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An alternative to sigma clipping in shaker testing.

Many shaker controllers for random vibration testing are based on the well-established FFT data processing algorithm. A test specification is given in the frequency domain in terms is given in the frequency domain in terms of power spectral density (PSD), and time histories are reconstructed from the prescribed PSD by the inverse FFT (IFFT). Consequently, the shaker is driven by a multifrequency signal with a large number of harmonics N

x(t) = [N summation over (n = l)][A.sub.n]sin(2[pi]n[DELTA]ft + [[phi].sub.n]) (1)

The amplitudes of the harmonics are determined

[A.sub.n] = [square root of 2[DELTA]fS(n[DELTA]f)] (2)

according to the prescribed PSD shape S(f) that is made of discrete lines with the frequency increment [increment of f]. The phases [phi][sub.n] are random and uniformly distributed from 0 to 2 [pi]. The excitation obtained is a pseudorandom signal with discrete spectrum; however, a technique of windowing and overlapping time history blocks can be implemented to make the resulting signal truly random with a continuous spectrum.

The discrete Fourier transform model is perfect for digital random controllers because it allows easy correction of the input PSD shape using the acceleration feedback signal from the UUT. A closed-loop iteration procedure is arranged for this purpose: If there is a difference between the heights of the same spectrum line at frequency f = n[DELTA]f in the feedback spectrum and the PSD specification profile, it can be compensated by decreasing or increasing the next iteration drive spectrum line compared to what it was on the pervious iteration.

Such an action repeated for each of N spectrum lines gives the full PSD of the drive signal for the next iteration. Then all discrete spectrum values S(n[DELTA]f obtained are substituted into Equation 2 for amplitudes [A.sub.n], and a new drive signal is generated according to Equation 1. The iteration can be continued until the acceleration feedback spectrum is close enough to the target profile.

This classic method for simulating random excitations on shakers can be updated to involve non-Gaussian characteristics such as kurtosis and to control them in the same closed-loop iteration manner as the PSD (1), (2). If the kurtosis of the drive signal is increased, the kurtosis of the acceleration feedback will follow or vice versa if a kurtosis decrease is needed. For this reason, the kurtosis can be adjusted in a way similar to that of the PSD line with only one such correction for the entire random signal, not N corrections as for the PSD.

Leading controller manufacturers such as Vibration Research, Unholtz-Dickie, MB Dynamics, and IMV have recognized non-Gaussian random testing with increased kurtosis and started using it to achieve a higher crest factor (CF) of the excitation time history. It is necessary because road vibration data typically contains occasional extreme peaks that are missing in the Gaussian random testing by the PSD/FFT approach.

However, is it possible to decrease kurtosis to generate a shaker excitation still with the given PSD but having a time history smoother than Gaussian? Will a smaller kurtosis really lead to shaker vibration with less distinctive peaks, such as with a lower CF? If yes, to what extent can be CF be lowered? And what applications could benefit from this new kind of non-Gaussian random testing?

Lower Kurtosis Allows Higher Power

To go non-Gaussian means to generate a much wider diversity of shaker excitation time history data. Various deviations from the Gaussian model can be considered, but we focus on how frequent and distinctive high peaks are. That is described by the kurtosis parameter

K = [M.sub.4]/[([M.sub.2]).sup.2] = 1/T[[integral].sub.0.sup.T][{x(t)}.sup.4]dt/[{1/T[[integral].sub.0.sup.T][{x(t)}.sup.2]dt}.sup.2] (3)

where the fourth [M.sub.4] and second [M.sub.2] distribution moments are the averages of the fourth and second powers of time history values in the data block of length T = 1/[DELTA]f.

Kurtosis is a useful alternative to characterizing signal peak heights by the CF expressed as a ratio between the absolute maximum of the signal and its rms value. For non-Gaussian random signals, the kurtosis and the CF can depart from their Gaussian values either up or down but both in the same direction.

However, kurtosis is more robust because, being an averaged characteristic, it summarizes the effect of various extreme peaks. CF is not as comprehensive and takes into account only the largest peak. Furthermore, in contrast to kurtosis, no strict theoretical value can be defined for the CF of a Gaussian signal because the magnitude of the largest peak depends on the length of a time history sample.

Reducing the CF by sigma clipping or sigma limiting is a practice that all shaker test engineers know about. A relevant option is available in standard FFT random controller menus.

