mmme2044 w0441 notes on linkages

uni/mmme/2xxx/2044_design_manufacture_and_project/linkages.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2044 // Linkages
+tags: [ mmme2044, linkages, uni ]
+uuid: ae955a7e-bc65-4055-a79d-38b57c5a00a5
+---
+
+# Freedom and Constraints
+
+A body that is free in space can move in three, independent perpendicular directions and rotation
+in three ways about those directions.
+It is said to have 6 degrees of freedom (DoF).
+
+The number of degrees of freedom are the number of components of motion that are required in order
+to generate the motion.
+
+If a joint is constrained to move along a line, it has 1 degree of freedom.
+If a joint is constrained to a plane with no rotation then it has two degrees of freedom.
+
+## Gruebler's Equation of Degrees of Freedom
+
+$$M = 3L - 2J - 3G$$
+
+where $M$ is degrees of freedom, $L$ is number of links, $J$ is number of joints, and $G$ is number
+of grounded links.
+
+![](./images/vimscrot-2022-10-17T16:33:52,261209255+01:00.png)
+
+- if DoF > 0, it is a mechanism with relative motions
+- if DoF = 0, it is a structure without motion
+- if DoF < 0, it is a preloaded (or overly constrained) structure without motion but possibly
+  additional stress
+
+![](./images/vimscrot-2022-10-17T16:36:24,581871695+01:00.png)
+
+## Grashof Condition
+
+Four bar linkage is the simplest possible pin-jointed mechanism.
+
+The Grashof condition is a simple relationship that predicts the linkage behaviour based only on
+the lengths of the links.
+A linkage is a Grashof linkage if at least one link is capable of making a full revolution and
+the following equation holds true:
+
+$$S + L < P + Q$$
+
+where
+
+- $S$ is the length of shortest link
+- $L$ is the length of longest link
+- $P$ is the length of one remaining link
+- $Q$ is the length of other remaining link
+
+### Triple Rockers
+
+If $S + L > P + Q$ then it is not a Grashof linkage and is known as a triple rocker.
+
+![](./images/vimscrot-2022-10-17T21:50:13,068568249+01:00.png)
+
+### Double Cranks / Crank Rockers
+
+$$S + L = P + Q$$
+
+![](./images/vimscrot-2022-10-17T21:50:29,856118739+01:00.png)
+
+## Limit Conditions
+
+Toggle is an important test to check if the linkage can reach all specified positions without
+encountering a limit or toggle position, also called a station configuration.
+
+Toggle positions may be determined by collinearity of two links
+
+![](./images/vimscrot-2022-10-17T21:53:02,839579659+01:00.png)
+
+## Transmission Angle
+
+Transmission angle, $\mu$, is the angle between the output link and the coupler.
+
+It is a useful test to check the quality of the force transmission in a linkage.
+
+Ideally $\mu = 90$ deg but at minimum $\mu \ge 40$ deg.