webinar_random_vibration

Unit 4

Vibrationdata

Random Vibration

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Random Vibration Examples

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Turbulent airflow passing over an aircraft wing

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Oncoming turbulent wind against a building

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Rocket vehicle liftoff acoustics

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Earthquake excitation of a building

Vibrationdata

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Random Vibration Characteristics

Vibrationdata

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

Vibrationdata 

Sinusoidal vibration is like a laser beam

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Random vibration is like “white light”

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White light passed through a prism produces a spectrum of colors

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Music Analogy

Vibrationdata 

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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

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Random vibration can be stationary or nonstationary

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Vibrationdata

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.

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Shaker table tests can be controlled to be stationary for the test duration

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Measured data is usually nonstationary

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White noise and pink noise are two special cases of random vibration

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White Noise

Vibrationdata 

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

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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

Vibrationdata 

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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 9

Sample Random Time History, Close-up View

Vibrationdata

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

Vibrationdata

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.

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Random Time History, Histogram

Vibrationdata

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. 14

RMS and Standard Deviation

Vibrationdata

 = 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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Vibrationdata

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

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Mean =

1 n

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Variance =

n

n

 Yi

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 Y i   

Skewness =

i 1

i 1

1

n

3

n

Yi     n

2

i 1

n



Kurtosis =

 Y i

 

4

i 1

n 

3

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 17

Statistics of Sample Time History

Vibrationdata

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Random vibration is often considered to have a 3 peak for design purposes

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Need to differentiate between input and response levels

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Response is more important for design purposes, fatigue analysis, etc.

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Both input and response can have peaks > 3 even for stationary vibration

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Probability Values for Random Signal

Vibrationdata

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

Vibrationdata

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

Vibrationdata

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). 21

Solving the Equation of Motion

Vibrationdata

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

Vibrationdata 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 23

SDOF Response, Close-up View

Vibrationdata

SDOF system tends to vibrate at its natural frequency. 60 peaks / 0.1 sec = 600 Hz. 24

Histogram of SDOF Response

Vibrationdata The response time history is narrowband random. The histogram has a normal distribution.

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Histogram of SDOF Response Peaks

Vibrationdata The histogram of the absolute response peaks has a Rayleigh distribution.

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Rayleigh Distribution 

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Vibrationdata

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

Vibrationdata R A Y L E IG H D IS T R IB U T IO N F O R  = 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

Vibrationdata

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. 29

Rayleigh Peak Response Formula

Vibrationdata

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 30

Unit 4 Exercise 1

Vibrationdata

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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