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MIT16_07F09_Lec26

MIT16_07F09_Lec26 - J Peraire S Widnall 16.07 Dynamics Fall...

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J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.1 Lecture L26 - 3D Rigid Body Dynamics: The Inertia Tensor In this lecture, we will derive an expression for the angular momentum of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor. Angular Momentum We start from the expression of the angular momentum of a system of particles about the center of mass, H G , derived in lecture L11, n n H G = ( r i × m i ( ω × r i )) = m i r i 2 ω (1) i =1 i =1 n n H G = ( r i × m i ( ω × r i )) = m i r i 2 ω = r × v dm (2) i =1 i =1 m H G = r × v dm . m Here, r is the position vector relative to the center of mass, v is the velocity relative to the center of mass. We note that, in the above expression, an integral is used instead of a summation, since we are now dealing with a continuum distribution of mass. For a 3D rigid body, the distance between any particle and the center of mass will remain constant, and the particle velocity, relative to the center of mass, will be given by v = ω × r . 1

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Thus, we have, H G = r × ( ω × r ) dm = [( r r ) ω ( r ω ) r ] dm . · · m m Here, we have used the vector identity A × ( B × C ) = ( A C ) B ( A B ) C . We note that, for planar bodies · · undergoing a 2D motion in its own plane, r is perpendicular to ω , and the term ( r · ω ) is zero. In this case, the vectors ω and H G are always parallel. In the three-dimensional case however, this simplification does not occur, and as a consequence, the angular velocity vector, ω , and the angular momentum vector, H G , are in general, not parallel . In cartesian coordinates, we have, r = x i + y j + z k and ω = ω x i + ω y j + ω z k , and the above expression can be expanded to yield, H G = ω x ( x 2 + y 2 + z 2 ) dm ( ω x x + ω y y + ω z z ) x dm i m m + ω y ( x 2 + y 2 + z 2 ) dm ( ω x x + ω y y + ω z z ) y dm j m m + ω z ( x 2 + y 2 + z 2 ) dm ( ω x x + ω y y + ω z z ) z dm k m m = ( I xx ω x I xy ω y I xz ω z ) i + ( I yx ω x + I yy ω y I yz ω z ) j + ( I zx ω x I zy ω y + I zz ω z ) k . (3) The quantities I xx , I yy , and I zz are called moments of inertia with respect to the x , y and z axis, respectively, and are given by I xx = ( y 2 + z 2 ) dm , I yy = ( x 2 + z 2 ) dm , I zz = ( x 2 + y 2 ) dm . m m m We observe that the quantity in the integrand is precisely the square of the distance to the x , y and z axis, respectively. They are analogous to the moment of inertia used in the two dimensional case. It is also clear, from their expressions, that the moments of inertia are always positive. The quantities I xy , I xz , I yx , I yz , I zx and I zy are called products of inertia . They can be positive, negative, or zero, and are given by, I xy = I yx = x y dm , I xz = I zx = x z dm , I yz = I zy = y z dm .
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MIT16_07F09_Lec26 - J Peraire S Widnall 16.07 Dynamics Fall...

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