{"product_id":"approx-straight-line-linkages-educational-model","title":"Approx. Straight-line Linkages - Educational Model","description":"\u003ch2\u003eMy Educational Mechanical Examples Series\u003c\/h2\u003e\u003cp\u003eThis model is one of my educational mechanical mechanism examples on 80mm x 80mm base plates.\u003cbr\u003eYou can find all models of the series in this collection =\u0026gt; \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/makerworld.com\/collections\/15048577-my-educational-mechanism-models\"\u003e[Mechanical Mechanism Examples]\u003c\/a\u003e\u003c\/p\u003e\u003cfigure class=\"image image_resized\" style=\"width: 61.78%\"\u003e\u003ca href=\"https:\/\/makerworld.com\/collections\/15048577-my-educational-mechanism-models\" target=\"_blank\" rel=\"noopener noreferrer ugc\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/2025-11-13_a0fbbb25e21458.png\"\u003e\u003c\/a\u003e\u003c\/figure\u003e\u003ch2\u003e \u003c\/h2\u003e\u003ch2\u003eThe present set\u003c\/h2\u003e\u003cp\u003eThis set contains educational models of four-bar linkages that realize approximate straight-line motions.\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eChebyshev lambda linkage\u003c\/li\u003e\n\u003cli\u003eWatt's Linkage\u003c\/li\u003e\n\u003cli\u003eGrasshopper Linkage (two types)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBy using such mechanisms, the motion of a \u003cstrong\u003ecoupler point\u003c\/strong\u003e can be constrained to an approximately straight-line path using only rigid links connected by revolute joints, without any linear guide or prismatic pair.\u003c\/p\u003e\u003cfigure class=\"image\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/b044c3769b18ba74.png\"\u003e\u003c\/figure\u003e\u003cfigure class=\"media\"\u003e\u003coembed url=\"https:\/\/youtu.be\/-MSl-yKxgP4\"\u003e\u003c\/oembed\u003e\u003c\/figure\u003e\u003cfigure class=\"media\"\u003e\u003coembed url=\"https:\/\/youtu.be\/7usGpc5gW2g\"\u003e\u003c\/oembed\u003e\u003c\/figure\u003e\u003cfigure class=\"media\"\u003e\u003coembed url=\"https:\/\/youtu.be\/FNHiJHZ3Xos\"\u003e\u003c\/oembed\u003e\u003c\/figure\u003e\u003ch2\u003eBrief Description\u003c\/h2\u003e\u003ch4\u003eWatt's Linkage\u003c\/h4\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eWatt's linkage was devised by \u003cstrong\u003eJames Watt\u003c\/strong\u003e, renowned for his pioneering work on the steam engine. Two long links of equal length connect the ends of a shorter central link to two fixed pivot points on the base plate. When viewed as a four-bar linkage, the imaginary line connecting the two fixed pivots serves as the fourth link. The fixed pivots are positioned such that their horizontal separation equals approximately twice the length of the long links and their vertical separation equals the length of the short link — geometry that places the mechanism in a nominally parallel configuration, with the short link perpendicular to both long links. In practice, the mechanism is often set slightly off this position to optimize the straightness of the output motion. The coupler point is located at the midpoint of the short central link. This point traces a distorted figure-eight path on the base plate, the approximately straight portion of which is what allows the mechanism to function as an approximate straight-line mechanism.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cfigure class=\"image image_resized\" style=\"width: 32.67%\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/efbd3ccca9164574.png\"\u003e\u003c\/figure\u003e\u003cp\u003e \u003c\/p\u003e\u003ch4\u003e\u003cstrong\u003eChebyshev Lambda Linkage\u003c\/strong\u003e\u003c\/h4\u003e\u003cp\u003eThe Chebyshev linkage is one of the earliest known approximate straight-line mechanisms, devised by Russian mathematician Pafnuty Chebyshev, who sought to improve upon earlier straight-line mechanisms such as Watt's linkage. It consists, as shown in orange in the figure below, of a short link of length 2\u003ci\u003ea\u003c\/i\u003e connected at both ends to two long links of length 5\u003ci\u003ea\u003c\/i\u003e, which in turn connect to fixed pivots on the base plate. The midpoint of the short link serves as the coupler point. The Chebyshev Lambda Linkage achieves the same motion through a different configuration, as shown in blue in the figure below.