[American Institute of Aeronautics and Astronautics 50th AIAA/ASME/ASCE/AHS/ASC Structures,...

Validation of Panel Damping Loss Factor Estimation Algorithms Using a Computational Model

Mark S. Ewing1, Himanshu Dande2 and Kranthi Vatti3

University of Kansas Aerospace Engineering, Lawrence, KS 66045, USA

Panel damping loss factors are estimated in full octave frequency bands with standard 1/3 octave band center frequency using a finite element model of a rectangular plate mechanically exited at a single point. The Impulse Response Decay Method (IRDM) and the Random Decrement (RD) techniques are studied for a range of loss factor from 0.001 to 0.1. The effect of considering varying numbers of response points was the primary focus. The effect of alternative time-domain filtering schemes for the RD method is also considered.

Nomenclature

A = Major dimension of the Finite Element. a = Major dimension of the Plate; Acceleration B = Minor Dimension of the Finite Element b = Major dimension of the Plate D = Diameter DR = Rate of Decrease (decibels per second) E = Young’s Modulus of the Material

nf = Natural Frequency (Hz)

cf = Center Frequency (Hz)

fΔ = Frequency Resolution (Hz) h = Impulse Response Function m = Mass of the Plate

dR = Effective Radius of the near Field t = Thickness of the Plate; time ζ = Critical Viscous Damping Factor η = Damping Loss Factor ρ = Density of the Material σ = Standard Deviation

dω = Damped Natural Frequency (rad/sec) ω = Frequency (rad/sec)

I. Introduction

ANEL damping loss factors can be computed using a number of techniques based on measured responses and, depending on the algorithm chosen, based on measured excitation forces as well. These

many techniques are based on a range of mathematical models, including both continuous and discretized formulations. Here, a linear finite element-based model of a free, uniform thin plate is used to simulate the

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1 Associate Professor & Chair, KU Aerospace Engineering, 1530 W. 15th St., Lawrence, KS 66045, Senior Member. 2 Graduate Research Assistant, KU Aerospace Engineering, 1530 W. 15th St., Lawrence, KS 66045. 3 Graduate Research Assistant, KU Aerospace Engineering, 1530 W. 15th St., Lawrence, KS 66045.

American Institute of Aeronautics and Astronautics

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50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2428

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

measurements needed for two techniques: the Impulse Response Decay Method (IRDM)1 and the Random Decrement (RD)2. In particular, the computational model is given a range of structural damping levels; then, the estimation techniques are used to “recover” the given level of damping. Presumably, if the validation is positive, the technique will be considered a valid process for measurements on actual structures.

II. Mathematical, Computational and Physical Models

The thin plate, of in-plane dimensions a by b has an aspect ratio, a/b, of 1.5, a thickness ratio, b/t, of 385 and a specific stiffness, E/ρ, of 2.59 * 107 N-m/kg. The damping loss factor, η, is assigned values of 0.001, 0.01 and 0.1. The plate is considered to be free of external supports and is loaded by a concentrated mechanical force.

Although an analytical model is undoubtedly available, a computational model has been developed using MSC/NASTRAN. QUAD4 elements (of dimensions, A by B) with a quadrilateral element fineness of approximately 80 (b/B or a/A) have been used to create a regular, rectangular mesh. As such, the spatial Nyquist frequency—at which modal half-wavelengths approach the element width (A or B)—has been shown to be approximately 10 kHz. The mechanical loads are applied in the center of one quadrant of the plate to a single node, modeling physical loading via the bottom surface of a small force gage. Responses to excitation at the single forcing point were computed at approximately the 17 points indicated in Figure 1, below, wherein each square region represents 100 elements.

4 possible excitations points

Figure 1. Computational model with a “regular” pattern of excitation and response points (17) indicated. There are 100 elements in each of the square regions, therefore there are 9600 points in the computational model.

In testing planned for the physical model, the external force will be exerted by an electrodynamic shaker via a thin stinger (tplate/Dstinger = 3) and a force gage attached to the plate. The diameter of the circular surface over which the force gage will be attached to the plate is approximately 1.5 cm, which is a good approximation of a point load until well over 10 kHz.


