Unit 4

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 Python GUI script: vibrationdata.py

10. Sample Random Time History, Close-up View

11. Random Time History, Standard Deviation Peak Absolute = 4.54 Std dev = 1 Crest Factor = (Peak Absolute / Std dev) = (4.54 / G) = 4.54 Kurtosis = 3.0

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.

14. Statistics of Sample Time History Consider limits: -4.53 to 4.53 Normal distribution Probability within limits 0.999994102 Probability of exceeding limits 5.89836865e-06 5.89836865e-06 * 200000 points = 1.2 Rounding to nearest integer . . . One point was expected to exceed 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 4.54 / 1 = 4.54

17. Statistical Formulas • Skewness = • Kurtosis = • Mean = • Variance = • Standard Deviation is the square root of the variance where Yiis 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 > 3 even for stationary vibration

19. Probability Values for Random Signal Normal Distribution, Instantaneous Amplitude

20. More Probability 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=900 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 = 2.62 G Peak Absolute = 11.6 G Crest Factor = 11.6 G / 2.62 G = 4.42 The theoretical Crest Factor from the Rayleigh Distribution is 4.40 Rice Characteristic Frequency = 886 Hz

24. SDOF Response, Close-up View SDOF system tends to vibrate at its natural frequency. 45 peaks / 0.05 sec = 900 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

29. Rayleigh Probability Table 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

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.py GUI script. Generate white noise vibrationdata > Miscellaneous Analysis > Generate Signal > white noise Statistics vibrationdata > Signal Analysis > Statistics Find probability from Normal distribution curve vibrationdata > Miscellaneous Analysis > Statistical Distributions > Normal

33. Unit 4 Exercise 2 (cont) SDOF Response vibrationdata > Signal Analysis > SDOF Response to Base Input Estimated Peak Response from Rayleigh distribution vibrationdata > Miscellaneous > SDOF Response: Peak Sigma