By Dan B. Marghitu

*Advanced Dynamics: Analytical and Numerical Calculations with MATLAB* offers an intensive, rigorous presentation of kinematics and dynamics whereas utilizing MATLAB as an built-in device to unravel difficulties. issues awarded are defined completely and directly,allowing primary ideas to emerge via purposes from parts reminiscent of multibody structures, robotics, spacecraft and layout of complicated mechanical units. This publication differs from others in that it makes use of symbolic MATLAB for either conception and functions. distinctive recognition is given to suggestions which are solved analytically and numerically utilizing MATLAB. The illustrations and figures generated with MATLAB strengthen visible studying whereas an abundance of examples provide extra aid.

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1,... 1,... ’V_1 x V_2’,’fontsize’,14,’fontweight’,’b’) A rotated MATLAB drawing of the vectors is shown in Fig. 16b. 52 1 Vector Algebra a b 5 z 4 y O z 3 x p z q 2 θ 1 p 0 20 y pxq 10 y O 0 q 10 5 0 -10 x -5 x Fig. 5. The vector p of magnitude |p| = p is located in the x − z plane and makes an angle θ with x-axis as shown in Fig. 17a. The vector q of magnitude |q| = q is situated along the x-axis. Compute the vector (cross) product v = p × q. Numerical application: |p| = p = 5, |q| = q = 4, and θ = 30◦ .

Im . For 2 elements, one can write 12 = 1, 21 = −1, 11 = 22 = 0. The symbol can also be interpreted as a tensor, in which case it is called the permutation tensor. 1 Operations with Tensors ⇒ ⇒ ⇒ ⇒ 1. The tensors X and Y are equal, and write X=Y if and only if Xi j = Yi j for all i = 1, 2, 3 and j = 1, 2, 3. ⇒ ⇒ 2. If X is a tensor and λ is scalar, a new tensor can be defined λ X, with the components λ Xi j . This is the product of a tensor with a scalar. ⇒ ⇒ 3. If X and Y are second-order tensors with the components Xi j and Yi j , one can define ⇒ ⇒ ⇒ ⇒ ⇒ Z = X + Y= Y + X, as the arithmetic sum of the tensors.

13b. 2. The coordinates of two points A and B relative to the origin O(0, 0, 0) are given by A(xA = 1, yA = 2, zA = 3) and B(xB = 3, yB = 3, zB = 3). Determine the unit vector of the line Δ that starts at point A(xA , yA , zA ) and passes through the point B(xB , yB , zB ). Solution The position vectors of the points A(xA , yA , zA ) and B(xB , yB , zB ) with respect to the origin O(0, 0, 0) are −→ −→ OA = rA = xA ı + yA j + zA k and OB = rB = xB ı + yB j + zB k. The symbolic expressions of the vectors rA and rB are introduced in MATLAB as syms x_A y_A z_A x_B y_B z_B real r_A = [ x_A y_A z_A ]; r_B = [ x_B y_B z_B ]; − → The vector AB = rAB is defined as − → AB = rAB = rB − rA = (xB − xA) ı + (yB − yA) j + (zB − zA ) k, or in MATLAB r_AB = r_B - r_A; The magnitude of the vector rAB is |rAB | = rAB = (xB − xA)2 + (yB − yA )2 + (zB − zA )2 or in MATLAB mr_AB = sqrt(dot(r_AB,r_AB)); The unit vector, uΔ , of the line Δ (line AB) is uΔ = = (xB − xA ) ı + (yB − yA ) j + (zB − zA ) k rAB = |rAB | (xB − xA)2 + (yB − yA )2 + (zB − zA )2 xB − x A (xB − xA )2 + (yB − yA )2 + (zB − zA )2 ı 40 1 Vector Algebra + + yB − y A (xB − xA )2 + (yB − yA)2 + (zB − zA )2 zB − zA 2 (xB − xA ) + (yB − yA)2 + (zB − zA )2 j k.