Numerical solution of the Duffing equation with random coefficients

Marcin Kamiński,Alberto Corigliano

doi:10.1007/s11012-015-0133-0

Marcin Kamiński, Alberto Corigliano

Open Access

https://doi.org/10.1007/s11012-015-0133-0

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Abstract

The main aim of this work is numerical solution of the nonlinear vibrations of micro-resonators exhibiting bounded and Gaussian uncertainty in their parameters. The mechanical response in deterministic situation is described by the Duffing equation, whose numerical solution is obtained with the Runge–Kutta–Fehlenberg algorithm, while probabilistic analysis is carried out using the generalized stochastic perturbation technique enriched with automatic optimization of the approximating polynomial. Basic solution to this nonlinear vibration in the deterministic context is obtained with the use of the computer algebra system MAPLE, where all additional probabilistic procedures are also implemented. We compare each time expectations, coefficients of variation, skewness and kurtosis for the structural response to show probabilistic sensitivity of the MEMS accelerometer with respect to its design parameter expectation and coefficient of variation. An additional comparison of the proposed technique with the traditional Monte-Carlo sampling for the first four probabilistic moments is also provided.

Highlights

Micro-electro-mechanical systems (MEMS) [5] are crucial nowadays for micro-gyroscopes and accelerometers [22], mobile communication [21], building and designing of new computers [25], precise detection of the vibrations and fatigue [8], in superconductors [24] as well as in various optics practical problems solutions, like image stabilization in digital photography [15]
The mechanical response in deterministic situation is described by the Duffing equation, whose numerical solution is obtained with the Runge–Kutta–Fehlenberg algorithm, while probabilistic analysis is carried out using the generalized stochastic perturbation technique enriched with automatic optimization of the approximating polynomial
Basic solution to this nonlinear vibration in the deterministic context is obtained with the use of the computer algebra system MAPLE, where all additional probabilistic procedures are implemented

Summary

Introduction

Micro-electro-mechanical systems (MEMS) [5] are crucial nowadays for micro-gyroscopes and accelerometers [22], mobile communication [21], building and designing of new computers [25], precise detection of the vibrations and fatigue [8] (structural inspection and monitoring), in superconductors [24] as well as in various optics practical problems solutions, like image stabilization in digital photography [15]. RMS error and correlation coefficient inherent in the least squares approximation; we computationally determine up to the first four probabilistic moments and coefficients of the desired structural response This strategy is tested on two different cases—the first one concerns damped vibrations of a linear oscillator and it serves rather for a comparison with the Monte-Carlo simulation scheme, the second one concerns a case study devoted to the forced vibration of a micro-beam exhibiting stochastic damping. An application of this strategy to coupled field Finite Element Method analysis has been provided before in [11]. A comparison of such a methodology against the Monte-Carlo simulation gives unique opportunity to initially confirm an applicability of such a higher order optimized stochastic perturbation technique in highly nonlinear transient problems

Governing equations

Perturbation method validation test

Stochastic MEMS modeling

Concluding remarks