a new approach to the modeling of distributed structures for control meirovitch

Journal of the Franklin Institute 338 (2001) 241–254

A new approach to the modeling of distributedstructures for control$

L. Meirovitch*, T.J. Stemple1

Department of Engineering Science and Mechanics (MC 0219), Virginia Polytechnic Institute

and State University, Blacksburg, VA 24061, USA

Abstract

Building structures represent complex distributed-parameter systems. The motion of such

systems is described by partial differential equations complemented by suitable boundaryconditions. For control design purposes, distributed-parameter systems must be discretized inthe spatial variables. But, if the discrete model is not sufficiently accurate, controls designed on

the basis of the discrete model can destabilize the actual distributed structure, in which case thecontrols are said to be sensitive to discretization errors. This paper presents a new approach tothe discretization of distributed structures yielding accurate discrete models of relatively low

order. A numerical example illustrates how controls can be designed for a complex structuresubjected to earthquake excitations. # 2001 The Franklin Institute. Published by ElsevierScience Ltd. All rights reserved.

1. Introduction

Structures are basically distributed-parameter systems with complex geometry.Assuming that base isolation is an indispensable part of seismic control design, abuilding structure can be modeled as an assemblage of elastic members, such ascolumns and beams, mounted on a rigid base, as shown in Fig. 1. Strictly speaking,the mathematical formulation for such a model consists of one ordinary differentialequation for the horizontal translation of the base and a certain number of partialdifferential equations for each of the elastic members, where the latter aresupplemented by a suitable number of boundary conditions to be satisfied at the

$Supported by the NSF Research Grant CMS-9423575.

*Corresponding author. Fax: +1-540-231-4574.

E-mail address: [email protected] (L. Meirovitch).1Now at Moxa Technologies Co. Shing Tien City, Taipei, Taiwan, ROC.

0016-0032/01/$20.00 # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 8 2 - X

interface between any two adjacent elastic members, or between the bottom columnsand the base. Clearly, such a mathematical formulation is not practical for controldesign, so that the distributed-parameter model must be approximated by a discretemodel.Distributed-parameter structures can be discretized in two distinct ways, namely,

through lumping of the distributed parameters, or through series discretization [1].Lumped-parameter models tend to be very inaccurate, which can cause problems incontrol of structures. It is well known that active controls designed improperly arecapable of destabilizing a system. A classical example is that in which the systemparameters are not known very well. When incomplete knowledge of the parameterscan lead to instability, the controls are said to be sensitive with respect to variations inthe system parameters [2]. Controls that perform well in spite of variations in thesystem parameters are said to be robust. A somewhat different situation can arisewhen control design based on a discrete model can destabilize the actual distributedstructure [3]. Whereas this type of control sensitivity can occur both in lumped-parameter models and in series-discretized models, the problem is much more criticalin the first than in the second. Hence, we rule out lumped-parameter models fromfurther consideration.The classical series discretization procedure is the Rayleigh–Ritz method, whereby

the elastic displacement is assumed to be a linear combination of known admissiblefunctions multiplied by undetermined coefficients. Then, the coefficients aredetermined by rendering Rayleigh’s quotient stationary [1]. The finite elementmethod, which is a variant of the Rayleigh–Ritz method, is by far the most popularseries discretization procedure. The basic difference between the two is that theclassical Rayleigh–Ritz method uses global functions as admissible functions,defined over the entire elastic member; and the finite element method uses localfunctions, defined over a finite element and referred to as interpolation functions.The Rayleigh–Ritz method improves accuracy by increasing the number ofadmissible functions, whereas the finite element method enhances accuracy byincreasing the number of finite elements. For single elastic members, the classicalRayleigh–Ritz method has the advantage that it can yield good accuracy with arelatively small number of degrees of freedom, but has the disadvantage that theadmissible functions tend to be complicated. By contrast, the finite element methodhas the advantage that the admissible functions tend to be simple, generally low-degree polynomials, but has the disadvantage that it requires a large number ofdegrees of freedom for good accuracy. Another discretization procedure is thehierarchical finite element method, which improves accuracy by keeping the numberof finite elements constant and increasing the number of interpolation functions perelement. The hierarchical finite element method combines many of the advantages ofboth the classical Rayleigh–Ritz method and the finite element method.In the case of structures as that shown in Fig. 1, the substructure synthesis

method, which is basically an extension of the Rayleigh–Ritz method to flexiblemultibody systems, has many advantages [1]. This paper develops a new approach tothe modeling of frame structures of the type shown in Fig. 1. The approachrepresents a combination of substructure synthesis and the hierarchical finite element

