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Research Papers

Experimental Identification of Generalized Proportional Viscous Damping Matrix

[+] Author and Article Information
S. Adhikari1

Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UKs.adhikari@bristol.ac.uk

A. Srikantha Phani

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UKskpa2@eng.cam.ac.uk

1

Corresponding author.

J. Vib. Acoust 131(1), 011008 (Jan 06, 2009) (12 pages) doi:10.1115/1.2980400 History: Received July 27, 2006; Revised February 28, 2007; Published January 06, 2009

A simple and easy-to-implement algorithm to identify a generalized proportional viscous damping matrix is developed in this work. The chief advantage of the proposed technique is that only a single drive-point frequency response function (FRF) measurement is needed. Such FRFs are routinely measured using the standard techniques of an experimental modal analysis, such as impulse test. The practical utility of the proposed identification scheme is illustrated on three representative structures: (1) a free-free beam in flexural vibration, (2) a quasiperiodic three-cantilever structure made by inserting slots in a plate in out-of-plane flexural vibration, and (3) a point-coupled-beam system. The finite element method is used to obtain the mass and stiffness matrices for each system, and the damping matrix is fitted to a measured variation of the damping (modal damping factors) with the natural frequency of vibration. The fitted viscous damping matrix does accommodate for any smooth variation of damping with frequency, as opposed to the conventional proportional damping matrix. It is concluded that a more generalized viscous damping matrix, allowing for a smooth variation of damping as a function of frequency, can be accommodated within the framework of standard finite element modeling and vibration analysis of linear systems.

Copyright © 2009 by American Society of Mechanical Engineers
Topics: Damping
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Figures

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Figure 7

Typical measured FRF on the three-cantilever system. Coherence is also shown on the same plot. Each pass band and flexural modes are labeled. Note that the peaks are clearly visible; hence modal identification can be performed with ease.

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Figure 8

Mode shapes corresponding to the three modes in the second pass band. It can be noticed that each of the cantilever beams deforms in its second mode in this band. Also notice that the second beam does not deform at all in the second mode of the pass band; i.e., it is a node. (a) Mode 1, (1, 1, 1); (b) Mode 2, (1, 0, –1); (c) Mode 3, (1, –2, 1).

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Figure 9

Mode shapes corresponding to the three modes in the third pass band. It can be noticed that each of the cantilever beams deforms in its third mode in this band. Also notice that the second beam does not deform at all in the second mode of the pass band; i.e., it is a node. (a) Mode 1, (1, 1, 1); (b) Mode 2, (1, 0, –1); (c) Mode 3, (1, –2, 1).

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Figure 10

Schematic representation of the finite element mesh of the clamped plate with slots shown in Fig. 6

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Figure 11

Modal damping factors and fitted generalized proportional damping function for the first nine modes of the clamped plate with slots

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Figure 12

Comparison of modal damping factors using different proportional damping matrix identification methods for the clamped plate with slots

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Figure 13

Experimental setup for the point coupled beams. (Courtesy to Cambridge University Engineering Department for allowing using this figure.)

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Figure 14

Schematic representation of the finite element mesh of the of the coupled beam system shown in Fig. 1

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Figure 15

Modal damping factors and fitted generalized proportional damping function for the coupled beam system. (a) Top beam; (b) bottom beam; (c) coupled system.

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Figure 1

Schematic representation of the experimental setup of the free-free beam

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Figure 2

Schematic representation of the finite element mesh of the of the free-free beam shown in Fig. 1

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Figure 3

Modal damping factors and fitted generalized proportional damping function for the first 11 modes

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Figure 4

Comparison of modal damping factors using different proportional damping matrix identification methods

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Figure 5

Geometric parameters of the plate with slots: B=50mm, L=400mm, S=10mm, and W=20mm. The source of damping in this test structure is the wedged foam between beams 1 and 2.

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Figure 6

Schematic representation of the experimental setup of the clamped plate with slots

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