Coupled Oscillator: horizontal Mass-Spring - Revision history

Gregory.peterson: /* Matrix Exponential */

2009-12-14T20:55:26Z

Matrix Exponential

← Older revision		Revision as of 13:55, 14 December 2009
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	0 & 0 & 0 & 3.0937		0 & 0 & 0 & 3.0937
	\end{bmatrix}</math>		\end{bmatrix}</math>

			created by Greg Peterson

Gregory.peterson: /* Matrix Exponential */

2009-12-10T23:17:53Z

Matrix Exponential

← Older revision		Revision as of 16:17, 10 December 2009
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	'''Taking the inverse of this we can solve for T'''		'''Taking the inverse of this we can solve for T'''
	:<math>T=\begin{bmatrix}		:<math>T=\begin{bmatrix}
	0 & 0 & 0 & 0 \\		-0.2914 & 0.0943 & -1.6996 & 0.5493 \\
	0 & 0 & 0 & 0 \\		- 1.2337i & -0.5770 & - 0.2117i & -0.0990 \\
	0 & 0 & 0 & 0 \\		1.2335i & -0.5770 & 0.2116i & -0.0990 \\
	0 & 0 & 0 & 0		0.2914 & 0.0943 & 1.6996 & 0.5493
	\end{bmatrix}</math>		\end{bmatrix}</math>

Gregory.peterson: /* Matrix Exponential */

2009-12-10T23:10:48Z

Matrix Exponential

← Older revision		Revision as of 16:10, 10 December 2009
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	:<math>T^{-1}=[k_1\|k_2\|k_3\|k_4]\,</math>		:<math>T^{-1}=[k_1\|k_2\|k_3\|k_4]\,</math>
	:<math>T^{-1}=\begin{bmatrix}		:<math>T^{-1}=\begin{bmatrix}
	0 & 0 & 0 & 0 \\		0.0520 & 0.4176i & - 0.4176i & -0.0520 \\
	0 & 0 & 0 & 0 \\		-0.1609 & -0.8928 & -0.8928 & -0.1609 \\
	0 & 0 & 0 & 0 \\		-0.3031 & - 0.0716i & 0.0716i & 0.3031 \\
	0 & 0 & 0 & 0		0.9378 & 0.1532 & 0.1532 & 0.9378
	\end{bmatrix}</math>		\end{bmatrix}</math>

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	0 & 0 & 0 & 0 \\		0 & 0 & 0 & 0 \\
	0 & 0 & 0 & 0		0 & 0 & 0 & 0
			\end{bmatrix}</math>

			'''So taking'''
			:<math>\dot{z}=TAT^{-1}z</math>
			'''We get the uncoupled matrix of'''
			:<math>\dot{z}=\begin{bmatrix}
			-3.0937 & 0 & 0 & 0 \\
			0 & 2.1380i & 0 & 0 \\
			0 & 0 & - 2.1380i & 0 \\
			0 & 0 & 0 & 3.0937
	\end{bmatrix}</math>		\end{bmatrix}</math>

Gregory.peterson: /* So then the answer is... */

2009-12-10T23:00:47Z

So then the answer is...

← Older revision		Revision as of 16:00, 10 December 2009
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	:<math>x=c_1k_1e^{\lambda_1 t}+c_2k_2e^{\lambda_2 t}+c_3k_3e^{\lambda_3 t}+c_4k_4e^{\lambda_4 t}</math>		:<math>x=c_1k_1e^{\lambda_1 t}+c_2k_2e^{\lambda_2 t}+c_3k_3e^{\lambda_3 t}+c_4k_4e^{\lambda_4 t}</math>

	<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0 \\0\\0\end{bmatrix}\,</math><math>e^{-3.0937}+ c_2</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,</math><math>e^{1}+ c_3</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,</math><math>e^{0}+ c_4</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,		<math>\ x=c_1</math><math>\begin{bmatrix}0.0520 \\-0.1609 \\-0.3031\\0.9378\end{bmatrix}\,</math><math>e^{-3.0937}+ c_2</math><math>\begin{bmatrix}0.4176i \\-0.8928\\- 0.0716i\\0.1532\end{bmatrix}\,</math><math>e^{2.1380i}+ c_3</math><math>\begin{bmatrix}- 0.4176i \\-0.8928\\0.0716i\\0.1532\end{bmatrix}\,</math><math>e^{- 2.1380i}+ c_4</math><math>\begin{bmatrix}-0.0520 \\-0.1609\\0.3031\\0.9378\end{bmatrix}\,
	</math><math>e^{0}\,</math>		</math><math>e^{3.0937}\,</math>

