

motion; PPf � ÿUsa2_XP is the input power by the semi-

active control force; P dissP � CP

_XP_XP � 2MPxPnP

_X 2P is the

power dissipated by the P-structure.

The energy-changing rate of the P-structure at any

instant is therefore

oEP

ot� MP

_XP

�ÿ �Xg ÿ Usa2

MP

ÿ 2nPxP_XP

�; �20�

where Usa2 is the control force by the semi-active control

in this case.

From Eq. (20), if the operating state of the interac-

tion element exists at any instant for which

PPf � ÿUsa2_XP < 0, then the semi-active control force,

Usa2, reduces the rate of energy increase of the P-struc-

ture from that using Usa2 � 0. And PPf < 0 means that

the semi-active controller absorbs power from the P-

structure. That is to say, oEP=ot for the controlled sys-

tem will be less than that for the uncontrolled system

when the semi-active controller absorbs power from the


P-structure. Since the semi-active control force at the

activated state is supplied by the passive controller, thus

the ``o�±on'' control algorithm can be obtained as fol-

lows:

o� state when

UP_XP6 0; Usa2 � 0; �21a�

on state when

UP_XP > 0; Usa2 � ÿKC-optZ ÿ CC-opt

_Z: �21b�

5. Numerical results

As stated above, the control forces of these semi-

active control strategies at the activated state are sup-

plied by the optimum passive controller. Thus, the

control e�ectiveness depends on the determination of

optimum parameters of passive coupling element.

For a given P-structure: MP � 1:50� 105 kg, xP �10:55 rad/s, DP � 0:422 1/s . The dynamic behavior of

the A-structure is fully characterized by specifying the

parameters: the mass ratio l, the natural frequency ratio

b1 and the damping ratio n1 � DA=DP. To clearly dem-

onstrate the in¯uence of structural parameters of the P±

A structure on the optimum parameters of the passive

coupling element and passive control e�ectiveness, a

control performance index is de®ned:

R1 � XPh icontrolled

XPh iuncontrolled

; �22�

where XPh icontrolled denotes RMS value of relative dis-

placement of the P-structure with optimum passive

control. XPh iuncontrolled denotes RMS value of relative

displacement of the P-structure without control.

Fig. 3 shows the in¯uence of structural parameters of

the P±A structure on the optimum sti�ness and damping

values of the passive coupling element. If the natural

frequency of the A-structure is equal to that of the P-

structure, the optimum sti�ness and damping values of

the passive coupling element are equal to zero. If the

natural frequency of the A-structure is higher than that

of the P-structure, the optimum sti�ness is equal to zero,

only the damping value exists, which means that the

mechanism of the control strategy is to dissipate the

energy by the passive coupling element. Finally, when

the natural frequency of the A-structure is lower than

that of the P-structure, there exist simultaneously opti-

mum sti�ness and damping values. The optimum sti�-

ness and damping values increase as the frequency ratio

b1 becomes smaller; optimum sti�ness also depends

largely on the mass ratio l, it increases as l decreases

(i.e. the mass of the A-structure increases) while the

mass ratio l has hardly any in¯uence on the optimum

damping value. When b1 < 1, the interaction between

the P- and the A-structures by the passive coupling ele-

ment becomes stronger as the mass ratio l and natural

frequency ratio b1 decrease. The control strategy is to

remove energy associated with vibration from the P-

structure in an optimum manner.

Fig. 4 shows the index R1 at various mass ratios l,

natural frequency ratios b1 and damping ratios n1. It is

apparent that the optimal control e�ectiveness depends

largely on the natural frequency ratio b1 of the A-

structure to the P-structure. The control e�ectiveness of

the passive control strategy increases as the natural

frequency of the A-structure is far from that of the P-

structure. Fig. 4(a) shows that the control e�ectiveness

can be improved by increasing the damping of the A-

structure when the natural frequency of the A-structure

is near the P-structure. We can see from Fig. 4(b) that

increasing the mass of the A-structure can e�ectively

reduce the vibration response of the P-structure. This

reminds one of the important aspects of choosing a

Fig. 3. Optimal parameters of coupling element with di�erent

mass ratios l and natural frequency ratios b1: (a) optimal

sti�ness parameter b2-opt; and (b) optimal damping parameter

DC-opt:


suitable A-structure for reducing the vibration responses

in the P-structure besides optimally designing the passive

coupling element.

In the next section, two examples corresponding to

two distinctly di�erent A-structures are given to illus-

trate the performance of these proposed control strate-

gies. The structural parameters of this P±A structure

shown in Fig. 2 are listed in Table 1.

In Example 1, the weighting matrices appearing in

the performance index (Eq. (15)) are assumed to be

Q1 � k1

1 0 0 00 0 0 00 0 1 00 0 0 0

26643775 and R1 � I: �23�

Here the state weighting matrix, Q1, is chosen to mini-

mize the mean squared relative displacement and ve-

locity.

