Kennedy's Theorem in Plane Motion

Kennedy's theorem is a fundamental theorem in the study of planar motion of rigid bodies. It relates the motion of three bodies in a plane to the instantaneous centers of rotation. According to Kennedy's theorem, if three bodies have plane motion, their instantaneous centers lie on a straight line.

Instantaneous Center of Rotation

The instantaneous center of rotation is the point about which a rigid body has pure rotation at a given instant. It is the point in the plane of motion that has zero velocity at a particular instant. For a rigid body with plane motion, every point on the body has a different instantaneous center of rotation.

Proof of Kennedy's Theorem

Consider three bodies A, B, and C in plane motion. Let P and Q be the instantaneous centers of rotation of A and B, respectively, with respect to C. Let the velocities of A and B with respect to C be vA and vB, respectively. Then, the velocity of A with respect to B is given by vA - vB. Similarly, the velocity of P with respect to B is vP - vB.

Since P is the instantaneous center of rotation of A with respect to C, the velocity of P with respect to A is zero. Therefore, the velocity of P with respect to B is equal to the velocity of A with respect to B, i.e., vA - vB = vP - vB. Simplifying this equation, we get vA = vP.

Similarly, since Q is the instantaneous center of rotation of B with respect to C, we can show that vB = vQ. Hence, vA = vP = vQ = vB. Therefore, P, Q, and C have the same velocity with respect to B. Since their velocities are collinear, they lie on a straight line.

Conclusion

Kennedy's theorem is a simple yet powerful tool for analyzing the planar motion of rigid bodies. It relates the motion of three bodies in a plane to the instantaneous centers of rotation, which lie on a straight line. This theorem has wide applications in many areas of engineering, including machine design, robotics, and biomechanics.