Research Papers

Shooting and Arc-Length Continuation Method for Periodic Solution and Bifurcation of Nonlinear Oscillation of Viscoelastic Dielectric Elastomers

[+] Author and Article Information
Fan Liu

Xi'an Institute of Space Radio Technology,
Xi'an 710100, China

Jinxiong Zhou

State Key Laboratory for Strength and
Vibration of Mechanical Structures,
Shaanxi Engineering Laboratory for Vibration
Control of Aerospace Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 28, 2017; final manuscript received October 29, 2017; published online November 16, 2017. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 85(1), 011005 (Nov 16, 2017) (7 pages) Paper No: JAM-17-1538; doi: 10.1115/1.4038327 History: Received September 28, 2017; Revised October 29, 2017

A majority of dielectric elastomers (DE) developed so far have more or less viscoelastic properties. Understanding the dynamic behaviors of DE is crucial for devices where inertial effects cannot be neglected. Through construction of a dissipation function, we applied the Lagrange's method and theory of nonequilibrium thermodynamics of DE and formulated a physics-based approach for dynamics of viscoelastic DE. We revisited the nonlinear oscillation of DE balloons and proposed a combined shooting and arc-length continuation method to solve the highly nonlinear equations. Both stable and unstable periodic solutions, along with bifurcation and jump phenomenon, were captured successfully when the excitation frequency was tuned over a wide range of variation. The calculated frequency–amplitude curve indicates existence of both harmonic and superharmonic resonances, soft-spring behavior, and hysteresis. The underlying physics and nonlinear dynamics of viscoelastic DE would aid the design and control of DE enabled soft machines.

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Grahic Jump Location
Fig. 2

Schematic of the rheological model used to represent a viscoelastic DE, which comprises two units in parallel: one with spring α and another with spring β and a dashpot

Grahic Jump Location
Fig. 1

Schematic of oscillation of a DE balloon under combined inflating pressure and applied voltage

Grahic Jump Location
Fig. 5

Limit cycle solution (solid line) obtained by the shooting method. The Poincaré map (filled circles) converges to and ceases at one fixed point on the limit cycle.

Grahic Jump Location
Fig. 3

An undamped DE balloon subjected to combined static inflation pressure and harmonic voltage exhibits pseudo-periodic oscillation. (a) Time history of the oscillation after along period of elapsed time to exclude the transient effect of initial conditions (lower line: applied sine voltage; upper line: circumferential stretch). Corresponding phase portrait (b) and Poincaré map (c) plotted in phase plane.

Grahic Jump Location
Fig. 4

Oscillation of a viscoelastic DE balloon subjected to static pressure and harmonic voltage. (a) Time history of the oscillation of a viscoelastic DE balloon. Starting from static equilibrium position caused by static pressure and static biased voltage, the stretch of the balloon relaxes gradually and approaches steady-state eventually. (b) Phase portrait of the oscillation of a viscoelastic DE balloon. After a period of transient process, the trajectory of phase portrait converges into one closed limit cycle.

Grahic Jump Location
Fig. 6

Frequency amplitude curve for nonlinear oscillation of a viscoelastic DE balloon. The multivaluedness of the response curve due to the nonlinearity of the system leads to jump phenomenon. The system behaves as having a soft spring; As the frequency is decreased, the response amplitude jumps to a lower amplitude.



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