mechanical vibration 13

8/13/2019 Mechanical Vibration 13

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1 Dr. Peter AvitabileModal Analysis & Controls Laboratory

22.457 Mechanical Vibrations - Random Vibrations

Mechanical Vibrations

Chapter 13

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell


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Random Vibrations

Up until now, everything analyzed wasdeterministic. Other loading conditions exist that

are not deterministic - random vibrations.The function below is irregular but may have somestatistical character.


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Random Vibrations

Each record is called a sample - the total set iscalled an ensemble.

If the function is evaluated at (t) and (t + )

and the averaged the function shows no differencethen the signal is stationary.


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Time Average - Expected Values

The expected value can be obtained if timeaveraging is performed over a long time record

Mean value

Variance

(13.2.1)

(13.2.4)

(13.2.5)

=== 0 dt)t(xT1

T

lim)t(x)t(x

___)]t(x[E

== 02

22 dt)t(x

T

1

T

lim

x

__)]t(x[E

( ) 22022 )x(

x

__dtxx

T

1

T

lim=

=


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Frequency Response Function

The linear input-output relationship also holds truefor random signals.

In the time domain, the response can bedetermined in terms of the impulse responsefunction using the convolution (Duhamel) integral as

(13.3.1) =t

0

d)t(h)(f)t(x


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Frequency Response Function

For the input-output problem, a simplier approachutilizes a frequency domain & frequency response

function under stationary or steady state conditionFor a sinusoidal excitation, the SDOF response is

or

Mean Squared Response is

(13.3.4)

tj

2

Fe

jcmk

1)t(x

+

=

tjFe)(H)t(x =

222F

__

)(Hx

__

=


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Probability Distribution

Many times it is desirable to know the probabilityof a certain value of a time signal.

The probabilitydensity function is

and thevariance

Gaussian andRayleigh

distributionswidely used

(13.4.2)

x

)x(P)xx(P

0x

lim)x(p

+

=

+

== 2222

)x(x

__

dx)x(p)xx( (13.4.7)


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Time Correlation Functions

Correlation is the measure of similatiry betweentwo signals. By time shifting one time signal

relative to another time signal, a correlationfunction can be obtained.


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Auto- and Cross-Correlation Function

Auto-Correlation

Cross-Correlation

(13.5.1)

(13.5.3)

)t(x)t(x)]t(x)t(x[E)(Rxx +=+=

+= 2/T

2/Txx dt)t(x)t(x

T

1

T

lim)(R

)t(y)t(x)]t(y)t(x[E)(Rxy +=+=

+= 2/T

2/Txy dt)t(y)t(xT

1

T

lim

)(R


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Fourier Transforms

Fourier Integral is used for the transformation

Fourier Transform Pair

or using

Fourier Transform Pair

(13.7.1)

Note: Thompson uses X(f) as a linear spectrum and S(f) as a power spectrumThese notes use S(f) as a linear spectrum and G(f) as a power spectrum

+

= dfe)f(S)t(x ft2j

+

= dte)t(x)f(S ft2j (13.7.2)

+

= de)(S

2

1)t(x tj

+

= dte)t(x)(S tj

(13.7.3)

(13.7.4)


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Fourier Transforms

Differentiation is simply

which is just multiplication by j +

= de)(Sj21

)t(x tj

&

[ ] [ ])t(xFTj)t(xFT =&

[ ] [ ])t(xFT)t(xFT 2=&&


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Fourier Transforms

Thus transforming the differential equation

or as more commonly written

Note the simple multiplication rather than the

convolution integral in the time domain

)t(fkxxcxm =++ &&&

)(F)(X)kjcm( 2 =++

)(F)(X)(H 1 =

)(F)(H)(X =


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Parsevals Theorum

Useful for converting time domain integration intofrequency domain integration

df)f(S)f(Sdt)t(x)t(x 2*

121 ++

=

df)f(S)f(S 21*

+

=


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+

+

= dt)t(x)t(x

T

1

T

lim)(R

xx

The auto-correlation function is

Applying Parseval, rearranging terms, simplifying

and the inverse

Auto-Correlation Function

These are the Wiener-Khintchine Equations

(13.7.9)

(13.7.10)

+

= dfe)f(S)f(S)(R ft2j*

xxxx

)f(S)f(S)f(G *

xxxx =

+ = de)(R)f(G ft2j

xxxx


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+

=

2/T

2/Txy

dt)t(y)t(xT

1

T

lim)(R

+ = de)(R)f(G ft2j

xyxy

)f(S)f(S)f(G *

yxxy =

+

= dfe)f(S)f(S)(R ft2j*

yxxy

Cross-Correlation Function

The cross-correlation function is

Applying Parseval, rearranging terms, simplifying

and the inverse

These are the Wiener-Khintchine Equations

(13.7.12)

(13.7.13)


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The Frequency Response Function (FRF) is theinput-output relation

Multiplying and dividing by the conjugate of theinput force spectrum yields

Fourier Response Technique

(13.8.1)

(13.8.2)

)]t(f[FT

)]t(x[FT

)(F

)(X)(H =

=

)(G

)(G

)(S)(S

)(S)(S)(H

ff

xf

f*

f

f*

x

=

=

)(S)(S)(H)(S)(S f*

ff*

x = )(G)(H)(G ffxf =


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Fourier Response Technique Schematic

INPUT TIME FORCE

INPUT SPECTRUM

OUTPUT TIME RESPONSE

OUTPUT SPECTRUM

f(t)

FFT

y(t)

IFT

f(j ) y(j )h(j )

FREQUENCY RESPONSE FUNCTION


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Using the frequency domain input-output relationships,

the response due to many forces can be computed

The frequency response function is needed for this response

Output Response = System Characteristic X Input Forces

==

oN

1j jiji

)j(f)j(h)j(y

=

+

=

m

1k*k

*k,ij

k

k,ijij

j

r

j

r)j(h


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Frequency domain input-output schematic

INPUT SPECTRUM

OUTPUT SPECTRUM

f(j )

y(j )

FREQUENCY RESPONSE FUNCTION

=

=oN

1j

jiji )j(f)j(h)j(y

=

+

=m

1k*k

* k,ij

k

k,ijij

pj

r

pj

r)j(h

mechanical vibration 13

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