Combining the Asymptotic Numerical Method with the Harmonic Balance Method to capture the nonlinear dynamics of spur gears

Abstract

The present work proposes the Asymptotic Numerical Method (ANM) combined to the Harmonic Balance Method (HBM) as a valuable approach to solve the nonlinear dynamics of gear pairs. The ANM is a continuation method based on high-order Taylor series expansion of the computed solution branch. The HBM is a periodic solution representation method based on high-order Fourier series. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. By employing this method, the periodic solutions may be easily expressed with respect to both the shaft rotation frequency and the gear mesh frequency as the adoption of a high number of harmonics has negligible effects on the computational burden. Effectiveness and reliability of the method are proven by comparing the numerical results with that obtained from the Runge–Kutta time integration scheme and experimental data from literature. Afterwards, a comparison in terms of computational efficiency is performed. Finally, some considerations are drawn in order to highlights the main differences between the two methods within gear dynamics computation.