3.2 Kinematics of a Rigid Body in Plane

Rigidity is an assumption. This asssumption simplifies the mathematical model to a very big extent. Due to rigidity assumption we can make the following important conclusions:

1. The plane motion of a rigid body is completely described by the motion of any two points within the rigid body.

Let us assume that the motions of points A and B are given. At any time instant, t, the position of the rigid body is then known. Take any point S. At every position of the rigid body the position of point S is completely known, since the distances AS, BS and AB are fixed. For example if the rigid body has moved from A, B and S to a new location given by A′, B′, using the constant dimensions of AS, BS one can locate S′. Hence every point on the rigid body can be determined one the position of any two points such as A and B are known.

The position of a rigid body can also be defined by a vector from one point of the rigid body to another such as A and B, which will be called AB vector.

The vector AB is a fixed vector on the rigid body. Since B can be selected anywhere, the length of the vector is arbitrary. Relative to a fixed reference frame, the position of the rigid body can as well be defined by giving the position of the origin of this vector (point A which is a point on the rigid body) and its orientation (angle θ).

2. Rigidity ensures that the particles lying on a straight line have equal velocity components in the direction of this line. Since the distance between any two points along this line remains constant, there can be no velocity difference for points along this line (otherwise they must come closer to each other or separate). Velocity difference may exist perpendicular to the line.

V′_A = V′_B

3. We are concerned with the kinematics of the rigid bodies only. It is sufficient to consider just a line on the rigid body (vector AB, for example). Since the actual boundaries of the body does not influence the kinematics, the rigid body in plane motion is to be regarded as a large plane which embraces any desired point in the plane.

These three conclusions are very important in the kinematic analysis and synthesis of mechanisms.

Another important point is that in real life points are never marked as A, B, C, etc. One can very easily call a point as A or B. The only reason why we are placing symbols to the points or numbers to the links is to communicate in simple terms. There is no need to fight, if I use a symbol Q for a point that you have called A. For simplicity, however, as a convention I will use 1 for the fixed link and I will start from the first letter of the alphabet from the input link and go towards the output link. For example the fixed revolute joint axis of the input link will be A₀ and the moving pivot will be A. If desired, one can label the links with letters and joints with numbers