Transcript SDOF Response to White Noise

Unit 4
Vibrationdata
Random Vibration
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Random Vibration Examples

Turbulent airflow passing over an aircraft wing

Oncoming turbulent wind against a building

Rocket vehicle liftoff acoustics

Earthquake excitation of a building
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Random Vibration Characteristics
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One common characteristic of these examples is that the motion varies randomly with
time. Thus, the amplitude cannot be expressed in terms of a "deterministic"
mathematical function.
Dave Steinberg wrote:
The most obvious characteristic of random vibration is that it is
nonperiodic. A knowledge of the past history of random motion is
adequate to predict the probability of occurrence of various acceleration
and displacement magnitudes, but it is not sufficient to predict the precise
magnitude at a specific instant.
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Optics Analogy
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
Sinusoidal vibration is like a laser beam

Random vibration is like “white light”

White light passed through a prism produces a
spectrum of colors
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Music Analogy
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

Playing a single piano key produces sinusoidal
vibration (fundamental + harmonics)
Playing all 88 piano keys at once produces a
signal which approximates random vibration
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Types of Random Vibration

Random vibration can be broadband or narrow band

Random vibration can be stationary or nonstationary


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Stationary random vibration is where the key statistical parameters remain
constant with each consecutive time segment
Parameters include: mean, standard deviation, histogram, power spectral
density, etc.

Shaker table tests can be controlled to be stationary for the test duration

Measured data is usually nonstationary

White noise and pink noise are two special cases of random vibration
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White Noise
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


Commercial white noise
generator designed to produce
soothing random noise which
masks household noise as a
sleep aid.

White noise and pink noise are two special cases
of random vibration
White noise is a random signal which has a
constant power spectrum for a constant
frequency bandwidth
It is thus analogous to white light, which is
composed of a continuous spectrum of colors
Static noise over a non-operating TV or radio
station channel tends to be white noise
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Pink Noise
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


Waterfalls and oceans waves
may generate pink noise

Pink noise is a random signal which has a constant
power spectrum for each octave band
This noise is called pink because the low
frequency or “red” end of the spectrum is
emphasized
Pink noise is used in acoustics to measure the
frequency response of an audio system in a
particular room
It can thus be used to calibrate an analog graphic
equalizer
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Vibrationdata
Sample Random Time History, Synthesized
WHITE NOISE
5
4
mean =0
3
std dev =1
ACCEL (G)
2
Sample rate =
20K samples/sec
1
0
Band-limited to 2 KHz via
lowpass filtering
-1
-2
Stationary
-3
-4
-5
0
2
4
6
8
10
TIME (SEC)
Synthesize time history with Matlab GUI script: vibrationdata.m
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Sample Random Time History, Close-up View
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WHITE NOISE
5
4
3
ACCEL (G)
2
1
0
-1
-2
-3
-4
-5
2.00
2.02
2.04
2.06
2.08
2.10
TIME (SEC)
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Vibrationdata
Random Time History, Standard Deviation
WHITE NOISE
5
Peak Absolute = 4.5 G
4
3
Std dev = 1 G
ACCEL (G)
2
1
Crest Factor
0
-1
= (Peak Absolute / Std dev)
-2
= (4.5 G/ 1 G)
-3
= 4.5
-4
-5
0
2
4
6
8
10
TIME (SEC)
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Histogram Comparison
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Sine Vibration has bathtub shaped histogram
 Sine vibration tends to linger at its extreme values
Random Vibration has a bell-shaped curve histogram
 Random vibration tends to dwell near zero
Thus, there is no real way to directly compare sine and random vibration.
But we can “sort of” make this comparison indirectly by taking a rainflow cycle count of
the response of a system to each time.
Rainflow fatigue will be covered in future units.
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Random Time History, Histogram
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Histogram of white noise
instantaneous amplitudes
has a normal distribution.
The amplitude is expressed
in bins with unit of G.
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Statistics of Sample Time History
Parameter
Value
Duration
10 sec
Sample Rate
20K sps
Samples
200K
Mean
0
Std Dev
1
RMS
1
Skewness
0
Kurtosis
3.0
Maximum
4.3
Minimum
-4.5
Vibrationdata
Consider limits: -4.49 to 4.49
Normal distribution
Probability within limits
0.99999288
Probability of exceeding limits
7.1223174e-06
7.1223174e-06 * 200000 points = 1.4
Rounding to nearest integer . . .
One point was expected to exceed 4.5 in terms of
absolute value.
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RMS and Standard Deviation
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 = standard deviation
RMS = root-mean-square
[ RMS ] 2 = [  ] 2 + [ mean ]2
RMS =  assuming zero mean
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Peak and RMS values
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Pure sine vibration has a peak value that is 2 times its RMS
value
Random vibration has no fixed ratio between its peak and RMS
values
Again, the ratio between the absolute peak and RMS values in
the previous example is
4.5 G / 1 G = 4.5
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Vibrationdata
Statistical Formulas

Mean =
n
n
1
 Yi
n

Skewness =
i 1

Variance =
1 n
Yi  2

n
i 1
 Yi  3
i 1
n 3
n

Kurtosis =
 Yi  4
i 1
n 4

Standard Deviation is the square root
of the variance
where Yi is each instantaneous amplitude, n is the total number of points,
 is the mean,  is the standard deviation
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Statistics of Sample Time History
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
Random vibration is often considered to have a 3 peak for design purposes

Need to differentiate between input and response levels
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Response is more important for design purposes, fatigue analysis, etc.

