5.2 Single Degree of Freedom: The Response to Random Loads

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Consider the second-order differential equation1 governing the linear motion of an oscillator

X€ ðtÞ þ 2zvn

X_ ðtÞ þv2

nXðtÞ ¼ FðtÞ ð5:1Þ

where the input force per unit mass is given by stationary random process FðtÞ and the output

displacement by random process XðtÞ: Note that it is customary to use upper case letters XðtÞ to represent

random processes, and lower case letters xðtÞ to represent the realizations of a random process.

The notation here is standard: 2zvn ¼ c=m where c ¼ viscous damping and m ¼ mass; and v2

n ¼ k=m

where k ¼ stiffness.

We assume that the reader understands the concepts of impulse response and convolution. We would

also like to present a priori the results which are derived in the subsequent section; the mean value of the

response mX and the spectral density of the output SXX ðvÞ:

mX ¼ Hð0ÞmF

SXX ðvÞ ¼ lHðvÞl2SFF ðvÞ

lHðvÞl2 ¼

1=v2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 2 v2=v2

nÞ2 þ ð2zv=vnÞ2

p

" #2

These equations tell us that the mean value of the response mx is proportional to the mean value of the

force mF, and that the response spectral density SXX ðvÞ is proportional to the force spectral density

SFF ðvÞ: For both results, the proportionality constants depend on the structural or system parameters.

Force

Time

FIGURE 5.1 Turbulent force history.

1In this instance, the system parameters z and vn are deterministic and, therefore, so is the governing equation of motion.

If z and vn are either random variables or processes then the governing equation is random. The case of random parameters

is much more complicated because the system itself is random rather than just the forcing. We would have to solve the

problem for the “many randomly prescribed systems” rather than for just one system with randomly prescribed forces.

5-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

By previewing the end results, the reader will hopefully be able to better follow the mathematical

manipulations which follow.

5.2.1 Formulation

Consider the linear system defined by Equation 5.1 and assume random process input FðtÞ to be

stationary, with mean mF and power spectrum SFF ðvÞ: The stationarity assumption for the forcing

means that transient dynamic behavior cannot be directly considered here.2 Thus, the initial loading

transients of an earthquake, a wind gust, or an extreme ocean wave cannot be considered as stationary.

Assuming that the character of the loading does not change, steady-state behavior can be assumed to be

statistically stationary.

5.2.2 Derivation of Equations

Begin with the convolution equation for the deterministic response of a linear oscillator

XðtÞ ¼

ð1

21

gðtÞFðt 2 tÞdt ð5:2Þ

where gðtÞ is the impulse response given by

gðtÞ ¼

1

vd

e2zvn t sin vd t

and stationary random load per unit mass FðtÞ is applied at t ¼ 21; that is, long before the present

time. This ensures that the impulse response is stationary. Beginning with Equation 5.2, take the

expected value of both sides and use the linear property of mathematical expectation to interchange it

with the integral:

E{XðtÞ} ¼

ð1

21

gðtÞE{Fðt 2 tÞ}dt ¼ E{FðtÞ}

ð1

21

gðtÞdt

¼ mF

ð1

21

gðtÞdt ¼ mF Hð0Þ

where the fact that FðtÞ is stationary was utilized in the second and third equations,3 and HðvÞ is the

frequency response function; Hð0Þ ¼ HðvÞlv¼0: Therefore, we arrive at the first important result

mX ¼ Hð0ÞmF ¼

1

v2

n

mF;

which shows us that since mF is time independent, then so must be E{xðtÞ}:

In order to derive the output spectral density, we must work through the intermediate results involving

the correlation function.

5.2.3 Response Correlations

For a stationary process, the autocorrelation function is given by

RXX ðtÞ ¼

ð1

21

xðtÞxðt þ tÞfX ðxÞdx

where fX ðxÞ is the probability density function of the process. We cannot take the Fourier transform

of this equation to find the response spectral density because we do not know the response density

2There are, however, clever ways by which stationary solutions can be utilized in nonstationary cases. One possibility is to

multiply the stationary process by a deterministic time function such that the product is an evolutionary, or nonstationary,

process. For example, use AðtÞFðtÞ as the forcing function where AðtÞ is a deterministic transient function and FðtÞ is

stationary.

