3.2. Pitch Circle Model of Double Arc Non-Circular Gears
A new method of double arc non-circular gear design is proposed in this paper, in which r′1 and r′2 are replaced by the original y1 and y2 to form a new double arc pitch curve. Given the preceding, the double arc pitch curve design is the critical issue of double arc non-circular gear design. The new method is the active treatment of the double arc, which ensures a smooth transition of the intersection position and the pitch curve radius of the integer teeth. Because y1 and y2 are shown to be numerically unstable during nonlinear solutions, a double arc non-circular gear pitch circle model design is proposed in this study. The benefit of the double arc non-circular gear pitch circle model is to effectively avoid the complex structure points y1 and y2, requiring just X = [x1 r′1 r1 x2 r′2 r2], which considerably increases design efficiency.
Through verification and research, it was determined that the non-integer tooth rounding on the pitch curve and the smooth transition of the double arc intersection are core to the design of the double arc pitch curve hydraulic motor. The model for the normal meshing gear pitch curve tangent can meet the tangency of the double arc at the intersection point, resulting in a smooth transition of the pitch curve while solving the non-integer tooth rounding problem. The approach satisfies the operating condition of maintaining the pitch curve tangent throughout precise non-circular gear operation, as shown in
Figure 5.
According to
Figure 5, the three engagement positions are shown, and the work processing has a significant impact on the final performance of the non-circular gear. Through these three engagement positions, the design of the double arc pitch curve can be replaced by pitch circle of integer teeth.
The first limit position satisfies the tangency of the pitch curve, and the second is at the point where the double arcs meet, which is not considered provisionally. As shown in
Figure 3, we find that the range of numerical fluctuations of the nonlinearly solved numerical
Figure 3d is the largest, and the most closely related is the third engagement position. Therefore, the third engagement position is prioritized in this paper.
The value of
y2 is taken multiple times with varying values, while the value of
y1 remains stable since the value of
y1 is more stable than that of
y2. We discovered that as the value of
y2 falls, the interference between the center wheel and the planetary wheel is reduced. When other values are constant, the value of
y2 declines to a certain value to ensure that the third limit position of the non-circular gear pitch curve is tangential to the others, as shown in
Figure 6.
We found that it changes the
y2 value from the initial interference to the end tangent by continuously repeating the geometric mapping of the third limit positions using different
y2 values, as shown in
Table 1. However, according to the graphical analysis of the data above, the value of
y2 should be limited between 400 and 1000. To avoid interference, the constraints of Equation (9) should be established by the principle of relative curvature consistency of the surface, which satisfies the requirement of the tangency of non-circular gear pitch curves [
28].
—The difference of the normal curvature of the curve along the two perpendicular directions (i = 1, 2).
—The difference in the short-range deflection rate of the curve along the two perpendicular directions.
Regarding gear design, gears are designed with integer teeth instead of non-integer teeth to reduce costs. The number of teeth on the arcs of the inner ring
r1 and
r′1 is actively rounded based on the value of the nonlinear solution. A part of an integer gear is used as a double arc pitch curve teeth profile, which requires rescaling based on known
Χ values and satisfying the constraints of the geometry of non-circular gear pitch curve design, as shown in
Figure 7.
Figure 7 depicts three parameters of the inner ring
x1,
y1, and
r1, redesigned by the pitch circle model, where
r11,
r11, and
x11 become the core data of the internal ring design.
—The difference after taking an integer number of teeth for the double arc pitch curve;
x11—The x1 of the pitch circle model;
r11—The r1 of the pitch circle model;
r′11—The r′1 of the pitch circle model;
Calculating the pitch circle model parameters depends on the geometric and transmission relationships.
The latter
x11,
r11, and
r′11 are replaced by
x1,
r1, and
r′1 to facilitate subsequent calculations and reading. First, the ray with an angle of 30° is made from the origin. Then, a circle is drawn with the connection point between
x1 and
r1 as the center,
O1O’1 is the sum of the radii of the double arcs
r1 and
r′1, as the radius makes an intersection with the ray, whose distance from the intersection point to the origin is the length of
y1. The meeting and source are connected from
Figure 7a to
Figure 7b. The three basic parameters
x1,
y1, and
r1 of the internal gear ring can be obtained through the pitch circle model, which can be used as the basis to achieve the design requirements of a smooth transition of the intersection point of the double arc pitch curve and the number of teeth to be rounded.
Similarly, the center wheel can be handled in the same way.
The final determination of Χ = [x1 r′1 r1 x2 r′2 r2] parameters through the model for the normal meshing gear pitch curve tangent and using the geometric constraints can be found for all the values of X = [x1 y1 r1 x2 y2 r2].
In the following procedure of the integer teeth based on the model for normal meshing gear, the problem of rounding up or down for non-integer teeth is verified using orthogonal experiments. The number of integer teeth for each section arc is determined by the orthogonal experiments, which will be addressed in the next section.
When compared to the original data, the largest change in the pitch circle model’s X value is 1.7%, while the minimum is 0.0332%. This assures that the nonlinear solution is correct. The tangency of the non-circular gear pitch curve is verified by Equation (9).
3.3. Treatment of Tooth Profile at the Point of Intersection of the Double Arc
The coordinate change’s aim is to use a coordinate change to transfer the tooth profile on the planetary wheel to the inner gear ring. The tooth profile at the intersection of the double arcs is corrected, which ensures a relatively stable mesh, as it is similar to the interpolation process in gear manufacturing. The known planetary wheel profile involute equation, through the equal arc length method, can determine the position of the planetary wheel and the rotation angle, which can be determined at the double arc intersection at the single side of the tooth profile [
29].
As shown in
Figure 8, the dashed circle at point
C indicates the initial position of the planetary wheel, while the solid circle at point
G indicates the end position of the planetary wheel. When the angle of planetary rotation is
θ3,
C’ indicates the angle of the planetary wheel after its rotation, and point
G indicates the point of tangency after the rotation.
—Pole diameter of the planetary wheel;
—Pole diameter of the inner gear ring;
—Planetary wheel angle;
—Angle of rotation of the planetary wheel on the inner gear ring;
Its planetary wheels are known to be standard involute equations and coordinates of the origin.
Considering that the planetary wheel is tangential to the inner gear ring everywhere in the process of rotation, there is both rotation and translation; therefore, by transforming the matrix:
—The angle of rotation of the planetary wheel, the angle of rotation is very small, able to form the side of a tooth;
—The difference between the coordinates of the origin of and in the coordinate system of , the variation of the axis X;
—The difference between the coordinates of the origin of and in the coordinate system of , the variation of the axis Y;
By translation of the coordinates, the involute is transformed from under the
coordinate system to under
, and the transformation matrix is shown below:
- angle of rotation required to rotate coordinate system to coincide with coordinate system ;
—The difference between the coordinates of the origin of and in the coordinate system of , the variation of the axis X;
—The difference between the coordinates of the origin of and in the coordinate system of , the variation of the axis Y;
The unilateral profile of the teeth profile of the inner gear ring:
Through this method, we can transform the planetary wheel’s involute to the inner gear ring to design the tooth profile at the point of intersection of the double arc and to ensure a smooth transition between the planetary wheel and the inner gear ring at the intersection of the double arc. The center wheel is also transformed by this kind of treatment.