Philippe Velex

Influence of the Nonlinear Dynamic Behavior of Journal Bearings on Gear-Bearing Assemblies

Journal bearings cannot be considered as passive elements in gear-bearing assemblies, and the lubricant is recognized as playing an important role in the interactions between the shafts and the bearings. In order to take this influence into account, bearings are usually modeled by means of eight dynamic coefficients, i.e., asymmetric stiffness and damping matrices. In this paper, a nonlinear approach is proposed enabling the behavior of a gear-shaft-bearing assembly to be analyzed. A discrete finite element model is used for the shafts, and a specific gear element is introduced which accounts for non-linear time-varying mesh stiffness as well as tooth shape deviations. The meshing forces are internal system forces whereas the effects of the bearings on the shafts are taken to be external. A combination of the Newmark time integration scheme and the Newton-Raphson algorithm is used to simultaneously solve the contact problem for the gear, and the Reynolds equations for the bearings. The resulting algorithm is applied to a single stage geared system with two shafts, four bearings, a pinion and a gear while taking mass unbalance, eccentricity and meshing excitations into account. Several examples are presented which demonstrate the influence of bearing nonlinearity and the efficiency of the proposed model and numerical procedure.Copyright

Windage Losses in High Speed Gears: Preliminary Experimental and Theoretical Results

Power losses in high-speed gears come from the friction between the teeth (sliding and rolling), the lubrication process (dip or jet lubrication), the pumping of a gas-lubricant mixture during the meshing and the losses associated with windage effects. The objective of this paper is to present a number of preliminary experimental and theoretical findings on the prediction of windage losses. Experiments were conducted on a test bench whose principle consists in driving a gear to a given speed and then measuring its deceleration once it has been disconnected from the motor. The transmission between the motor and the rotor is ensured by a friction wheel which also plays the role of speed multiplier. A pneumatic jack either imposes a sufficient contact pressure between the driving wheel and the rotor (transmission of rotation), or is used for separating the parts when the maximum speed is reached. A disk and 4 different gears were tested in the absence of a lubricant at speeds ranging from 0 to 12 000 rpm. Two different theoretical approaches have been developed: i) a dimensional analysis based upon the dimensionless groups of terms which account for the flow characteristics (Reynolds number), the gear geometry (tooth number, pitch diameter, face width) and the speed, ii) a quasi-analytical model considering in detail the fluid flow on the gear faces and inside the teeth. It is found that both approaches give good results in comparison with the experimental evidence and two analytical formulae aimed at predicting windage losses in high-speed gears are proposed.© 2003 ASME

Power Transmission with Gears

Gears are considered as one of the oldest machine elements. Early concepts used by the Chinese can be traced back to more than 2000 years ago in the BC time period. However, major development into precision elements did not occur until the first industrial revolution in the late 18th century. Since then, even though gear design has improved drastically, the fundamental concept has remained practically the same and there have been no other precision machine elements seriously challenging the important role of gears in transmitting power. In the era of the fourth industrial revolution, the reliance of modern industry on power transmission with gears is becoming ever more important. The needs of industry are currently dominated by increased output power to weight ratio, refinement of parasitic losses, robustness and improved noise, vibration and harshness behaviour. Systems transmitting power with gears are continuously improved in all aspects to meet the above requirements. New materials and designs are introduced by engineers; numerical methods are developed to decrease the time to market and more sophisticated computational and experimental techniques are employed. This Special Issue on Power Transmission with Gears aims to communicate to colleagues in Industry and Academe recent advances in the design of such precision systems. The objective is to inform the Community about methods, analysis, design advances and new materials that contribute to the improvement in the performance of geared systems. The subsequent benefits are longer product/component life, less energy consumption and reduction in the product development time and cost. To this end, we have a showcase of research studies ranging from gear geometry to wear and dynamics. This collection of papers covers a variety of precision gear elements including spur, helical, planetary, bevel, hypoid and other related machines. The papers have been authored and co-authored by gear researchers in all major industrial regions around the world. It is our hope that seasoned and new gear researchers will find this Special Issue useful, interesting and practical. We aspire that the collection of papers will highlight the importance of power transmission with gears and the role they play in modern society.

On the Correlation Between Dynamic Transmission Error and Dynamic Tooth Loads in Three-Dimensional Gear Systems

A three-dimensional dynamic model is presented to simulate the dynamic behavior of single stage gears by using a combination of classic shaft, lumped parameter and specific 2-node gear elements. The mesh excitation formulation is based on transmission errors whose mathematical grounding is briefly described. The validity of the proposed methodology is assessed by comparison with experimental evidence from a test rig. The model is then employed to analyze the relationship between dynamic transmission errors and dynamic tooth loads or root stresses. It is shown that a linear dependency can be observed between the time variations of dynamic transmission error and tooth loading as long as the system can be assimilated to a torsional system but that this linear relationship tends to disappear when the influence of bending cannot be neglected.Copyright

A torsional dynamic model of multi-stage geared systems submitted to internal and external excitations

A simplified torsional model is presented which can be used to simulate the dynamic behaviour of multi-stage spur and helical gears. The time-varying mesh stiffness functions are estimated from the formulae of Weber & Banaschek and their relative phases are determined based on the gear geometry and relative positioning. A time-varying external torque can also be inserted at any node. The resulting differential system is solved by combining a Newmark’s numerical scheme and a normal contact algorithm. A number of simulation results are presented on the influence of the combined effect of errors and shape deviations and internal excitation sources on dynamic effort.

Simulation of the Dynamic Behavior of a Multi-stage Geared Systems with Tooth Shape Deviations and External Excitations

In this paper, a torsional dynamic model of multi-stage idler spur and helical gears is presented which combines time-varying internal and external excitations such as time-varying external torques. Each contact line in the various base planes is discretized in elemental cells which are all attributed a time-varying mesh stiffness element and an initial separation to account for tooth shape deviations from ideal involute flanks. The corresponding non-linear differential system is solved by combining a Newmark’s numerical scheme and a normal contact algorithm. A number of simulation results are presented on the influence of the combined effect of errors and shape deviations along with external excitation sources on dynamic tooth loads.