Internal resonance in nonlinear structures—and what it can be used for

Abstract

AbstractIn vibrating structures, nonlinear effects can play a decisive role. For example, stiffening responses can occur in integral components with low damping, but softening effects with a shift of the resonance to lower values by an increase of the vibration energy can be observed. Often these systems are considered only in a characteristic nonlinear vibration mode, allowing well‐known phenomena to be investigated. However, it may be the case that more than one nonlinear mode has a significant effect on the structural response of multi‐degree‐of‐freedom systems. If these modes take on integer multiples of frequency values, we speak of an internal resonance. In this work, the phenomenon of internal resonance is explained on the basis of a two‐mass oscillator. For this purpose, nonlinear modal analysis (NMA) is used to determine the amplitude‐dependent behavior of the resonant frequencies, referred to as nonlinear modes. These nonlinear modes already display the internal resonance in the form of characteristic trajectories, which are examined and explained in more detail here. A computational framework consisting of harmonic balance and continuation is used to calculate the nonlinear mode curves. In addition, nonlinear frequency responses are determined and their trajectories and properties in the internal resonance region are shown. To further understand the oscillatory behavior time plots are considered. Finally, a possible use for the observed properties is discussed as well and current approaches in research for the use of internal resonances are pointed out.