Coupled Oscillator: Double Pendulum - Revision history

Jimmy.apablaza: /* Matrix Exponential */

2009-12-13T19:46:41Z

Matrix Exponential

← Older revision		Revision as of 12:46, 13 December 2009
Line 319:		Line 319:

	Hence,		Hence,
	:<math>
			: <math>
	\dot{\underline{x}}		\dot{\underline{x}}
	=		=
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	\underline{x}		\underline{x}
	</math>		</math>

			so, the matrix exponential can be solved using matrix multiplication.

2009-12-13T19:38:13Z

Matrix Exponential

@@ Line 339: / Line 339: @@
 .853913 \mathbf{i} &  0.73951  &  0.426956 \mathbf{i}  &  0.369755 \\
 -0.853913 \mathbf{i} &  0.73951  & -0.426956 \mathbf{i}  &  0.369755 \\
 \end{bmatrix}
 \underline{x}
 </math>

2009-12-13T18:47:21Z

Matrix Exponential

@@ Line 339: / Line 339: @@
 .853913 \mathbf{i} &  0.73951  &  0.426956 \mathbf{i}  &  0.369755 \\
 -0.853913 \mathbf{i} &  0.73951  & -0.426956 \mathbf{i}  &  0.369755 \\
 \end{bmatrix}
 \underline{x}
 </math>

2009-12-13T18:40:45Z

Matrix Exponential

2009-12-13T18:40:04Z

← Older revision		Revision as of 11:40, 13 December 2009
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	\dot{\underline{x}}		\dot{\underline{x}}
	=		=
			\begin{bmatrix}
			-0.2 \mathbf{i} & 0.2 \mathbf{i} & -0.29277 \mathbf{i} & 0.29277 \mathbf{i} \\
			0.4 & 0.4 & 0.33806 & 0.33806 \\
			0.4 \mathbf{i} & -0.4 \mathbf{i} & 0.58554 \mathbf{i} & 0.58554 \mathbf{i} \\
			-0.8 & -0.8 & -0.67621 & 0.67621 \\
			\end{bmatrix}
			\begin{bmatrix}
	e^{2 \mathbf{i} t} & 0 & 0 & 0 \\		e^{2 \mathbf{i} t} & 0 & 0 & 0 \\
	0 & e^{-2 \mathbf{i} t} & 0 & 0 \\		0 & e^{-2 \mathbf{i} t} & 0 & 0 \\
	0 & 0 & e^{1.1547 \mathbf{i} t} & 0 \\		0 & 0 & e^{1.1547 \mathbf{i} t} & 0 \\
	0 & 0 & 0 & e^{-1.1547 \mathbf{i} t} \\		0 & 0 & 0 & e^{-1.1547 \mathbf{i} t} \\
			\end{bmatrix}
			\begin{bmatrix}
			1.25 \mathbf{i} & 0.625 & -0.625 \mathbf{i} & -0.3125 \\
			-1.25 \mathbf{i} & 0.625 & 0.625 \mathbf{i} & -0.3125 \\
			0.853913 \mathbf{i} & 0.73951 & 0.426956 \mathbf{i} & 0.369755 \\
			-0.853913 \mathbf{i} & 0.73951 & -0.426956 \mathbf{i} & 0.369755 \\
	\end{bmatrix}		\end{bmatrix}
	</math>		</math>

2009-12-13T18:38:26Z

Matrix Exponential

@@ Line 239: / Line 239: @@
 : <math>
-\hat{A}
+=
 \begin{bmatrix}
@@ Line 297: / Line 297: @@
 \end{bmatrix}
 \bar{z}
 </math>

2009-12-13T18:22:00Z

Standard Equation

← Older revision		Revision as of 11:22, 13 December 2009
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	Now, we plug the eigenvalues and eigenvectors to produce the standar equation,		Now, we plug the eigenvalues and eigenvectors to produce the standar equation,

	: <math>\underline{x} = c_1 k_1 e^{\lambda_1 t} + c_2 ~~k_2~~ e^{\lambda_2 t} + c_3 k_3 e^{\lambda_3 t} + c_4 k_4 e^{\lambda_4 t}</math>		: <math>\underline{x} = c_1 \underline{k_1} e^{\lambda_1 t} + c_2 \underline{k}_2 e^{\lambda_2 t} + c_3 \underline{k_3} e^{\lambda_3 t} + c_4 \underline{k_4} e^{\lambda_4 t}</math>

	: <math>		: <math>

2009-12-13T18:19:11Z

Standard Equation

← Older revision		Revision as of 11:19, 13 December 2009
Line 188:		Line 188:
	Now, we plug the eigenvalues and eigenvectors to produce the standar equation,		Now, we plug the eigenvalues and eigenvectors to produce the standar equation,

	: <math>x = c_1 k_1 e^{\lambda_1 t} + c_2 k_2 e^{\lambda_2 t} + c_3 k_3 e^{\lambda_3 t} + c_4 k_4 e^{\lambda_4 t}</math>		: <math>\underline{x} = c_1 k_1 e^{\lambda_1 t} + c_2 k_2 e^{\lambda_2 t} + c_3 k_3 e^{\lambda_3 t} + c_4 k_4 e^{\lambda_4 t}</math>

	: <math>		: <math>

	x		\underline{x}
	=		=
	c_1		c_1

2009-12-13T00:22:09Z

Problem Statement

← Older revision		Revision as of 17:22, 12 December 2009
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	* the system oscillates vertically under the influence of gravity.		* the system oscillates vertically under the influence of gravity.
	* the mass of both rod are negligible		* the mass of both rod are negligible
	* no ~~dumping~~ forces act on the system		* no damping forces act on the system
	* positive direction to the right.		* positive direction to the right.

2009-12-13T00:21:49Z

Problem Statement

← Older revision		Revision as of 17:21, 12 December 2009
Line 11:		Line 11:
	'''Assumptions:'''		'''Assumptions:'''
	* the system oscillates vertically under the influence of gravity.		* the system oscillates vertically under the influence of gravity.
	* the mass of both rod are ~~neligible~~		* the mass of both rod are negligible
	* no ~~dumpung~~ forces act on the system		* no dumping forces act on the system
	* positive direction to the right.		* positive direction to the right.