Dynamic analysis of the response of Duffing-type oscillators subject to interacting parametric and external excitations

• Published in: Nonlinear dynamics Vol. 107; no. 1; pp. 99 - 120
• Main Authors: Aghamohammadi, Mehrdad, Sorokin, Vladislav, Mace, Brian
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.01.2022; Springer Nature B.V
• Subjects: Accuracy; Amplitudes; Approximation; Automotive Engineering; Classical Mechanics; Control; Damping; Dynamical Systems; Energy harvesting; Engineering; Equations of motion; Exact solutions; Excitation; Linear damping; Mechanical Engineering; Methods; Multiscale analysis; Nonlinear response; Nonlinear systems; Original Paper; Oscillators; Parameters; Upper bounds; Vibration; Nonlinear forced Mathieu equation; Bounded response; Method of multiple scales; Nonlinear dynamical systems; Interacting parametric and external excitations; Method of varying amplitudes
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-021-06972-5

Prediction of the response of nonlinear dynamical systems under interacting parametric and external excitations is important in designing systems such as sensors, amplifiers or energy harvesters, to achieve the desired performance. This paper concerns the nonlinear forced Mathieu equation, with linear damping and a 2:1 ratio between the parametric and external excitation frequencies. The Method of Varying Amplitudes (MVA) is employed to derive approximate analytical expressions for the response of the system. Both single-term and double-term solutions are developed: it is seen that, employing the double-term approximation, the MVA can accurately predict the response of the system over a wide range of frequencies and system parameters, showing a maximum of 0.2% deviation from numerical results obtained by direct integration of the equation of motion. This is in contrast with most of the available theoretical approaches such as the conventional Method of Multiple Scales, which can predict the response accurately only for a narrow range of system parameters and excitation frequencies. Furthermore, it is seen that the response is bounded, and analytical expressions for the frequency and amplitude of the upper bound are developed: this is unlike other methods which predict unbounded response, unless nonlinear damping is considered. Analytical expressions for the response are developed, and results are verified with numerical results obtained from direct integration of the equation of motion. Numerical examples are presented, showing good agreement with results obtained by the MVA.