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Research Papers

Transmissibility of Bending Vibration of an Elastic Beam

[+] Author and Article Information
Hu Ding

Shanghai Institute of Applied
Mathematics and Mechanics;
Shanghai Key Laboratory of Mechanics
in Energy Engineering,
Shanghai University,
Shanghai 200072, China
e-mail: dinghu3@shu.edu.cn

Earl H. Dowell

Pratt School of Engineering,
Duke University,
Durham, NC 27708

Li-Qun Chen

Shanghai Institute of Applied
Mathematics and Mechanics;
Shanghai Key Laboratory of Mechanics
in Energy Engineering,
Shanghai University,
Shanghai 200072, China;
Department of Mechanics,
Shanghai University,
Shanghai 200444, China

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 18, 2017; final manuscript received October 25, 2017; published online January 25, 2018. Assoc. Editor: Alper Erturk.

J. Vib. Acoust 140(3), 031007 (Jan 25, 2018) (13 pages) Paper No: VIB-17-1268; doi: 10.1115/1.4038733 History: Received June 18, 2017; Revised October 25, 2017

This paper proposes an isolation transmissibility for the bending vibration of elastic beams. At both ends, the elastic beam is considered with vertical spring support and free to rotate. The geometric nonlinearity is considered. In order to implement the Galerkin method, the natural modes and frequencies of the bending vibration of the beam are analyzed. In addition, for the first time, the elastic continuum supported by boundary springs is solved by direct numerical method, such as the finite difference method (FDM). Moreover, the detailed procedure of FDM processing boundary conditions and initial conditions is presented. Two numerical approaches are compared to illustrate the correctness of the results. By demonstrating the significant impact, the necessity of elastic support at the boundaries to the vibration isolation of elastic continua is explained. Compared with the vibration transmission with one-term Galerkin truncation, it is proved that it is necessary to consider the high-order bending vibration modes when studying the force transmission of the elastic continua. Furthermore, the numerical examples illustrate that the influences of the system parameters on the bending vibration isolation. This study opens up the research on the vibration isolation of elastic continua, which is of profound significance to the analysis and design of vibration isolation for a wide range of practical engineering applications.

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Figures

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Fig. 1

A uniform elastic beam with vertical spring support

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Fig. 2

The natural mode shapes of the linear elastic beam with vertical spring support: (a) the first-order mode, (b) the second-order mode, (c) the third-order mode, and (d) the fourth-order mode

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Fig. 3

The natural frequencies of the linear beam versus the stiffness of vertical support springs: (a) the first-order frequency, (b) the second-order frequency, (c) the third-order frequency, and (d) the fourth-order frequency

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Fig. 4

The time histories of the boundary support spring forces and excitation force: the Galerkin truncate calculations: (a) symmetric support: ωb = ω1, (b) asymmetric support: ωb = ω1, and (c) asymmetric support: ωb = ω2

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Fig. 5

The stable steady-state amplitude of three different locations along the elastic beam versus the excitation frequency: the Galerkin truncate calculations: (a) the first-order resonance and (b) the third-order resonance

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Fig. 6

The steady-state amplitude of the elastic beam versus the excitation frequency: the FDM: (a) the first-order resonance and (b) the third-order resonance

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Fig. 7

Comparisons between four-term Galerkin truncation and the FDM: the stable steady-state response amplitude: (a) the middle point and (b) the spring support end

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Fig. 8

Effects of nonlinearity on the first three order primary resonances: (a) the stable steady-state response and (b) the vibration transmissibility

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Fig. 9

Effects of nonlinearity on the third-order primary resonance: (a) the stable steady-state response: X = L, (b) the stable steady-state response: X = L/2, (c) the stable steady-state response: X = 0, and (d) the vibration transmissibility

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Fig. 10

Effects of the stiffness of the support springs on the vibration transmissibility: (a) elastic spring support and rigid support and (b) various boundary spring stiffness

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Fig. 11

Comparisons of linear and nonlinear responses around the first natural frequency of the beam for various boundary spring stiffness: (a) the stable steady-state response and (b) the vibration transmissibility

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Fig. 12

Comparisons between one-term and four-term Galerkin truncation: (a) the stable steady-state response and (b) the vibration transmissibility

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Fig. 13

Comparisons between the downward force and the composite force transmissibility: (a) transmissibility and (b) time histories: ωb = 66.845

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Fig. 14

Effects of the stiffness of the right side support spring on the vibration transmissibility

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Fig. 15

Effects of the external damping of the system on the stable steady-state response and the vibration transmissibility: (a) the stable steady-state response and (b) the vibration transmissibility

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Fig. 16

Effects of the external excitation intensity on the stablesteady-state response and the vibration transmissibility: (a) the stable steady-state response and (b) the vibration transmissibility

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