File:Asymmetric Epicyclic Gearing Stationary Sun.gif
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Asymmetric_Epicyclic_Gearing_Stationary_Sun.gif (335 × 335 pixels, file size: 2.85 MB, MIME type: image/gif, looped, 450 frames, 18 s)
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| Description |
Deutsch: Unsymmetrisches Umlaufrädergetriebe; Zähne: zSonne=22, zPlanet_1=14, zPlanet_1=26, zPlanet_1=38, zHohl=74.
English: Eccentric Planetary Gears. Asymmetric epicyclic gearing; Teeth: zsun=22, zplanet_1=14, zplanet_2=26, zplanet_3=38, zring=74. |
| Date | |
| Source | Own work |
| Author | Jahobr |
| Other versions |
|
| GIF development | |
| Source code | MATLAB codefunction Asymmetric_Epicyclic_Gearing()
% Source code for drawing epicyclic gearing.
% The shape of the gears is not precise, it creates a decent GIF and a SVG.
%
% 2017-06-21 Jahobr
teethSun = 22;
teethPlan = [14 26 38];
teethRing = 74;
modul = 16;
carrierCol = round([0.1 0.7 0.1].*255)./255; % green
sunCol = round([0.95 0.65 0 ].*255)./255; % yellow (obviously)
palnetCol = round([0.2 0.2 1 ].*255)./255; % blue (obviously)
ringCol = round([1 0.2 0.2].*255)./255; % red
xySize = 335; % size in pixel
scaleReduction = 3; % the size reduction: adds antialiasing
diameterPlan = modul.*teethPlan;
diameterRing = modul.*teethRing;
[pathstr,fname] = fileparts(which(mfilename)); % save files under the same name and at file location
figHandle = figure(15674455); clf
set(figHandle,'Units','pixel');
set(figHandle,'ToolBar','none');
set(figHandle,'GraphicsSmoothing','on') % requires at least version 2014b
set(figHandle,'position',[1 1 [xySize xySize]*scaleReduction]); % big start image for antialiasing later [x y width height]
axesHandle = axes;
hold(axesHandle,'on')
set(axesHandle,'position',[-0.05 -0.05 1.1 1.1]); % stretch axis bigger as figure, easy way to get rid of ticks [x y width height]
axis equal; drawnow;
% Rotation offsets to align teeth
angleOffsetRing = pi/teethRing; % ALLIGNMENT; THIS MAY NEED MANUAL ADJUSTMENT
angleOffsetSun=0.1972; % ALLIGNMENT; THIS MAY NEED MANUAL ADJUSTMENT
angleOffsetPlan(1)=0; % ALLIGNMENT; THIS MAY NEED MANUAL ADJUSTMENT
angleOffsetPlan(2)=0.043; % ALLIGNMENT; THIS MAY NEED MANUAL ADJUSTMENT
angleOffsetPlan(3)=0.094; % ALLIGNMENT; THIS MAY NEED MANUAL ADJUSTMENT
% polar coordinates of gears
radSun = 224.4037; % found by try and error
angSun = 0.4078; % found by try and error
radPlan = (diameterRing-diameterPlan)./2; %
angPlan(1) = 0;
angPlan(2) = pi*2/3-0.415; % 1/3 of a circle , but not quite (found by try and error)
angPlan(3) = pi*4/3+0.075; % 2/3 of a circle , but not quite (found by try and error)
for currentCase = 1:4; %
xlim([-diameterRing diameterRing]*0.72); % default
ylim([-diameterRing diameterRing]*0.72); % default
switch currentCase
case {1,2} % Stationary & Nonrotating Sun
nFrames = 450;
reducedRGBimage = uint8(ones(xySize,xySize,3,nFrames)); % allocate
angleCarrier = -linspace(0,pi*2,nFrames+1); % define gear position in frames
angleCarrier = angleCarrier(1:end-1); % remove last frame, it would be double
anglePlan = (( teethSun./teethPlan)'+1) *angleCarrier; % gear ratio
anglePlan = anglePlan + angleOffsetPlan(:)*ones(size(angleCarrier)); %
angleRing = angleCarrier.* (teethSun+teethRing) / teethRing; % gear ratio
angleRing = angleRing + angleOffsetRing;
angleSun = zeros(size(anglePlan))+angleOffsetSun;
if currentCase == 1
saveName = [fname '_Stationary_Sun'];
xlim([-diameterRing diameterRing]*0.95); % bigger range
ylim([-diameterRing diameterRing]*0.95); % bigger range
elseif currentCase == 2
saveName = [fname '_Nonrotating_Sun'];
end
case 3 % Stationary_Ring
