MS09.8: Nonlinear Dynamics in Engineering Systems

This paper explores stability and instability in periodic responses within nonlinear dynamic systems under forced vibrations using the Efficient Path Following Method (EPFM). The method, known for its cost efficiency with a 60% reduction in calculations through an updating formula, is analyzed for its accuracy in computing Floquet multipliers. By investigating two 4-degree-of-freedom systems and a nonlinear beam, the analysis involves calculating periodic solutions and determining Floquet multipliers to comprehensively discuss stability and instability. The study includes examining a discrete 4-degree-of-freedom system, identifying stable branch locations based on frequency, and analyzing changes in phase space due to energy variations. Additionally, the research extends to a simply supported nonlinear beam with in-plane stress, accounting for substantial out-of-plane deflections. The paper introduces the Efficient Path Following Method (EPFM) as a solution to the computational limitations of the Pseudo-arclength continuation method in computing periodic solutions of nonlinear systems. While EPFM addresses the challenge of generating the monodromy matrix with an innovative updating formula, its accuracy in predicting solution stability is yet to be fully explored. The study investigates EPFM's capability in forecasting stability by analyzing Floquet multipliers in various system setups, demonstrating satisfactory accuracy despite utilizing auxiliary formulas. Additionally, EPFM significantly improves computation speed, marking a notable advancement in stability analysis methodologies.

This work presents a new mathematical model of wheel hop for a vehicle with independent suspension, and performs a bifurcation analysis to investigate its nonlinear response under changes in tire inflation pressure and vertical load. Specifically, a three degree-of-freedom (DOF) model of a rear wheel is established that includes the coupling mechanism between longitudinal and vertical modes, along with an extended tire model that considers the influence of inflation pressure and vertical load on tire friction. Numerical continuation is used to study the effect of input speed on the wheel’s hop response. This analysis yields two oscillation modes: wheel hop at 1.86 Hz, with a longitudinal and vertical displacement ratio of 1: 3.1, which may occur over a range of 5~108 rad/s (strongly depending on initial conditions); and a high frequency longitudinal mode identified from 5~231 rad/s. The tire’s stick-slip effect is found to be the cause of wheel hop, and across one period of oscillation the wheel experiences a conversion from sticking to slipping. Finally, the effect of the inflation pressure and vertical load on wheel hop is revealed: the range where wheel hop occurs expands first to maximum and then shrinks to zero at fold bifurcations as the pressure and load are increased. The results of this study can not only provide a theoretical reference for hop attenuation, but also demonstrate the effectiveness of the analyzed model, which can be used in further dynamic analysis of wheel vibrations with suspension geometry considered.

In this paper, a recently proposed continuous dynamic second-order friction model is implemented in the commercial multibody system package Simpack and compared with impack's discontinuous friction model for stick-slip applications. The comparison, performed on a model of a festoon cable system, focuses on ease of use and runtime behavior.