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INVESTIGATION OF TARGETED ENERGY TRANSFER IN STOCHASTICALLY EXCITED SYSTEM WITH NONLINEAR ENERGY SINK
Last modified: 2017-05-22
Abstract
The phenomena of targeted energy transfer (TET) from a linear oscillator (LO) to a nonlinear attachment and associated complex transitions in the dynamics of a two degree-of-freedom coupled oscillator (Fig. 1) are investigated in a stochastic framework. The system under investigation comprises of a general damped LO strongly coupled to an essentially damped, lightweight nonlinear energy sink (NES) [1]. First, the deterministic dynamics of the underlying Hamiltonian (undamped) systems are studied. Subsequently, vibration suppression of NES is investigated by solving the corresponding four dimensional Fokker Planck (FP) equation using the finite difference method [2]. Interaction of the Gaussian white noise with the strongly modulated response regime is investigated in details. Effects of noise intensity as well as system properties such as stiffness and damping on the optimal regime of TET are considered.
Further by applying complexification-averaging combined with diffusion approximation methodology, the Ito stochastic differential equation corresponding to the slow dynamics of the system is derived. Stochastic transitions in the optimal TET regimes are obtained by solving the reduced transient FP equation by using the Crank–Nicholson time integration scheme. Effect of stochastic initial conditions on energy dissipation (probability of escape) is effectively employed through finite element solution. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis. Numerical evidence for exciting the strongly modulated burst due to noise in the oscillator response is also presented. The work also presents optimization of the NES parameters in order to mitigate efficiently the optimal energy from the linear system to nonlinear attachment under random excitation. Fairly good agreement between FP solution and numerical simulation is observed.
REFERENCES
1. A. F. Vakakis , O. V. Gendelman , L. A. Bergman , D. M. McFarland , G. Kerschen ,Y. S. Lee, Nonlinear targeted energy transfer in mechanical and structural systems, Springer Science & Business Media 2009.
2. P. Kumar, S. Narayanan, Solution of Fokker–Planck equation by finite element and finite difference methods for nonlinear system, Sadhana 31 (4) (2006) 455–473.
Further by applying complexification-averaging combined with diffusion approximation methodology, the Ito stochastic differential equation corresponding to the slow dynamics of the system is derived. Stochastic transitions in the optimal TET regimes are obtained by solving the reduced transient FP equation by using the Crank–Nicholson time integration scheme. Effect of stochastic initial conditions on energy dissipation (probability of escape) is effectively employed through finite element solution. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis. Numerical evidence for exciting the strongly modulated burst due to noise in the oscillator response is also presented. The work also presents optimization of the NES parameters in order to mitigate efficiently the optimal energy from the linear system to nonlinear attachment under random excitation. Fairly good agreement between FP solution and numerical simulation is observed.
REFERENCES
1. A. F. Vakakis , O. V. Gendelman , L. A. Bergman , D. M. McFarland , G. Kerschen ,Y. S. Lee, Nonlinear targeted energy transfer in mechanical and structural systems, Springer Science & Business Media 2009.
2. P. Kumar, S. Narayanan, Solution of Fokker–Planck equation by finite element and finite difference methods for nonlinear system, Sadhana 31 (4) (2006) 455–473.