# Plotting path of rods on a circular disk along a surface to deduce rack gear profile.

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##### 0 Comments

### Accepted Answer

Mathieu NOE
on 5 Jul 2024

ok, I had some fun today with your exercise ...

time to grind a bit the sharp edges though !

% a disk fitted with 4 rods

d = 2; % central disk diameter

% rods

w = 0.75; % width

l = 1.5; % length

%% main code

N = 100; % disk geometry is created with 4N points, rods with N/2 points for each side segment

[xx,yy] = geometry(d,w,l,N);

% let's plot the geometry once

figure(1)

plot(xx,yy,'*-');

title('The disk / rods geometry');

axis square

% let it roll half a turn (in the x positive direction

figure(2)

plot(xx,yy,'-');

hold on

N = 100; % iterations

% store all points for future use (boundary)

xx_all = [xx];

yy_all = [yy];

for k = 1:N

alpha = -pi*k/N;

[xxr,yyr] = roll(xx,yy,alpha,d,l);

plot(xxr,yyr,'-');

% pause(0.05)

% store all points for future use (boundary)

xx_all = [xx_all xxr];

yy_all = [yy_all yyr];

end

% axis square

% extract now the rack profile using

k = boundary(xx_all(:),yy_all(:),1);

x_rack = xx_all(k);

y_rack = yy_all(k);

% remove some points that shall not belong to the rack

% remove points above the top flat sections of the rack

ind = y_rack>-d/2*0.9;

x_rack(ind) = [];

y_rack(ind) = [];

% remove negative x points

ind = x_rack<0;

x_rack(ind) = [];

y_rack(ind) = [];

% remove trailing x points

ind = find(y_rack<=y_rack(1),1,'last');

x_rack(ind:end) = [];

y_rack(ind:end) = [];

plot(x_rack,y_rack,'r*-');

title('rolling !');

hold off

figure(3)

% shift the y coordinates so the bottom appears at y = 0

y_rack = y_rack - min(y_rack);

plot(x_rack,y_rack,'r*-');

title('The rack geometry');

% % smooth a bit the top of the rack

% y_racks = smoothdata(y_rack,'movmean',5);

% plot(x_rack,y_rack,'r*-',x_rack,y_racks,'b*-');

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

function [xxr,yyr] = roll(xx,yy,alpha,d,l)

% rotation

% alpha = -pi/10;

c = cos(alpha);

s = sin(alpha);

R = [c -s;s c];

tmp = R*[xx;yy];

xxr = tmp(1,:);

yyr = tmp(2,:);

% and x translation as if the disk is rolling to the right

xshift = -(d/2 + l)*alpha;

xxr = xxr + xshift;

end

%%

function [xx,yy] = geometry(d,w,l,N)

% disk

% d = 2;

n = 4*N;

theta = (0:n-1)/n*2*pi;

xd = d/2*cos(theta);

yd = d/2*sin(theta);

% add the four legs (rods)

% for more control over the rectangle shape , you can use this fex submission :

% https://fr.mathworks.com/matlabcentral/fileexchange/85418-rectangle2?s_tid=ta_fx_results

% first rod at position 3 o' clock

x1 = xd(1);

x2 = x1 + l;

y1 = yd(1)-w/2;

y2 = y1 + w;

% create some dots for the rods

k = round(N/2);

xx = linspace(x1,x2,k);

yy = linspace(y1,y2,k);

xBox = [xx x2*ones(1,k) xx(end:-1:1)];

yBox = [y1*ones(1,k) yy y2*ones(1,k)];

% remove some points of the circle

ind = (abs(xd)<w/2) | (abs(yd)<w/2);

xd(ind) = [];

yd(ind) = [];

% concatenate all coordinates

xx = [xd xBox yBox -xBox -yBox];

yy = [yd yBox xBox -yBox -xBox];

% put all those points in ascending theta order

[TH,R] = cart2pol(xx,yy);

[TH,ind] = sort(TH);

R = R(ind);

[xx,yy] = pol2cart(TH,R);

% close the curve

xx(end+1) = xx(1);

yy(end+1) = yy(1);

end

##### 8 Comments

Mathieu NOE
on 16 Jul 2024

hello again

simple modification below so you can now slect any number of rods you want

% a disk fitted with Nr rods

d = 3; % central disk diameter

rd = 0.75; % rods diameter

Nr = 8; % number of rods

%% main code

N = 30; % geometry is created with N points

[xx,yy] = geometry(d,rd,N,Nr);

% let's plot the geometry once

figure(1)

plot(xx,yy,'*-');

title('The disk / rods geometry');

axis square

% let it roll half a turn (in the x positive direction

figure(2)

plot(xx,yy,'-');

hold on

N = 100; % iterations

% store all points for future use (boundary)

xx_all = [xx(~isnan(xx))];

yy_all = [yy(~isnan(yy))];

for k = 1:N

alpha = -pi*k/N;

[xxr,yyr] = roll(xx,yy,alpha,d,rd);

plot(xxr,yyr,'-');

% store all points for future use (boundary)

xx_all = [xx_all xxr(~isnan(xxr))];

yy_all = [yy_all yyr(~isnan(yyr))];

end

% extract now the rack profile using boundary

k = boundary(xx_all(:),yy_all(:),1);

x_rack = xx_all(k);

y_rack = yy_all(k);

% select only the bottom section of the boundary

tf = find(islocalmin(y_rack, 'MinSeparation', 10) & (y_rack<min(y_rack)*0.9));

ind = tf(1):tf(end);

x_rack = x_rack(ind);

y_rack = y_rack(ind);

plot(x_rack,y_rack,'r*-');

axis equal

title('rolling !');

hold off

figure(3)

% shift the y coordinates so the bottom appears at y = 0

y_rack = y_rack - min(y_rack);

plot(x_rack,y_rack,'r-');

title('The rack geometry');

axis equal

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

function [xxr,yyr] = roll(xx,yy,alpha,d,rd)

% rotation

c = cos(alpha);

s = sin(alpha);

R = [c -s;s c];

tmp = R*[xx;yy];

xxr = tmp(1,:);

yyr = tmp(2,:);

% and x translation as if the disk is rolling to the right

xshift = -(d/2 + rd/2)*alpha;

xxr = xxr + xshift;

end

function [xx,yy] = geometry(d,rd,N,Nr)

% one peg

theta = (0:N)/N*2*pi;

xd = rd/2*cos(theta)+d/2;

yd = rd/2*sin(theta);

xx = [xd NaN];

yy = [yd NaN];

% add the remaining pegs

% peg are positioned on the pitch circle of diameter d

% first peg at position 3 o'clock

for k = 1:Nr-1

alpha = k*2*pi/Nr;

c = cos(alpha);

s = sin(alpha);

R = [c -s;s c];

tmp = R*[xd;yd];

xxr = tmp(1,:);

yyr = tmp(2,:);

xx = [xx xxr NaN];

yy = [yy yyr NaN];

end

end

%%

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