Vibration and stability of high-speed spinning cantilevered beams with shear deformation effects and an attached rigid body at its tip and and nonlinear hyperelastic behavior of circular dielectric elastomer membranes

Abstract

Instability of components and systems causes large-amplitude deformations or vibrations that limit performance, increase fatigue, and cause overload failures. In spinning systems, gyroscopic effects drive instabilities at high speeds. Instability in soft actuators occurs due to nonlinear material behavior. This work examines the dynamic stability of spinning structures with shear deformation effects and static instability in dielectric elastomer actuators.The vibration behavior, critical speeds, and high-speed instabilities in thick beams, spinning about their longitudinal axis, that have a rigid body attached to their tip are analyzed analytically and using finite element models. The analytical beam model includes shear deformation effects, incorporated by assuming the beams transverse displacements are independent of its cross-sectional rotations. A model is derived using Hamilton’s principle. The equations are cast into extended operator form, which exemplifies the system’s gyroscopic structure and facilitates Galerkin discretization for numerical solution. Numerical results are calculated for a system with identical inertia and stiffness properties in the two bending directions (called a symmetric system) and non-identical inertia and stiffness properties in the two bending directions (called an asymmetric system). Both systems have forward- and backward-orbit vibrations in single-mode, free response. Symmetric systems have material points that move in circular orbits along the span. The material point orbits become elliptical for asymmetric systems. Symmetric systems have degenerate stationary-system natural frequencies that split and become distinct for non-zero speeds. All eigenvalues cross, without interaction, as the rotation speed varies. The symmetric system eigenvalues are purely imaginary except at high speeds where critical speeds occur. Asymmetric systems, due to the differing inertia and stiffness properties in the two bending directions, have distinct stationary-system eigenvalues that increase and decrease with increasing speed from vanishing speed. Because of shear deformation effects, eigenvalue veering occurs when any decreasing forward-orbit eigenvalue comes into close proximity with an increasing backward-orbit eigenvalue. Within veering regions the forward- and backward-orbit modes couple, creating a single-mode free response that has forward-orbit vibrations in some regions along the span and backward-orbit vibrations in others. Asymmetric beams have distinct critical speeds and regions of divergence instability. Shear deformation effects lead to flutter instability at high speeds. Parametric studies reveal atypical high-speed instability behavior, including immediate transitions from divergence to flutter instability as speed varies. The mathematical structure observed for symmetric beam-rigid body systems is leveraged for closed-form solutions for their vibration. Finite element analyses are used to confirm the speed-dependent natural frequencies and identify axially- and torsionally-dominated modes. The results from this work could improve the high-speed performance of resonator devices like MEMS gyroscopes. The equilibria and stability of voltage-activated, pre-stretched circular dielectric elastomer membrane actuators under equilibrium conditions are studied using nonlinear analytical and coupled electro-mechanical finite elements simulations. The analytical model includes only the active region of the actuator; the passive region is replaced by a uniform outer circumferential load intended to capture the membranes initial pre-stretch. The model is derived in terms of a general strain energy density so that the voltage-stretch behavior of common hyperelastic material models can be examined. The Gent hyperelastic model is used as a baseline because of its ability to accurately predict the large-stretch mechanical behavior of a common acrylic elastomer used in dielectric elastomer transducers. The Gent model predicts one equilibrium at low voltages, three equilibria at moderate voltages, and one equilibrium at high voltages. The membrane experiences snap-through and snap-back bifurcations between small- and large-stretch equilibria. Moderate stretch models can capture the loss of stability, but snap-through to large-stretches are not possible. Hyperelastic models that include strain stiffening can capture large-stretch deformations, provided sufficient numbers of terms and material parameters are used. The finite element model includes both active and passive regions of the membrane actuator, inhomogeneity of deformations, the potential for material compressibility, and electric fringe fields at the boundaries of the electrodes. When the passive region is neglected, the finite element model predicts similar voltage-stretch behavior as the analytical model. When the passive region is included, the stretch meaningfully decreases, highlighting the potential importance of the passive region in actuators. Whereas the finite element model shows nearly homogeneous deformations within the active region, the deformations in the passive region are highly inhomogeneous. The finite element model is compared to available experiments.