Fig.2 The involute spline on the right side

In the process of rolling, the revolution surface on the roller and the helicoids on the involute spline are always tangent when they make relative motion. The motion of roller can be seen as the tangent line motion along the roller surface. The helicoids and its dimensional position are constant. The revolution of the tangent line about the roller axis makes the revolution surface on the roller. The spiral motion of the line about the spline axis makes the helicoids on the involute spline. So the simultaneous equations of the tooth side and the conditional expression of tangent line can be made to get the expression of tangent line. Let the line revolves about the roller axis and the roller model can be built. According to the theoretical analysis, design model of roller can be expressed as follow:

 
The design model of axial section in the left roller can beshown as follow:
 
where, rb represents the radius of base circle. The xoy coordinate system is the same as the involute spline . The δ0 , the start point of the involute, rest with the parameter of the involute spline. θ means the motion angle when the spline revolves the z axis. U is a variable, p is the parameter of the helix.
p=P*z/2π
p*z means the lead of the helicoids. R is the distance between the point on the axial section and the Z axis. Σ is the angle between the Z axis on the spline and the rotational axis on the roller. A is the distance between the axis mentioned above.
In the rolling process, the amount of elastic and plastic deformation will be accumulated gradually .It will affect the precise of diameter and the spline’s teeth shape. So on the base of model mentioned above, the amendment method is brought
forward according to the experimental data.
III. DEVELOP CAD SYSTEM OF ROLLER
A. The Arithmetic of the CAD System
The fact that can be inferred from the rolling principle and roller model as follows:
The parameter p, Σ and a are setting parameter of roller. Rb and δ0 can be inferred from the given spline parameter. The starting and final value of variable u can be inferred from the end face parameters of involute spline. Corresponding to each value of u( the value of u vary according to a certain step), the value of θ can be solved from the ( 2) and (6). And the value of x,y and z can be get if the value of u and θ be put into the equation (1) and (5). Further the X,Y and Z can be get and the R and Z will be solved finally. And the curve fit to the roller’s axial section can be obtained. Each part of the curve can be gotten according to the model and can be fit to get the whole axial section’s shape of roller. From the solving process mentioned above, the value of θ is most difficult to get. The (2) can be expressed as follows：
 
To make sure the interval with the equation resolution, select two point x1 and x2 at random and solve f(x1) and f(x2). If f(x1)*f(x2)=0,one answer of equation have been found. If f(x1)*f(x2)<0,[x1,x2] is one resolution interval of equation f(x)=0. If f(x1)*f(x2)>0, the point x1 will be reselect along the degressive direction. In other words, if |f(x1)|< |f(x2)|,then the new point x1 will be reselected , according to the tendency from x1 to x2,to replace the old one. If not, the new point x2 will be reselected , according to the tendency from x2 to x1. Suppose f(x) has resolution in the  inteval [x1,x2]. If the interval be unbiased partitioned into n parts, the equation can be get as follows:
 
And suppose these smaller intervals {[xi-1,xi]} have max intervals at most. So the intervals can be scanned according to the step h. And the values of the adjacent functions are checked too. If f(xk)*f(xk-1)<0,the resolution interval is be fixed on and the scanning will stop. To get the resolution of the equations,the resolution can be gotten by the method of interval midpoint in each intervalwe get from (2). The curve fit can be realized by least square method. 加工渐开线花键辊英文文献和中文翻译(2):http://www.youerw.com/fanyi/lunwen_4689.html