Jérôme Bruyère

Some Analytical Results on Transmission Errors in Narrow-Faced Spur and Helical Gears: Influence of Profile Modifications

Some theoretical developments are presented, which lead to approximate analytical results on quasi-static transmission errors valid for low and high contact ratio spur and helical gears. Based on a multidegree-of-freedom gear model, a unique scalar equation for transmission error is established. The role of profile relief is analyzed by using Fourier series and it is shown that transmission error fluctuations depend on a very limited number of parameters representative of gear geometry and profile relief definition. An original direct solution to the optimum relief minimizing transmission error fluctuations is presented, which is believed to be helpful for designers. The analytical results compare well with the numerical results provided by a variety of models and it is demonstrated that some general laws of evolution for transmission error fluctuations versus profile modifications can be established for spur and helical gears.

Analytical Investigations on the Mesh Stiffness Function of Solid Spur and Helical Gears

Approximate formulae are presented which give the time-varying mesh stiffness function for ideal solid spur and helical gears. The corresponding results compare very well with those obtained by using two-dimensional (2D) finite element (FE) models and specific benchmark software codes thus validating the proposed analytical approach. More deviations are reported on average mesh stiffness which, to a large extent, are due to the modeling of gear body deflections.

Derivation of Optimum Profile Modifications in Narrow-Faced Spur and Helical Gears Using a Perturbation Method

A perturbation method is presented which makes it possible to obtain approximate closed-form expressions for profile relief that minimize the fluctuations of quasi-static transmission errors under load. A number of results are displayed which prove the theoretical effectiveness of the proposed solutions for low and high-contact ratio spur and helical gears. It is also shown that the corresponding relief performance is not significantly downgraded by centre-distance variations. Finally, a number of practical considerations are brought up and commented.

Set based robust design of mechanical systems using the quantifier constraint satisfaction algorithm

Embodiment design is an important phase of the design process where the initial design parameters and their feasible solution spaces with design configurations are decided for the design problem. This article presents a new approach of embodiment design space exploration of the product based on set based design with integration of robustness for the mechanical systems. The approach presented addresses the initial design phase of the mechanical systems design and provides a three step approach based on a formal expression syntax, transformation and evaluation engine and a computational algorithm for performing a domain search for sets of robust solutions for the product designs by taking into the account the variations and uncertainties related to the manufacturing process and material. The approach is based on the design domain exploration and reduction techniques. This is achieved by the utilization and integration of existential and universal quantifiers from the quantifier constraint satisfaction problem (QCSP) for the expression of the parameters and variables related to the product design and robustness. The quantifier notion has been used to develop the consistency check for the existence of a design solution and existence of a robust design solution. In order to compute the developed quantifier approach, an algorithm based on the transformation of the quantifier with interval arithmetic has also been developed. In order to demonstrate the capability of the developed approach, this article includes three examples of mechanical systems from earlier research works that apply the quantifier model and the resolution algorithm to successfully explore the design domain for robust solutions while taking into account different types of variations such as variations in mechanical/material properties, manufacturing variations or variations in geometric dimensions which may be of continuous or discrete type.

A Contribution to the Design of Robust Profile Modifications in Spur and Helical Gears by Combining Analytical Results and Numerical Simulations

This paper addresses the definition of robust profile modifications in spur and helical gears. An original methodology is introduced which relies on closed-form analytical results on transmission errors combined with a gradient descent algorithm and a Gauss quadrature (GQ) based full factorial method. The results compare very well with those delivered by using classic Monte Carlo simulations with a considerable gain in computational time. The influence of the probability distribution law for the design parameters (depth and extent of modification) is analyzed along with the contribution of gear quality grade and load variation. Some optimum robust linear relief is presented which minimizes transmission error fluctuations over a broad range of loads even in the presence of significant geometrical tolerances.

Optimization of Profile Modifications With Regard to Dynamic Tooth Loads in Single and Double-Helical Planetary Gears With Flexible Ring-Gears

This paper is mostly aimed at analyzing optimum profile modifications (PMs) in planetary gears (PGTs) with regard to dynamic mesh forces. To this end, a dynamic model is presented based on 3D two-node gear elements connected to deformable ring-gears discretized into beam elements. Double-helical gears are simulated as two gear elements of opposite hands which are linked by shaft elements. Symmetric tip relief on external and internal gear meshes are introduced as time-varying normal deviations along the lines of contact and time-varying mesh stiffness functions are deduced from Wrinckler foundation models. The equations of motion are solved by coupling a Newmark time-step integration scheme and a contact algo-rithm to account for possible partial or total contact losses. Symmetric linear PMs for helical and double-helical PGTs are optimized by using a genetic algorithm with the objective of minimizing dynamic tooth loads over a speed range. Finally, the sensitivity of these optimum PMs to speed and load is analyzed.

An Alternative Approach to the Definition of Profile Modifications in High-Contact-Ratio Spur Gears

Two original analytical formulations are presented which explicitly give the depths and extents of symmetric profile modifications minimising the fluctuations of quasi-static transmission error in narrow-faced spur and helical gears. Numerous comparisons with quasi-static and dynamic simulation results are presented which prove that the proposed theory is sound. The proposed formulae can therefore help define adapted reliefs in terms of transmission error and dynamic tooth loading with minimum effort.

Set Based Robust Design of Systems – Application to Flange Coupling

A set-based approach to design of mechanical systems is presented in the following text. Set-based technique allows keeping multiple alternatives alive during the design process while narrowing through the possibilities towards the most optimal solution. Using the Quantifier notion from QCSP (Quantified Constraint Satisfaction Problem), a formal expression for the problem has been developed. An algorithm using QCSP transformation through interval analysis has also been developed. In order to demonstrate the approach, an example of design of rigid flange coupling with a variable number of bolts and a choice of bolts from ISO M standard has been resolved and demonstrated.

Optimization of Profile Modifications With Regard to Dynamic Tooth Loads in Planetary Gears With Flexible Ring-Gears

This paper deals with the optimization of tooth profile modifications in planetary gears. A dynamic model is proposed based on 3D two-node gear elements connected to a deformable ring-gear discretized into beam elements. Symmetric tip relief on external and internal gear meshes are introduced as normal deviations along the lines of contact superimposed on a stiffness distribution aimed at simulating position- and time-varying mesh stiffness functions. The equations of motion are solved by the combination of a Newmark’s time-step integration scheme and a contact algorithm to account for possible partial or total contact losses. Symmetric linear profile modifications are then optimized by using a genetic algorithm with the objective of minimizing dynamic tooth loads over a speed range. Finally, the interest of the corresponding optimum profile modifications with regard to speed and torque variations is analyzed.Copyright

Analytical Investigations on the Mesh Stiffness Function of Solid Narrow Faced Spur and Helical Gears

Approximate formulae are presented which give the time-varying mesh stiffness function for ideal solid narrow-faced spur and helical gears. The corresponding results compare very well with those obtained by using 2D finite element models and specific benchmark software codes thus validating the proposed analytical approach. More deviations are reported on average mesh stiffness which, to a large extent, are due to the modelling of gear body deflections.Copyright