MODAL ANALYSIS OF NONCLASSlCALLY DAMPED ...,11111 1111.---- MODAL ANALYSIS OFNONCLASSICALLY DAMPED...
Transcript of MODAL ANALYSIS OF NONCLASSlCALLY DAMPED ...,11111 1111.---- MODAL ANALYSIS OFNONCLASSICALLY DAMPED...
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
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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
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Technical Report
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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)