Power Spectral Density
By Tom Irvine
SECTION 1
Decibels & Slopes
Differences

Power spectral density functions are sometimes specified in terms of decibels. The dB difference between two levels A & B in units of G2 or G2/Hz is

(1.1)

A 6 dB increase would quadruple a G2/Hz level. A 6 dB decrease would reduce the G2/Hz level by a ratio of one-fourth.

The dB difference between two levels in units of G or GRMS is

(1.2)

A 6 dB increase would double a G level. A 6 dB decrease would reduce the G level by a ratio of one-half.

Slopes

Consider a frequency domain plot with a straight-line segment in log-log format with coordinates (f1, y1) and (f2, y2).

The slope N between the coordinate pair is

(1.3)

The line equation for this pair is

(1.4)

The dB/octave slope is for units of G2 or G2/Hz is

(1.5)

The dB/octave slope is for units of G or GRMS is

(1.6)
SECTION 2
PSD Specifications
Figure 2.1. Navmat P-9492 Acceleration PSD

The power spectral density in Figure 16.1 is taken from Reference [27]. The lowercase g convention is carried over from this source, but uppercase G is used elsewhere in the present document This level is intended as a base input for a shaker table test to screen components for parts and workmanship defects. It is representative of similar levels found in various military and NASA standards. The Y-axis unit is more properly grms2/Hz. The specification consists of three straight line segments in log-log format. The overall grms value is the “square root of the area under the curve.” The integration approach is to take the grms2 area under each of the three segments. Then sum the areas and take the square root. This requires special integration formulas due to the log-log format which depend on the slope from equation (1.3).

The area ai for a segment is

(2.1)

The overall level for m total segments is then

(2.2)

Performing these steps for the Navmat P-9492 specification in Figure 1.1 requires some preliminary work, because the specification did not explicitly give the first and last amplitudes. The first coordinate amplitude is 0.01 g2/Hz by inspection of the graph. Identifying last coordinate amplitude requires a two-step calculation. The slope is

(2.3)

The final coordinate amplitude is

(2.4)

The final amplitude is whimsically referred to as “Bond.” The overall grms calculation is summarized in the following table.

Table 2.1. Navmat P-9492 Overall Level Calculation
Frequency Domain Slope N grms2
20 to 80 1 1.5
80 to 350 0 10.8
350 to 2000 -1 24.4
  Total 36.7

The overall level is the square root of 36.7 grms2 which is 6.06 grms.

SECTION 3
PSD Integration in the Frequency Domain

An acceleration PSD has a corresponding velocity PSD and a displacement PSD. Let

   APSD = Acceleration PSD

   VPSD = Velocity PSD

   DPSD = Displacement PSD

The integration formulas are

(3.1)
(3.2)

The resulting velocity and displacement PSDs for the Navmat specification are shown in Figure 1.2 and Figure 1.3, respectively. These plots are useful to determine whether a given acceleration PSD test can be performed on a certain shaker table given the table’s velocity and displacement limits.

Figure 3.1. Navmat P-9492 Velocity PSD
Figure 3.2. Navmat P-9492 Displacement PSD
SECTION 4
PSD Calculation Methods

Power spectral density functions of measured data may be calculated via three methods:

  • Measuring the RMS value of the amplitude in successive frequency bands, where the signal in each band has been bandpass filtered
  • Taking the Fourier transform of the autocorrelation function
  • Taking the limit of the Fourier transform X(f) times its complex conjugate divided by its period T as the period approaches infinity.

The filtering option is most instructive method and is covered in Section 20. The autocorrelation option is the Wierner-Khintchine approach and is omitted from this document. A modified form of the third option is covered in this section and is the preferred method for computational efficiency.

The one-sided power spectral density function is calculated from the discrete Fourier transform as

(4.1)

The Fourier transform in extends from zero frequency to the Nyquist frequency, which is one-half the sample rate. Each Fourier transform has peak dimension. The one-half factor is needed to convert the amplitude from peak2/Hz to rms2/Hz.

The frequency step is finite in practice and is the inverse of the total measured duration.

(4.2)

This frequency step is the smallest sine wave frequency which can be resolved, where this sine wave has one period equal to the total duration. The frequency step is linked to the number of statistical-degrees-of-freedom. The reliability of the power spectral density data is proportional to the number of degrees. A wider gives greater PSD confidence in terms of smoothing the spectral components.

The number of degrees is defined as

(4.3)

The variable B is the bandwidth of an ideal rectangular filter. It is equal to the frequency step Δf for an ideal rectangular filter. The product BT is unity and the number of degrees-of-freedom is always equal to two for a given record assuming a rectangular filter. This number may be increased for a given record by subdividing it into a set of small segments, where each segment has two degrees. The total number of degrees is then equal to twice the number of segments. This subdivision approach tends to smooth the PSD. The penalty, however, is that the frequency resolution widens as the record is subdivided which could smear narrow peaks. These tradeoffs are shown by example in Table 1.2.