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For the computational studies, the best frequency resolution for a frequency response function (FRF) attempted is 1Hz over a 10 kHz frequency range. These FRFs were inverted using the inverse Fourier transform, resulting in the analogous impulse response functions (IRFs). Based on the frequency resolution, Δf =1 Hz, and the number of “lines” in the FRF, N=20,000, the length of the IRF is 1/ Δf=1 second. Since there were “N” lines in the FRF, the number of response points in the IRF is N=20,000, which corresponds to the temporal resolution of 1/20,000=50 microseconds. As such, the temporal resolution in the computational studies allows resolution up to 10 kHz, but only good responses up to about 4 kHz.

III. Damping Loss Factor Estimation

A. Single degree-of-freedom model The simplest damping loss factor estimation algorithms are based on a single degree of freedom, spring-mass-viscous damper (k-m-c) system, for which the impulse response is:

td

d

tem

th ζωωω

−= sin1)( (1)

where ωd is the damped natural frequency, m is the mass and ζ is the critical viscous damping factor. Denoting η = 2ζ as the damping loss factor, the decay component of the impulse response function (and the associated acceleration) is e-ηωt. Defining DR as the decrease in acceleration amplitude, a (t), in terms of decibels per second, i.e., DR = 20 log10 {a(t2)/a(t1)}/{t2- t1}, the damping loss factor can be expressed as1,2 :

nfDR

3.27=η (2)

where fn is the natural frequency. For such a simple system, the decay rate may be observed directly. However, this same decay rate may be observed by first computing, for instance, the accelerance (the acceleration-to-force) frequency response function (FRF). This FRF can be based on response to an actual impulse or some other form of excitation, for instance, random or burst excitation. Then, the inverse Fourier transform of the FRF can be computed, yielding the impulse response function—subject to biases injected due to discretization. B. Damping Loss Factor Estimation for Structures The damping loss factor, whether for a structural element or an entire structure, is known to have frequency dependence. This frequency-dependence is partly due to frequency-dependent material properties of the constituent materials, and partly due to the distribution of material strain as a function of frequency. In particular, a plate-like structure has many distinct frequency bands within which the structure’s modal response is more or less evident, and the damping characteristics of the material are, accordingly, more or less pronounced2,3.

Assigning a single damping loss factor for a structure requires specifying some sort of process to account for the spatially-variable response, mobility or accelerance, upon which the damping loss factor is based. Analytically, an integrated response can be used, allowing the prediction of a frequency- and spatially-dependent loss factor to be expressed. Experimentally, of course, only a finite number of points are available, which suggests averaging the response from multiple points. Some researchers have shown


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that responses from a small number of randomly-selected points seem to show convergence to a common value, as long as the modal density is “high enough”. In this study, convergence is studied with regard to the number and location of response measurements. It is also known that response is significantly affected by the excitation point as well.

IV. Impulse Response Decay Method

The impulse response decay method is based on the estimation of the decay rate in a spring-mass-damper system. But, for a real structure, there are many modes of vibration excited by an impulse (or any other type of excitation). A common means of estimating damping loss factor uses frequency-domain filtering, that is, band-limited segments of the FRF, which, when inverted in the Fourier sense, yield a decaying response with rather narrow frequency content. The assumption is that the decay observed in these frequency bands tends to represent the frequency-dependent damping loss factor. In this study, center frequencies, fc, for the analysis bands are standard 1/3 octave frequencies (i.e., 20, 25, 32, 40, etc.). However, the bandwidth of each band, or “bin”, is a full octave. Alternative bin bandwidths are discussed in sections A, below. The frequency-dependent damping loss factor is then estimated to be:

cc f

DRf3.27

)( =η (3)

where fc is the band center frequency and DR is as described for Equation 2. Figure 2 is the accelerance FRF for a selected point in the computational model based on a frequency resolution of 10 Hz (Δf) over a frequency range of 10 kHz. Note that this corresponds to a temporal resolution of 50 microseconds.

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Full Octave FRF for a 1250 Hz Center Frequency

Figure 2. Accelerance FRFs for the thin plate, for the full spectrum (top) And in the octave band centered on 1250 Hz (bottom).


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Figure 3, below, shows the inverse Fourier transform of the full octave, band-limited FRF with a

1250 Hz center frequency. The value of η is estimated from the slope of the envelope of either of these decay signatures. Figure 4 is a composite of average acceleration free decay signatures for a set of 16 regularly-spaced response points over a range of frequencies. From these average decay signatures, the frequency-dependent damping loss factors were estimated.