nFrames = 450;
reducedRGBimage = uint8(ones(xySize,xySize,3,nFrames)); % allocate
angleCarrier = -linspace(0,pi*2,nFrames+1); % define gear position in frames
angleCarrier = angleCarrier(1:end-1); % remove last frame, it would be double
anglePlan = (( teethRing./teethPlan)'-1)*angleCarrier; % gear ratio
anglePlan = -anglePlan + angleOffsetPlan(:)*ones(size(angleCarrier)); %
angleSun = angleCarrier.* (1+teethRing/teethSun); % gear ratio
angleSun = angleSun + angleOffsetSun; %
angleRing = zeros(size(anglePlan)) + angleOffsetRing; %
saveName = [fname '_Stationary_Ring'];
case 4 % Stationary_Carrier
nFrames = 20;
reducedRGBimage = uint8(ones(xySize,xySize,3,nFrames)); % allocate
angleSun = -linspace(0,pi*2/teethSun,nFrames+1); % define gear position in frames
angleSun = angleSun(1:end-1); % remove last frame, it would be double
angleRing = -angleSun.* (teethSun/teethRing); % gear ratio
angleRing = angleRing + angleOffsetRing; %
anglePlan = ((teethSun./teethPlan )')*angleSun; % gear ratio
anglePlan = -anglePlan + angleOffsetPlan(:)*ones(size(angleSun)); %
angleCarrier = zeros(size(anglePlan));
angleSun = angleSun + angleOffsetSun; %
saveName = [fname '_Stationary_Carrier'];
end
for iFrame = 1:nFrames
cla(axesHandle) % fresh frame
%% ring
drawRingGear(teethRing,modul,ringCol,angleRing(iFrame))
%% sun
[X,Y] = pol2cart(angSun+angleCarrier(iFrame),radSun);
drawCogWheel([X,Y],teethSun ,modul,sunCol,angleSun(iFrame));
if currentCase == 1 % viewbox shift for "_Stationary_Sun"; improvised, but it works
xlim([-diameterRing diameterRing]*0.92+X); % bigger range
ylim([-diameterRing diameterRing]*0.92+Y); % bigger range
end
%% planets
for iPlan = 1:numel(angPlan)
[X,Y] = pol2cart(angPlan(iPlan)+angleCarrier(iFrame) ,radPlan(iPlan));
drawCogWheel([X,Y],teethPlan(iPlan),modul,palnetCol,anglePlan(iPlan,iFrame)); % planetary gear
end
%% carrier
% the carrier is adapted to the gears, to avoid obstruction of the sun gear
% % two extra nodes one extra node
angCarr = [angPlan(1) (angPlan(1:2)*[2;1])/3 (angPlan(1:2)*[1;2])/3 angPlan(2) angPlan(3) mean(angPlan([1,3])+[0 2*pi]) angPlan(1)];
radCarr = [radPlan(1) (radPlan(1:2)*[2;1])/3 (radPlan(1:2)*[1;2])/3 radPlan(2) radPlan(3) mean(radPlan([1,3])+[0 2*pi]) radPlan(1)];
angCarr(2) = angCarr(2)/1.4; % manual shift to avoid obstruction
radCarr(2) = radCarr(2)*1.1; % manual shift to avoid obstruction
angCarr(3) = angCarr(3)/1.3; % manual shift to avoid obstruction
radCarr(3) = radCarr(3)*1.1; % manual shift to avoid obstruction
angCarr(end-1) = angCarr(end-1)*1.12; % manual shift to avoid obstruction
radCarr(end-1) = radCarr(end-1)/1.05; % manual shift to avoid obstruction
[X,Y] = pol2cart(angCarr+angleCarrier(iFrame) ,radCarr);
plot(X,Y,'color',[0 0 0],'LineWidth',33) % black outline
plot(X,Y,'color',carrierCol,'LineWidth',30) % carrierCol "filling"
for iPlan = 1:numel(angPlan)
[X,Y] = pol2cart(angPlan(iPlan)+angleCarrier(iFrame) ,radPlan(iPlan));
circlePatch(X,Y,diameterPlan(iPlan)*0.25,carrierCol,1.5);
circlePatch(X,Y,diameterPlan(iPlan)*0.15,palnetCol,1.5);
end
%% save animation
f = getframe(figHandle);
reducedRGBimage(:,:,:,iFrame) = imReduceSize(f.cdata,scaleReduction); % the size reduction: adds antialiasing
if iFrame == 1 % SVG
if ~isempty(which('plot2svg'))
plot2svg(fullfile(pathstr, [fname '_Stationary.svg']),figHandle) % by Juerg Schwizer
else
disp('plot2svg.m not available; see http://www.zhinst.com/blogs/schwizer/');
end
end
end
map = createImMap(reducedRGBimage,16,[0 0 0;1 1 1;carrierCol;sunCol;palnetCol;ringCol]); % colormap
im = uint8(ones(xySize,xySize,1,nFrames)); % allocate
for iFrame = 1:nFrames
im(:,:,1,iFrame) = rgb2ind(reducedRGBimage(:,:,:,iFrame),map,'nodither');
