Kinematics of rigid bodies

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2. KINEMATIC ANALYSIS OF RIGID BODIES A rigid body that can move in a 3D space has six degrees of freedom (d.o.f.s), that is the three components of the translational displacements, x-y-z, evaluated along the axes of Cartesian coordinate system and the th ree rotations, θx-θy-θz, about the above mentioned axes x-y-z. Conversely, a rigid body that can be only subjected to planar motions has three degrees of freedom (d.o.f.s), that is , the two components of the translational displacements, x-y, evaluated along the axes of a 2D Cartesian coordinate system and the rotation θz about an axis, z, orthogonal to the plane in which the body can move. Let us consider a rigid body (Figure 2.1) that is moving in a plane, for the sake of simplicity. The Cartesian x-y reference system, having origin in O, is fixed. The straight line r is fixed to the rigid body. The position of the rigid body can be completely determined by assigning the coordinates, x and y, of a generic point A of the body and the angular position of the straight line r, with respect to a reference axis (e.g. horizontal). In a general case study the rigid body can be subjected to a rotor -translating instantaneous motion, that, at any time t , can be always decomposed into an infinitesimal (or finite) rigid translation and an infinitesimal (or finite) pure rotation about an axis orthogonal to the plane x-y. Let us consider two points, A and B of this rigid body (Figure 2.2). The distance ()= −BA B A is constant. The absolute position vectors of the points A and B, defined with respect to a fixed Cartesian reference system x-y are given by: AA() = () ( ) ie xyα− −= +AO AO i j () = () ie β−−BO BO (2.1) The relative position of B, with respect to A, is given by the vector: () = () () = () () iieeβα− −−− − − −BA BO AO…