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Unit 22

Unit 22. Integration and Differentiation of Time Histories & Spectral Functions. PSD Types. PSDs can be calculated for. Acceleration Velocity Displacement Force Pressure or Stress Strain. Acceleration PSDs are very common in the aerospace industry.

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Unit 22

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

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

  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. 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. 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. 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. 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. 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. Fourier Transform Integration & Differentiation Fourier Transforms Referenced to Displacement Fourier Transform j = sqrt(-1) ω = angular frequency (rad/sec) Fourier Transforms Referenced to Acceleration Fourier Transform

  10. PSD Integration & Differentiation PSDs Referenced to Displacement PSD ω = angular frequency (rad/sec) PSDs Referenced to Acceleration PSD

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

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

  14. Navmat P9492 Velocity

  15. Navmat P9492 Displacement

  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. SDOF Response to Navmat Base Input File: navmat_spec.psd vibrationdata > power spectral density > SDOF Response to Base Input 60 second duration

  18. SDOF Response Summary Maximum expected peak from Rayleigh distribution = 4.5-sigma Absolute Acceleration = 11.18 GRMS = 33.53 3-sigma G = 49.93 4.5-sigma G Absolute Displacement = 0.008532 inches RMS Pseudo Velocity = 3.416 inch/sec RMS = 10.25 inch/sec 3-sigma = 15.26 4.5-sigma inch/sec Relative Displacement = 0.002719 inches RMS = 0.008156 inches 3-sigma Acceleration Rate of Up-zero crossings = 199.7 Hz Rate of peaks = 231.3 Hz Calculated from acceleration PSD. Useful for fatigue analysis.

  19. MIL-STD-810G, Two-Wheeled Trailer Vertical, Acceleration File: two_wheeled_trailer_vertical.txt

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

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

  22. 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 = 307.2 Hz

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