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Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 1 Part 6

Tham khảo tài liệu 'crc press mechatronics handbook 2002 by laxxuss episode 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Product of Inertia. The product of inertia for a differential element dm is defined with respect to a set of two orthogonal planes as the product of the mass of the element and the perpendicular or shortest distances from the planes to the element. So with respect to the y - z and x - z planes z common axis to these planes the contribution from the differential element to I is dI and is given by dIxy xydm. As for the moments of inertia by integrating over the entire mass of the body for each combination of planes the products of inertia are Ixy Iyx Iyz Izy Ixz Izx I xy dm m I yz dm m I xz dm m 9.23 The product of inertia can be positive negative or zero depending on the sign of the coordinates used to define the quantity. If either one or both of the orthogonal planes are planes of symmetry for the body the product of inertia with respect to those planes will be zero. Basically the mass elements would appear as pairs on each side of these planes. Parallel-Axis and Parallel-Plane Theorems. The parallel-axis theorem can be used to transfer the moment of inertia of a body from an axis passing through its mass center to a parallel axis passing through some other point see also the section Kinetic Energy Storage . Often the moments of inertia are known for axes fixed in the body as shown in Fig. 9.33 b . If the center of gravity is defined by the coordinates xG yG zG in the x y z axes the parallel-axis theorem can be used to find moments of inertia relative to the x y z axes given values based on the body-fixed axes. The relations are Ixx Ixx a m yG zG Iyy Iyy a m xG zG Izz Izz a m xG yG where for example Ixx a is the moment of inertia relative to the xa axis which passes through the center of gravity. Transferring the products of inertia requires use of the parallel-plane theorem which provides the relations Ixy Ixy a mxGyG Iyz Iyz a myGỈG Izx Izx a mzGxG Inertia Tensor. The rotational dynamics of a rigid body rely on knowledge of the inertial properties which are .