16.07 Dynamics Fall 2004 ecture D23-3D Rigid Body Kinematics: The Inertia Tensor In this lecture, we will derive expressions for the angular momentum and kinetic energy of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor Angular momentum We start form the expression of the angular momentum of a system of particles about the center of mass HG, derived in lecture D17 H=/ Here, T'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 am 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 Thus, we have e=/ r3×(×r)dhm=/I(r We note that, for planar bodies undergoing a 2D motion in its own plane, r' is perpendicular to w, and the term(r. w)is zero. In this case, the vectors w and HG are always parallel. In the three dimensional case however, this simplification does not occur, and as a consequence, the angular velocity vector, w, and the angular momentum vector, HG, are in general, not parallel In cartesian coordinates, we have, r'sr'i+yj+ak and w=wi+w,+w, k, and the above expression can be expanded to yieldJ. Peraire 16.07 Dynamics Fall 2004 Version 1.0 Lecture D23 - 3D Rigid Body Kinematics: The Inertia Tensor In this lecture, we will derive expressions for the angular momentum and kinetic energy of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor. Angular Momentum We start form the expression of the angular momentum of a system of particles about the center of mass, HG, derived in lecture D17, HG = Z m r ′ × v ′ dm . 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 ′ . Thus, we have, HG = Z m r ′ × (ω × r ′ ) dm = Z m [(r ′ · r ′ )ω − (r ′ · ω)r ′ ] dm . 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 HG 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, HG, are in general, not parallel. In cartesian coordinates, we have, r ′ = x ′ i + y ′ j + z ′k and ω = ωxi + ωyj + ωzk, and the above expression can be expanded to yield, 1