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Geometric Simulation of Involute Gear Hobbing in Transverse Section *

*Maple 6

B.Laczik Technical University Budapest, Hungary

> restart; with(plots); with(geometry)

Warning, the name changecoords has been redefined

> CC := draw(circle(C,x^2+y^2 = (24.5/2)^2,[x, y]),co...

Basic geometric data for involute gear:

z||i - number of teeth

m - modulus in mm (modulus = Diametral Pitch/25.4)

m||n - normal modulus in transverse section of manufactured gear

xx||i - addendum modification coefficient of gear i

h||t - height factor of teeth between pitch and bottom circle

h||k - height factor of teeth between pitch and addendum circle

c -clearance factor for depths

A - distance of axes

alpha - pressure angle in grad, min and sec

beta - helix angle in grad, min and sec

N - number of position of tool by process of hobbing animation

s - number of teeth of tool

delta - increment angle of position of tool

> z || 1 := 8; z || 2 := 11; m := 1; xx || 1 := .4; x...

> alpha || 1 := 20; alpha || 2 := 0; alpha || 3 := 0;...

> alpha := (alpha || 1+alpha || 2/60+alpha || 3/3600)...

> N := 30; s := 5; delta := -.7e-1

Generating corner points of one tooth on basic rack

> m || n := m/cos(beta); d || 1 := z || 1*m || n; R |...

> w || x1 := h || t*m || n*tan(alpha)+m || n*Pi/4; w ...

> w || y1 := h || t*m; w || y2 := (h || k+c)*m; Delta...

Generating corner points of all teeth on basic racks

> Q[0] := -w || x1+I*w || y1-Delta; Q[1] := -w || x2-...
Q[0] := -w || x1+I*w || y1-Delta; Q[1] := -w || x2-...
Q[0] := -w || x1+I*w || y1-Delta; Q[1] := -w || x2-...
Q[0] := -w || x1+I*w || y1-Delta; Q[1] := -w || x2-...

> for p from 4 to (4*s-5) do
Q[p]:=Q[p-4]+m||n*Pi:
Q[p+4]:=Q[p]+m||n*Pi:
od:
Q[4*s]:=Q[0]:

> QQ[0] := Q[2]; QQ[1] := Q[3]; QQ[2] := Q[4]; QQ[3] ...

> for p from 4 to (4*s-5) do
QQ[p]:=QQ[p-4]-m||n*Pi:
QQ[p+4]:=QQ[p]-m||n*Pi:
od:
QQ[4*s]:=QQ[0]:

Simulation of hobbing process

> for k from 0 to 4*s do
for j from 0 to N do
beta||j:=delta*j+.52: gamma||j:=(R.2/R.1)*beta.j:
h||1[j,k]:=evalf(((Q[k]+R||1*gamma||j)+I*(R||1+xx||1*m||n))*exp(I*gamma||j)):
h||2[j,k]:=evalf(((QQ[k]-R||2*beta||j)-I*(R||2+xx||2*m||n))*exp(I*beta||j)):
for p from 1 to 2 do
H||p[j,k]:=[Re(h||p[j,k]),Im(h||p[j,k])]: od; od;
od;

> q||1:=pointplot([seq(seq([H||1[j,k][1],H||1[j,k][2]],k=0..4*s),j=0..N)],style=line,color=blue,connect=true):

> q||2:=pointplot([seq(seq([-H||2[j,k][1],H||2[j,k][2]+A+(xx||1+xx||2)*m||n],k=0..4*s),j=0..N)],style=line,color=magenta,connect=true):

> q||3:=pointplot([seq(seq([H||2[j,k][1],H||2[j,k][2]+A+(xx||1+xx||2)*m||n],k=0..4*s),j=0..N)],style=line,color=magenta,connect=true):

> for i from 4 to 8 do
q||i:=pointplot([seq(seq([H||1[j,k][1],H||1[j,k][2]],k=i+1),j=0..N)],style=line,color=red,thickness=5,connect=true):
qq||i:=pointplot([seq(seq([-H||2[j,k][1],H||2[j,k][2]+A+(xx||1+xx||2)*m||n],k=i+1),j=0..N)],style=line,color=blue,thickness=5,connect=true):
od:

> display([q||1, q||2, q||3],scaling = constrained,title = `Hobbing of involute gears without undercutting line`);

[Maple Plot]

> display([q||1,q||4,q||5,q||8],view=[-5*m||n..5*m||n,(R||1-3*m||n)..(R||1+2*m||n)],scaling=constrained,title=`The undercutting lines`);

[Maple Plot]

> display([qq||5,qq||4,qq||8,q||2],scaling=constrained,view=[-mn..7*mn,R||2-2*mn..R||2+4*mn],title=`The undercutting lines`);

[Maple Plot]

>