Electromechanical Dynamics (Part 1).0068

Electromechanical Dynamics (Part 1).0068 - izz is the...

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2.2. Mechanics ' and rig. Z.2.11, wnicn were aennea tor a point mass, hold equally well for describing the motion of the center of mass of a rigid body. Our rotational examples involve rotation about a fixed axis only. Thus we treat only the mechanics of rigid bodies rotating about fixed axes.* For this purpose we consider the system of Fig. 2.2.13. The body has mass density p that may vary with space in the body but at a point p fixed in the material is constant. We select a rectangular coordinate system whose z-axis co- incides with the axis of rotation. The instan- taneous angular velocity of the body is dO Fig. 2.2.13 Rigid-body rota- - = i dt . (2.2.16) tion. At the point p with coordinates (x, y, z) the element of mass p dV will have the instantaneous velocity v = w x r, (2.2.17) where r = ixx + iy +
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Unformatted text preview: izz is the radius vector from the origin to the mass element. The acceleration force on this mass element is dfdt V pdVdw x r + x ( xr) , (2.2.18) dt dt II where df, contains both internal and external forces [see (2.2.14)] and the last term has been written by using (2.2.17). We use the identity for the triple vector product a x (b x c) = b(a - c) -c(a - b) (2.2.19) to write (2.2.18) in the form df, = p dV -x r +± (o . r) -r(t -w) . (2.2.20) Ldtd To find the acceleration torque on this mass element we write dT, = r x df,. (2.2.21) * For a treatment of the general case of simultaneous translation and rotation in three dimensions, see, for example, Long, op. cit., Chapter 6. 2.2.1 Mechanics A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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This note was uploaded on 02/10/2012 for the course MECE 4371 taught by Professor Liu during the Fall '11 term at University of Houston.

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