By Dan B. Marghitu

Advanced Dynamics: Analytical and Numerical Calculations with MATLAB presents a radical, rigorous presentation of kinematics and dynamics whereas utilizing MATLAB as an built-in instrument to unravel difficulties. issues awarded are defined completely and directly,allowing basic ideas to emerge via purposes from parts comparable to multibody structures, robotics, spacecraft and layout of advanced mechanical units. This ebook differs from others in that it makes use of symbolic MATLAB for either thought and purposes. targeted consciousness is given to strategies which are solved analytically and numerically utilizing MATLAB. The illustrations and figures generated with MATLAB toughen visible studying whereas an abundance of examples supply extra aid.

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Example text

8 Cauchy’s Inequality, Lagrange’s Identity, and Triangle Inequality 21 or a2 b2 − (a · b)2 = (a × b) · (a × b). 26). (a x b) LHS = (dot(a,b))ˆ2; RHS = dot(a,a)*dot(b,b)-dot(cross(a,b),cross(a,b)); expand(LHS)==expand(RHS). If a and b are nonzero vectors, the following relation can be obtained: |a + b| ≤ |a| + |b| . 30) is known as triangle inequality. Proof : It is obvious that (a + b) · (a + b) = a · a + a · b + b · a + b · b = |a|2 + a · b + b · a + |b|2. 31) The following relation exists; a · b + b · a ≤ 2 |a · b| ≤ 2 |a||b| .

10 Tensors 29 Using a matrix form, the unity tensor can be written as ⎡ ⎤ 1 0 0 E= ⎣ 0 1 0 ⎦ . 0 0 1 ⇒ ⇒ The transpose of the tensor Z given as ⎡ ⎤ Z11 Z21 Z31 Z = ⎣ Z12 Z22 Z32 ⎦ . Z13 Z23 Z33 ⇒ ⇒ ⇒T is obtained by interchanging the subscripts of each element. If Z = Z , the tensor ⇒ Z is called a symmetric tensor. ⇒ ⇒ Two tensors Z and W are added: ⎡ ⎤ Z11 + W11 Z12 + W12 Z13 + W13 Z + W= ⎣ Z21 + W21 Z22 + W22 Z23 + W32 ⎦ Z31 + W31 Z32 + W32 Z33 + W33 ⇒ ⇒ or multiplied by scalars ⎡ ⎤ sZ11 sZ21 sZ31 s Z = ⎣ sZ12 sZ22 sZ32 ⎦ .

The solution could also be obtained by expressing the vector product v = p × q of the given vectors p and q in terms of the their rectangular components. Resolving p and q into components, one can write v = p × q = (px ı + py j + pz k) × (qx ı + qy j + qzk) ı j k = px py pz qx qy qz = (py qz − pz qy ) ı + (pz qx − px qz ) j + (px qy − py qx ) k. The components px , py , and pz of the vector p are √ √ 3 5 3 ◦ = , py = 0, and px = |p| cos θ = p cos θ = 5 cos 30 = 5 2 2 5 1 = . pz = |p| sin θ = p sin θ = 5 2 2 The components qx , qy ,and qz of the vector q are qx = q = 4, qy = 0 and qz = 0.

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