<Text-field style="Heading 1" layout="Heading 1">Maple Preliminaries</Text-field>

The Relationship between Pole Locations and Time-Domain Performance for a Second Order System

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To use this worksheet:

When first opening, click the 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 button to re-execute the worksheet.

If you want to make changes, you can click 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to restart the Maple server, which clears all internal variables, then click 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again.

The triangles in front of the section headings may be clicked to expand or collapse a section or subsection.

A command that looks like this: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEocGFja2FnZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRKWZ1bmN0aW9uRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJ0ZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY2NiQtRiM2JS1GLDYlUStwYXJhbWV0ZXJzRidGL0YyRj1GQEZARj1GQA== is calling the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpZnVuY3Rpb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvRjNRJ25vcm1hbEYn located in package LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocGFja2FnZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy9GM1Enbm9ybWFsRic= with input parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVErcGFyYW1ldGVyc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy9GM1Enbm9ybWFsRic=. A package is a group of related functions. An alternative to this notation is to type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKHBhY2thZ2VGJ0YvRjIvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvRjNRJ25vcm1hbEYnRkBGPUZA prior to calling any of the functions within this package which will load them all into memory. Then you can simply type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpZnVuY3Rpb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUStwYXJhbWV0ZXJzRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJ0ZARj1GQA==.

The assignment operator, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGLw==, takes whatever is to the right of this symbol and assigns it to the variable on the left. For example, 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 assigns the equation 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 to the name LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkZXExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJw==. Now, when Maple encounters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkZXExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJw==, it replaces it with 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.

An alternative way to reference equations is using the equation label. For example, equation (2.1) can be referenced by typing Ctrl-L and typing "2.1" into the dialog box.

Terminating a statement with a colon suppresses the Maple output. Terminating a statement with a semicolon or nothing allows the Maple output. When the equation is inline with text (as in the previous bullet points), it will not execute unless it contains either a colon or semicolon.

Dot notation can be used for time derivatives by typing Ctrl-Shift-" to move the cursor above a variable, the type period (.) to add a dot. Prime notation implies differentiation with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy9GM1Enbm9ybWFsRic= and is accomplished by using a single-quote marks (') after a variable.

Variables with assumptions have a tilde (~) after their name. For example, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy9GM1Enbm9ybWFsRic= has no assumptions, but LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RInxpckYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZMLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjk= has assumptions. Maple adds the tilde in its output, but you are not required to put it in your input.

Right-clicking on an expression brings up numerous operations and functions that can be performed on that expression.

<Text-field style="Heading 1" layout="Heading 1">A Second Order System</Text-field>

A second order systems is a system whose governing differential equation involves a second derivative. An electrical example would be an LRC circuit. The example we will look at in detail is a mechanical mass-spring-damper system, as shown in Figure 1.

Figure 1 - A mass spring damper system that results in a second order governing equation.

In the figure, the mass, spring, and damper have parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, respectively, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is an externally applied force. To derive the equation of motion for the system, we require Newton's Second Law, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJtRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRImFGJ0YvRjJGOQ== , Hooke's Law for the spring, 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, and linear viscous damping for the damper, 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. We note that 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 and 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. In the absence of an external applied force (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRj1GPQ==), the force applied to the mass in Newton's Second Law is equal to the spring-damper forces, so that 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, or alternatively 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. With a non-zero applied force, the time-domain governing equation is

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JImZHNiI2I0kidEdGJSwoKiZJIm1HRiUiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYtNiQtSSJ5R0YlRiZGJ0YnRitGKyomSSJiR0YlRitGMEYrRisqJkkia0dGJUYrRjJGK0Yr

Now we load the Integral Transformation (inttrans) toolbox, which allows us to perform Laplace transforms.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKWludHRyYW5zRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZAIn the Laplace domain, this equation becomes

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JKGxhcGxhY2VHNiI2JS1JImZHRiU2I0kidEdGJUYqSSJzR0YlLC4qKEkibUdGJSIiIilGKyIiI0YvLUYkNiUtSSJ5R0YlRilGKkYrRi9GLyomRi5GLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGNTYjIiIhRi8hIiIqKEYuRi9GK0YvLUY1Rj5GL0ZAKihJImJHRiVGL0YrRi9GMkYvRi8qJkZERi9GQkYvRkAqJkkia0dGJUYvRjJGL0Yv

which, assuming zero initial conditions and using the usual corresponding Laplace-domain variable names becomes,