L. Meirovitch, T.J. Stemple / Journal of the Franklin Institute 338 (2001) 241–254242

method, and it exhibits superior accuracy, thus avoiding questions of controlsensitivity. Then, the paper shows how controls can be designed on the basis ofdiscrete models so derived.

2. Structural model

The system shown in Fig. 1 represents a model of a base-isolated structureconsisting of an N-story elastic frame clamped to a rigid slab. The base of thebuilding is capable of moving horizontally relative to the ground, and is connected tothe ground by an elastic spring and a viscous damper. In addition, horizontal controlforces act on the base and throughout the structure.The structural members of the frame are distributed-parameter beams and

columns, modeled as Euler–Bernoulli beams. Beams and columns connected at agiven joint undergo the same displacement and rotation at that joint. The axialextension of both the beams and columns is neglected, and consequently the twoends of a beam undergo no transverse displacement, and the tips of columnscorresponding to the same story undergo the same horizontal displacement.However, the effect of axial forces working throughout the shortening of thecolumns due to bending is included in the model.

3. Derivation of hybrid equations of motion

In the first place, it is assumed that the motion of the ground and the motion of thebase are in the horizontal direction alone. Moreover, the motion of a typical point on

Fig. 1. Model of a base-isolated structure.

L. Meirovitch, T.J. Stemple / Journal of the Franklin Institute 338 (2001) 241–254 243

the structure shown in Fig. 1 can be regarded as a superposition of the motion of thebase and the elastic displacement of the point relative to the base. In view of this, adistinction must be made between points on beams, which undergo only elasticdisplacements in the vertical direction, and points on the columns, which undergoboth rigid-body and elastic displacements in the horizontal direction. Moreover, weassume that the elastic members do not deform axially.To describe the motion of the system, we denote the absolute displacement of the

ground by ugðtÞ, the absolute displacement of the base by ubðtÞ and the rigid-bodydisplacements of the floors relative to the base by uiðtÞ (i ¼ 1; 2; . . . ;N), all threetypes of displacement taking place in the horizontal direction. Moreover, we denotethe elastic displacement of a typical point on member j relative to the base bywjðx; tÞ, where j ¼ 1; 4; . . . ; 3N � 2 for the left columns, j ¼ 2; 5; . . ., 3N � 1 for thebeams and j ¼ 3; 6; . . . ; 3N for the right columns.We propose to derive the equations of motion by means of the extended

Hamilton’s principle, which requires the kinetic energy T , potential energy V andvirtual work dWnc of the nonconservative forces, such as the control forces.Damping forces can be generated by means of a Rayleigh’s dissipation function andcan be included in the virtual work. The extended Hamilton’s principle can be statedin the form [1]Z t2

t1

ðdT � dV þ dWncÞ dt ¼ 0; dub ¼ 0; du1 ¼ du2 ¼ � � � ¼ duN ¼ 0;

dw1 ¼ dw2 ¼ � � � ¼ dw3N ¼ 0; t ¼ t1; t2: ð1Þ

For the structure of Fig. 1, the kinetic energy has the expression

T ¼ 1

2mb _u

2b þ

1

2

XNi¼1

mið _ub þ _uiÞ2 þ1

2

X3N�1

j¼2;5;...

Z ‘j

0

rj _w2j dx

þ 1

2

X3N�2;3N

j¼1;4;...3;6;...

