Pd88-18]ij51

NATIONAL CENTER FOR EAJn'HQUAKEENGINEERING RESEARCH

State Universit;' of New York at Buffalo

MODAL ANALYSIS OF NONCLASSlCALLYDAMPED STRUCTURAL SYSTEMS

USING CANONICAL TRANSFORMATION

by

J.N. Yang, s. Sarkani and EX. LongDept. of Civil, Mechanical and Environmental Engineering

The George Washington UniversityWashington, D.C. 20052

Technical Report NCEER-87-oot9

September 'Zl, 198'7REPRODUCED BYU.S. DEPARTMENT OF COMMERCENational Technical Information 5ervIceSPRINGFIELD, VA. 22161

NOI'ICEThia N?<;.t :NU prepared by The Geotp Waahinpm UnMnityu a lftU1t 01 reeearch sponsored by the NatioIW center forEarthquake £nsineerblI ReIearch (NCEER) and the NatioIIalSdence Foundlltlon. NeitMNCEER,~01 NCEER, itssponIOI'I, The GeoqJe Washington U , nor any penonacting 01\ their behalf:

a. makes any WaDld)', ecpn!U or Unplied, with retpect to the11M 01 any information, appanlUI, methods, or procelldiIdoeed in thiI report or that IUII:h UIe IN)' not~uponprivaIIeIy owned ....i or

b. ...... any u.bIitieI 01 wh.tIIoewr kind with retpect to the11M 01, or ror ...... resuItina &om the ue 01, myiNonNltIoft, apJ*IIhII, method or procell 6doeecl in thJI

NPOd· ,,

,11111 1111.----

MODAL ANALYSIS OF NONCLASSICALLY DAMPEDSTRUCTURAL SYSTEMS USING CANONICAL TRANSFORMATION

by

J.N. Yangl , S. Sarkani1 and F.x. Long3

September 27, 19&7

Technical Report NCEER-87-Q019

NCEER Contract Number 86-3022

NSF Master Contract Number ECE 86-07591

Professor, Dept. of Civil, Mechanical and Environmental Engineering, The GeorgeWashington University

2 Visiting Assistant Profefsor, Dept. of Civil, Mechanical and Environmental Engineering,The George Washington University

3 Visiting Scholar, Dept. cf Civil, Mechanical and Environmental Engineering, The GeorgeWashington University

NATIONAL CENTER FOR EARlHQUAKE ENGINEERING RESEARCHState University of New York at BuffaloRed Jacket Quadrangle, Buffalo, NY 14261

511272·1111.. RltClpI.nt'. Ac:ces_ No.

__ September..21.....nB.I. _s.

~J:~MENTA:~~ t:~=-NO·NC~R_=~~-00194. Title and Subl/tla

Modal Analysis of Nonc1assica11y Damped StructuralSystems Using Canonical Transformation

_'6 JI5. Raport Data

/371-S'/

.- -------._--------I. Performl,.. O..anlz.tlon R.pt. No.

11. Cant,acUC) or Gram(G) No.

~CE-86-0759186-3022

7. Authar(s)

~_~J!..!.•.!.!N.!...-:Y~a~ncliQ.L' ,--"S~.~Sa~r~k~a.!:!-n:l:-i L'..!.F......aX-,-. ...!.L..I.!oC!.!n~g-------------- _t. P.rforml". O...nlzatlan N..... and Add,... -IO~-P",j"'Ct/T••k/W_ Unit No.

National Center for Earthquake Engineering ResearchSUNY/BuffaloRed Jacket QuadrangleBuffalo, NY 14261

12. Sponwrl". a..-nlullon N..... and Address IS. Type 01 R.port • Perlocl Covered

Technical Report

14.

1---------------.-------------..-. ---- .-.~--...L-------------___l15. Supplementary Not..

This research was conducted at The George Washington University and was partiallysupported by the National Science Foundation under Grant No. ECE 86-07591.

I" AbstfltCl(lImlt: 200 words) An alternate modal decomposition method for dynamic analysis of non­classically damped structural systems is presented. The resulting decoup1ed equationscontain only real parameters. Hence. the solution can be obtained in the real field.Several procedures are outlined to solve these equations for both deterministic andnonstationary random ground excitations. Prior work has shown that the effect ofnonclassical damping may be significant for the response rf light equipment attachedto a structure. Therefore. the proposea solution technique is applied to find theresponse of a light equipment that is attached to a multi-degree-of-freedom structure.

Numerical results obtained from determi~istic and nonstationary random vibration analy­ses indicate that the effect of nonclassical damping on the response of tuned equipmentis significant only when the mass ratio and damping ratio of the equipment ure small.Under this circumstance. the approximate classically damped solution, i.e .• the solu­tion obtained using the undamped modal matrix and disregarding the off-diagonal termsof the resulting damping matrix, is usually unconservative.

For detuned equipment. neglecting the effect of nonclassical damping generally resultsin an equipment response that is close to the exact results. However, for certainequipment detuned at high frequency, neglecting nonclassical damping results in con-servative eguipment resporsf.S.....- H____ ___

17. Document AMIysIs •. OHenptors

SEISMIC ANALYSISCANONICAL TRANSFORMATIONSEISMIC EXCITATION

EARTHQUAKE ENGINEERINGMODAL DECOMPOSITIONDAMPED STRUCTURAL ANALYSIS

:zo. Sac:urlty c.... (Thl. Pap)

unclassifiedIv.-Release unlimited

c. COSATI FI.ld/Group~----------_._-----_._------_._--_._----_._--------,----_._-~I" A".I..blllty Itetement I', Sac:urlty C.... (Thl. Report) 210 No. of p....

_u~~lass i fied!l1._.EokJPrlh I 'I. f~0Pn0IUlL .... m (4-77)(Formerly NTI~35)

oep._t of com_

AlSDACT

An alternate modal decomposition method for dynamic analysis of

nonr.lassically damped structural systems is presented. The resulting

decoupled equations contain only real parameters. Hence, the solution can

be obtained in the real field. Several procedures are outlined to solve

these equations for both deterministic and nonstationary random ground

excitations. Prior work has shown that the effect ot nO'lclassical damping

may b. significant for the response of light equipment attached to a

structure. Therefore, the proposed solution technique is applied to find

the response of a light equipment that is attached to a Ir•• lti-degree-of­

freedom structure.

Numerical results obtained from deterministic and nonsttitionary random

vibration analyses indicate that the effect of nonclassical damping on the

response of tuned equipment is significn, only when the mass ratio and

damping ratio of the equipment are small. Under this circumstance, the

approximate classically damped solution, i.e., the solution obtained using

the undamped modal matrix and disregarding the off-diagonal terms of the

resulting damping matrix, is usually unconservative.

For detuned equipment, neglecting the effect of nonclassical damping

generally results in an equipment response that is close to the exact

results. However, for certain equipment detuned at high frequency, ne­

glecting nonclassical damping results in conservative equipment responses.

- i - C

~

This work i. supported by the National Center for Eart~quake

~n&ineering research, State University of New York at Buffalo under grant

NCEER-86-3022. The authors are grateful to Y. K. Lin and J. HoLung of

Florida State University for valuable discussions.

11

SKC'l:IOIf 'lI'lL8

'lULl& or COiiIIftW'l'S

PAGE

1

2

33.1

44.14.24.3

5

6

7

9

APPENDIX

:IM'1'JtODUC2':IOIf. . . . . . . • . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1-1

a!lC'KGAOOllD . . . . . . . . . . . . . • . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . .. 2-1

CANONICAL MODAL ARaLYSIS 3-1Reduction to Classically Damped Structures 3-4

DI:'lI:RN:IH:IS'lIC RESPOIfSI: 'to SPECIrIC I:XCI'lA'lION....•.......... 4-1Direct Numerical Integration 4-1Impulse Response Function 4-1Frequency Response Function 4-2

S'lOCBAS'lIC RESPOHSI: 'to RARDON I:XCI'lA'lION 5-1

N1JNI:RICAL KDHPL8 . .........•................................ 6-1

COIICLUS IONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 7-1

unRmlCKS 9-1

CANORICAL ~SraRNA'lIOIf rca EQUA~:ION 2 A-1

iii

rIGOU 'l'1'l'r..

LIS'l' or ILLUSTRA'l'IONS

PAGE

12

3

4

5

6

7,8,9

10

11

12

13

14a-c14d-f

Simulated Ground Acceleration 6-18Primary Structure and Single Degree-of-Freedom Equipment

(a) Two Story Primary Structure, (b) Eight StoryPrimary Structure 6-19

Maximum Relative Displacement of Tuned Equipment Attachedto Classically Damped Two Story Primarr Structure asFunction of Equipment Damping, ~ - ~e/~c"'" .••...•...•• 6-20

Maximum Relative Displacement of Detuned EquipmentAttached to Classically Damped Two Story PrimaryStructure as Function of Equipment Damping, ~ - ~e/~ec" .. 6-21

Maximum Relative Displacement of Tuned Equip~ent Attachedto Classically Damped Eight Story Primary Structure asFunction of Equipment Damping, ~ ~ ~e/~c" 6-22

Maximum Relative Displacement of Detuned EquipmentAttached to Eight Story Primary Structure as Functionof Equipment Damping, '1'\ - ~e/~c •...••..••...•...•........ 6-23

Maximum Relative Displacement of Tuned Equipment Attachedto Nonclassically Damped Two Story Primary Structure asFunction of Equipment Damping, '1'\' - ~e/~c""'" 6-26

Root Mean Square Response of T~ned Equipment Attachedto Classically Damped Two Story Primary Structure as aFunction of Time, t .............•......................... 6-29

Root Mean Square Response of Detuned E:quipr,.ent Attachedto Classically Damped Two Story Primary Structure as aFunction of Time, t 6-30

Maximum RHS Response of Tuned Equipment Attached to TwoStory Classically Damped Primary Structure as aFunction of Equipment Damping, ~ - ;e/~c 6-31

Maximum RMS Response of Detuned Equipment Attached toEight Sto~y Classically Damped Primary Structure asa Function of Equipment Damping, ~ - ~e/~c""""""" .6-32

~S as a Function of Time 6-33RMS of Story Acceleration 6-34

iv

Maximum Floor Displacement 6-15

Maximum Story Deformation " ....•.... , 6-16

Eight Story Classically Damped 6-17

TABU

1

2

3

'UTLC

LIST or 'USLIIS

PAGB

v

SECTIOR 1

IN'l'llODUCTIOH

In seismic analysis of linear multi-degree-of-freedom visco~sly damped

structures, it is quite common to assume that the damping matrix is of the

classical (proportional) form (i.e., the form specified by Caughey and

O'Kelly [1]), This assumpt~on enables one to decouple the equations of

motion by using the undamped eigenvectors of the system, After the

equations are decoup led , the response quantities of interest can be obtained

by either the response spe~trum approach or numerical integration of the

decoupled equations (2). Solution of each decoupled equation represents the

contribution of a particular mode of vibration of the structure to the total

response. Furthermore, the response in most situations can be approximated

by contributions from only a few dominant modes.