\u003c\/p\u003e\u003cp\u003eIn the lambda variant, three links of lengths 5\u003ci\u003ea\u003c\/i\u003e, 2.5\u003ci\u003ea\u003c\/i\u003e, and \u003ci\u003ea\u003c\/i\u003e are used. The 2.5\u003ci\u003ea\u003c\/i\u003e link connects to the midpoint of the 5\u003ci\u003ea\u003c\/i\u003e link, and the a link connects to one end of the 5\u003ci\u003ea\u003c\/i\u003e link; the free ends of both shorter links attach to fixed pivots on the base plate. The 5\u003ci\u003ea\u003c\/i\u003e link and the 2.5\u003ci\u003ea\u003c\/i\u003e link extending from its midpoint resemble the Greek lowercase letter λ, giving this variant its name. The two fixed pivots are separated by a distance of 2\u003ci\u003ea\u003c\/i\u003e, and the pivot of the 2.5\u003ci\u003ea\u003c\/i\u003e link lies at the midpoint between the two fixed pivots of the corresponding standard Chebyshev linkage. In this configuration, the 5\u003ci\u003ea\u003c\/i\u003e and \u003ci\u003ea\u003c\/i\u003e links are parallel to their counterparts in the standard linkage, and the free end of the 5\u003ci\u003ea\u003c\/i\u003e link coincides with the coupler point. In this model, the \u003ci\u003ea\u003c\/i\u003e link is replaced by a disk for ease of manipulation.\u003c\/p\u003e\u003cp\u003eThe coupler point traces a path similar to that of \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/makerworld.com\/models\/2078221-theo-jansen-mechanism-educational-model\"\u003ethe Theo Jansen mechanism\u003c\/a\u003e. However, unlike the Theo Jansen mechanism, the trajectory is traced above the mechanism, making it unsuitable for walking on a surface. Instead, it lends itself to suspended locomotion, where the mechanism travels along an overhead track.\u003c\/p\u003e\u003cp\u003e\u003cimg class=\"image_resized\" style=\"width: 33.48%\" src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/314839eee492d097.gif\"\u003e  \u003cimg class=\"image_resized\" style=\"width: 32.9%\" src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/7d21c1b59d5c4616.png\"\u003e\u003c\/p\u003e\u003ch4\u003eGrasshopper Linkage\u003c\/h4\u003e\u003cp\u003eThe Grasshopper linkage is based on a configuration in which one end of a link of length \u003ci\u003ea\u003c\/i\u003e is connected to the midpoint of a link of length 2\u003ci\u003ea\u003c\/i\u003e. In this model, the other end of the \u003ci\u003ea\u003c\/i\u003e link is fixed to the upper left of the base plate. The geometry becomes clear by imagining a rectangle whose three corners are the fixed point and the two ends of the 2\u003ci\u003ea\u003c\/i\u003e link, with its sides running horizontally and vertically. In this rectangle, the 2\u003ci\u003ea\u003c\/i\u003e link forms one diagonal, and the \u003ci\u003ea\u003c\/i\u003e link forms half of the other. It follows that when the right end of the 2\u003ci\u003ea\u003c\/i\u003e link moves horizontally along the same height as the fixed point, the left end — which serves as the coupler point — travels along a vertical straight line passing through the fixed point.\u003c\/p\u003e\u003cp\u003eThis would be perfect straight-line motion, but in the Grasshopper linkage the right end of the 2\u003ci\u003ea\u003c\/i\u003e link is constrained to move along an arc rather than a straight line, by a third link whose lower end is fixed to the base plate. The longer this third link, the closer its arc approximates a straight line, so longer is better — though there is no single optimal length. Any deviation of the arc from a straight line causes a corresponding deviation of the coupler point from its ideal vertical path. The base plate is engraved with both the ideal straight line and the actual coupler point trajectory, allowing the difference to be seen directly.\u003c\/p\u003e\u003cp\u003eThis set includes two variants of the Grasshopper linkage. In the left model, the third link is vertical in the nominal configuration, giving better approximation for small displacements. In the right model, the third link is tilted slightly to the right, trading some accuracy near the center for a wider range of near-linear motion. Moving the coupler point by hand allows the difference between the two variants to be felt directly.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cfigure class=\"image\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/ad4a296a78c338a5.jpg\"\u003e\u003c\/figure\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch3\u003eRelated Models\u003c\/h3\u003e\u003cp\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/makerworld.com\/models\/2078221-theo-jansen-mechanism-educational-model\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/471b6145345178f4.png\"\u003e\u003c\/a\u003e\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch2\u003eCase\u003c\/h2\u003e\u003cp\u003eThis model is compatible with the case included \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/makerworld.com\/models\/1983088-educational-gear-examples-1\"\u003ein my first set\u003c\/a\u003e.\u003c\/p\u003e\u003cfigure class=\"image\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/2025-12-05_3b935e2548d228.jpg\"\u003e\u003c\/figure\u003e\u003cp\u003e \u003c\/p\u003e\u003ch2\u003ePrinting\u003c\/h2\u003e\u003cul\u003e\n\u003cli\u003eUse the models named \u003ci\u003e\u003cstrong\u003e???-printable.stl \u003c\/strong\u003efor printing\u003cstrong\u003e.