1. Vibrationdata 1 Unit 4 Random Vibration

2. Vibrationdata 2 Random Vibration Examples n Turbulent airflow passing over an aircraft wing n Oncoming turbulent wind against a building n Rocket vehicle liftoff acoustics n Earthquake excitation of a building

3. Vibrationdata 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. Vibrationdata 4 Optics Analogy n Sinusoidal vibration is like a laser beam n Random vibration is like “white light” n White light passed through a prism produces a spectrum of colors

5. Vibrationdata 5 Music Analogy n Playing a single piano key produces sinusoidal vibration (fundamental + harmonics) n Playing all 88 piano keys at once produces a signal which approximates random vibration

6. Vibrationdata 6 Types of Random Vibration n Random vibration can be broadband or narrow band n Random vibration can be stationary or nonstationary n Stationary random vibration is where the key statistical parameters remain constant with each consecutive time segment n Parameters include: mean, standard deviation, histogram, power spectral density, etc. n Shaker table tests can be controlled to be stationary for the test duration n Measured data is usually nonstationary n White noise and pink noise are two special cases of random vibration

7. Vibrationdata 7 White Noise n White noise and pink noise are two special cases of random vibration n White noise is a random signal which has a constant power spectrum for a constant frequency bandwidth n It is thus analogous to white light, which is composed of a continuous spectrum of colors n 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. Vibrationdata 8 Pink Noise n Pink noise is a random signal which has a constant power spectrum for each octave band n This noise is called pink because the low frequency or “red” end of the spectrum is emphasized n Pink noise is used in acoustics to measure the frequency response of an audio system in a particular room n It can thus be used to calibrate an analog graphic equalizer Waterfalls and oceans waves may generate pink noise

9. Vibrationdata 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. Vibrationdata 10 Sample Random Time History, Close-up View

11. Vibrationdata 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. Vibrationdata 12 Histogram Comparison Sine Vibration has bathtub shaped histogram u Sine vibration tends to linger at its extreme values Random Vibration has a bell-shaped curve histogram u 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. Vibrationdata 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. Vibrationdata 14 Statistics of Sample Time History ParameterValue Duration10 sec Sample Rate20K sps Samples200K Mean0 Std Dev1 RMS1 Skewness0 Kurtosis3.0 Maximum4.3 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. Vibrationdata 15 RMS and Standard Deviation  = standard deviation RMS = root-mean-square [ RMS ] 2 = [  ] 2 + [ mean ] 2 RMS =  assuming zero mean

16. Vibrationdata 16 Peak and RMS values n Pure sine vibration has a peak value that is  2 times its RMS value n Random vibration has no fixed ratio between its peak and RMS values n Again, the ratio between the absolute peak and RMS values in the previous example is 4.5 G / 1 G = 4.5

17. Vibrationdata 17 Statistical Formulas n Skewness = n Kurtosis = n Mean = n Variance = n Standard Deviation is the square root of the variance where Y i is each instantaneous amplitude, n is the total number of points,  is the mean,  is the standard deviation

18. Vibrationdata 18 Statistics of Sample Time History n Random vibration is often considered to have a 3  peak for design purposes n Need to differentiate between input and response levels n Response is more important for design purposes, fatigue analysis, etc. n Both input and response can have peaks > 3  even for stationary vibration

19. Vibrationdata 19 Probability Values for Random Signal Normal Distribution, Instantaneous Amplitude StatementProbability RatioPercent -  < x < +  0.682768.27% -2  < x < +2  0.954595.45% -3  < x < +3  0.997399.73%

20. Vibrationdata 20 More Probability StatementProbability RatioPercent | x | >  0.317331.73% | x | > 2  0.04554.55% | x | > 3  0.00270.27% Normal Distribution, Instantaneous Amplitude

21. Vibrationdata 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. Vibrationdata 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. Vibrationdata 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. Vibrationdata 24 SDOF Response, Close-up View SDOF system tends to vibrate at its natural frequency. 60 peaks / 0.1 sec = 600 Hz.

25. Vibrationdata 25 Histogram of SDOF Response The response time history is narrowband random. The histogram has a normal distribution.

26. Vibrationdata 26 Histogram of SDOF Response Peaks The histogram of the absolute response peaks has a Rayleigh distribution.

27. Vibrationdata 27 Rayleigh Distribution n Consider a lightly damped, single-degree-of-freedom system subjected to broadband random excitation n The system will tend to behave as a bandpass filter n The bandpass filter center frequency will occur at or near the system’s natural frequency. n 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 n The absolute values of the system’s response peaks, however, will have a Rayleigh distribution

28. Vibrationdata 28 Rayleigh Distribution

29. Vibrationdata 29 Rayleigh Probability Table Rayleigh Distribution Probability Prob [ A >  ] 0.588.25 % 1.060.65 % 1.532.47 % 2.013.53 % 2.54.39 % 3.01.11 % 3.50.22 % 4.00.034 % Thus, 1.11 % of the peaks will be above 3 sigma for a signal whose peaks follow the Rayleigh distribution.

30. Vibrationdata 30 Rayleigh Peak Response Formula 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 Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations.

31. Vibrationdata 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. Vibrationdata 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. Vibrationdata 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