Rigid Body Dynamics - Planar Rigid Body Dynamics

Planar Rigid Body Dynamics

If a rigid system of particles moves such that the trajectory of every particle is parallel to a fixed plane, the system is said to be constrained to planar movement. In this case, Newton's laws for a rigid system of N particles, P_i, i=1,...,N, simplify because there is no movement in the k direction. Determine the resultant force and torque at a reference point R, to obtain

where r_i denotes the planar trajectory of each particle.

The kinematics of a rigid body yields the formula for the acceleration of the particle P_i in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as,

For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors e_i from the reference point R to a point r_i and the unit vectors t_i=kxe_i, so

This yields the resultant force on the system as

and torque as

where e_ixe_i=0, and e_ixt_i=k is the unit vector perpendicular to the plane for all of the particles P_i.

Use the center of mass C as the reference point, so these equations for Newton's laws simplify to become

where M is the total mass and I_C is the moment of inertia about an axis perpendicular to the movement of the rigid system and through the center of mass.