Table 4.1. Time History with 4096 Samples Taken over 16 Seconds, Rectangular Filter
Number of Records
NR
Number of Time Samples per Record Period of Each Record Ti
(sec)
Frequency Resolution
Bi = 1 / Ti
(Hz)
dof per Record
= 2Bi Ti
Total dof
1 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

Additional rows could be added to the above table but are omitted for brevity. Selecting the optimum processing strategy for a given time history is a matter of engineering judgment. Continuing with the table example, a frequency step of 4 Hz with 128 dof might be a good choice for demonstrating that the broadband base input for a shaker table test complied with the specified PSD’s tolerance bands. On the other hand, a narrow frequency resolution from the first or second row may be needed if the purpose of analyzing the PSD it to identify lightly-damped, low-frequency resonant peaks.

Recall that a rectangular filter leaves the time history data unmodified. Leakage error may be a concern, however, as discussed in Section 15.3. A Hanning window can be applied to each data segment to reduce leakage, but it is a non-ideal filter which reduces the number of degrees-of-freedom. The key to applying the Hanning window is to subdivide the data into overlapping segments. Nearly 90% of the degrees-of-freedom are recovered with a 50% overlap, as shown in Figure 1.4.

Figure 4.1. White Noise and Subdivision into Overlapped Segments with Hanning Windows
SECTION 5
PSD Calculation Examples
White Noise PSD

Recall the white noise time history in Figure 15.25. A PSD is calculated for this signal with Hanning window and 50% overlap for each of two frequency resolution cases. The PSDs are shown in Figure 1.5.

Figure 5.1. White Noise PSDs

The blue curve has a finer frequency resolution and more variation than the red curve. The red curve is smoother due to its wider frequency resolution and higher dof number. The red curve effectively averages the PSD points into wider bands. Note that pure white noise would have a flat PSD curve. The time history in Figure 13.5 departs from this ideal because it was taken over a short, 10 second duration. It was also band-limited via low-pass filtering such that it has a roll-off beginning at about 85 Hz. Filtering is covered in Section 20.

Taurus Auto PSD
Figure 5.2. 2003 Ford Taurus Auto

The author mounted an accelerometer in his auto’s console as shown in Figure 1.6. The vehicle was driven on a highway at 65 mph. An accelerometer time history from this road test is given in Figure 1.7. The PSD is shown in Figure 16.8, with a narrow frequency resolution to aid in spectral peak identification. The fundamental mode damping is calculated via the half-power bandwidth method in Figure 1.9.

Figure 5.3. Taurus Time History
Figure 5.4. Taurus PSD, Df = 0.3 Hz, 6 dof
Figure 5.5. Taurus PSD, Half-Power Bandwidth
(5.1)

The Taurus auto spring-mass Frequency is 1.5 Hz with an amplification factor equivalent to 33% damping, which is typical damping for shock absorbers. Common automobile natural frequencies are given in the following table.

Table 5.1. Automobile Spring-Mass Frequencies
Vehicle Fundamental Frequency
Passenger Car 1 to 1.5 Hz
Sports Car 2 to 2.5 Hz
Hummer 4.5 Hz

The spectral peaks in Figure 1.8 at 14.6 and 29.1 Hz can be explained as tire imbalance frequencies per the following calculations steps.

  • a. Assume 25 inch tire outer diameter at 65 mph
  • b. Circumference = π (25 inch) = 78.5 inch
  • c. 65 mph = 1144 in/sec
  • d. Speed/Circumference = (1144 in/sec) / 78.5 in = 14.6 Hz
  • e. 2X harmonic = 29.1 Hz
SECTION 6
PSD Time History Synthesis
Table 6.1. Time History Synthesis Steps to Meet PSD Specification
Step Description
1 Generate a white noise time history
2 Take the FFT
3 Scale the FFT amplitude per the PSD for each frequency
4 The time history is the inverse FFT
5 Use integration, polynomial trend removal, fade in and out, and differentiation so that corresponding mean velocity and mean displacement are both zero
6 Scale the time history so that its GRMS value matches the specification’s overall GRMS value
7 Take a PSD of the synthesized time history to verify that it matches the PSD specification

The method for synthesizing a time history to satisfy a PSD are shown in Table 1.4 . These steps are used to synthesize an acceleration time history to satisfy the Navmat P-9492 specification. The resulting broadband random time history is shown in Figure 16.10, with its normal distribution histogram. It began as white noise but was modified such that the final time history is no longer white noise. Its final, corresponding PSD is shaped and defined over a finite frequency domain, as shown in Figure 16.11. White noise would have a flat PSD in contrast.

