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Coordinate Transformations

  • Peter W. Likins
  • Robert E. Roberson
  • Jens Wittenburg
Part of the International Centre for Mechanical Sciences book series (CISM, volume 103)

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.

Keywords

Coordinate Transformation Order Differential Equation Generalize Coordinate Transformation Flexible Spacecraft Coordinate Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Wien 1971

Authors and Affiliations

  • Peter W. Likins
    • 1
  • Robert E. Roberson
    • 1
  • Jens Wittenburg
    • 2
  1. 1.University of CaliforniaUSA
  2. 2.University of HannoverGermany

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