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