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2I0kic0dGJSwoKihJIm1HRiUiIiIpRiciIiNGKy1JIllHRiVGJkYrRisqKEkiYkdGJUYrRidGK0YuRitGKyomSSJrR0YlRitGLkYrRis=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExaXNvbGF0ZX5mb3J+WShzKUYnRi9GMkZD

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

Ly1JIllHNiI2I0kic0dGJSwkKiYtSSJGR0YlRiYiIiIsKComSSJtR0YlRiwpRiciIiNGLCEiIiomSSJiR0YlRixGJ0YsRjJJImtHRiVGMkYyRjI=

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

Ly1JIllHNiI2I0kic0dGJSomLUkiRkdGJUYmIiIiLCgqJkkibUdGJUYrKUYnIiIjRitGKyomSSJiR0YlRitGJ0YrRitJImtHRiVGKyEiIg==

Using the right-click menu, we were able to isolate the expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, which puts the equation in the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRIlRGJ0YvRjJGNS1GLDYlUSJGRidGL0YyRjVGPQ==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is the output (position), LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is the transfer function, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is the input (applied force). Then, choosing "Simplify \342\206\222 Normalize" eliminateds the negative signs. The transfer function is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVExVHJhbnNmZXJGdW5jdGlvbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRJHJoc0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiNRIUYnLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRX3EvLUkiWUc2IjYjSSJzRzYiKiYtSSJGRzYiNiNJInNHNiIiIiIsKComSSJtRzYiIiIiKUkic0c2IiIiIyIiIiIiIiomSSJiRzYiIiIiSSJzRzYiIiIiIiIiSSJrRzYiIiIiISIiRicvJSthY3Rpb250eXBlR1E6bWFwbGVzb2Z0LmNvbTpsYWJlbChMMzg5KUYnLyUldmlld0dRJmxhYmVsRidGZm4vJStleGVjdXRhYmxlR0Y9RjlGOUZmb0Y5LUYjNiYtRiw2JVEiRkYnRi9GMi1GWDYkLUYjNiUtRiw2JVEic0YnRi9GMkYvRjJGOUYvRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmlwLyUpYmV2ZWxsZWRHRj1GZm9GOQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiQsKComSSJtRzYiIiIiKUkic0dGJiIiI0YnRicqJkkiYkdGJkYnRilGJ0YnSSJrR0YmRichIiI=

Using the following substitutions, we are able to generalize the transfer function to be in terms of the system's natural resonant frequency, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1, damping ratio, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and gain, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 is the frequency at which the system would resonate if the damping, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, were zero. The damping ratio describes whether the step response of the system will oscillate (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy near 0) or settle smoothly (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy near 1). The gain, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, determines the final value of the system in response to a constant input.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Jy1GLzYlUSJtRidGMkY1RjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0Y1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZILyUpYmV2ZWxsZWRHRj8tSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGXW8tSS5tbXVsdGlzY3JpcHRzR0YkNigtRiM2Ji1GLzYjUSFGJy1JKG1zdWJzdXBHRiQ2KC1GLzYlUSgmb21lZ2E7RicvRjNGP0ZRLUYjNiYtRi82JVEibkYnRjJGNUY9RkBGUS1GIzYnLUkjbW5HRiQ2JFEiMkYnRlFGMkY9RkBGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGXHEvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnRj1GUS1JJW5vbmVHRiRGJ0ZicS1JLG1wcmVzY3JpcHRzR0YkRidGYnEtRiM2J0Zlb0YyRj1GQEY1RmVvRj1GUQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LyomSSJrRzYiIiIiSSJtR0YlISIiKiQpJkkmb21lZ2FHRiU2I0kibkdGJSIiI0Ym

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyomSSJiRzYiIiIiSSJtR0YlISIiLCQqKCIiI0YmSSN4aUdGJUYmJkkmb21lZ2FHRiU2I0kibkdGJUYmRiY=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kiS0c2IiokSSJrR0YkISIi

We rearrange these equations and assign the results to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, so that they are replaced in the transfer function.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

PCUvSSJiRzYiLCQqKiIiIyIiIkkjeGlHRiVGKSZJJm9tZWdhR0YlNiNJIm5HRiUhIiJJIktHRiVGL0YpL0kia0dGJSokRjBGLy9JIm1HRiUqJilGK0YoRi9GMEYv

The transfer function is now of the more generalized form:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiQsKCooKSZJJm9tZWdhRzYiNiNJIm5HRigiIiMhIiJJIktHRihGLClJInNHRihGKyIiIkYwKixGK0YwSSN4aUdGKEYwRiZGLEYtRixGL0YwRjAqJEYtRixGMEYs