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Inverse Fourier transform of the Decay

Logarithmic Decay of the Absolute Value of the Decay

Logarithmic Decay of the Absolute Value of the Hilbert Transform

Figure 3. Inverse Fourier Transform of the FRF in a one octave band centered on 1250 Hz (top); the

same decay shown, in dB, as the absolute value of response (middle) and the magnitude of the Hilbert transform of the response.


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Eta= 0.0107 Eta=0.0102 Eta=0.01

Eta=0.01 Eta=0.01 Eta=0.01

Eta=0.01 Eta=0.01 Eta=0.01

Figure 4. Averaged decay signatures for a range of full octave bands, with the linear estimate of slope and the resulting loss factor (Eta) shown.

A. Effect of narrower frequency bands The loss factors for standard 1/3 octave center frequencies have been found using full octave bins. To investigate this choice, a study has been conducted using 1/3, 1/6 and 1/12 octave bins. Figure 5 shows the predicted loss factors. Apparently, there is no reason to choose any one bandwidth over another. So, a full octave bin will be used in all subsequent work.

Figure 5. Effect of Different Octave bands on the Loss Factor ( 4 Output Points )


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B. Variable numbers of measurement points One of the goals of the study reported here is to determine the variance of estimated loss factors as a function of the number of response points averaged. Although more information is typically better, knowledge of how many response points needed to get an acceptable estimation error would be useful. Figure 6 shows the damping loss factor estimates (as a function of frequency) resulting from averages of 20, 15, 10 and 5 sets of 4, 8, 12 and 16 responses, respectively, due to excitation at a single excitation point. [This keeps the number of data considered constant, at 80]. For this study, the sets of 4, 8, 12 and 16 response points were selected at random from the set of 16 responses considered. Note that in this case, data from 8 response points are needed to get acceptable estimates of damping loss factor (within 5% of the known value of 0.01) in the 40-8000 Hz range. Also, very little improvement is apparent when more points are considered. This can be seen in figures 6 , 7 and 8, below.

Figure 6. Damping loss factor estimates based on averages of 4, 8, 12 and 16 input-output pairs.



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The variation in estimates of loss factor has also been studied for the case of η=0.01. Figure 9

shows the effect on the variation in the estimated loss factor as a function of the number of randomly-selected sets of 4 response points considered. As expected, for larger numbers of randomly-selected sets of response data (from the available 16), the variation in the estimates decreases. Note that the minimum and maximum estimates are reported from across the frequency range. The average of the estimates (for all frequencies and the specified number of times the estimation was performed) is also shown. As expected, the average seems to be converging to 0.01.

Figure 9. Variation in the range of estimates of the loss factor for a range of numbers of randomly-selected sets of 4 response points.


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C. Effect of Frequency Resolution The effect of frequency resolution has been studied for η=0.001, 0.01 and 0.1. This study was based on the FRFs from the computational model, for which the frequency resolution is 1 Hz. Lower levels of frequency resolution were created through simple decimation.

Figures 10, 11 and 12 show the estimated loss factors for the three levels of loss factors in the computational model for 5 different frequency resolutions. Figure 10, for the case of η=0.001, shows that only the 1 Hz frequency resolution allows successful estimation, say within 5% above 3000 Hz (except at 5000 Hz). Figure 11, for the case of η=0.01, shows that very successful estimation, well within 2%, is achieved above 50 Hz for all frequency resolutions. Finally, and most importantly, Figure 12 shows the estimation for the case of η=0.1, there is a poor estimation for all frequency resolution cases. This failure is believed to be due to the fact that responses used in the estimation which happened to be in the “near field” response region—where the magnitude of response is systematically larger than the “reverberant field” region—skewed the estimation to a high loss factor, especially at the higher frequencies.

Figure 10. Effect of Frequency Resolution , Eta = 0.001


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Figure 11. Effect of Frequency Resolution, eta = 0.01

Figure 12. Effect of Frequency Resolution, eta = 0.1

The effective radius of the near field in a plate of finite size has been studied by Lyon and DeJong4. Using their analysis, the effective radius of the near field, Rd, has been computed using the physical properties of the plate in the computational study. As shown in Figure 13, the effective radius is approximately 0.4 meters or 1.3 feet for the 800 Hz frequency band. This is the radius at which the mean square velocity of the near field equals that of the far field, or reverberant field, where the response is assumed to be uniform.


