Planar Kinematics Of A Rigid Body

Rigid Body Planar Motion:

The motion during which path is limited in a plane which are always parallel and equidistant to a fixed plane.

• We focus on geometric aspect of motion.
• It is complex than particle kinematics because it involves translation + rotation.
• System of particles which cannot deform under load is called Rigid body.
• Assumption of rigidity is acceptable because change in shape is very small compared with motion of whole body.

Types Of Rigid Body Planar Motion:

There are three types of rigid body planar motion which are described below:

1. Translation:

Motion along straight line or around a curve path is called Translation. There are two types of translation:

1. Rectilinear Motion  →  Due to straight line motion.
2. Curvilinear Motion  →  Due to congruent curve motion.

Analysis of Translation Motion:

Consider a rigid body which is subjected to translation in X-Y plane.

The position of rigid body is given by:

Differentiate the above equation with respect to time is given by:

• Kinematic analysis is used to specify the kinematics of points located in a translating rigid body.

2. Rotation About A Fixed Axis:

The angular motion of a body about its own axis is called rotation. During the rotation about a fixed axis, path of motion of any point particle in rigid body is circular.

• Angular Position  →  Angle θ measured from a fixed axis.
• Angular Displacement  →  Change in angular position ( 1 rev = 2π radians ).
• Angular Velocity  →  Rate change in angular displacement.

• Angular Acceleration  →  Rate change in angular velocity.

3. General Planar Motion ( Absolute Motion Analysis ):

• Simultaneous translation and rotation.
• Behavior is completely specified as: Angular rotation about fixed axis and Motion of point on body.

There are two methods for analysis which are:

1. Geometric or Kinematic Analysis
2. Analysis by Relative motion

A. Relative Motion Analysis For Velocity:

The general plane motion of a rigid body can be described as a combination of translation and rotation.

The position of body in position vector is given by:

Consider the rotation of body about A, which is given by:

So the difference in position vector is given by:

Differentiate the above equation with respect to time, we get:

B. Relative Motion Analysis For Acceleration:

Differentiate with respect to time above equation, we get:

The acceleration of body in relative motion analysis is equal to the normal and tangential acceleration components. So,

Instantaneous Center Of Zero Velocity:

If body rolls without slipping, the point at which body rotates is called Instantaneous Center of Zero velocity.

The velocity of body is given by:

There are three methods of finding Instantaneous Center of Zero velocity which are described as below:

1. For Non-Parallel Velocity Vectors:

The method of finding IC for non-parallel velocity vectors is:

• Draw perpendiculars from velocity vectors.
• Extend these vectors and make them intersect.
• The intersecting point is known as Instantaneous Center of Zero Velocity.

2. For Parallel & Opposite Velocity Vectors:

The method of finding IC for parallel and opposite velocity vectors is:

• Draw perpendiculars from velocity vectors.
• Draw line from the head of both velocity vectors intersecting line AB.
• The intersecting point is known as Instantaneous Center of Zero Velocity.

2. For Parallel Velocity Vectors:

The method of finding IC for parallel velocity vectors is:

• Draw perpendiculars from velocity vectors.
• Draw line from the head of both velocity vectors intersecting extended line AB.
• The intersecting point is known as Instantaneous Center of Zero Velocity.

Relative Motion Analysis Using Rotating Axes:

There are following scenario through which we analyze relative motion analysis:

1. Rigid bodies or mechanisms are such that sliding along with rotation.
2. One of the particle moves relative to other along a rotating path.
3. Two or more points on a mechanism are not located in the same body.

Kinematic Analysis:

Kinematic analysis uses coordinates system for translating and rotating any body.

The position of rigid body in a translating and rotating frame is given by:

Differentiate the above equation with respect to time, we get:

Again; differentiate with respect to time, we get:

• If you wanna have fun with rigid bodies, then have a go at it:

References:

• Material from Class Lectures + Book named Engineering Mechanics Statics & Dynamics by R.C. Hibbeler (12th Edition) + from YouTube channel named Yiheng Wang for Engineering Dynamics Mechanics + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.