Sigma clipping is most commonly used to get more power from the same shaker system. Assume that a controller drives a shaker to its power limit of producing vibration with rms of [sigma] m/[s.sup.2], but a further rms increase is not possible because the peak value of 4[sigma] m/s[sup.2] is what corresponds to the amplifier/shaker abort limits.

If the controller could operate the shaker so that the CF is reduced, then the same limiting peak height of 4[sigma] m/[s.sup.2] an rms larger than [sigma] m/[s.sup.2] in exactly the same proportion by which the CF was reduced. Such an increase of the rms value means that the power delivered by the shaker system would be greater.

How Successful Is Sigma Clipping?

The basic sigma clipping procedure simply removes everything from the Gaussian drive signal that is higher than the prescribed limit. However, is clipping free from unpleasant side effects? Furthermore, being performed on the drive signal input, how efficient is it in decreasing CF of the acceleration output?

If any alterations are made to the time history after a drive signal is IFFT-generated, the drive signal spectrum no longer corresponds exactly to the given PSD. The resulting harmonic distortions are spread over the entire frequency interval including both the test specification bandwidth and the out-of-band frequencies.

Some harmonic distortions can be compensated in the decreasing the corresponding PSD values S(n[DELTA]f) for the next iteration. But this does not work at the out-of-band frequencies or in the vicinity of sharp resonances. For the out-of-band frequencies, all spectrum line amplitudes [A.sub.n] are further. If any PSD content is present at the out-of-band frequencies, it is caused only by harmonic distortions and will stay there as an uncontrollable noise floor in the PSD.

The harmonic distortions and the increased noise floor they produce affect more than out-of-band frequencies. The shaker armature, test fixture, or the UUT may have resonances where the controller needs very low values in the drive signal PSD to match the required acceleration PSD at these frequencies.

Obviously, the controller cannot go deeper than the noise floor and consequently cannot reduce the resonance level further. If, as a result of sigma clipping, the PSD noise floor rises, then the controller's dynamic range will be reduced. This is the price to pay for sigma clipping. The wider the dynamic range, the better the controller reproduces test specifications with bigger ratios between the amplitudes of the highest peak and the lowest notch in the prescribed PSD shape.

Shaker controller manufacturers regard the dynamic range as one of the most important performance characteristics and report it to be in the region of 90 dB in the latest products if no clipping is used. Controller manuals and manufacturers' websites warn that a loss be expected if clipping is introduced.

However, what is the fraction of the 90 dB that disappears for various clipping levels? To answer this, an experiment was carried out on a Derritron electrodynamic shaker controlled by a PC with Lab VIEW software and a National Instruments data acquisition board.

Kurtosis and CF were evaluated using samples of 80 blocks of 4,096 data points each. The power spectral analysis was carried out with 2,000 spectrum lines and frequency increment [DELTA]f = 2 Hz. The target PSD profile was a uniform spectrum from 20 Hz to 180 Hz.

Figure la depicts a Gaussian drive signal with no clipping and CF = 4.4. As shown in Figure 1b, the specified PSD profile was realized with the drive signal dynamic range DR = 80 dB. This drive input has produced an acceleration output signal (Figure lc) also with CF = 4.4. The acceleration PSD obtained is shown in Figure Id. The dynamic range of the output signal equal to DR = 33 dB is less than in commercial controllers but still enough for the purposes of this experiment.

[FIGURE 1 OMITTED]

With clipping added, the DR values change. First, 3[sigma] clipping was tried, but no noticeable change was observed in the acceleration output CF. This is not a surprise for those who have ever checked the actual CF of the shaker table acceleration after setting up a certain clipping level for the drive signal. They know that the shaker system partially restores peak heights. This phenomenon also has been reported by shaker manufacturers. (3)

Harder clipping down to 2[sigma] (Figure le) was required to reduce the acceleration CF to 3.5 (Figure lg). However, at the out-of-band frequencies, the acceleration PSD (Figure lh) has a much higher level than with no clipping (Figure Id). Half of the control dynamic range has been lost. This is the result of the drive signal dynamic range in Figure If being only 32 dB rather than DR = 80 dB for no clipping in Figure lb.

Clipping as low as 1. 2[sigma] (Figure 2a) was required to contain the acceleration output within [+ or -]3[sigma] limits (Figure 2c). However, in this case, only 9 dB remains as an available acceleration dynamic range (Figure 2d). The drive signal clipped at 1.2[sigma] actually had CF = 1.5 because the signal rms was reduced by removing all time history points higher than the clipping level.

