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Research Papers

Highly Reduced Lumped Parameter Models Representing Impedance Functions at the Interface of One-Dimensional Viscoelastic Continua

[+] Author and Article Information
Masato Saitoh

Graduate School of Science and Engineering,  Saitama University, 255 Shimo-Okubo, Sakura-Ku, Saitama, Japansaity@mail.saitama-u.ac.jp

J. Vib. Acoust 134(3), 031008 (Apr 24, 2012) (11 pages) doi:10.1115/1.4005826 History: Received October 02, 2010; Revised October 24, 2011; Published April 23, 2012; Online April 24, 2012

In recent dynamic problems dealing with high-frequency excitations, such as ultrasonic vibrations, a proper representation of rods transmitting kinetic energy from the interface attached to the vibrating system to the other end is strongly demanded for effectively reducing computational time and domain. A highly reduced lumped parameter model that properly simulates the dynamic characteristics of a uniform, isotropic, homogeneous, and viscoelastic rod subjected to excitations at its end is proposed in this paper. The model consists of springs, dashpots, and so called “gyro-mass elements.” The gyro-mass element generates a reaction force proportional to the relative acceleration of the nodes between which it is placed. This model consists of units arranged in series, each unit consisting of a spring, a dashpot, and a gyro-mass element arranged in parallel. A formula is proposed for determining the properties of the elements in the units based on the modal expansion. The results show that a notable reduction of 90% in the degrees of freedom is accomplished with high accuracy by using the proposed model consisting of a set of units associated with modes in a target frequency region and a supplemental unit associated with residual stiffness, which is advantageous for efficient numerical computations in recent dynamic problems.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Viscoelastic rod studied: (a) longitudinal vibration mode, (b) transverse shear vibration mode, and (c) torsional vibration mode

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Figure 2

Lumped parameter model with gyro-mass elements (GLPM) for simulating impedance function at the interface of a viscoelastic rod

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Figure 3

Mechanical analogy of gyro-mass element (a), and its symbol (b), proposed by Saitoh in 2007 [8] (revised in parts, with permission, from the American Society of Civil Engineers)

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Figure 4

Conventional LPM studied

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Figure 5

Normalized impedance functions at the interface of a lightly damped rod (ζ=0.02) using the GLPM with various maximum numbers of units (J). Also shown for comparison are results obtained with the rigorous solution and with the CLPM with various numbers of units (m) each consisting of mass − dashpot − spring elements.

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Figure 6

Normalized impedance functions at the interface of a strongly-damped rod (ζ=0.20) using the GLPM with various maximum numbers of units (J). Also shown for comparison are results obtained with the rigorous solution and with the CLPM with various numbers of units (m) each consisting of mass − dashpot − spring elements.

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Figure 7

Comparison of the response displacement at the interface of the rod evaluated by GLPMs, CLPMs, and rigorous solutions when subjected to an impulsive force

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Figure 8

Normalized Fourier amplitude of the transient displacement (ζ=0.02)

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Figure 9

Normalized impedance functions at the interface of a rod with damping constant ζ=0.005 using the CLPMs with various maximum numbers of DOF (m = 250 and 500). Results obtained with the rigorous solution are also shown for comparison.

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Figure 10

Normalized impedance functions at the interface of a rod with damping constant ζ=0.005 in a low-frequency region (0≤a0≤20) and a high-frequency region (15≤a0≤35), using GLPMs with various combinations of units and a GLPM with residual stiffness (RS). Results obtained with the rigorous solution are also shown for comparison.

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Figure 11

GLPMs with residual stiffness for approximating impedance functions

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Figure 12

A massive structure supported by a number of rods subjected to a high-frequency excitation

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Figure 13

Comparison of the response displacement at the center of gravity of a massive structure supported by a number of rods evaluated by a GLPM with residual stiffness, CLPMs (m = 250 and 500), and the rigorous solution when subjected to a high-frequency harmonic force (a0=30)

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