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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L27- 3D Rigid Body Dynamics: Kinetic Energy; Instability; Equations of Motion 3D Rigid Body Dynamics In Lecture 25 and 26, we laid the foundation for our study of the three-dimensional dynamics of rigid bodies by: 1.) developing the framework for the description of changes in angular velocity due to a general motion of a three-dimensional rotating body; and 2.) developing the framework for the effects of the distribution of mass of a three-dimensional rotating body on its motion, defining the principal axes of a body, the inertia tensor, and how to change from one reference coordinate system to another. We now undertake the description of angular momentum, moments and motion of a general three-dimensional rotating body. We approach this very dicult general problem from two points of view. The first is to prescribe the motion in term of given rotations about fixed axes and solve for the force system required to sustain this motion. The second is to study the free motions of a body in a simple force field such as gravitational force acting through the center of mass or free motion such as occurs in a zero-g environment. The typical problems in this second category involve gyroscopes and spinning tops. The second set of problems is by far the more dicult. Before we begin this general approach, we examine a case where kinetic energy can give us considerable insight into the behavior of a rotating body. This example has considerable practical importance and its neglect has been the cause of several system failures. Kinetic Energy for Systems of Particles In Lecture 11, we derived the expression for the kinetic energy of a system of particles. Here, we derive the expression for the kinetic energy of a system of particles that will be used in the following lectures. A typical particle, i , will have a mass m i , an absolute velocity v i , and a kinetic energy T i = (1 / 2) m i v i v i = (1 / 2) m i v i 2 . The total kinetic energy of the system, T , is simply the sum of the kinetic energies for each particle, n n n 1 2 1 2 1 2 T = T i = m i v i = mv G + m i v i , . 2 2 2 i =1 i =1 i =1 1 where v is the velocity relative to the center of mass. Thus, we see that the kinetic energy of a system of particles equals the kinetic energy of a particle of mass m moving with the velocity of the center of mass, plus the kinetic energy due to the motion of the particles relative to the center of mass, G . We have said nothing about the conservation of energy for a system of particles. As we shall see, that depends upon the details of internal interactions and the work done by the external forces. The same is true for a rigid body. Work done by internal stresses, or energy lost due to the complexities of a system described as a rigid body, but which in reality may have internal modes which drain energy, will act to decrease the kinetic energy. Thus, although we can confidently...
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- Fall '09