View source for Coupled Oscillator: horizontal Mass-Spring

=Problem Statement=

'''Write up on the Wiki a solution of a coupled oscillator problem like the coupled pendulum.  Use State Space methods.  Describe the eigenmodes and eigenvectors of the system.'''

  [[Image:horizontal spring.jpg]]
'''Initial Conditions:'''
:<math>m_1= 10 kg\,</math>

:<math>m_2 = 10 kg\,</math>

:<math>k1=25 N/m\,</math>

:<math>k2=75 N/m\,</math>

:<math>k3=50 N/m\,</math>

'''Equations for M_1'''

:<math>\begin{alignat}{3}
                                                        F & = ma \\
                                                        F & = m\ddot{x} \\
                               -k_{1}x_{1}-k_{2}(x_1x_2)  & = m_1\ddot{x_1} \\
         -{k_1x_1 \over {m_1}}-{k_2(x_1-x_2) \over {m_1}} & = m_1\ddot{x_1} \\
         -{k_1x_1 \over {m_1}}-{k_2(x_1-x_2) \over {m_1}} & = \ddot{x_1} \\
           -{k_1+k_2 \over {m_1}}x_1+{k_2 \over {m_1}}x_2 & = \ddot{x_1} \\
\end{alignat}</math>

'''Equations for M_2'''
:<math>\begin{alignat}{3}
                                                        F & = ma \\
                                                        F & = m\ddot{x} \\
                                            -k_2(x_2-x_1) & = m_2\ddot{x_2} \\
                              {-k_2(x_2-x_1) \over {m_2}} & = \ddot{x_2} \\
               -{k_2 \over {m_2}}x_2+{k_2 \over {m_2}}x_1 & = \ddot{x_2} \\
\end{alignat}</math>

'''Additional Equations'''
:<math>\dot{x_1}=\dot{x_1}</math>
:<math>\dot{x_2}=\dot{x_2}</math>

==State Equations==

<math>
\begin{bmatrix} 
\dot{x_1} \\
\ddot{x_1} \\
\dot{x_2} \\
\ddot{x_2}
\end{bmatrix}\,
</math>
=
<math>
\begin{bmatrix}
0&1&0&0 \\
\frac{(k_1-k_2)}{m_1}&0&\frac{-k_1}{m_1}&0 \\
0&0&0&1 \\
\frac{k_1}{m_2}&0&\frac{(k_1+k_2)}{m_2}&0 
\end{bmatrix}

\begin{bmatrix}
x_1 \\
\dot{x}_1 \\
x_2 \\
\dot{x}_2
\end{bmatrix}

+

\begin{bmatrix}
0&0&0&0 \\
0&0&0&0 \\
0&0&0&0 \\
0&0&0&0
\end{bmatrix}

\begin{bmatrix}
0\\
0\\
0\\ 
0
\end{bmatrix}

</math>

'''With the numbers...'''


<math>
\begin{bmatrix} 
\dot{x_1} \\
\ddot{x_1} \\
\dot{x_2} \\
\ddot{x_2}
\end{bmatrix}\,
</math>
=
<math>
\begin{bmatrix}
0&1&0&0 \\
\frac{(-50 N/m)}{10 kg}&0&\frac{-25 N/m}{10 kg}&0 \\
0&0&0&1 \\
\frac{25 N/m}{10 kg}&0&\frac{(100 N/m)}{10 kg}&0 
\end{bmatrix}

\begin{bmatrix}
x_1 \\
\dot{x}_1 \\
x_2 \\
\dot{x}_2
\end{bmatrix}


</math>
== Eigen Values ==
Once you have your equations of equilibrium in matrix form you can plug them into a calculator or a computer program that will give you the eigen values automatically. This saves you a lot of hand work. Here's what you should come up with for this particular problem given these initial conditions.
:'''Given'''
:<math>m_1=10kg\,</math>
:<math>m_2=10kg\,</math>
:<math>k_1=25\,{N\over {m}}</math>
:<math>k_2=50\,{N\over {m}}</math>

We now have
:<math>\begin{bmatrix}
\dot{x_1} \\
\ddot{x_1} \\
\dot{x_2} \\
\ddot{x_2}
\end{bmatrix}
=
\begin{bmatrix}
0 & 1 & 0 & 0 \\
-5 & 0 & -2.5 & 0 \\
0 & 0 & 0 & 1 \\
2.5 & 0 & 10 & 0
\end{bmatrix}

\begin{bmatrix}
{x_1} \\
\dot{x_1} \\
{x_2} \\
\dot{x_2}
\end{bmatrix}
+
\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}</math>

From this we get
:<math>\lambda_1=\,</math>
:<math>\lambda_2=\,</math>
:<math>\lambda_3=\,</math>
:<math>\lambda_4=\,</math>
== Eigen Vectors ==
Using the equation above and the same given conditions we can plug everything to a calculator or computer program like MATLAB and get the eigen vectors which we will denote as <math>k_1,k_2,k_3,k_4\,</math>.
:<math>k_1=\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}</math>

:<math>k_2=\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}</math>

:<math>k_3=\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}</math>

:<math>k_4=\begin{bmatrix}
0 \\
0 \\
0 \\
0
\end{bmatrix}</math>