\u003c\/strong\u003e\u003c\/i\u003e\u003cbr\u003eThe models named \u003ci\u003e\u003cstrong\u003e???-assembled.stl\u003c\/strong\u003e\u003c\/i\u003e are provided just to show how they should be assembled.\u003cbr\u003e \u003c\/li\u003e\n\u003cli\u003eUse \u003cstrong\u003ewell-dried PETG\u003c\/strong\u003e to have better dimensional accuracy.\u003c\/li\u003e\n\u003cli\u003eUse \u003cstrong\u003e0.1 mm or 0.08 mm layer height\u003c\/strong\u003e to have smoother surfaces.\u003c\/li\u003e\n\u003cli\u003eUse slow printing speed for overhangs.\u003c\/li\u003e\n\u003cli\u003eSelect \u003cstrong\u003e“Random” seam position\u003c\/strong\u003e to have smoother rotation.\u003cbr\u003eRandomly distributed seam should be easily worn out after some wearing.Printing\u003c\/li\u003e\n\u003c\/ul\u003e\u003ch2\u003eSanding and Filing\u003c\/h2\u003e\u003cp\u003eNote that, in this model, the rotation of the bases for bearings is intentionally made not too smooth.\u003c\/p\u003e\u003cp\u003eSometimes, the gears suffer from the stringing effect and\/or elephant foot effect, resulting in a too tight fit to the shafts (they are designed with a 0.15 mm radial clearance). \u003c\/p\u003e\u003cp\u003eIf you see rough surface on the shafts due to stringing, sand off the roughness with a small piece of sand paper.\u003c\/p\u003e\u003cfigure class=\"image\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/b748bc3e-0db4-48d9-8e7e-374a48abd806.webp\"\u003e\u003c\/figure\u003e\u003cp\u003eIf you feel the gears do not rotate smoothly due to an elephant effect, widen the hole slightly by using a thin round bar file.\u003c\/p\u003e\u003cfigure class=\"image image_resized\" style=\"width: 75%\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/c721d770-6619-492e-8792-c8696130b106.webp\" width=\"800\"\u003e\u003c\/figure\u003e\u003cp\u003eWithout those issues, the parts should rotate very smoothly with minimal friction.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch2\u003eAssembly\u003c\/h2\u003e\u003cp\u003eJust secure the parts by the retaining rings.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch2\u003eOther examples\u003c\/h2\u003e\u003cp\u003eYou may also be interested in the models in my educational mechanical mechanism examples.\u003c\/p\u003e\u003cp\u003eFind them in this collection:\u003cbr\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/makerworld.com\/collections\/15048577-my-educational-mechanism-models\"\u003ehttps:\/\/makerworld.com\/collections\/15048577-my-educational-mechanism-models\u003c\/a\u003e\u003c\/p\u003e\u003cfigure class=\"image image_resized\" style=\"width: 61.78%\"\u003e\u003cimg src=\"https:\/\/makerworld.bblmw.com\/makerworld\/model\/DSM00000002656748\/design\/2025-11-13_a0fbbb25e21458.png\" width=\"964\"\u003e\u003c\/figure\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eHappy printing!\u003c\/p\u003e\u003ch3\u003eAcknowledgement\u003c\/h3\u003e\u003cp\u003eI got into gears thanks to \u003cstrong\u003eK.$uzuki\u003c\/strong\u003e's amazing \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/karakurist.jp\/\"\u003earticles\u003c\/a\u003e and \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/www.youtube.com\/@karakurist\"\u003eYouTube videos\u003c\/a\u003e. Many of the mechanisms shown in this series came from the introductions on \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/karakurist.jp\/\"\u003ehis website\u003c\/a\u003e. He also makes \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/oishiiosushi.thebase.in\/\"\u003eexcellent gear models\u003c\/a\u003e himself. This series wouldn’t have existed without his inspiration.\u003c\/p\u003e\u003cp\u003eI learned a lot about technical detail of designing gear tooth profiles from \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/involutegearsoft.hatenablog.com\/\"\u003eHaguruma-No-Hanashi website\u003c\/a\u003e. I’m truly grateful for that.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003ch2\u003eLicense (2026-03-13 updated)\u003c\/h2\u003e\u003cul\u003e\n\u003cli\u003eThe 3D model(s) are licensed under \u003ca target=\"_blank\" rel=\"noopener noreferrer ugc\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\"\u003eCreative Commons Attribution 4.0 International\u003c\/a\u003e. (unchanged)\u003c\/li\u003e\n\u003cli\u003eHowever, the text and images on this page are copyright reserved. (added on 2026-03-13)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eDesign by osamutake on MakerWorld (license: BY).\u003c\/p\u003e","brand":"Mymadmanlab","offers":[{"title":"Default Title","offer_id":67397086937392,"sku":null,"price":9.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0987\/0809\/5280\/files\/0_66d2738e-c3d6-49bb-be68-f60b5e4184a2.png?v=1784860934","url":"https:\/\/mymadmanlab.com\/products\/approx-straight-line-linkages-educational-model","provider":"Mymadmanlab.com","version":"1.0","type":"link"}