Laplace transforms: Critically Damped Motion - Revision history

Mark.bernet: /* State Space */

2009-11-18T23:58:56Z

State Space

← Older revision		Revision as of 16:58, 18 November 2009
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	<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k/m & -C/m \end{bmatrix} \begin{bmatrix} x \\ dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u(t)</math>		<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k/m & -C/m \end{bmatrix} \begin{bmatrix} x \\ \dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u(t)</math>

2009-11-18T23:58:34Z

State Space

← Older revision		Revision as of 16:58, 18 November 2009
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	=State Space=		=State Space=

			<math>\text {Using state equatons is just another way to solve a system modeled by an ODE }\,</math>

			<math>\text {First we need to add an applied force so u(t)=2N }\,</math>

			<math>m=\frac{8}{32}=\frac1 4 slugs</math>

			<math>\text {k=4}\,</math>

			<math>\text {C=2}\,</math>

			<math>\text {x(0)=0}\,</math>

			<math>\dot{x}(0)=-3</math>

			<math>\ddot{x}(0)=0</math>


			<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ -k/m & -C/m \end{bmatrix} \begin{bmatrix} x \\ dot{x} \end{bmatrix} + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}u(t)</math>









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2009-11-18T23:54:07Z

Things we know

2009-11-18T23:53:43Z

Things we know

← Older revision		Revision as of 16:53, 18 November 2009
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	<math>\text {k=4}\,</math>		<math>\text {k=4}\,</math>

	<math>\text {C=4}\,</math>		<math>\text {C=4}\,</math>

	<math>\text {x(0)=0}\,</math>		<math>\text {x(0)=0}\,</math>
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	<math>\text {Standard equation: }\,</math>		<math>\text {Standard equation: }\,</math>
	<math>m\frac{d^2x}{dt^2}+C\frac{dx}{dt}+khx=0</math>		<math>m\frac{d^2x}{dt^2}+C\frac{dx}{dt}+khx=0</math>


	===Solving the problem===		===Solving the problem===

2009-11-18T23:53:30Z

Using the Laplace Transform to solve a spring mass system that is critically damped

2009-11-18T23:46:38Z

Using the Laplace Transform to solve a spring mass system that is critically damped

← Older revision		Revision as of 16:46, 18 November 2009
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	<math>\text {k=4}\,</math>		<math>\text {k=4}\,</math>

	~~<math>\text {Damping constant C=2}\,</math>~~

	<math>\text {x(0)=0}\,</math>		<math>\text {x(0)=0}\,</math>

	<math>\dot{x}(0)=-3</math>		<math>\dot{x}(0)=-3</math>



	<math>\text {Standard equation: }\,</math>		<math>\text {Standard equation: }\,</math>

2009-11-18T23:19:25Z

Convolution

← Older revision		Revision as of 16:19, 18 November 2009
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	<math>\text {Not exactly the same but remember initial conditions arnt used}\,</math>		<math>\text {Not exactly the same but remember initial conditions arnt used}\,</math>



			=State Space=
	---		---

2009-11-18T23:18:43Z

Using the Laplace Transform to solve a spring mass system that is critically damped

← Older revision		Revision as of 16:18, 18 November 2009
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	<math>\text {A break point is located at any value where s = what is being added to it. }\,</math>		<math>\text {A break point is located at any value where s = what is being added to it. }\,</math>
	<math>\text {So for this transfer function its at s=4 (that is also the asymptotes location). }\,</math>		<math>\text {So for this transfer function its at s=4 (that is also the asymptotes location). }\,</math>


	=Convolution=		=Convolution=
	<math>\text {The convolution equation is as follows: }\,</math>		<math>\text {The convolution equation is as follows: }\,</math>

2009-11-18T16:56:39Z

Solving the problem

← Older revision		Revision as of 09:56, 18 November 2009
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	<math>\mathcal{L}^{-1}[-\frac{3}{(s+4)^2}]</math><br /><br />		<math>\mathcal{L}^{-1}[-\frac{3}{(s+4)^2}]</math><br /><br />

	<math>\text {and we get}\,</math>		<math>\text {and we get, by using the table of transforms on p. 515 of the text bokk}\,</math>

	<math>\mathbf {x}(t)=-3te^{-4t}</math><br /><br />		<math>\mathbf {x}(t)=-3te^{-4t}</math><br /><br />

2009-11-08T23:45:58Z

Convolution

← Older revision		Revision as of 16:45, 8 November 2009
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	==Convolution==		=Convolution=
	<math>\text {The convolution equation is as follows: }\,</math>		<math>\text {The convolution equation is as follows: }\,</math>