19 Motion of Rigid Body II

Ashok Goyal

1. Introduction

We have seen that rigid body has six degrees of freedom. We thus require six independent generalized coordinates to specify its configuration. There may be additional constraints on the body which will further reduce the number of degrees of freedom and hence the number of generalized coordinates required to specify the orientation and position of the rigid body. In order to describe the motion of the rigid body we need to derive the equations of motion from the Lagrangian appropriate to the rigid body. How to assign these coordinates ? This can be done by setting a coordinate system in an inertial frame as will be explained below.

2. Coordinates of a Rigid Body.

Let us identify the body fixed coordinate system  with coordinates  and an inertial coordinate system  with coordinates .

After a translation motion of the inertial frame, we let the origins of the two systems coincide and show the inertial axis  by dotted lines. There are several ways in which the Cartesian axis with a common origin can be specified. A straight forward way is through the direction cosines of the primed system with respect to the unprimed system. If we denote the unit vectors along the coordinate axis of the unprimed and primed systems by  and  respectively we can express the  coordinates in terms of unprimed coordinates using the direction cosines i.e. for example, we can write

Since the coordinate system is Cartesian we have

There are nine possible products between the unit vectors in primed and unprimed systems. Rotation of the coordinate axis leaves the magnitude of the radius vector of any point in the rigid body unaltered. We thus have six relations between the products of direction cosines namely

We can thus use a set of three independent functions of direction cosines as generalized coordinates. A very convenient choice of three generalized coordinates for describing the motion of a rigid body is through what are known as Euler Angles.

3. Euler Angles

A most common and useful choice of the three generalized coordinates is to define the three Euler Angles. A transformation from a given Cartesian system to another can be performed by three successive rotations through these angles in a specific sequence which will take the inertial system  to the body fixed system . The transformation can be represented in terms of a matrix equation.

where R is a 3 x 3 orthogonal matrix.

We will define R though a sequence of three rotations on  as follows:

(1) The first rotation is by angle about the z axis resulting into a coordinate system labeled by the

(3) Third rotation is counter clock wise through angle about the axis which results in the final coordinate system  The three Euler’s angles completely specify the orientation of the two coordinate systems

4. Rotational velocity about the body fixed Axis

Body fixed axis turn out to be most useful for the discussion of rigid body motion. In order to write down equations of motion for the rigid body when Euler’s angles have been used as generalized coordinates to fix the orientation of the body fixed axis we would require to express the rotational velocities expressed as time derivatives of Euler’s angles. This is done as follows:

We see from the above discussion in particular from equations (19.7) – (19.11) that the angular velocities

5. Euler’s Equations for a Rigid Body

We will first consider the motion of rigid body in the absence of any external force. In this situation there is no potential energy associated with the body and the Lagrangian reduces to the kinetic energy term for the rigid body. We choose the body fixed axis along the principal axis of the body in which case the kinetic energy

We will choose the Eulerian angels  as generalized coordinates and can then write down the Lagrange’s equation for these coordinates.

The Lagrange’s equation for the coordinate  for example is given by

T is a function of ’s which in turn are given in terms of generalized coordinates and velocities as in equation (19.15), we can thus express (19.17) as

is the set of Euler’s equation of motion of a rigid body. The motion of the rigid body depends on the shape of the body through its moments of inertia about the principal axis.

Example:

We will illustrate the dynamics of a rigid body by calculating the kinetic energy of a homogeneous cone. We will consider the case when the base of the cone rolls without slipping on a horizontal plane when its apex is fixed at a height equal to the radius of the base as shown. The cone has a height h, half angle-, mass M and radius R. Let be the coordinate system of the inertial frame and  be the coordinate system of the body-fixed frame. The origin of the body-fixed axis is chosen at the C.M. of the cone which lies at a distance of  from the apex.

Summary:

• Three generalized coordinates are required to fix the orientation of a rigid body.
• We can use a set of three independent functions of direction cosines as generalized coordinates.
• A very convenient choice of three generalized coordinates for describing the motion of a rigid body is through Euler Angles.
• The motion of a rigid body in the absence of any external force is described by a set of Euler’s equation with the choice of axis along the principal axis of the body.