5.3 Response to Two Random Loads

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Previously, the system responses were due to a single randomly varying force. In general, the situation is

more complicated, because more than one load may act on a system and the resulting response depends

not only on the properties of each force, but also on the correlation between the two forces.

TABLE 5.3 Output Spectral Density

SXX ðvÞ ¼ lHðvÞl2 SFF ðvÞ

lHðvÞl2 from structural parameters

SFF ðvÞ is the input force power spectrum

FIGURE 5.3 Correlation functions and corresponding power spectral densities.

Random Vibration 5-7

© 2005 by Taylor & Francis Group, LLC

Consider the response of the system to two

random loadings, PðtÞ and QðtÞ; acting simultaneously

but at different points on the system, as

shown in Figure 5.4. We are interested in

calculating the response statistics of the displacement

XðtÞ at an arbitrary point on the system.

Assume that E{PðtÞ} ¼ 0 and E{QðtÞ} ¼ 0: Also,

by utilizing available data, we are able to estimate

RPP ðtÞ and RQQðtÞ: Our interest is in evaluating

RXX ðtÞ and its Fourier transform SXX ðvÞ: Using linear superposition and the convolution integral, the

response due to both forces is given by

XðtÞ ¼

ð1

21

􀀑

gXP ðt1ÞPðt 2 t1Þ þ gXQðt1ÞQðt 2 t1Þ

􀀜

dt1

Similarly, for Xðt þ tÞ:

Xðt þ tÞ ¼

ð1

21

􀀑

gXP ðt2ÞPðt þ t 2 t2Þ þ gXQðt2ÞQðt þ t 2 t2Þ

􀀜

dt2

where gXP is the impulse response function at coordinate X due to force P and gXQ is the impulse response

function at X due to Q: Then,

RXX ðtÞ ¼ E{XðtÞXðt þ tÞ} ¼ E

ð1

21

􀀑

gXP ðt1ÞPðt 2 t1Þ þ gXQðt1ÞQðt 2 t1Þ

􀀜

dt1

􀀘

􀀐

ð1

21

􀀑

gXP ðt2ÞPðt þ t 2 t2Þ þ gXQðt2ÞQðt þ t 2 t2Þ

􀀜

dt2

􀀙

Now expand the product, and then move the expectation operator to the random processes, as follows:

RXX ðtÞ ¼

ð1

21

gXP ðt1Þ

ð1

21

gXP ðt2ÞE{Pðt 2 t1ÞPðt þ t 2 t2Þ}dt2

􀀒 􀀓

dt1

þ

ð1

21

gXP ðt1Þ

ð1

21

gXQðt2ÞE{Pðt 2 t1ÞQðt þ t 2 t2Þ}dt2

􀀒 􀀓

dt1

þ

ð1

21

gXQðt1Þ

ð1

21

gXP ðt2ÞE{Qðt 2 t1ÞPðt þ t 2 t2Þ}dt2

􀀒 􀀓

dt1

þ

ð1

21

gXQðt1Þ

ð1

21

gXQðt2ÞE{Qðt 2 t1ÞQðt þ t 2 t2Þ}dt2

􀀒 􀀓

dt1

In this expression

E{Pðt 2 t1ÞPðt þ t 2 t2Þ} ¼ RPP ðt þ t1 2 t2Þ

E{Qðt 2 t1ÞQðt þ t 2 t2Þ} ¼ RQQðt þ t1 2 t2Þ

The expectations in the second and third terms are cross-correlations of the form RPQðtÞ ¼

E{PðtÞQðt þ tÞ}: Therefore, the autocorrelation of the response becomes

RXX ðtÞ ¼

ð1

21

gXP ðt1Þ

ð1

21

gXP ðt2ÞRPP ðt þ t1 2 t2Þdt2

􀀒 􀀓

dt1

þ

ð1

21

gXP ðt1Þ

ð1

21

gXQðt2ÞRPQðt þ t1 2 t2Þdt2

􀀒 􀀓

dt1

þ

ð1

21

gXQðt1Þ

ð1

21

gXP ðt2ÞRQP ðt þ t1 2 t2Þdt2

􀀒 􀀓

dt1

þ

ð1

21

gXQðt1Þ

ð1

21

gXQðt2ÞRQQðt þ t1 2 t2Þdt2

􀀒 􀀓

dt1

P(t)

Q(t)

X(t)

FIGURE 5.4 Response to two random loads.

5-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The importance of this result is primarily in the observation that we cannot derive RXX ðtÞ unless the

cross-correlations RPQðtÞ and RQP ðtÞ are also known. Using the Fourier transform relation between

RXX ðtÞ and SXX ðvÞ; that is

SXX ðvÞ ¼

1

2p

ð1

21

RXX ðtÞe2ivt dt

we obtain

SXX ðvÞ ¼ Hp

XP ðvÞHXP ðvÞSPP ðvÞ þ Hp

XP ðvÞHXQðvÞSPQðvÞ

þ Hp

XQðvÞHXP ðvÞSQP ðvÞ þ Hp

XQðvÞHXQðvÞSQQðvÞ ð5:8Þ

where

Hp

XP ðvÞHXP ðvÞ ¼ lHXP ðvÞl2

Hp

XQðvÞHXQðvÞ ¼ lHXQðvÞl2

and

SPQðvÞ ¼

1

2p

ð1

21

RPQðtÞe2ivt dt

As expected, the evaluation of the output spectral density requires knowledge about the cross-spectra

SPQðvÞ and SQP ðvÞ: If there are more than two forces, then we will have additional cross-spectra between

each pair of forces.

