2D Kinematics of a Rigid Body | Sahithyan's Notes

Rigid Body

A solid body that doesn’t deform.

Degrees of Freedom

In the motion of a rigid body in 2D kinematics, there are $3$ degrees of freedom.

Movement along $x$ direction
Movement along $y$ direction
Rotation about $z$ direction

In 3D, there are $6$ degrees of freedom: movement and rotation along each direction.

Translation

Movement that changes the position of an object. Translation can be done through a rectilinear or curvilinear path. Axes of the body always stays parallel.

Rotation

Circular movement of an object about a fixed axis that is perpendicular to the plane.

General 2D Motion

General 2D motion

Mixture of translation and rotation.

v_\text{B}= v_\text{A}+v_{\text{B}/\text{A}}= v_\text{A}+\underline{\dot{\theta}}\times r_{\text{B}/\text{A}}

a_\text{B}= a_\text{A}+a_{\text{B}/\text{A}}= a_\text{A}+\underline{\ddot{\theta}}\times r_{\text{B}/\text{A}}+\underline{\dot{\theta}}\times (\underline{\dot{\theta}}\times r_{\text{B}/\text{A}})

Here:

$\underline{\dot{\theta}}$: angular velocity of $\text{B}$ relative to $\text{A}$
$v_{\text{B}/\text{A}}$: velocity of $\text{B}$ relative to $\text{A}$
$a_{\text{B}/\text{A}}$: acceleration of $\text{B}$ relative to $\text{A}$
$r_{\text{B}/\text{A}}$: constant position vector of $\text{B}$ relative to $\text{A}$

In general motion, each particle of the body has a different velocity at every instance.

Instantaneous Centre of Rotation

The point that has $0$ velocity at a particular instant of time. This point might be changing throughout the motion. Denoted by $I$ .

It can be imagined that the object is momentarily having a pure rotation about this centre $I$ .

Finding instantaneous centre of rotation

$I$ can be found by drawing a line perpendicular at the velocity vectors at 2 different points and finding their intersection point.

Centrode

The locus of instantaneous centres during the motion.