Gear design Optimized tooth engagement through targeted deformed gearwheels

In order to optimize the load-bearing image of the teeth for all load conditions with the gearwheel, a load-dependent, targeted deformation of the wheel rim is generated so that the gearwheel remains perpendicular to the centerline of the unformed shaft.

Current practice in mechanical engineering is to optimize the load distribution, and thus the performance of the gearswheel, for a specific load point. However, the load to be transferred or the torque to be transferred fluctuates over time in almost all applications. As a result, the gear stage is operated predominantly in non-optimal operating points. It was recognized that there is potential for improvement in a joint project conducted two years ago by drive technology manufacturer Adrian Technologies GmbH and the Faculty of Mechanical Engineering and Supply Technology at TH Nürnberg (all Germany). The participants set themselves the goal of achieving an optimized load distribution of the gears over a wide range of application. The idea was to improve the gear meshing by deliberately deformed gearswheels. The analytical calculations and simulations are promising.

In gearwheel design, the load-bearing verification [5] is of particular importance. The ideal engagement of two meshing teeth with a large contact area over the entire tooth width does not always occur even with straight-toothed gears. The load-bearing image is negatively influenced by a multitude of factors:

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• Firstly, a manufacturing deviation should be mentioned, which already generates a partially stronger load.

• Additionally, deformations during operation often have a negative impact.

• Additionally, optimization is carried out for only one load condition. When loads vary, different deformations occur on the gearwheel, which generally have a negative impact on the tooth load distribution.

• In addition to this, the applied load leads not only to the deformation of the gearwheel due to the radial force component, but also to a deformation of the shaft, which also influences the tooth load distribution of the spur gear pair.

These effects of an uneven force distribution over the tooth width are taken into account in the load capacity analysis via the flank load K_Hβ. The deviations from production are taken into account via the manufacturing-related flank line deviation f_ma and the assembly as well as the elastic deformations via the flank line deviation due to deformation f_sh. The pinion position to the bearings also flows into this via the factor K'.

The aim is now to generate a load-dependent and targeted deformation of the wheel rim of the gears, so that the gear always remains perpendicular to the centerline of the undeformed shaft. This way, an optimal load distribution of the teeth in the tooth engagement can be generated for all load conditions. The deformation is determined by two factors: On the one hand, the stiffness is determined by the thickness of the wheel rim receptacle, on the other hand, the axial force for deforming the support is adjusted via the helix angle β. It should be noted that this method can only be applied for torsional moments with a constant direction of rotation. In the case of changing torque directions, a significantly negative state would be created after the reversal of the direction of rotation.

Under the premise of the conventional roller bearing of gear shafts and the maximum allowable inclination of the shafts in the bearing points, in this use case according to [3], thin discs with small deflections can be assumed. This allows the application of the Kirchhoff model to the gear rim.

This minor deformation makes the structural theory of the first order applicable to plate theory, which allows a combination of disc theory for the radial and tangential force in helically cut spur gears with the plate theory of the axial force of the helically cut gear.

Determining the tilt angle of the gear

The process begins with the calculation of the bending line of the shaft. At the position of the gear, the inclination angle γ of the shaft is to be determined, which determines the inclination angle of the gear to be compensated. The approach is determined analytically according to [6] using a simplified shaft. The shaft is examined in a simplified manner, undercuts, grooves for retaining rings or even the transverse pressed joint are not considered in the examination of the bending line.

Radial and tangential force component

The radial force and the tangential force of the tooth engagement act in the median plane of the examined gear body. Since the loads occur within the median plane of the gear body, in this case the application of disk theory is sufficient. All stress and strain components are assumed to be constant over the gear body thickness t.

For both disc theories, the outer radius r_a is defined as fixed in place. The radial beam from the centerline of the gear body to the point of force application is defined with the coordinate axis r. This results in the coordinate system in polar coordinates for the disc theories.

Axial force component of the gear engagement

The axial force occurring in helical gears acts perpendicular to the median plane of the wheel body and will cause a bending of the wheel body. This bending should be adjusted so that the centerline of the deformed tooth rim is parallel to the centerline of the unformed shaft. Thus, the inclination angle with positioning between the bearing locations will be less than the allowable inclination in the roller bearing locations. Due to the restrictive limitation of the inclination angle in bearing locations, in the applied plate theory, small deflections compared to the wheel body thickness must be assumed and thus a superposition of the results with the results of the disk consideration is possible.

The gear mesh forces are considered separately and finally the deformations are superimposed by superposition.

Comparison with FEM simulation

In the FEM simulation, the force components of the axial, radial, and tangential force are simulated separately. In addition to comparing these results with the calculation, the von Mises comparative stress is also calculated. This must not exceed the permissible stress of the wheel body. The optimization is achieved by a thicker wheel rim in case of too high comparative stress and an adjustment of the inclination angle β of the teeth to optimize the elastic deformation of the wheel rim.

This is accompanied by the determination of the deformation of the wheel rim. As can be seen in the table, the analytical calculation results in an axial displacement of the tooth contact point (rolling point C) of 30 µm, and the simulation results in a compensatory displacement in the opposite direction of 29 µm. This means that the negative influence on the load distribution caused by the inclination of the shaft can be almost completely compensated for.

The results show a sufficient agreement of the analytical calculation with the FEM simulation, thus suggesting a realistic representation of the analytical calculations. Tests that are planned as a second step are still pending. (dm)

Sources:

[1] Göldner, H. and others; "Textbook Higher Strength Theory"; Leipzig: Fachbuchverlag; Volume 1 Principles of Elasticity Theory; 3rd edition; ISBN 3-343-00495-2

[2] Göldner, H. and others; "Exercise Book Higher Strength Theory"; Leipzig: VEB Fachbuchverlag; Elasticity Theory, Plasticity Theory, Viscoelasticity Theory; 1978; ISBN 3-87664-037-7

[3] Altenbach, H. and others; "Flat Surface Structures"; Berlin: Springer-Vieweg-Verlag; Basics of modelling and calculation of disks and plates; 2nd edition; 2016; ISBN 978-3-662-47229-3

[4] Muschelischwili, N. I.; "Some Basic Tasks of Mathematical Elasticity Theory"; Munich: Carl Hanser Verlag; 1971; ISBN 3-446-10089-X

[5] DIN 3990-1 - Load capacity calculation of spur gears - Introduction and general influencing factors; Beuth-Verlag; December 1987

[6] Wittel, H. and others; "Roloff/Matek Machine Elements"; Springer-Verlag; 24th edition; ISBN 978-3-658-26279-2