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THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 33, NUMBiJR I NOVEMBER, 196t

Effect of Damping on the Natural Frequencies of Linear Dynamic SystemsT. K. CAIIGHE AI'qDM. E. J. O'Ksnnv

California nstiht f Technology, asadena, Californi(Received July 10, 1961)

An analysis s presented I the effect of weak damping nthe natural requencies f inear dynamic ystems.It is shown hat the highest natural frequency s always decreased y damping, but the lower natural fre-quencies may either ncrease r decrease, epending n the form of the damping matrix.

INTRODUCTION

Nhis octoralhesis,ergconsideredhe ibrationf a dynamic system with generalized inear damp-ing, and showed umerically hat the damped natural[requency of the lowest mode was larger than thecorresponding requency of the undamped system.This phenomenon s probably well known o workers n

vibration and circnit analysis; however, the authorshave been unable to find any systematic reatment ofthis problem n the literature.

It is well known that in a single-degree-of-freedomsystem, he damped natural frequency s always essthan the undamped atural frequency. n the case ofmulti-degree-of-freedom ystems with classical ormalmodes? it may be shown hat the damped naturalfrequencies re always ess han, or equal o, the corre-sponding ndamped requencies.

The purpose f this paper s to study the effects ofweak damping on the natural frequencies of lineardynamic systems and to show under what conditionsthe natural frequencies may be increased y damping.

ANALYSIS

The equations of motion of an X-degree-of-freedomlinear dynamic system with lumped parameters maybe written in matrix notation as

[M]IX"I +[C]lX'l +[K]{XI = {l(t)}. (1)

For passive ystems he NXN matrices M] and [K]are symmetric nd positive efinite, nd the matri,r C]is symmetric and nonnegative efinite.Consider the homogeneous ystem obtained bysetting If(t)}=0 in (1)

[M]lX"l+[C]lX'l+[K]lX}=o. (2)

CLASSICAL NORMAL MODES

The system defined by (2) possesses lassical ormalmodes? f and only if the matrix [C] is diagonalizedby the same transformation which simultaneouslydiagonalizes I] and [K-].

Let

IXl = [*]l l, (3)t O. V. Berg, Ph.D. Thesis, University of Michigan, Ann

Arbor, Michigan, 1958.2T. K. Caughey, . Appl. Mech. 27E. 269-271 (1960).a For the definition of Classical Normal Modes see Appendix.

where [I,] is the normalized matrix which simul-taneously iagonalizes M] and [K].

If [C] is such hat classical ormal modes xist, hen[]rFC][cb]=[0]--a diagonal matrix with elements

o,= {,,} '[c]{,,}. o)If (3) is substituted nto (2) and then premultiplied by

I-P , there results he system of equations:ffl.if'+Oii'+Iii=O, i=1, 2, -.-, N, (5)

where

Let

Then

0i: l'}T[C]{ }'= 13

(6)

i= i*ex". (7)

7,=2/+ - . (8)Hence, the damped natural frequency s given by:

F , / 0i \21;o,=/,'-'-/--I / _

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I)AM PING IFFECT ()N NATI'R.\I. FRI,;()I'I,]NC[I-;. OF S'fS'I'f,]MS 1459

Let

I x } = {&} expL.. (ll)With substitution of (11) into (10)

et&="+,+,o"+. ., (3)

where " aud X,, re the nth eigenvector nd eigenvMuefor the undmped roblem, =0. Inserting 13) nd(14) into (12) leads o the following ystem of equa-tions n separating ut the vrious orders n :

(XEM]+[]){I =0,

(X,SEM]+EK]){ }

From these equations, he perturbations f variousorders may be calculated.

ZEROTH ORDER SOLUTION

The zeroth order solution s obtained rom Eqs. (15):

(X,?-[M]+ K]){ 'q 0 ,,=1, 2, ..., Y. (18)Siuce HI and EK] are symmetric ndpositive efinite:

(1) X,-

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DAMPING EFFECT ON NATURAL FREQUENCIES OF SYSTEMS 1461

CONCLUSIONS

From the above analysis he following conclusionsmay be drawn:

(1) In a linear dynamic system with weak damping,

the damped natural frequency of the highest mode salways ess han or equal to the undamped requency,no matter what form of damping matrix is used.

(2) The damped natural frequency of tile lowestmode may be higher han the corresponding ndampedfrequency, epending n the choice of damping matrixand the mode separation.

(3) In a system with classical ormal modes, hedamped natural frequencies re always less than orequal to the corresponding ndamped requencies.

Example. To illustrate the results of lhe aboveanalysis, consider he following system:

[M]lX"l + [']lX'l + [g]lX'l =0, (54)where

EX]= - 2 ---1

Ec'] = oo

=0.1.

Unda.ped System. or the undamped ystem,

=0.765366;

{-} o -, 1.414214;

aa= 1.847759.

With use of Eq. (47), the damped atural requenciesare

coa0.765687 co,co2a1.413993 co-,, (57)

waa-'- .846696 < cos.The exact values obtained by solving Eq. (54) are

cola 0.765688,

co.,a= .413990, (58)cOaa= .846698.

Comparison f (S7) and (58) shows xcellent umericalagreement. t should be noted hat the damped naturalfrequency f the first mode s higher than that for theundamped system, while the damped frequencies orthe second and third modes are lower than the corre-

sponding alues or the undamped ystem.

APPENDIX

ClasMcal Normal Modes

It is well known that undamped inear dynamicsystems ossess ormal modes, nd that in each normal

(55) mode hevarious arts f the system ass hroughtheir maximum or minimum positions at the sameinstant of time. Since this type of normal mode wasthe subject of Lagrange's lassic reatise on mechanics,*the author has defined such normal modes as "ClassicalNormal Modes."

In damped systems n general, t is found that in anormal mode of oscillation, the various parts of thesystem o not pass hrough heir maximum or minimumposition t the same nstant of time. In such cases hemore general treatment of F. A. Foss must be used.Rayleigh showed hat if the damping matrix is a linearcombination of the sliffness and inertia matrices, thedamped system possesses lassical ormal modes. Morerecently, Caughey has shown that a necessary nd

(56) sufficient ondition or the existence f classical ormal

modes s that the damping matrix be diagonalized bythe same transformation which uncouples he un-damped system.

s . L. Lagrange, Mect, nique Analytique Gauthier-Villar, Paris,1811), Nouvelle edition.

Lord Rayleigh, Tkeory of Sound. (Dover Publications, NewYork, 1945), Vol. I.