end
imwrite(im,map,fullfile(pathstr, [saveName '.gif']),'DelayTime',1/25,'LoopCount',inf) % save gif
disp([saveName '.gif has ' num2str(numel(im)/10^6 ,4) ' Megapixels']) % Category:Animated GIF files exceeding the 50 MP limit
end
%%
function drawCogWheel(center,toothNumber,modul,colFilling,startOffset)
% DRAWTOOTHEDWHEEL - draw a simple Toothed Wheel
%
% Input:
% axesHandle:
% center: [x y]
% toothNumber: scalar
% modul: scalar tooth "size"
% colFilling: color of filling [r g b]
% startOffset: start rotation (scalar)[rad]
effectiveRadius = modul*toothNumber/2; % effective effectiveRadius
outsideRadius = effectiveRadius+1* modul; % +---+ +---+
upperRisingRadius = effectiveRadius+0.5*modul; % / \ / \
% effective Radius % / \ / \
lowerRisingRadius = effectiveRadius-0.5*modul; % I I I I
rootRadius = effectiveRadius-1.1*modul; % + - - - + + - - - + +
angleBetweenTeeth = 2*pi/toothNumber; % angle between 2 teeth
angleOffPoints = (0:angleBetweenTeeth/16:(2*pi));
angleOffPoints = angleOffPoints+startOffset; % apply rotation offset
angleOffPoints( 7:16:end) = angleOffPoints( 7:16:end) + 1/toothNumber^1.2; % hack to create smaller tooth tip
angleOffPoints(11:16:end) = angleOffPoints(11:16:end) - 1/toothNumber^1.2; % hack to create smaller tooth tip
angleOffPoints( 8:16:end) = (angleOffPoints( 7:16:end) + angleOffPoints(9:16:end))/2; % shift the neighbouring tip point in accordingly
angleOffPoints(10:16:end) = (angleOffPoints(11:16:end) + angleOffPoints(9:16:end))/2; % shift the neighbouring tip point in accordingly
angleOffPoints( 6:16:end) = angleOffPoints( 6:16:end) + 1/toothNumber^1.7; % hack to create slender tooth
angleOffPoints(12:16:end) = angleOffPoints(12:16:end) - 1/toothNumber^1.7; % hack to create slender tooth
radiusOffPoints = angleOffPoints; % allocate with correct site
radiusOffPoints(1:16:end) = rootRadius; % center bottom I
radiusOffPoints(2:16:end) = rootRadius; % left bottom I
radiusOffPoints(3:16:end) = rootRadius; % left bottom corner +
radiusOffPoints(4:16:end) = lowerRisingRadius; % lower rising bottom \
radiusOffPoints(5:16:end) = effectiveRadius; % rising edge \
radiusOffPoints(6:16:end) = upperRisingRadius; % upper rising edge \
radiusOffPoints(7:16:end) = outsideRadius; % right top corner +
radiusOffPoints(8:16:end) = outsideRadius; % right top I
radiusOffPoints(9:16:end) = outsideRadius; % center top I
radiusOffPoints(10:16:end) = outsideRadius; % left top I
radiusOffPoints(11:16:end) = outsideRadius; % left top corner +
radiusOffPoints(12:16:end) = upperRisingRadius; % upper falling edge /
radiusOffPoints(13:16:end) = effectiveRadius; % falling edge /
radiusOffPoints(14:16:end) = lowerRisingRadius; % lower falling edge /
radiusOffPoints(15:16:end) = rootRadius; % right bottom corner +
radiusOffPoints(16:16:end) = rootRadius; % right bottom I
[X,Y] = pol2cart(angleOffPoints,radiusOffPoints);
X = X+center(1); % center offset
Y = Y+center(2); % center offset
patch(X,Y,colFilling,'EdgeColor',[0 0 0],'LineWidth',1.5)
% plot(axesHandle,X,Y,'-x','linewidth',2,'color',[0 0 0]);
% %% effective Radius
% [X,Y] = pol2cart(angleOffPoints,effectiveRadius);
% X = X+center(1); % center offset
% Y = Y+center(2); % center offset
% plot(axesHandle,X,Y,'-.','color',[0 0 0]);
function drawRingGear(toothNumber,modul,colFilling,startOffset)
% subfunction for the outer static gear
effectiveRadius = modul*toothNumber/2; % effective effectiveRadius
outsideRadius = effectiveRadius-1* modul; % +---+ +---+
upperRisingRadius = effectiveRadius-0.5*modul; % / \ / \
% effective Radius % / \ / \
lowerRisingRadius = effectiveRadius+0.5*modul; % I I I I