Z ‘j

0

rjð _ub þ _wjÞ2 dx; ð2Þ

where mb is the mass of the base, mi the total mass of a floor, rj a mass density and ‘jan elastic member length. The potential energy is assumed to have the form

V ¼ 1

2kbðub � ugÞ2 þ

1

2

X3N�1

j¼2;5

Z ‘j

0

EIjðw00j Þ

2 dx

þ 1

2

X3N�2;3N

j¼1;4;...3;6;...

Z ‘j

0

½EIjðw00j Þ

2 � Pjðw0jÞ2 dx ð3Þ

in which kb is the spring constant of the isolation system, EIj is a bending stiffnessand Pj an axial force; as usual, primes denote differentiations with respect to the


spatial variable x. The virtual work of the nonconservative forces can be written as

dWnc ¼Fbdub þXNi¼1

Fiðdub þ duiÞ þX3N�1

j¼2;5;...

Z ‘j

0

fjdwj dx

þX3N�2;3N

j¼1;4;...3;6...

Z ‘j

0

fjðdub þ dwjÞ dx; ð4Þ

where FbðtÞ is the horizontal force on the base, FiðtÞ the horizontal force on floor iand fjðx; tÞ the force density on the elastic member j. As indicated earlier in thissection, damping forces can be generated by means of a Rayleigh’s dissipationfunction and accounted for separately in the virtual work. To this end, we assumethat the Rayleigh dissipation function has the form

F ¼ 1

2cbð _ub � _ugÞ2 þ

1

2

X3Nj¼1

Z ‘j

0

cj _w2j dx ð5Þ

in which cb is the coefficient of viscous damping of the isolation system and cj arecoefficients of viscous damping per unit length of the elastic members j. Then, thedamping forces can be generated from F by writing

F *b ¼ �@F

@ _ub; f *j ¼ �@F

@ _wj; j ¼ 1; 2; . . . ; 3N; ð6Þ

where F is the integrand in Eq. (5). With the understanding that the forces in Eq. (4)are due to controls alone, we can rewrite the virtual work in the form

dWnc ¼ � @F

@ _ubþ Fb

� �dub þ

XNi¼1

Fiðdub þ duiÞ �X3Nj¼1

Z ‘j

0

@F

@ _wjdwj dx

þX3N�1

j¼2;5;...

Z ‘j

0

fjdwj dxþX3N�2;3N

j¼1;4;...3;6;...

Z ‘j

0

fjðdub þ dwjÞ dx: ð7Þ

Inserting Eqs. (2)–(4) and (7) into Eq. (1) and carrying out the usual steps, wecan obtain a set of hybrid equations of motion, ordinary differential equations forubðtÞ; u1ðtÞ; u2ðtÞ; . . . ; uNðtÞ and boundary-value problems for w1ðx; tÞ;w2ðx; tÞ; . . . ;w3Nðx; tÞ for the elastic displacements of the beams and columns, where theboundary-value problems consist of partial differential equations and suitableboundary conditions. Because hybrid sets of equations cannot be used for controldesign, it is necessary to spatially discretize the boundary-value problems; this isdone in the next section.

4. Discretization of the boundary-value problems

Spatial discretization of the boundary-value problems can be carried out by meansof the finite element method or by substructure synthesis [1]. The former tends to


yield discrete models with extremely large numbers of degrees of freedom and thelatter has difficulty in handling systems with closed structural loops, such as theframe structure of Fig. 1. A method combining the advantages of both the finiteelement method and substructure synthesis, but avoiding some of their disadvan-tages, is the hierarchical finite element method [1]. The hierarchical finite elementmethod represents a special version of the finite element method in the sense thataccuracy of the discrete model is improved by increasing the number of interpolationfunctions per element rather than increasing the number of elements, as in theordinary finite element method. For this reason, the hierarchical is referred to as the‘‘p-version’’ and the ordinary as the ‘‘h-version’’ of the finite element method [1]. Itshould be noted that hierarchical interpolation functions are such that the elementmass and stiffness matrices associated with adjacent elements are not affected.At this point, we abandon generalities to some extent and consider a four-story

