Research On The Dynamic Characteristics Of The Single Stage Spur Gear Systems Considering Temperature Factor

Gear system is an important element in mechanical engineering and other engineering fields and is widely used because of its advantages such as high transmission efficiency, compacted structure, stable transmission and so on. Vibration characteristic of the gear pair system has important influence on the working performances and reliability of machine. The research of the mechanism and the mechanics influence factors of the gear vibration have important scientific meaning and engineering value in order to reduce vibration and noise and improve the service life and reliability of the machine.A single stage spur gear transimission system used in a vehicle system is the research object in the dissertation when the factors such as backlash, time-varying stiffness, comprehensive transmission error, meshing damping and so on are comprehensively considered. The nonlinear dynamics and the relationship and matching law between dynamic characteristic and dynamical parameters of the four gear systems are studied in two cases whether the temperature is considered from the simple to the complex that are the singledegree-of-freedom torsion-vibration gear system, gear-rotor system, three-degree-of-freedom gear system considering the supporting and gear-rotor-bearing system. The main research contents are as follows.Firstly, a developed algorithm is designed according to the characteristics of the gear system. The researched continuous state space can be divided into discrete cell spaces according to the simple cell to cell mapping idea. The mapping relation of cell to cell is established to group the cells according to the simple cell to cell mapping idea primarily. The sink cell is identified according to the escape time algorithm and is grouped again according to its periodicity. And the periodic cell is judged according to the point mapping method. Because of the existence of spurious solution and cellular flow expansion in the non-smooth dynamic system, the simple cell mapping method and the escape time algorithm are used to deal with every state cell to decrease the computation time of the final state of every cell. The computation precision of the domain of attraction obtained by the improved cell mapping method in the paper is almost equal to the precision obtained by the escape time algorithm when an adaptive number or a fixed escape time number can be adopted to increase the computing precision and the same iterative strategy can be adopted for the periodic cell to save the computing time in actual programming. The spurious solution and the cell flow expansion can be avoided as system switching in the non-smooth dynamic system.A nonlinear dynamic model of a single-stage spur gear system is established when the nonlinear factors such as the time-varying stiffness, the tooth surface friction, the backlash and the comprehensive transmission error are considered. The dynamic characteristics of the system in the parameters plane are analyzed by the fluctuation nephogram of the largest amplitude and pcolor of displacement-time via the change of the parameters. And the influences of the different parameters on the local and global dynamics of the system by the developed cell mapping algorithm integrated the single primary value bifurcation diagrams and the multiple primary value one, the top Lyapunov exponents spectrum are analyzed, too.The flash temperature of tooth surface of driving gear and driven gear is calculated based on Blok’s flash temperature theory. The mathematic expression of the contact temperature changed with time of the tooth surface is derived and the tooth profile deformation caused by the change of contact temperature of the tooth surface is computed. The expression of the mesh stiffness changed by the contact temperature of the tooth surface is derived based on Hertz contact theory. The temperature stiffness is constructed so as to lead the influence of the temperature on the dynamics into the dynamic model of the gear system.A nonlinear dynamic model of a single-stage spur gear system is established when the nonlinear factors such as the time-varying stiffness, the tooth surface friction, the backlash, the comprehensive transmission error and the contact temperature of the tooth surface especially are considered. The influences of the contact temperature on the dynamic characteristics of the system are analyzed by calculated the maximum flash temperature nephograms of the teeth faces under the relationship and matching law between dynamical parameters and the trend charts of flash temperature via points of engagement. And the influences of the different parameters on the local and global dynamics of the system considering temperature factor by the developed cell mapping algorithm integrated the single primary value bifurcation diagrams and the multiple primary value one, bifurcation diagrams, phase portraits and the diagrams of Poincaré maps are analyzed, too. The dynamical characteristics of the two systems whether the temperature is considered are constructive analyzed.In order to investigate the dynamic response of a single stage gear system coupled with rotors, the torsional vibration equations of the gear-rotor system are derived when the effect of gear backlash, time-varying mesh stiffness and static transmission errors are considered. The variables of the system are derived. Based on the bifurcation diagrams, top Lyapunov exponents spectrum, fall diagrams of FFT spectrum and pcolor of displacement-time, the periodic and aperiodic motions of the system are analyzed via the change of parameters, and the characteristics on frequency and time domains of the system are discussed. The Hopf bifurcations and the phenomenon of gradual changed instability amplitude result from these transformations are obtained. The temperature factor is added to the nonlinear dynamic equation of the gear-rotor system and its bifurcation, chaos and dynamic characteristics are researched.A single-stage spur gear pair system with three degree-of-freedom is studied wherein the backlash, the time-varying stiffness, the torsion motion and the transmission error are considered. The nonlinear three-degree-of-freedom equations are solved by employing variable step size Runge-Kutta methods and the bifurcation diagrams are obtained. The nonlinear dynamic characteristic of the system is discussed for the varying of the exciting frequency and classified based on bifurcation diagrams, top Lyapunov exponents spectrum, Poincaré maps and FFT spectrum. The Hopf bifurcation, saddle-node bifurcation and doubling bifurcation are found in the different value of the exciting frequency and their bifurcation points are given. There are two routes from periodic motion to chaos motion. The one is from period-4 motion to chaos by Hopf bifurcation, and the other is from period-3 motion and period-3 by saddle-node bifurcation to chaos. And the temperature stiffness is added to the model to study the influence of temperature on the dynamic characteristics of the three-degree-of freedom gear system.Two nonlinear dynamic gear-rotor-bearing models whether the temperature is considered are established according to the single stage spur gear transmission system used in a vehicle system. The bifurcation, chaos, attractor coexisting and the dynamic characteristics of the systems are analyzed. Some bifurcation forms are found such as doubling bifurcation, Hopf bifurcation, grazing bifurcation and so on. Here the grazing bifurcation includes grazing of gear surfaces and gear contact surfaces. Many attractor coexistings are found in the systems. The torsional vibration magnitude will increase with the increasing of the transmission and it can be decreased by the valuable backlash. There is a strong nonlinear coupling relationship between the comprehensive transmission error and the time-varying stiffness.The conclusion can be drawn that the contact temperature can not be neglected when the dynamics of the gear system is studied. The calculated results showed that the maximum of the flash temperature of tooth surface appeared when the gear meshes at teeth top and teeth root and it reached to zero near pitch point. It indicated that the calculated method of the flash temperature of gear tooth surface is correct and it can reflect the rules for the change of gear meshing temperature and sliding of gear tooth surface. The results in the dissertation will be helpful to enhance and improve the dynamic behaviors of system and provide theoretical basis of parameter optimization design of the gear pair systems.