An analysis of gearing

The article has presented a numerical method for determining the gear ratio function and the gearing line of the toothed gear. The described algorithm, explained on the example of planar gearing, can be used also for the analysis of spatial gearings. Assembly errors have been allowed for in the gear model.


Introduction
The quality of a gear is determined by the analysis of the gearing [1][2][3], the key element of which is the determination of the gear ratio function [4]. The gearing analysis is used chiefly for determining the kinematic errors that result from toothing cutting errors and gear assembly errors, as well as for optimizing the gear synthesis (especially for the analysis of spatial gearing) [5,6]. Generally, this is a process involving numerical computation by successive approximation methods.
The method of planar gearing analysis was first formulated by Litvin (1968) [4]. This method can be used not only for toothed wheels and gears, but also for the majority of simple mechanisms (such as cam mechanisms). A similar discussion as for the planar gearing can be conducted for the spatial gearing. The gearing analysis, as supplemented by the analysis of the gear contact trace, is called Tooth Contact Analysis (TCA) [4]. Programs for spatial TCA were first developed by Gleason Works (1960) and by Litvin and Gutman (1981).
It is assumed that the tooth profiles are given, and the distance between the gear wheel axes is also given. The gear ratio function (relationship between gear wheel rotations) and the gearing lines need to be determined.
is given, which satisfies equations (9). In the next step, the successive point is determined, which satisfies equations (9) 1  where the superscript denotes the successive approximation.
A system of two equations is chosen from the system of equations (9). It is assumed that the following equations are chosen: 0 , , ,  (14) should be solved again, and then the condition (15) needs to be rechecked.
The above algorithm involves splitting the system (9) of three equations into two subsystems, one of them containing two equations and the other one, and solving them independently.

The numerical algorithm
The parameters 1 I and 2 I are the angles of rotations of gear wheels 1 and 2, while the parameters 1 T and 2 T are the parameters of the gear wheel profiles 1 6 and 2 6 .
In the reference system f 6 , in the system of coordinates where the subscript f The algorithm described above is simple, which constitutes its great asset. It can also be used for the analysis of spatial gearing.
In the case of spatial gearing analysis, that is the analysis of toothing surface mating, the surfaces should be determined as sets of profiles in parallel cutting planes - Figure 4b. The above-described algorithm should be repeated for successive cutting planes fg z w g ,..
The surface equations can be written as where ig z defines the position of the cutting plane, while being also a surface parameter.
The surface equations in the reference system f 6 are as follows where the subscript g identifies the cutting plane, while the subscript l , the ordinate (with the abscissa fgi x ) in that plane.
As a result, the surfaces will be determined in the form of sets of normalized points lying on ordinate lines in cutting planes, which enables the examination of their mutual position and determination of the gear ratio function. Whereas, the surfaces may only contact linearly or be tangent in one point only. Therefore, for a fixed value of angle r

Example
A spur gear with an involute tooth profile is taken - Figure 5. The The gear ratio function is a linear function, which means that the gear ratio is constant. The error of the gear wheel axis distance has caused a parallel shift in the gear ratio function - Figure 6a. On the other hand, the modification of the profile by either rounding off or truncating the tooth tip - Figure 7, affects the gear ratio function - Figure 6b. The result is a gear ratio error and uneven gear operation.

Conclusions
The presented algorithm does not require the solving of systems of equations, in contrast to the analytical method using the Newtonian method for solving the system of equations. There is no need for allowing for the condition of normal unit vector collinearity, either.
The program consists of a module, in which the user enters profiles to be examined, and a universal module, in which the profile contact points are determined. The effectiveness of the Newtonian method depends largely on the accuracy of the first approximations of roots, while in the proposed method, root variability ranges can be taken with a greater approximation.
The developed algorithm and computation program can also be used for the analysis of spatial gearing.
The computation accuracy depends on the number of iteration steps, the number of profile points and the number of grid lines, and can be easily changed.