3The force is stationary and has a constant mean value.

Random Vibration 5-3

© 2005 by Taylor & Francis Group, LLC

function fX ðxÞ: There are two other equivalent ways to proceed: (i) utilize the ergodic definition of the

autocorrelation,4 or (ii) use Equation 5.2, utilizing available information on FðtÞ; as follows.

First, derive the cross-correlation between FðtÞ and XðtÞ: Multiply both sides of Equation 5.2 by

Fðt 2 a1Þ and take the expected values of both sides:

E{XðtÞFðt 2 a1Þ} ¼

ð1

21

gðt1ÞE{Fðt 2 t1ÞFðt 2 a1Þ}dt1

where E{Fðt 2 t1ÞFðt 2 a1Þ} ¼ RFF ðt1 2 a1Þ and E{XðtÞFðt 2 a1Þ} ¼ RXF ða1Þ; the cross-correlation

between loading FðTÞ and response XðtÞ: Thus,

RXF ða1Þ ¼

ð1

21

gðt1ÞRFF ðt1 2 a1Þdt1 ð5:3Þ

and RFF is known from experimental data. Next, multiply both sides of Equation 5.2 by Xðt þ a2Þ and

take expected values of both sides:

E{Xðt þ a2ÞXðtÞ} ¼

ð1

21

gðt2ÞE{Xðt þ a2ÞFðt 2 t2Þ}dt2

RXX ða2Þ ¼

ð1

21

gðt2ÞRXF ðt2 þ a2Þdt2 ð5:4Þ

Substitute Equation 5.3 into Equation 5.4 to find

RXX ðtÞ ¼

ð1

21

ð1

21

gðaÞgðbÞRFF ðt þ b 2 aÞda db ð5:5Þ

which is a double convolution. To evaluate the variance, we use

s2

X ¼ E{XðtÞ2} 2 E2{XðtÞ} ¼ RXX ð0Þ 2 ½Hð0ÞE{FðtÞ}􀀉2

While this information on the correlation is of interest, a more important result is the response spectral

density that we derive in the next section.

5.2.4 Response Spectral Density

Begin with the Fourier transform relation between power spectrum and correlation function

SXX ðvÞ ¼

1

2p

ð1

21

RXX ðtÞe2ivt dt;

and substitute Equation 5.5 for RXX ðtÞ; with l ¼ t þ b 2 a :

SXX ðvÞ ¼

1

2p

ð1

21

e2ivt

ð1

21

ð1

21

gðaÞgðbÞRFF ðlÞda db

􀀒 􀀓

dt

¼

ð1

21

gðaÞe2iva da

ð1

21

gðbÞeþivb db £

1

2p

ð1

21

RFF ðlÞe2ivl dl

¼ HðvÞHpðvÞSFF ðvÞ

by definition, where p denotes complex conjugate and therefore

SXX ðvÞ ¼ lHðvÞl2SFF ðvÞ ð5:6Þ

This is the fundamental result for random vibration and linear systems theory that allows us to

evaluate the output spectral density, given the input spectral density and the system frequency response.

4

Rxx ðtÞ ¼ limT!1

1

2T

ðþT

2T

xðtÞxðt þ tÞdt:

5-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

It is emphasized here that the derivation of Equation 5.6 made use of the convolution equation that is

valid for linear systems and structures. Any generalization for nonlinear behavior requires problemspecific

approaches.

Example 5.1 Oscillator Response to White Noise

Consider a simple application of the above ideas to an oscillator. What is the response of a damped

oscillator to a force with white noise density?

