第十一章 齒輪(4)--兩齒輪之滾動

10.4 兩齒輪之滾動

若有兩個不同大小之齒輪在不同位置下相互契合，此時之滾動情形可利用下面程式示範。此程式稱為move2_gear.m，其呼叫方式與move2_rack.m幾乎相同。呼叫格式如下：

function move2_gear(Dpitch,nn1,nn2,phi,omega1)

輸入參數之定義如下：

Dpitch:節矩
nn1,nn2:兩齒輪之齒數
phi:壓力角，degrees
omega1: 齒輪１之角速度，rad/s

程式內容

function move2_gear(Dpitch,nn1,nn2,phi,omega1)
%    move2_gear(Dpitch,nn1,nn2,phi,omega1)
% To draw a whole gear
% Inputs:
%  Inputs:
%    Dpitch:dimetral pitch
%    nn1,nn2: no. of teeth for both gears
%    phi:pressure angle, degrees
%    omega1: angular velocity of gear 1
% Example move2_gear(10,15,20,20,10)
clf;
d2r=pi/180;delt=0.01;
[coord1,r1,rb1]=one_tooth(Dpitch,nn1,phi,360,0,0);
[coord2,r2,rb2]=one_tooth(Dpitch,nn2,phi,360,0,0);
st=180/nn2;if nn1+nn2>2*fix((nn1+nn2)/2),st=0;end
coord2=rotate2D(coord2,180+st,0,0);
xc1=coord1(:,1);yc1=coord1(:,2);
xc2=coord2(:,1);yc2=coord2(:,2);
height=max(r1,r2)*1.2;
ar=min(abs(r1),abs(r2));
coord=bushing(ar/5,0,0); % Get the coordinates of  1st bushing
xb1=coord(:,1)-r1;yb1=coord(:,2);
xb2=coord(:,1)+r2;yb2=coord(:,2);
coord=bushing(-r1,-r1,0);%Get the 1st pitch circle
xp1=coord(:,1);yp1=coord(:,2);
coord=bushing(-r2,r2,0);% Get the 2nd pitch circle
xp2=coord(:,1);yp2=coord(:,2);
plot(xb1,yb1,'r-');hold on;
plot(xb2,yb2,'k-');
plot(xp1,yp1,'r:');
plot(xp2,yp2,'k:');
plot([-r1,r2]',[0,0]','r:');
xx1=min([r1,r2])/2;phir=(90-phi)*d2r;
plot([0,0]',[-xx1*2,xx1*2]','b:');
plot([-xx1  xx1]',[-xx1*tan(phir), xx1*tan(phir)]','b:');

cir1=line('xdata',[],'ydata',[],'erasemode','xor','linewidth',1,'color','r');
cir2=line('xdata',[],'ydata',[],'erasemode','xor','linewidth',1,'color','k');
line1=line('xdata',[],'ydata',[],'erasemode','xor','linewidth',2,'color','r');
line2=line('xdata',[],'ydata',[],'erasemode','xor','linewidth',2,'color','k');
lx1=[0 -r1]';ly1=[0,0]';
lx2=[r2,0]';ly2=[0,0]';
axis([-2.5*r1 2.5*r2  -height height]);
axis equal;
title('Press Ctl-C to stop');
theta1=180;theta2=180;s1=omega1*delt/d2r;
while 1,
z1=rotate2D([xc1,yc1],theta1,-r1,0);
z2=rotate2D([xc2,yc2],theta2,r2,0);
L1=rotate2D([lx1,ly1],theta1,-r1,0);
L2=rotate2D([lx2,ly2],theta2,r2,0);
set(cir1,'xdata',z1(:,1),'ydata',z1(:,2)); % For 1st circle moving
set(cir2,'xdata',z2(:,1),'ydata',z2(:,2)); % For 2nd circle moving
set(line1,'xdata',L1(:,1),'ydata',L1(:,2)); % For 1st line
set(line2,'xdata',L2(:,1),'ydata',L2(:,2)); % For 2nd line
drawnow;
pause(1/s1);      %Stop for a while so we can see the graph
theta1=theta1+s1;
theta2=theta2-s1*r1/r2;
if theta1>360, theta1=theta1-360;end;   %Reverse the direction at bondary line
if theta2>360,theta2=theta2-360;end;
end




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [coords,rp,rb]=one_tooth(Dp,N,phi,range,x0,y0)
% [coords]=draw_gear(Dp,N,phi,range,x0,y0)
% To draw a whole gear
% Inputs:
%     Dp: Diametrial pitch
%     N: no of teeth in a gear
%     phi: pressure angle, degrees
%     range: the section range to be drawn
% Example one_tooth(10,10,20,360)
nn=10;
d2r=pi/180;
phir=phi*d2r;
rp=N/Dp/2;
pc=pi/Dp;
ra=rp+1/Dp;
rb=rp*cos(phir);
rd=rp-1.25/Dp;
thpb=pc/rp;% angle respect to one pitch
tp=pc/2;
rr=linspace(rb,ra,nn)';
invphi=tan(phir)-phir;
for i=1:nn
beta=acos(rp/rr(i)*cos(phir));
tt(i)=(tp/rp/2+invphi-tan(beta)+beta);
end
coord1=[rr.*cos(tt') rr.*sin(tt')];
coord3=reverse(coord1);
th1=linspace(thpb/2,thpb/2-tt(nn),nn)';
coord0=[cos(th1) sin(th1)]*rd;
th2=linspace(tt(nn),-tt(nn),nn)';
coord2=[cos(th2) sin(th2)]*ra;
coord4=reverse(coord0);
coord=[coord0;coord1;coord2;coord3;coord4];
theta=thpb/d2r;
coords=[];i=0;
while i<range
coord1=rotate2D(coord,-i,x0,y0);
coords=[coords;coord1];
i=i+theta;
end

function [B]=reverse(A)
nn=length(A); B=A;
for i=1:nn,B(i,:)=A(nn-i+1,:);end
B=[B(:,1) -B(:,2)];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [coords] = bushing(rr,x0,y0)
d2r=pi/180;
theta=[360:-10:0]*d2r;
r=abs(rr);
rx=r*cos(theta);ry=r*sin(theta);
if rr<0,
rx=rx+x0;
ry=ry+y0;
coords=[rx' ry'];
return;
end;
rx1=rx/2;rx2=-rx1;
ry1=ry/2;ry2=-ry1;
r4=r+r/4;r3=r/3;
bx=[ 0   0 0 -r -r -r4 -r4  r4 r4 -r  r r];
by=[r3 -r3 0  0 -r -r  -r4 -r4 -r -r -r 0];
coords(:,1)=[bx rx rx1 rx2]'+x0;
coords(:,2)=[by ry ry1 ry2]'+y0;

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

function [coords]=rotate2D(coord,theta,x0,y0)
th=theta*pi/180;
c=cos(th);s=sin(th);fact=[c s;-s c];
coords=coord*fact;
coords(:,1)=coords(:,1)+x0;
coords(:,2)=coords(:,2)+y0;

執行範例

>> move2_gear(10,15,20,20,10)

兩齒輪相互喫合運動之情形