	== Matrix Exponential ==		== Matrix Exponential ==

Gregory.peterson: /* So then the answer is... */

2009-12-10T22:57:43Z

So then the answer is...

← Older revision		Revision as of 15:57, 10 December 2009
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	:<math>x=c_1k_1e^{\lambda_1 t}+c_2k_2e^{\lambda_2 t}+c_3k_3e^{\lambda_3 t}+c_4k_4e^{\lambda_4 t}</math>		:<math>x=c_1k_1e^{\lambda_1 t}+c_2k_2e^{\lambda_2 t}+c_3k_3e^{\lambda_3 t}+c_4k_4e^{\lambda_4 t}</math>

	<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0~~\sqrt(0)~~ \\0 \\0~~\sqrt(0)~~\end{bmatrix}\,</math><math>e^{~~0\sqrt{0}~~}+ c_2</math><math>\begin{bmatrix}0 \\0~~\sqrt(0)~~ \\0 \\0~~\sqrt(0)~~\end{bmatrix}\,</math><math>e^{~~00\sqrt{0}~~}+ c_3</math><math>\begin{bmatrix}0 \\0~~\sqrt(0)~~ \\0 \\0~~\sqrt(0)~~\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_4</math><math>\begin{bmatrix}0 \\0~~\sqrt(0)~~ \\0 \\0~~\sqrt(0)~~\end{bmatrix}\,</math><math>e^{0*0\sqrt{0}}\,</math>		<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0 \\0\\0\end{bmatrix}\,</math><math>e^{-3.0937}+ c_2</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,</math><math>e^{1}+ c_3</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,</math><math>e^{0}+ c_4</math><math>\begin{bmatrix}0 \\0\\0\\0\end{bmatrix}\,
			</math><math>e^{0}\,</math>

	== Matrix Exponential ==		== Matrix Exponential ==

Gregory.peterson: /* Eigen Vectors */

2009-12-10T22:50:05Z

Eigen Vectors

← Older revision		Revision as of 15:50, 10 December 2009
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	'''Using the equation above and the same given conditions we can plug everything into MATLAB and get the eigen vectors which we will denote as <math>k_1,k_2,k_3,k_4\,</math>.'''		'''Using the equation above and the same given conditions we can plug everything into 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}		:<math>k_1=\begin{bmatrix}
	0 \\		0.0520 \\
	0 \\		-0.1609 \\
	0 \\		-0.3031 \\
	0		0.9378
	\end{bmatrix}</math>		\end{bmatrix}</math>

	:<math>k_2=\begin{bmatrix}		:<math>k_2=\begin{bmatrix}
	0 \\		0.4176i \\
	0 \\		-0.8928 \\
	0 \\		- 0.0716i \\
	0		0.1532
	\end{bmatrix}</math>		\end{bmatrix}</math>

	:<math>k_3=\begin{bmatrix}		:<math>k_3=\begin{bmatrix}
	0 \\		- 0.4176i \\
	0 \\		-0.8928 \\
	0 \\		0.0716i \\
	0		0.1532
	\end{bmatrix}</math>		\end{bmatrix}</math>

	:<math>k_4=\begin{bmatrix}		:<math>k_4=\begin{bmatrix}
	0 \\		-0.0520 \\
	0 \\		-0.1609 \\
	0 \\		0.3031 \\
	0		0.9378
	\end{bmatrix}</math>		\end{bmatrix}</math>

	==So then the answer is...==		==So then the answer is...==
	'''We can now plug these eigen vectors and eigen values into the standard equation'''		'''We can now plug these eigen vectors and eigen values into the standard equation'''