In general, however, real structures are not classically damped.

Therefore, the damping matrix can not be diagonalized by the eigenvectors of

the undamped system. Under this circumstance, one can use a step by step

Integr&tion procedure to evaluate the response of structures. This,

however, may involve numerical difficulties unless the time steps are

sufficiently small. In general, the use of this procedure can be justified

when the structure is behaving nonlinearly.

ane may still decouple the equations of motion of a nonclassically

damped structural system using the eigenvectors of the damped system.

However, for such structures, the damped eigenvectors are complex valued.

Thi¥ procedure, first proposed by Foss (4) and Traill-Nash [11), decouples

the equations of aotio~ of an n degree-of-freedom nonclassically damped

aystem into a s.t of 2n first order equations. These decoupled equations

contain complex parameters. Recently, Singh and Ghafory-Ashtiany [10)

1-1

developed a step by step numerical integration algorithm to solve these

decoupled complex valued equations. 19usa. Der Kiureghian, and ~acKman [6),

used r.ndom vibration approach to obtain statistical moments of the response

of nonclassically damped system subjected to stationary white noise

excitations. Veletsos and Ventura (12) made a review of the dynamic

properties of nonclassically damped structures, and co,pared the exact

solutions ~f such systems with those computed using the undamped eigen

vectors and disregarding the off-diagonal terms of the resulting damping

matrix under deterministic excitations.

In this paper an alternate modal der-omposition approach employing

canonical transformation is presented. The resulting decoupled equations

involve only real parameters. thus avoiding computatlons to be carried out

in the complex field. Several procedures are outlined to solve these

equations for both deterministic and nonseationary random ground exci­

tations. Prior work [6] haa shown that the effect of nonclassical damping

may be significant for the response of light equipment attached to a

structure. The proposed canonical modal analysis technique is employed to

carry out a par..etric study for the effect of nonclassical damping on the

response of a secondary system that is attached to a primary structure.

Numerical r.aulta indicate that for tuned light equip.ents, neglecting

nonclassical damping can result in a significantly unconservative equipment

response.

Finally, the effect of nonclassical damping on the response of primary

syste. is studied. An eight story shear be.. type structure is considered

and the distribution of the da.ping is varied so that a variety of

noncla.sically damped syste.. can ba studied.

1-2

SECTIOIi 2

The response of an n degree-of-freedom structure subjected to a ground

excitation, x . can be obtained by solving the following differential equa­g

tions of motion

MX+CX+KX M r xg (1)

in which ~, g. and ~ denote the (nxn) mass, damping, and stiffness matrices

of the structure, respectively, and X is an n vector denoting the displace-

ments of th9 structure relative to the moving base. The vector r is a

unit vector, E - [1,1, ... ,I]'. A super dot (0) represents differentiation

with respect to time and an under bar (-) denotes a vector or a matrix. In

Eq. (1), the argument of time, t, for X and xg have been omitted for simpli­

city.

Caughey and O'Kelly [1] showed that if the damplng matrix statisfies

-1 -1the identity f ~ ~ - ~ ~ g. the eigenvectors of the undamped system can

be used to transform the equations of motion, Eq. (1), into a set of n

decoupled equations. The system with damping ~atrix satisfying this

condition is said to be classically damped. However, the eigenvectors of

the undamped Iystem will no longer diagonalize the dampins matrix that is

not of the cla•• ical form.

For the noncla••ically damped system, the approach proposed by Foss

[4] and Traill·N••h [11] can be used. In this approach. the n second-order

equations of .otion are converted into 2n first-order equations as follows:

(2)

2-1

in which ~l and 6 are (2nx2n) symmetric matrices and P is a 2n vector

defined as

~l- [-H+J 8- [WJ !-{.. ~.}-M r

(3)

and! is a 2n vector, referred to as the state vector,

y-ID (4)

Th~ eigenvalue problem of Eq. (2), IA ~l + !I - 0, can be expressed as

follows

(5)

in which ~j and !j are the jth eigenvalue and eigenvector, respectively.

From the definition of the state vector! given by Eq. (4), the jth

eigenvector, !j' has the following form

(6)

in which ~j represents the displlce.ent eigenvector. Since ..trice. ~1 and

B do not commute, i.e., ~l ! ~ ! ~l' the re.ulting eigenvalue. and eigenvec­

tors are complex valued (3). Furtheraore, if the syst•• i. underdamped, the

2-2

eigenvalues and eigenvectors are n pairs of complex conjugate. The jth pair

of eigenvalues can be written as

(7)

and wj

is different from the frequency

system. The above notations was first

~ 1/2in which! - J-l, wDj - wj (l-ej 2)

of the corresponding undamped

introduced by Singh [9].

The response state vector Y can be expressed as a linear combination

of the eigenvectors

Y - ! z (8 )

where f - [fl' f2" .. , f2n] is a (2nx2n) complex modal ~atrix. Substituting

Eq. (8) into Eq. (2) and premultlplylng it by the inverse of the f matrix,

-I! ,one obtains a set of 2n decoupled differential equations

j - 1, 2, ... , 2n (9)

in which Qj is the jth element of the 9 vector where 9 - i-I {~~.}.

The solutions of Eqs. (9) together with the transformation of Eq. (8)

yield the response state vector !(t) of the structural system. Note that

Eq. (9) involves complex coefficients and the solutions for Zj (j - 1, 2,

... ) are complex. The solution for the state vector !. however. is real.

In order to avoid the numerical solution for the complex differential

equations, we propose the following alternate approach [13]. In this

2-3

approach, the only complex calculations are the determination of the

eigenvalues and eigenvectors.

2-4

SECTIOR 3

CABORICAL MODAL ARALYSIS

The equations of motion for the state vector, Eq. (2). are rewritten

as follows:

Y-AY+Wx- - - - g

(10)

where

A- [_~~lEI-~~~] (11)

The eigenvalue. and eigenvectors of matrix A are identical to those of

Eq. (5). denoted by ~j And !j' for j - 1. 2•...• 2n. FULt;~ler the jth pair

of eigenv.ctors can be expre.sed as

(12)

j-l.2, .... n

in which !j and 2j are 2n r.al v.ctors

The (2nx2n) r.al matrix ! co~tructed in the following

(13)

vill tr.~form the maerix A into a canonical form A [81. i .•.•

(14)

3-1

in which T- 1 is the inverse of the T matrix and

where

(15)

, j - 1, 2, ...• n (16)

Let the transformation of the state vector be

(17)

Substituting Eq. (17) into Eq. (10) and premu1tip1ying it by the inverse of

-1the! matrix,! ,one obtain.

~ - A II + F it- g

in which A ia given by Eq. (15) and

·1 {-! }r - T ...- - 2

Equation (18) con.iat. of n paira of dacoup1ed equations.

(18)

(19)

Each pair

of equatlona repre.ent. one vibrational IIOda, and it is uncoupled "ith other

pairs. Ho"ever, the two equation. in .ach pair are coupled. The

3-2

transformation given in Eq. (17) is referred to as the canonical

transformation (13. 14]. All the parameters in Eq. (18) are real.

The jth pair of coupled equations in Eq. (18), corresponding to the

jth vibrational mode, is given as follows:

(20a)

(20b)

in which F2j _l and F2j are the 2j-lth and the 2jth elements of the f vector,

respectively. Solutions of Eqs. (20) together with the transformation of

Eq. (17) yield the response state vector ~(t) of the structural system.

The advantage of the formulation given above is that the computations

for the solutions are all in the real field. The mo~al decomposition

approach described above is referred to as the canonical modal

decomposition.

Equation (20) can be solved easily using either impulse response

function approach or frequency response function approach to be described

later. Likewise. it can further be decoupled, if one wishes to obtain one

equation in terms of each unknown. although this procedure is unnecessary.

The deeoupled differential equations are second order, and the pair

corresponding to mode j is

xg (2la)

3-3

(21b)

The solutions of the above decoupled equations represent the response of the

jth mode. However, from the computational standpoint. Eq. (20) appears to

be more advantageous than Eq. (21).

The canonical transformation given by Eq. (17) is applied to Eq. (10)

in which the ~ matrix is , in general, nonsymmetric. Since the equation of

motion given by Eq. (2) is identical to Eq. (10), the same canonical

transformation can be applied to Eq. (2) to arrive at the modal decomposi-

tion equations given by Eqs. (18) and (20). As such. some orthogonality

conditions can be used advantageously, since ~l and B matrices are symme­

tric. Detail derivations for the canonical transformation of Eq. (2) are

presented in the Appendix.

By virtue of the canoniclal transformation presented in the Appendix.

elements of the F vector in Eqs. (19) and (20) can be expressed as follows

(a'-1

(b'-j

(22a)

in which a prime indicat•• the tran.po.e of a vector or matrix.

3.1 .....rUn to C1U.iGllly Derpesl St;ryc;qar..