Examining Equation 5.8 closely, we find that we can write SXX ðvÞ as

SXX ðvÞ ¼

h

Hp

XP ðvÞ Hp

XQðvÞ

i SPP ðvÞ SPQðvÞ

SQP ðvÞ SQQðvÞ

" #

HXP ðvÞ

HXQðvÞ

" #

ð5:9Þ

Example 5.3 Conjugates of Cross Spectra

It was briefly mentioned that RPQðtÞ ¼ RQP ð2tÞ: How are SPQðvÞ and SQP ðvÞ related?

Solution

By the definition of spectral density

SPQðvÞ ¼

1

2p

ð1

21

RPQðtÞexpð2ivtÞdt

Replacing RPQðtÞ with RQP ð2tÞ

SPQðvÞ ¼

1

2p

ð1

21

RQP ð2tÞexpð2ivtÞdt

Letting 2t ¼ t

SPQðvÞ ¼

1

2p

ð1

21

RPQðtÞexpðivtÞdt

Then,

SPQðvÞ ¼ SQP ð2vÞ ¼ Sp

QP ðvÞ

Example 5.4 Response Spectrum due to Two Random Loads

Consider a mass – spring – damper system in Figure 5.5 subject to two random forces PðtÞ and QðtÞ: Find

the response spectrum SXX ðvÞ assuming that

SPP ðvÞ ¼ SP ; SPQðvÞ ¼ 0; SQQðvÞ ¼ SQ

Random Vibration 5-9

© 2005 by Taylor & Francis Group, LLC

Solution

The equation of motion for this system is given by

mX€ þ cX_ þ kX ¼ PðtÞ þ QðtÞ

First, assume that QðtÞ ¼ 0 in order to first obtain

HXP ðvÞ: Taking the Fourier transform, the equation

of motion is given by

ð2mv2 þ civ þ kÞXðvÞ ¼ PðvÞ

Then, the frequency response function HXP ðvÞ is

HXP ðvÞ ¼

1

ð2mv2 þ civ þ kÞ

or

HXP ðvÞ ¼

1

mð2v2 þ 2vnziv þ v2

nÞ

Similarly, HXQðvÞ is obtained by setting PðtÞ ¼ 0 and is also given by

HXQðvÞ ¼

1

mð2v2 þ 2vnziv þ v2

nÞ

Then, the spectral density of the response is given by

SXX ðvÞ ¼

h 􀀄

m

􀀄

v2

n 2 v2 2 2vnziv

􀀅􀀅21 􀀄

m

􀀄

v2

n 2 v2 2 2vnziv

􀀅􀀅21

i SPP ðvÞ SPQðvÞ

SQP ðvÞ SQQðvÞ

2

4

3

5

􀀐

ðmðv2

n 2 v2 þ 2vnzivÞÞ21

ðmðv2

n 2 v2 þ 2vnzivÞÞ21

2

4

3

5 ¼

SPP ðvÞ þ SPQðvÞ þ SQP ðvÞ þ SQQðvÞ

m2

􀀑

ðv2

n 2 v2Þ2 þ ð2vnzÞ2

􀀜

In our case, the spectral density is reduced to

SXX ðvÞ ¼

SP þ SQ

m2

􀀑􀀄

v2

n 2 v2

􀀅2 þ

􀀄

2vnz

􀀅2􀀜

Spectral densities are rather complicated expressions to evaluate in general. A number of special cases

will help better understand the effects of the cross-terms:

1. PðtÞ and QðtÞ arise from independent sources and are, therefore, uncorrelated.7 Then, RPQðtÞ ¼ 0;

RQP ðtÞ ¼ 0 and SPQðvÞ ¼ 0; SQP ðvÞ ¼ 0:

2. PðtÞ and QðtÞ are directly correlated; that is, QðtÞ ¼ kPðtÞ where k is a constant.

3. PðtÞ and QðtÞ are exponentially correlated, E{PðtÞQðt þ tÞ} ¼ kPQ exp{2at} where kPQ is a

constant.

4. PðtÞ and QðtÞ are correlated in a “simplified” exponential; that is, with a triangular correlation

defined by E{PðtÞQðt þ tÞ} ¼ k􀀊PQð1 2 t=t1Þ; 2t1 # t # t1:

X(t)

P(t)

m

k

Q(t)

c

FIGURE 5.5 Single-DoF system subjected to two

random loads.

