Massimo Guiggiani

Optimization of the Loaded Contact Pattern in Hypoid Gears by Automatic Topography Modification

Systematic optimization of the tooth contact pattern under load is an open problem in the design of spiral bevel and hypoid gears. In order to enhance its shape and position, gear engineers have been assisted by numerical tools based on trial-and-error approaches, and/or they have been relying on the expertise of skilled operators. The present paper proposes a fully automatic procedure to optimize the loaded tooth contact pattern, with the advantage of eventually reducing design time and cost. The main problem was split into two identification subproblems: first, to identify the ease-off topography capable of optimizing the contact pattern; second, to identify the machine-tool setting variations required to obtain such ease-off modifications. Both of them were formulated and solved as unconstrained nonlinear optimization problems. In addition, an original strategy to quickly approximate the tooth contact pattern under load was conceived. The results obtained were very satisfactory in terms of accuracy, robustness, and computational speed. They also suggest that the time required to optimize the contact pattern can be significantly reduced compared with typical time frames. A sound mathematical framework ensures results independent of the practitioner’s subjective decision-making process. By defining a proper objective function, the proposed method can also be applied to affect other contact properties, such as to improve the motion graph or to decrease the sensitivity of the transmission to assembly errors. Furthermore, it can be easily adapted to any gear drive by virtue of its systematic and versatile nature.

The evaluation of cauchy principal value integrals in the boundary element method-a review

In this paper several methods of dealing with Cauchy Principal Value integrals in advanced boundary element methods are discussed and compared. An attempt is made to present a comprehensive description of these methods in a unified, systematic manner. It is shown that the methods can be grouped into two basic approaches, the (more classical) indirect approach, such as the rigid-body motion technique in elastostatics, and the (more recent) direct approach, that allows any Cauchy Principal Value integral to be evaluated by standard quadrature formulae.

Nonlinear Identification of Machine Settings for Flank Form Modifications in Hypoid Gears

This paper presents a new systematic method for identifying the values of the machine-tool settings required to obtain flank form modifications in hypoid gears. The problem is given a nonlinear least-squares formulation, and it is solved by the Levenberg-Marquardt method with a trust-region strategy. To test the method, the same ease-off topography was obtained by means of very different sets of machine-tool settings, including a set of only kinematic parameters and a highly redundant set of 17 parameters. In all cases, the goal was achieved in a few iterations, with residual errors well below machining tolerances and, even more importantly, with realistic values of all parameters. Therefore, significant improvements in practical gear design can be achieved by employing the overall proposed procedure.

Multi-Objective Ease-Off Optimization of Hypoid Gears for Their Efficiency, Noise, and Durability Performances

Micro-geometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, micro-geometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneously minimized (or maximized), in the presence of constraints. This matter naturally lends itself to be framed as a multi-objective optimization problem. This paper proposes a general algorithmic framework for ease-off multi-objective optimization, with special attention to computational efficiency. Its implementation is fully detailed. A simulation model for loaded tooth contact analysis is assumed to be available. The proposed method is tested on a face-hobbed hypoid gear set. Three objectives are defined: maximization of mechanical efficiency, minimization of loaded transmission error, minimization of maximum contact pressure. Bound constraints on the design variables are imposed, as well as a nonlinear constraint aimed at keeping the loaded contact pattern inside a predefined allowable contact region. The results show that the proposed method can obtain optimal ease-off topographies that significantly improve the basic design performances. It is also evident that the method is general enough to handle geometry optimization of any gear type.© 2011 ASME

Robust Optimization of the Loaded Contact Pattern in Hypoid Gears With Uncertain Misalignments

This paper presents an automatic procedure to optimize the loaded tooth contact pattern of face-milled hypoid gears with misalignments varying within prescribed ranges. A two-step approach is proposed to solve the problem: in the first step, the pinion tooth microtopography is automatically modified to bring the perturbed contact patterns (as the assembly errors are varied within the tolerance limits) match a target area of the tooth while keeping them off the edges; in the second step, a subset of the machine-tool settings is identified to obtain the required topography modifications. Both steps are formulated and solved as unconstrained nonlinear optimization problems. While the general methodology is similar to the one recently proposed by the same authors for the optimization at nominal conditions, here, the robustness issues with respect to misalignment variations are considered and directly included in the optimization procedure: no a posteriori check for robustness is therefore required. Numerical tests show that nominally satisfactory and globally robust hypoid pairs can be designed by a direct process and within a unified framework, thus avoiding tiresome trial-and-error loops. DOI: 10.1115/1.4001485

