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Analysis of Parametric Resonances in In-Plane Vibrations of Electrostrictive Hyperelastic Plates

[+] Author and Article Information
Astitva Tripathi

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: astitva.tripathi@gmail.com

Anil K. Bajaj

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: bajaj@purdue.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 1, 2017; final manuscript received February 25, 2018; published online July 26, 2018. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 13(9), 090911 (Jul 26, 2018) (11 pages) Paper No: CND-17-1348; doi: 10.1115/1.4040261 History: Received August 01, 2017; Revised February 25, 2018

Electrostriction is a recent actuation mechanism which is being explored for a variety of new micro- and millimeter scale devices along with macroscale applications such as artificial muscles. The general characteristics of these materials and the nature of actuation lend itself to possible production of very rich nonlinear dynamic behavior. In this work, principal parametric resonance of the second mode in in-plane vibrations of appropriately designed electrostrictive plates is investigated. The plates are made of an electrostrictive polymer whose mechanical response can be approximated by Mooney Rivlin model, and the induced strain is assumed to have quadratic dependence on the applied electric field. A finite element model (FEM) formulation is used to develop mode shapes of the linearized structure whose lowest two natural frequencies are designed to be close to be in 1:2 ratio. Using these two structural modes and the complete Lagrangian, a nonlinear two-mode model of the electrostrictive plate structure is developed. Application of a harmonic electric field results in in-plane parametric oscillations. The nonlinear response of the structure is studied using averaging on the two-mode model. The structure exhibits 1:2 internal resonance and large amplitude vibrations through the route of parametric excitation. The principal parametric resonance of the second mode is investigated in detail, and the time response of the averaged system is also computed at few frequencies to demonstrate stability of branches. Some results for the case of principal parametric resonance of the first mode are also presented.

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Figures

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Fig. 1

The starting base structure (rectangular plate with cut-out and clamped at two ends) for topology optimization using MMA. Red lines indicate the fixed vertical sides (all dimensions are in meters).

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Fig. 2

The final structure obtained from the base structure given in Fig. 1 after topology optimization using MMA (all dimensions are in meters)

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Fig. 3

The mode shapes of the final structure shown in Fig. 2 (all dimensions are in meters)

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Fig. 4

The net displacement in the X-direction of the mode shapes of the final structure shown in Fig. 2 (all dimensions are in meters)

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Fig. 5

The net displacement in the Y-direction of the mode shapes of the final structure shown in Fig. 2 (all dimensions are in meters)

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Fig. 6

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 2 to an externally applied electric field leading to parametric excitation of the second mode. The plots are for the amplitudes of the first mode. Part (b) expands the display of the marked region in part (a).

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Fig. 7

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 2 to an externally applied electric field leading to parametric excitation of the second mode. The plots are for the amplitudes of the second mode. Part (b) expands the display of the marked region in part (a).

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Fig. 8

Eigenvalues of the Jacobian of Eq. (23) for the trivial solution. Part (a) shows the region around the imaginary axis, whereas part (b) shows the eigenvalue λ3 becoming unstable.

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Fig. 9

The solution branch corresponding to a12 growing out of the trivial Solution. Part (a) shows the region around the point Ω1, whereas part (b) shows the region around the point Ω2.

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Fig. 10

Result of time integration of Eq. (23) for an excitation frequency Ωa = 6.93 × 104 starting close to the trivial solution of the system: (a) mode 1 and (b) mode 2. The response converges to the stable trivial solution.

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Fig. 11

Result of time integration of Eq. (23) for an excitation frequency Ωa = 6.93 × 104 starting significantly away from the trivial solution: (a) mode 1 and (b) mode 2. The response converges to the stable nonzero equilibrium solution.

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Fig. 12

Result of time integration of Eq. (23) for an excitation frequency Ωb = 6.944 × 104 starting close to the trivial solution: (a) mode 1 and (b) mode 2. The response converges to the stable trivial solution.

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Fig. 13

Result of time integration of Eq. (23) for an excitation frequency Ωb = 6.944 × 104 starting significantly away from the trivial solution: (a) mode 1 and (b) mode 2. The steady-state response is an oscillatory motion.

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Fig. 14

Result of time integration of Eq. (23) for an excitation frequency Ωc = 6.97 × 104 starting close to the trivial solution: (a) mode 1 and (b) mode 2.

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Fig. 15

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 2 to an externally applied electric field leading to parametric excitation of the first mode and Ξ1 > 0. The plots are for the amplitudes of the first and the second mode. Part (b) expands the display of the marked region in part (a).

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Fig. 16

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 2 to an externally applied electric field leading to parametric excitation of the first mode and Ξ1 < 0. The plots are for the amplitudes of the first and the second mode. Part (b) expands the display of the marked region in part (a).

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