structure. In the context of the hierarchical finite element method, columns andbeams are assumed to consist of one finite element each, and each element is assignedsix hierarchical degrees of freedom. Because the beams and columns areinextensional, the nodal translational displacements coincide with the horizontaldisplacements u1ðtÞ; u2ðtÞ; u3ðtÞ and u4ðtÞ of the floors, and the nodal rotationaldisplacements are y1ðtÞ; y2ðtÞ; . . . ; y8ðtÞ, as shown in Fig. 1. The displacement of atypical point on a beam or a column is obtained by means of Hermite cubics used asinterpolation functions between nodal displacements, in addition to the contributionfrom the hierarchical functions. Hence, we express the displacement of a point onelement j in the form

wjðx; tÞ ¼ uTj ðxÞqeðtÞ; j ¼ 1; 2; . . . ; 12; ð8Þ

where the components of the vector uj are either zero or appropriately chosen shapefunctions and qe ¼ ½u1 u2 u3 u4 y1 y2 . . . y8 q1;1 q2;1 . . . q6;1 q1;2 q2;2 . . . q6;2 . . .q1;12 q2;12 . . . q6;12T is an 84-dimensional elastic displacement vector for the wholestructure, in which the subscript i in qi; j identifies the hierarchical function and j thestructural member. As indicated, the interpolation functions for the nodaldisplacements are Hermite cubics (Fig. 2). In terms of a nondimensional localcoordinate x they are given by

h1ðxÞ ¼ 1� 3x2 þ 2x3; h2ðxÞ ¼ x� 2x2 þ x3;

h3ðxÞ ¼ 3x2 � 2x3; h 4ðxÞ ¼ �x2 þ x3;04x41: ð9Þ

Moreover, as hierarchical functions, we use the eigenfunctions of a uniform fixed–fixed Euler–Bernoulli beam (Fig. 3), which can be expressed in the computationally

Fig. 2. Hermite cubics.


useful form as

crðxÞ ¼ ðcos lr þ sin lr � e�lrÞsin lrxþ ðcos lr � sin lr � e�lrÞ cos lrxþ ðe�lr cos lr � 1Þe�lrð1�xÞ þ sin lr e�lrx; r ¼ 1; 2; . . . ; 04x41; ð10Þ

where lr satisfies the characteristic equation cos lr cosh lr ¼ 1. The six lowestcharacteristic values are given by l1 ¼ 4:730041; l2 ¼ 7:853205; l3 ¼ 10:99561;l4 ¼ 14:13717; l5 ¼ 17:27876; l6 ¼ 20:42035. Now we can be more specific aboutthe nonzero components of ujðxÞ. In particular, for the lower left column, we have

j1;1ðxÞ ¼ h1ðx=‘1Þ; j5;1ðxÞ ¼ ‘1h2ðx=‘1Þ; j12þr;1ðxÞ ¼ crðx=‘1Þ;r ¼ 1; 2; . . . ; 6 ð11aÞ

the first-story beam is characterized by

j5;2ðxÞ ¼ ‘2h2ðx=‘2Þ; j6;2ðxÞ ¼ ‘2h4ðx=‘2Þ; j18þrðxÞ ¼ crðx=‘2Þ;r ¼ 1; 2; . . . ; 6 ð11bÞ

and the lower right column by

j1;3ðxÞ ¼ h1ðx=‘1Þ; j6;3ðxÞ ¼ ‘1h2ðx=‘1Þ; j24þrðxÞ ¼ crðx=‘1Þ;r ¼ 1; 2; . . . ; 6: ð11cÞ

Moreover, for the second from bottom left column we write

j1;4ðxÞ ¼ h3ðx=‘1Þ; j2;4ðxÞ ¼ h1ðx=‘1Þ; j5;4ðxÞ ¼ ‘1h4ðx=‘1Þ;

j7;4ðxÞ ¼ ‘1h2ðx=‘1Þ; j30þrðxÞ ¼ crðx=‘1Þ; r ¼ 1; 2; . . . ; 6: ð11dÞ

The vectors uj for the remaining elements can be determined by following the samepattern.

Fig. 3. Fixed–fixed Euler–Bernoulli eigenfunctions.


