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In this paper, nonlinear dynamic model of spur gear pairs with fractional-order damping under the condition of time-varying stiffness, backlash and static transmission error is established. The general formula of fractional-order damping term is derived by using the incremental harmonic balance method (IHBM), and the approximate analytical solution of the system is obtained by use of the iterative formula. The correctness of the results is verified by comparing with the numerical solutions in the existing literature. The effects of mesh stiffness, internal excitation amplitude and fractional order on the dynamic behavior of the system are analyzed. The results show that changing the fractional order can effectively control the resonance position and amplitude in the meshing process. Both the mesh stiffness and internal excitation can control the collision state and the stability.

In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.

The nonlinear torsional model of a multistage gear transmission system which consists of a planetary gear and two parallel gear stages is established with time-varying meshing stiffness, comprehensive gear error and multi-clearance. The nonlinear dynamic responses are analyzed by applying the reference of backlash bifurcation parameters. The motions of the system on the change of backlash are identified through global bifurcation diagram, largest Lyapunov exponent (LLE), FFT spectra, Poincaré maps, the phase diagrams and time series. The numerical results demonstrate that the system exhibits rich features of nonlinear dynamics such as the periodic motion, nonperiodic states and chaotic states. It is found that the sun-planet backlash has more complex effect on the system than the ring-planet backlash. The motions of the system with backlash of parallel gear are diverse including some different multi-periodic motions. Furthermore, the state of the system can change from chaos into quasi-periodic behavior, which means that the dynamic behavior of the system is composed of more stable components with the increase of the backlash. Correspondingly, the parameters of the system should be designed properly and controlled timely for better operation and enhancing the life of the system.

A dynamic modeling method for Multistage Planetary Gear Transmission (MPGT) is proposed based on the concept of integral planetary gearbox modeling. The integrated interaction of multiple nonlinear parameters is considered in the dynamic model. The time-varying mesh stiffness of each gear pair is calculated by the energy method. The effects of input torque, gear backlash, and meshing damping on the chaos and impact characteristics of the system are analyzed in detail. The results show that the dynamic behavior of the system is closely related to the Dynamic Meshing Force (DMF). When the system is in the states of chaos, bifurcation, and jumping, the DMF fluctuates violently, and the stability and reliability of the system are seriously affected. With the increase of input torque and meshing damping, the system exits chaos through the inverse period-doubling bifurcation path, which indicates that increasing the input torque and meshing damping can suppress the chaotic motion and enhance the stability of the system. The backlash has a significant effect on the nonlinear behavior and meshing impact characteristics of the system. When the backlash is small, the system is in bilateral impact, and the meshing impact tends to be stable as the backlash increases. In order to improve the vibration characteristics of the system, a slightly larger backlash is necessary. The results can be used to guide the dynamic characteristics design and vibration control of the MPGT.

In the present study a single degree of freedom oscillator with clearance type non-linearity is considered. Such oscillator represents the simplest model able to analyze a single teeth gear pair, neglecting: bearings and shafts stiffness and multi mesh interactions. One of the test cases considered in the present work represents an actual gear pair that is part of a gear box of an agricultural vehicle; such gear pair gave rise to noise problems. The main gear pair characteristics (mesh stiffness and inertia) are evaluated after an accurate geometrical modelling. The meshing stiffness of the gear pair is piecewise linear and time varying (in particular periodic); it is evaluated numerically using nonlinear finite element analysis (with contact mechanics) for different positions along one mesh cycle, then it is expanded in Fourier series. A direct numerical integration approach and a smoothing technique have been considered to obtain the dynamic scenario. Bifurcation diagrams of Poincaré maps are plotted according to some sample case study from literature. Optimization procedures are proposed, in order to find optimal involute modifications that reduce gears vibration.

The existence of backlash greatly reduces the performance of the multi-stage gear transmission system and even leads to system instability. In order to solve the problems, the backlash is modeled by the nonlinear function and the lumped-parameter method is applied to establish the equations of the system. The influence of backlash impact on multi-stage gear transmission system is simulated by Matlab/Simulink and the control method to eliminate the influences of backlash is mainly studied. The compensating method is based on ideal model control theory to reduce backlash effect on the system. Instead of using a typical PID controller, the fuzzy controller with angle difference feedback is designed. Simulations results show the control method obviously effective in compensating the backlash and enhance the response and the stable precision of the multi-stage gear transmission system.