Damped response - Revision history

Jpye at 08:19, 29 March 2011

2011-03-29T08:19:19Z

← Older revision		Revision as of 08:19, 29 March 2011
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			{{missingimages}}
	{{experimental}}		{{experimental}}

Jpye at 01:27, 3 August 2010

2010-08-03T01:27:48Z

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	<math>s^2 Y + 2 s Y + 2 Y = A X</math>		<math>s^2 Y + 2 s Y + 2 Y = A X</math>

	Now, let <math>s Y = Z</~~span~~> so that we have a pair of equations		Now, let <math>s Y = Z</math> so that we have a pair of equations

	<math>		<math>

Jpye at 01:27, 3 August 2010

2010-08-03T01:27:35Z

← Older revision		Revision as of 01:27, 3 August 2010
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	We can model a system like this fairly easily with ASCEND. First we need to reduce the model to first order differential equations:		We can model a system like this fairly easily with ASCEND. First we need to reduce the model to first order differential equations:

	<~~span class="texhtml"~~>''s~~''<sup>~~2~~</sup>''~~Y'' + 2''s~~''''~~Y'' + 2''Y'' = ''A~~''''~~X''</~~span~~>		<math>s^2 Y + 2 s Y + 2 Y = A X</math>

	Now, let <~~span class="texhtml"~~>''s~~''''~~Y'' = ''Z''</span> so that we have a pair of equations		Now, let <math>s Y = Z</span> so that we have a pair of equations

	<math>		<math>

Jpye at 00:37, 30 July 2010

2010-07-30T00:37:29Z

← Older revision		Revision as of 00:37, 30 July 2010
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	Here is the output from the above system, with A = 2, x(t) = 1, plotted using OpenOffice. Different values of the coefficients in the <math>\dot{z}</math> equation give different levels of damping, resonant frequency, etc.		Here is the output from the above system, with A = 2, x(t) = 1, plotted using OpenOffice. Different values of the coefficients in the <math>\dot{z}</math> equation give different levels of damping, resonant frequency, etc.

	~~<div class="thumb tnone"><div class="thumbinner" style="width:402px;">~~[[Image:Dampedresponse.png~~]] <div class="thumbcaption"><div class="magnify">[[File:Dampedresponse.png~~\|~~<img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" />~~]]~~</div></div></div></div>~~		[[Image:Dampedresponse.png\|thumb\|420px\|none]]

	This plot shows that the above transfer function has roots at <math> -1 \pm j </math> as expected:		This plot shows that the above transfer function has roots at <math> -1 \pm j </math> as expected:

	~~<div class="thumb tnone"><div class="thumbinner" style="width:402px;">~~		[[Image:Simpleroots.png\|thumb\|420px\|none]]
	~~{\| class="wikitable"~~
	\|-
	~~\| Error creating thumbnail: convert~~: ~~unable to open image `/var/www/wiki/images/2/25/~~Simpleroots.png~~': No such file or directory.<br /> convert: unable to open file `/var/www/wiki/images/2/25/Simpleroots.png'.<br /> convert: missing an image filename `/var/www/wiki/images/~~thumb~~/2/25/Simpleroots.png/400px-Simpleroots.png'.<br />~~ \|~~} <div class="thumbcaption"><div class="magnify">[[File:Simpleroots.png~~\|~~<img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" />~~]]~~</div></div></div></div>~~

	Here is the model that was used to produce these solutions:		Here is the model that was used to produce these solutions:

	<source lang="a4c">REQUIRE ~~"~~ivpsystem.a4l~~"~~;		<source lang="a4c">REQUIRE "ivpsystem.a4l";

	REQUIRE ~~"~~atoms.a4l~~"~~;		REQUIRE "atoms.a4l";
	IMPORT ~~"~~johnpye/extpy/extpy~~"~~;		IMPORT "johnpye/extpy/extpy";

	IMPORT ~~"~~johnpye/roots~~"~~;		IMPORT "johnpye/roots";

	(*		(*
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	METHODS		METHODS
	METHOD values;		METHOD values;
	x[7]~~ ~~:= 0;		x[7] := 0;

	x[8]~~ ~~:= 0;		x[8] := 0;
	END values;		END values;

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	See also [[IDA]]		See also [[IDA]]




	~~[[Category:Experimental]]~~
	[[Category:Documentation]]		[[Category:Documentation]]
	[[Category:Examples]]		[[Category:Examples]]

