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TECHNICAL PAPERS

A Stochastic Model for the Random Impact Series Method in Modal Testing

[+] Author and Article Information
W. D. Zhu1

Department of Mechanical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250

N. A. Zheng

Department of Mechanical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250

C. N. Wong

Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409

1

Corresponding author.

J. Vib. Acoust 129(3), 265-275 (Oct 25, 2006) (11 pages) doi:10.1115/1.2730529 History: Received January 13, 2006; Revised October 25, 2006

A novel stochastic model is developed to describe a random series of impacts in modal testing that can be performed manually or by using a specially designed random impact device. The number of the force pulses, representing the impacts, is modeled as a Poisson process with stationary increments. The force pulses are assumed to have an arbitrary, deterministic shape function, and random amplitudes and arrival times. The force signal in a finite time interval is shown to consist of a wide-sense stationary part and two nonstationary parts. The expectation of the force spectrum is obtained from two approaches. The expectations of the average power densities associated with the entire force signal and the stationary part of it are determined and compared. The analytical expressions are validated by numerical solutions for two different types of shape functions. A numerical example is given to illustrate the advantages of the random impact series over a single impact and an impact series with deterministic arrival times of the pulses in estimating the frequency response function. The model developed can be used to describe a random series of pulses in other applications.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

The amplitude or the averaged amplitude of the force spectrum with (a) τi=i, and ψi=1N (dotted line) or normally distributed random ψi (solid line); (b) normally distributed τi around i, and ψi=1N (solid line) or normally distributed random ψi (dotted line); and (c) the frequency of the impacts increasing uniformly from 7Hzto13Hz, and ψi=1N (dotted line) or normally distributed random ψi (solid line). In (a) and (b), N=20, and in (c), N=200; 100 samples are averaged for the cases with random variables.

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Figure 2

Schematic of a random series of pulses with the same deterministic shape and random amplitudes and arrival times. The last pulse shown arrives at time τN(T) and ends at a time between T and T+Δτ.

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Figure 3

The domain of integration for the double integral in Eq. 50 shown by the shaded area. The order of integration is changed from integrating with respect to k first to integrating with respect to t1 first.

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Figure 4

The averaged, normalized shape function of the force pulses from the random impact test on a four-bay space frame; Δτ=0.15625s

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Figure 5

Analytical (solid line) and numerical (dotted line) solutions of E[x(t)] with the shape function in Fig. 4, T=8s, λ=4.14∕s, E[ψ1]=0.8239N, and E[ψ12]=0.7163N2

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Figure 6

Analytical (dotted line) and numerical (solid line) solutions of 20log∣E[X(jω)]∣ with the same shape function and parameters as those for Fig. 5

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Figure 7

Analytical (solid line) and numerical (dotted line) solutions of 10log{E[S1(jω)]} with the same shape function and parameters as those for Fig. 5. The analytical solution of 10log{E[S1(jω)]} with λ=1∕s and the shape function and other parameters unchanged is shown as a dashed line.

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Figure 8

Analytical solutions of 10log{E[S1(jω)]} (solid line) and 10log{E[S2(jω)]} (dotted line) with the same shape function and parameters as those for Fig. 5. The solid and dotted lines are virtually indistinguishable.

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Figure 9

Analytical (solid line) and numerical (dotted line) solutions of E[x(t)] with the shape function in Eq. 69, T=8s, Δτ=0.15625s, λ=4.14∕s, E[ψ1]=0.8239N, and E[ψ12]=0.7163N2

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Figure 10

Analytical (dotted line) and numerical (solid line) solutions of 20log∣E[X(jω)]∣ with the same shape function and parameters as those for Fig. 9

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Figure 11

Analytical (solid line) and numerical (dotted line) solutions of 10log{E[S1(jω)]} with the same shape function and parameters as those for Fig. 9. The solid and dotted lines are virtually indistinguishable.

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Figure 12

The true magnitude function ∣G(jω)∣2 (solid line) and the estimated magnitude function ∣Gm(jω)∣2 using a single impact (dashed-dotted line), a random impact series with T=16s (dotted line), and a random impact series with T=11.39s (dashed line)

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Figure 13

The true magnitude function ∣G(jω)∣2 (solid line) and the estimated magnitude function ∣Gm(jω)∣2 using a random impact series with T=11.39s (dashed line) and an impact series with deterministic arrival times of the pulses and T=11.39s (dotted line)

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