Watches and clocks traditionally use cycloidal gearing. This section will focus on generation of the cycloidal gear profile. More details on the design can be found in British Standard 978 Part 2: Cycloidal Type Gears..
Development of the mathematical equations for the cycloidal curve uses simple
rotation. The equations can be obtained referring to the following figure. The epicycloid starts from the pitch circle on which the generating circle rolls.
Figure 184.108.40.206.1 Development of an epicycloidal curve
Considering two points, P and Q, on the generating circle, while the point Q moves to the postion Q’, the point P moves to the position P’. Since the arc lengths of PQ and P’Q’ are same, the position P’ can be determined by rotating the position Q’.
where r and rg are radii of the pitch circle and generating circle, respectively. Note that with negative value of rg the curve becomes hypocycloid.
The gear profile below pitch circle can be straight radial lines or can be generated by the hypocycloid.
Let’s take a case with N = 26 and DP = 4 as in the case of involute gear.
1. Sketch Curve > create the outside circle
2. Repeat this for the pitch, base and root circles
3. Change the dimension name for outside circle to OD, the first offset dimension to ADDENDUM, the second offset dimension to DEDENDUM, and the last offset dimension to CLEARANCE
5. Tools > Program > in the RELATIONS section, enter the relations as below > save and exit
PD = OD - 2*ADDENDUM
BD = PD - 2*DEDUNDUM
RD = BD - 2*CLEARANCE
6. Datum Curve > From Equations / Done > pick the default coordinates > Cartesian > enter the equation as below > File > Save > Exit > OK
7. Let’s create the hypocycloid below the pitch circle using the datum curve with same equation as below. The curves appear as
Edit > Feature
Operations > Copy > Move / Done > select both curves > OK (select)
> Rotate > Crv/Edge/Axis
> pick the axis at the center > select the direction and
9. Pick both curves in Model Tree > Mirror > pick the horizontal datum plane > OK (feature). Now, the curves appear as
10. Extrude > create a circular plate with UseEdge and outside diameter, and then the thickness of 1.0
11. Extrude > using UseEdge again, select the curves for boundary of the cut > use Trim to Entity to trim the corners (Dynamic Trim may leave stranded curves giving error messages at exit) > To Next for depth > Remove Materials > selection the directions of cut properly > Done (feature). One cut appears
12. Pick the cut (if not highlighted) > Pattern > select Axis for type in dropdown list > enter 26 for the number of patterns and the angle 360/26 > Done (feature). The final cycloidal gear appears as
Figure 220.127.116.11.2 Cycloidal gear and its profile
13. Complete the relations and inputs in the program as below > save and exit > Yes (to incorporate the changes) > Enter > select all values and then enter 26, 4, and 2.
The following shows the profiles for various diameters of generating circle. Note that as DG increases, the profile below pitch circle becomes almost straight just like real teeth that is often approximated by straight radial lines.
Figure 18.104.22.168.3 Cycloidal gear profiles for various diameter of generating circle