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To check this analysis, the mean square response per unit force in the 800 Hz full-octave band was calculated and is plotted in Figure 14. These were calculated by integrating the FRFs in the frequency band. The levels of response are seen to be substantially larger within 1.5 feet of the excitation point—which is consistent with the predicted effective radius of the near field. Therefore, the decay of the response near the excitation point is not as high as elsewhere, which tends to cause under-prediction of the loss factor.

Figure 13. The Size of the Direct field for the mentioned excitation point

Figure 14. Mean Square Response per Unit Force at 800 Hz center frequency


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V. Random Decrement

The Random Decrement technique, unlike the IRDM, is based solely on the response of a system to random excitation. As such, the expectations for loss factor estimation are somewhat less than those for the IRDM.

For this computational study, the analytical FRF is convolved with a random force, producing an arbitrarily-long response signal. This signal is then used for the decay signature extraction. The response is filtered in specified (typically octave-wide) frequency bands. Then, the response is decomposed into an ensemble of responses after a triggering event—in this case the attainment of response in a pre-selected amplitude band and a zero slope. This ensemble of responses is then averaged, which, theoretically, removes the effect of the randomness of the excitation. The averaged response, then, is the decay response of the triggering impulse, upon which the loss factor is based (c.f., equation 3, section IV). Figure 15, below, shows 3 (of 250 possible) representative response records (the top 3 records) and the averaged response (bottom record) for the 2500 Hz frequency band . Note that the averaged response shows a clear decay.

A. Triggering Scheme

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Figure 16, below, is a schematic of a triggering scheme based on the definition of a “lower trigger

level”, above which the response must be to be a candidate trigger point. If the response “peaks” above this level and decreases below this level before exceeding an “upper trigger level”, this constitutes a triggering event. This event, then, is characterized by a non-zero (positive, in this case) acceleration and zero slope in the acceleration. The baseline lower trigger level is taken to be σ√2, where σ is the standard deviation of the presumed random response. A study was performed in which the lower triggering level and the width of the triggering band were varied, these being the first two of five parameters the RD user must select. Depending on the value of the lower trigger level and the width of the triggering band, more or less decaying events were available to average for the computed loss factor. For a response record of fixed length, the following is true:

• Increasing the lower triggering level results in less triggers, but produces a better decay curves

(averaged response) • Increasing the width of the triggering band results in more triggers, and produces better decay curves.

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Figure 15. Three representative triggered responses (top 3 curves) and the average of 250 responses (bottom), showing the free decay signature.


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The data records used to extract the damping loss factor were created starting at a trigger event and—

ideally—span the length of time required for the response to decay to the noise floor. The length of these records, then, constitutes the third of five parameters the user must select. For a loss factor near 0.01, 100 cycles of response of a sinusoid at the 1/3 octave center frequency has been chosen, and it seems to work well. The derived loss factors from the first 100 cycles of 800 Hz, full octave data from the computational model with a 0.01 loss factor have been computed. The results show that the best results (derived loss factor nearest 0.01) occurred for rather high lower trigger levels (over 2√2σ) and rather big trigger band widths (over 25% of the lower trigger level).

Figure 16. Triggering Scheme Based on Definition of the Lower Trigger Level


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B. Slope calculation Figure 17 shows the averaged free decay signature of Figure 15 (bottom curve) on a logarithmic scale along with the magnitude of the Hilbert transform. The straight line is an “eye-balled” slope upon which a loss factor can be based. But, note that the choice of slope is a subjective one, unlike for a perfect single degree-of-freedom oscillator, for which the Hilbert transform actually describes a straight line. There is no unique slope, indicating some automated process to get the slope would be helpful.

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Time

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Envelope by HilbertLog Decrement

Figure 17. Free decay signature in the 2500 Hz frequency band.

The choice of the free decay signature for either real or simulated data has a number of possible

processes. Three were studied here, the difference being the point in the process at which the averaging takes place, i.e., before or after taking the log10 or the Hilbert Transform. Table 5.1 shows the processes used for these three cases.

