Abstract
This article presents the advantages and limitations of a harmonic balance method applied for solving non-linear equations of monition. This method provides an opportunity to find stable and unstable periodic solutions, which was demonstrated for a few equations. An error of solution decreases rapidly with increase of number of harmonics for smooth time history of acceleration, which shows convergence; whereas, for discontinuous time histories, this method is not effective.
Highlights
Non-linear equations of motion are usually solved with implicit and explicit numerical methods, which integrate time histories
The Harmonic Balance Method (HBM) provides an opportunity to find stable and unstable periodic cycles, which play a significant role in the bifurcation theory
Unstable solutions are difficult to obtain with numerical methods, because the numerical solution escapes to an attractor
Summary
Non-linear equations of motion are usually solved with implicit and explicit numerical methods, which integrate time histories. The Runge–Kutta, polynomial or direct methods are used to solve the equations [1] These numerical methods provide an opportunity to simulate chaotic and transient vibration; whereas stable periodic vibrations, are obtained after certain time, when transient component decreases to negligible value. There appears to be some problems with simulation undamped and unstable vibrations, because it is known that numerical methods change the energy of simulated system, sometimes vibration amplitude rises and transient component remains large. These phenomena are coupled with stability of numerical methods [2]. Whereas the Harmonic Balance Method (HBM) can be used to find stable and unstable cycles and their spectrums without these problems [3]
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