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5.6 Continuous System Random Vibration

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Continuous models of vibrating systems are viewed as more realistic because the structural properties

are distributed rather than being concentrated at discrete points. The price to pay for increased realism

is increased complexity. The governing equations of motion go from being ordinary differential

equations for the discrete models, to partial differential equations for the continuous ones, since

displacement is a function of both time and position. For our elementary applications, deriving and

solving the partial differential equations does not pose a great challenge. But, when we need to model

nonuniform structures with varying cross sections and material properties, approximate techniques

will be needed. Here our introduction to continuous systems will focus on strings and vibrating

beams. These are found in most structures and machines, and understanding how they behave is of

great use to modeling more complex systems. We will approach these studies using the direct and

modal solutions.

5.6.1 Transverse Vibration of Beams

Begin with the equation of motion of a damped, forced, simply-supported beam with uniform

properties

›4y

›x4 þ c

›y

›t þ m

›2y

›t2 ¼ pðx; tÞ ð5:41Þ

where it is assumed the natural frequencies and the mode shapes have already been evaluated.

To solve this problem, we will use the normal mode method, which depends on the modal

orthogonality properties. First, expand the applied force pðx; tÞ in terms of the modes,11 then do the

11The modes Yj ðxÞ are the eigenfunctions of the problem

›4 y

›x4 þ m

›2 y

›t2 ¼ 0

That is, they satisfy

d4 Yj

dx4 2 v2mYj ¼ 0; j ¼ 1; 2; …

and the boundary conditions at hand.

Random Vibration 5-29

same with the structural displacement

pðx; tÞ ¼

j¼1

pjðtÞYjðxÞ

Multiply both sides of this equation by mYkðxÞ and then integrate over the beam span

ðL

mpðx; tÞYkðxÞdx ¼

j¼1

pjðx; tÞ

ðL

mYjðxÞYkðxÞdx

On the right-hand side of this equation, we note that, by the orthogonality properties of the modes, the

integral equals zero for k – j; and otherwise is normalized to 1 for k ¼ j: Therefore, we are left with

ðL

mpðx; tÞYjðxÞdx ¼ pjðtÞ ð5:42Þ

Apply the same procedure to the response

yðx; tÞ ¼

j¼1

yjðtÞYjðxÞ ð5:43Þ

and multiply and integrate as above to find

ðL

myðx; tÞYjðxÞdx ¼ yjðtÞ

yðx; tÞ can be differentiated with respect to x and t so that all the terms in the equation of motion

(Equation 5.41) can be put into the modal expansion form

j¼1

EIyjðtÞ

d4Yj

dx4 þ c

dyj

Yj þ m

d2yj

dt2 Yj

" #

j¼1

pjðtÞYjðxÞ ð5:44Þ

We know that the modes are defined by and satisfy the equation EI d4Yj=dx4 ¼ v2j

mYj: Substituting this

relation into Equation 5.44 leads to

j¼1 ½v2j

myjðtÞ þ cy_jðtÞ þ my€jðtÞ 2 pjðtÞ􀀉YjðxÞ ¼ 0

Since the eigenfunctions YjðxÞ are not zero except for unrestrained motion, the expression in the square

brackets must vanish for every j;

y€jðtÞ þ

y_jðtÞ þv2j

yjðtÞ ¼

pjðtÞ; j ¼ 1; 2; … ð5:45Þ

As for single-DoF systems, let c=m ¼ 2zjvj; that is,

zj ¼

2mvj

Equation 5.52 is the damped, forced, harmonic oscillator, with the well-known convolution solution,

hjðtÞ ¼

ðt

pjðtÞgjðt 2 tÞdt ð5:46Þ

where gjðtÞ is the impulse response function for a damped oscillator. Once solved, yjðtÞis substituted

into the expansion equation

yðx; tÞ ¼

j¼1

yjðtÞYjðxÞ ¼

j¼1

YjðxÞ

ðt

pjðtÞgjðt 2 tÞdt

5-30 Vibration and Shock Handbook

and substituting Equation 5.42 for pjðtÞ results in

yðx; tÞ ¼

j¼1

YjðxÞ

ðt

ðL

pðj; tÞYjðjÞdj

􀀒 􀀓

gjðt 2 tÞdt ð5:47Þ

This is the complete steady-state time-domain solution.

5.6.2 Random Transverse Vibration

Assume that the loading Pðx; tÞ is random and represents an ensemble of functions of space and time. We

need only the first two statistics of the load to proceed with the analysis of the response. Its mean value is

denoted

E{Pðx; tÞ} ¼ mP ðx; tÞ

with autocorrelation

E{Pðx1; t1ÞPðx2; t2Þ} ¼ RPP ðx1; x2; t1; t2Þ

These statistics are available through data-gathering experiments performed on the loading to be

experienced by the structure. Our goal in this analysis is to derive the statistics for the response, mY and

RYY ; which are functions of the statistics of the loading and the structural parameters.

