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Research Papers

Mitigating Vibration Along an Arbitrarily Supported Elastic Structure Using Multiple Two-Degree-of-Freedom Oscillators

[+] Author and Article Information
Philip D. Cha1

Department of Engineering, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA 91711philip_cha@hmc.edu

Michael Chan

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

1

Corresponding author.

J. Vib. Acoust 131(3), 031008 (Apr 22, 2009) (10 pages) doi:10.1115/1.3085891 History: Received April 21, 2008; Revised December 04, 2008; Published April 22, 2009

Simple spring-mass systems are often deployed as vibration absorbers to quench excess vibration in structural systems. In this paper, multiple two-degree-of-freedom oscillators that translate and rotate are used to mitigate vibration by imposing points of zero displacement, or nodes, along any arbitrarily supported elastic structure during harmonic excitations. Nodes can often be enforced along an elastic structure by attaching suitably chosen two-degree-of-freedom oscillators. In application, however, the actual selection of the oscillator parameters also depends on the tolerable translational and rotational vibration amplitudes of the attached oscillators, because if these vibration amplitudes are large, then theoretically feasible solutions would not be practical to implement. In this paper, an efficient approach is developed that can be used to tune the oscillator parameters that are required to induce nodes, while satisfying the tolerable vibration amplitudes of the oscillators. Instead of solving for the oscillator parameters directly, the restoring forces exerted by the springs are computed instead. The proposed approach is simple to apply, efficient to solve, and more importantly, allows one to easily impose the tolerable translational and rotational vibration amplitudes of the two-degree-of-freedom oscillators. A design guide for choosing the required oscillator parameters is outlined, and numerical experiments are performed to validate the proposed scheme of imposing nodes along a structure at multiple locations during harmonic excitations.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

An arbitrary supported elastic structure carrying S two-degree-of-freedom oscillators

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Figure 2

The system of Fig. 1, where the S two-degree-of-freedom oscillators are replaced by the 2S restoring forces they exert on the arbitrarily supported elastic structure

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Figure 3

Free body diagram of a two-degree-of-freedom oscillator that translates and rotates

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Figure 4

The steady state deformed shapes of a uniform simply supported Euler–Bernoulli beam with (solid line) and without (dotted line) an attached oscillator. The horizontal line represents the configuration of the undeformed beam. The system parameters are ω=220EI/(ρL4), xf=0.40L, xa1=0.60L, and xa2=0.65L. The attachment and node locations are collocated.

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Figure 5

Design plots for m1 (solid line) and J1 (dotted line) for the system described by Fig. 4

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Figure 6

Design plots for k1 and z¯1 for the system described by Fig. 4

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Figure 7

The steady state deformed shapes of a uniform cantilevered Euler–Bernoulli beam with (solid line) and without (dotted line) two attached oscillators. The horizontal line represents the configuration of the undeformed beam. The system parameters are ω=150EI/(ρL4), xf=L, xa1=0.50L, xa2=0.60L, xa3=0.65L, and xa4=0.75L. The attachment and node locations are collocated.

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Figure 8

Design plots for m1 (solid line) and J1 (dotted line) for the system described by Fig. 7

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Figure 9

Design plots for k1 and z¯1 for the system described by Fig. 7

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Figure 10

Design plots for m2 (solid line) and J2 (dotted line) for the system described by Fig. 7

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Figure 11

Design plots for k2 and z¯2 for the system described by Fig. 7

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Figure 12

The steady state deformed shapes of a uniform cantilevered Euler–Bernoulli beam with (solid line) and without (dotted line) an attached oscillator. The horizontal line represents the configuration of the undeformed beam. The system parameters are ω=130EI/(ρL4), xf=0.75L, xa1=0.55L, xa2=0.65L, xn1=0.80L, and xn2=L. The attachment and node locations are noncollocated.

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Figure 13

Design plots for m1 (solid line) and J1 (dotted line) for the system described by Fig. 1

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Figure 14

Design plots for k1 and z¯1 for the system described by Fig. 1

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Figure 15

The steady state deformed shapes of a uniform simply supported Euler–Bernoulli beam with (solid line) and without (dotted line) two attached oscillators. The horizontal line represents the configuration of the undeformed beam. The system parameters are ω=190EI/(ρL4), xf=0.45L, xa1=0.40L, xa2=0.50L, xa3=0.80L, xa4=0.90L, xn1=0.20L, xn2=0.65L, xn3=0.67L, and xn4=0.69L. The attachment and node locations are noncollocated.

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Figure 16

Design plots for m1 (solid line) and J1 (dotted line) for the system described by Fig. 1

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Figure 17

Design plots for k1 and z¯1 for the system described by Fig. 1.

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Figure 18

Design plots for m2 (solid line) and J2 (dotted line) for the system described by Fig. 1.

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Figure 19

Design plots for k2 and z¯2 for the system described by Fig. 1

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