Equations of motion of rigid bodies

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1. DYNAMICS OF A RIGID BODY This chapter deals with the dynamic analysis of single rigid bodies. 1.1 Planar motion Let us consider a rigid body subjected to a planar motion. Let us denote by m and JG the mass and the mass moment of inertia evaluated about an axis passing through the centre of mass G of the body. Let us denote by ω and ω the absolute angular speed and absolute angular acceleration of the body, while aG is the absolute acceleration of the barycentre G. If the rigid body is translating, any point of the body has the same acceleration, aP, and velocity, vP. If the rigid body is rotating or roto- translating, each point P of the body is characterized by a specific acceleration vector a P. The refore, the infinitesimal inertia force associated with the infinitesimal mass Vdm d ρ= is: Pid dm= −Fa . Then, all the infinitesimal inertia forces id F form a distribution of vectors that are parall el only in the case of a translational motion. Therefore, in a general case of planar motion, the distribution of all the inertia forces acting on the rigid body is equivalent to an inertial force, GiF , applied to the centre of mass G, and an inertial torque, GiM , applied to the body. The barycentrical inertial force is given by: GGi m= −Fa (1.1) The inertial torque, evaluated using a mass moment of inertia, JG, about a barycentrical axis, is given by: GG Ji = −M ω (1.2) where ω is the absolute angular acceleration of the body. Now, let us consider an isolated rigid body, that is a body for which any possible external constraint has been removed, and substituted with the corresponding reaction forces and torques, vjR and vkM , respectively, with j = 1, 2, …, n1, and k = 1, 2, …, n 2 . Besides, if the rigid body is connected to other bodies, in order to isolate this…