Numerical solution of the Duffing equation with random coefficients

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.