For steady-state vibration problems, we may reasonably assume that the statistics of the loading are

stationary. Begin with Equation 5.47 but now, for random loading Pðx; tÞ

yðx; tÞ ¼

j¼1

YjðxÞ

ðt

ðL

Pðj; t 2 tÞYjðjÞdj

􀀒 􀀓

gjðtÞdt

where the shift t has been placed within pðx; tÞ and take the expectation of both sides

E{Y ðx; tÞ} ¼

j¼1

YjðxÞ

ðt

ðL

E{Pðj; t 2 tÞ}YjðjÞdj

􀀒 􀀓

gjðtÞdt

The impulse response function is gðtÞ, and for stationary loading, the expected value is a constant,

E{Pðj; t 2 tÞ} ¼ mP ðjÞ: In order to simplify the following analysis, we assume that the loading has zero

mean. This does not overly simplify the problem since a nonzero mean can be introduced later by simply

shifting the response by the (constant) mean value. For zero mean loading, mP ðxÞ ¼ 0, and the response is

also zero mean, E{Y ðx; tÞ} ¼ mY ¼ 0:

We need to relate the autocorrelation of the response to that for the loading. Begin with

Equation 5.42:

RPj Pk ðt1; t2Þ ¼ E{Pjðt1ÞPkðt2Þ} ¼ m2

ðL

E{Pðx1; t1ÞPðx2; t2Þ}Yjðx1ÞYkðx2Þdx1 dx2

¼ m2

ðL

RPP ðx1; x2; t1; t2ÞYjðx1ÞYkðx2Þdx1 dx2 ð5:48Þ

where, for a stationary loading, RPP ðx1; x2; t1; t2Þ ¼ RPP ðx1; x2; tÞ; where t ¼ t2 2 t1 and therefore,

RPj Pk ðt1; t2Þ ¼ RPj Pk ðtÞ: Using Equation 5.43, the response autocorrelation is

RYY ðx1; x2; t1; t2Þ ¼ E{Y ðx1; t1ÞY ðx2; t2Þ} ¼

j¼1

k¼1

E{Yjðt1ÞYkðt2Þ}Yjðx1ÞYkðx2Þ

j¼1

k¼1

RYj Yk ðt1; t2ÞYjðx1ÞYkðx2Þ

Random Vibration 5-31

where, using Equation 5.46, with upper and lower limits extended to ^1 without changing the

final value of the integral due to the fact that the impulse response is zero outside the original

limits

RYj Yk ðt1; t2Þ ¼

m2 E

ð1

Pjðt1 2 uÞgjðuÞdu

ð1

Pkðt2 2 kÞgkðkÞdk

􀀘 􀀙

ð1

E{Pjðt1 2 uÞPkðt2 2 kÞ}gjðuÞgkðkÞdu dk

ð1

RPj Pk ðt1 2 u; t2 2 kÞgjðuÞgkðkÞdu dk

Since Pðx; tÞ is stationary, RPj Pk ðt1 2 u; t2 2 kÞ ¼ RPj Pk ðt2 2 t1 þ u 2 kÞ and RYj Yk ðt1; t2Þ ¼ RYj Yk ðt2 2

t1Þ: Therefore,

RYj Yk ðt1; t2Þ ¼

ð1

RPj Pk ðt2 2 t1 þ u 2 kÞgjðuÞgkðkÞdu dk ð5:49Þ

and

RYY ðx1; x2; t2 2 t1Þ ¼

j¼1

k¼1

RYj Yk ðt2 2 t1ÞYjðx1ÞYkðx2Þ

This last equation is a relation between the forcing and response autocorrelations. The spectral

density of the response, SYY ðvÞ; is given by the Fourier transform of RYY ðtÞ

SYY ðvÞ ¼

j¼1

k¼1

ð1

RYj Yk ðtÞe2ivt dt

􀀒 􀀓

Yjðx1ÞYkðx2Þ ð5:50Þ

where t ¼ t2 2 t1; and the term in the square brackets equals SYj Yk ðvÞ; which now needs to be

evaluated in terms that have already been derived.

