Efficient model reduction and prediction of superharmonic resonances in frictional and hysteretic systems

Modern engineering structures exhibit nonlinear vibration behavior as designs are pushed to reduce weight and energy consumption. Of specific interest here, joints in assembled structures introduce friction, hysteresis, and unilateral contact resulting in nonlinear vibration effects. In many cases, it is impractical to remove jointed connections necessitating the understanding of these behaviors. This work focuses on superharmonic and internal resonances in hysteretic and jointed systems. Superharmonic resonances occur when a nonlinear system is forced at an integer fraction of a natural frequency resulting in a large (locally maximal) response at an integer multiple of the forcing frequency. When a second vibration mode simultaneously responds in resonance at the forcing frequency, the combined phenomena is termed an internal resonance. First, variable phase resonance nonlinear modes (VPRNM) is extended to track superharmonic resonances in multiple degree of freedom systems exhibiting hysteresis. Then a novel reduced order model based on VPRNM (VPRNM ROM) is proposed to reconstruct frequency response curves faster than utilizing the harmonic balance method (HBM). The VPRNM ROM is demonstrated for a 3 degree of freedom system with a 3:1 internal resonance and for the jointed Half Brake-Reuß Beam (HBRB), which exhibits a 7:1 internal resonance. For the HBRB, new experimental results are used to validate the modeling approaches, and a previously developed physics-based friction model is further validated, achieving frequency predictions within 3%. For the considered cases, VPRNM ROM construction is up to 4 times faster than HBM, and the evaluation of the VPRNM ROM is up to 780,000 times faster than HBM. Furthermore, the modeling framework provides insights into the mechanisms of superharmonic resonances in jointed structures, showing that both tangential slipping and normal direction clapping of the joint play important roles in exciting the superharmonic resonances in the HBRB.