1. Vibrationdata 1 Unit 22 Integration and Differentiation of Time Histories & Spectral Functions

2. Vibrationdata 2 PSD Types Acceleration Velocity Displacement Force Pressure or Stress Strain PSDs can be calculated for Acceleration PSDs are very common in the aerospace industry. But some other examples are given in the following slides.

3. Vibrationdata 3 Tohoku-oki Earthquake, 11 March 2011 Displacement PSD Velocity time histories measured by geophones can be integrated to displacement. http://www.mdpi.com/1424-8220/13/11/14261/htm

4. Vibrationdata 4 Ocean Wave Height Measurement A 'Waverider buoy' is a surface following buoy anchored to the sea bed by means of an elastic mooring An accelerometer mounted within the buoy registers the rate at which the buoy is rising or falling as it follows the pattern of waves The acceleration signal can be converted to vertical displacement by double integration The displacement values are relayed to a recording station on the shore http://www.mhl.nsw.gov.au/www/wave_data_cut.htmlx https://www.bodc.ac.uk/data/online_request/waves/waves_recording_processing.html

5. Vibrationdata 5 Displacement Power Spectral Density The wind speed is 12 m/sec at 19.5 m above the mean ocean surface level. vibrationdata > Miscellaneous > Wind & Waves > Pierson-Moskowitz Spectrum

6. Vibrationdata 6 Fluctuating Wind Velocity Measurement Hot-wire Pulse-width modulation Laser Doppler Sonic Acoustic resonance Anemometer types Sonic anemometers ultrasonic sound waves to measure wind velocity. They measure wind speed based on the time of flight of sonic pulses between pairs of transducers.

7. Vibrationdata 7 Horizontal Gustiness in Strong Winds The wind speed is 12m/sec at 10 m above the ground. vibrationdata > Miscellaneous > Wind & Waves > Davenport-King Spectrum

8. Vibrationdata 8 Fourier Transforms The Fourier transform X(f) for a continuous time displacement series x(t) is defined as The velocity Fourier transform V(f) is The acceleration Fourier transform A(f) is

9. Vibrationdata 9 Fourier Transform Integration & Differentiation Fourier Transforms Referenced to Displacement Fourier Transform ParameterValue displacementX velocity -j  X acceleration  2 X Fourier Transforms Referenced to Acceleration Fourier Transform ParameterValue accelerationA velocity A/(-j  ) displacement A/  2 j = sqrt(-1) ω = angular frequency (rad/sec)

10. Vibrationdata 10 PSD Integration & Differentiation PSDs Referenced to Displacement PSD ParameterValue displacementX velocity  2 X acceleration 4 X4 X PSDs Referenced to Acceleration PSD ParameterValue accelerationA velocity A/  2 displacement A/  4 ω = angular frequency (rad/sec)

11. Vibrationdata 11 PSD Spectral Moments These relationships are useful for fatigue, which will be covered in a future webinar. The spectral moment M j is The index j may be an integer or non-integer. W(f) is a one-sided PSD.

12. Vibrationdata 12 PSD Spectral Moments (cont ) The rate of zero up-crossings, or effective frequency, can be estimated as The rate of peaks, or rainflow fatigue cycle rate, is

13. Vibrationdata 13 Navmat P9492 Acceleration File: navmat_spec.psd vibrationdata > power spectral density > overall RMS

14. Vibrationdata 14 Navmat P9492 Velocity

15. Vibrationdata 15 Navmat P9492 Displacement

16. Vibrationdata 16 Navmat P9492 Summary Overall levels Acceleration = 6.06 GRMS = 18.2 G (3-sigma) Velocity = 2.07 in/sec RMS = 6.21 in/sec (3-sigma) Displacement = 0.00765 inch RMS = 0.0229 inch (3-sigma) Input PSD Statistics (from Spectral Moments) Rate of Up-zero crossings = 869 Hz Rate of peaks = 1421 Hz

17. Vibrationdata 17 SDOF Response to Navmat Base Input File: navmat_spec.psd vibrationdata > power spectral density > SDOF Response to Base Input

18. Vibrationdata 18 SDOF Response Summary SDOF Acceleration Response = 11.2 GRMS = 33.5 G 3-sigma = 49.9 G 4.47-sigma (maximum expected) SDOF Pseudo Velocity Response = 3.42 inch/sec RMS = 10.2 inch/sec 3-sigma = 15.3 inch/sec 4.47-sigma (maximum expected) SDOF Relative Displacement Response = 0.00272 inch RMS = 0.00816 inch 3-sigma = 0.0121 inch 4.47-sigma (maximum expected) Rate of Up-zero crossings = 199.7 Hz Rate of peaks = 231.3 Hz Calculated from acceleration PSD. Useful for fatigue analysis.

19. Vibrationdata 19 MIL-STD-810G, Two-Wheeled Trailer Vertical, Acceleration File: two_wheeled_trailer_verti cal.txt vibrationdata > power spectral density > SDOF Response to Base Input

20. Vibrationdata 20 MIL-STD-810G, Two-Wheeled Trailer Vertical, Velocity

21. Vibrationdata 21 MIL-STD-810G, Two-Wheeled Trailer Vertical, Displacement

22. Vibrationdata MIL-STD-810G, Two-Wheeled Trailer Vertical Overall levels Acceleration = 4.45 GRMS = 13.4 G (3-sigma) Velocity = 11.7 in/sec RMS = 35.2 in/sec (3-sigma) Displacement = 0.268 inch RMS = 0.804 inch (3-sigma) Input PSD Statistics Rate of Up-zero crossings = 200.2 Hz Rate of peaks = 307.1 Hz 22