The fixed–fixed shape functions given by Eq. (10) satisfy the orthogonalityproperties:Z 1

0

cmðxÞcnðxÞ dx ¼ 0;

Z 1

0

c00mðxÞc00

nðxÞ dx ¼ 0; m 6¼ n: ð12aÞ

Moreover, the second derivative of the fixed–fixed shape functions are orthogonal tothe second derivative of the Hermite cubics, orZ 1

0

h00mðxÞc00nðxÞ dx ¼ 0; m ¼ 1; 2; 3; 4; n ¼ 1; 2; . . . : ð12bÞ

The hierarchical finite element method in conjunction with the interpolationfunctions given by Eqs. (9) and (10) permit accurate modeling of the structure withfar fewer degrees of freedom than the h-version of the finite element method. In thecase at hand, the structure is modeled by 84 degrees of freedom, for a total of 85 forthe whole system, including the motion of the base.

5. Derivation of discrete equations of motion

Inserting Eq. (8) into Eq. (2), the discretized kinetic energy takes the form

T ¼ 1

2mb _u

2b þ

1

2

X4i¼1

mið _ub þ _uiÞ2 þ1

2

X11j¼2;5;...

Z ‘j

0

rj _qTe uju

Tj _qe dx

þ 1

2

X10;12j¼1;4;...3;6;...

Z ‘j

0

rjð _ub þ uTj _qeÞ

Tð _ub þ uTj _qeÞ dx ¼ 1

2_qM _q; ð13Þ

where

qðtÞ ¼ ½ubðtÞ qTe ðtÞT ð14Þ

is the 85-dimensional overall configuration vector,

M ¼m aT

a Me

" #ð15Þ

is the system mass matrix, in which m is the total mass of the system,

a ¼m

0

" #þ

X10;12j¼1;4;...3;6;...

Z ‘j

0

rjuj dx ð16Þ

in which m ¼ ½m1 m2 m3 m4T and

Me ¼ diag½m1 m2 m3 m4 0 0 . . . 0 þX12j¼1

Z ‘j

0

rjujuTj dx: ð17Þ


Similarly, using Eq. (3), the discretized potential energy is

V ¼ 1

2kbðub � ugÞ2 þ

1

2

X11j¼2;5;...

Z ‘j

0

qTe EIju00j u

00Tj qe dx

þ 1

2

X10;12j¼1;4;...3;6;...

Z ‘j

0

qTe ðEIju00j u

00Tj � Pju

0ju

0Tj Þ dx ¼ 1

2qTKq� kbugub þ

1

2kbu

2g;

ð18Þ

where

K ¼kb 0T

0 Ke

" #ð19Þ

in which

Ke ¼X11

j¼2;5;...

Z ‘j

0

EIju00j u

00Tj dxþ

X10;12j¼1;4;...3;6;...

Z ‘j

0

ðEIju00j u

00Tj � Pju

0ju

0Tj Þ dx: ð20Þ

Finally, using Eq. (7) in conjunction with Eqs. (5) and (8), the discretized virtualwork becomes

dWnc ¼ �cbð _ub � _ugÞ þ Fb þX4i¼1

Fi þX10;12

j¼1;4;...3;6;...

Z ‘j

0

fj dx

2664

3775dub

þX4j¼1

Fidui þX12j¼1

� _qTe

Z ‘j

0

cjujuTj dxþ

Z ‘j

0

fjuTj dx

� �dqe ¼ QTdq;

ð21Þ

where

Q ¼ Qd þQc þD ð22Þ

in which

Qd ¼�cb _ub

�P12

j¼1R ‘j0 cjuju

Tj dx _qe

" #ð23Þ

is a generalized damping force vector,

Qc ¼ Fb þX4i¼1

Fi þX10;12

j¼1;4;...3;6;...

