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Research Papers

An Efficient Method for Tuning Oscillator Parameters in Order to Impose Nodes on a Linear Structure Excited by Multiple Harmonics

[+] Author and Article Information
Philip D. Cha

Professor
Department of Engineering,
Harvey Mudd College,
Claremont, CA 91711

Kenny Buyco

Department of Engineering,
Harvey Mudd College,
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 10, 2014; final manuscript received January 5, 2015; published online February 20, 2015. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 137(3), 031018 (Jun 01, 2015) (13 pages) Paper No: VIB-14-1341; doi: 10.1115/1.4029612 History: Received September 10, 2014; Revised January 05, 2015; Online February 20, 2015

Undamped oscillators are often used to attenuate and control excess vibration in elastic structures. In this paper, vibration absorbers are used to impose nodes, i.e., points of zero vibration, along an arbitrarily supported linear structure externally forced by multiple steady-state harmonic excitations. An efficient approach is proposed to tune the absorber parameters based on the active force method. Using the active force approach, the oscillators are first replaced with the unknown restoring forces they exert. These restoring forces are found by enforcing the required node locations, and they correspond to the solution of a set of linear algebraic equations, which can be obtained using Gauss elimination. These restoring forces are subsequently used to tune the sprung masses. Because of the computational efficiency of the proposed method, design plots can be easily generated, from which specific sets of oscillator parameters can be selected. Even if the input consists of multiple frequencies, it is possible to induce multiple nodes anywhere on the structure by attaching properly tuned spring–mass oscillators. An efficient procedure to tune the oscillator parameters necessary to impose nodes at the desired locations is outlined in detail, and numerical case studies are presented to verify the utility of the proposed scheme to impose multiple nodes along an arbitrarily supported elastic structure subjected to external excitations consisting of multiple harmonics.

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Figures

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Fig. 1

An arbitrarily supported elastic structure that is subjected to the qth localized force and carrying multiple sets of spring–mass oscillators

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Fig. 2

The oscillators in the system shown in Fig. 1 are replaced with corresponding active forces

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Fig. 3

Steady-state response of the beam defined in Example 1. The solid and dashed lines represent the beam deflection with and without the absorber. The system parameters are shown in Table 1.

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Fig. 4

Design space for Example 1. Black region indicates solutions for which all oscillator parameters are within their tolerable bounds. This plot is created using Eqs. (24) and (26).

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Fig. 5

Design plot for Example 1 relating k11,k21,m11, and m21 to (z¯21)2 for specified (z¯11)1 = -0.05FL3/EI. From this plot, the designer can pick one oscillator parameter to specify a solution.

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Fig. 6

A plot of active force amplitudes as a function of attachment location for Example 1

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Fig. 10

Steady-state response of the beam defined in Example 1 with (f¯f1)1 = 2F (instead of F). The solid and dashed lines represent the beam deflection with and without the absorber. The system parameters are shown in Table 4.

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Fig. 13

Set of attachment location combinations that will yield feasible solutions for Example 2 with |z¯max| = 0.01FL3/(EI)

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Fig. 14

Set of attachment location combinations that will yield feasible solutions for Example 2 with |z¯max| = 0.009FL3/(EI)

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Fig. 15

Steady-state response of the beam defined in Example 3. The solid and dashed lines represent the beam deflection with and without the absorber. The system parameters are shown in Table 6.

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Fig. 7

Design space for Example 1 with xa1=0.30L (instead of 0.40L). Black region indicates solutions for which all oscillator parameters are within their tolerable bounds.

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Fig. 8

Steady-state response of the beam defined in Example 1 with xa1 = 0.30L (instead of 0.40L). The solid and dashed lines represent the beam deflection with and without the absorber. The system parameters are shown in Table 3.

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Fig. 9

Design space for Example 1 with (f¯f1)1 = 2F (instead of F). Black region indicates solutions for which all oscillator parameters are within their tolerable bounds.

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Fig. 11

Steady-state response of the beam defined in Example 2. The solid and dashed lines represent the beam deflection with and without the absorber. The system parameters are shown in Table 5.

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Fig. 12

Set of attachment location combinations that will yield feasible solutions for Example 2 with |z¯max| = 0.05FL3/(EI)

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