第十一章 齒輪(2)-齒輪曲線

11.2 齒輪曲線之形成

由前面齒列之程式可以瞭解齒輪雖然僅有齒牙相接觸，但其曲線之形成仍然必須依據特定的曲線規則。漸開線是最常用的曲線，為此可以使用齒列作為切刀，修正曲線之外廓。程式gear_curve.m除可產生齒輪之基本曲線外，可以依所需之齒數建立一個完整的齒輪，並且使其作適當的迴轉運動。程式中另呼叫onegear.m、rotate2xy.m。前者是將一個完整的齒輪的外形座標組合，後者則是座標旋轉的程式。程式gear_curve.m之呼叫格式如下：

　　function [rack,gear]=gear_curve(Dp,N,phi,ar,br,rt,rf,ag,ptregion)

Dp      = 節矩
ar      = 齒列之齒冠
br      = 齒列之齒根
phi     = 壓力角， degrees
rt      = 頂部圓角半徑
rf      = 根部圓角半徑
r       = 齒輪之節圓半徑
ag      = 齒輪之齒冠
bg      = 齒輪之齒根
ptregion= 齒形之五區點數，列矩陣，例如：[5 15 5 15  5]
N       = 齒輪之齒數

function [rack,gear]=gear_curve(Dp,N,phi,ar,br,rt,rf,ag,ptregion)
%
% Matlab program to draw a gear generated by a straight flanked hob.
%
%Some of the important variables used are:
%
% Example  gear_curve(10,10,20,1.25,1,0.02,0.04,1.0,[30 30 40 30 30],10)
%          gear_curve(10,10)
%
%Dp      = Diametral pitch
%ar      = Addendum constant for rack
%br      = Dedendum constant for rack
%phi     = pressure angle in degrees
%rt      = tip radius of rack
%rf      = fillet radius of rack
%r       = pitch radius of gear
%ag      = Addendum constant for gear
%bg      = Dedendum constant for gear
%ptregion= No. of points for five regions, row matrix
%N       = Number of gear teeth
if nargin<4,
ar=1.25;br=1;rt=0.02;rf=0.04;ag=1;
%ptregion=[10 15 20 15 15];
ptregion=[10 15 10 15 10];
end
th=linspace(0,2*pi,50);
if nargin<3,phi=20;end ans="'y';" d2r="pi/180;" ansp="'y';" pc="pi/Dp;" r="pc*N/(pi*2);%" ro="r+ag/Dp;" phir="phi*d2r;" rackline="line('xdata'," rackpitchline="line('xdata'," xmax="max(rack(1,:));xmin="min(rack(1,:));xrange="xmax-xmin;" ymax="max(rack(2,:));ymin="min(rack(2,:));yrange="ymax-ymin;" ymin="ymin-0.05*yrange;" ymax="ymax+0.05*yrange;" dtheta="2*pi/N/d2r;" gear0="gear;" i="1:N-1," gear="[gear," gearline="line('xdata'," gearpitchline="line('xdata'," xtemp="gear(1,:);ytemp="gear(2,:);" xmax="max(xtemp);xmin="min(xtemp);xr="xmax-xmin;" ymax="max(ytemp);ymin="min(ytemp);yr="ymax-ymin;" s="0.1;" xgear="xtemp';ygear="ytemp';" ainc="1;" theta="0;i="0;">360,theta=theta-360;i=i+1; end
set(gearline, 'xdata', xgear, 'ydata', ygear);
[xgear,ygear]=rotate2xy(xgear,ygear,ainc);
theta=theta+ainc;
drawnow;
pause(0.01);
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [rack,n]=rackm(phir,Dp,A,B,rt,rf,ptregion)
%
%Calculate a rack profile, with ptregion specifies 5-region points
% phir:pressure angle in radian
% Dp:diametral pitch
% ar: addendium
% br: bottom reference,dedendum of rack
% rt  = tip radius of rack
% rf  = fillet radius of rack
gamma=pi/2-phir;
lt=pi/(2*Dp)-2*A*tan(phir)-2*rt*tan(gamma/2);
lb=pi/(2*Dp)-2*B*tan(phir)-2*rf*tan(gamma/2);
if lt < 0
lt=0;
rt=(pi/(2*Dp)-2*A*tan(phir))/(2*tan(gamma/2));
end
if lb <0
lb=0;
rf=(pi/(2*Dp)-2*B*tan(phir))/(2*tan(gamma/2));
end
i=0;
for k=1:length(ptregion) %for 5 regions
beta=0.0;nx=ptregion(k);dbeta=1/nx;
for j=1:nx
i=i+1; 
beta=beta+dbeta;
switch k
case 1
rack(:,i)=[beta*lt/2;A];n(:,i)=[0;1.0];
case 2
rack(:,i)=[lt/2+rt*sin(beta*gamma);...
A-rt*(1-cos(beta*gamma))];
n(:,i)=[sin(beta*gamma);cos(beta*gamma)];
case 3
sg=sin(gamma);cg=cos(gamma);
rack(:,i)=[(1-beta)*(lt/2+rt*sg)+beta*(pi/(2*Dp)-lb/2-rf*sg);...
(1-beta)*(A-rt*(1-cg))+beta*(-B+rf*(1-cg))];
n(:,i)=[cos(phir);sin(phir)];
case 4
sb=sin((1-beta)*gamma); cb=cos((1-beta)*gamma);
rack(:,i)=[pi/(2*Dp)-lb/2-rf*sb;-B+rf*(1-cb)];
n(:,i)=[sb;cb];
case 5
rack(:,i)=[pi/(2*Dp)-lb/2*(1-beta);-B];
n(:,i)=[0;1];
end
end
end

