Linearly Perturbed Frequency Equation, New Frequency Formula, and a Linearized Galerkin Method for Nonlinear Vibrational Oscillators

A new frequency–amplitude formula by improving an ancient Chinese mathematics method results in a modification of He’s formula. The Chinese mathematics method that expresses via a fixed-point Newton form is proven to be equivalent to the original nonlinear frequency equation. We modify the fixed-point Newton method by adding a term in the denominator, and then a new frequency–amplitude formula including a parameter is derived. Upon using the new frequency formula with the parameter by minimizing the absolute error of the periodicity condition, one can significantly raise the accuracy of the frequency several orders. The innovative idea of a linearly perturbed frequency equation is a simple extension of the original frequency equation, which is supplemented by a linear term to acquire a highly precise frequency for the nonlinear oscillators. In terms of a differentiable weight function, an integral-type formula is coined to expeditiously estimate the frequency; it is a generalized conservation law for the damped nonlinear oscillator. To seek second-order periodic solutions of nonlinear oscillators, a linearized residual Galerkin method (LRGM) is developed whose process to find the second-order periodic solution and the vibrational frequency is quite simple. A hybrid method is achieved through a combination of the linearly perturbed frequency equation and the LRGM; very accurate frequency and second-order periodic solutions can be obtained. Examples reveal high efficacy and accuracy of the proposed methods; the mathematical reliability of these methods is clarified.