Forced Response Analysis of Dynamic Systems With Inertia Nonlinearity by Applying the Multiharmonic Balance Method

ABSTRACT Nonlinear mechanical vibrations under harmonic forcing can be well approximated by Fourier series. For a finite number of harmonics, the error is minimized over one period of vibration. This technique, known as multiharmonic balance method (MHBM), is today widely used in academics as well as industrial applications, e.g., for friction‐damped system in turbomachinery. The idea is often motivated by the original harmonic balance approach using the fundamental harmonic only, which is often referred to as single‐term harmonic balance method. One can show analytically that a single term approach applied to a one degree‐of‐freedom system results in an oscillator with equivalent stiffness and equivalent damping changing with vibration amplitude. However, an equivalent mass, which changes for different energy levels has not been considered within harmonic balance analysis so far. In this work, we will show that the harmonic balance procedure can be applied to systems with energy‐dependent inertia behavior. A famous nonlinear pendulum invented by Christiaan Huygens is used to demonstrate the applicability of MHBM. As stability information is crucial in nonlinear system design, this work will also point out how stability can be assessed in the frequency domain in such cases. By extending Hill's theory, Floquet exponents are obtained from which stable and unstable solutions can be distinguished.