Chapter 16. Plane Motion of Rigid Bodies: Forces and Accelerations
Chapter 16. Plane Motion of Rigid Bodies: Forces and Accelerations
Introduction Equations of Motion of a Rigid Body Angular Momentum of a Rigid Body in Plane Motion Plane Motion of a Rigid Body: d'Alembert's Principle
Axioms of the Mechanics of Rigid Bodies Solution of Problems Involving the Motion of a Rigid Body
Systems of Rigid Bodies
Constrained Plane Motion
Rigid Body Kinetics The forces and moments applied to a robotic arm control the resulting kinematics, and therefore the end position and forces of the actuator at the end of the robot arm.
Introduction
? In this chapter and in Chapters 17 and 18, we will be concerned with the kinetics of rigid bodies, i.e., relations between the forces acting on a rigid body, the shape and mass of the body, and the motion produced.
? Results of this chapter will be restricted to: ? plane motion of rigid bodies, and ? rigid bodies consisting of plane slabs or bodies which are symmetrical with respect to the reference plane.
? Our approach will be to consider rigid bodies as made of large numbers of particles and to use the results of Chapter 14 for the motion of systems of particles.
F = ma and M G = H G
16.1 Kinematics of Rigid Body
16.1A Equations of Motion for a Rigid Body
?
? Consider a rigid body acted upon by several external forces.
? Assume that the body is made of a large number of particles.
? For the motion of the mass center G of the body with respect to the Newtonian frame Oxyz,
F = ma
M G = H G
For the motion of the body with respect to the centroidal frame Gx'y'z',
? System of external forces is equipollent to the system consisting of ma and H G .
16.1B Angular Momentum of a Rigid Body in Plane Motion
? ? Angular momentum of the slab may be computed by
n
H G = (ri? vimi )
i =1
n
= [ri? ( ? ri)mi ]
i =1
( ) = ri2mi
?
= I
? After differentiation,
H G = I = I
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