Coupled Oscillator: Hellie - Revision history

Andrew.hellie: /* Problem Statement */

2009-12-14T06:28:34Z

Problem Statement

2009-12-13T22:29:57Z

Problem Statement

@@ Line 1: / Line 1: @@
 ===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 of the system.'''
    [[Image:Coupled_Oscillator.jpg]]
@@ Line 192: / Line 193: @@
 .17764 \\
 .94046
-\end{bmatrix},
+\end{bmatrix}\,
 </math><math>e^{0}\,</math>
 '''Eigenmodes'''
@@ Line 207: / Line 206: @@
-'''Solve Using the Matrix Exponential'''
@@ Line 228: / Line 276: @@
-<math>[SI-A]^{-1} = \,</math>
+<math>[SI-A]^{-1} =\,</math> (something too large for my calculator to display or that I want to type out)
 Written by: Andrew Hellie

2009-12-13T20:03:56Z

Problem Statement

← Older revision		Revision as of 13:03, 13 December 2009
Line 15:		Line 15:

	:<math>k3=100 N/m\,</math>		:<math>k3=100 N/m\,</math>
			'''F=ma'''
			:<math>\ddot{x_1}=\frac{x_1(k_1-k_2)}{m_1}-\frac{x_2*k_1}{m_1}\,</math>

			:<math>\ddot{x_2}=\frac{x_2(k_1+k_2)}{m_2}-\frac{x_1*k_1}{m_2}\,</math>

	'''State Equations'''		'''State Equations'''

2009-12-13T19:54:31Z

Problem Statement

@@ Line 121: / Line 121: @@
 '''Eigenvalues'''
-<math>\lambda_1=-5.29412\,</math>
+:<math>\lambda_1=-5.29412\,</math>
-<math>\lambda_2=2.83333i\,</math>
+:<math>\lambda_2=2.83333i\,</math>
-<math>\lambda_3= -2.83333i\,</math>
+:<math>\lambda_3= -2.83333i\,</math>
-<math>\lambda_4=0\,</math>
+:<math>\lambda_4=0\,</math>
@@ Line 162: / Line 162: @@
 .94046
 \end{bmatrix}</math>

2009-12-13T19:38:44Z

Problem Statement

@@ Line 6: / Line 6: @@
 '''Initial Conditions:'''
-:<math>m_1= 15 kg\,</math>
+:<math>m_1= 10 kg\,</math>
-:<math>m_2 = 15 kg\,</math>
+:<math>m_2 = 10 kg\,</math>
 :<math>k1=100 N/m\,</math>
@@ Line 75: / Line 75: @@
 \begin{bmatrix}
 &1&0&0 \\
-\frac{(-50 N/m)}{15 kg}&0&\frac{-100 N/m}{15 kg}&0 \\
+\frac{(-50 N/m)}{10 kg}&0&\frac{-100 N/m}{10 kg}&0 \\
 &0&0&1 \\
-\frac{100 N/m}{15 kg}&0&\frac{(250 N/m)}{15 kg}&0
+\frac{-100 N/m}{10 kg}&0&\frac{(250 N/m)}{10 kg}&0
 \end{bmatrix}
@@ Line 89: / Line 89: @@
 </math>

2009-12-13T19:11:04Z

Problem Statement

2009-12-01T04:26:14Z

Problem Statement

← Older revision		Revision as of 21:26, 30 November 2009
Line 93:		Line 93:
	'''Eigenmodes'''		'''Eigenmodes'''

	:There are ~~three~~ eigenmodes for the system		:There are two eigenmodes for the system

	::1) m1 and m2 oscillating together		::1) m1 and m2 oscillating together

2009-12-01T04:25:06Z

Problem Statement

← Older revision		Revision as of 21:25, 30 November 2009
Line 120:		Line 120:

	</math>		</math>


			<math>[SI-A]^{-1} = \,</math>


			<math>\mathcal{L}^{-1}\left\{[SI-A]^{-1}\right\} = \,</math>


	Written by: Andrew Hellie		Written by: Andrew Hellie

2009-12-01T04:05:38Z

Problem Statement

← Older revision		Revision as of 21:05, 30 November 2009
Line 105:		Line 105:


	<math>e^{At}=\mathcal{L}^{-1}\left\{(SI-A)^{-1}\right\},\,</math>		<math>e^{At}=\mathcal{L}^{-1}\left\{[SI-A]^{-1}\right\}\,</math>



			<math>[SI-A]\,</math>
			=
			<math>
			\begin{bmatrix}
			S&1&0&0 \\
			\frac{(-50 N/m)}{15 kg}&S&\frac{-100 N/m}{15 kg}&0 \\
			0&0&S&1 \\
			\frac{100 N/m}{15 kg}&0&\frac{(250 N/m)}{15 kg}&S
			\end{bmatrix}

			</math>

	Written by: Andrew Hellie		Written by: Andrew Hellie

2009-12-01T03:20:25Z

Problem Statement

← Older revision		Revision as of 20:20, 30 November 2009
Line 1:		Line 1:
	===Problem Statement===		===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 of the system.		'''Write up on the Wiki a solution of a coupled oscillator problem like the coupled pendulum. Use State Space methods. Describe the eigenmodes of the system.'''

	[[Image:Coupled_Oscillator.jpg]]		[[Image:Coupled_Oscillator.jpg]]
	Initial Conditions:		'''Initial Conditions:'''

	:<math>m_1= 15 kg\,</math>		:<math>m_1= 15 kg\,</math>
Line 16:		Line 16:
	:<math>k3=100 N/m\,</math>		:<math>k3=100 N/m\,</math>

	State Equations		'''State Equations'''

	<math>		<math>
Line 60:		Line 60:
	</math>		</math>

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


Line 91:		Line 91:


	Eigenmodes		'''Eigenmodes'''

	:There are three eigenmodes for the system		:There are three eigenmodes for the system
Line 102:		Line 102:


	Solve Using the Matrix Exponential		'''Solve Using the Matrix Exponential'''



			<math>e^{At}=\mathcal{L}^{-1}\left\{(SI-A)^{-1}\right\},\,</math>

	Written by: Andrew Hellie		Written by: Andrew Hellie