A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Background Since combinations of the Newton’s method and the harmonic balance (HB) method require, at each iteration step, calculating the first or the first- and second-order derivatives of the restoring force function, and expanding the function, its first- and second-order derivatives into Fourier series, the procedural costs are high and sometimes difficult to achieve algebraically. It is thus preferable to avoid expensive re-linearization or computation of the second-order derivative.
Purpose A new approach is proposed to construct accurate analytical approximation solutions to strongly nonlinear conservative oscillators with odd nonlinearities.
Methods The approach is based on a combination of a modified Newton method and the HB method. For the modified Newton method, two simplified Newton steps are taken between each Newton step where only one linearization of the restoring force function is required. The resulting equations are solved by applying the HB method appropriately.
Results Using only one modified Newton iteration step may achieve highly accurate analytical approximation solutions to the strongly nonlinear oscillators. Three examples with physical implications are used to illustrate the proposed method.
Conclusion Through the modified Newton iteration step, the multiple cumbersome linearizations of the restoring force function are replaced by only one linearization, and the corresponding governing equations can be properly solved by the HB method. The current work is expected to extend to the study of other nonlinear oscillations.