Both input and response can have peaks > 3 even for stationary vibration
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Probability Values for Random Signal
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Normal Distribution, Instantaneous Amplitude
Statement
Probability Ratio
Percent
- < x < +
0.6827
68.27%
-2 < x < +2
0.9545
95.45%
-3 < x < +3
0.9973
99.73%
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More Probability
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Normal Distribution, Instantaneous Amplitude
Statement
Probability Ratio
Percent
|x|>
0.3173
31.73%
| x | > 2
0.0455
4.55%
| x | > 3
0.0027
0.27%
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SDOF Response to White Noise
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The equation of motion was previously derived in Webinar 2.
Apply the white noise base input from the previous example as a base
input to an SDOF system (fn=600 Hz, Q=10).
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Solving the Equation of Motion
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A convolution integral is used for the case where the base input acceleration is
arbitrary.
The convolution integral is numerically inefficient to solve in its equivalent digitalseries form.
Instead, use…
Smallwood, ramp invariant, digital recursive filtering relationship!
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SDOF Response
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mean =0
std dev =2.16 G
Peak Absolute = 9.18 G
Crest Factor
= 9.18 G / 2.16 G
= 4.25
The theoretical Crest Factor from
the Rayleigh Distribution is 4.31
Rice Characteristic Frequency
= 595 Hz
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SDOF Response, Close-up View
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SDOF system tends to vibrate at its natural frequency. 60 peaks / 0.1 sec = 600 Hz.
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Histogram of SDOF Response
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The response time history is
narrowband random.
The histogram has a normal
distribution.
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Histogram of SDOF Response Peaks
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The histogram of the absolute
response peaks has a Rayleigh
distribution.
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Rayleigh Distribution
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Consider a lightly damped, single-degree-of-freedom system subjected to
broadband random excitation
The system will tend to behave as a bandpass filter
The bandpass filter center frequency will occur at or near the system’s natural
frequency.
The system response will thus tend to be narrowband random. The
probability distribution for its instantaneous values will tend to follow a
Normal distribution, which the same distribution corresponding to a
broadband random signal
The absolute values of the system’s response peaks, however, will have a
Rayleigh distribution
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Rayleigh Distribution
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RAYLEIGH DISTRIBUTION FOR  = 1
0.7
0.6
0.5
p(A)
0.4
0.3
0.2
0.1
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
A
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Rayleigh Probability Table
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Rayleigh Distribution Probability

Prob [ A >  ]
0.5
88.25 %
1.0
60.65 %
1.5
32.47 %
2.0
13.53 %
2.5
4.39 %
3.0
1.11 %
3.5
0.22 %
4.0
0.034 %
Thus, 1.11 % of the peaks will be above 3 sigma for a signal whose
peaks follow the Rayleigh distribution.
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Rayleigh Peak Response Formula
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Consider a single-degree-of-freedom system with the index n.
The maximum response can be estimated by the following equations.
cn 
2 ln fn T 
Cn  cn 
Maximum Peak
fn
T
ln
n
0.5772
cn
 Cn n
is the natural frequency
is the duration
is the natural logarithm function
is the standard deviation of the oscillator response
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Unit 4 Exercise 1
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Consider an avionics component. It is powered and monitored during a bench
test. It passes this "functional test."
Nevertheless, it may have some latent defects such as bad solder joints or bad
parts. A decision is made to subject the component to a base excitation test on a
shaker table to check for these defects. Which would be a more effective test:
sine sweep or random vibration? Why?
Reference: NAVMAT P9492, Section 3.1
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Unit 4 Exercise 2
Vibrationdata
Repeat the pervious examples on your own. Use the vibrationdata.m GUI script.
Generate white noise
vibrationdata > Miscellaneous Functions > Generate Signal > white noise
Statistics
vibrationdata > Signal Analysis Functions > Statistics
Find probability from Normal distribution curve
vibrationdata > Miscellaneous Functions > Statistical Distributions > Normal
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Unit 4 Exercise 2 (cont)
Vibrationdata
SDOF Response
vibrationdata > Signal Analysis Functions > SDOF Response to Base Input
Estimated Peak Response from Rayleigh distribution
vibrationdata > Miscellaneous Functions > SDOF Response: Peak Sigma
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