Mechanisms

Mechanism

An assembly of rigid bodies or links designed to obtain a desired motion from an available motion while transmitting appropriate forces and moments. Motion of the links have definite relative motion with other links.

Simple mechanisms

• Lever
• Pulley
• Gear trains
• Belt and chain drive
• Four bar linkage

Other complex mechanisms

• Lock stitch mechanism (used in sewing machine)
• Geneva mechanism
  Constant rotational motion to intermittent rotational motion. mostly used in watches.
• Scotch yoke mechanism
  Constant rotational motion to linear motion (vice versa.). Mainly used as valve actuators in high pressure gas pipelines.
• Slider crank mechanism
  Used in internal combustion engines

2D link mechanisms

Method of instantaneous centre of rotation

• Find the instantaneous centre of the rotation from known velocities at known points
• Use the instantaneous centre to find velocities at other points

Kinematic chain

An arbitrary collection of links (forming a closed link) that is capable of relative motion and that can be made into a rigid structure by an additional single link.

Lower and Higher pairs

Lower Pair

Aka. primary pair. A pair of kinematic elements which share a surface of contact.

When a rigid body is constrained by a lower pair, which allows only rotational or sliding movement. It has $1$ degree of freedom, and the $2$ degrees of freedom are lost.

Some examples:

• Turning pair
• Sliding pair
• Helical thread

Higher Pair

A pair of kinematic elements which share only a line or a point of contact.

When a rigid body is constrained by a higher pair, it has $2$ degrees of freedom: translating along the curved surface and turning about the instantaneous contact point. $$

Gear is an example.

When 2 independent objects are brought together to create a link, some degree of freedom will be lost.

Grubler’s Equation

Suppose $N$ kinematic elements are brought together. $1$ of them is fixed. The remaining elements have $3(N-1)$ degrees of freedom. Each lower pairs loses $2$ degrees of freedom. Each higher pairs loses $1$ degree of freedom. For a workable mechanism, resultant degrees of freedom must be $1$.

$$F=3(N-1)-2L-H=1 \implies 3N-2L+H=4$$

Here:

• $F$ - degree of freedoms
• $N$ - number of kinematic elements
• $L$ - number of lower pairs
• $H$ - number of higher pairs