Transcript Unit 6e
Vibrationdata Unit 11 Power Spectral Density Function PSD 1 PSD Introduction Vibrationdata • A Fourier transform by itself is a poor format for representing random vibration because the Fourier magnitude depends on the number of spectral lines, as shown in previous units • The power spectral density function, which can be calculated from a Fourier transform, overcomes this limitation • Note that the power spectral density function represents the magnitude, but it discards the phase angle • The magnitude is typically represented as G2/Hz • The G is actually GRMS 2 Sample PSD Test Specification 0.1 Vibrationdata Overall Level = 6.0 grms 2 0.04 g / Hz -3 dB / octave 2 PSD ( g / Hz ) +3 dB / octave 0.01 0.001 20 80 350 2000 FREQUENCY (Hz) 3 Calculate Final Breakpoint G^2/Hz 0.1 Overall Level = 6.0 grms 2 0.04 g / Hz -3 dB / octave +3 dB / octave Number of octaves between two frequencies 2 PSD ( g / Hz ) Vibrationdata n 0.01 ln (f 2 / f1 ) ln (2) Number of octaves from 350 to 2000 Hz = 2.51 0.001 20 80 350 2000 FREQUENCY (Hz) The level change from 350 to 2000 Hz = -3 dB/oct x 2.51 oct = -7.53 dB For G^2/Hz calculations: dB 10 logA / B The final breakpoint is: (2000 Hz, 0.007 G^2/Hz) 4 Vibrationdata Vibrationdata > Miscellaneous > dB Calculations for log-log Plots > Separate Frequencies 5 Overall Level Calculation Vibrationdata Note that the PSD specification is in log-log format. Divide the PSD into segments. The equation for each segment is y1 n y (f ) f f n 1 The starting coordinate is (f1, y1) 6 Vibrationdata Overall Level Calculation (cont) The exponent n is a real number which represents the slope. The slope between two coordinates y log 2 y1 n f log 2 f1 The area a1 under segment 1 is f y a1 2 1 f n df f1 n f1 7 Overall Level Calculation (cont) Vibrationdata There are two cases depending on the exponent n. a1 y1 1 f n 1 f n 1 , for n 1 1 f n n 1 2 1 f 2 y1f1 ln , for n 1 f1 8 Overall Level Calculation (cont) Vibrationdata Finally, substitute the individual area values in the summation formula. The overall level L is the “square-root-of-the-sum-of-the-squares.” L m ai i 1 where m is the total number of segments 9 Vibrationdata dB Formulas dB difference between two levels If A & B are in units of G2/Hz, A dB 10log B If C & D are in units of G or GRMS, C dB 20log D 10 dB Formula Examples Vibrationdata Add 6 dB to a PSD The overall GRMS level doubles The G^2/Hz values quadruple Subtract 6 dB from a PSD The overall GRMS level decreases by one-half The G^2/Hz values decrease by one-fourth 11 Vibrationdata vibrationdata > Overall RMS Input File: navmat_spec.psd 12 dB Formula Examples Vibrationdata Add 6 dB to a PSD The overall GRMS level doubles The G^2/Hz values quadruple Subtract 6 dB from a PSD The overall GRMS level decreases by one-half The G^2/Hz values decrease by one-fourth 13 PSD Calculation Method 3 Vibrationdata The textbook double-sided power spectral density function XPSD(f) is lim X(f)X * (f) XPSD (f) T T The Fourier transform X(f) has a dimension of [amplitude-time] is double-sided 14 PSD Calculation Method 3, Alternate Vibrationdata Let be the one-sided power spectral density function. lim G(f) G * (f) ˆ X (f) PSD Δf Δf 0 The Fourier transform G(f) has a dimension of [amplitude] is one-sided ( must also convert from peak to rms by dividing by 2 ) 15 Recall Sampling Formula Vibrationdata The total period of the signal is T = Nt where N is number of samples in the time function and in the Fourier transform T is the record length of the time function t is the time sample separation 16 More Sampling Formulas Vibrationdata Consider a sine wave with a frequency such that one period is equal to the record length. This frequency is thus the smallest sine wave frequency which can be resolved. This frequency f is the inverse of the record length. f = 1/T This frequency is also the frequency increment for the Fourier transform. The f value is fixed for Fourier transform calculations. A wider f may be used for PSD calculations, however, by dividing the data into shorter segments 17 Statistical Degrees of Freedom Vibrationdata • The f value is linked to the number of degrees of freedom • The reliability of the power spectral density data is proportional to the degrees of freedom • The greater the f, the greater the reliability 18 Statistical Degrees of Freedom (Continued) Vibrationdata The statistical degree of freedom parameter is defined follows: dof = 2BT where dof is the number of statistical degrees of freedom B is the bandwidth of an ideal rectangular filter Note that the bandwidth B equals f, assuming an ideal rectangular filter The BT product is unity, which is equal to 2 statistical degrees of freedom from the definition in equation 19 Trade-offs Vibrationdata • Again, a given time history has 2 statistical degrees of freedom • The breakthrough is that a given time history record can be subdivided into small records, each yielding 2 degrees of freedom • The total degrees of freedom value is then equal to twice the number of individual records • The penalty, however, is that the frequency resolution widens as the record is subdivided • Narrow peaks could thus become smeared as the resolution is widened 20 Example: 4096 samples taken over 16 seconds, rectangular filter. Vibrationdata Period of Each Record Ti (sec) Frequency Resolution Bi=1/Ti (Hz) dof per Record =2Bi TI Total dof 1 Number of Time Samples per Record 4096 16 0.0625 2 2 2 2048 8 0.125 2 4 4 1024 4 0.25 2 8 8 512 2 0.5 2 16 16 256 1 1 2 32 32 128 0.5 2 2 64 64 64 0.25 4 2 128 Number of Records NR 21 Summary Vibrationdata • Break time history into individual segment to increase degrees-of-freedom • Apply Hanning Window to individual time segments to prevent leakage error • But Hanning Window has trade-off of reducing degrees-of-freedom because it removes data • Thus, overlap segments • Nearly 90% of the degrees-of-freedom are recovered with a 50% overlap 22 Original Sequence Vibrationdata Segments, Hanning Window, Non-overlapped 23 Original Sequence Vibrationdata Segments, Hanning Window, 50% Overlap 24