Random Vibration

1
Unit 4

• Random Vibration

2
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

3
Random Vibration Characteristics

• 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.

4
Optics Analogy

• Sinusoidal vibration is like a laser beam
• Random vibration is like white light
• White light passed through a prism produces a
  spectrum of colors

5
Music Analogy

• Playing a single piano key produces sinusoidal
  vibration (fundamental harmonics)
• Playing all 88 piano keys at once produces a
  signal which approximates random vibration

6
Types of Random Vibration

• Random vibration can be broadband or narrow band
• Random vibration can be stationary or
  nonstationary
• 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

7
White Noise

• 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

Commercial white noise generator designed to
produce soothing random noise which masks
household noise as a sleep aid.
8
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

Waterfalls and oceans waves may generate pink
noise
9
Sample Random Time History, Synthesized
mean 0 std dev 1 Sample rate 20K
samples/sec Band-limited to 2 KHz via lowpass
filtering Stationary
Synthesize time history with Matlab GUI script
vibrationdata.m
10
Sample Random Time History, Close-up View
11
Random Time History, Standard Deviation
Peak Absolute 4.5 G Std dev 1 G Crest
Factor (Peak Absolute / Std dev)
(4.5 G/ 1 G) 4.5
12
Histogram Comparison

• 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 history.
• Rainflow fatigue will be covered in future units.

13
Random Time History, Histogram
Histogram of white noise instantaneous amplitudes
has a normal distribution. The amplitude is
expressed in bins with unit of G.
14
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
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.
15
RMS and Standard Deviation

• ? standard deviation
• RMS root-mean-square
• RMS 2 ? 2 mean 2
• RMS ? assuming zero mean

16
Peak and RMS values

• 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

17
Statistical Formulas

• Skewness
• Kurtosis

• Mean
• Variance
• Standard Deviation is the square root of the
  variance

where Yi is each instantaneous amplitude, n is
the total number of points, m is
the mean, s is the standard deviation
18
Statistics of Sample Time History

• Random vibration is often considered to have a 3?
  peak for design purposes
• Need to differentiate between input and response
  levels
• Response is more important for design purposes,
  fatigue analysis, etc.
• Both input and response can have peaks gt 3?
  even for stationary vibration

19
Probability Values for Random Signal
Normal Distribution, Instantaneous Amplitude
Statement Probability Ratio Percent
-? lt x lt ? 0.6827 68.27
-2? lt x lt 2? 0.9545 95.45
-3? lt x lt 3? 0.9973 99.73
20
More Probability
Normal Distribution, Instantaneous Amplitude
Statement Probability Ratio Percent
x gt ? 0.3173 31.73
x gt 2? 0.0455 4.55
x gt 3? 0.0027 0.27
21
SDOF Response to White Noise
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 (fn600 Hz, Q10).
22
Solving the Equation of Motion
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 digital-series
form. Instead, use Smallwood, ramp invariant,
digital recursive filtering relationship!
23
SDOF Response
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
24
SDOF Response, Close-up View
SDOF system tends to vibrate at its natural
frequency. 60 peaks / 0.1 sec 600 Hz.
25
Histogram of SDOF Response
The response time history is narrowband
random. The histogram has a normal distribution.
26
Histogram of SDOF Response Peaks
The histogram of the absolute response peaks has
a Rayleigh distribution.
27
Rayleigh Distribution

• Consider a lightly damped, single-degree-of-freedo
  m 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 systems 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 systems response
  peaks, however, will have a Rayleigh distribution

28
Rayleigh Distribution
29
Rayleigh Probability Table
Rayleigh Distribution Probability Rayleigh Distribution Probability
? Prob A gt ??
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.
30
Rayleigh Peak Response Formula
Consider a single-degree-of-freedom system with
the index n. The maximum response can be
estimated by the following equations.
Maximum Peak
fn is the natural frequency
T is the duration
ln is the natural logarithm function
is the standard deviation of the oscillator response
31
Unit 4 Exercise 1

• 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

32
Unit 4 Exercise 2

• Repeat the pervious examples on your own. Use
  the vibrationdata.m GUI script.
• Generate white noise
• vibrationdata gt Miscellaneous Functions gt
  Generate Signal gt white noise
• Statistics
• vibrationdata gt Signal Analysis Functions
  gt Statistics
• Find probability from Normal distribution curve
• vibrationdata gt Miscellaneous Functions gt
  Statistical Distributions gt Normal

33
Unit 4 Exercise 2 (cont)

• SDOF Response
• vibrationdata gt Signal Analysis Functions
  gt SDOF Response to Base Input
• Estimated Peak Response from Rayleigh
  distribution
• vibrationdata gt Miscellaneous Functions gt
  SDOF Response Peak Sigma