Bistable dynamics analysis using Padé approximation and resultant theory

Abstract

The bistable structures are widely used for energy harvesting and have also received great attention for vibration isolation recently. The bistable dynamics analysis is a key step for the practical mechanism design, and the Taylor series is usually deployed to approximate the nonlinear irrational structure stiffness. However, the truncation order must be sufficiently high to receive a desirable accuracy since bistable structures can undergo large-amplitude inter-well oscillations. Alternatively, we apply the Padé approximation, which gives a higher accuracy but also a more involved dynamical equation. Combined with the harmonic balance method, a resultant-based analysis framework is accordingly proposed such that we are allowed to handle purely univariate polynomial equations to facilitate calculation. The analysis considers a classic nonlinear isolator structure equipped with three springs and two links without loss of generality, and the theoretical solutions of both the inter- and intra-well frequency responses and the associated jump frequencies are obtained agreeing well with the numerical results. Furthermore, the improved accuracy of the nonlinear dynamic predictions using the Padé approximation and the efficiency of the solving process using the proposed resultant-based algorithms are proved by numerical comparisons. This work presents a distinctive approach to analyzing bistable dynamics with higher accuracy while keeping the computational cost at a desirably low level, which provides a theory for the calculations and analysis of the more advanced and complicated bistable structures.