Z ‘j

0

fj dx F1 F2 F3 F4

X12j¼1

Z ‘j

0

fjuTj dx

2664

3775T

ð24Þ


is a generalized control vector and

D ¼ ½kbug þ cb _ug 0 0 . . . 0T ð25Þ

is a disturbance vector due to the earthquake.Inserting Eqs. (13), (18), (21)–(23) into Eq. (1) and carrying out the indicated

operations, we obtain the discrete equations of motion

M�qþ C _qþ Kq ¼ Qc þD; ð26Þ

where M is the mass matrix, Eq. (15),

C ¼cb 0T

0 Ce

" #ð27Þ

is a damping matrix in which

Ce ¼X12j¼1

Z ‘j

0

cjujuTj dx ð28Þ

and K is the stiffness matrix, Eq. (19).

6. Model reduction

The hierarchical finite element method yields a discrete model with far fewerdegrees of freedom than the ordinary finite element method. Still, the number ofdegrees of freedom tends to be large in relation to the useful information containedin such a discrete model. Indeed, it is a well-known fact that higher modes aredifficult to excite, as they require a great deal of energy. Moreover, higher modestend to be inaccurate, which is a characteristic of discretized models [1].Hence, a model reduction designed to eliminate the effect of higher modes seems in

order. To this end, we consider the eigenvalue problem corresponding to theundamped structure alone, which can be obtained by assuming that the base is heldfixed and by ignoring damping. The eigenvalue problem has the form

KeU ¼ MeUL; ð29Þ

where U is the modal matrix and L the diagonal matrix of eigenvalues of the elasticstructure clamped at the base. The modal matrix is orthonormal with respect to boththe mass matrix and stiffness matrix, or

UTMeU ¼ I ; UTKeU ¼ L ð30Þ

in which I is the identity matrix.We assume that the structure alone has n degrees of freedom, so that U and L are

n� n matrices. Consistent with the above discussion, we propose to retain only Nr

modes, Nr5n. To this end, we denote by Utr the submatrix of U containing the firstNr columns alone and introduce the linear transformation

qeðtÞ ¼ UtrqeðtÞ; ð31Þ


where qe is an Nr-dimensional vector of elastic modal coordinates. IntroducingEq. (31) into Eq. (26) and premultiplying through by UT

br, we obtain the truncatedequations of motion

M�qþ C _qþ K q ¼ Qc þ D ð32Þ

in which q ¼ ½ub qTe T is the displacement vector of the truncated system,

M ¼m aTUtr

UTtra I

" #; C ¼

cb 0T

0 Ce

" #; K ¼

kb 0T

0 Ltr

" #ð33Þ

are the truncated mass, damping and stiffness matrices, respectively, where Ce ¼UT

trCeUtr;Ltr is the truncated diagonal matrix of the lowest Nr eigenvalues and

Qc ¼ UTtrQc; D ¼ UT

trD ð34Þ

are truncated control and disturbance vectors, respectively. We observe that thetruncated discrete system, Eq. (32), has only Nr þ 1 degrees of freedom, as opposedto nþ 1 degrees of freedom of the original discrete system.

7. Control design

It is clear from the nature of the structure that control must be carried out by morethan one actuator. In view of the fact that the actuator forces are likely to be verylarge, physical considerations dictate that the actuators be located so that thestructure suffers no damage. Hence, we assume that there are N þ 1 actuators actinghorizontally on the base and at each floor. The control implementation is by meansof direct feedback controls, whereby the actuators and sensors are arranged incollocated sensor, actuator pairs, and the actuator at a given location responds to thesignal from the sensor at the same location. The control law is on–off with a two-tiered dead zone, which is nonlinear. The control law for a typical actuator force is

FðtÞ ¼

F0; vðtÞ4�v2;

0; jvðtÞj4v1;

�F0; v25vðtÞ;Fðt�Þ; v14jvðtÞj4v2

8>>><>>>:

ð35Þ

and is depicted in Fig. 4, where vðtÞ is the inertial velocity as measured by thecollocated sensor. Note that the notation FðtÞ ¼ Fðt�Þ means that the control attime t remains the same as at the time immediately preceding t. This control law byitself can cause the actuator to operate when the inertial acceleration a0 of the pointcoinciding with the sensor is relatively small. To prevent this, the actuator isactivated only if the acceleration of the base exceeds some minimum value a0. Thecontrol design amounts to selecting optimal values for the control parameters F0;v1; v2 and a0. This new control law is more efficient than the on–off control usedin [4].