7Independence implies that

E{Pðt1 ÞQðt2 Þ} ¼ E{Pðt1 Þ}E{Qðt2 Þ}

They are uncorrelated if

CovðPðt1 ÞQðt2 ÞÞ ¼ E{Pðt1 ÞQðt2 Þ} 2 E{Pðt1 Þ}E{Qðt2 Þ} ¼ 0

Independent processes are always uncorrelated whereas uncorrelated processes may not be independent.

5-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

We consider in more detail the first two cases listed above. Where the loads are independent, because

the cross-correlations are identically zero, the output spectral density is just the sum of the two respective

spectral densities obtained with the forces acting separately:

SXX ðvÞ ¼ HXP ðvÞHp

XP ðvÞSPP ðvÞ þ HXQðvÞHp

XQðvÞSQQðvÞ

¼ lHXP ðvÞl2SPP ðvÞ þ lHXQðvÞl2SQQðvÞ ð5:10Þ

We note that the output spectral density for a linear system follows a strict interpretation of the principle

of linear superposition only when the forces are uncorrelated.

For the case where QðtÞ ¼ kPðtÞ, we have

RPQðtÞ ¼ E{PðtÞkPðt þ tÞ} ¼ kRPP ðtÞ

RQP ðtÞ ¼ E{kPðtÞPðt þ tÞ} ¼ kRPP ðtÞ

RQQðtÞ ¼ E{kPðtÞkPðt þ tÞ} ¼ k2RPP ðtÞ

Then, we can obtain the respective spectral densities

SPQðvÞ ¼ SQP ðvÞ ¼ kSPP ðvÞ

SQQðvÞ ¼ k2SPP ðvÞ

leading to the spectral density of the response

SXX ðvÞ ¼ Hp

XP ðvÞHXP ðvÞSPP ðvÞ þ Hp

XP ðvÞHXQðvÞkSPP ðvÞ þ Hp

XQðvÞHXQðvÞkSPP ðvÞ

þ Hp

XQðvÞHXQðvÞk2SPP ðvÞ ¼ ðHXP þ kHXQÞðHp

XP þ kHp

XQÞSPP ðvÞ

¼ lHXP ðvÞ þ kHXQðvÞl2SPP ðvÞ

ð5:11Þ

This last expression is related to the relative phase between the two functions HXP ðvÞ and HXQðvÞ: The

addition of two frequency response functions HXP þ kHXQ is shown graphically in Figure 5.6.

Suppose SPP ðvÞ ¼ SQQðvÞ ¼ SðvÞ Then, from Equation 5.10, for independent loadings

SXX ðvÞ ¼ ½lHXP l2 þ lHXQl2􀀉SðvÞ ð5:12Þ

If the forces are directly correlated with parameter k ¼ 1; Equation 5.11 yields

SXX ðvÞ ¼ lHXP þ HXQl2SðvÞ ¼ ½lHXP l2 þ lHXQl2 þ 2lHXP llHXQlcos f􀀉SðvÞ ð5:13Þ

where f is the phase difference between HXP and HXQ as shown in Figure 5.6. The law of cosines is

used for the second relation. Therefore, a comparison of Equation 5.12 with Equation 5.13

shows that the results of an uncorrelated loading

will be identical to those that are correlated where

cos f ¼ 0; that is, f ¼ ^p=2: This is when the two

vectors in Figure 5.6 are perpendicular to each

other. For other values of f, the spectral density in

the correlated case may have any value in the

range defined by ½lHXP l2 ^ lHXQl2􀀉SðvÞ depending

on the value of f:

If, at some frequency, HXP ¼ 2HXQ, the spectral

density at that frequency for the correlated case

with k ¼ 1 will be zero. For any case where

HXP ¼ HXQ, the spectral density with correlation

will be twice that obtained without correlation.

Another specialized result is where QðtÞ follows

PðtÞ after a lag of t0 so that QðtÞ ¼ Pðt þ t0Þ:

Imaginary

Real

HXP

kHXQ

φ

HXP+kHXQ

FIGURE 5.6 Addition of frequency-response

functions.

Random Vibration 5-11

© 2005 by Taylor & Francis Group, LLC

Then,

RPQðtÞ ¼ E{PðtÞPðt þ t0 þ tÞ} ¼ RPP ðt0 þ tÞ

with the respective spectral density

SPQðvÞ ¼

1

2p

ð1

21

RPQðtÞe2ivt dt ¼

1

2p

ð1

21

RPP ðt0 þ tÞe2ivt dt

¼

1

2p

eivt0

ð1

21

RPP ðt0 þ tÞe2ivðt0 þtÞ dðt0 þ tÞ ¼ eivt0 SPP ðvÞ

Since SPQðvÞ ¼ SQP ðvÞ are complex conjugates

SQP ðvÞ ¼ e2ivt0 SPP ðvÞ

Also, SPP ðvÞ ¼ SQQðvÞ: Then,

SXX ðvÞ ¼

􀀑

HXP ðvÞHp

XP ðvÞ þ eivt0 HXP ðvÞHp

XQðvÞ þ e2ivt0 HXQðvÞHp

XP ðvÞ þ HXQðvÞHp

XQðvÞ

􀀜

SPP ðvÞ

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