By definition, we know the Fourier transform relation

SYj Yk ðvÞ ¼

ð1

RYj Yk ðtÞe2ivt dt

We can evaluate RYj Yk by beginning with Equation 5.49

RYj Yk ðtÞ ¼

ð1

RPj Pk ðt þ u 2 kÞgjðuÞgkðkÞdu dk

Take the Fourier transform of both sides, letting l ¼ t þ u 2 k; to find

SYj Yk ðvÞ ¼

ð1

e2ivðl2uþkÞ 1

ð1

RPj Pk ðlÞgjðuÞgkðkÞdu dk

􀀒 􀀓

dl ð5:51Þ

where t has been replaced by l 2 u þ k and dt by dl: Rewrite Equation 5.51 in a more useful form by

separating the integrals according to dummy variables

SYj Yk ðvÞ ¼

ð1

gjðuÞeivu du

ð1

gk ðkÞe2ivk dk

ð1

RPj Pk ðlÞe2ivl dl ð5:52Þ

The Fourier transform of the impulse response function gðtÞ is the frequency response function HðvÞ:

Therefore, Equation 5.52 becomes

SYj Yk ðvÞ ¼

m2 Hp

j ðvÞHkðvÞSPj Pk ðvÞ ð5:53Þ

5-32 Vibration and Shock Handbook

where HjðvÞ ¼ ½v2j

2 v2 þ 2zjvjv􀀉21 and SPj Pk ðvÞ; the spectral density of the modal force components,

is derived assuming that the Fourier transform for PjðtÞ exists, as it does for most physical processes

PjðtÞ ¼

ð1

PjðvÞeivt dv

Take the Fourier transform of Equation 5.48, to find

SPj Pk ðvÞ ¼ m2

ðL

SPP ðvÞYjðx1ÞYkðx2Þdx1 dx2 ð5:54Þ

where SPP ðvÞ is the spectral density of the loading, a quantity that is estimated from data. Substituting

Equation 5.54 into Equation 5.53, we now have the expression for SYj Yk ðvÞ; which can be substituted into

Equation 5.50, resulting in

SYY ðx1; x2; vÞ ¼

j¼1

k¼1

j ðvÞHkðvÞSPj Pk ðvÞYjðx1ÞYkðx2Þ

One value of having such an equation is that the mean-square displacement can be evaluated

YMSðxÞ ¼ RYY ðx; x; 0Þ ¼

ð1

SYY ðx; x; vÞdv

Recall that if mY ðxÞ ¼ 0 then yMSðxÞ ¼ s2

Y ðxÞ; the variance.

The derivations are now complete, but what do they mean and what do they do for us? One of the

functions of a probabilistic analysis is to help us bound our uncertainties so that we can understand how

randomness in the forcing results in a scatter of possible structural responses. Furthermore, this scatter is

not haphazard, but is defined by a standard deviation and possibly a density function. It is the variance

that is used to bound the mean-value response.

Acknowledgments

I am pleased to acknowledge my collaboration over the past few years with Dr. Seon Mi Han of Wood-

Hole Oceanographic Institute. Seon has had input to this document and has prepared the figures. It is

also a great pleasure to acknowledge the support provided by the Office of Naval Research and by our

program manager Dr. Thomas Swean under Grant No. N00014-97-10017. Finally, I appreciate the

invitation by Professor Clarence W. de Silva to prepare this chapter.

Bibliography

The discipline of random vibration includes thousands of references. Applications include wind

and earthquake engineering, aerospace engineering, and ocean engineering to name the broad areas.

Nomenclature

Symbol Quantity

E{·} mathematical expectation

HðvÞ frequency response function

RXX ðtÞ autocorrelation function

RXY ðtÞ cross-correlation function

SXX ðvÞ power spectral density

SXY ðvÞ power cross-spectral density

Symbol Quantity

{u}i structural mode

W0 one-sided spectral density

zðtÞ modal coordinates

m mean value (first moment)

s2 variance (second moment)

Random Vibration 5-33

More recently, applications in materials engineering and biomechanical engineering have also broadened

to include random effects. Therefore, we provide here a brief list of references.

Augusti, G., Baratta, A., and Casciati, F. 1984. Probabilistic Methods in Structural Engineering, Chapman &

Hall, London.

Benaroya, H. 1998. Mechanical Vibration: Analysis, Uncertainties, and Control, Prentice Hall, Upper

Saddle River, NJ.

Bolotin, V.V. 1969. Statistical Methods in Structural Mechanics, Holden-Day, San Francisco, CA.

Bolotin, V.V. 1984. Random Vibrations of Elastic Systems, Martinus Nijhoff, The Hague.

Dimentberg, M.F. 1988. Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies

Press, England.

Ibrahim, R.A. 1985. Parametric Random Vibration, Research Studies Press, England.

Lin, Y.K. 1976. Probabilistic Theory of Structural Dynamics, Krieger, Malabar, FL.

Madsen, H.O., Krenk, S., and Lind, N.C. 1986. Methods of Structural Safety, Prentice Hall, Englewood

Cliffs, NJ.

To, C.W.S. 2000. Nonlinear Random Vibration: Analytical Techniques and Applications, Swets & Zeitlinger,

Lisse, The Netherlands.

Vanmarcke, E. 1983. Random Fields: Analysis and Synthesis, MIT Press, Cambridge, MA.

Wirsching, P.H., Paez, T.L., and Ortiz, K. 1995. Random Vibrations: Theory and Practice, Wiley-

Interscience, New York.

5-34 Vibration and Shock Handbook

Computer Techniques

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