Integration and Differentiation of Time Histories

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.
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Tohoku-oki Earthquake, 11 March 2011
Displacement PSD
Velocity time histories measured by geophones can
be integrated to displacement. http//www.mdpi.co
m/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
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Displacement Power Spectral Density
The wind speed is 12 m/sec at 19.5 m above the
mean ocean surface level. vibrationdata gt
Miscellaneous gt Wind Waves gt Pierson-Moskowitz
Spectrum
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Fluctuating Wind Velocity Measurement
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.
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Horizontal Gustiness in Strong Winds
The wind speed is 12m/sec at 10 m above the
ground. vibrationdata gt Miscellaneous gt Wind
Waves gt Davenport-King Spectrum
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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
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Fourier Transform Integration Differentiation
Fourier Transforms Referenced to Displacement
Fourier Transform
Parameter Value
displacement X
velocity -j ? X
acceleration ?2 X
j sqrt(-1) ? angular frequency (rad/sec)
Fourier Transforms Referenced to Acceleration
Fourier Transform
Parameter Value
acceleration A
velocity A/(-j?)
displacement A/?2
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PSD Integration Differentiation
PSDs Referenced to Displacement PSD
Parameter Value
displacement X
velocity ?2 X
acceleration ?4 X
? angular frequency (rad/sec)
PSDs Referenced to Acceleration PSD
Parameter Value
acceleration A
velocity A/?2
displacement A/?4
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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 index j may be an integer or non-integer.
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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
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Navmat P9492 Acceleration
File navmat_spec.psd vibrationdata gt power
spectral density gt overall RMS
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Navmat P9492 Velocity
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Navmat P9492 Displacement
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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
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SDOF Response to Navmat Base Input
File navmat_spec.psd vibrationdata gt power
spectral density gt SDOF Response to Base Input
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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.
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MIL-STD-810G, Two-Wheeled Trailer Vertical,
Acceleration
File two_wheeled_trailer_vertical.txt vibrationd
ata gt power spectral density gt SDOF Response to
Base Input
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MIL-STD-810G, Two-Wheeled Trailer Vertical,
Velocity
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MIL-STD-810G, Two-Wheeled Trailer Vertical,
Displacement
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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