Efficient Model Reduction and Prediction of Superharmonic Resonances in Frictional and Hysteretic Systems

TL;DR

This work tackles nonlinear vibration in frictional and hysteretic jointed structures by extending Variable Phase Resonance Nonlinear Modes (VPRNM) to multi-DOF systems and coupling it with Extended Periodic Motion Concept (EPMC) based backbones to form a fast, accurate reduced-order modeling framework. The proposed VPRNM ROM efficiently captures superharmonic resonances and internal resonances, achieving speedups up to $4\times$ in ROM construction and up to $7.8\times10^{5}$ in ROM evaluation compared to the Harmonic Balance Method (HBM), with validation on a 3-DOF system and a Half Brake-Reuß Beam (HBRB). Experimental measurements using instrumented bolts, shaker ring-down, and stepped-sine tests validate the modeling approach, showing good agreement in linear frequencies and nonlinear trends, including a $7:1$ SR in the HBRB. The study reveals that both tangential slipping and normal clapping at the joint contribute to SR excitation and highlights the importance of modal filtering and amplitude- or phase-control strategies in large-scale nonlinear jointed systems. Overall, the VPRNM ROM provides a practical and efficient tool for predicting and understanding complex nonlinear resonances in real-world structures, with potential extensions to extrema-tracking approaches and broader jointed-applications.

Abstract

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. The modeling shows that both tangential slipping and normal direction clapping of the joint play important roles in exciting the superharmonic resonances in the HBRB.

Figures (53)

• Figure 1: Frequency responses for DOF 1 of the 3 DOF system described in \ref{['sec:3dof_sys']} for force levels of 1.6 N, 8.0 N, and 16 N from darkest to lightest. The subplots are (a) peak response amplitude for a cycle normalized by force magnitude, (b) magnitude of the first harmonic response, (c) magnitude of the third harmonic response, and (d) phase difference between the third and first harmonics. Note that the forcing frequency ranges are identical for all plots, but plots (c) and (d) plot the superharmonic response frequency, which is three times the forcing frequency. Only plot (a) is normalized by force.
• Figure 2: Frequency responses for DOF 1 of the 3 DOF system described in \ref{['sec:3dof_sys']} for controlled first harmonic response amplitude levels of 20 m, 30 m, and 40 m from darkest to lightest. The subplots are (a) peak response amplitude for a cycle normalized by force magnitude, (b) magnitude of the first harmonic response, (c) magnitude of the third harmonic response, and (d) phase difference between the third and first harmonics. Note that the forcing frequency ranges are identical for all plots, but plots (c) and (d) plot the superharmonic response frequency, which is three times the forcing frequency. Only plot (a) is normalized by force.
• Figure 3: Schematic of 3 DOF system with nonlinear element.
• Figure 4: Overview of the 3 DOF system: (a) mass-normalized mode shapes around a linear state including half of the linearized frictional force (representative of partial slip) and (b) example hysteresis for the four-parameter Iwan model used in this study. EPMC is used to calculate backbones of the system for (c) frequency and (d) damping of the first two modes.
• Figure 5: First harmonic response magnitude of the 3 DOF system. Lighter gray lines correspond to higher controlled amplitudes at DOF 1 calculated with HBM. The proposed VPRNM ROM (dashed) lines approximates HBM truth solution lines (solid gray). The y-axes are shared between all three plots. The EPMC and VPRNM backbones track the primary and superharmonic resonance frequencies respectively.