Unit 22 Vibrationdata Integration and Differentiation of Time

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

PSD Types Vibrationdata 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. 2

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

Ocean Wave Height Measurement Vibrationdata • 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_recording_processing. html 4

Displacement Power Spectral Density Vibrationdata The wind speed is 12 m/sec at 19. 5 m above the mean ocean surface level. vibrationdata > Miscellaneous > Wind & Waves > Pierson-Moskowitz Spectrum 5

Fluctuating Wind Velocity Measurement Vibrationdata Anemometer types • Hot-wire • Pulse-width modulation • Laser Doppler • Sonic • Acoustic resonance 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. 6

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

Fourier Transforms Vibrationdata 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 8

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

PSD Integration & Differentiation Vibrationdata PSDs Referenced to Displacement PSD Parameter displacement Value X velocity 2 X acceleration 4 X ω = angular frequency (rad/sec) PSDs Referenced to Acceleration PSD Parameter acceleration Value A velocity A/ 2 displacement A/ 4 10

PSD Spectral Moments Vibrationdata 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 index j may be an integer or non-integer. 11

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

Navmat P 9492 Acceleration Vibrationdata File: navmat_spec. psd vibrationdata > power spectral density > overall RMS 13

Navmat P 9492 Velocity Vibrationdata 14

Navmat P 9492 Displacement Vibrationdata 15

Navmat P 9492 Summary Vibrationdata 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 16

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

SDOF Response Summary Vibrationdata 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. 18

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

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

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

MIL-STD-810 G, Two-Wheeled Trailer Vertical Vibrationdata 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