---
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.