Table 5.1 Description of alternative decay curve determination processes

LF1:-DB Decay from Avg Free Decay 1) Consider 16 free decays from each position 2) Compute the average of these free decays 3) Compute the magnitude of the Hilbert transform of the average decay 4) Use the dB decay slope to get the loss factor LF2:-Avg DB Decay of All Output Points 1) Consider 16 free decays from each position 2) Compute the magnitude of the Hilbert transform for each decay curve 3) Compute the decay curves in decibels 4) Average the dB decay slopes to get the loss factor LF3:-DB Decay for Avg Envelope of All Output Points 1) Consider 16 free decays from each position 2) Compute the magnitude of the Hilbert transform for each decay curve 3) Average the Hilbert decay curves 4) Use the dB decay to get the loss factor


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The averaged decay curves and resulting decay slopes representing the three averaging strategies

have been studied for an ensemble of 16 decays in the (full octave) 2000Hz band. Figure 18 shows the log magnitude of the Hilbert transform of the decay curve for all three averaging schemes. Figures 19, 20 and 21 show the LF1, LF2 and LF3 curves, respectively, with an automatically-calculated slope shown. In all of these cases, the slope-calculation algorithm was based on a least-squares fit to the local maxima in a portion of the decay curves. Note that the “early” maxima are ignored, as are the maxima past a certain point in time. The “start” and “end” times of the decay record, are identified in the discussion to follow in terms of the percentage of the selected record length. The record length is variable, depending on the level of damping, and must be adjusted by the user for each situation at hand.

Figure 18. Log Magnitude of the Hilbert transform


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Figure 19. Based on the LF1 Approach



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Based on the study of loss factor extractions for the 800, 1250, 2000 and 3150 Hz (full octave) bands, the LF1 & LF2 averaging schemes have shown the best results; still all three averaging schemes will be considered in the studies reported here. The study considered varying lengths of response records, 1000 cycles for η = 0.001, 100 cycles for η = 0.01 and 10 cycles for η = 0.1. Further, the number of cycles skipped at the beginning of the decay record to start the slope-fitting and the number of cycles until the end of the slope-fitting were varied. The following was found: • For the lower frequency bands, larger numbers of cycles of response produced better results, especially

for the lower loss factor cases, because the decay event lasts longer

• For the higher frequency bands, the algorithm under-predicts loss factor when more cycles of response are considered, presumably because of the influence of the noise floor

• The best number of cycles to skip at the beginning of the record is generally small, except for the lower

frequencies, where a skip of up to 20% of the record yields the best results

• The algorithm performs better in general for the lower loss factors, presumably because the decay lasts longer, and it can be observed for a longer period of time.

C. Filter selection For experimental studies in which response to random noise is used to derive the loss factor, the responses must be filtered into the desired frequency bands. This can be accomplished either with analog filters or by computation with a digital record of the response. The result, in either case, is a narrow-band response record which can be used to derive the loss factor.


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For studies with the computational model, there are two potential strategies. Since only the FRFs are available from the computational model, the time-domain responses can be generated by inverting the FRFs, then convolving the result with a random force, thereby creating a response of arbitrary length. This response record is then digitally filtered in the time domain to create the narrow band response records necessary for decay curve generation. Alternatively, the FRF could be “filtered” in the frequency domain by simply selecting octave-width bands of the FRF, followed by computation of the impulse response functions, which are convolved with a random force to produce band-limited responses. These are then used to extract the loss factor. 1. Time-domain filtering The time-domain filtering steps are:

• Get the FRF from the computational model • Generate the inverse of the FRF • Convolve the inverse of the FRF with a pseudo-random force to generate the response • Time-domain filter the response

In this study, various filtering schemes were studied, and finally the Kaiser window was selected. The basis of the filter selection scheme is the determination of the type and order of filter necessary to get a good measure of loss factor for a one-octave frequency band. A computational experiment was conducted based on the superposition of decaying sine wave signals wherein the signals represented a plausible combination of frequency content both within and just outside a full octave band.