[FIGURE 2 OMITTED]

There is a limit to the acceleration CF decrease that can be achieved by clipping. In our experiment,, the result shown in Figure 2c was the best. Increasing clipping beyond CF = 1.2[sigma] gave no improvement. The acceleration CF stayed around 3 because the harmonic distortions from severe clipping are so strong that the noise generated by them becomes comparable with the useful control signal.

When the noise component with Gaussian probability distribution is increasing, its higher peaks dominate the opposite tendency in the clipped main component. This prevents the CF of the entire signal from following the controller's command for further peak limiting.

These results were obtained with basic straight clipping. Commercial shaker controllers provide some kind of rounding-off to smooth the clipped peaks back to a bell-shaped form like in the original Gaussian signal. Smoothing of clipped peaks helps, but there is no full time history repair capable of bringing the PSD noise floor back to where it was before clipping.

It is best to avoid any modification of the signal after it has been generated. There is a way of securing sigma limiting not after, but in the process of, the IFFT signal generation.

Suggestion: Use Phase Selection

Two methods of non-Gaussian closed-loop IFFT generation of the shaker drive signal have been suggested and realized experimentally. (1), (2) The method of Gaussian-to-non-Gaussian polynomial transform initially was recommended (4) for preparing non-Gaussian time histories off-line before running a test and then reproducing these time histories on the shaker in the Time Waveform Replication mode.

It was only later that the polynomial transform was extended and converted (2) into a method that could be used in the closed-loop frequency-domain control mode. With such an approach, the polynomial transform becomes just a post-IFFT signal transformation addition, still in the frame of the regular closed-loop FFT/PSD shaker testing.

The polynomial transform is easy to implement, but it significantly reduces the controller's dynamic range compared to the traditional Gaussian random testing. Insufficient dynamic range disrupts the PSD shape if any resonances of the UUT or the shaker armature are present in the excitation frequency band. There is no such difficulty with the special phase selection method since nothing is done to the signal after the IFFT generation. (1), (2) In particular, small PSD values are not disturbed at all.

Because the power spectrum does not depend on the phases of the harmonics, the phases can be manipulated to achieve non-Gaussian simulation. In the phase selection method, the variables are separated: Amplitudes still are responsible for the PSD according to Equation 2 while the phases [[phi].sub.n] are not all random anymore. Instead, some of them are used to adjust kurtosis.

As a result, both characteristics, PSD and kurtosis, are controlled independently in the best possible way. It is important that the PSD and kurtosis corrections can be done simultaneously in the same iteration process to save the controller loop time.

To develop a methodology for kurtosis control by phase selection, a general equation was needed for the kurtosis expressed via amplitudes [A.sub.n] and phases [[phi].sub.n] in Equation 1. For the second moment, integration in the denominator of Equation 3 poses no problem as an integral over period T of a product of two sine functions from different harmonics

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

is zero for any values of two phases involved. Also, an integral of

[sin.sup.2](2[pi]n[DELTA]ft + [[phi].sub.n])

is equal to 0.5 no matter what the [[phi].sub.n]value is. Consequently, the second moment does not depend on phases [[phi].sub.n] and is a function only of the amplitudes of the harmonics that appears to be

[M.sub.2] = 1/2[N summation over(n = 1)][A.sub.n.sup.2] (5)

For the fourth moment, the fourth power in the integrand in the numerator of Equation 3 produces sine function products with more diverse combinations of indices of harmonics [n.sub.1], [n.sub.2], ... than that in the denominator with the second power. Of these products for [M.sub.4], many vanish similarly to the product of two sine functions in Equation 4.

Also, some other combinations of indices result in sum functions involving only [A.sub.n.sup.2] but no dependence on phases as in the equation for [M.sub.2] in Equation 5. However, distinct from the second moment, there will be non-zero integrals resulting in additional components in [M.sub.4] that are functions of both the amplitudes [A.sub.n] and the phases [[phi].sub.n].

If the phases of the harmonics [[phi].sub.n] are random, as in the classical Gaussian technique, then the phase-dependent components in [M.sub.n] also produce random values that compensate each other, bringing the result of summation close to zero. The non-Gaussian phase selection method alters this randomness by prescribing some of the phases in a certain deterministic way.

Because only some of the phases become deterministic with the rest remaining random, each new data block generated according to Equation 1 comes with a different set of phases. Accordingly, the blocks in the signal sequence are all different, and the non-Gaussian time history never repeats itself as is the case for the common Gaussian pseudorandom generation.

