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Research Papers

Enforcing Nodes to Suppress Vibration Along a Harmonically Forced Damped Euler-Bernoulli Beam

[+] Author and Article Information
Philip D. Cha1

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

Jennifer M. Rinker

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

1

Corresponding author.

J. Vib. Acoust 134(5), 051010 (Jun 05, 2012) (10 pages) doi:10.1115/1.4006375 History: Received June 06, 2011; Revised January 25, 2012; Published June 04, 2012; Online June 05, 2012

In this paper, properly tuned damped absorbers are used to suppress excess vibration anywhere along an arbitrarily supported, damped Euler-Bernoulli beam during forced harmonic excitations. This vibration suppression is achieved by enforcing distinct nodes, or points of zero vibration, at desired locations along the beam. Instead of directly solving for the absorber parameter, which is highly computationally intensive, an efficient method is developed whereby the restoring forces exerted by the damped absorbers are first determined using Gaussian elimination. These restoring forces are then used to tune the parameters of the damped oscillators. Numerical experiments show that by inducing nodes at the appropriate locations, a region of nearly zero vibration amplitudes can be enforced, effectively quenching vibration in that segment of the beam.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 3

The free body diagram of the ith damped vibration absorber

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Figure 4

A set of design plots for the system of Example 1. The subplots show the required k/(EI/L3), c/(γL), and |z¯|/(FL3/EI)) as a function of m/(ρL).

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Figure 5

Real and imaginary parts of the steady-state deformed shapes of the beam of Example 1. The solid and dashed lines represent the deflection of the beam with and without an absorber, respectively. The system parameters are xf  = 0.71L (arrow), ω=32EI/(ρL4), xa  = 0.76L (square), xn  = 1.0L (circle), and m = 4.688 × 10−2 ρL, c = 6.904 × 102 γL, and k=5.051×101 EI/L3.

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Figure 7

The feasible attachment locations that can be used to induce nodes for the system of Example 2. The system parameters are xf  = 0.31L, ω=101EI/(ρL4), xn1=0.51L, and xn2=0.65L.

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Figure 8

The design plots of the feasible spring stiffnesses and damping coefficients versus xa2, for xa1=0.59L. The system parameters are identical to those of Fig. 7.

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Figure 1

An arbitrarily supported beam that is subject to R localized harmonic excitations and carrying S damped vibration absorbers

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Figure 2

The system of Fig. 1, where the S absorbers are replaced by their restoring forces

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Figure 6

Steady-state displacement at xn versus the normalized forcing frequency ω/32EI/(ρL4) for Example 1. The solid and dashed lines indicate the steady-state deflection of the beam with and without an absorber, respectively.

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Figure 9

Real and imaginary parts of the steady-state response of the beam defined in Example 2. The solid and dashed lines represent the deflection of the beam with and without absorbers, respectively. The system parameters are xf  = 0.31L (arrow), ω=101EI/(ρL4), (xn1,xn2)=(0.51L,0.65L) (circle), (xa1,xa2=(0.59L,0.41L) (square), (m1,c1,k1)=(1.249 × 10-3ρL,3.272×10-5γL,1.274×101EI/L3), and (m2,c2,k2)=(5.264 × 10-3ρL,3.703×10-3γL,5.344×101EI/L3).

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Figure 10

Steady-state displacements at xni versus the normalized forcing frequency ω/101EI/(ρL4) for Example 2. The solid and dashed lines indicate the steady-state deflection of the beam with and without absorbers, respectively.

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Figure 11

The feasible attachment locations that can be used to induce nodes for the system of Example 3. The system parameters are xf  = 0.80L,ω=77EI/(ρL4), xn1=0.23L, and xn2=0.60L.

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Figure 12

The design plots of the feasible spring stiffnesses and damping coefficients versus xa2, for xa1=0.37L. The system parameters are identical to those of Fig. 1.

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Figure 13

Real and imaginary parts of the steady-state response of the beam defined in Example 3. The solid and dashed lines represent the deflections of the beam with and without absorbers, respectively. The system parameters are xf  = 0.80L (arrow), ω=77EI/(ρL4), (xn1,xn2)=(0.23L,0.60L) (circle), (xa1,xa2)=(0.37L,0.69L) (square), (m1,c1,k1)=(9.369 × 10-4ρL,2.592×10-5γL,5.548×100EI/L3), and (m2,c2,k2)=(1.153 × 10-2ρL,4.464×10-3γL,6.798×101EI/L3).

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Figure 14

Steady-state displacements at xni versus the normalized forcing frequency ω/77EI/(ρL4) for Example 3. The solid and dashed lines indicate the steady-state deflection of the beam with and without absorbers, respectively.

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Figure 15

The feasible attachment locations that can be used to induce nodes for the system of Example 4. The system parameters are xf1=0.80L, xf2=0.90L, ω=68EI/(ρL4), xn1=0.59L, and xn2=0.60L.

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Figure 16

The design plots of the feasible spring stiffnesses and damping coefficients versus xa2, for xa1=0.65L. The system parameters are identical to those of Fig. 1.

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Figure 17

Real and imaginary parts of the steady-state deflection of the beam defined in Example 4. The solid and dashed lines represent the deflections of the beam with and without absorbers, respectively. The system parameters are (xf1,xf2)=(0.80L,0.90L) (arrow), ω=68EI/(ρL4), (xn1,xn2)=(0.59L,0.60L) (circle), (xa1, xa2) = (0.65L, 0.83L) (square), (m1,c1,k1)=(1.253× 10-2ρL, 5.479×10-5γL, 5.792×101EI/L3), and (m2,c2,k2)=(3.761× 10-2ρL,1.232×10-1γL,1.710×102EI/L3).

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Figure 18

Steady-state displacements at xn versus the normalized forcing frequency ω/68EI/(ρL4) for Example 4. The solid and dashed lines indicate the steady-state deflection of the beam with and without absorbers, respectively.

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