User:Oh Isaac/R3-11

=R*3.11=

Problem 11: Free Vibration of Coupled Pendulums
Report problem 3.11 from Pea1.f12.sec15-5.

Given: Variables
Pendulums:

No applied forces:

Initial conditions:

Find: Show the Process of Verification

 * 1) Use MATLAB's ode45 command to integrate the system (1)(2)p.14-5 for $$t \in [0,7]$$. Can also use equivalent Octave.
 * 2) Use (2)p.15-2to find the solution at the same time stations as in Q1.
 * 3) Plot $$\theta_1(t)$$ from Q1 and from Q2.
 * Plot $$\theta_2(t)$$ from Q1 and from Q2.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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 * style="width:92%; padding:10px; border:2px solid #8888aa" |
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Integration
If we arrange the ($$) in the preparing form of matrices, we get:

Define:

$$ \mathbf{x}=\begin{Bmatrix} x_1\\x_2\\x_3\\x_4 \end{Bmatrix}=\begin{Bmatrix} \theta_1\\ \dot\theta_1\\\theta_2\\ \dot \theta_2 \end{Bmatrix} \mbox{ and } \mathbf{\dot x}=\begin{Bmatrix} x_5\\x_6\\x_7\\x_8 \end{Bmatrix}=\begin{Bmatrix} \dot\theta_1\\\ddot\theta_1\\\dot\theta_2\\ \ddot \theta_2 \end{Bmatrix} $$

Obviously

Conbining $$, we can easily write $$into a matrix form:

detailed with

in which

As we know $$g=9.8$$. Substitute $$with $$and$$, when $$t\in[0,7]$$then we get

With $$,coding on MATLAB, and get the codes.

With the MATLAB programs above, we get the plot of $$ \mathbf{x}=\begin{Bmatrix} x_1\\x_2\\x_3\\x_4 \end{Bmatrix}=\begin{Bmatrix} \theta_1\\ \dot\theta_1\\\theta_2\\ \dot \theta_2 \end{Bmatrix} $$ below.

Find Solution
According to (2)p.15-2, we get:

On the condition of $$, $$turns to be

in which $$\mathbf{A}$$ as obtained above. Substitute the relevent coefficiants with $$. When $$t\in[0,7]$$ and $$\mathbf{A}$$ as obtained above, and

we get the solution of this model:

Then we get the MATLAB codes below:

So we get the figure of $$ \mathbf{x}=\begin{Bmatrix} x_1\\x_2\\x_3\\x_4 \end{Bmatrix}=\begin{Bmatrix} \theta_1\\ \dot\theta_1\\\theta_2\\ \dot \theta_2 \end{Bmatrix} $$ shows their changes in given time space.

Notice
 By examing the previous reports, I notice that some people got incorrect plots as show below.The reason of the error comes from the misusing of MATLAB command. The command "exp(n)" is used for figure,while"expm(A)" is used for matrix.

by Hao Lin

Draw the plot
We plot $$\theta_1(t)\mbox{ and }\theta_2$$from Q1 and Q2 as below accoring to the former data.

And it is obvious that the two plots are identical because the two methords can all get the correct answer.