By Shabana A.A.
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Extra resources for Computational Continuum Mechanics
These equations are given as ÿ T Á €i ¼ BT ðQe þ Qv ÿ MB_ q_ i Þ B MB q ð1:157Þ ÿ Á In this equation, BT MB is the generalized inertia matrix associated with the independent coordinates, BT ðQe þ Qv ÿ MB_ q_ i Þ is the vector of generalized forces that include applied, centrifugal, and Coriolis forces. The procedure described in this section for writing the dynamic equations in terms of the independent coordinates or the degrees of freedom is called the embedding technique. The dynamic equations can also be formulated in terms of redundant coordinates using the technique of Lagrange multipliers.
5 POLAR DECOMPOSITION THEOREM The polar decomposition theorem states that any square nonsingular matrix can be decomposed as the product of an orthogonal matrix and a symmetric matrix. According to this theorem, the square matrix J can have one of the following two decompositions: J ¼ RU, J ¼ VR ð1:86Þ where R is an orthogonal matrix and U and V are nonsingular symmetric matrices. Note that if the decomposition in the first equation is proved, the proof of the decomposition of the second equation follows because V ¼ JRT ¼ RURT .
Using this condition of rigidity, the number of degrees of freedom of a continuum can be significantly reduced. In the case of spatial analysis, a rigid body has six degrees of freedom that describe three independent translations and three independent rotations. In the case of planar motion, the rigid body has only three degrees of freedom: two describe the body translation and one describes the rotation of the body. For instance, as shown in Figure 1, the configuration of the rigid body in planar motion can be described using the vector rO and the angle h.
Computational Continuum Mechanics by Shabana A.A.