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RIj1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRi1JJm10ZXh0R0YkNiVRK35zaW1wbGlmeX5GJ0YvRjJGQg==

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

KigpJkkmb21lZ2FHNiI2I0kibkdGJiIiIyIiIkkiS0dGJkYqLCgqJClJInNHRiZGKUYqRioqKkYpRipJI3hpR0YmRipGL0YqRiRGKkYqKiRGI0YqRiohIiI=

<Text-field style="Heading 1" layout="Heading 1">System Step Response</Text-field>

The step response of a system is an important performance characteristic because it determines how quickly and accurately a system responds to changing inputs. As detailed in the previous section, a second order system can be described using its natural frequency, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1, its damping ratio, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and its gain, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, which we will assume to be unity. Maple will assume that all variables are complex unless it is told otherwise. A few hints to Maple on the domain of these parameters will often result in more compact and intuitive solutions later. We consider only positive frequencies,

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

and set the damping ratio between 0 and 1

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiktSSNtbkdGJDYkUSIwRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiPEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUScmIzk1ODtGJy9GMEZGRj5GQC1GOzYkUSIxRidGPi8lK2V4ZWN1dGFibGVHRkZGPkY+LUYsNiNRIUYnRmluRj4=

If the damping ratio were greater than 1, then the second order system would simplify to two first order poles, so for the time being we will assume that it is less than one. The transfer function of the system, as shown in the previous section, is (with unit gain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=)

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYpJkknb21lZ2F8aXJHNiI2I0kibkdGJiIiIyIiIiwoKiQpSSJzR0YmRilGKkYqKipGKUYqSSR4aXxpckdGJkYqRi5GKkYkRipGKiokRiNGKkYqISIi

Multiplying by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjI= (the Laplace transform of a unit step as the input) and taking the inverse Laplace transform gives the time-domain step response.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCgiIiJGIyomLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKCksJiokKUkkeGl8aXJHRioiIiNGIyEiIkYjRiMjRiNGMkYjSSJ0R0YqRiMmSSdvbWVnYXxpckdGKjYjSSJuR0YqRiNGIy1JJGV4cEdGJzYjLCQqKEY1RiNGMUYjRjZGI0YzRiNGMyoqRi1GM0YxRiMtSSRzaW5HRidGK0YjRjpGI0Yz

Since the arguments of the sine and cosine functions are identical, we can convert them into a single trigonometric function with a new magnitude and phase offset. This gives a more intuitive form of the step response.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVE1cGhhc2VBbXBTdGVwUmVzcG9uc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUShjb252ZXJ0RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNi0tRiw2JVEtc3RlcFJlc3BvbnNlRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZobi9GTkZpbi1GNjYtUSInRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRl9vLUY2Ni1RIzotRidGOUY7Rj5GQEZCRkRGRkZIRmhuRl9vLUYsNiVRKXBoYXNlYW1wRidGL0YyRmBvRlotRiw2JVEidEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5Rl5wRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYiIiJGIyooKSwmRiNGIyomLCYqJClJJHhpfGlyRzYiIiIjRiMhIiJGI0YjRi5GKkYjRiMjRiNGLUYjLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YsNiMsJCooSSJ0R0YsRiNGK0YjJkknb21lZ2F8aXJHRiw2I0kibkdGLEYjRi5GIy1JJGNvc0dGMjYjLCYqKClGKEYvRiNGOEYjRjlGI0YjLUknYXJjdGFuR0YyNiMqJkZCRi5GK0YjRi5GI0Yu

<Text-field style="Heading 1" layout="Heading 1">Pole Locations and Time-Domain Performance</Text-field>

Click and drag the poles on the left to see the impact on the step response on the right. As the poles move further to the left, the rise-time of the step response improves. As the poles move closer to the real axis, the response becomes less oscillatory.

S-plane (Laplace domain)

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use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. realPart := GetProperty('PolesPlot','clickx'); imaginaryPart := GetProperty('PolesPlot','clicky'); UpdatePlots(); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. realPart := GetProperty('PolesPlot','endx'); imaginaryPart := GetProperty('PolesPlot','endy'); UpdatePlots(); end use; 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

Step response (time-domain)

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The time-domain step response:

Damping Ratio:

Natural Frequency (rad/s):

Natural Frequency (Hz):

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy can be determined geometrically from the s-plane location of the pole, as illustrated in Figure 2. The length of the vector between the origin and the pole is the natural frequency, and the slope of the vector relative to the negative real axis is related to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy .

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Figure 2 - Geometric interpretation of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y1 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy

JSFHJSFH

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JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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