In Example 2, by selecting the weighting matrices as

shown in Eq. (24), one may view the design problem as

minimizing a performance index which is proportional

to the weighted kinetic and potential energies of the

structure:

Q2 � k2

x2P 0 0 0

0 0 0 00 0 1 00 0 0 0

26643775 and R2 � I : �24�

The variables ki (i� 1,2) in Eqs. (23) and (24) are

introduced to allow for adjustment in the strength of

control. A larger value of k leads to a stronger control.

The weighting variables are given as

k1 � 3M2P

50; k2 � M2

P

40x2P

: �25�

The gain control matrices as de®ned by Eq. (14) were

obtained by solving the Riccati Eq. (17), and substitut-

ing the symmetric matrix, P, into Eq. (16).

The optimum values of sti�ness and damping of the

passive coupling element can be obtained according to

Eq. (13a,b).

Example 1: KC-opt � 2:936� 106 N/m, CC-opt �2:54� 105 kg/s.

Example 2: CC-opt � 2:53� 105 kg/s.

It is instructive to examine the control e�ect on

structural behavior. The system control parameters re-

sulted from the application of passive and active cou-

pling element are compared with those without control

in Table 2. In Example 1, the passive and active control

systems only slightly change the natural frequency of

the P-structure from xP � 10:55 to 11.34 rad/s and

xP � 10:14 rad/s. On the other hand, the corresponding

damping ratio can be largely increased from nP � 0:02 to

Fig. 4. Structural control index R1 with di�erent mass ratios l,

natural frequency ratios b1 and dampind ratio n1: (a) l � 5=3;

and (b) n1 � 1.

Table 1

Structural parameters of the P±A structure

Primary structure Auxiliary structure in Example 1 Auxiliary structure in Example 2

MP � 1:50� 105 kg MA � 0:75� 105 kg MA � 0:90� 105 kg

KP � 1:67� 107 N/m KA � 2:087� 106 N/m KA � 2:044� 107 N/m

CP � 6:33� 104 kg/s CA � 3:165� 104 kg/s CA � 5:425� 104 kg/s

nP � 0:02 nA � 0:04 nA � 0:02

xP � 10:55 rad/s xA � 5:275 rad/s xA � 15:07 rad/s


nP � 0:1521 and nP � 0:1255. Similarly, in the second

example, the natural frequency of the P-structure

changes little for the control case, but the passive and

active controls signi®cantly increase the damping ratio

of the P-structure from nP � 0:02 to nP � 0:1001 and

nP � 0:1115. Thus, the main e�ect of passive and active

controls is to signi®cantly increase the damping of the P-

structure while only slightly altering the natural fre-

quency, and consequently the associated sti�ness. Since

the semi-active controller can e�ectively adjust the

damping of the P±A structure, from the above section, it

can be predicted that it can achieve the same control

e�ectiveness as passive and active controls.

The seismic response of the P-structure is also nu-

merically simulated. Table 3 lists absolute acceleration

and relative displacement subjected to El Centro 1940

NS with normalized peak acceleration of 140.0 cm/sec2.

In the same table, the corresponding peak values of

control force are also listed. From this table, the two

semi-active control strategies based on the optimum

passive coupling element are always superior to the

optimum passive control. Speci®cally, for semi-active

control, the instantaneous power absorption control

strategy shows much improvement in reducing absolute

acceleration and relative displacement of the P-struc-

ture, except for the RMS in Example 1. Furthermore,

the passive and semi-active control strategies are com-

parable to the active control strategy with the same

degree control force.

Referring to Table 3, the passive control and semi-

active control strategies based on optimum passive

coupling element are more e�ective in reducing the rel-

ative displacement of the P-structure if there exists a

rigid A-structure (Example 2). On the other hand, they

are more e�ective for absolute acceleration of the P-

structure if the A-structure is much ¯exural than the P-

structure (Example 1).

Fig. 5 shows the relative displacement of the P-

structure in Example 1 versus time with di�erent kinds

of control strategies under the El-Centro 1940 NS

earthquake. Fig. 5(a) shows that a moderate relative

displacement reduction for the P-structure can be

achieved via optimum passive control with a ¯exural A-

structure. Fig. 5(b) and (c) show that there is an im-

provement with the semi-active control strategies based

on the optimum passive coupling element. The instan-

taneous power absorption algorithm is always superior

to the conventional linear quadratic case, as is shown in

Fig. 5(d).

Fig. 6 shows the absolute acceleration of the P-

structure versus time under passive and semi-active

controls. The results over a period of time of 40 s dem-

onstrate that a signi®cant reduction of the absolute ac-

celeration of the P-structure can be achieved by di�erent

kinds of control strategy with a ¯exural A-structure. Fig.

6(b)±(d) also indicate that the control e�ectiveness can

be improved by using the semi-active control strategies

based on the optimum passive coupling element, and the

instantaneous power absorption algorithm is better than

the linear quadratic optimum algorithm.

