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

I am trying to design a custom rack gear similar to ones used in a rack and pinion gear system. Instead of a pinion i am using a disk with 4 rods equally spaced on the disk that almost act as the teeth. I want to be able to plot the path of the rods as the disk rotates over a flat surface by simulating the movement. With this path I can design the rack.

Answer 1

[Mathieu NOE]

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

[DGM]

It's up to OP, but I had interpreted the description as something like

https://en.wikipedia.org/wiki/Gear#Cage_gear

% a disk fitted with 4 rods/pegs on its face
d = 2; % pitch circle of rods
w = 0.75; % rod diameter

% main code 
N = 100; % each rod is composed of N points
[xx,yy] = geometry(d,w,N);

% let's plot the geometry once 
figure(1)
plot(xx,yy,'-');
title('The disk / rods geometry');
axis square

function [xx,yy] = geometry(d,w,N)
	% one peg
	theta = (0:N)/N*2*pi;
	xd = w/2*cos(theta);
	yd = w/2*sin(theta);
	
	% add the four pegs
	% peg are positioned on the pitch circle of diameter d
	% first peg at position 3 o'clock
	xx = [xd+d/2 NaN xd NaN xd-d/2 NaN xd];
	yy = [yd NaN yd+d/2 NaN yd NaN yd-d/2];
end

... though trying to get a discontiguous shape to behave with alphashape tools is confusing me at the moment.

[Benjamin]

Yes this is more the part I was going for.

[Benjamin]

@Mathieu NOE i don't see how that part would mesh with the gear profile you have derived.

[Mathieu NOE]

@Benjamin

ok maybe I did'nt see the rods oriented the same way as you

I thought there is a central disk from which would four legs (rods) extend radially

.

[Mathieu NOE]

Now this is an update of my suggestion, based on the "new" geometry as suggested by @DGM (tx to him !!)

% a disk fitted with 4 rods
d = 3; % central disk diameter
rd = 1; % rods diameter

%% main code 
N = 50; % geometry is created with N points
[xx,yy] = geometry(d,rd,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(~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
% axis square

% 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 
    % 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 + rd/2)*alpha;
    xxr = xxr + xshift;

end

function [xx,yy] = geometry(d,rd,N)
	% one peg
	theta = (0:N)/N*2*pi;
	xd = rd/2*cos(theta);
	yd = rd/2*sin(theta);
	
	% add the four pegs
	% peg are positioned on the pitch circle of diameter d
	% first peg at position 3 o'clock
	xx = [xd+d/2 NaN xd NaN xd-d/2 NaN xd];
	yy = [yd NaN yd+d/2 NaN yd NaN yd-d/2];
    
end

%%

[Benjamin]

This is close to what i was looking for thanks!

[Benjamin]

What would i need to change to do this for 6 or 8 rods?

[Mathieu NOE]

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

%%