(22b)

If the structural .yste. i. cla•• ically damped, then the di.placement

part of the eigenvector i. real, 1 .•.. fj In Eq. (6) i. real.

part of the eig.nvector, ~j fj' ia aillply the .igenvalue,

by the dl.place.ent part of the eig.nv.ctor!j ( ••• Eq. 6).

2j can b. .xpr••••d a.

3-4

Th. v.locity

~j' multipU.d

H.ne., !j and

(23 )

in which !j is the lower half of !j vector representing tloe displacement e i·

genvector. Thus, for classically damped systems, ~j is simply the j th

eigenvector of the undamped system.

The ele~ents F2j _l and F2j given by Eq. (22) can be simplified as fol·

lows [see the Appendix]

F2j _l - 0

Finally, the simplified form of Eqs. (2la·b) become

(24a)

(24b)

(25a)

(25b)

Substituting Eq. (23) into Eq. (13). and then into Eq. (17). one

obeains the displace.ent response vector !. that is the lower half of Y

vector as follows

x-.u- -- (26)

in which ~ is a (nxn) modal matrix of the undamped system and U is_an n

vector with v 2j _l as the jth element, representing the generalized

nate of the undamped system, i.e"

coordi-

(27)

The solution for the displacement vector ~ given by Eqs. (26) and (27)

along with Eq. (25a) is simply the solution one would have obtained from the

modal analysis of a classically damped system. It can also be shown that

the solution of the velocity response! also reduces to that of the classi­

cally damped system using Eqs. (25a) and (25b).

In the followi'lg sections different apprClaches that can be used to

solve Eql. (20a-b) for both deterministic and non-stationary stochastic

ground excitations are described.

SECTlOB 4

DE'IIIKIBISTlC IlESl'OlISE TO SPEClnc EXClTAnOR

4.1 Direst eerisa1 lutlnatlon

Given the record of ground excitation, Eqs. (20a-b) can be numerically

integrated and the response state vector !(t) can be obtained by using the

transformation of Eq. (17). Again, all the parameters in these equations

are real.

4.2 Iwzul•• RepoNe Punctfpp

One can obtain the impulse response vector h (t) for mode j. ~_j(t) ­-~j

,[~2j-l' v2j ] I due to the ground acceleration xg(t) - 6(t), where Set) is

the Dirac delta function. The jth modal impulse response vector is given by

F2j -1 +-~j"'j t

Bin ",Djt)(e

F2j -1 +-~j"'jt

cos wDj t)(e

(28)

and the response vector, ~j(t), corresponding to mode j at time t is

(29)

Alain Eq. (29) can be integrated nuaerically and the respo~se state vector

!(t) can be obtained through the transforaation of Eq. (17).

4-1

4. 3 frequencY KelpoNe Function

Let H (w) be the complex frequency response vector for mode j due to-v

j

ground acceleration, x - eg

Eqs. (20a-b) as follows

iwt Then, H (w) can be obtained easily from-v

j

H (w)­-v

j(30)

Note that the vectors H (w) and h (t) are related through the Fourier-v

j-vj

transform pair, i.e.,

...H (w) - f h (t) e·!wt dt (31)-IIj ....-11 j

If ~j(w) denotes the Fourier tran.form of the re.ponae vector of mode j,

!::j (t), then

Vj(W) - H (w) x (w)- -vj g

(32)

in which x (w) i. the Fourier tranefor. of the ground aceeleration, x (t).g g

Finally I the j th modal n.ponee vector in the ti.e domain can be obtained a.

follow.

• •!::j(t) - ~ f ~j(w) e !wtdw - ~ J!~(w) x,(w) e !wtdw (33)

•• .• j

4-2

the above calculation can be earried out efficiently using the Fast Fourier

transform (FFT) algorithm.

SECTION 5

STOCHASTIC USPOlJSE 1'0 IlAlfDOII EXCITATION

The canonical modal decomposition method presented above can be used

conveniently to obtain the response of a nonclassically damped system to a

nonstationary random ground acceleration [14. 15]. The expressions for mean

squares of the response state vector. ~(t), will be derived in the

following.

Often. the earthquake ground excitation, xg(t), can be modeled as a

~niformly modulated nonstationary random process with zero mean

(34)

In which aCt} - a deterministic non-negative modulating or envelope function

and xO(t} is a stationary random process with zero mean and a power spectral

density, ~xx(w}, A commonly us.d functional form for the spectral density

is

(35)1 + 4 ~ 2 (w/w }2 52

g I(w/w }2]2 + 4 ~ 2 (w/w )2

g g g

in which ~ • W , and S are parameters depending on the intensity and theg g

characteristics of the earthquake at a particular geological location.

Various types of the envelope function o(t) have been suggested in the

literature to introduce the nonatationarity of the ground acceleration into

Eq. (34). One possible form of aCt) is: aCt} - (t/t1}2 for 0 ~ t ~ t l ,

aCt) - 1 for t 1 ~ t ~ t 2 , and aCt} - exp [-~(t-t2») for t > t 2 , Note that

t l , t 2 and ~ can be selected to reflect the shape and duration of the earth­

quake ground acceleration, When aCt) - I, it follows from Eq. (34) that the

5-1

ground acceleration is a stationary random process. The stationary

assumption is reasonable when the duration of the strong shaking of the

earthquake ground motion is much longer than the natural period of the

structure.

Since the ground acceleration xg(t) is a zero mean random process, the

mean value of all the response quantities are zero as well. Let ~v(t) be

the impulse response vector of v(t), i.e., h (t) - [h'l(t),h' 2(t),- -v -v - v

~~n(t)] '. Then, the response state vector ~(t) is given by

~(t) - jt! ~v(~) Xg(t-~) d~o

(36)

The covariance matrix ~(t,t) of the response state vector, ~(t), is

obtained from Eq. (36) as

~(t,t) - E[ jt! ~v(~l) Xg(t·~l)d~l Jt~~(F2) T' Xg(t-F 2)dF 2]

o 0

(37)

in which Eq. (34) ha. been u.ed, and ~iX(t) i. the autocorrelation function

of the .tationary random proce" xO(t), which i. related to the .pectral

den.ity .xx(w) throulh the Wiener-Khlnchin'. re~ation

•(38)

5-2

Substituting Eq. (38) into Eq. (37), one obtains

•

J,*

L(t,t) - H..._(t,w) M.._ (t,w) .;.... (w)dc.I~~ ~~ ~~ xx•

in which a star denotes the complex conjugate and

Jt -iw~

~(t,w) - ! ~v(~) a(t-~) e - d~

o

(39)

(40)

The variance vector ~y2(t) of !(t), which is equal to the mean square

response vector of !(t), consists of the diagonal el•••nts of ~(t.t) and it

can be .xpr••••d a. follows

•2 I 2!y (t) - • I !y(t,w) I ';xx(w) dw (41)

i. the evolutionary power .pectral density cf2in which I !y(t,w) I ';xx(w)

the .tate vector, !(t), and I 12!!y(t,w) i. a vector in which it. el.ments

are the .quare. of the ab.olute value of the corr••ponding el••ent. of

!y(t,w) given by Eq. (40).

The nonstationary mean square re.pon•• liv.n by Eq. (41) can b.

computed ea.ily a. follow•. Fir.tly, the impul.e re.ponse vector h (t) can-.,either be computed directly from Eq. (28) or indirectly from the frequence

r ••ponae vector !.,(w) given by Eq. (30) using the FFT technique. Secondly,

the vector ~(t,w) i. evaluated from Eq. (40) using the FFT techniqua again.

Finally, the ti.. dependent root ...n .quare re.ponse, ~y(t), i. obtained by

nuaerically integrating Eq. (41) and taking the .quare root.

5-)

SICTIOR 6

lUIIIlICAL DAIIPLES

The canonlcal modal analysis presented in this paper will be used to

study the response behavior of nonclassically damped structural systems

subjected to earthquake-type excitations. Emphasis will be placed on the

response behavior of primary-secondary structural syste~. Parametric

studies will be conducted for the response of secondary system, such as a

light equipment attached to a structure. In particular, under what condi­

tion the nonclassical damping may be signlficant. Frequently, it may be

assumed that the structure and equipment are classically damped individ­

ually. However, because of different damping characteristics of the

equipment and the structure, the combined equipment-structure system

generally is not classically damped.

Igusa, Der Klureghian, and Sackman (6) considered a single-degree-of­

freedom equipment attached to a single-degree-of-freedom structure and

subjected to a stationary white noise ground excitation. They showed that

at tuning (when frequencies of the equipment and structure coincide) the

effect of nonclassical damping on the response of light equipment becomes

significant. Here we investigate the effect of nonclassical damping on

somewhat complex equipment-structure syste.s excited by an earthquake ground

acceleration. Young and Lin [16J and HoLung, Cai and Lin [5J conducted

extensive para.etrie studie. for the frequency response function of the

primary-secondary structural system. Here we examine the response of the

equip.ent-structure syste. to both deter.inistie and nonstatlonary

stochaatic ground excitation•.

Of particular interest is the ca.e in which the primary structure

it.elf is n~nclassleally damped. Hence, the .econd clas. of proble••

studied herein deals with the effect of nonclassical damping on the

6-1

structure itself. An eight story shear beam type building is considered and

the distribution of the damping within the structure is slightly varied to

study the effect of nonclassical damping on che response.

All the example problems studied are subjected to either a simulated

sample ground motion or the ground motion that is modeled as a nonstationary

random process described in Eqs. (31-32). The parameters that define the

envelope function, a(t), and the spectral density, ~xx(w), of the earth­

quake model are: t l - 3sec., t 2 - 13 sec., ~ - 0.26, wg - 3.0 Hz, ~g - 0.652 -4 2 3and S -74.7xlO m /sec./rad. A sample function of the earthquake ground

acceleration is simulated and shown in Fig. 1.