Direct evaluation of double singular integrals and new free terms in 2D (symmetric) Galerkin BEM

In this paper a new general algorithm is developed for the direct evaluation of all singular double integrals arising in the 2D Galerkin BEM, including those with hypersingular kernels. A distinguishing feature of the proposed method is that double singular integrals are treated as a whole, that is, not as inner integrals followed by outer ones. Therefore, when applied to the symmetric Galerkin BEM, the proposed technique is strictly symmetry preserving. Moreover, a careful analysis of the limiting process is performed which shows that some new free terms may arise.

Self-adaptive boundary elements with h- hierarchical shape functions

Abstract A self-adaptive boundary element method as implemented in the new computer program Sherpa is presented. Error indicators are obtained by comparing two BEM solutions obtained from exactly the same discretization but with (partly) different sets of collocation points. The code Sherpa has some advanced features, including fully automatic mesh refinement and user-friendly interfaces. h -heirarchical quadratic shape functions are employed to improve the computational efficiency. Numerical results for elastic problems are presented. Pointwise as well as global convergence is always reached in a few adaptive steps.

Critical review of handling diagram and understeer gradient for vehicles with locked differential

The steady-state cornering behaviour of rear-wheel drive vehicles fitted with locked differential is critically analysed by means of simple, albeit carefully formulated, vehicle models, which allow for a rigorous theoretical analysis. Results obtained for some classical manoeuvres, with either constant forward speed, steer angle or turning radius, clearly show that, in the case of locked differential, the vehicle cornering behaviour is strongly affected by the manoeuvre. As an important consequence, the handling diagram is not unique and the understeer gradient is no longer dependent only upon the lateral acceleration, as in vehicles equipped with an open differential. Accordingly, this study shows that some typical tools and concepts of vehicle dynamics are indeed inadequate in the case of locked differential.

Robust Optimization of Cylindrical Gear Tooth Surface Modifications Within Ranges of Torque and Misalignments

Tooth surface modifications are small, micron-level intentional deviations from perfect involute geometries of spur and helical gears. Such modifications are aimed at improving contact pressure distribution, while minimizing the motion transmission error to reduce noise excitations. In actual practice, optimal modification requirements vary with the operating torque level, misalignments, and manufacturing variance. However, most gear literature has been concerned with determining optimal flank form modifications at a single design point, represented by fixed, single load and misalignment values. A new approach to the design of tooth surface modifications is proposed to handle such conditions. The problem is formulated as a robust design optimization problem, and it is solved, in conjunction with an efficient gear contact solver (Load Distribution Program (LDP)), by a direct search, global optimization algorithm aimed at guaranteeing global optimality of the obtained microgeometry solutions. Several tooth surface modifications can be used as microgeometry design variables, including profile, lead, and bias modifications. Depending on the contact solver capabilities, multiple performance metrics can be considered. The proposed method includes the capability of simultaneously and robustly handling several conflicting design objectives. In the present paper, peak contact stress and loaded transmission error amplitude are used as objective functions (to be minimized). At the end, two example optimizations are presented to demonstrate the effectiveness of the proposed method.

On the Identification of Machine Settings for Gear Surface Topography Corrections

In this paper we set out to investigate the performances of some of the algorithms proposed in the gear literature for identifying the machine-settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For the ease of comparison, the problem is formulated as a nonlinear least-squares minimization, and the most widely employed algorithms are derived as particular cases. The algorithms included in the analysis are: (i) one-step methods; (ii) iterative methods; (iii) iterative methods with step control. The performance index is devised in their ability of returning practical solutions in the presence of: (i) strong model nonlinearities, (ii) ill-conditioning of the sensitivity matrix, (iii) demanding topographic shapes purposely selected. Instrumental here is an original classification of topographic modifications as either “simple” or “complex”, based on the SVD analysis of the sensitivity matrix. On the basis of the numerical tests documented, iterative techniques with step control seem the most convenient, due to reliability and robustness of the solutions produced. The generation process here considered is face-milling of hypoid gears, even though the methodology is general enough to cope with any gear cutting method requiring only some minor technical changes.Copyright