Complete course notes

Page 1

Fundamental properties and the dynamics of a Rigid body Rigid Body dynamics Here we will consider N to be a fixed inertial frame and B a body fixed frame with its origin O at the centre of mass of a rigid body as depicted in the Figure. 1 2 3r xb yb zb= + + is the position vector of a small mass element dm from the origin of the orthonormal frame. This point could also be expressed with respect to the origin of the inertial frame. It is critical that we are able to compute the derivative of arbitrary vectors with respect to different frames in order to perform analysis and control design. One of the theorems that is used throughout this course is the Thransport Theorem and here to derive some fundamental quantities expressed in the body frame. It relates the derivative of an arbitrary vector x taken with respect to one orthonormal frame N to a derivative taken with respect to another frame B where  is the relative angular velocity vector between the frames where: NB dx dx xdt dt     = +        Angular momentum of a rigid body The angular momentum of an infinitesimal point mass dm is defined as: odh r Rdm= Therefore, for the whole rigid body “B”: k v B r R RC j N i o B h r R dm=   (in discrete form we have 11 mm oi i i i nn h r v m == = where 1 m oi o n hh = → as t → ) The velocity is expressed as: CR R r=+ Then from the Transport theorem CR R r = +  where  is the angular velocity of the rigid body in the body frame with respect to the inertial frame. Evaluating the integral yields: ( ) ( )CCo B B B h r R r dm R rdm r r dm=  +  = −  +     The first term of the integral can be expressed as a function of the first moment of mass So, that is a function of the distance between the origin of the reference system and the center of mass: o…