mechanical vibration 13
TRANSCRIPT
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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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Fourier Response Technique
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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Fourier Response Technique
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