rootRadius = effectiveRadius+1.1*modul; % + - - - + + - - - + +
angleBetweenTeeth = 2*pi/toothNumber; % angle between 2 teeth
angleOffPoints = (0:angleBetweenTeeth/16:(2*pi));
angleOffPoints = angleOffPoints+startOffset; % apply rotation offset
%% outerEdge
maxRadius = rootRadius*1.2; % definition of outer line
[X,Y] = pol2cart(angleOffPoints,maxRadius);
patch(X,Y,colFilling,'EdgeColor',[0 0 0],'LineWidth',1.5) % full outer disc
% plot(axesHandle,X,Y,'linewidth',2,'color',[0 0 0]); % draw outer circle
%% inner teeth
radiusOffPoints = angleOffPoints; % init
angleOffPoints(7:16:end) = angleOffPoints(7:16:end) + 1/toothNumber^1.2; % hack to create smaller tooth tip
angleOffPoints(11:16:end) = angleOffPoints(11:16:end) - 1/toothNumber^1.2; % hack to create smaller tooth tip
angleOffPoints(8:16:end) = (angleOffPoints(7:16:end) + angleOffPoints(9:16:end))/2; % shift the neighbouring tip point in accordingly
angleOffPoints(10:16:end) = (angleOffPoints(11:16:end) + angleOffPoints(9:16:end))/2; % shift the neighbouring tip point in accordingly
angleOffPoints(6:16:end) = angleOffPoints(6:16:end) + 1/toothNumber^1.7; % hack to create slender tooth
angleOffPoints(12:16:end) = angleOffPoints(12:16:end) - 1/toothNumber^1.7; % hack to create slender tooth
radiusOffPoints(1:16:end) = rootRadius; % center bottom I
radiusOffPoints(2:16:end) = rootRadius; % left bottom I
radiusOffPoints(3:16:end) = rootRadius; % left bottom corner +
radiusOffPoints(4:16:end) = lowerRisingRadius; % lower rising bottom \
radiusOffPoints(5:16:end) = effectiveRadius; % rising edge \
radiusOffPoints(6:16:end) = upperRisingRadius; % upper rising edge \
radiusOffPoints(7:16:end) = outsideRadius; % right top corner +
radiusOffPoints(8:16:end) = outsideRadius; % right top I
radiusOffPoints(9:16:end) = outsideRadius; % center top I
radiusOffPoints(10:16:end) = outsideRadius; % left top I
radiusOffPoints(11:16:end) = outsideRadius; % left top corner +
radiusOffPoints(12:16:end) = upperRisingRadius; % upper falling edge /
radiusOffPoints(13:16:end) = effectiveRadius; % falling edge /
radiusOffPoints(14:16:end) = lowerRisingRadius; % lower falling edge /
radiusOffPoints(15:16:end) = rootRadius; % right bottom corner +
radiusOffPoints(16:16:end) = rootRadius; % right bottom I
[X,Y] = pol2cart(angleOffPoints,radiusOffPoints);
patch(X,Y,[1 1 1],'EdgeColor',[0 0 0],'LineWidth',1.5) % overlay white area for inner teeth
% plot(axesHandle,X,Y,'-','linewidth',2,'color',[0 0 0]); % teeth line
function circlePatch(x,y,r,col,linW)
% x coordinates of the center
% y coordinates of the center
% r is the radius of the circle
% col patch color
% linW LineWidth
angleOffPoints = linspace(0,2.001*pi,200);
xc = x + r*cos(angleOffPoints);
yc = y + r*sin(angleOffPoints);
patch(xc,yc,col,'EdgeColor',[0 0 0],'LineWidth',linW);
function im = imReduceSize(im,redSize)
% Input:
% im: image, [imRows x imColumns x nChannel x nStack] (unit8)
% imRows, imColumns: must be divisible by redSize
% nChannel: usually 3 (RGB) or 1 (grey)
% nStack: number of stacked images
% usually 1; >1 for animations
% redSize: 2 = half the size (quarter of pixels)
% 3 = third the size (ninth of pixels)
% ... and so on
% Output:
% imNew: unit8([imRows/redSize x imColumns/redSize x nChannel x nStack])
%
% an alternative is : imNew = imresize(im,1/reduceImage,'bilinear');
% BUT 'bicubic' & 'bilinear' produces fuzzy lines
% IMHO this function produces nicer results as "imresize"
[nRow,nCol,nChannel,nStack] = size(im);
if redSize==1; return; end % nothing to do
if redSize~=round(abs(redSize)); error('"redSize" must be a positive integer'); end
if rem(nRow,redSize)~=0; error('number of pixel-rows must be a multiple of "redSize"'); end
