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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 3.0 Lecture L28- 3D Rigid Body Dynamics: Equations of Motion; Euler’s Equations 3D Rigid Body Dynamics: Euler’s Equations We now turn to the task of deriving the general equations of motion for a three-dimensional rigid body. These equations are referred to as Euler’s equations. The governing equations are those of conservation of linear momentum L = M v G and angular momentum, H = [ I ] ω , where we have written the moment of inertia in matrix form to remind us that in general the direction of the angular momentum is not in the direction of the rotation vector ω . Conservation of linear momentum requires L ˙ = F (1) Conservation of angular momentum, about a fixed point O , requires H ˙ = M (2) or about the center of mass G H ˙ G = M G (3) In our previous application of these equations, we specified the motion and used the equations to specify what moments would be required to produce the prescribed motion. In this more general formulation, we allow the body to execute free motions, possibly under the action of external moments. We consider the general motion of a body about its center of mass, first examining this in a inertial reference frame. 1 At an instant of time, we can calculate the angular momentum of the body as H = [ I ] ω . One possible method to obtain the moments and the motion of the body is to perform our analysis in this inertial coordinate system. We would of course align our coordinate system initially with the principal axes of the body. We could then write M G = H ˙ G = d/dt ([ I ] ω ) = [ I ˙ ] ω + [ I ] ω ˙ (4) This would be a appropriate approach but the diﬃculty is keeping track of [ I ˙ ] in the inertial coordinate system. The initial inertial axis, even if principal axis, will not remain principal axis, and the inertia ”seen” in this coordinate system will vary with time. So unless we are considering the motion of a sphere, for which all axis are principal and the inertia tensor is constant about all axis, we cannot get very far with this approach. Body-Fixed Axis We formulate the governing equations of motion in an axis system fixed to the body, paying the price for keeping track of the motion of the body in order to have the inertia tensor remain independent of time in our reference frame. Given our earlier discussion of terms added to the description of motion in a rotating and accelerating coordinate system, it may seem surprising...
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This note was uploaded on 05/02/2011 for the course DYNAM 101 taught by Professor Matuka during the Spring '11 term at MIT.
- Spring '11