User:Rjorjorjo/HW6

HW 6 Problem Set  =Find the equation of motion $$\mathbf{Z}$$ when local coordinate vector $$ s = \frac{1}{2}$$=

Problem Statement
Prove $$\begin{align} \mathbf{Z}_{i+\frac{1}{2}}& = \ \mathbf{Z}(s = \frac{1}{2}) & = \ \frac{1}{2}(\mathbf{Z}_{i}+\mathbf{Z}_{i+1}+\frac{h}{8}(f_{i}-f_{i+1}))\end{align}$$

Given :
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C_{i}s^{i}$$
 * (a): (2) [[media:Egm6341.s10.mtg35.djvu|p.35-2]] $$\mathbf{Z}(s) = \ \sum_{i=0}^{3}
 * (a): (2) [[media:Egm6341.s10.mtg35.djvu|p.35-2]] $$\mathbf{Z}(s) = \ \sum_{i=0}^{3}
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C_{0} \\ C_{1} \\ C_{2} \\ C_{3} \\ \end{bmatrix} = \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ -3&-2&3&-1 \\ 2&1&-2&1 \\ \end{bmatrix} \cdot \!\, \begin{bmatrix} \mathbf{Z}_{i} \\ \mathbf{Z}_{i}^{'} \\ \mathbf{Z}_{i+1} \\ \mathbf{Z}_{i+1}^{'} \\ \end{bmatrix} $$
 * (b): (1) [[media:Egm6341.s10.mtg35.djvu|p.35-4]] $$\begin{bmatrix}
 * (b): (1) [[media:Egm6341.s10.mtg35.djvu|p.35-4]] $$\begin{bmatrix}
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 * (c): (1) [[media:Egm6341.s10.mtg36.djvu|p.36-1]] $$ \mathbf{Z}^{'} = h \dot{\mathbf{Z}}$$
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Ref: Lecture Notes [[media:Egm6341.s10.mtg35.djvu|p.35-2&4]]  [[media:Egm6341.s10.mtg36.djvu|p.36-1]]

Solution
$$\begin{align} \mathbf{Z}(s = \frac{1}{2})& = \ \sum_{i=0}^{3}C_{i}s^{i} \\ & =\ C_{0}(\frac{1}{2})^{0}+C_{1}(\frac{1}{2})^{1}+C_{2}(\frac{1}{2})^{2}+C_{3}(\frac{1}{2})^{3} \\ & =\ \mathbf{Z}_{i}+\mathbf{Z}_{i}^{'}(\frac{1}{2})+(3\mathbf{Z}_{i+1}-3\mathbf{Z}_{i}-2\mathbf{Z}_{i}^{'}-\mathbf{Z}_{i+1}^{'})(\frac{1}{4})+(\mathbf{Z}_{i+1}^{'}-2\mathbf{Z}_{i+1}+2\mathbf{Z}_{i}+\mathbf{Z}_{i}^{'})(\frac{1}{8}) \\ & = \ (1-\frac{3}{4}+\frac{1}{4})\mathbf{Z}_{i}+(\frac{1}{2}-\frac{1}{2}+\frac{1}{8})\mathbf{Z}_{i}^{'}+(\frac{3}{4}-\frac{1}{4})\mathbf{Z}_{i+1}+(-\frac{1}{4}+\frac{1}{8})\mathbf{Z}_{i+1}^{'} \\ & = \ \frac{1}{2} \mathbf{Z}_{i}+\frac{1}{8}\mathbf{Z}_{i}^{'}+\frac{1}{2}\mathbf{Z}_{i+1}-\frac{1}{8}\mathbf{Z}_{i+1}^{'} \\ & = \ \frac{1}{2}(\mathbf{Z}_{i}+\mathbf{Z}_{i+1})+\frac{1}{8}(\mathbf{Z}_{i}^{'}-\mathbf{Z}_{i+1}^{'}) \\ & = \ \frac{1}{2}(\mathbf{Z}_{i}+\mathbf{Z}_{i+1})+\frac{h}{8}(\frac{1}{h}\mathbf{Z}_{i}^{'}-\frac{1}{h}\mathbf{Z}_{i+1}^{'}) \\ & = \ \frac{1}{2}(\mathbf{Z}_{i}+\mathbf{Z}_{i+1})+\frac{h}{8}(\dot{\mathbf{Z}_{i}}-\dot{\mathbf{Z}_{i+1}}) \\ & =\ \frac{1}{2}(\mathbf{Z}_{i}+\mathbf{Z}_{i+1})+\frac{h}{8}(f_{i}-f_{i+1}) \end{align}$$

=Run the circumference formula with the complete elliptic integral codes of Team 3, and compare to the results of our own code=

Problem Statement
verify the results circumference formula with the complete elliptic integral codes

Solution
1Composite Trapezoidal Rule (a)team 3's codes :

subfunction is

(b)Ours

2Use Clenshaw-Curtis quadrature :

(a)Team 3

(b)Team1(ours)

3Use Romberg Table:

(a)Team 3

(b)Team 1 (ours)

EGM6341.s10.Team1.Ya-Chiao Chang 02:53, 7 April 2010 (UTC)