Research On Global Bifurcation And Multiple Attractor Coexistence Of Mechanical Collision Systems With Dry Friction

In engineering practice,the operation of mechanical equipment is often affected b y various nonlinear factors such as collisions,friction,and clearance,which are limite d by manufacturing capabilities.These factors interact with each other,giving rise to dynamic phenomena in mechanical systems,such as vibration and stick-slip motion.T hese phenomena not only decrease the efficiency of mechanical equipment but also ha ve the potential to shorten its service life.Therefore,it is essential to investigate the dynamic behavior resulting from nonlinear factors in mechanical systems.to address thi s,this study proposes the utilization of diamond-shaped embedded connecting rod cons traints,which closely resemble real working conditions,in the context of collision osci llators and linear constraints within mechanical systems.Two types of mechanical vibr ation systems are established,incorporating dry friction and nonlinear constraints.Thro ugh numerical simulation analysis,the dynamic characteristics of these two system typ es are examined,along with their correlation to system parameters.The findings deriv ed from this research provide valuable insights into the selection of appropriate param eters for mechanical systems,thereby contributing to a solid theoretical foundation in t his field.To explore the effects of nonlinear dynamics,a single-and two-degree-of-freedom mechanical collision system with nonlinear constraints was established in a frictional environment.Through force analysis,the dynamic differential equations were derived.To investigate the global dynamic characteristics of the two systems and their cyclic transition process in the low-frequency region,numerical calculations and a multi-parameter co-simulation method were employed.Results indicate that under low-frequency conditions,the transfer of p/1 basic impact motion is predominant for the mechanical vibration system with single and double degrees of freedom.However,the p/1 periodic motion is transformed to p/1^—motion containing friction viscosity due to the influence of the friction model.Nonetheless,its motion window is short.As bifurcation parameters decrease,the p/1 grazing bifurcation shifts to（p+1）/1 cycle,resulting in the generation of（np+1）/n movement during the transition process.A chatting-impact phenomenon occurs when the collision frequency p is high.Due to the distinct locations of the saddle-node bifurcation and the grazing bifurcation,the coexistence region of periodic motions,known as the hysteresis domain,emerges in adjacent periodic motions.Using the principles of nonlinear dynamics,this paper employs the concept of cell mapping to investigate the distribution of the attractor domain in the coexistence region of periodic motions.Additionally,it analyzes the stability of coexisting motions in the hysteresis domain by examining the changes in bifurcation parameters through the multi-parameter co-simulation method.The results indicate that,in a given initial state domain,the attractor domain of coexisting motions is highly dependent on the bifurcation parameters,resulting in an irregularly shaped domain with an increasing area proportion as the parameters converge.The impact of key parameters such as:mass distribution ratioα_m,stiffness distribution ratioα_k,damping distribution ratioα_c,damping coefficientζ,excitation force distribution ratio f₂₀,gapδ₁,conveyor belt speed v₀on the coexistence zone of periodic motion and the impact velocity of the mechanical system is studied and analyzed.δ₁,α_m,v₀ andαc have no obvious effect on the coexisting region of periodic motion.The increase of f₂₀,α_k andζmakes the coexisting region transition to the high frequency region and makes the motion state of the system become stable.With the increase ofδ₁,f₂₀,α_c,v₀andαm,the resonance peak value of the impact velocity of the system has no obvious change and tends to be stable,increases rapidly with the increase ofα_k,and decreases rapidly with the increase ofζ.Compared with the single degree of freedom mechanical vibration system,the selection of conveyor speed v₀is different,and the influence of damping coefficientζon the impact velocity is almost the same.Finally,taking the single-degree-of-freedom mechanical vibration system as an example,an equivalent circuit of the module in the single-degree-of-freedom system is built by using TL074CN integrated operational amplifier,NONLINEAR—DEPENDENT module,capacitor and resistor in the Multisim software,and a circuit model equivalent to the dynamic model is established.The simulation results are consistent with the numerical simulation results.Multisim equivalent circuit simulation provides a more convenient and efficient method for nonlinear dynamics simulation research.