Laplace transforms:Mass-Spring Oscillator - Revision history

David.steinweg: /* Part 4 */

2009-11-23T19:54:52Z

Part 4

← Older revision		Revision as of 12:54, 23 November 2009
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	==Part 4==		==Part 4==
	From the transfer function ~~in the [[#Laplace Transform\|Part 1]] section~~,		From the transfer function derived earlier,

	<math>		<math>

David.steinweg: /* Part 4 */

2009-11-23T19:54:02Z

Part 4

← Older revision		Revision as of 12:54, 23 November 2009
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	==Part 4==		==Part 4==
	From the transfer function in the [[#Laplace Transform\|~~Laplace Transform~~]] section,		From the transfer function in the [[#Laplace Transform\|Part 1]] section,

	<math>		<math>

David.steinweg: /* Part 6 */

2009-11-23T19:52:53Z

Part 6

← Older revision		Revision as of 12:52, 23 November 2009
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	==Part 6==		==Part 6==

	~~Choose~~ the state ~~variable~~ to be <math> x\frac{}{} </math> and <math> \dot{x} </math>, then ~~following~~ the ~~example from class, the state~~ equation is		If we choose the state variables to be <math> x\frac{}{} </math> and <math> \dot{x} </math>, then the equation is the matrix we have below


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	\begin{bmatrix}		\begin{bmatrix}
	0 & 1 \\		0 & 1 \\
	\frac{-k}{m} & \frac{-~~\lambda~~}{m}		\frac{-b}{m} & \frac{-k}{m}
	\end{bmatrix}		\end{bmatrix}
	\,		\,
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	\begin{bmatrix}		\begin{bmatrix}
	0 \\		0 \\
	1		\frac {1}{m}
	\end{bmatrix}		\end{bmatrix}
	\, g \~~, \sin{\theta}~~		\,\delta(t)
	</math>		</math>
	----		----

David.steinweg: /* Solution */

2009-11-23T19:43:10Z

Solution

← Older revision		Revision as of 12:43, 23 November 2009
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	</math>		</math>

			==Part 6==

			Choose the state variable to be <math> x\frac{}{} </math> and <math> \dot{x} </math>, then following the example from class, the state equation is


			<math>
			\begin{bmatrix}
			\dot{x} \\
			\ddot{x}
			\end{bmatrix}
			=
			\begin{bmatrix}
			0 & 1 \\
			\frac{-k}{m} & \frac{-\lambda}{m}
			\end{bmatrix}
			\,
			\begin{bmatrix}
			x \\
			\dot{x}
			\end{bmatrix}
			+
			\begin{bmatrix}
			0 \\
			1
			\end{bmatrix}
			\, g \, \sin{\theta}
			</math>
	----		----

David.steinweg: /* Problem Statement */

2009-11-23T19:42:48Z

Problem Statement

← Older revision		Revision as of 12:42, 23 November 2009
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	Use convolution to find the output.		Use convolution to find the output.

			==Part 6==

			Make a the state equations for this problem

	=Solution=		=Solution=

David.steinweg: /* Part 5 */

2009-11-17T23:32:15Z

Part 5

← Older revision		Revision as of 16:32, 17 November 2009
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	</math>		</math>

	~~To solve the integral, one must do two integration by parts, or alternatively plug it into a calculator, which yields~~

	~~<math>~~
	x(t)= \frac{-3}{2120} \, g \, \sin{\theta} \, e^{\frac{-1}{6}t} \left( 159 \cos{\left( \frac{\sqrt{159} \, t}{6} \right)} + \sqrt{159} \sin{\left( \frac{\sqrt{159} \, t}{6} \right)} - 159 \, e^{\frac{1}{6}t} \right)
	~~</math>~~

	----		----

David.steinweg at 23:34, 3 November 2009

2009-11-03T23:34:24Z

@@ Line 198: / Line 198: @@
 ==Part 4==
 ==Part 5==
 ----

David.steinweg at 19:11, 27 October 2009

2009-10-27T19:11:57Z

← Older revision		Revision as of 12:11, 27 October 2009
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	==Part 4==		==Part 4==

	Find the break points and asymptotes on the ~~bode plot~~.		Find how to use the break points and asymptotes to obtain the magnitude frequency response.

	==Part 5==		==Part 5==

	~~Find~~ the ~~convolution~~		Use convolution to find the output.

David.steinweg at 19:03, 27 October 2009

2009-10-27T19:03:57Z

← Older revision		Revision as of 12:03, 27 October 2009
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	===Problem Statement===		=Problem Statement=

	==Part 1==		==Part 1==
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	Make a bode plot.		Make a bode plot.

	==~~=Solution=~~==		==Part 4==

			Find the break points and asymptotes on the bode plot.

	~~'''~~Part 1~~'''~~		==Part 5==

			Find the convolution



			=Solution=


			==Part 1==


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	~~'''~~Part 2~~'''~~		==Part 2==


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	~~'''~~Part 3~~'''~~		==Part 3==


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	[[Image:HW-7.png‎\|560px\|thumb\|center\|Fig. 2]]		[[Image:HW-7.png‎\|560px\|thumb\|center\|Fig. 2]]

			==Part 4==

			==Part 5==
	----		----

David.steinweg at 18:58, 27 October 2009

2009-10-27T18:58:27Z

← Older revision		Revision as of 11:58, 27 October 2009
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	===Problem Statement===		===Problem Statement===

	~~'''~~Part 1~~'''~~		==Part 1==

	An ideal mass ''m''=10kg is sitting on a plane, attached to a rigid surface via a spring. The spring with ''k''=500N/m is exerting zero force when the mass is centered at ''x=0''. Between the mass and plane there is a 1 mm layer of a viscous fluid and the block has an area of .1m^2 in contact the plane. Find the equation of motion that the spring mass system follows if there is an initial impulse applied, and then find the viscosity value of the fluid so that the mass comes to a stop within three seconds with the initial impulse used. Below is a picture/FBR of the system.		An ideal mass ''m''=10kg is sitting on a plane, attached to a rigid surface via a spring. The spring with ''k''=500N/m is exerting zero force when the mass is centered at ''x=0''. Between the mass and plane there is a 1 mm layer of a viscous fluid and the block has an area of .1m^2 in contact the plane. Find the equation of motion that the spring mass system follows if there is an initial impulse applied, and then find the viscosity value of the fluid so that the mass comes to a stop within three seconds with the initial impulse used. Below is a picture/FBR of the system.
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	As set up, the mass is sitting at x=0 with positive being to the right.		As set up, the mass is sitting at x=0 with positive being to the right.

	~~'''~~Part 2~~'''~~		==Part 2==

	Apply the Initial Value and Final Value Theorems to this problem.		Apply the Initial Value and Final Value Theorems to this problem.

	~~'''~~Part 3~~'''~~		==Part 3==

	Make a bode plot.		Make a bode plot.