Damping identification from non-linear random responses using a multi-triggering random decrement technique

Mechanical Systems and Signal Processing (1987) l(4), 389-397

DAMPING

IDENTIFICATION

RESPONSES

USING

FROM

NON-LINEAR

A MULTI-TRIGGERING

DECREMENT

RANDOM

RANDOM

TECHNIQUE

SAMIR R. IBRAHIM Department of Mechanical Engineering and Mechanics, Old Dominion University, Norfolk, Virginia 23508, U.S.A.

K.R.

WENTZ

AND J.LEE

Flight Dynamics Laboratory, Air Force Wright Aeronautical Laboratories, Dayton, Ohio 45433, U.S.A. (Received January 1987, accepted April 1987)

The concept of multi-triggering random decrement technique is introduced. Like the single triggering technique, it reduces multi-mode multi-measurement stationary random responses to free decay responses but has the advantage of increasing the apparent number of the resulting free decay time response functions. The maximum number of these free decay responses is equal to the square of the number of random measurements. These free decay responses are then used in a linear time domain modal identification algorithm to extract frequencies, damping factors and mode shapes of a structure under test. The quasi-linear modal approach is used to deal with non-linearities by repeating the linear identification process at different levels of inputs/responses. The procedure is applied to rectangular panels subjected to acoustic random input ranging from 130 to 157 dB. The changes in frequencies and damping factors with input level are reported. This application is part of a sonic fatigue research program.

1.

INTRODUCTION

Modal identification of structures, to experimentally determine natural frequencies, mode shapes and damping factors, is critical to the accurate mathematical modeling of structures. These extracted modal parameters, besides being directly related to the structure’s physical parameters, are very useful in applications such as response and loads predictions, trouble shooting excessive vibrations, stability and control, verification and or modification of analytical dynamic models, structural integrity monitoring and incipient failure detection, fatigue design, among others. While undamped natural frequencies and normal mode shapes can be approximately estimated from analytical models, damping characteristics of structures are determined mainly from experimentation. Even with the tremendous advances in the state-of-the-art of modal identification in both frequency and time domains, the accuracy of damping identification, unlike frequencies and mode shapes, is an area of active current research efforts. Damping identification accuracy becomes more critical in applications such as incipient failure detection and structural integrity monitoring, flutter margin predictions and fatigue design. To further complicate the problem of damping measurements, non-linearities can cause unacceptable scatter of “error” bounds. Actually, the damping level may drastically vary with input and or response levels; a situation where the concept of a unique modal damping value can be erroneous and misleading. 389 0888-3270/87/040389+09

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@ 1987 Academic Press Limited

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Another application where damping, as well as modal, identification becomes more difficult is when a structure has to be tested under normal operating conditions with unmeasurable or unknown random inputs. The two main issues in this study to identify damping are: 1. The structure under consideration is highly non-linear and 2. The input to the structure is unmeasurable or unknown stationary random force(s). The approach under consideration is to deal with non-linearities by using a quasi-linear identification technique and with unmeasurable random input(s) by using the random decrement method. A linear time domain modal identification technique, which implements the free decay responses of the structure under test, is used for the identification process which is performed at different levels of input to study the variation of modal parameters with the input level. The free decay time response functions are obtained from the random responses by using the random decrement technique [2-41. The concept of multi-triggering in random decrement computation is introduced in this paper and promises to be advantageous for improving identification accuracy when the number of measurements on the structure is limited. The main thrust of the paper is to introduce this new concept of multi-triggering in random decrement computation. Even though it is not quantitatively shown here, such an approach promises to increase the identification accuracy when the resulting random decay response functions are used in time domain identification. The reason is that the additional free decay response functions, resulting from the multi-triggering process, reduce the need to use pseudo-measurements, which are measurements shifted in time. Since these are decaying responses where the noise to signal ratio increases with time, using more measurements and less pseudo-measurements reduces the overall noise to signal ratio in the identification model. Thus it is expected to result in higher identification accuracy. Comparative and statistical studies of the identification error analysis are required to confirm the characteristics of multi-triggering, as compared to single triggering, in improving identification accuracy. However, and even though relevant, such a study is outside the scope of this paper and is to be reported in future work. The entire approach is applied to strain and acceleration measurements of a rectangular panel subjected to acoustical input of varying level. 2. BACKGROUND 2.1.

TIME

DOMAlN

MODAL

IDENTIFICATION

linear time domain modal identification technique [l] used in this study uses the free decay responses of a structure and converts the identification procedure to solving the eigenvalue problem The

(1) where the matrix of eigenvalues and eigenvectors, [A], is computed from the measured free decay time response functions of the structure. The eigenvalues cz are directly related to the characteristic roots of the system from which frequencies and damping factors are determined. The eigenvectors + relate to, or are, the mode shapes at the measured co-ordinates. 2.2.

