Study On Dynamics Of Cycloid Ball Planetary Transmission With The Cross-slot Equi-speed Mechanism

The cycloid ball planetary transmission is a new type of precision transmission mechanism that uses steel ball as an intermediary to transfer motion and force.The mechanism has the advantages of no backlash,strong bearing capacity,high efficiency,wide range of transmission ratio and so on.But in actual production work,the transmission mechanism will produce vibration and noise during operation,which will affect the precision and service life of the transmission mechanism.Therefore,the research on the dynamics of cycloid ball planetary transmission system is of great significance to improve the precision,service life and structural parameters of the transmission mechanism.According to the nonlinear relationship between the meshing force and the contact deformation in the Hertz theory,the four-point contact mechanical models of the reduction speed engagement and the equi-speed engagement for cycloid ball planetary transmission are established respectively.Through the axial force balance equation and the torque balance equation,the normal force at the meshing point of the reduction speed engagement and equi-speed engagement is obtained.Based on the stress results of elasticity mechanics,the normal stress and shear stress at different depths under the contact surface are obtained,and the depth corresponding to the maximum shear stress is obtained.The influence of the parameters on the normal force at the meshing point and the stress under the contact surface is analyzed.Considering the influence of the variation of the curvature of the cycloidal groove and the normal force,the time-varying meshing stiffness models of the reduction speed engagement and the equi-speed engagement are established respectively,and the numerical solution of the time-varying meshing stiffness is obtained by solving the nonlinear equations.And the meshing stiffness is expressed as the sum of the mean value of meshing stiffness and the increment of meshing stiffness by using the Fourier transform.The time-varying meshing stiffness increment of meshing pairs is analyzed in frequency domain,and the influence of parameters on the average value and increment of meshing stiffness of the meshing pairs is analyzed.Considering the influence of four point contact of meshing pair,steel ball motion and time-varying mesh stiffness,a coupled translational torsion dynamic model of cycloidal ball planetary transmission is established.According to the Newton's second law,the differential dynamic equations of the transmission mechanism are derived,and then the characteristic equation of the transmission mechanism of undamped free vibration is obtained.And the natural frequency of the transmission mechanism and its corresponding main mode shape are obtained by MATLAB,and modal analysis is performed.Using the Dukhami integral formula,the time-domain displacement response formulas of cycloid ball planetary transmission system are deduced respectively under the external excitation,the meshing stiffness excitation and the compound excitation.And the steady-state time-domain displacement response formulas under the original coordinate is obtained by the mode superposition method.The time-domain response curve of each component in the cycloid ball planetary transmission system is plotted by MATLAB,and the time-domain displacement response of each component is analyzed.Without considering the influence factors such as external force and damping,the parametrically excited vibration model of cycloid ball planetary transmission system is established,and the dynamic equations of parametric vibration of the transmission system are deduced,and the canonical equation of the transmission system is obtained by decoupling it.The stability analysis of transmission mechanism is carried out by using multi-scale method,and the parameter unstable region of cycloidal ball planetary transmission under the excitation of meshing stiffness is obtained.The formula of steady-state response of cycloidal ball planetary transmission mechanism under the excitation of meshing stiffness is derived by using the Linzi Ted-Poincaré perturbation method.