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By Dan B. Marghitu
Advanced Dynamics: Analytical and Numerical Calculations with MATLAB presents an intensive, rigorous presentation of kinematics and dynamics whereas utilizing MATLAB as an built-in device to resolve difficulties. subject matters awarded are defined completely and directly,allowing primary rules to emerge via purposes from parts resembling multibody structures, robotics, spacecraft and layout of advanced mechanical units. This booklet differs from others in that it makes use of symbolic MATLAB for either conception and purposes. detailed awareness is given to options which are solved analytically and numerically utilizing MATLAB. The illustrations and figures generated with MATLAB make stronger visible studying whereas an abundance of examples supply extra aid.
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Extra info for Advanced Dynamics: Analytical and Numerical Calculations with MATLAB
Sample text
The permutation symbol i jk is defined by the equations 111 = 222 = 333 = 123 = 231 = 312 = 1, 213 = 321 = 132 = −1, 112 = 121 = 211 = 221 = 331 = ... 58) that is, i jk is a tensor antisymmetric in all indices (in any pair of indices) also called alternating tensor of third order. A tensor Ti j is said to be symmetric in a pair of indices (consider the indices i and j for this case) if Tij = T ji , and antisymmetric in i and j if Tij = −T ji . A tensor is called completely antisymmetric if changes sign under the exchange of any pair of indices.
55). The tensor notion was introduce by A. L. Cauchy (1789–1857) as a part of the deformable continuous medium. The matrix attached to a tensor of rank two in a Cartesian reference frame Oxyz can be written as ⎛ ⎞ X11 X21 X31 X = ⎝ X12 X22 X32 ⎠ , X13 X23 X33 28 1 Vector Algebra and in the Cartesian reference frame, Ox y z is ⎛ ⎞ X11 X21 X31 X = ⎝ X12 X22 X32 ⎠ . 56), it is obtained X = A−1 X A. 55). Let X = (X1 , X2 , X3 ) and Y = (Y1 ,Y2 ,Y3 ) be two nonzero vectors written in a Cartesian reference frame Oxyz.
45), one can write 3 3 3 3 3 3 3 ∑ Xii = ∑ ∑ ∑ αik αil Xkl = ∑ ∑ Xkl ∑ αik αil i=1 i=1 k=1 l=1 = 3 k=1 l=1 3 i=1 3 ∑ ∑ Xkl δkl = ∑ Xll , k=1 l=1 l=1 thus the scalar X11 + X22 + X33 = X11 + X22 + X33 stays invariant if the reference frame is changed. The result says that the trace of the matrix Xi j ( j = 1, 2, 3 and j = 1, 2, 3), that is, the sum of its diagonal elements, stays invariant under orthogonal transformation. 1. In Fig. 13a, the rectangular component of the vector F on the OA direction is f, with the magnitude |f| = f .