Jpye at 00:35, 30 July 2010

2010-07-30T00:35:40Z

← Older revision		Revision as of 00:35, 30 July 2010
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	~~<div class="pageexperimental">''This page documents an '''~~experimental~~''' feature. You can help out by '''testing it''' and <span class="plainlinks">[http://ascendwiki.cheme.cmu.edu/index.php?title=Damped_response&action=edit recording]</span> your experiences.''</div>~~		{{experimental}}

	In classical control theory, a second-order system has a transfer function like		In classical control theory, a second-order system has a transfer function like

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2010-05-13T13:42:30Z

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<div class="pageexperimental">''This page documents an '''experimental''' feature. You can help out by '''testing it''' and <span class="plainlinks">[http://ascendwiki.cheme.cmu.edu/index.php?title=Damped_response&action=edit recording]</span> your experiences.''</div>

In classical control theory, a second-order system has a transfer function like

<math>
G(s) = \frac{Y(s)}{X(s)} = \frac{A}{s^2+2s+2}
</math>

We can model a system like this fairly easily with ASCEND. First we need to reduce the model to first order differential equations:

<span class="texhtml">''s''<sup>2</sup>''Y'' + 2''s''''Y'' + 2''Y'' = ''A''''X''</span>

Now, let <span class="texhtml">''s''''Y'' = ''Z''</span> so that we have a pair of equations

<math>
\begin{matrix}
sZ + 2Z +2Y = AX \\
sY = Z
\end{matrix}
</math>

Taking the inverse Laplace transform and rearranging,

<math>
\begin{matrix}
\dot{z} &=& Ax-2z -2y \\
\dot{y} &=& z
\end{matrix}
</math>

This can be coded into ASCEND in the usual syntax (see below).

Here is the output from the above system, with A = 2, x(t) = 1, plotted using OpenOffice. Different values of the coefficients in the <math>\dot{z}</math> equation give different levels of damping, resonant frequency, etc.

<div class="thumb tnone"><div class="thumbinner" style="width:402px;">[[Image:Dampedresponse.png]] <div class="thumbcaption"><div class="magnify">[[File:Dampedresponse.png|<img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" />]]</div></div></div></div>

This plot shows that the above transfer function has roots at <math> -1 \pm j </math> as expected:

<div class="thumb tnone"><div class="thumbinner" style="width:402px;">
{| class="wikitable"
|-
| Error creating thumbnail: convert: unable to open image `/var/www/wiki/images/2/25/Simpleroots.png': No such file or directory.<br /> convert: unable to open file `/var/www/wiki/images/2/25/Simpleroots.png'.<br /> convert: missing an image filename `/var/www/wiki/images/thumb/2/25/Simpleroots.png/400px-Simpleroots.png'.<br /> |} <div class="thumbcaption"><div class="magnify">[[File:Simpleroots.png|<img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" />]]</div></div></div></div>

Here is the model that was used to produce these solutions:

<source lang="a4c">REQUIRE "ivpsystem.a4l";

REQUIRE "atoms.a4l";
IMPORT "johnpye/extpy/extpy";

IMPORT "johnpye/roots";

(*
'Test problem'
We create a set of models with artificially-selected roots

and then check that the roots plot is as expected.
*)
MODEL simpleroots;

range IS_A set OF integer_constant;

range :== [7..8];
x[range], dx_dt[range] IS_A solver_var;

(* roots at -1 +/i j *)
dx_dt[7] = 2.0 * 1.0 - 2.0 * x[7] - 2.0 * x[8];

dx_dt[8] = x[7];

t IS_A time;

METHODS
METHOD values;
x[7] := 0;

x[8] := 0;
END values;

METHOD ode_init;
FOR i IN [range] DO

x[i].ode_type := 1;
dx_dt[i].ode_type := 2;

x[i].ode_id := i;
dx_dt[i].ode_id := i;

x[i].obs_id := i;
END FOR;

t.ode_type := -1;
t := 0 {s};

END ode_init;

METHOD on_load;
RUN default_self;

RUN reset; RUN values;
RUN ode_init;

RUN roots;
END on_load;

METHOD roots;

EXTERNAL roots(SELF);
END roots;
END simpleroots;</source>

See also [[IDA]]

[[Category:Experimental]]
[[Category:Documentation]]
[[Category:Examples]]