Effects of variable loading conditions on the dynamic behaviour of planetary gear with power recirculation

Variable loads to which gearboxes are subjected are considered as one of the main sources of non-stationarity in these transmissions. In order to characterise their dynamic behaviour in such conditions, a torsional lumped parameter model of a planetary gear with power recirculation was developed. The model included time varying loading conditions and took into account the non-linearity of contact between teeth. The meshing stiffness functions were modelled using Finite Element Method and Hertzian contact theory in these conditions. Series of numerical simulations was conducted in stationary conditions, with different loading conditions. Equation of motion was solved using Newmark algorithm. Numerical results agreed with experimental results obtained from a planetary gear test bench. This test bench is composed of two similar planetary gears called test planetary gear set and reaction planetary gear set which are mounted back-to-back so that the power recirculates through the transmission. The external load was applied through an arm attached to the free reaction ring. Data Acquisition System acquired signals from accelerometers mounted on the rings and tachometer which measured instantaneous angular velocity of the carrier's shaft. The signal processing was achieved using LMS Test.Lab software. Modulation sidebands were obtained from the ring acceleration measurements as well as a non-linear behaviour in case of variable loading resulted by a transfer of the spectral density from the fundamental mesh stiffness to its second harmonic.This work was financially supported by the Tunisian-Spanish Joint Project No. A1/037038/11. The authors would like also to acknowledge the project funded by the Spanish Ministry of Science and Technology and called ‘‘Development of methodologies for the simulation and improvement of the dynamic behavior of planetary transmissions DPI2013-44860”

Nonlinear time-varying dynamic analysis of a spiral bevel geared system

In this paper, a nonlinear time-varying dynamic model of a drivetrain composed of a spiral bevel gear pair, shafts and bearings is developed. Gear shafts are modeled by utilizing Timoshenko beam finite elements, and the mesh model of a spiral bevel gear pair is used to couple them. The dynamic model includes the flexibilities of shaft bearings as well. Gear backlash and time variation of mesh stiffness are incorporated into the dynamic model. Clearance nonlinearity of bearings is assumed to be negligible, which is valid for preloaded rolling element bearings. Furthermore, stiffness fluctuations of bearings are disregarded. Multi-term harmonic balance method (HBM) is applied on the system of nonlinear differential equations in order to obtain a system of nonlinear algebraic equations. Utilizing receptance method, system of nonlinear algebraic equations is grouped in nonlinear and linear sets of algebraic equations where the nonlinear set can be solved alone decreasing the number of equations to be solved significantly. This reduces the computational effort drastically which makes it possible to use finite element models for gear shafts. In the calculation of Fourier coefficients, continuous-time Fourier transform as opposed to the gear dynamics studies that utilize discrete Fourier Transform is used. Thus, convergence problems that arise when the number of nonlinear DOFs is large are avoided. Moreover, analytical integration is employed for the calculation of Fourier coefficients rather than numerical integration in order to further reduce the computational time required. Nonlinear algebraic equations obtained are solved by utilizing Newton's method with arc-length continuation. Direct numerical integration is employed to verify the solutions obtained by HBM. Several case studies are carried out, and the influence of backlash amount, fluctuation of gear mesh stiffness and variation of bearing stiffness are investigated. In addition to these, the response of the coupled gear system model is compared with that of gear torsional model in order to study the influence of the coupling on dynamics of the system