The first equipment-structure example conslsts of a two-degree-of

freedom .hear be.m type structure with a single-degree-of-freedom light

equipment attached to it as shown in Fig. 2(a). This primary structure is

cl•• sically damped if cllkl - c2lk2 and the combined equipment-structure

system is classically damped if cllkl - c2lk2 - celke in which the subscript

e refer. to the equipment. The a.ss and stiffne.s of each .tory unit of

the prilll&ry structure are: ml - 2--30 tons; kl -k2-k-19,379 kN/m. The na­

tural frequencies of the primary strueture £re 2.5 and 6.5 Hz, respectively.

Parametric studi.J will be carried out by verying the distribution of the

damping of the primary structure and the equipment damping.

Let the value. of c l and c2 be equal to 123.4 kN/m/.ec., so that the

primary .tructure i. clas.ically damped with the fir.t modal damping ratio

of apprOXimately 5.. Given a .... ratio, the damping ratio of the equip.ent

i. varied and the re.ponse of the equipment, i.e., di.place.ent relativa to

the attachment point, i. evaluated by the exact method pre.ented in this

pap.r. The re.ult. are compared agaln.t tho.a obtalnad u.lng the

approx1aate cla••lcally damped approach. Th. approximate cla•• lcally damped

approach 1. to decouple the .econd order equatlon. of .otlon u.ing

6-2

eigenvectors of the undamped system and disregard the off diagonal terms of

the !' 2 ! matrix, where ~ is the (nxn) modal matrix of the undamped primary

-secondary system, Eq. (27).

Let the equipment frequency, we' be tuned to the fundamental frequency

of the primary system, i.e., we - 2.5 Hz. The response of the equipment to

the simulated deterministic ground acceleration shown in Fig. 1 is presented

in Fig. 3 for different values of equipment damping ratio, ee' and mass

ratio, 7. The mass ratio. 7. of the equipment is the ratio of the eq~ip-

ment mass to first modal mass of the primary structure that is equal to 30

tons. In Fig. 3. the ordinate is the maximum r.esponse of the equipment

relative to the attachment point in 30 seconds of earthquake episode, and

the abscissa. q. is the ratio of the equipment damping ratio, ~ , to thee

unique damping ratio of the equipment, e ,that would make theec combined

equipment-structure system classically damped. For the classically damped

shear beam type primary system of Fig. 2(a), the damping ratio of the equip-

ment, eec' that results in a classically damped primary-secondary system

can be obtained using the Caughey-O'Kelly identity. eec - (we /2) (cifki ), in

which i refers to any of the degr••s of freedom of the primary system.

Likewi•• , when the primary system is classically damped, e is equal to theec

jth modal damping ratio of the primary structure if the equipment is tuned

to the jth mode of the primary .tructure. Hence in the present example, eec

is equal to the first modal damping ratio of the primary structure which is

Fig. 3 pre.ents the results for three different value. of the

equipmen~ .... ratio. 1. From this figure it is clear that the equip.ent

re.pon.. increa.e. as its damping ratio {e decreases. The results based on

the approximate clas.ieally damped approach start to deviate from the exact

6-3

solution only when €e < {ec' The deviation increases as the equipment dam­

ping ratio or the mass ratio decreases. Further, the solutions obtained

using the approximate classically damped approach in the region € < € aree ec

nonconservative. In other words, the effect of nonclassir.al damping becomes

signiffcant only when the equipmen~ damping ratio, { . is smaller than ee ~

that results in a classical damping for the combined equipment-structure

system. It is also evident that the smaller the mass ratio or the quipment

damping, the more pronounced is the effect of neglecting the off-diagonal

terms of the ~' C ~ matrix.

Figures 4(a), (b) and (c) show the effect of nonclassical damping on

t.he response of the equipment that is not tuned to any of the frequencies of

the primary structure. The equipment fr~quency we is chosen to be the

average of the first two frequencies of the primary structure, i.e., we ­

(wl +w2)/2 - 4.5 Hz. From these figures, it is clear that for detuned equip­

ment attached to the two-story primary structure of Fig. 2(a), the effect of

nonclassical damping may be ignored without causing any proble~. Likewise.

the maximum equipment response is not sensitive to the mass ratio.

The observations made from Figs. 3 and 4 above hold for a SDOF

equipment attached to a two-story shear beam type primary structure. The

second example consists of an eight-degree-of-freedom shear beam type

primary structure with a single-degree-of-fre.dom equipment attached to the

.eventh story unit .s shown in Fig. 2(b). The ma.s and stiffness of each

.tory unit of the primary structure are: ml - m2- ... - ma - m - 345.6 tons

5and k1-k2 - ... -ka-k-3.4xlO kN/m. The undamped natural frequencies of the

primary structure are wl - 0.92, w2 - 2.73, ~3 - 4.45, w4 - 6.02, Ws - 7.38,

w6 - 8.49, w7 - 9.32, wa - 9.a2 Hz and the first modal m••s is 345.6 ton•.

Let the values of c l through c a be equal to 2,937 kN/m/s.c, .uch that the

primary structure Is cl••• ically damped with the fir.t modal d••pinS ratio

6-4

of approximately 2.5'. In this example the equipment frequency, we' is

tuned to the third natural frequency of the primary structure, i.e .• we ­

4.45 Hz. This results in the value of e of approximately 12.. Again. theec

simulated earthquake ground acceleration shown in Fig. I i~ used as the

input excitation.

Figure 5 shows the response of the equipment for various values of

equipment damping ratio, e , and mass ratio, 1. Similar to the results fore

the ewo-degree-of-freedom primary system. the effect of nonclassical damping

is generally significant only when the equipment mass ratio is small and its

damping ratio is smaller than {ec' However. the results S~em to indicate

that for certain tuned equipment structure systems, ignoring the nonelas-

sical damping may indeed result in conservative equipment response, see Fig.

5(c)

Figure 6(a) shows the effect of nonclassical damping on the response

of a de tuned equipment attached to the eight story primary structure of Fig.

2(b). The frequency of the equipment is set to be equal to the average of

the first and second frequency of the primary structure. i.e .• we -(wl +w2)/2

- 1.83 Hz. which r ••ult. in a value of {.c of approximately 5t. Fig. 6(b)

shows the effect of nonclassical damping for the saae equipment-st~cture

system except that the frequency of the equipment i. now set to be equal to

the average of the second and third frequency of the primary system. This

results in an equipment frequency of 3.6 Hz and a ( value of approxi-ec

aately 9.7t. Fro. the.e figure. it is clear that the effect of nonclassical

~lng is insignificant at .n.

However, as the frequency of the d.tuned .quipment incre••••• the

situation may b. diff.r.nt. Figur. 6(c) pr.s.nts the equipment respon.e

when the .quipm.nt fr.quency is .qual to the average of the fourth and fifth

frequencies of the primary structur•• i .•.• we - (w4+w5)/2 - 6.70 Hz. Not.

6-5

that for this equipment-structure system, the value of ~ec is approximately

lS\. When the equipment frequency is set to be equal to the average of the

seventh and eighth frequencies of the primary structure, i.e., we- (w7+wS)/2

- 9.57 Hz, the effect of nonclassical damping is shown in Fig. 6(d).

Figure 6(e) presents the same results as those of Fig. 6(c) except that the

mass rat~o of the equipment is now increased by a factor of hundred.

Figures 6(c) through 6(e) indicate that the effect of nonclassical

damping becomes significant for detuned equipments when (i) the frequency of

the detuned equipment is high, and (ii) the equipment damping ratio is

small. Under this circumstance the approximate classically damped proeelure

results in a higher equipment response than the exact solution. This

phenomenon can be explained in the following. The spacing between natural

frequencies of the primary system reduces in the high frequency range. In

other words, higher natrual frequencies tend to be closely spaced as

evidenced by the primary structure considered herein. When the equipment

has a high frequency, although detuned, it will interact with its

neighboring structural frequencies due to its cl~senes. to them. Because of

such modal coupling and interaction, the equipment response tends to

de~rease as obser.ed also in Ref. 5. Such a trend become. stronger as the

damping of the primary system increase.. Hence, the observation made above

holds for high frequency equip.ent. However, at sueh a high fraqueney, the

damping of the primary Itructure is also pretty high, and it is questionable

whether the viacoul damping assumption ia atill valid for the primary

structure.

SuppO•• the damping of eaeh story unit of the primary struetur. 1s

r.duead by a factor of five. Thi. rasults in a value of (ec five ti•• s

a..Uer. When the equip••nt la datuned at a frequency of we - (w4+wS)/2,

the response of the equipment 1. displayed in Fil. 6(£). Thi. filure

presents the .ame results as those of Fig. 6(c) except that the damping of

the primary structure is now five times smaller. From this figure, the

effect of nonclassical damping for this primary-secondary system is not

nearly a. significant as that of Fig. 6(c). As a result, damping of the

primary structure also plays an important role for the response of high

frequency eqUipment. In other words, the modal coupling and interaction

increases not only as the equipment frequency increases, but also as the

structural damping increases. Therefore, it is reasonable to infer that

for detuned eqUipment at high frequency, the approximate classically damped

procedure results in significantly higher Equipment response than the exact

procedu~e, if the modal damping of the primary structure adjacent to the

equipment node is high and the damping of the equipment is smaller than ~ .ec

When the primary system is nonclassically damped, the response of the

secondary system to the simulated ground motion in Fig. 1 will be investi-

gated. Again, consider the equiment-structure system of Fig. 2(a). in which

values of c l and c2 are selected such that the primary structure itself is

nonclassic.lly damped. Two different damping distributions for the primary

strucure are considered. In the first case, all damping of the primary

structure of Fig. 2(a) is placed in the first story unit; with the results

cl -246.8 kN/m/sec and c2-o.0. In the second case, all damping of the struc­

ture 1. placed in the second story unit, leading to the results cl-D.O and

c 2-246.8 kN/m/.ec. The damping ratio of the equipment is varied and the re­

.ponae behavior of the equipment will be examined. It has been demonstrated

above that ignoring the effect of nonclas.ical damping results in unconser-

v.tiv. re.pon.e. for tuned equip.ent with small •••• ratio 7 and small

daap1na ratio, ee' Equipment with .u,~ characteristics will be considered

in the following.

