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Technical Briefs

Modal Representation of Second Order Flexible Structures With Damped Boundaries

[+] Author and Article Information
Lea Sirota

e-mail: lsirota@technion.ac.il

Yoram Halevi

Faculty of Mechanical Engineering,
Technion-Israel Institute of Technology,
Haifa 32000, Israel

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 30, 2011; final manuscript received July 2, 2013; published online September 12, 2013. Assoc. Editor: Bogdan Epureanu.

J. Vib. Acoust 135(6), 064508 (Sep 12, 2013) (6 pages) Paper No: VIB-11-1290; doi: 10.1115/1.4025161 History: Received November 30, 2011; Revised July 02, 2013

The problem of obtaining a modal (i.e., infinite series) solution of second order flexible structures with viscous damping boundary conditions is considered. In conservative boundary systems, separation of variables is well established and there exist closed form modal solutions. However, no counterpart results exist for the damped boundary case and previous publications fall short of providing a complete solution for the series, in particular, its coefficients. The paper presents the free response of damped boundary structures to general initial conditions in the form of an infinite sum of products of spatial and time functions. The problem is attended via Laplace domain approach, and explicit expressions for the series components and coefficients are derived. The modal approach is useful in finite dimension modeling, since it provides a convenient framework for truncation. It is shown via examples that often few modes suffice for approximation with good accuracy.

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References

Figures

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Fig. 1

A schematic diagram of a damped-damped rod

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Fig. 2

The real (top) and imaginary (bottom) parts of the mode shapes H1 (solid) and H2 (dashed)

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Fig. 3

The response of the system, a modal approximation with two modes, and an FE model with two elements to g(x) = cos(πx/L)

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Fig. 4

The response of the system, a modal approximation with two modes, and an FE model with two elements to g(x) = xcos(πx/L)

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Fig. 5

The response of the system, a modal approximation with four modes, and an FE model with four elements to g(x) = xcos(πx/L)

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Fig. 6

The response of the system, a modal approximation with two modes, and an FE model with four elements to v(x) = sin(πx/L)

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