Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs

We establish and analyze an analytical framework by accounting for both the piezoelectric and flexoelectric effects in bimorph cantilevers. The focus is placed on the development of governing electroelastodynamic piezoelectric–flexoelectric equations for the problems of resonant energy harvesting, sensing, and actuation. The coupled governing equations are analyzed to obtain closed-form frequency response expressions via modal analysis. The combined piezoelectric–flexoelectric coupling coefficient expression is identified and its size dependence is explored. Specifically, a typical atomistic value of the flexoelectric constant for barium titanate is employed in the model simulations along with its piezoelectric constant from the existing literature. It is shown that the effective electromechanical coupling of a piezoelectric material, such as barium titanate, is significantly enhanced for thickness levels below 100 nm. The electromechanical coupling coefficient of a barium titanate bimorph cantilever increases from the bulk piezoelectric value of 0.065 to the combined piezoelectric–flexoelectric value exceeding 0.3 toward nanometer thickness level. Electromechanical frequency response functions for resonant power generation and dynamic actuation also capture the size-dependent enhancement of the electromechanical coupling. The analytical framework given here can be used for parameter identification and design of nanoscale cantilevers that can be used as energy harvesters, sensors, and actuators.

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Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48446 ◽

2003 ◽

Cited By ~ 2

Author(s):

Alexander F. Vakakis ◽

Richard H. Rand

Keyword(s):

Normal Modes ◽

Nonlinear Normal Modes ◽

Resonance Capture ◽

Periodic Motions ◽

Internal Resonances ◽

Dynamical Mechanism ◽

Elliptic Orbits ◽

Weakly Coupled ◽

Resonant Dynamics ◽

Nonlinear Normal

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes — NNMs), as well as, asynchronous periodic motions (elliptic orbits — EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

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