6-7

Figures 7 and 8 show the maximum response in 30 seconds of the time

history for an equipment that is attached to a nonclassically damped primary

structure described above. The equipment frequency, w , is tuned to the un­e

damped fundamental frequency ~f the primary structure. In these figures,

the ordinate is the maximum displacement of the equipment relative to the

attachment point and the abscissa, ~' - € /~' , is the ratio of the dampinge ec

ratio of the equipment to the approximate € ,denoted by €' , as describedec ec

in the following.

For tuned equipment attached to classically damped primary structures,

the damping ratio {ec of the equipment which results in a classically damped

primary-secondary system is the same as the modal damping ratio of the

primary structure in which the equipment is tuned to. However, when the

primary structure is nonc1assically damped, one can obtain an approximate

value for {ec' denoted by {~c' by treating the primary system as L'eing

an equivalent classically damped system. An equivalent classically darlped

primary system is obtained from the original primary system by neglecting

the off-diagonal terms of the i C ! matrix, where C is the damping matrix of

the primary structure and i is the modal matrix of the undamped primary

structure. Thus, the jth equivalent classical modal damping ratioa of the

primary atructure, denoted by {j , ia obtained by dividing the diagnonal

terma of the i C i matrix by twice the corr.aponding jth undamp.d modal fre-

queney. Finally, {~c can be obtained from the equivalent classical modal

damping ratios of the primary atructure (a.e Ref. (1). For the two

nonclaaaically damped primary-aecondary atructures conaidered above, the

approximate firat modal damping ratios are ei - e~c - 7' (cl -2c, C2-O) and

ei - e~c - 2.8. (cl-o, c2-2c), reapectively. Recall that for this primary

atructure, distributing tha damping aqually between the two-atory unit.

6-8

results in a classically damped structure with the first modal damping ratio

of approximately St, i.e., ei - eec - 5•.

As expected, the results shown in Figs. 7 and 8 indicate that the

distribution of damping in the primary structure has a significant effect on

the response of the tuned equipment. Likewise, the effect of nonclassical

damping is significant only when the equipment damping is small. When the

damping of the equipment ee is equal to ~~c' i.e , ~' - ee/e~c-l, it is ob­

served from Figs. 7 and 8 that the exact equipment response and the

approximate classically damped results are almost identical. Such a

solution at ~e/e~c is denoted by (Uec)max. To examine whether e' is a use­ec

ful parameter for measuring the effect of nonclassical damping, results in

Figs. 7 and 8 are replotted in Fig. 9 in a different form. In this figure

the ordinate is the exact maximum equipment response, denoted by

(Ue)max, normalized by the value (Vec)max. Also plotted in Fig. 9 are the

corresponding results when the primary stucture is classically damped

(cl -c2-c). From this figure i.t is clear that the approximate modal damping

ratiol obtained from the diagonal terms of the !' £ ! matrix of the primary

.tructure can be used as a useful measure in determining the effect of

nonclassical damping on the response of tuned equipment even if che primary

structure is not classically damped.

A nonstationary stochastic ground acceleration with a power-spectral

densiCy, ~xx (w), and an envelopa function, Q(t), described previously i.

considered as the input excitation. The primary structure i. a two-degree-

of·fre.do. cla.slcally damped structure shown in Fig. 2(a). Since the mean

value of .arthquake ground acceleration is zero, che ••an valu. of response

quantities are all zero. Therefore, the root mean square (ras) of the

response Is equal to the standard deviation. The time dependent root mean

square (rms) re.pon.a of an equipment tuned to the first mod., 1.e., we

Wl

' is shown in Fig. 10 for several values of equipment damping ratio. The

corresponsing results for a de tuned equipment with we - (wl +w2)/2 are dis­

played in Fig. 11. As expected, the rrns response increases as the equip-

ment damping reduces and the response for a tuned equipment is at least one

order of magnitude larger than that of a detuned equipment. Further, a com-

parison between Figs. 10 and 11 indicates that the root mean square response

of a tuned equipment is quite sensitive to the equipment damping ee' where­

as this is not the case for a de tuned equipment.

Extensive numerical results show that the effect of nonclassical

damping on the rms response of a detuned equipment with we (wl +w2)/2 is

insignificant. For tuned equipment with we- wl' the effect of nenclassical

damping on the maximum rms reponse in 30 seconds is shown in Fig. 12 for

three differant values of mass ratio, 7. In Fig. 12 the ordinate is the

maximum rrns of the equipment response in 30 seconds of earthquake episode,

and the abscissa is ~ - ~e/~ec - ~e/ 0.05 as described in the first example.

The results based on exact solution and approximate classically damped

approach are presented in the figure. Similar to the response of equipment

subjected to a simulated deterministic excitation presented previously, the

effect of nonclassical damping for tuned equipment attached to the two story

primary structure is pronounced only when the equipment is light and its

damping ratio e is smaller than e ,i.e., q - e Ie < 1. Further, thee ec e ec

approximate classically damped solutions are unconservative.

Next, the effect of nonclassical damping on the maximum rm. response

of the equipment that is detuned at the higher frequency is investigated.

Again, consider the eight-story primary-secondary structure of Fig. 2(b) in

which the equipment frequency is equal to the average of the fourth and

fifth frequencies of the primary structure, i.e., we - (w4+wS)/2 - 6.7 Hz.

Figs. IJ(a)-(b) show the maximum rm. of the equipment response, and F1g.

&-10

l3(c) shows the same results as those of Fig. l3(a) except that the damping

of the primary structure is reduced by a factor of 5. From these figures,

it is observed that the effect of nonclassical damping is significant for

the detuned equipment, when it satisfies the following conditions

simultaneously: (i) the equipment is detuned at high frequency with a high

€ec value, (ii) the equipment damping is small, i.e., ~e/€ec < I, and (iii)

the mass ratio is small. Under this circumstance, the rms response obtained

using approximate classically damp~d procedures is always higher than the

exact solution. Again, this is due to the modal coupling and interaction of

the equipment and the primary system. These conclusions are identical to

those obtained previously, when the excitation is a simulated deterministic

sample earthquake ground motion.

The intensity of earthquake ground acceleration usually consists of

three segments as shown by the envelope function a(t), Eq. (34). The

intensity builds up in the first segment (0, t 1) and reaches a stationary

magnitude in the second segment (tl , ( 2), representing the most intense

portion of earthquake. The response history of structures also consists of

three segments and its stationary rms value in the second .egment may be

obtained using the stationary random vibration analysis. The transient

response in the first segment resulting from zero initial conditions as well

as transient earthquake excitations is usually s..ller than the stationary

response in the second .egment. However, under suitable conditions, such as

flight vehieles subjected to atmospheric turbulences, the transient response

at so.e point in time in the first seg.ent .ay exce.d the stationary

response in the second seg.ent [7J. Such an overshooting phenomenon is

iaportant in the design of structures. The overshooting pheno.enon ••y

occur if the excitation 18 appUed (or bulldll up) suddenly. Under ordinary

conditions where the initlal condltio~ for the structure are zero and where

6-11

the earthquake excitation also builds up from zero at time zero, such an

overshooting phenomenon is unlikely. For tuned equipment, however. the

displacement or acceleration response may overshoot. Parametric studies

will be made to examine the possibility of overshooting, in particular, the

rate of increase of the envelope function will be varied.

The two-degree-of-freedom primary structure with a single-degree-of

freedom equipment illustrated in the first example ia considered. The

equipment is tuned to the first natural frequency of the primary structure,

i.e., we - w l- 2.5 Hz and the damping ratio of the equipment, is set at 5\.

The first and second modal damping ratios of the primary structure are 2\

and 5.2\. respectively. The power spectral density, ; •..• (w). of the sta­xx

tionary ground acceleration is identical to that described previously except

the value of wg is now at 2.5 Hz. The envelope function o(t), is given as

4follows: Bet) - (t/tl ) for 0 ~ t s t l , oCt) - 1 for t l < t < t 2 and Bet) -

exp [·P(t-t 2 ) J for t < t 2' This envelope function results in a ground

acceleration that builds up toward a stationary value a(t) - 1 at a faster

rate then the envelope function used in the previous example. The time

dependent nas responses of the primary and secondary structure, including

the r.s of the relative displacement and acceleration, are shown in Figs.

14(a) through 14(f). From th.s. figures and extensive results obtained from

para.etric study, including variation of envelope function, variation of

primary-secondary syste., tuned or detuned equipments, etc., it is observed

that the overshooting pheno.enon for the displacement or acceleration

response do.s not occur for either the primary or secondary system.

Flnally, the effect of nonclassical damping on the r.spons. of a

structure without any equipm.nt is examined. An .i,ht-story prlmary

structure shown In Fig. 2(b) i. consid.red and the effect of the distri-

butlon of damping on the r.spons. vil1 b. exa.ined.

6-12

Six different

distributions of damping for this structure are considered. In all cases,

the distribution of damping among the story units are varied but the total

damping in the structure is kept the same as the classically damped case

(i.e., c l -c2- ... -cS-2,937 KN/m/sec.) In case 1, the damping in each story

unit is proportional to the height of the story, i.e., c i - icO' in which i

is the st.ory number and i - 1 is the ground level. The second case is the

opposite of the first case, i.e., the damping in each story unit is in­

versely proportional to its height, i.e., c1 - (9-i)cO' In the third case,

the largest damping is placed in the fourth story unit and the damping in

other story units decreases linearly with respect to their distances from

the fourth story unit. The fourth case is similar to the third case except

that the largest damping is placed in the fifth story unit. In the fifth

case, total damping is distributed equally among the first, third, fifth and

seventh story units. The last case is similar to the fifth case, except

that total damping is equally distributed among the second, fourth, sixth,

and eighth story units.

