1-node coordinate calculation principle The profile curve of the cycloidal gear (as shown in 1) is the equidistant line of the short external cycloid. The equation is:
x=Rzsin-Asin(zb 1) r3sin(-)y=Rzcos-Acos(zb 1)-r3cos(-)(1) where Rz is the center circle radius of the pin wheel; A is the eccentricity; r3 is the pin wheel Radius; zb is the number of teeth of the cycloidal wheel; it is the parameter angle, that is, the angle of the cycloidal wheel; (-) is the angle between the normal of a point on the cycloidal profile and the Y-axis.
=arctansin(zb)1/k1-cos(zb)(2)
Where k1 is the short-width coefficient; k1=Azk/Rz=r1/Rz; zk is the number of pinwheel teeth, zk=zb 1; r1 is the pitch radius of the pin wheel.
According to the differential theory, the radius of curvature of any point on the profile of the cycloidal gear can be obtained from equation (1): =[1 k21-2k1cos(zb)]3/2Rzk1(zb 2)cos(zb)-[ 1 k21(zb 1)] r3(3)
If the given approximation error is, the radius of curvature of the current node on the tooth profile curve is, the shape of the neighborhood of this node can be replaced by the arc of radius. To ensure that the approximation error is less than, you can estimate the step size of the current node to the next node:
l=42-28(4)
The step length is the chord length of this micro-curve segment, which can be expressed as:
l=dxd2 dyd2 (5)
maximum. It can also be seen from the figure that the excessive pressure of the progressive pressure is unfavorable for the improvement of the ultrasonic processing efficiency.
b 1) r3cos(-)d-1dyd=-Rzsin A(zb 1)sin(zb 1) r3sin(-)d-1d=zb[1/k1cos(zb)-1][1/k1-cos(zb )] 2cos2 substituting the above results into equation (5), after shifting the term, the increment of the parameter angle can be found: =ldxd2 dyd2
Summarizing the above results, the steps for finding the nodes on the profile of the cycloidal gear are as follows.
1) Given the initial angle ni (i = 1) of the parameter angle i = 0 on the tooth profile curve, find xi = 0; yi = Rz - r3 according to the equation (1). Find i from equation (2). Find i from equation (3). Li is obtained from the formula (4). Find dxidi; dyidi at this point according to equation (5). Find i from equation (6).
2) Find the coordinates of the next node ni 1 and the actual step size i 1 = i i into the equation (1) to find xi 1 and yi 1. The coordinate increments xi, yi and the actual step length lir at this time are: xi=xi 1-xiyi=yi 1-yilir=x2i y2i(7)
3) Let i 1=i, repeat steps 1) and 2), and get the coordinates of the third point and the actual step size. Repeat this way until i=/zb. Thus, the node coordinates on the contour curve on the 1/2 circumferential pitch on the cycloidal wheel have been found. Due to the repeated and symmetrical tooth shape, the node coordinates of the half tooth are obtained according to the above steps, and a numerical control machining subroutine is programmed. It is easy to obtain the subroutine calling function, mirror programming function and rotating coordinate function of the milling machine numerical control system. The NC program for machining the entire cycloidal wheel is machined with a general end mill on CNC milling, and the tooth profile of the cycloidal wheel is obtained. It should be noted that the diameter of the milling cutter is reasonably selected to avoid interference.
2 Calculation example The main parameters of the cycloidal pinion drive pair are: zb=59, zk=60, A=2mm, r3=85mm.Rz=165mm, k1=07272727,=002. Find the error straight line approaching the cycloidal tooth The node coordinates of the profile.
According to the method described in this paper, the coordinates of all the tooth nodes on one tooth side are calculated and listed in the middle. It can be seen from the tabular data that the node coordinates fall on the tooth profile curve of the cycloidal wheel without any error. The actual step size is close to the estimated step size. Only the 5, 6, 7, and 8 points have large differences. This is because the radius of curvature of the curve has a large change, but the actual step size is less than the estimated step size. Therefore, the approach error must be within the allowable range and does not affect the machining accuracy.
In summary, the algorithm proposed in this paper can be used in the actual CNC machining of the cycloidal wheel due to its simple calculation and reliable accuracy.
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