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 software to unravel difficulties. issues offered are defined completely and directly,allowing primary ideas to emerge via purposes from parts similar to multibody platforms, robotics, spacecraft and layout of complicated mechanical units. This ebook differs from others in that it makes use of symbolic MATLAB for either conception and purposes. specific awareness is given to options which are solved analytically and numerically utilizing MATLAB. The illustrations and figures generated with MATLAB toughen visible studying whereas an abundance of examples provide extra help.
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Extra info for Advanced Dynamics: Analytical and Numerical Calculations with MATLAB
14. 3. The vectors V1 , V2 , V3 , and V4 with the magnitude |V1 | = V1 , |V2 | = V2 , |V3 | = V3 , and |V4 | = V4 are concurrent at the origin O(0, 0, 0) and are directed through the points of coordinates A1 (x1 , y1 , z1 ), A2 (x2 , y2 , z2 ), A3 (x3 , y3 , z3 ), and A4 (x4 , y4 , z4 ), respectively. Determine the resultant vector of the system. Numerical application: V1 = 10, V2 = 25, V3 = 15, V4 = 40, A1 (3, 1, 7), A2 (5, −3, 4), A3 (−4, −3, 1), and A4 (4, 2, −3). Solution The magnitudes, Vi , of the vectors Vi and the coordinates, xi , yi , zi , of the points Ai , i = 1, 2, 3, 4 are introduced with MATLAB as V(1)=10; x(1)= 3; x(2)= 5; x(3)=-4; x(4)= 4; V(2)=25; y(1)= 1; y(2)=-3; y(3)=-3; y(4)= 2; V(3)=15; z(1)= 7; z(2)= 4; z(3)= 1; z(4)=-3; V(4)=40; % magnitudes V_i % A_1 % A_2 % A_3 % A_4.
Combining the previous results, one can conclude d da db d da db (a · b) = ·b+a· and (a × b) = ×b+a× . ’ diff(cross(a, b), t) == cross(diff(a, t), b) ... ’ is the array transpose of A. The general derivative of a vector v can be expressed as dv dı dvy dj dvz dk d dvx = (vx ı + vy j + vz k, ) = ı + vx + j + vy + k + vz , dt dt dt dt dt dt dt dt and if the reference basis or reference frame [ı, j, k] is unchanging, then dvy dv dvx dvz = ı+ j+ k. 8 Cauchy’s Inequality, Lagrange’s Identity, and Triangle Inequality The vectors a and b are nonzero vectors.
2. Skew-Symmetry. Let Xi j be the components of the tensor X in the Cartesian reference frame Oxyz. ⇒ If Xi j = −X ji , the second-order tensor X is said to be skew-symmetric. Its matrix is skew-symmetric about the principal diagonal, having zeros along this diagonal. In general, a second-order tensor is neither symmetric nor skew-symmetric, but any second-order tensor may be expressed as the sum of a symmetric and skew-symmetric tensor, in the following way: Xi j = 1 1 (Xi j + X ji ) + (Xi j − X ji ) .
Advanced Dynamics: Analytical and Numerical Calculations with MATLAB by Dan B. Marghitu