The six nonclassically damped structures are subjected to the

simulated ground acceleration of Fig. 1. Tables 1 and 2 present the results

of floor displacements and story deformations for these structurel. Also

presented in these tables are the re.ult. obtalned by neglecting the off

diagonal tarma of !' £ ! matrix, where! is the modal matrix of the undamped

structure. The relults for the cl.ssically damped structure, i .•. ,

(i.e., c l -c2- ... -eS-2,937 KN/m/s.c.) are pr.sented in Table 3. An examina­

tion of these tables indicates that ~he effect of neglecting the off­

diagonal terma of !'£! matrix is insignificant for all the cas.s

considered, where the di.tribution of damping along the building height

varies slowly. The maximum error using the approximate .olution was le••

than 4' for story displacement and le•• than 2' for Itory deformation.

6-13

Therefore, it may be concluded that for primary structures. the effect of

nonclassical damping may be ignored if the damping distribution within the

structure does not change drastically. Such a conclusion does not hold if

extra high dampings are added to one or two story units, such as active or

passive control devices.

6-14

uu.& 1

IIAZllGII FLDDI. DISPlAC!IIERT (ca.)

Xl X2 X3X4 Xs X6 X7 X8

EXACT 3.33 6.55 9.58 12.71 14.51 16.23 17.40 17.99

APPllOX.CASE 1 ClASS.

DAKI'ED 3.31 6.53 9.55 12.24 14.49 16.22 17. 38 17.96

EXACT 2.66 5.23 7.65 9.80 11.61 13.00 13.92 14.38

APPROX.CASI2 ClASS.

DAMPED 2.66 5.24 7.66 9.81 11.60 12.97 13.90 14.37

EXAct 2.90 5.72 8.34 10.65 12.59 14.09 15.10 15.61

APPROX.CASI3 Cu.sS.

DAMPED 2.89 5.70 8.32 10.64 12.58 14.07 15.08 15.58

EXACT 3.00 5.90 8.62 11.03 13.03 14.58 15.63 16.16

APPllOX.C:U14 Cu.sS.

DAKlED 2.99 5." 1.59 11.00 13.02 14.56 15.60 16.13

D.\C'f 2.14 5.59 8.16 10.50 12.42 13.90 14.90 15.40

APPROX.CAR 5 CLASS.

DNIPID 2.13 5.57 8.14 10.43 12.34 13.81 14.12 15.32

IIACT J.08 6.06 8.90 11.38 13.41 15.08 16.15 16.69

AlrROX.CAR • CLASS.

DMPID 3.0' 6.06 '.14 11.)2 13.39 14.91 16.04 16.57

6-15

'L\BUl 2

M'IDII'II FLOOR DlftlRIIAnOlf (CII.)

U1 U2 UJ U4 U5 U6 U7 U8

EXACT 3.33 3.22 3.03 2.70 2.25 1. 72 1.18 0.61

APPROX.CASI 1 CUSS.

DAKPED 3.31 3.22 3.02 2.70 2.25 1.73 1.18 0.61

EXACT 2.66 2.58 2.42 2.16 1.80 1.J9 0.95 0.49

APPROX.C' - CUSS.

DAKPED 2.66 2.58 2.42 2.15 1.79 1.38 0.95 0.49

EXACT 2.90 2.81 2.63 2.33 1.95 1.50 1.02 0.52

APPROX.CASI 3 CUSS.

DAKPED 2.89 2.80 2.62 2.34 1.94 1.49 1.01 0.52

EXACT 3.00 2.91 2.72 2.42 2.01 1.55 1.06 0.54

APPllOX.CAS•• CLASS.

DAKPED 2.99 2.90 2.71 2.42 2.02 1.55 1.06 0.54

EXACT 2.14 2.76 2.51 2.33 1.93 1.41 1.01 0.52

APPROX.CASI5 CLASS.

DAKPED 2.13 2.74 2.57 2.30 1.91 1.47 1.01 0.51

EXACT 3.01 2.99 2.14 2.51 2.10 1.60 1.10 0.56

APPROX.CAS. , CLASS.

DAKPED 3.01 2.91 2.79 2.49 2.07 1.59 1.01 0.55

6-16

TABU 1

1-lml8.Y CLASSICALLY DMCPID

SIOBX Nt1MBP ll) 1 2 3 4 5 6 7 8

IlAXIKtlM FLOORDISPLACEMENT (Xi) 2.94 5.80 8.47 10.85 12.83 1t..35 15.39 15.91

(CII. )

IlAXIKtlM FLOORDISPLACEMENT CUi) 1.94 2.85 2.61 2.39 1.99 1.53 1.04 0.53

(CII).

6-17

CD. o CO. N•

(Of)I

0(W)

10 CN .2

!en .!!

§0 C

~

C\I Z 'V

0 C~

0 2W CJ

10 en i'P"

Z 'SE- 0

0 W'P" ~ ?"'- W~ II:

je"

10Ii:

~Oi8/W

tNOIJ.VH31300V ONnOHe

6-18

G'I

I-'-0

FIGURE 2

ke

28

2b

Primary Structure and Single Degret"-\lf-Freedom Equipment.

(a) Two Story Primary Structure, (b) Eight Story Primary Structure.

i~30

TWO STORY PRNARY STflJCT\.IE <;=C2

,~ IUCl CASS1CIlUY 0-eDlIlI.UTICN T\ND EOUI'rtefT

r20

MASS RAm .. 10-2

e 38t; 10

w~

i; 00 Q5 1.5 2 2.5 3

FRACTION OF CLASSICAL DAM'I"G. " = l. I tee

I~60

TWO STORY PRNARY SlllUClUlIE <;= C,

~~Q.UIIC;UY_

~TlJ'S) EQUI'M3'n'

a~ MASS RATIO =10·'re3bt; 20

w~ ~a Q.,ASSlCAL.Ll

~H!o-tD SClUIf1lIN

ii 00 05 1.5 2 2.5 3

FRACTION OF CLASSICAL DMf>Nj. " = leI l..

80

!~ TWO STORY PflIMNlY STflUCT\.IE <;=CZ

~ 60 ~Y-

n.ND EOl.-.MENTa~ MASS RAllO .. 10-4

Ii40

3C

; II! 20

ii0

0 Q!l 1.& 2 2.!l 3

fRACTION OF CLASSICAl DMt"Ni. " .. le I lu

FIGURE 3 Maximum Relative Displacement of Tuned equipment Attached To Classi­cally Damped Two Story Prlmary Structure •• Function of EquipmentDamping, " =V~_.

6-20

6r---------------------,

3

4a

2.52Ui0.5

Ol-__"--__~__~___'_____'___---J

o

TWO STORY PRNARY STftUClLfIE C.=C 2

lQ.ASSICALlY DAW'BII

2

FRACTKlN Of ClASSICAL CAMPI«;, 1'I =., I {u

4

Ell-'CTSOLUTION

3

4b

4c

2.152

~x a.ASSlCAU.r-

1.5

0E1\HD~

MASS RAm : 10-4

0.5

TWO STORY PRIMARY SlRUCT\.AE C,= C 2

ItIoJl$liICAU.Y llMoI"EDI

OEnIoE>EQ~

MASS RATIO =10-3

1WO STORY PRNARY S1l'.JC1LfIE C,=C 2lQ.ASICAU.Y DNoftIlI

2

2

4

01--_"--__"'-__"--__"'-__"-_--'

o 0.5 15 2 2.5 3

O'----"----"'---"--__"--__"--_---J

o

6...------------- ---,

6,...-------------------,

Maximum Reldve Dlapl8cement of Detuned Equipment Attllched to Classi­cally o.mped Two Story PrImary Structure •• Function of EquipmentDamping, " • V~,

(.) Mus Ratio Y.10-2, (b) Ma.. RatIo y. 10-a, (c) Mau Ratio Y.10",

6-21

I~3

EIGiT STORY PRIMARY STRUCTlH

~~IQ....S5lCAI.LY 0..-

2 TLND EOlFMENT

Ci& MASS RATIO :: 10-2

H 5a

S~X QASSICAI.I.'

~lI! DAWBJ SCl..VTlON

i~0

0 05 15 2 25 3

FRACTION OF ClASSICAL. DAMON>. 11:: {II (IC

I ~.3

EIGIT STORY PAMARY STRUCnH

~~CLASSlCAU.'O~

2 TLND EOlM'MENT

0& MASS RATIO :: 10-3

H 5b

fil~~ll QASSICAI.I.'

~~ 0MftIl SClWTIQN

0x E 0 05 1.5 2 2.5 3i~

FRACTION OF ClASSICAL DAM'NG, 11:(I/(u

I~3

EIGiT STORY PRIMARY STRUCTlH

~~lCI&SSlC.ou., 0-..

2 TLND EOlJIIt.eIT

i5P5 MASS RATIO :: 10-4

H 5c

w~ ~. CLASSICMU

!~_ SCIUITlON

00 0,5 1.5 2 2.5 3

i FRACTION OF ClASSICAL DAM'NlI1::(./("

FIGURES Maximum Reldve Displacement of Tuned Equipment AttBched to Claul·cally Damped Eight Story Primary Structure .. Function of Equipment

Damping, " • ~J~'

(a) Ma.. Ratio y. 10.2, (b) Mus RatIo y. 104 , (c) ..... Ratio Y• 10".

6-22

10,----------------------,

2L-__.J...-__....I..-__-L..__

o 0.5 1.5

6a

32.52

EIGHT STORY PRIMAAY STRUC~E

!CLASSICAU.Y DAM'fD1

DETlJIED EOUIPPJ8ff w,= IWI • WZl/2

MASS RATIO = 10-4

EllACT

5a.Ul1CN

N'f'ROX a.A~SICAlLYDAM"EO sa-unON

4

6

8

FRACTI~ OF a.ASSICAL DAI*N:;. /I ={,I {"

.,.(8) EQUIPMENT fREQUENCY ~ - <w1+w2)/2, MASS RATIO 7 - 10 .