Figure 6.1. Acceleration Time History and Histogram
Table 6.2. Acceleration Time History Statistics
Parameter Value   Parameter Value
Mean 0 G Kurtosis 3.01
Std Dev 6.06 G Crest Factor 5.375
RMS 6.06 G Maximum 32.56 G
Skewness 0.011 Minimum -30.93 G
Figure 6.2. PSD Verification

The synthesized time history satisfies the specification well within the tolerance bands. The frequency step is 2.84 Hz with 1024 statistical degrees-of-freedom.

Figure 6.3. Synthesized Time History, Acceleration, Velocity and Displacement

Each of the three response time histories has a stable oscillation about its zero baseline. Each also has a brief fade in and out, which could be seen in a close-up view of the start and finish.

Now assume an SDOF system with (fn=200 Hz, Q=10) subjected to the Navmat P-9492 PSD base input, using the model in Figure 12.12. Solve for the acceleration response in the time domain using the synthesized base input. The numerical engine is the Smallwood ramp invariant digital recursive relationship. The response time history is shown in Figure 1.13.

Figure 6.4. SDOF Response to Synthesized Base Input
Table 6.3. Acceleration Response Time History Statistics
Parameter Value   Parameter Value
Mean 0 G Kurtosis 3.02
Std Dev 11.3 G Crest Factor 4.548
RMS 11.3 G Maximum 51.36 G
Skewness 0.003 Minimum -50.95 G

The overall response of 11.3 GRMS is very close to the theoretical values from the frequency domain methods which will be shown in Sections 16.7 and 16.8. The theoretical crest factor for this case per equation (13.18) is 4.71, slightly higher than the 4.548 value in the table.

The time domain synthesis method could whimsically be referred to as a Rube Goldberg approach, after the famous inventor and cartoonist.

Figure 6.5. Input and Response Acceleration including Response Velocity, Close-up View

The base input is broadband random vibration. The response acceleration and velocity time histories are narrowband random. The SDOF system prefers to oscillate at its natural frequency. The positive slope zero cross rate is 199.6 Hz.

Figure 6.6. Input and Response PSDs for Navmat P-9492 Synthesis

The response PSD tracks the input at the low frequency end with nearly unity gain. The resonant response occurs at and near the 200 Hz natural frequency. The energy above √2 times the natural frequency is attenuated. Compare Figure 1.15 with the same set of PSD curves derived from the frequency domain method which will be shown in Figure 1.18.

Figure 6.7. Power Transmissibility for Input and Response PSDs

The peak power transmissibility reaches nearly 100 G2/ G2 at 200 Hz which is equivalent to Q2 where Q=10. But this is a special case of an SDOF system subjected to base excitation. A more robust method for estimating the Q value, as if this were experimental data, is to use the half-power bandwidth method shown in Figure 1.17.

Figure 6.8. Power Transmissibility with Half-Power Bandwidth Points

The half-power bandwidth method yields the following amplification factor.

(6.1)
SECTION 7
Miles Equation for Base Excitation

Consider an SDOF system subjected to base excitation where the input PSD is white noise over the frequency domain from 0 to infinity Hz, using the model in Figure 12.12. The resulting acceleration GRMS can be determined by Miles equation from Reference [10]. The constant power spectral density amplitude is represented by A with unit of (G^2/Hz).

(7.1)

An equivalent form is

(7.2)

The Miles equation is widely used due to its simplicity, but its assumption of white noise over an infinite domain does not exist in physical reality. A rule-of-thumb states that it may be used if the input PSD is flat within one-octave on either side of the natural frequency. In practice, this rule is used with some compromise.

Assume an SDOF system with (fn=200 Hz, Q=10) subjected to the Navmat P-9492 PSD base input. The overall response is

(7.3)

A more robust method for performing the response calculation is given in Section 16.8.

SECTION 8
General Method for Base Excitation

Real-world PSD specifications are shaped and have lower and upper frequency limits. Miles’ equation cannot account for a PSD with ramps and plateaus. The problem is exacerbated if the PSD is from narrowband measured data with peaks and dips. These practical characteristics can be readily accounted for by applying the power transmissibility function to the base input PSD and then by doing a point-by-point multiplication calculate the response PSD. The overall response is then the square root of the area under the response PSD curve.

Recall the SDOF transmissibility function from equation (12.47).

(8.1)

The power transmissibility is equal the transmissibility squared. The response PSD is calculated from the power transmissibility to the input PSD.

(8.2)
(8.3)

Equation (16.23) is cumbersome but is readily implement via a software program. It does not appear to have a name in the literature but is referred to as the “general method” in this document.

Figure 8.1. General Method SDOF Response to Navmat P-9492 Base Input PSD

Repeat the example of an SDOF system with (fn=200 Hz, Q=10) subjected to the Navmat P-9492 PSD base input using the general method. The overall response in Figure 1.18 agrees with the Miles result in equation. It is also very close to the time domain overall response in Table 1.6. The general method would yield a more accurate result if the natural frequency fell on either of the input PSD slopes.

Compare Figure 1.18 with the same set of PSD curves derived from the time domain synthesis in Figure 1.15.

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