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Research Papers

Design for 1:2 Internal Resonances in In-Plane Vibrations of Plates With Hyperelastic Materials

[+] Author and Article Information
Astitva Tripathi

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

Anil K. Bajaj

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

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 23, 2013; final manuscript received August 7, 2014; published online September 1, 2014. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 136(6), 061005 (Sep 01, 2014) (10 pages) Paper No: VIB-13-1329; doi: 10.1115/1.4028268 History: Received September 23, 2013; Revised August 07, 2014

With advances in technology, hyperelastic materials are seeing use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. Development of such complex devices is leading to increased use of hyperelastic materials in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot made of hyperelastic polymers. Since the strain energy potentials of various hyperelastic material models have nonlinearities present in them, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney–Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the Mooney–Rivlin material's strain energy potential has quadratic nonlinearities, a possibility of 1:2 internal resonance is explored. A finite element method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the 1:2 ratio. Once a topology of the structure is achieved, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete nonlinear Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode dynamic model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the nonlinear strain energy potential for the Mooney–Rivlin material makes it possible for internal resonance to occur in such structures. The effect of nonlinear material parameters on the dynamic response is investigated.

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Figures

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Fig. 1

Structure used for verification of the Matlab FEM formulation with Ansys. The bottom side indicated by the thick line is fixed (all dimensions are in meters).

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Fig. 2

The starting base structure (rectangular plate with cut-out and clamped at one end) for topology optimization using MMA. The bottom side indicated by the thick line is fixed (all dimensions are in meters).

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Fig. 3

The final structure obtained from the base structure given in Fig. 2 after topology optimization using MMA (all dimensions are in meters)

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Fig. 4

The lowest two-mode shapes of the final structure shown in Fig. 3 (all dimensions are in meters)

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Fig. 5

The base structures used for optimization using the simple iterative procedure (all dimensions are in meters)

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Fig. 6

Illustration of the simple iterative procedure for optimizing the base structure (all dimensions are in meters)

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Fig. 7

Final structure obtained after optimization of the base structure by the simple iterative procedure. For the structure shown, ω2/ω1 = 2.0069 (all dimensions are in meters).

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Fig. 8

The lowest two-mode shapes of the final structure in Fig. 7, obtained after optimization started with the base structure in Fig. 5(d) (all dimensions are in meters)

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Fig. 11

Time response of the point A on the optimal structure in Fig. 7

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Fig. 10

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 3 to a vertical harmonic excitation. The plots are for the amplitudes of the two interacting modes.

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Fig. 9

Nonlinear in-plane response of the hyperelastic structure shown in Fig. 7 to a vertical harmonic excitation. The plots are for the amplitudes of the two interacting modes.

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