~ _~, eec

4 ,.------------.--------,

oLO----'---............,---,:-a:.5:---~2"---2.~5~---:30.5

6b

EIGHT STORY PRIMARY STRUC~E

I<1ASSICAU.Y OAM'8JI

DElU'ED EQUlPPJENT w,=(w2+ w3!/2

MASS RATID =10-4

1 ~ACT5Q.UTION

2 \ ~X CUlSSICAU 'f

I V~srAUTOt

3

FAACT1a'4 OF Q..ASSICAL OAM'ING. ,,= (t/(,c

(b) EQUIPMENT FREQUENCY w. - (w2+w])/2, IlASS RATIO., _ 10.4 ,

(.c - 9.7t.

FIGURE 6 Maximum Relative DlaplKement of Detuned Equipment Attached to EightStory PrtmllY Structure .. Function of equipment Damping. 11 • ~~.'

6-23

11,------------ ---,

~• ~. a.ASSlCAl.lY

f2 . y-rm SQ.UTlON/

II,-.

10

8

4

EIGiT STORY PRIMAAY STRUCT\ftICI.ASSICAlLY O-aJI

~~T ...,=(........S]/Z

MASS RAno =10-4

6c

2'---"'-----"'------'---_~___L__----'o 0.5 1.5 2 25 3

FRACTION OF CLASSICAl.. DAJIIPING. ,,= ltl {Ie

(c) EQUIPHENT FREQUENCY w - (w4+wS)/2, HASS RATIO T - 10. 4 ,e - 18'. eec

4.--------------------,

OL-__"'O"-__....L...-__~--~--_="::::--~

o 0.5 1.5 2 25 3

3

2

EXACTSOLUTION

EIGtT STORY PRIMARY STRUCTlHta.ASSlCJIU.Y DAMI'H'l

CET\JIE)EO~Wt=(wr+ w .lf2MASS RATIO = 10-4

6d

FIGURES

-4Cd) EQUIPltENT FREQUElfCY w - (w7+wS)/2. HASS RATIO T - 10 ,e.c - 26'. •

Maximum ReIaUve DI.placement of Detuned Equipment Attached to EightStory Primary Structure a. Function of Equipment Damping. 11 • ~.'~

(Cont'd).

6-24

5~---------------------,

26e

DE1\HD eol.JlP'JBfl' Wt=lw 4+lIIsl/2MASS RATIO = 10-2

4

~N'f'ftOX Ct"SSICALlY

OAM'HI SOlUTION

3L.-

\EXACT

SOlU1lON

EIGHT STORY PRIMARY STRUCTtH

ta."SSlCALUO~

1 L __--I-. ·I-__....I-__--' -'-__--J

o 0.5 1.5 2 2.5 3

FRACTION Of ClASSICAL OAtJPING. 1'1 ={t/{u

-2(e) EQUIPlfENT FREQUENCY w - (w4+w

S)/2, MASS RATIO., - 10 •

E - 18'. eec

11,---------------------,

EIGHT STORY PRIMARY smucn.n:ICUSSlCALlY DAM'l!OI

0ET\..f'S) EOUIPM:NT w, =Iw, +11I51/2

MASS RAno = 10-·

10

B

6

4

~ll QASSICAllY/NIPf!O SOlUOON

6f

2 LO---O......5----'----.&1.5-----'21---2.....5---J3

FRACTION OF ClASSICAL DANFI'IG. 1'1 =(.1 (.e

FIGURE 6 Maximum Relative DI.plllCement of Detuned equipment Attached to EightStory Primary Structure a. Function of Equipment Damping, Tl • V~(Cont'd),

,,-25

80 , i

3

EXACT,UnDN

TlND EOlFMENTMASS RATIO =10-4

TWO STORY PRNARY STRJC1l.R C1: 2e. C2=0f«)NCI ASSICAU.y DM BlL

0' t • I I I ,

o 05 1 1.6 2 25l

FRACTION OF APPROXNATE CLASSICALD~ II' =T'tc

40,

60

20

I;i5~...~

a­I..,

0'

FIGURE 7 Mulmum Relative Displacement of Tuned Equipment Attached to Nonclas­slcally Damped Two Story Prlm..ry Structure as Function of Equipment

Damplna.'ll' =~J~~c'

160 • I

3

1\J\B) EOUFMENT

MASS RATIO =10-4

TWO STORY PRIMARY STflJCTlH <;= 2C. C1=0NlNCLASSICALLy DAJlI 1iA

0' I • I I , ,

o Q5 1 1.5 2 2.5(

FRACTION OF APPROXMATE a.ASSICAL DAM"Ni. .... =-+­(ec

40

80

120!;Zi~~

IiIi

0­I..,

......

FIGURE 8 Maximum Relative Displacement of Tuned Equipment Attached to Nonclas­slcally Damped Two Story Primary Structure as Function of Equipment

Damping, 11' =~/~o'

