By Harman et al.

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**Additional resources for Advanced engineering mathematics with Matlab**

**Sample text**

Denote the acceleration of the object by a. The rod is subjected to a distributed load q = ama = ail ma , a = lal whose magnitude and direction with respect to the basis {ij same during deformation. (1) el + al(l) 21 e2 + a 31 e3 . Xz Fig. 13. As another example, let us consider a rod mounted on a rotating disk (Fig. 13). The disk is referred to a Cartesian coordinate system such that the axis X3 keeps its direction in space while the axes Xl and X2 are rigidly attached to the disk. The distributed load applied to the rod is 28 1.

The following relations take place: q = qo + ~q, p(i) = Pbi) + ~p(i) 11 = 110 + ~J1" and T(v) = Tbv) + ~T(v). The subscript '0' is used to indicate parameters that correspond to the case of an absolutely rigid rod (the vectors qo, Pbi ) , 110, and Tbv) are known). The vectors ~q, ~p(i), ~I1, and ~ T(v) are increments of the external loads. These increments depend on the displacements of the axial points and the angles of rotation of the attached coordinate system. 56): + C(3)u' + C(1){); ~p(i) = B(2)U + B(3)U' + B(ll{).

1 Vector Equilibrium Equations in the Attached Coordinate System To derive the equilibrium equations in a certain basis, we should represent all the vectors in this basis, say, {ed, which is the basis related to the principal axes of a cross section. Note that the unit base vectors themselves are functions of "I. This fact is emphasized by the notation ei(T/). 35) in terms of local derivatives 3 : 2 3 The prime is used to designate the differentiation with respect to the axial coordinate. The local derivatives are marked by a tilde.

### Advanced engineering mathematics with Matlab by Harman et al.

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