By Derek F. Lawden
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Extra info for A Course in Applied Mathematics, Vols 1 & 2
Nevertheless, the definitions and basic principles used in rigid-body dynamics can be generalized to the case of a general continuum by considering the continuum to consist of an infinite number of points that can move relative to each other. For example, the inertia force of a material point of an infinitesimal mass ρdv on the continuum in the current configuration can be written as (ρdv) r¨ , where r¨ is the absolute acceleration vector of the material 30 Introduction point and ρ and v are, respectively, the mass density and volume in the current deformed configuration.
Therefore, these coordinates depend on time. If these coordinates are determined, the global position of any point on the body, or equivalently the body configuration, can be determined using the preceding equation. An equation in the same form as Equation 92 can be obtained in the case of spatial motion of rigid bodies, as will be demonstrated in Chapter 7. In the case of spatial motion, three-dimensional vectors instead of two-dimensional vectors are used, and the transformation matrix A is expressed in terms of three independent rotation parameters instead of one parameter.
Note that if the decomposition in the first equation is proved, the proof of the 22 Introduction decomposition of the second equation follows because V = JRT = RURT . Therefore, it is sufficient to prove the first decomposition. Although the proof of the polar decomposition theorem is outlined in this section for 3 × 3 matrices, the generalization to square matrices with higher dimensions is straightforward. 79) Because C is symmetric, its eigenvalues are real and its eigenvectors are orthogonal.
A Course in Applied Mathematics, Vols 1 & 2 by Derek F. Lawden