To test full octave band pass filters centered at 100 Hz frequency, a composite signal generated by the addition of four decaying sine waves of 10, 50, 100 and 150 Hz was studied. Therefore the 10, 50 and 150 Hz signals are outside the filter bandwidth. The decay was consistent with a viscous damping coefficient of 0.01, i.e. a loss factor of 0.02. A single octave filter with a 100 Hz center frequency nominally passes signals with frequency content between 70.7 Hz and 141.4 Hz. However, the “roll-off” of the filter below 70.7 Hz and above 141 Hz will vary with the order of the filter. This roll-off is typically specified by the selection of the level of attenuation at selected frequencies outside the primary (in this case, octave) bandwidth. All of the filters used in this paper had the side-band attenuation specified at frequencies at 90% of the lower roll-off frequency and 110% of the upper roll-of frequency, in this case 63.6 and 155.6 Hz, respectively. Figure 22, below, shows the composite signal after filtering by Kaiser filters of 10th, 100th, 167th and 415th order, the first three of which had 6 dB roll-off attenuation, while the last had a 40 dB attenuation. The 167th and 415th order filters represent an optimal filter.

Note the group delay, which is especially prominent for the three higher order filters. For finite impulse response (FIR) filters, such as the Kaiser filter, a systematic time delay is imposed on the signal, which equals the group delay multiplied by the time between samples, dt. The group delay in terms of samples is equal to half of the filter order. Therefore, for a 100 Hz decaying sinusoid sampled at 1000 Hz, a time delay of approximately 0.2 seconds is expected (415/2/1000=0.2075). A delay of approximately 0.2 seconds is readily seen in Figure 22. The time delay for the 167th order filter is a bit less than half as long, and for the 100th order filter of about one fourth as long, or 0.05 seconds is also observed. The time delay for the 10th order filter is difficult to observe. At any rate, the presence of group delay does not affect the observance of decay rate, as it simply “time-shifts” the initiation of the decay.


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Figure 22. Variation of Response for different Orders of Filters

All of the filters used were generated with the MatLab Filter Design Assistant (FDA) tool. Figure 23 shows a “screen shot” of the FDA tool for an optimal filter with side-band “roll-off” of 50 dB over 10% of the bandwidth, that is, between 63.6 and 70.7 Hz and between 141.4 and 155.6 Hz. This Optimal Order (“Minimum Order” in FDA Tool parlance) filter is 415th order.

Figure 23. The FDA tool at a Center Frequency of 100 Hz (Full Octave Band )


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Figure 24, below, shows the effect of filter order on the decay curves after they have been subjected to the Hilbert transform and the decay of the magnitude of the Hilbert transform has been plotted in decibels.

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10th Order F ilter (6dB Atten)100th Order Filter (6dB Atten)415th Order Filter (50dB Atten: Min Order MATLAB)Orignal S ignal

Figure 24. Decay Curve Comparison for Different Order Filters.

From the figure, above, it is observed that there are two dominant slopes apparently available to determine the loss factor. The initial slopes for the filtered data appear to be the same, but are difficult to resolve unless a higher order filter is used. Each of the decaying sinusoids produces a distinct slope, since decay rate is linearly proportional to the frequency. Considering the in-band, 100 Hz signal, the lower-order filters do not sufficiently reject the out-of-band signals, each of which “contaminate” the decay rate for the 100 Hz sinusoid. Clearly, the in-band decay rate estimate is superior when a high order filter is used. The “gentler” slope for longer times in each curve is NOT appropriate for loss factor determination at in the 100 Hz band, as it represents the long-lived, “out of band” 10 Hz signal (with a slope exactly one tenth that of the 100 Hz signal). This signal’s coherence “lasts a long time.” Butterworth infinite impulse response (IIR) filters were also studied, but for higher frequency bands. For tight filtering in both full octave and one-third octave bands, the computation time increased substantially as the filter order increased. Hence, focus was shifted towards the use of FIR filters. The ease and better control over the FIR filters made them a preferred choice for rest of the study. A four-frequency comparison of loss factors from filters generated for frequencies centered on full octave and one-third octave bands was conducted. Figure 25 shows the loss factor estimates for 800, 1250, 2000 and 3200 Hz using both full and 1/3 octave filters and both a good (LF1) and a poorer (LF3) averaging scheme. It can be concluded that results are better with full octave frequency bands used for filter generation and the LF1 averaging scheme produce the best results. In all the subsequent studies, filters based on full octave frequency bands centered at one-third octave central frequencies are used.


