{"id":506,"date":"2018-11-06T09:39:32","date_gmt":"2018-11-06T09:39:32","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=506"},"modified":"2019-04-29T06:12:35","modified_gmt":"2019-04-29T06:12:35","slug":"motion-of-rigid-body-ii","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/motion-of-rigid-body-ii\/","title":{"rendered":"Motion of Rigid Body II"},"content":{"raw":"
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  1. Introduction<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    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.<\/p>\r\n\r\n

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    1. Coordinates of a Rigid Body.<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      Let us identify the body fixed coordinate system \u00a0with coordinates \u00a0and an inertial coordinate system \u00a0with coordinates .<\/p>\r\n \r\n\r\n\"\"\r\n

      After a translation motion of the inertial frame, we let the origins of the two systems coincide and show the inertial axis \u00a0by 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 \u00a0and \u00a0respectively we can express the \u00a0coordinates in terms of unprimed coordinates using the direction cosines i.e. for example, we can write<\/p>\r\n \r\n\r\n\"\"\r\n

      Since the coordinate system is Cartesian we have<\/p>\r\n\"\"\r\n

      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<\/p>\r\n \r\n\r\n\"\"\r\n

      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.<\/strong><\/p>\r\n\r\n

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      1. Euler Angles<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        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 \u00a0to the body fixed system . The transformation can be represented in terms of a matrix equation.<\/p>\r\n \r\n\r\n\"\"\r\n

        where R is a 3 x 3 orthogonal matrix.<\/p>\r\n \r\n

        We will define R though a sequence of three rotations on \u00a0as follows:<\/p>\r\n \r\n

        (1) The first rotation is by angle about the z axis resulting into a coordinate system labeled by the<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

        (3) Third rotation is counter clock wise through angle about the axis which results in the final coordinate system \u00a0The three Euler\u2019s angles completely specify the orientation of the two coordinate systems<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

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        1. Rotational velocity about the body fixed Axis<\/strong><\/li>\r\n<\/ol>\r\n \r\n

          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\u2019s 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\u2019s angles. This is done as follows:<\/p>\r\n \r\n

          We see from the above discussion in particular from equations (19.7) \u2013 (19.11) that the angular velocities<\/p>\r\n\"\"\r\n

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          1. Euler\u2019s Equations for a Rigid Body<\/strong><\/li>\r\n<\/ol>\r\n \r\n

            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<\/p>\r\n \r\n\r\n\"\"\r\n

            We will choose the Eulerian angels \u00a0as generalized coordinates and can then write down the Lagrange\u2019s equation for these coordinates.<\/p>\r\n \r\n

            The Lagrange\u2019s equation for the coordinate \u00a0for example is given by<\/p>\r\n \r\n\r\n\"\"\r\n

            T is a function of \u2019s which in turn are given in terms of generalized coordinates and velocities as in equation (19.15), we can thus express (19.17) as<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

            is the set of Euler\u2019s 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.<\/p>\r\n \r\n\r\nExample:<\/strong>\r\n\r\n \r\n

            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\u00a0be the coordinate system of the inertial frame and \u00a0be 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 \u00a0from the apex.<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nSummary:<\/strong>\r\n