Gregory.peterson: /* Eigen Values */

2009-12-10T22:46:14Z

Eigen Values

← Older revision		Revision as of 15:46, 10 December 2009
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	'''From this we get'''		'''From this we get'''
	:<math>\lambda_1=\,</math>		:<math>\lambda_1=-3.0937,</math>
	:<math>\lambda_2=\,</math>		:<math>\lambda_2=2.1380i,</math>
	:<math>\lambda_3=\,</math>		:<math>\lambda_3=- 2.1380i,</math>
	:<math>\lambda_4=\,</math>		:<math>\lambda_4=3.0937,</math>

	== Eigen Vectors ==		== Eigen Vectors ==

Gregory.peterson: /* Eigen Vectors */

2009-12-10T22:09:44Z

Eigen Vectors

← Older revision		Revision as of 15:09, 10 December 2009
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	0		0
	\end{bmatrix}</math>		\end{bmatrix}</math>
	~~'''~~So then the answer is...'''		==So then the answer is...==
			'''We can now plug these eigen vectors and eigen values into the standard equation'''
			:<math>x=c_1k_1e^{\lambda_1 t}+c_2k_2e^{\lambda_2 t}+c_3k_3e^{\lambda_3 t}+c_4k_4e^{\lambda_4 t}</math>

	<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0\sqrt{0}}+ c_2</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_3</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_4</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0*0\sqrt{0}}\,</math>		<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0\sqrt{0}}+ c_2</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_3</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_4</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0*0\sqrt{0}}\,</math>

Gregory.peterson: /* Eigen Vectors */

2009-12-10T22:08:03Z

Eigen Vectors

← Older revision		Revision as of 15:08, 10 December 2009
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	'''So then the answer is...'''		'''So then the answer is...'''

	<math>\ x=c_1</math><math>\begin{bmatrix}-1 \\-2\sqrt(10) \\1 \\2\sqrt(10)\end{bmatrix}\,</math><math>e^{2\sqrt{10}}+ c_2</math><math>\begin{bmatrix}-1 \\2\sqrt(10) \\1 \\-2\sqrt(10)\end{bmatrix}\,</math><math>e^{2-2\sqrt{10}}+ c_3</math><math>\begin{bmatrix}1 \\2\sqrt(5) \\1 \\2\sqrt(5)\end{bmatrix}\,</math><math>e^{32\sqrt{5}}+ c_4</math><math>\begin{bmatrix}1 \\-2\sqrt(5) \\1 \\-2\sqrt(5)\end{bmatrix}\,</math><math>e^{4*-2\sqrt{5}}\,</math>		<math>\ x=c_1</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0\sqrt{0}}+ c_2</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_3</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{00\sqrt{0}}+ c_4</math><math>\begin{bmatrix}0 \\0\sqrt(0) \\0 \\0\sqrt(0)\end{bmatrix}\,</math><math>e^{0*0\sqrt{0}}\,</math>

	== Matrix Exponential ==		== Matrix Exponential ==

Gregory.peterson: /* Eigen Vectors */

2009-12-10T22:06:22Z

Eigen Vectors

← Older revision		Revision as of 15:06, 10 December 2009
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	0		0
	\end{bmatrix}</math>		\end{bmatrix}</math>
			'''So then the answer is...'''

			<math>\ x=c_1</math><math>\begin{bmatrix}-1 \\-2\sqrt(10) \\1 \\2\sqrt(10)\end{bmatrix}\,</math><math>e^{2\sqrt{10}}+ c_2</math><math>\begin{bmatrix}-1 \\2\sqrt(10) \\1 \\-2\sqrt(10)\end{bmatrix}\,</math><math>e^{2-2\sqrt{10}}+ c_3</math><math>\begin{bmatrix}1 \\2\sqrt(5) \\1 \\2\sqrt(5)\end{bmatrix}\,</math><math>e^{32\sqrt{5}}+ c_4</math><math>\begin{bmatrix}1 \\-2\sqrt(5) \\1 \\-2\sqrt(5)\end{bmatrix}\,</math><math>e^{4*-2\sqrt{5}}\,</math>

	== Matrix Exponential ==		== Matrix Exponential ==