Vibrationdata 1 Unit 4 Random Vibration. Vibrationdata 2 Random Vibration Examples n Turbulent...

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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

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10

TIME (SEC)

AC

CE

L (G

)

WHITE NOISE

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

-5

-4

-3

-2

-1

0

1

2

3

4

5

2.00 2.02 2.04 2.06 2.08 2.10

TIME (SEC)

AC

CE

L (G

)

WHITE NOISE


11

Random Time History, Standard Deviation

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10

TIME (SEC)

AC

CE

L (G

)

WHITE NOISE

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 =

4

n

1i

4i

n

Y

3

n

1i

3i

n

Y

Mean =

Variance =

Standard Deviation is the square root of the variance

n

1i

2iY

n

1

n

1iiY

n

1

where Yi is each instantaneous amplitude, n is the total number of points,

is the mean, 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 > 3 even for stationary vibration


19

Probability Values for Random Signal

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%


20

More Probability

Statement Probability Ratio Percent

| x | > 0.3173 31.73%

| x | > 2 0.0455 4.55%

| x | > 3 0.0027 0.27%

Normal Distribution, Instantaneous Amplitude


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 (fn=600 Hz, Q=10).


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-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


28

Rayleigh Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

A

p(A

)

RAYLEIGH DISTRIBUTION FOR = 1


29

Rayleigh Probability Table

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.


30

Rayleigh Peak Response Formula

Tfnln2nc

nc

5772.0ncnC

nnC Maximum Peak

fn is the natural frequencyT is the durationln is the natural logarithm function

is the standard deviation of the oscillator responsen

Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations.


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 > Miscellaneous Functions > Generate Signal > white noise

Statistics vibrationdata > Signal Analysis Functions > Statistics

Find probability from Normal distribution curve vibrationdata > Miscellaneous Functions > Statistical Distributions > Normal


33

Unit 4 Exercise 2 (cont)

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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