Speaker
Description
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 Fourier-Galerkin-Method (FGM) or Multi-Harmonic-Balance Method (MHBM) is today widely used in academics as well as industrial applications, e.g. friction damping 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 or Describing Function 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 so far. In this work we will present how the standard Harmonic Balance procedure can be applied to system with energy dependent inertia behaviour. First, examples for this kind of dynamic system are provided to motivate the problem. Three different real systems will be shown to exhibit inertia nonlinearity, namely a pendulum with varying length, a slender cantilever beam at moderately large deflection and a particle container. For fundamental investigations two academic systems are chosen. The first one is a reference system with nonlinear acceleration term in cubic form. It can be seen as an inertia version of the Duffing oscillator and will be used to point out numerical challenges for both time integration and Harmonic Balance. A second reference system is linear in acceleration but the leading factor representing the oscillators inertia is a function of generalized coordinates. The latter system is found to be a suitable minimal model for the above mentioned examples. Single-term Harmonic Balance is applied to obtain an equivalent mass and related nonlinear eigenfrequency. It will be shown how MHBM can be modified to tackle vibration problems with amplitude dependent inertia and the characteristic vibration behaviour is analysed. Reference results are computed by time integration which may be challenging in cases where nonlinear behaviour is directly expressed in the accelerations. As stability information is crucial in nonlinear system design, this work will point out how stability can be assessed in the frequency domain by extending Hill’s theory for computing Floquet exponents in the case of inertia nonlinearity.