8. Numerical example

We propose to control a four-story framed structure, so that we use five sensor,actuator pairs, one on the base and one at each floor. All elastic members, whether abeam or a column, have the mass density 1160 kg/m and the flexural stiffness2:6042� 108 N=m2. The length of the columns is 3.5 m and that of the beams is 7 m.The base parameters are: mb ¼ 8000 kg, cb ¼ 65:7 kN s=m and kb ¼ 2:63MN=m.The properties of the isolation system (assuming the structure is rigid) are as follows:total mass ¼ 72960 kg, natural frequency ¼ 3:5 rad=s and damping ratio ¼ 0:1. Thecontrol parameters for the base and all four floors are as given in Table 1. Inaddition, the minimum base acceleration for control is a0 ¼ 50 cm=s2.As indicated earlier, the original discretized elastic model has n ¼ 84 degrees of

freedom. The reduced elastic model involves only Nr ¼ 10 elastic modes. Hence, thenumber of degrees of freedom of the reduced system, including the base motion isNr þ 1 ¼ 11.The newly modified on–off control scheme, Eq. (35), has been used to control the

motion of the four-story building. As an input for the computer simulation, thedisplacement and velocity of the ground due to the El Centro 1940 earthquake havebeen used. Moreover, the acceleration of the ground has been used for comparison

Fig. 4. Modified on–off control law.

Table 1

Control parameters

F0 ðkNÞ v1 ðcm=sÞ v2 ðcm=sÞ

Base 5.500 7 8

Floor 1 5.125 7 8

Floor 2 4.800 7 8

Floor 3 4.800 7 9

Floor 4 5.015 7 9


purposes. Indeed, in control of buildings in earthquakes, the acceleration plays theimportant role. Table 2 shows the maximum acceleration magnitudes for threedifferent cases, as well as the number of control activations.Fig. 5 shows the acceleration time histories of the base and of the top floor, with

the gray line representing the response without base isolation and without control,the dashed line the response with base isolation and without control and the blackline the response with base isolation and control. Of course, there is no gray line forthe base response, because in this case the base has the same motion as the ground.Furthermore, the gray line is not included in the plots for the top of the structure,because it would fill the plots completely. As can be seen from Fig. 5, the accelerationof the top floor is kept relatively low, so that the control design must be consideredquite satisfactory. In this regard, it must be stressed that base isolation playsan indispensable part in mitigating earthquake effects. Fig. 6 shows time historiesof the actuator forces on the base and all four floors. The conclusion is that theseforces are also relatively low, making practical implementation feasible. Indeed, theratio of all actuator forces combined to the dead weight of the structure is less than3.5%.

Table 2

Maximum acceleration magnitudes ðcm=s2Þ

Fixed Isolation Isolation Actuations

frame alone and control number

Base 341 162 121 18

Floor 1 622 157 133 18

Floor 2 987 151 124 19

Floor 3 1063 160 124 16

Floor 4 1597 170 123 17

Fig. 5. Acceleration time histories.


9. Conclusions

This paper uses a different type of base isolation (than the one in common use) inconjunction with multi-input direct feedback control to achieve both goals ofstructural control, namely, to prevent injury of the occupants and to prevent damageto the structure and its contents. These goals are achieved with low control forcesrelative to the weight of the structure, less than 3.5%.

References

[1] L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall, Englewood Cliffs, NJ, 1997.

[2] L. Meirovitch, Dynamics and Control of Structures, Wiley, New York, 1990.

[3] L. Meirovitch, M.A. Norris, Sensitivity of distributed structures to model order in feedback control,

J. Sound Vib. 144 (3) (1991) 365–380.

[4] L. Meirovitch, T.J. Stemple, Nonlinear control of structures in earthquakes, ASCE J. Engng. Mech.

123 (10) (1997) 1090–1095.

Fig. 6. Control force time histories.


a new approach to the modeling of distributed structures for control meirovitch

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