Fig. 7 shows the comparison between the relative

displacement and absolute acceleration time-histories of

Table 2

The comparison between structure behavior with and without control

Uncontrolled Passive control Active control

Example 1 x1;2 5.275, 10.55 7.375, 11.34 5.27, 10.14

n1;2 0.04, 0.02 0.1296, 0.1521 0.2605, 0.1255

Example 2 x1;2 10.55, 15.07 10.73, 14.64 10.68, 14.74

n1;2 0.02, 0.02 0.1001, 0.1156 0.1115, 0.0892

Table 3

Peak values and RMS of the P-structure subject to El Centro 1940 NS

Noncontrol Case 1 Case 2 Case 3 Case 4

Example 1 Acceleration

(gal)

Peak 339.41 225.31 220.94 223.92 208.52

RMS 100.60 45.992 48.002 44.222 44.737

Displacement

(cm)

Peak 3.0398 2.5532 2.4155 2.4565 2.2009

RMS 0.90315 0.54615 0.49182 0.51933 0.52279

Control force Peak 13.064 10.667 13.375 26.260

Example 2 Acceleration

(gal)

Peak 339.41 293.07 265.40 292.09 283.48

RMS 100.60 59.917 55.65 59.625 57.282

Displacement

(cm)

Peak 3.0398 2.3385 2.1245 2.3291 2.2461

RMS 0.90315 0.5007 0.46718 0.49857 0.48368

Control force Peak 7.1779 7.0913 7.1859 7.3475


Fig. 5. Relative displacement of the P-structure versus time using passive or semi-active control in Example 1.


the P-structure with passive control and those without

control under the El Centro 1940 NS earthquake. From

Fig. 7(a), we can see that a control system with a sti� A-

structure is favorable to reduce the relative displacement

of the P-structure. Fig. 7(b) shows that passive control

with a sti� A-structure can be e�ective in reducing the

absolute acceleration of the P-structure, but is not so

e�ective as that in reducing relative displacement.

The ®gures for the relative displacement and absolute

acceleration of the P-structure in Example 2 versus time

Fig. 6. Absolute acceleration of the P-structure versus time using passive or semi-active control in Example 1.


under the El Centro 1940 NS earthquake would give us

the same results as those in Example 1, which are not

given in detail due to limited space.

6. Conclusions

A new and e�ective control approach which sets up

passive, semi-active or active control systems between

two parallel structures was proposed. The strategy of

this control approach is to reduce the vibration response

of the P-structure by using the interaction of the two

parallel structures during earthquake action.

Firstly, the optimum values of sti�ness and damping

of a passive-coupling element between the two parallel

structures were derived for a simpli®ed model of the P±

A structure with two coupled single-degree-of-freedom

oscillators. The general expressions of optimum sti�ness

and damping include the following: mass ratio of the P-

structure to the A-structure; natural frequency ratio and

damping ratio of the A-structure to the P-structure. The

in¯uence of structural parameters of a P±A structure

such as mass ratio l, frequency ratio b1 and damping

ratio n1 on the optimum parameters and control e�ec-

tiveness was discussed in detail.

The e�ectiveness of this strategy depends not only on

the determination of structural parameters of the passive

coupling element but also the structural parameters of

the P±A structure. For a suitable P±A structure, the

proposed vibration control method can be quite robust

and e�ective for the vibration reduction of the P-struc-

ture. The control e�ectiveness increases as the mass of

the A-structure increases, and the natural frequency of

the A-structure is further from that of the P-structure. A

rigid A-structure is more useful in decreasing the relative

displacement of the P-structure, while the ¯exural A-

structure is more e�ective in reducing the absolute ac-

celeration of the P-structure.

Low cost, reliability, stability, and simplicity of de-

sign were the advantages that make semi-active control

an option for response reduction of structures under

seismic loads. A semi-active on±o� control algorithm

based on instantaneous power-absorption was then

presented and compared with the traditional linear

quadratic control case. This on±o� algorithm for semi-

active control was particularly attractive and feasible

because of its simplicity and e�ectiveness.

Finally, numerical results including the RMS values

and time histories of relative displacement and absolute

acceleration of the P-structure due to El-Centro 1940 NS

excitation were presented to demonstrate the dramatic

e�ectiveness of these strategies in controlling structural

vibration responses of the P-structure under earthquake

excitation. The results show that the control perfor-

mance of the proposed semi-active control strategy is

better than the optimum passive control strategy and

comparable to the fully active control case. The superi-

ority of the instantaneous power absorption semi-active

Fig. 7. Relative displacement and absolute acceleration of the P-structure versus time using passive control in Example 2.


control to the conventional linear quadratic control law

is also numerically illustrated.

Acknowledgements

The work was carried out when the ®rst author was a

research fellow at Kyoto University (Japan). This re-

search was funded by the National Natural Science

Foundation of China (Grant no. 59908003) and the

Japan Society for Promotion of Science (JSPS). The

supports are greatly appreciated and helpful suggestions

from the reviewers are also acknowledged. The authors

wish to thank Mr. Terry Clayton, Asian Institute of

Technology, for his help in checking the manuscript.

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A study on interaction control for seismic response of parallel structures