if rem(nCol,redSize)~=0; error('number of pixel-columns must be a multiple of "redSize"'); end
nRowNew = nRow/redSize;
nColNew = nCol/redSize;
im = double(im).^2; % brightness rescaling from "linear to the human eye" to the "physics domain"; see youtube: /watch?v=LKnqECcg6Gw
im = reshape(im, nRow, redSize, nColNew*nChannel*nStack); % packets of width redSize, as columns next to each other
im = sum(im,2); % sum in all rows. Size of result: [nRow, 1, nColNew*nChannel]
im = permute(im, [3,1,2,4]); % move singleton-dimension-2 to dimension-3; transpose image. Size of result: [nColNew*nChannel, nRow, 1]
im = reshape(im, nColNew*nChannel*nStack, redSize, nRowNew); % packets of width redSize, as columns next to each other
im = sum(im,2); % sum in all rows. Size of result: [nColNew*nChannel, 1, nRowNew]
im = permute(im, [3,1,2,4]); % move singleton-dimension-2 to dimension-3; transpose image back. Size of result: [nRowNew, nColNew*nChannel, 1]
im = reshape(im, nRowNew, nColNew, nChannel, nStack); % putting all channels (rgb) back behind each other in the third dimension
im = uint8(sqrt(im./redSize^2)); % mean; re-normalize brightness: "scale linear to the human eye"; back in uint8
function map = createImMap(imRGB,nCol,startMap)
% createImMap creates a color-map including predefined colors.
% "rgb2ind" creates a map but there is no option to predefine some colors,
% and it does not handle stacked images.
% Input:
% imRGB: image, [imRows x imColumns x 3(RGB) x nStack] (unit8)
% nCol: total number of colors the map should have, [integer]
% startMap: predefined colors; colormap format, [p x 3] (double)
imRGB = permute(imRGB,[1 2 4 3]); % step1; make unified column-image (handling possible nStack)
imRGBcolumn = reshape(imRGB,[],1,3,1); % step2; make unified column-image
fullMap = double(permute(imRGBcolumn,[1 3 2]))./255; % "column image" to color map
[fullMap,~,imMapColumn] = unique(fullMap,'rows'); % find all unique colores; create indexed colormap-image
% "cmunique" could be used but is buggy and inconvenient because the output changes between "uint8" and "double"
nColFul = size(fullMap,1);
nColStart = size(startMap,1);
disp(['Number of colors: ' num2str(nColFul) ' (including ' num2str(nColStart) ' self defined)']);
if nCol<=nColStart; error('Not enough colors'); end
if nCol>nColFul; warning('More colors than needed'); end
isPreDefCol = false(size(imMapColumn)); % init
for iCol = 1:nColStart
diff = sum(abs(fullMap-repmat(startMap(iCol,:),nColFul,1)),2); % difference between a predefined and all colores
[mDiff,index] = min(diff); % find matching (or most similar) color
if mDiff>0.05 % color handling is not precise
warning(['Predefined color ' num2str(iCol) ' does not appear in image'])
continue
end
isThisPreDefCol = imMapColumn==index; % find all pixel with predefined color
disp([num2str(sum(isThisPreDefCol(:))) ' pixel have predefined color ' num2str(iCol)]);
isPreDefCol = or(isPreDefCol,isThisPreDefCol); % combine with overall list
end
[~,mapAdditional] = rgb2ind(imRGBcolumn(~isPreDefCol,:,:),nCol-nColStart,'nodither'); % create map of remaining colors
map = [startMap;mapAdditional];
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Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 22:41, 29 October 2017 | | 335 × 335 (2.85 MB) | Jahobr (talk | contribs) | graph smoothing |
| 20:35, 24 June 2017 |  | 334 × 334 (2.89 MB) | Jahobr (talk | contribs) | more fitting to the name | |
| 15:28, 21 June 2017 |  | 334 × 334 (3.74 MB) | Jahobr (talk | contribs) | User created page with UploadWizard |
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