Download Cartesian Tensors in Engineering Science by L. G. Jaeger and B. G. Neal (Auth.) PDF

By L. G. Jaeger and B. G. Neal (Auth.)

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6). The reader may now see the reason for the reservation made above in parentheses concerning the moments of inertia of the cylindrical bar shown in Fig. 23. For any given diameter of bar, there is one particular length for which h = h = h \ in this one case the moment of inertia tensor is isotropic with respect to all three axes, and the xi direction is no longer uniquely defined. In fact any axis through G is then a principal axis. Some further discussion of the principal axes of a symmetric tensor will be found in Chapter 4 under the discussion of eigenvectors.

The vector C is called the vector product, or cross product of A with B. In a familiar notation C = Α χ Β. 11) SpqrApBql', where i is a unit vector in the x direction. It will be noted that the order in which A and Β are written down is important because if, in eqn. 11) the roles of A and Β are reversed, the associated e changes sign owing to the fact that the subscripts ρ and q change places. r r pqr Hence Α χ Β = - Β χ Α. 12) Important particular examples of eqn. 11) are the cross products of unit vectors ii,i2 and 13.

The reader may now see the reason for the reservation made above in parentheses concerning the moments of inertia of the cylindrical bar shown in Fig. 23. For any given diameter of bar, there is one particular length for which h = h = h \ in this one case the moment of inertia tensor is isotropic with respect to all three axes, and the xi direction is no longer uniquely defined. In fact any axis through G is then a principal axis. Some further discussion of the principal axes of a symmetric tensor will be found in Chapter 4 under the discussion of eigenvectors.

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