Research On The Time-delayed Dynamics Of A Mechanical Centrifugal Governor System

The outline of "Made in China 2025" pointed out that the active control technology of mechanical system is the key technology for the high-end equipment in the fields of aerospace,marine engineering,ships,rail transit,robots and agricultural machinery.Among them,the time-delay effect in the control system is an important factor restricting their development,and this also put forward a higher requirement for basic research in the field of mechanical engineering.It is inevitably that the time-delay effects arise in the actual control loops for the reaction time of the sensors,filters,controllers and actuators.And the dynamics models of such controlled systems are described by the delayed differential equations.The infinite dimensional nature of the solution space of the delayed differential equations makes the theoretical and numerical study of such systems challenging.Therefore,the study of nonlinear time-delay dynamics of the mechanical system has important value on the basic theory and engineering application.At present,there is relatively little work on the matching law between the parameters and the response of the delayed systems,and the multi-stability.In view of this,this paper takes a class of mechanical governor system as a research object aiming to reveal the time-delay dynamics of this system and to discuss the mechanism of the time-delay on the system response.Meanwhile,this paper aims to refine the numerical simulation method for complex nonlinear systems with time-delay,the detailed contents are as follows:(1)The dynamical governing equations of the mechanical governor system are modified,and the stability and co-dimension one Hopf bifurcation of the equilibrium are analyzed as well.In this paper,a novel governor model with two time-delay is established by considering the time-delay caused by the elastic force of the mass-spring structure and the reaction time caused by the working principle of the engine in the control loop.Furthermore,based on the two time-delay governor model,the stability and the instability mechanism of the equilibrium are analyzed in a specified two-parameter plane under different time-delay working conditions.The results show that,compared with the working condition without time-delay,the time-delay caused by the working principle of the engine tends to reduce the stability domain area of equilibrium.Moreover,under this type of time-delay,the equilibrium will lose its stability mainly by the supercritical Hopf bifurcation,which generates a stable periodic solution.(2)The other instability mechanisms of the equilibrium are investigated,including the Bautin bifurcation and the double Hopf bifurcation.For the governor system,in addition to the co-dimension one Hopf bifurcation,the equilibrium could lose its stability by the high co-dimension bifurcations as well,including the Bautin bifurcation and the double Hopf bifurcation.In order to investigate the system response where the equilibrium lose its stability in these cases,the method of multiple scales and the numerical integration method are applied to obtain the unfolding structures of these high co-dimension bifurcations.The results show that the time-delay can not only increase the probability of Bautin bifurcation of the equilibrium,but also induce other types of high co-dimension bifurcation,including the supercritical Bautin bifurcation,weak resonance double Hopf bifurcation and non-resonant double Hopf bifurcation.Correspondingly,in the neighborhood of certain parameter points,the time-delay governor system will behave the coexistence of stable equilibrium and stable periodic solution,or the coexistence of two independent stable periodic solutions.(3)The secondary bifurcation law of the simple periodic solution and the transition law of the two-parameter dynamics of the governor system are analyzed.Firstly,the numerical tool,i.e.,DDE-BIFTOOL is applied to perform the continuation analysis of the simple periodic solution generated from the co-dimension one Hopf bifurcation of equilibrium.The results show that the increase of the time-delay caused by the elastic force makes the secondary evolution process of the simple periodic solution more complicated,while the increase of the time-delay caused by the working principle of the engine makes the evolution of the periodic solution simple first and then complex.Furthermore,in order to explore the matching relationship between system parameters and system response,a two-parameter co-simulation strategy based on GPU parallel is proposed to obtain the two-parameter dynamics distribution map of the system.The results show that,under the influence of the time-delay caused by the working principle of the engine,the governor system may not exhibit complex periodic motion and chaotic motion in the two-parameter plane.This discovery is expected to provide a theoretical reference for motion control of mechanical governor system.(4)A numerical scheme is proposed to calculate the steady-state response of time-delay nonlinear system under the large sample initial disturbances,and the multistable coexistence mechanism of the governor system is analyzed.The key points of this numerical strategy are the dimension reduction processing of the time-delay system and the integration of cell mapping method into probability calculation of basin stability of attractor to realize the acceleration.For the governor system,the coexistence mode of bistability to the coexistence of eight attractors can be observed,where the coexistence of eight attractors is a unique coexistence mode under time delay conditions.By studying the variation laws of probability of basin stability of attractors with system parameter,it shows that four main types of multi-stability mechanisms can induce the above mentioned coexistence patterns,including the continuous bifurcation of the periodic solution generated from the equilibrium,the co-dimension two bifurcation of equilibrium,the bifurcation of simple periodic solution combined with the boundary crisis of chaos,and the bifurcation of two branches of independent periodic solutions.Compared to the single-parameter bifurcation diagram combined with the path-following technique,the method of the probability of basin stability of attractor can provide a more overall information of the multi-stability behavior,such as the latter method can detect a larger number and type of attractors under the same conditions.