| In modern engineering structures,non-smooth factors are commonly present and significantly influence the dynamic mechanics of the structure.Nonlinear systems are extensively utilized in mathematical modeling and describing dynamic behavior,playing a crucial role in addressing practical problems.In recent years,there has been extensive research on nonlinear dynamical systems under non-smooth conditions,revealing complex phenomena such as oscillations,chaos,and bifurcations,which better meet practical engineering requirements.Therefore,exploring the nonlinear characteristics of non-smooth dynamical systems,including global bifurcations and chaotic motion,contributes to understanding and controlling the vibration of engineering structures.This provides theoretical support for the design and optimization of engineering applications.The dissertation primarily investigates nonlinear piecewise-smooth systems featuring three types of non-smooth factors: dry friction,rigid impact,and elastic impact.The Melnikov theory is used to reveal the complex dynamic phenomena of the system,such as chaotic motion and subharmonic bifurcation.The main research work of this dissertation includes the following aspects.(1)The global dynamics of a class of nonlinear conveyor belt systems are studied.The parameter thresholds of homoclinic orbit bifurcation and Smale horseshoe chaos are predicted by the Melnikov method.Through a detailed analysis of the Melnikov function,the relationship between system parameters and chaotic characteristics is determined,and three parameter scenarios are proposed.Finally,theoretical proofs are provided to demonstrate the influence of system parameters on homoclinic orbits and chaos thresholds under these three scenarios.(2)A class of conveyor belt systems with bilateral rigid constraints connected by oblique springs and dampers is considered.The chaotic motion and subharmonic bifurcations of the system are investigated from the perspective of overall dynamics.For friction impact systems,the complexity of homoclinic and subharmonic bifurcations is higher due to the influence of two non-smooth factors: impact and friction.By applying the Melnikov method to the non-smooth conveyor belt system with friction and impact,the piecewise Melnikov function of homoclinic orbits,which depends on the conveyor belt speed,is analytically obtained.The threshold conditions for chaos and subharmonic bifurcation in the system are derived.Additionally,critical regions are plotted in multiple parameter spaces based on these threshold conditions to differentiate between chaotic and non-chaotic regions.The critical parameter region for the occurrence of subharmonic bifurcation in the friction impact system is determined using the Melnikov method,and the relationship between subharmonic bifurcation and chaos is discussed.The effects of parameters such as the damping coefficient,excitation frequency,excitation amplitude,impact coefficient of restitution,dry friction,and belt speed on chaotic motion and subharmonic bifurcation are studied based on the derived threshold conditions.(3)Homoclinic bifurcation in a class of SD(Smooth and Discontinuous)oscillators under external quasi-periodic excitation with two frequencies is investigated.By using the Melnikov method,the Melnikov function of the system under two-frequency excitation is derived,and the threshold condition of chaos in the system is obtained.Based on the threshold condition,then a complete description of the bifurcation sets and the chaotic regions in the parameter space are presented.The chaotic motion is verified by calculating the largest Lyapunov exponent of the system.Since the geometric strong nonlinearity of the SD oscillator,there are infinitely many extreme points in the frequency-dependent function of the Melnikov function.The previous research on the frequency-dependent function of the Melnikov function is mainly focused on one or two extremes.In this work,the dynamical system with infinitely many extreme points in the frequency-dependent function is considered,and a conjecture of chaotic region under this condition is given.(4)The chaotic motion of a class of non-smooth hybrid constrained systems perturbed by two-frequency external excitation is discussed.The Melnikov method is applied to a bilateral rigid-elastic hybrid constraint system with twofrequency excitation,and the threshold conditions for chaos occurrence in the system are obtained.Due to different collision constraints,two distinct Melnikov functions are derived.It has been discovered that the frequencydependent function of the Melnikov function for the right-homoclinic orbit exhibits singularities,leading to infinite values of the function at certain frequencies.Furthermore,the frequency-dependent function possesses infinitely many extremal points.Finally,the bifurcation sets and chaotic regions of the frequency variation of the left and right homoclinic orbits are analyzed,and the effects of the parameters such as frequency,amplitude,and recovery coefficient on the bifurcation sets and chaotic regions are discussed. |
