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

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

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
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.

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
Parameter
Value
displacement X
velocity
-j  X
acceleration
2 X
j = sqrt(-1)
ω = angular frequency (rad/sec)
Fourier Transforms Referenced to Acceleration Fourier Transform
Parameter
Value
acceleration A
velocity
A/(-j)
displacement
A/2

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

11. PSD Spectral Moments W(f) is a one-sided PSD.
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.

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 P Velocity

15. Navmat P Displacement

16. Navmat P9492 Summary Overall levels Acceleration = 6.06 GRMS
= G (3-sigma)
Velocity = in/sec RMS
= in/sec (3-sigma)
Displacement = inch RMS
= inch (3-sigma)
Input PSD Statistics (from Spectral Moments)
Rate of Up-zero crossings = Hz
Rate of peaks = 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 Calculated from acceleration PSD.
Maximum expected peak from Rayleigh distribution = sigma
Absolute Acceleration = GRMS
= sigma G
= sigma G
Absolute Displacement = inches RMS
Pseudo Velocity = inch/sec RMS
= inch/sec 3-sigma
= sigma inch/sec
Relative Displacement = inches RMS
= inches 3-sigma
Acceleration
Rate of Up-zero crossings = Hz
Rate of peaks = 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 = inch RMS = inch (3-sigma) Input PSD Statistics Rate of Up-zero crossings = Hz Rate of peaks = Hz