%%%%%%%%%%%%%%%%   

function [gear,plx,ply]=onegear(rack,n,r,ro,N)
%
%find coordinates of one gear
% rack:coordinates of a rack
% n: normal vector of each point on the rack
% r: pictch radius
% ro:outside radius
% N: no. of teeth
d2r=pi/180;
ang=(rack(1,:).*n(2,:)-rack(2,:).*n(1,:))./(r*n(2,:));
geart(1,:)=-cos(ang).*(rack(1,:)-r*ang)+sin(ang).*(rack(2,:)-r);
geart(2,:)=-sin(ang).*(rack(1,:)-r*ang)-cos(ang).*(rack(2,:)-r);

% Define the pitch circle arc

maxang=max(ang);
minang=min(ang);
dk=(maxang-minang)/20;
i=0;
for k=minang:dk:maxang
i=i+1;
plx(i)=r*sin(k);
ply(i)=r*cos(k);
end 

% Rotate the coordinate system by -pi/N so that the y axis is along the
% centerline of the gear tooth instead of the centerline of the gap.

[xx,yy]=rotate2xy(geart(1,:)',geart(2,:)',-pi/N/d2r);
xx=xx';yy=yy';
% Test for the undercut points which cannot be on the gear
itotal=length(rack(1,:));
jstop=itotal+1;
istop=itotal+1;
for i = 1:itotal-2
for j=i+1:itotal-1
if istop==itotal+1 & jstop==itotal+1
ra=[xx(i) yy(i)];
rb=[xx(i+1) yy(i+1)];
rc=[xx(j) yy(j)];
rd=[xx(j+1) yy(j+1)];
rba=rb-ra;rca=rc-ra;
rac=ra-rc;rda=rd-ra;
rdc=rd-rc;rbc=rb-rc;
rbd=rb-rd;
Aabc=rba(1)*rca(2)-rba(2)*rca(1);
Aabd=rba(1)*rda(2)-rba(2)*rda(1);
Acda=rdc(1)*rac(2)-rdc(2)*rac(1);
Aadb=rda(1)*rba(2)-rda(2)*rba(1);
Acdb=rdc(1)*rbc(2)-rdc(2)*rbc(1);
test1=Aabc*Aabd;
test2=Acda*Acdb;   

% Check for the intersection of two line segments

if test1<0 & test2<0
istop=i;
jstop=j+1;
alpha=abs(Acdb)/(abs(Acda)+abs(Acdb));
re(1)=alpha*ra(1)+(1-alpha)*rb(1);
re(2)=alpha*ra(2)+(1-alpha)*rb(2);
j=itotal;
i=itotal;
end 
end  
end
end

% If two lines intersect, delete the points in the undercut region.

j=0;
for i=1:itotal
if i<=istop | i >= jstop
j=j+1;
geart(:,j)=[xx(i);yy(i)];
if i==istop
j=j+1;
geart(1,j)=re(1);
geart(2,j)=re(2);
end
end
end
itotal=j;

% Check that the points do not extend beyond the addendum circle (ro).

j=0;
rtest1=0;
istop='n';
i=0;
for i=1:itotal
if istop=='n'
rtest2=sqrt(geart(1,i)^2+geart(2,i)^2);
if rtest2<=ro
j=j+1;
gear(:,j)=geart(:,i);
xx(j)=gear(1,j);
yy(j)=gear(2,j);
end

% If the points extend beyond the addendum circle, discard the
% points beyond ro and find the intersection of the gear with the
% ro circle.

if rtest2>ro
alpha=(ro-rtest1)/(rtest2-rtest1);
j=j+1;
gear(:,j)=geart(:,i-1)+alpha*(geart(:,i)-geart(:,i-1));
xx(j)=gear(1,j);
yy(j)=gear(2,j);
istop='y';
end
rtest1=rtest2;
end
end

% if the addendum is reached replace the points beyond the
% addendum with points on a circle arc.

if j<itotal
beta = atan2(gear(1,j), gear(2,j));
dbeta=beta/5;
betak=beta;
for k=1:5
j=j+1;
betak=betak-dbeta;
sbetak=sin(betak);
cbetak=cos(betak);
gear(1,j)=ro*sbetak;
gear(2,j)=ro*cbetak;
end
end
jtotal=j;

% Reflect the points to draw the rest of the gear tooth

for i=1:jtotal-1
j=j+1;
gear(1,j)=-gear(1,jtotal-i);
gear(2,j)=gear(2,jtotal-i);
end 
itotal=j;

%%%%%%%%%%%%%%%%%%%%%%%%

function [x,y]=rotate2xy(x,y,theta,x0,y0)
if nargin<4, x0=0;y0=0;end;
th=theta*pi/180;
c=cos(th);s=sin(th);fact=[c s;-s c];
coords=[x y]*fact;
x=coords(:,1)+x0;y=coords(:,2)+y0;

執行例：

>>gear_curve(10,10);

齒列之外形

單齒之外形

整個齒輪之齒形