Results of Sigma Limiting by Kurtosis Control

The phases [[phi].sub.n] in the IFFT procedure can be manipulated so that the kurtosis of the signal generated by Equation 1 becomes lower. The amplitudes [A.sub.n] remain responsible for the PSD as in the common Gaussian technique. The results obtained by the phase selection method for the same experimental setup are as follows:

* A non-Gaussian drive signal with kurtosis K = 1.7 is shown in Figure 2e and its magnified fragment in Figure 3. The signal was generated for the same target PSD profile as that used in the Gaussian experiment (Figure la,b,c, d). Now, with the non-Gaussian random control (Figure 2e), the CF = 2.0, down from the value of CF = 4.4 for the Gaussian drive signal (Figure 1a). Despite such a drastic time-history modification, the PSD of non-Gaussian signal (Figure 2f) has exactly the same low tail for out-of-band frequencies as that of the PSD of Gaussian drive signal (Figure 1b).

[FIGURE 3 OMITTED]

* The PSD noise floor for the non-Gaussian simulation by phase selection (Figure 2f) remains where it was for the traditional Gaussian simulation (Figure 1b), maintaining the drive signal dynamic range of 80 dB. After the drive signal sigma limiting was achieved without compromising the out-of-band frequencies, we expect the dynamic range of the shaker acceleration signal to be as good as for the Gaussian simulation. This is confirmed by comparison of Figure Id and Figure 2h.

* The CF of the non-Gaussian acceleration time history (Figure 2g) appears to be CF = 2.8, much better than CF =3.5 obtained with clipping at the same 2[sigma] level (Figure lg) and even lower than the best clipping result (CF - 3) observed in our experiment with 1.2 [sigma] clipping (Figure 2c). The difference between CF = 3 and CF = 2.8 might look nonessential, but remember that the former comes with the input and output dynamic ranges of 23 dB and 9 dB while the latter has 80 dB and 33 dB, which are exactly the same as in the Gaussian control mode.

Conclusion

When doing non-Gaussian simulation by phase selection, the multiharmonic character of the pseudorandom signal remains intact with absolutely no alterations from the combination of pure sinusoids. As a result, there are no harmonic distortions passed to other frequencies, in contrast to the distortion caused by using clipping or the polynomial transform.

With the phase selection method, we have managed to preserve the dynamic range and the PSD precision while decreasing kurtosis to substantially reduce the CF. This means that more powerful excitation can be obtained from the same shaker because an extra space is created between the peaks of reduced height and the system abort limit.

If the power restriction comes from the amplifier capabilities, then what has been achieved with CF of the drive signal truly matters. The fact that CF was reduced from 4.4 to 2.0 allows the drive signal rms to increase twice with the entire signal still staying comfortably within the amplifier input peak limits. This new drive signal will be four times more powerful than before. In the case of the shaker armature overstroke limit being the power restriction, the acceleration feedback CF decrease from 4.4 to 2.8 means that the shaker power can be doubled in tests where the PSD is the only test specification prescribed.

References

(1.) Steinwolf, A., "Shaker Simulation of Random Vibrations With a High Kurtosis Value," Journal of the IEST, Vol. XL, No. 3, 1997, pp. 33-43.

(2.) Steinwolf, A., "Closed-Loop Shaker Simulation of Non-Gaussian Random Vibrations," Test Engineering and Management, June/July 2006, pp. 10-13, and October/November 2006, pp. 14-19.

(3.) Rogers, P., "Aggressive Low-Frequency Random Vibration," Sound and Vibrations, October, 2005, pp. 8-9.

(4.) Smallwood, D.O., "Generating Non-Gaussian Vibration for Testing Purposes," Sound and Vibrations, October 2005, pp. 18-24.

by Alexander Steinwolf, AST Consulting Ltd.

About the Author

Alexander Steinwolf, Ph.D., is president and founder of AST Consulting and a faculty member at the University of Auckland. He has guest lectured at and conducted joint research with the U.S. Air Force Research Laboratory, the U.S. Army Aberdeen Test Center, and the NASA Langley Research Center. Dr Steinwolf received the 1998 M. Simpson Award from the IEST for his work on random vibration testing with kurtosis control. His work has influenced a number of shaker controller manufacturers. AST Consulting, Auckland, New Zealand, e-mail: astc@ihug.co.nz
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Title Annotation:ENVIRONMENTAL TEST APPLICATIONS
Author:Steinwolf, Alexander
Publication:EE-Evaluation Engineering
Geographic Code:1USA
Date:Oct 1, 2008
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