Table 1. Formula for a gear surface
Concave
UCC=(w*w+(dgear/2.0)*(dgear/2.0)-2.0*(dgear/2.0)*(Wcutter/2.0)*sin(spiralang))/(2.0*w);
oo=2*atan((VG-sqrt(HG*HG+VG*VG-UCC*UCC))/(HG+VG));
o=oo/sin(GPitchConeAng);
Xg (u, w, δb1)= cos (δb1)*w*cos (u*sin(δb1));
Yg (u, w, δb1)= cos(o)*(sin(u)*sin(δb1)*w*cos (u*sin(δb1)) –cos (u)*w*sin(u*sin(δb1))) –
sin(o)*(cos(u)*sin(δb1)*w*cos (u*sinδb1)) +sin(u)*w*sin(u*sin(δb1)));
Zg(u,w,δb1)=sin(o)*(sin(u)*sin(δb1)*w*cos (u*sin(δb1)) -cos(u)*w*sin(u*sin(δb1)))+
cos(o)*(cos(u)*sin(δb1)*w*cos (u*sin(δb1)) +sin(u)*w*sin(u*sin(δb1)));
Convex
Xg1= Xg (-u, w, δb1);
Yg1= Yg (-u, w, δb1)*cos(base) - Zg (-u, w, δb1)*sin(base) ;
Zg1= Yg (-u, w, δb1) *sin(base) +Zg (-u, w, δb1)*cos(base);
where dgear is diameter of gear, Wcutter is gear milling cutter diameter, HG and VG
represent the locations of the centres of the circular cutter, base is the rotation angle along
X-axis on base circle from concave surface to convex surface on the adjacent tooth. Spiralang
is spiral angle and GPitchConeAng is gear pitch cone angle. The detailed introduction was in
(Suh S.H. 2001).
UCCp = (w*w+(dpinion/2.0)*(dpinion/2.0)-2.0*(dpinion/2.0)*Pcuttercave*sin(spiralang))/(2.0*w);
oop=2.0*atan((VPcave-sqrt(HPcave*HPcave+VPcave*VPcave-UCCp*UCCp))/(HPcave+VPcave));
op=oop/sin(PPitchConeAng);
Xp= cos(δb)*w*cos(u*sin(δb))
Yp= cos(op)*(sin(u)*sin(δb)*w*cos (u*sin(δb)) –cos (u)*w*sin(u*sin(δb)))-
sin(op)*(cos(u)*sin(δb)*w*cos (u*sinδb)) +sin(u)*w*sin(u*sin(δb)));
Zp= sin(op)*(sin(u)*sin(δb)*w*cos (u*sin(δb)) -cos(u)*w*sin(u*sin(δb)))+
cos(op)*(cos(u)*sin(δb)*w*cos (u*sin(δb)) +sin(u)*w*sin(u*sin(δb)));
UCCp1= (w*w+ (dpinion/2.0)*(dpinion/2.0) -2.0*(dpinion/2.0)*Pcuttervex *sin (spiralang))/(2.0*w);
oop1=2.0*atan ((VPvex-sqrt(HPvex*HPvex+VPvex*VPvex-UCCp1*UCCp1))/ (HPvex+VPvex));
op1=oop1/sin (PPitchConeAng);
Xp1= cos(δb)*w*cos(ur*sin(δb));
Yp1= cos(op1)*(sin(ur)*sin(δb)*w*cos ( ur*sin(δb))-cos (ur)*w*sin( ur*sin(δb)))-
sin(op1)*(cos(ur)*sin(δb)*w*cos(ur*sin(δb))+sin(ur)*w*sin(ur*sin(δb)));
Zp1= sin(op1)*(sin(ur)*sin(δb)*w*cos(ur*sin(δb ))-cos(ur)*w*sin(ur*sinδb)))+
cos(op1)*(cos(ur)*sin(δb)*w*cos(ur*sin(δb))+sin(ur)*w*sin(ur*sin(δb)));
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