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Loss Factor Comparison

0.001

0.01

0.1100 1000 10000

Frequency

Loss

Fac

tor

One-Third Octave LF1 One-Third Octave LF3 Full Octave LF1 Full Octave LF3

RD:

Figure 25. Effect of Filters Based on Full Octave and One-Third Octave Ban 2. Frequency-domain filtering Frequency-domain filtering is accomplished with the following steps: • Get the FRF from the computational model • Filter the FRF by selecting only the response in the selected frequency band • Generate the impulse response from the frequency-domain filtered FRF • Convolve the impulse response function with random force to generate the band-limited response

For frequency bands centered at very low frequencies, such as 20 Hz, the frequency-domain approach is better than the time-domain filtering approach, because convolution is faster than filtering. “Tight” lower frequency band filters have higher order, hence longer filtering times. But for higher frequency bands filtering is faster, since minimum filter order decreases with frequency. Meanwhile, the time consumed for the convolution for the frequency-domain approach is the same as for lower frequencies. Since most of the research deals with frequencies above 100 Hz, the time-domain filtering approach is adopted. D. Results The results obtained for time-domain filtering (TDF) & frequency-domain filtering (FDF) are compared in following figures for the LF1, LF2 & LF3 averaging schemes, all for a loss factor of 0.01. The slope estimation starts at 5% of the time required for 100 cycles of the center frequency, and ends at 50 % of the record length. These results are the best results of a study, documented in the Appendix, of the best start and stop percentages of the record length.


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RD: Loss Factor Comparison ( TDF & FDF; LF1; 5-50%)

0.001

0.01

0.1100 1000 10000

Frequency (Hz)

Loss

Fac

tor

TDF LF1 FDF LF1

Figure 26. TDF & FDF compared for LF1 averaging scheme.

RD: Loss Factor Comparison (TDF & FDF; LF2; 5-50%)

0.001

0.01

0.1100 1000 10000

Frequency (Hz)

Loss

Fac

tor

TDF LF2 FDF LF2


RD: Loss Factor Comparison (TDF & FDF; LF3; 5-50%)

0.001

0.01

0.1100 1000 10000

Frquency (Hz)

Loss

Fac

tor

TDF LF3 FDF LF3



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Figures 29 and 30, below are loss factor estimations for frequency-domain and time-domain filtering, respectively, for a loss factor of 0.01, comparing the best results for the three averaging schemes.

RD: FDF; LF=0.01; 100 Cycles (Start 5% End 50%)

0.001

0.01

0.1100 1000 10000

Frequency (Hz)

Loss

Fac

tor

LF1 (5-50) LF2 (5-50) LF3 (5-50)

Figure 29. FDF results for different averaging schemes and 0.01 loss factor.

RD:TDF; LF =0.01; 100 Cycles (Start 5% End 50%)

0.001

0.01

0.1100 1000 10000

Frequency (Hz)

Loss

Fac

tor

LF1 (5-50) LF2 (5-50) LF3 (5-50)

Figure 30. TDF results for different averaging schemes and 0.01 loss factor.

Based on these results, the LF2 and LF3 averaging schemes appear to be the best, in almost all cases substantially better than the LF1 scheme.

Figure 31 shows a comparison of loss factor estimations for a loss factor of 0.1 for the three averaging schemes. Note that for this high damping loss factor, the LF1 scheme is, once again, an inferior technique. Further, the LF2 technique seems to perform very slightly better that the LF3 technique.


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RD: TDF; LF =0.1; 10 Cycles (Start 5% End 80%)

0.01

0.1

1100 1000 10000

Frequency (Hz)

Loss

Fac

tor

LF1 (5-80) LF2 (5-80) LF3 (5-80)

Figure 31. TDF results for different averaging schemes and 0.1 loss factor.

VI. Comparison of IRDM and RD Techniques The loss factor comparison between the IRDM and RD techniques are shown below for both 0.01 and 0.1 loss factors. In Figure 32, for the lower loss factor, the IRDM technique produces clearly better results, but RD performs very well.

Figure 32. Comparison of loss factors estimated with IRDM and RD for 0.01 loss factor.


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In Figure 33, for the 0.1 loss factor, the RD technique performs almost as well as for the 0.01 loss factor, but the IRDM results are much worse.