RANDOM

DECREMENT

TECHNIQUE

The concept of the random decrement technique was initially presented by Cole [2]. The presentation or proof of the theoretical basis was imperical and intuitive but nevertheless, acceptable. The technique, when first presented, was designed to obtain a single mode signature of one single measurement.

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Because of its simplicity and effectiveness, randomdec has subsequently gained widespread use by the aerospace industry. Randomdec was later generalised to multi-measurements multi-mode responses [3], not to generate signature but rather to produce free decay response time functions that can be used in time domain modal identification of structures. An excellent paper on randomdec by Vandiver et al. [4] offers a more rigorous proof on the theory of randomdec, analyses errors and residuals and compared different methods of triggering ensembles. Since the introduction of the concept of random decrement technique, efforts have been directed to investigating the theoretical grounds for accuracy and convergence and any inherent assumptions, conditions, or limitations. One main assumption in these theoretical proofs was that the random input force(s) must be white noise. The assumption of white noise input(s) is very convenient for theoretical proofs. The fact is, such white noise rarely exists in real life applications. The strict condition of having white noise, or flat power spectrum density, it not necessary, at least for the purpose of modal identification of structures. The only condition on the input random force(s) is that it is of zero mean, stationary random [S]. Considering a multi-degree-of-freedom linear system with [Ml, [K], and [C] as mass, stiffness and damping matrices and {y} as the response vector to a random force input vector {f), such a system is governed by the equation

(2) The above equation is valid for any time t and any set of initial conditions. Replacing time t with the expression tr + T where t,‘s are selected according to the method triggering the start of ensembles for the randomdec computations, then equation (1) can be written as [M]{j;(ti+T)}+[C]~(ti+7)+[K]{~(ti+7)=Cf(ti+T)}(i=1,2,~~~~N)

(3)

where N is the number of averages intended for use in randomdec computation. By summing all of the N equations, dividing by N and replacing l/N C {y( t, + T)} by {X(T)}, the resulting summation can be written as

[MI{~(T)}+[CII~(T)}+[~I{X(T)I=~

f!, {f(ti+T))*

(4)

Noting that since ti’s were selected according to a specific randomdec triggering criterion, the resulting response {X(T)} will not average to zero. Now considering the right-hand side of equation (4), if {f} is a stationary random signal, then

and equation (4) will be [M]{jr(T)}+[C]{a(T)}+[K]{X(7))

=o.

(5)

Equation (4) implies that {X(T)} is a free-decay response that resulted from applying randomdec to random responses due to a force input vector, {f}. where {f} is assumed to be only some stationary random signals with zero mean. Thus if {y( r)} are the p measured random responses due to a stationary random input(s), then {X(T)} are the randomdec, or free decay, time response functions where

Xj(T)=$ ki,Yj(rk+T),

j=l,2

, . . . p.

S. R.

392

IBRAHIM

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AL

In equation (6) the times, ti are determined according to an arbitrary measurement, a leading station, y,( 1) satisfying a selected triggering criteria; t = t, when y,(r) satisfies one of the following: (a) constant level; (b) zero crossings and positive slopes; (c) zero crossings and negative slopes; (d) positive; (e) negative.

3. MULTI-TRIGGERING

RANDOM

DECREMENT

The leading, triggering, measurement of equation (6) is totally arbitrary as long as all measurements are simultaneously recorded. For that reason yl can be any one or every one of the measurements. In such a case equation (6) becomes

j=l,2

,...

p

and

1=1,2 ,...

(7)

p

where fk is determined when measurement /, I= 1,2,. . . p, satisfies the specified triggering criterion. It is obvious that equation (6) yields p free decay time response functions as compared to p2 for equation (7). Considering that the size of identification model of equation (1) is usually larger than the number of measurements and or the number of modes in the responses, obtaining more randomdec functions becomes advantageous. It increases the apparent number of measurements thus reduces the need to use more pseudo measurements (time delayed measurements). The delayed measurements, since they are free decay responses, contain higher noise to signal ratio than the original measurements. This multi-triggering random decrement approach becomes increasingly effective when the number of measurements is limited especially with higher modal density and or higher noise levels. The approach has the advantages of 1. Increasing the apparent number of measurements which is likely to improve the identification accuracy. 2. When automated decreasing the effect of selecting one bad measurement as the leading measurement. 4. EXPERIMENTS

AND

RESULTS

The test structure in this series of experiments is a rectangular aluminium plate 25.4 x 50.8 x 0.16 cm. The panel is tested with all of its four edges clamped to the wall of the wideband noise test facility of the Sonic Fatigue Group at the Air Force Wright Aeronautical Laboratories. A sketch of the panel with instrumentation is shown in Fig. 1. The panel in test configuration is shown in Fig. 2 while Fig. 3 shows the horne system ACM

..-

.-

Figure

1. Schematic

sketch of panel with instrumentation.

