Presentation on theme: "Vibrationdata 1 Unit 22 Integration and Differentiation of Time Histories & Spectral Functions."— Presentation transcript:
Vibrationdata 1 Unit 22 Integration and Differentiation of Time Histories & Spectral Functions
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.
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
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
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
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.
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
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
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)
Vibrationdata 10 PSD Integration & Differentiation PSDs Referenced to Displacement PSD ParameterValue displacementX velocity 2 X acceleration 4 X4 X PSDs Referenced to Acceleration PSD ParameterValue accelerationA velocity A/ 2 displacement A/ 4 ω = angular frequency (rad/sec)
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.
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
Vibrationdata 13 Navmat P9492 Acceleration File: navmat_spec.psd vibrationdata > power spectral density > overall RMS