Solution

The governing equation of motion is

X€ ðtÞ þ 2zvn

X_ ðtÞ þv2

nXðtÞ ¼ FðtÞ

where FðtÞ is the external force per unit mass, the system transfer function is given by

HðvÞ ¼

1

v2

n þ i2zvnv þ ðivÞ2

and the squared magnitude of HðvÞ is given by

lHðvÞl2 ¼

1

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2

Therefore, given any input spectral density SFF ðvÞ; the response spectral density is

SXX ðvÞ ¼ lHðvÞl2SFF ðvÞ ¼

SFF ðvÞ

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2

Suppose, for mathematical simplicity, that the forcing is white noise, SFF ðvÞ ¼ S0: Then,

SXX ðvÞ ¼

S0

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2 ð5:7Þ

and the mean-square response is given by

E{XðtÞ2} ¼

ð1

21

SXX ðvÞdv ¼

pm2S0

kc ¼

pS0

2v3

nz

The mean-square response can also be written in terms of a one-sided spectrum

E{XðtÞ2} ¼

pm2S0

kc ¼

m2W0

4kc

where the one-sided density W0 is related to the two-sided density by S0 ¼ pW0=4:

This integral is not standard, but can be found in texts on random vibration.5 Even though infinite

mean-square energy is input to the system,6 it responds with finite mean-square energy. See Figure 5.2 for

plots of the components of Equation 5.7. Only the positive frequency ranges are plotted as they are

5For example, the integral of this example problem is a specialized version of

ð1

21

B0 þ ivB1

A0 þ ivA1 2 v2 A2

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

2

dv ¼

pðA0 B21

þ A2 B20

Þ

A0 A1 A2

where

A0 ¼ v2n

; A1 ¼ 2zvn ; A2 ¼ 2; B0 ¼ 1; B1 ¼ 0:

6The energy input equals to the area under the spectral density, which for white noise is

ð1

21

S0 dv ¼ 1:

Random Vibration 5-5

© 2005 by Taylor & Francis Group, LLC

symmetric about the abscissae. White noise is

useful and frequently used, even though it is

nonphysical, because it leads to good approximate

results.

Example 5.2 Response to Colored

Noise

Suppose the same system as in the last

example is subjected to more complex loading,

where the spectral density of the forcing is not a

constant, but a function of v: How would the above

analysis change?

Solution

The output spectral density becomes a more

complex function of frequency, for example, if

the loading density is similar to those found for

wind loads. Then, the mean-square response must

be evaluated numerically.

The applied problems are always solved numerically,

although hopefully after some significant

analytical exposition. There are numerical

methods that are specifically geared to handling

uncertainties. Of particular note is the group

known as Monte Carlo methods. These methods

utilize the massive computational power available

today to account for uncertainties. This is

accomplished by generating random numbers that are used to represent random parameters. For each

of these generated random values, the program recalculates the problem. After running enough cycles to

ensure the convergence of the statistics, these numerical realizations are averaged to find the mean value

and variance of the relevant parameters.

We summarize the key results for the random vibration of a single-DoF linear oscillator in Table 5.1 to

Table 5.3. Figure 5.3 shows some of the most important correlation and spectral density pairs.

ω

SFF

So

Input Spectrum

ω

|H(iω)|2

System function

ωn

ωn

ω

Response spectrum

SXX

FIGURE 5.2 SFF ðvÞ; lHðivÞl2 ; SXX ðvÞ:

TABLE 5.1 Mean-Value Response

mX ¼ Hð0ÞmF

lHðvÞl2 ¼ 1=v2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 2 v2 =v2n

Þ2 þ ð2zv=vn Þ2

q

2

64

3

75

2

mF is known from force data

TABLE 5.2 Output Correlation Function/Variance

RXX ðtÞ ¼

Ð

12

1

Ð

12

1 gðaÞgðbÞRFF ðt þ b 2 aÞda db

s2

X ¼ RXX ð0Þ 2 ½Hð0ÞE{FðtÞ}􀀉2

RFF ðt þ b 2 aÞ is known from force data

5-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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