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Research Papers

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

[+] Author and Article Information
Philip D. Cha

Professor
Department of Engineering,
Harvey Mudd College,
301 Platt Boulevard,
Claremont, CA 91711
e-mail: philip_cha@hmc.edu

Siyi Hu

Department of Engineering,
Harvey Mudd College,
301 Platt Boulevard,
Claremont, CA 91711

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 2, 2016; final manuscript received November 29, 2016; published online March 16, 2017. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 139(3), 031005 (Mar 16, 2017) (15 pages) Paper No: VIB-16-1436; doi: 10.1115/1.4035382 History: Received September 02, 2016; Revised November 29, 2016

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

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Figures

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Fig. 1

Arbitrarily supported rod carrying various lumped translational elements

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Fig. 2

Arbitrarily supported beam carrying various translational and torsional lumped elements

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Fig. 4

The eigenfunction of a fixed–fixed rod evaluated at xa=3/8 squared

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Fig. 3

The eigenfunction of a fixed–fixed rod evaluated at xa=1/5 squared

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Fig. 5

The eigenfunction of a fixed-free rod evaluated at xa=4/7 squared

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Fig. 6

The slope of the eigenfunction of a simply supported beam evaluated at xa=2/7 squared over the natural frequency of the simply supported beam

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Fig. 7

The slope of the eigenfunction of a cantilever beam evaluated at xa=1 squared over the natural frequency of the cantilever beam

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Fig. 8

The product of the eigenfunction and the slope of a cantilever beam evaluated at xi=xj=1 over the square root of the natural frequency of the cantilever beam

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Fig. 9

ω4 versus n′ for Example 7

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