a­t

'"<Xl

!~..

~~~o

4

3

2

1

TWO STORY PRMARY STRUCTlH

T\.N:D~

MASS RATIO =10-4

Ie, :: 2C. Cz :: Ot

0' , , , , I I

o 0.5 1 1.5 2 2.5 3{

FRACTION OF APPROXIMATE QASSICALD~ ". =~"'It

FIGURE 9 MaxImum RelaUve Dlspl8cement of Tuned equipment Attached to Nonclss­..cally Damped Two Story Primary Structure 8S Function of Equipment

Damping, TI' =~'/'c·

15I

TWO STORY PRNARY STRUCTlH C, =C2

'1=05 IClASSICAU.Y DAM'EDI

12~ / - (,=2.5"\ 1\H:D EOlFMENT

I~MASS RATIO =10-·

9Q=10

(,=5"

I 1/- 6

'"

I~,'"'"

~. 3~ 1/ Q=3.0

(,=15"

00 6 10 15 20 26 30

TNE IN SEaNJS

FIGURE 10 Root Mean Square of Response of Tuned Equipment Attached to Classi­cally Damped Two Story Primary Structure as • Function of Time, t.

0.75I

TWO STCIlY PRNARY STflJCTlH C, =C2tClASSICAUy OAM'EOI

0.6 ~ q=o.5. (,=4.5"DETl.f6) EOUIPNENT

MASS RATIO =10-4

~.r-" _ __ . __ AI'

~0.45

0.3'"

i~I....

0

B. Q15

00 5 10 15 20 25 30

TNE IN SECCJN)S

FIGURE 11 Root Mean Square of Response of Detuned Equipment Attached to Classi­cally Damped Two Story PrImary Structure as a Function of Time. t.

40 r-------------.-----,

TWO STORY PRIMAIlY STlIUC~ C, =C,

12a

Ol.----'---........--~-~--~-~o 0-5 15 2 2.5 3

FRACTION a: Q.A$SIC.AL DAM'nG. II =t, I t..

4Or------------------,

TWO STORY PRIMARY STftUCTUIE C, =C,

12b

oL-_~--':_-_:':"-~:__-~-____:o Q!l US 2 2.!l 3

10

la 'IWO S1CR'f'''''''''''' snu::nN ~.C2

10 _.-il

1\ND IClUfttNf

.. ""1'10 • '0'"40

Uzg 12c-c:uquu,.--00 0.& , ,.. a a.a 3

I'MCT1ON 01 QASSK"AL DAM'tG.". le' lee

~uUAE 12 Maximum AMS A_pon.. of Tuned equipment Attached to Two StoryCI....C1111y Damped Primary Structure .. Function of equipment Damping,11- ~~eo'

6-31

II

IEIlJfr SfOIlY PMoWlY S1IlUC1\IlE

4a-..._

I'Dr1\HI) EOL..-n ..=Iw.'~2

MASS ....TIO = 10·'

3

;1 _0.-.., 138-U ....-

00 Oil UI 2 :lll 3

4

EIGHT STORY PRNARY S1'RlJC'lUle1l1A1SICAU' QAMICII

Z; 3 0E'l\JIIEC EOlJPf,ENT ...=1.....~1/2

~ZMASS RAM = 10-'

2 1O~ 13b......~ ~

_. CUSSIC....'

"'51 ~.-.o ....UTICH

ia~~ \JlACT~ ~

_ut_:l!

00 O.S U 2 2.5 3

bEIGfT STORY _ S'TlllCnH

I~--,-

4 0ET\.fe)~ ...f"'~I/Z

MASS llAno. 10·'

MI~~i 3ii5 13c

~~i~

2 PACYIl1<mllN

,0 011 1.11 2 2.5 3

fl'IACTION OF Q.A~ D.UiPlNG. ...(./(..

FIGURE 13 Maximum RMS Response of Detuned Equipment AttaChed to Eight StoryClassically Damped Primary Structure as Function of equipment Damping,

1'1 =V~·

(a) Mau Ratio Y.10~, ;. .18%, (b) Ma.. Ratio Y.10-2,~.1~

(c) Mau Ratio Y.10~,;.. 3.8%.6-32

".VI

i

~3

~

~ 2 TIIO _I nJIIAI'l nauc:na&. <:,<.U lCUIIIc:au.I _I~ _1IlV1_>- ......~ Mil IAIIO - 10-'l-en

InII;; 14dIL

~ 0

I0 3 6 9 12 15

,..n~ 4

~«

~3

'MI~~~~. ('.<.-1IIUl_U ......,«>- 2 ..... 1a1l0 - 10'"

~In

~uIII~ 148eni 0

0 3 6 9 12 15

". 40VI

~

~ 30

I 'MI ~c:J::::k~.1:,<.

~ 20 -1IlVI-I

-.-'_ lanD - 10'"

10

~ 14fI 00 3 8 9 12 15

TM III SECXJN)S

FIGURE 14 RMS of Story Acceleration.

(d) Firat Story, (e) second Story, (f) Equipment DI8PI-=emenl

6-34

SEtTlOR 7

CONCUJSIORS

A modal analysis approach, referred to as the canonical modal

decomposition procedure, for seismic analysis of nonclassically damped

structural system is presented. The main advantage of this procedure is

th~t the resulting decoupled equations of motion contain only real

parameters. Procedures are outlined to solve the decoupled equations for

deterministic ground excitations. Also presented is a procedure to solve

these decoupled equations when the ground excitation is a nonstationary

random process.

The canonical modal decomposition procedure is used to obtain the

response of primary-secondary structural system and to perform parametric

studies for the effect of nonclassical damping on the response of both

primary and secondary structures. In parametric studies several examples

were considered under both deterministtic and nonstationary stochastic

ground accelerations. A single-degree-of·freedom equipment attached to a

classically damped multi-degree-of-freedom primary structure was considered.

Using the canonical modal decomposition procedures, the response of the

equipment for various equipment dampings, mass ratio., and tunning and

detuning have been calculated.

Based on both deterministric and .tocha.tic earthquake ground .otion

inputs, the following conclusions are obtained from our sensitivity studies

for the re.pon.e of primary-.econdary system, where the primary .tructure is

cla.sically damped. (1) The effect of noncla•• ical damping on the equipment

re.ponse i •• ignificant when the following condition. are .ati.fied

.imultaneou.ly, (1) the frequency of the equipment i. tuned to that of any

mode of th~ primary structure, (ii) the ratio i •••all. and (iii) the

damping ratio E of the equipment is ller than the dallping ratio e thate ee

7-1

results in a classically damped primary-secondary system. Under this

circumstance, the approximate classically damped solutions, i.e., the

solutions obtained using the undamped modal matrix and disregarding the off-

diagenal terms of the resulting damping matrix, are usually unconservative.

(2) When the equipment is detuned at low frequency, the effect of

nonclassical damping on the equipment response is negligible. Hence, the

approximate classically damped approach can be used. However, under the

following conditions, the effect of nonclassical damping on the detuned

equipment response can be significant. (i) The equipment is detuned at high

frequency, (ii) the mass ratio is small, (iIi) the damping ratio of the

equipment. (ec' that results in a classically damped primary-secondary

system is high, and (iv) the ratio of equipment damping ratio e to e ise ec

smaller than unity. Under this circumstance. the approximate classically

damped solutions for the equipment response are higher than the exact

solutions.

Also studied were small equipment-structure systems in which the

primary structure is nonclassically damped. Limited results indicate that

for such primary-secondary systems, the effect of nonclassical damping on

the equipm.nt response can b. estimated by approximating the primary

structure a. being classically damped. This is accomplished using the

undamped modal matrix of the primary structure and disregarding the off­

diagonal ter.s of the resulting damping matrix. Then the conclusions

described above hold. Corresponding to e for cla.sically damped primaryec

structure, a .eaningful measure of equip.ent da.ping for noncl•••ically

damped primary sy.te., denoted by e~c' is detemined using the

approximate cla••ically damped primary system.

With the consideration of nonstationary earthquake ground

acceleration, exten.ive para.etric .tudies indicate that the overshooting

7-2

phenomenon doesn't occu~ for the response of either primary or secondary

system. In other words, the stationary response is always larger than the

transient response. This may be attributed to the fact that both the

earthquake excitation and the stI~ctural response are zero at time zero.

An eight story structure was considered and the distribution of

damping in the structt:re was varied. Results indtcdte that the effect of

nonclassical damping on the response is not significant if the damping

distribution within the structure does not change drastically.

Finally, the equipment location may be an important factor for the

behavior of the equipment response [5]. Such a problem is currently being

i.nvestigated. In general, the conclusions obtained herein are consistent

with those of Refs. 5 and 16 in which parametric studies for the frequency

response function of equipment responses were conducted.

7-3

SECTION 9REFER£NCES

1. Caughey, T.H., and O'Kelly, E.J., ~Classical Normal Modes in DampedLinear Dynamic Systems," S, Applied Mechanics, ASME 32, 583-588, 1965.

2. Clough, R.W., and Penzien, J., "Dynamics of Structures," McGraw Hill,New York, 1975.

3. Edelen, D.G.B., and Kydoniefs, f •. D., "An Introduction to LinearAlgebra for Science and Engineering, Elsevier, New York, 1976.

4. Foss, K,A., "Coordinates Which Uncouples the Equations of Hotion ofDamped Linear Dynamic Systems," J. Applied Mechanics, ASHE 25, 361­364, 1958.

5. HoLung, J.A., Cai, J" and Lin, Y. K., "Frequency Response ofSecondary Systems Under Seismic Excitation," National Center ForEarthquake Engineering Research, Technical Report No, NCEER·87-0013,July 1987, SUNY, Buffalo, N. Y.

6. Igusa, T., Der Kiureghian, A., and Sackman, J.L" "Kodal DecompositionHethod for Stationary Reponse of Non-classically Damped Systems,"Earthquake Engineering and Structural Dynamics, Vol. 12, 121-136,1984.

7. Lin, Y.K., and Howell, L.J., "Response of Flight Vehicles toNonstatlonary Turbulence," AIAA Journal, Vol. 9, No. 11, 1971, pp.2201·2207.

8. Perlis, 5., "IheotY of Hatrices," Addision wesley, Reading, MA, 1958,

9. Singh, M.P., "Seismic Response by SRSS for Proportional Damping,"Journal of the Engineeriog Mechanics Division, ASeE, Vol. 106, 1980,pp. 1405·1419.

10. Singh, M.P" and Ghafory.Ashtiany, M., "Modal Time History Analysis ofNon-Cla•• ically Damped Structure. for Sei.mic Motion.," EarthquakeEngineering and Structural RxnIm1C" Vol, 14, 133·146, 19~6.

11. Traill·Nalh, R.W., "Modal Methods in the Dynamics of Systems with Non·Cla•• ical Damping," EarthqUAke Engineering and StructurAL DYnamics.Vol. 9, 153-169, 1983,

12. Veletol, A,S., and Ventura, C.E., "Modal Analy.i. of Non·ClassicallyDamped Linear System.," Earthquake Engine,ring And StructuralDynagic., Vol. 14, 217·243, 1986.

13. Yang, J.N., and Lin, M.J., "Optimal Critical·Mode Control of TallBuilding Under Seismic Load,- Journal of the Engin.ering Mechanic.Diyilion, ASCE, Vol. 108, No. EM6, 1982, pp. 1167·1185.

8-1

14. Yang, J.N .• Lin, M.J., -Building Critical-Mode Control: NonstationaryEarthquake,- Journal of the Enlineerinl Mechanic. Diyision, ASCE,Paper 18429, Vol. 109. No. EM6. 1983. pp. 1375-1389.

15. Yang, J.N .• Lin. Y.K .• and Sae Ung. S., -Tall Building Re.ponse toEarthquake Excitation.,- Journal of the Enlineerinl MechanicsPiyi.ion, ASCE, Vol. 106. No. EM4, 1980, pp. 801-817.

16. Yong. Y., and Lin, Y.K., -Parametric Studies of Frequency Response ofSecondary Syste.. Under Ground Acceleration Excitations,- NationalCenter For Earthqu.ke Engin••ring R••••rch. Technic.l R.port No.NCEER-87-00l2, Jun. 1987, SUNY, Buffalo, N.Y.

8-2

The equations of motion given by Eq. (2) is as follows

in which ~l and ~ are (2nx2n) symmetric matrices. The jth pair of eigen­

vectors, t2j-l and t2j' are expressed aa

(1-2)

j - 1, 2, .... n

in which !j and 2j are 2n real vectors.

Let

Y - T v

in which T is a (2nx2n) real matrix constructed in the following

(1-3)

(1-4)

Substituting Eq. (1-3) into Eq. (1·1) and pr••ultiplying it by the

transpose of 1 matrix, 1', one obtains

r ; + A v - T' P x-- -- - - I

A-l

(1-5)

in which £ and ~ are of canonical form

where

r - T' ~l T 6 - T' B T (1-6)

(I-7)

and £j and ~j are (2x2) matrices given by

[ !j ~, !Ja' ~l

bJ 1 [.' ! !j !j ! ~j

1-j -j

!:j - 6 - (I-8)

~j ~j !j b' ~l ~j-j b' ! !j ~j ! ~j-j -j

Equation (l-S) consists of n pairs of equations in which each pair

repr•••nta one vibrational ..ode, and it b uncoupled with other pair.. The

jth pair of coupled equations in Eq. (l-S) i. given a. follows

•j -1 -j 2j -j - -j 2j-1 -j

Sine. ~ and Bare .yaaetric, the orthogonality condition. are given

by

A-2

.. ,ok (1-10)

Substituting Eq. (1-2) into Eq. (1-10) with .. - 2j-l and k - 2j. one obtains

(1-11)

Hence the following orthogonality conditions are obtained from Eq. (1-11),

(1·12)

Eliminating ;2j and; 2j-l re.pectively. u.inS Iq. (1-9) and the or­

thogonality propertie. of Eq. (1-12), one obtain. the following two

equat10M for the j th vibrational mode

(I -13a)

A-3

(I·13b)

A comparison between Eq•. (20) and (1-13) le.da to the following

alternate expres.ions for the ele.ent. of the! vector.

(1-14)

'1!Pi'7<11 TO soumQl lQI ClASSICALLy IWIPIQ nIl.

If the .tructura1 .y.t.. 1. cla••ically ~.d. then the ! .atrix 1.

greatly a1mplified. For .uch a .yate., the di.plac••ent part of the

eig.nvector ia r.al. Since the velocity part of the .ig.nvector is the

eig.nvalue multiplied by the displacement part of the eigenvector, Eq. (6),

!j and ~j can b••xpr••••d a.:

(1-15)

ment eigenvector.

is simply the jth

in which ! j is the lower half of the !j vector representing the displace­

Note that for classically damped structural systems, !j'

eigenvector of the undamped system. Hence, for a

classically damped system, the following simplifications can be made

- O. (1-16)

Furthermore, .xpre.sio~ for F2j _1 and F2j can b. simplified using the

orthogonality condition, !j ~1 !j - ~j ~1 ~j'

- !; AJ, 2, 2 {~jQ' .~~} {.'-KO_,.J.(~j ~1 !j)

- 0, (1-17)

A-S

and

e' P-) -

- 2j ~l !j

•"!j !! !

loIDj !j ~ !j

A-6

(1-18)