DAMPING

IDENTIFICATION

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Figure 2. Panel in test configuration.

Figure

3. Air modulation

system.

RANDOM

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IBRAHIM

t-7 AL.

for air modulation. The input level was changed from 130 to 160 dB in steps of 3 dB. Figure 4 shows a typical input spectrum. For this study only five sensors were used. These are strain gauges 1, 2, 3 and 8 and accelerometer 4. Four levels of input were used with rms of 130, 139, 148 and 157 dB. Three seconds of random data were sampled at 4000 Hz. The accelerometer fell off at 154 dB input, thus only the four strain gauge responses were used for the 157 dB input level. Typical output spectra for the four different levels are shown in Fig. 5(a)-(d). These spectra indicate the non-linearity as well as the difficulties associated with any attempts to measure modal damping especially at higher levels of input.

0

20

40

60

80

10 0

Figure

4. Input spectra.

0

20

40

60

80

100

Frequency (Hz X 10’)

Frequency (Hz X 10’)

(a) Input

130 dB; (b) input

157 dB.

To prepare the random responses for use in the time domain identification algorithm, the multi-triggering randomdec technique was used to convert them to free decay response time functions. By using five random responses, 25 free decay time functions were computed (16 for the 157 dB input, no accelerometer). The record length of the randomdec functions was selected to be 0.25 set (1000 samples) and the number of averages was limited to 500 using all positive points triggering. Thus the actual random record length used is about 2000 samples (0.5 set). Figure 6(a)-(d) shows the random and randomdec functions (1000 samples each) for the five measurements. The resulting free decay responses were used in the time domain identification with model sizes of 25, 25, 35 and 50 modes for the four levels of inputs. Higher number of modes for higher input levels were needed in order to deal with increasing non-linearities. Identification results indicated high modal densities of the system. Several of the identified modes were those of the acoustic chamber and suspension wall. To correlate the same identified modes at different levels to each other, neither frequencies nor damping factors can be used since they vary with input levels. For this purpose each mode shape at one input level was correlated to all mode shapes at each different input level. High correlation was used to indicate the same mode at different input levels. Figure 7 shows the changes in the frequencies and damping factors of two modes denoted as modes m and n. Since the purpose of this paper is to only suggest a procedure, and not to study mechanisms of non-linearities, the authors at this stage have no comments on the pattern of frequencies and damping changes with input level.

DAMPING

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NON-LINEAR 102

RANDOM

395

RESPONSES

: 1 (b)

I IU

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0

20’

40

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IU -‘o

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1’ 0

Frequency (Hz X 10’)

Frequency (Hz X 10’) 1021 (d)

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Frequency (Hz X 10’) Figure 5. Output

spectra. (a) Input

20

40

60

60

100

Frequency (Hz X 10’)

130 dB; (b) input 139 dB; (c) 148 dB; (d) input 157 dB.

5. CONCLUSION The concept of multi-triggering in random decrement computation is introduced. By increasing the apparent number of free decay response measurements, up to the square of the number of random measurements used, this approach is likely to increase the modal identification acccuracy of time domain techniques. The use of a linear identification technique to identify, in a quasi-linear sense, non-linear system is shown to work successfully with highly non-linear structures as indicated by the reported experiment of a panel subjected to a random acoustic excitation with varying level. Further studies are to be conducted to investigate the effects of multi-triggering on the identification accuracy.

ACKNOWLEDGEMENT

This work is part of a research contract from Wright Patterson Laboratory, K. R. Wentz is the technical monitor.

Flight Dynamics

396

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ET Al

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Figure 6. Random and randomdec 148 dB; (d) input 157 dB.

time response

functions.

(a) Input

130dB;

(b) input

139dB;

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7. Changes

I 130

in identified

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frequencies

I

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and damping

factors

- 100

with input level.

(c) input

DAMPING

IDENTIFICATION

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RANDOM

RESPONSES

397

REFERENCES 1. S. R. Ibrahim 1984 AIAA Journal, 22, 817-823. Time domain quasi linear identification of nonlinear dynamic systems. 2. H. A. Cole, Jr. 1973 NASA CR 2205. On-line failure detection and damping measurement of aerospace structures by random decrement signature. 3. S. R. Ibrahim 1977 Journal ofSpacecraft and Rockets 14,696-700. Random decrement technique for modal identification of structures. 4. J. K. Vandiver, A. B. Dunwoody and R. B. Campbell 1982 Journal Mechanical Design 104, 307-313. A mathematical basis or the random decrement signature analysis technique. 5. S. R. Ibrahim 1984 ASME Book No. G00255, pp. 69-81. Incipient failure detection from random decrement time functions.