Figure 33. Comparison of IRDM and Rd , for 0.1 loss factor VII. Conclusions The Impulse Response Decay Method and Random Decrement techniques for panel damping loss factor estimation have been evaluated using acceleration responses from a computational model over a range of loss factors. Essential agreement has been observed between the known values of the loss factor (specified in the computational model) and those “extracted” from the simulated accelerations using the IRDM for a loss factor of 0.01. At the much higher loss factor of 0.1, the IRDM technique systematically under-predicts the loss factor by as much as 60% (at 2000 Hz). However, the RD technique performs only slightly worse than for the lower loss factor.

Further study is needed to determine the effects of noise and to study the effect of exciting the panel at more than one location.


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Appendix

The loss factor estimation results for frequency-domain filtering in the 800 Hz band are shown in Figure A1, below. Each graph addresses a different averaging scheme (LF1, LF2 and LF3) for a loss factor of 0.01, as a function of the start and stop percentages used when averaging the decay signatures.

RD: FDF 800 Hz (LF=0.01; 100 Cycles; LF1)

0

0.005

0.01

0.015

30 40 50 60 70 80 90 100

End (%)

Loss

Fac

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Start 1% Start 5% Start 10% Start 15% Start 20%


00.0050.01

0.015

30 40 50 60 70 80 90 100End (%)

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Fac

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0

0.005

0.01

0.015

30 40 50 60 70 80 90 100

End (%)

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Fac

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Figure A1. FDF results for three averaging schemes and 0.01 loss factor. The following conclusions can be drawn: • Loss Factor with LF1 is best when start from 1-15 % and end at 60-90%. • Loss Factor with LF2 is best when start from 1-5 % and end at 50-70%. • Loss Factor with LF3 is best when start from 1-10 % and end at 50-70%.


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The loss factor estimation results for time domain filtering in the 800 Hz band are shown in Figure A2, A3 and A4, below. Each set of three graphs addresses a different averaging scheme (LF1, LF2 and LF3). Within each set of graphs, the estimations are given for the 0.01, 0.1 & 0.001 loss factors, as a function of the start and stop percentages used when averaging the decay signatures.

RD: TDF 800 Hz (LF=0.1; 10 Cycle; LF1)

00.050.1

0.150.2

30 40 50 60 70 80 90 100

End (%)

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RD: TDF 800 Hz (LF=0.01; 100 Cycles; LF1)

00.0050.01

0.0150.02

30 40 50 60 70 80 90 100

End (%)

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800 Hz (Lf=0.001; 1000 Cycles; LF1)

00.00050.001

0.00150.002

30 40 50 60 70 80 90 100

End (%)

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Fac

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Figure A2. TDF results for three averaging schemes and 0.001 loss factor.


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RD: TDF RD: TDF 800 Hz (LF=0.1; 10 Cycle; LF2)

00.050.1

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30 40 50 60 70 80 90 100

End (%)

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RD: TDF 800 Hz (LF=0.01; 100 Cycle; LF2)

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800 Hz (LF=0.001; 1000 Cycle; LF2)

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RD: TDF RD: TDF 800 Hz (LF=0.1; 10 Cycle; LF3)

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End (%)

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00.0050.01

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Acknowledgements The authors acknowledge the technical assistance of Mark Moeller and Albert Allen of Spirit Aero Systems. The work reported here has been accomplished as an element of a research grant provided by Spirit Aero Systems to the University of Kansas.

References

1 Bloss, B. C. and Rao, M. D., “Estimation of frequency-averaged loss factors by the power injection and the impulse response decay methods”, Journal of the Acoustical Society of America, Vol. 117, Issue 1, January 2005. 2 Wu, L., Agren, A., and Sundback, U., “A study of the initial decay rate of two-dimensional vibrating structures in relation to estimates of loss factor”, Journal of Sound and Vibration, Vol. 206, Issue 5 (1997), pp 663-684. 3 Liu, W. and Ewing, M.S., “Predicting Damping Loss Factors for Beams and Plates with Constrained Layer Damping”, Proceedings, 48th AIAA Structures, Structural Dynamics and Materials Conference, April 2008, Shaumberg, IL. 4 Lyon, R.H., and DeJong, R.G., Theory and Application of Statistical Energy Analysis, 2nd Edition, R H Lyon Group, 1998.


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