Addendum modification of spur gears with equalized efficiency at the points where the meshing stars and ends

When determining the geometric dimensions of cylindrical spur gears, most often, specific addendum modifications are used. Their purpose is to obtain a well defined distance between the axes at the same time ensuring the correct meshing of teeth flanks over a longer time. The x1 and x2 addendum modification coefficients have influence on tooth shape and tooth thickness. Different assumptions or criteria can be used to determine their values as shown in [1] [7]. Balancing or equalization of the sliding velocity [1], specific sliding [1, 2] or power lost by friction [3] are used to influence friction losses to obtain better efficiency, lower operating temperature, noise and wearing to increase the life of the gears. Instead of equalizing the power losses the paper presents a new method to establish the specific addendum modification coefficients using the equalization of the efficiencies, at the A and E points, where the meshing begins and ends (see Fig. 1). Compared to [2] this method is considering friction coefficients with distinct values between the teeth flanks at beginning and the end of the contact allowing to study the gear efficiency in a general case. Compared to [3] the method is dealing directly with the efficiency the most important objective in the field of transmissions – and allows the study of the balanced gear efficiency for various friction coefficients and addendum modifications.


Introduction
When determining the geometric dimensions of cylindrical spur gears, most often, specific addendum modifications are used.Their purpose is to obtain a well defined distance between the axes at the same time ensuring the correct meshing of teeth flanks over a longer time.The x1 and x2 addendum modification coefficients have influence on tooth shape and tooth thickness.Different assumptions or criteria can be used to determine their values as shown in [1] - [7].Balancing or equalization of the sliding velocity [1], specific sliding [1,2] or power lost by friction [3] are used to influence friction losses to obtain better efficiency, lower operating temperature, noise and wearing to increase the life of the gears.Instead of equalizing the power losses the paper presents a new method to establish the specific addendum modification coefficients using the equalization of the efficiencies, at the A and E points, where the meshing begins and ends (see Fig. 1).Compared to [2] this method is considering friction coefficients with distinct values between the teeth flanks at beginning and the end of the contact allowing to study the gear efficiency in a general case.Compared to [3] the method is dealing directly with the efficiency -the most important objective in the field of transmissionsand allows the study of the balanced gear efficiency for various friction coefficients and addendum modifications.

Efficiencies at the beginning and end of the teeth meshing
The efficiency of the teeth meshing is determined using the following general relation: where T1 is the torque of the driving forces; T2 the torque of the useful resistance forces; ω1 the angular velocity of the driving wheel; ω2 the angular velocity of the driven wheel; z1 the number of teeth of the driving wheel; z2 the number of teeth of the driven wheel.Considering Fig. 1 the torque at the beginning of the teeth meshing, in point A, can be written with the following expressions: Using the same methodology the torque is determined in the E point, where the meshing ends, thus: where FnE is the normal force in the E meshing point, μE the friction coefficient between the teeth flanks, in the E point, eE the EC segment length from Fig. 1.From ( 5) and ( 6) the meshing efficiency in the E point becomes: In order to check the possibility of making equal the (4) and ( 6) expressions we will compute and plot these surfaces depending on x1 and x2 addendum modifications.

Computation and graphical representation of the efficiency
The numerical results for the (4) and ( 7) expressions are obtained using the following known formulae [6] from the spur gear geometry: where αw is computed from the following equation: Fig. 2 The graphical representation of the ( 4) and ( 7) surfaces for z1 = 19, z2 = 19 and μ = 0.05 In order to obtain the graphical representation of the ( 4) and ( 6) surfaces the following input data were considered: μA = μE = μ = 0.05.The x1 and x2 values are in range [-1, 1] and the z1 and z2 values are fixed and the same for each surface representation.As seen in Fig. 2 the two surfaces intersect, so the equalization of the efficiencies, at the A and E points, have solutions.The curve from Fig. 3 was obtained using a Matlab code extracting the isoline of the intersection and then interpolating on surface (7).The results from these figures are however rough because:  the solutions are obtained using Matlab facilities (surface intersection extraction and interpolation) instead of solving directly the (4) = ( 7) equation;  in order to obtain results for the surfaces as Matlab requires, no limitations on the obtained results were applied; these concern the values of the meshing angle and the x1 and x2 values to avoid tooth undercut and sharpening.

Numerical results of the efficiency equalization
To avoid the tooth undercut and sharpening the following limitations were applied to the obtained x1, x2 results [1], [3]:  for tooth undercut:  for tooth sharpening by putting the following conditions: we obtain: with da1 the outside circle diameter of wheel 1, da2 the outside circle diameter of wheel 2, dv1 the sharp tip circle di-ameter of wheel 1, dv2 the sharp tip circle diameter of wheel 2. Equalization of the ( 4) and ( 7) relations give the following equation: The results from Table 1 are showing the raw data produced by the Matlab code.The [-1, 1] domain is divid-ed into 41 points for x2.Each solution is numbered from 1 to 41 in the i column of Table 1.Missing lines are representing a lack of solution for the (23) equalization equation.Some of the lines from Table 1 are eliminated in Table 2 as αw must be over 14°.Table 2 to Table 5 are condensed versions of valid solutions and together with the plots from Fig. 4 to Fig. 7 allow making the following conclusions:  higher μ friction coefficient will increase the number of valid solutions, however because the variation of the friction coefficient is very limited its influence is low;  higher number of teethe as high ratio of numbers of teeth favor the number of solutions;  the equalized efficiencies are maximum for the highest values of the positive of the x1, x2 pair solution;  high ratio of numbers of teeth favors a higher value of the equalized efficiencies.The x1 and x2 values are chosen independently under normal circumstances.Imposing the equalization condition a relationship between these to independent variables is defined and only one, of the two, will remain independent.As seen in the algorithm, the x2 values are given, while the x1 values result from the equalization condition if (23) has valid solutions.

Conclusions
The values of the specific addendum modifications x1 and x2 can be determined by this equalization procedure.A balanced efficiency will lead to balanced efficiency loss.As the efficiency loss is connected to sliding and rolling frictional losses the equalization will lead to a uniform wearing of teeth flanks at the points where the meshing ends and begins during load transmission.The algorithm allows finding a better balanced efficiency where the power losses are reduced.Reduced power losses lead to better operating conditions (lower operating temperature, noise and wear).The paper gives a new method for obtaining the geometrical dimensions for involute spur gears based on the equalization conditions of the efficiencies, at the A and E points, where the meshing begins and ends.Numerical results and plots for the x1 and x2 addendum modifications are obtained, using the Matlab programming environment.The solutions are obtained for different pairs of teeth numbers, when the equalization condition stands, together with the limitations given by the teeth undercut, sharpening and operating pressure angle.

Fig. 1
Fig. 1 Meshing of involute spur gears where FnA is the normal force in the A meshing point, rb1 and rb2 the base circles radii of the meshing gears with involute profiles (rb2 z1 = rb1 z2), μA the friction coefficient between the teeth flanks, in the A point, where the meshing begins, αw the meshing angle, eA the AC segment length from Fig. 1.From (2) and (3) the meshing efficiency in the A point becomes: