Abstract

Computational Analysis of Non-Periodic Oscillations Using the Harmonic Balance Method

Payam Azadian

This thesis investigates different approaches for applying the harmonic balance (HB) method to study steady-state non-periodic motion. The HB method is based on approximating the response of a nonlinear system using a truncated Fourier series. It is an established, powerful technique for analyzing and predicting the steady-state response of nonlinear systems.
The HB method is very popular in engineering because it reduces the cost of computations compared to direct numerical integration of the governing equations. This reduction is particularly important in nonlinear systems because the computational cost for these systems is typically much higher. Furthermore, the HB method can be used to approximate the steady-state quasi-periodic response of different types of systems. The quasi-periodic motion is characterized by two or more harmonic components with incommensurate frequencies. This type of motion can emerge in a mechanical system in response to two harmonic excitations with an irrational frequency ratio. The excitations can be external, internal (parametric), or a combination of the two. Since quasi-periodic motion is not periodic, it cannot be represented by a simple Fourier series. Therefore, specialized formulations are required for using the HB method for the analysis of quasi-periodic motion.
The objective of this work is to investigate different approaches for approximating the steady-state quasi-periodic response of mechanical systems using the HB method. In particular, we focus on two techniques: the multidimensional HB method and the trained HB method. The results are verified with direct numerical analysis and published literature results to show that the approximation error is acceptable. The problem formulations are discussed in the non-dimensional form to keep the results general. Various case studies are used to better demonstrate how the algorithms work.