Coordinate Transformations

Abstract

There remains the critical task of packaging Eqs. (53) and (62), with substitutions from Eq. (46), in a form convenient for the generation of coordinate transformations. To this end, let

$$q\mathop = \limits^\Delta {\left[ {u_1^1u_2^1u_3^1\beta _1^1\beta _2^1\beta _3^1u_2^1...\beta _3^n} \right]^T}$$

(63)

be the (6n × 1) matrix of nodal deformation coordinates, and rewrite the 6n second order differential equations implied by Eqs. (46), (53), and (62) in the form

$$M'\ddot q + D'\dot q + G'\dot q + K'q + A'q = L'$$

(64)

where M′, D′ and K′ are (6n × 6n) symmetric matrices and where G′ and A′ are (6n × 6n) skew-symmetric matrices, with L′ a (6n × 1) matrix not involving the deformation variables in q. Since Eqs. (53), (62), and (46) are all linear in the variables u^j ,β ^j. and ${\bar y^j}$ contained within q, and since any square matrix can be written as the sum of symmetric and skew-symmetric parts, the possibility of expression of these equations in the form of Eq. (64) is guaranteed by the symmetric character of the coefficients of ${\ddot u^j},{\ddot \beta ^j}\,and\,{\ddot \bar y^j}$ in the constituent equations.