Gear transmission dynamic: Effects of tooth profile deviations and support flexibility

In this work a non-linear dynamic model of spur gear transmissions previously developed by the authors is extended to include both desired (relief) and undesired (manufacture errors) deviations in the tooth profile. The model uses a hybrid method for the calculation of meshing forces, which combines FE analysis and analytical formulation, so that it enables a very straightforward implementation of the tooth profile deviations. The model approach handles well non-linearity due to the variable meshing stiffness and the clearances involved in gear dynamics, also including the same phenomena linked to bearings. In order to assess the ability of the model to simulate the impact of the deviations on the transmission dynamics, an example is presented including profile deviations under different values of transmitted torque. Several results of this example implementation are presented, showing the model's effectiveness.This paper has been developed in the framework of Project DPI2006-14348 funded by the Spanish Ministry of Science and Technology

Enhanced model of gear transmission dynamics for condition monitoring applications: Effects of torque, friction and bearing clearance

Gear transmissions remain as one of the most complex mechanical systems from the point of view of noise and vibration behavior. Research on gear modeling leading to the obtaining of models capable of accurately reproduce the dynamic behavior of real gear transmissions has spread out the last decades. Most of these models, although useful for design stages, often include simplifications that impede their application for condition monitoring purposes. Trying to filling this gap, the model presented in this paper allows us to simulate gear transmission dynamics including most of these features usually neglected by the state of the art models. This work presents a model capable of considering simultaneously the internal excitations due to the variable meshing stiffness (including the coupling among successive tooth pairs in contact, the non-linearity linked with the contacts between surfaces and the dissipative effects), and those excitations consequence of the bearing variable compliance (including clearances or pre-loads). The model can also simulate gear dynamics in a realistic torque dependent scenario. The proposed model combines a hybrid formulation for calculation of meshing forces with a non-linear variable compliance approach for bearings. Meshing forces are obtained by means of a double approach which combines numerical and analytical aspects. The methodology used provides a detailed description of the meshing forces, allowing their calculation even when gear center distance is modified due to shaft and bearing flexibilities, which are unavoidable in real transmissions. On the other hand, forces at bearing level were obtained considering a variable number of supporting rolling elements, depending on the applied load and clearances. Both formulations have been developed and applied to the simulation of the vibration of a sample transmission, focusing the attention on the transmitted load, friction meshing forces and bearing preloads.The authors would like to acknowledge Project DPI 2013-44860 funded by the Spanish Ministry of Science and Technology and Project PRX14/00451 funded by the Spanish Ministry of Education, Culture and Sports

Advanced model for the calculation of meshing forces in spur gear planetary transmissions

This paper presents a planar spur gear planetary transmission model, describing in great detail aspects such as the geometric definition of geometric overlaps and the contact forces calculation, thus facilitating the reproducibility of results by fellow researchers. The planetary model is based on a mesh model already used by the authors in the study of external gear ordinary transmissions. The model has been improved and extended to allow for the internal meshing simulation, taking into consideration three possible contact scenarios: involute–involute contact, and two types of involute-tip rounding arc contact. The 6 degrees of freedom system solved for a single couple of gears has been expanded to 6 + 3n degrees of freedom for a planetary transmission with n planets. Furthermore, the coupling of deformations through the gear bodies’ flexibility has been also implemented and assessed. A step-by-step integration of the planetary is presented, using two typical configurations, demonstrating the model capability for transmission simulation of a planetary with distinct pressure angles on each mesh. The model is also put to the test with the simulation of the transmission error of a real transmission system, including the effect of different levels of external torque. The model is assessed by means of quasi-static analyses, and the meshing stiffness values are compared with those provided by the literature.The authors would like to acknowledge Project DPI2013-44860 funded by the Spanish Ministry of Science and Technology

Some Experimental and Simulation Results on the Dynamic Behaviour of Spur and Helical Geared Transmissions with Journal Bearings

Some interactions between the dynamic and tribological behaviour of geared transmissions are examined, and a number of experimental and simulation results are compared. A model is introduced which incorporates most of the possible interactions between gears, shafts and hydrodynamic journal bearings. It combines (i) a specific element for wide-faced gears that includes the normal contact conditions between actual mating teeth, that is, with tooth shape deviations and mounting errors, (ii) shaft finite elements, and (iii) the external forces generated by journal bearings determined by directly solving Reynolds' equation. The simulation results are compared with the measurement obtained on a high-precision test rig with single-stage spur and helical gears supported by hydrodynamic journal bearings. The experimental and simulation results compare well thus validating the simulation strategy both at the global and local scales

Journal off Mechanical Design 1& l|i>««'M sJBM l^^A J Contributed by the Power Transmission and Gearing Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received

This paper presents an analytical expression of the time-varying contact length between perfect involute spur and helical gears. It is shown that contact lengths can be expressed as where AQ (r) and Bo ( T ) are 2 periodic piecewise linear functions represented in AO(T) and BO(T) can be easily decomposed in Fourier series as: AO(T) = -+ X «n • cos (InnT BO(T) * ao contact zone with: sin (2nn(ea + tp)IP)'\ Ea: transverse contact ratio eg: overlap contact ratio The length of a contact line starting at r = (' for either spur or helical gears can be deduced from (1) by: li(T) = lo(T -i) and the total contact length at T becomes: Lir) = S Zo(r -0 1=0 (5) 58