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TECHNICAL PAPERS

Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach

[+] Author and Article Information
B. Kang

Department of Mechanical Engineering, Indiana University – Purdue University Fort Wayne, Fort Wayne, Indiana 46805-1499kang@engr.ipfw.edu

J. Vib. Acoust 129(2), 193-201 (Sep 29, 2006) (9 pages) doi:10.1115/1.2424972 History: Received July 18, 2005; Revised September 29, 2006

An alternative analysis technique, which does not require eigensolutions as a priori, for the dynamic response solutions, in terms of the transfer function, of one-dimensional distributed parameter systems with arbitrary supporting conditions, is presented. The technique is based on the fact that the dynamic displacement of any point in a waveguide can be determined by superimposing the amplitudes of the wave components traveling along the waveguide, where the wave numbers of the constituent waves are defined in the Laplace domain instead of the frequency domain. The spatial amplitude variations of individual waves are represented by the field transfer matrix and the distortions of the wave amplitudes at point discontinuities due to constraints or boundaries are described by the wave reflection and transmission matrices. Combining these matrices in a progressive manner along the waveguide using the concepts of generalized wave reflection and transmission matrices leads to the exact transfer function of a complex distributed parameter system subjected to an externally applied force. The transient response solution can be obtained through the Laplace inversion using the fixed Talbot method. The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the system’s transfer function. This wave-based analysis technique is applicable to any one-dimensional viscoelastic structure (strings, axial rods, torsional bar, and beams), in particular systems with multiple point discontinuities such as viscoelastic supports, attached mass, and geometric/material property changes. In this paper, the proposed approach is applied to the flexural vibration analysis of a classical Euler–Bernoulli beam with multiple spans to demonstrate its systematic and recursive formulation technique.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Flexural waves traveling along a multi-span beam with arbitrary intermediate point supports and boundary conditions. The subscripts il and ir denote the left and right sides of station i, respectively, and ξi denotes the local coordinate in span i.

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Figure 2

Flexural wave motion in a multi-span beam due to a point excitation force applied at x=x0 (Station 1). Waves traveling away from x=x0 are measured positive.

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Figure 3

The first four transverse modeshapes of the uniformly damped nine-span beam; first mode (◯), second mode (◻), third mode (●), and fourth mode (∎). The markers also indicate the locations of the intermediate constraints. Refer to Table 1 for the numerical values of the system parameters.

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Figure 4

Transient responses of the three-span beam due to the impulse applied at x=0.4, where M=64 is used: (a) schematic of the system; (b) time response at x=0.15; and (c) time response at x=0.825. The solid (——) and dashed curves (- - -) represent the results from the present analysis and the method of normal mode expansion, respectively. Refer to Table 2 for the numerical values of the system parameters.

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Figure 5

Transient responses of the damped six-span beam due to the impulse applied at x=0.39, where M=64 is used: (a) schematic of the system; (b) time response at x=0.06; (c) time response at x=0.92; and (d) flexural waves developed due to the impulse. Refer to Table 3 for the numerical values of the system parameters.

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Figure 6

Frequency response of the damped six-span beam due to harmonic excitation applied at x=0.39: (a) schematic of the system; (b) amplitudes sampled at x=0.325 (solid curve, ——); and at x=0.92 (dashed curve, - - -); and (c) corresponding phase angles in radian. Refer to Table 4 for the numerical values of the system parameters.

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Figure 7

Unit impulse response of the single-span taut string near the point of application at t=0.001. The thick solid (—— ) and dashed (- - -) curves represent the results from the present analysis with M=64 and M=32, respectively; the thin dashed-dot-dot (-∙∙-), dashed-dot (-∙-∙), dashed (- - -), and solid (——) curves represent the results from the method of normal mode expansion with N=2500, N=5000, N=10,000, and N=20,000, respectively.

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