<Text-field style="Heading 1" layout="Heading 1">Deriving the Equations of the 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the sum of the forces, the DE for the system can be found: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:. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VG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transfer function can now be assigned to a DynamicSystems object for use throughout the labJSFH
<Text-field style="Heading 2" layout="Heading 2">Initialization</Text-field>QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJL0R5bmFtaWNTeXN0ZW1zR0YlISIiQyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiJSFHQyQ+SSV0ZmVxRzYiKiQsKComSSJtR0YlIiIiSSJzR0YlIiIjRioqJkkiY0dGJUYqRitGKkYqSSJrR0YlRiohIiJGMA==JSFHQyQ+SSN0Zkc2Ii1JMVRyYW5zZmVyRnVuY3Rpb25HRiU2I0kldGZlcUdGJSEiIg==JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Part 1</Text-field>The first case has parameter values m=1 kg, k=2 N/m, and c=3 N s/m. The transfer function for the system is:JSFHLUklZXZhbEclKnByb3RlY3RlZEc2JEkldGZlcUc2IjwlL0kiY0dGJyIiJC9JImtHRiciIiMvSSJtR0YnIiIiprint(); # input placeholder
<Text-field style="Heading 2" layout="Heading 2">Unit step input</Text-field>The position as a function of time for a unit step response can be calculated by using an inverse Laplace transform:JSFHLSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSomSSJzR0YoISIiLCgqJEYtIiIjIiIiRi0iIiRGMUYyRi5GLUkidEdGKA== = print(); # input placeholderJSFHPlotting the response plot for a 10 second simulation can be seen below:JSFHLUktUmVzcG9uc2VQbG90RzYiNihJI3RmR0YkLUklU3RlcEdGJEYkL0krcGFyYW1ldGVyc0dGJDwlL0kiY0dGJCIiJC9JImtHRiQiIiMvSSJtR0YkIiIiL0kmdGl0bGVHRiRRPlN0ZXB+cmVzcG9uc2V+d2l0aH5jPX4zfk5+cy9tRiQvSSdsYWJlbHNHRiQ3JFEpdGltZX4ocylGJFExZGlzcGxhY2VtZW50fihtKUYkL0kwbGFiZWxkaXJlY3Rpb25zR0YkNyRRK2hvcml6b250YWxGJFEpdmVydGljYWxGJA==%;As the plot shows, the system reaches steady-state value of 0.5 m at approximately 6 seconds.
<Text-field style="Heading 2" layout="Heading 2">Unit impulse</Text-field>The position as a function of time for a unit impulse response can be calculated by using an inverse Laplace transform:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSokLCgqJEkic0dGKCIiIyIiIkYvIiIkRjBGMSEiIkYvSSJ0R0Yo = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Plotting the response plot for a 10 second simulation can be seen below:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUk0SW1wdWxzZVJlc3BvbnNlUGxvdEc2IjYnSSN0ZkdGJC9JK3BhcmFtZXRlcnNHRiQ8JS9JImNHRiQiIiQvSSJrR0YkIiIjL0kibUdGJCIiIi9JJnRpdGxlR0YkUUFJbXB1bHNlfnJlc3BvbnNlfndpdGh+Yz1+M35OfnMvbUYkL0knbGFiZWxzR0YkNyRRKXRpbWV+KHMpRiRRMWRpc3BsYWNlbWVudH4obSlGJC9JMGxhYmVsZGlyZWN0aW9uc0dGJDckUStob3Jpem9udGFsRiRRKXZlcnRpY2FsRiQ=%;As the plot shows, the system peaks at 2.5 m and reaches steady-state value of 0 m at approximately 7 seconds.JSFHJSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Part 2</Text-field>The first case has parameter values m=1 kg, k=2 N/m, and c=2 N s/m. The transfer function for the system is:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklZXZhbEclKnByb3RlY3RlZEc2JEkldGZlcUc2IjwlL0kiY0dGJyIiIy9JImtHRidGKy9JIm1HRiciIiI=print(); # input placeholder
<Text-field style="Heading 2" layout="Heading 2">Unit step input</Text-field>The position as a function of time for a unit step response can be calculated by using an inverse Laplace transform:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSomSSJzR0YoISIiLCgqJEYtIiIjIiIiRi1GMUYxRjJGLkYtSSJ0R0Yo = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Plotting the response plot for a 10 second simulation can be seen below:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUktUmVzcG9uc2VQbG90RzYiNihJI3RmR0YkLUklU3RlcEdGJEYkL0krcGFyYW1ldGVyc0dGJDwlL0kiY0dGJCIiIy9JImtHRiRGLi9JIm1HRiQiIiIvSSZ0aXRsZUdGJFE+U3RlcH5yZXNwb25zZX53aXRofmM9fjJ+Tn5zL21GJC9JJ2xhYmVsc0dGJDckUSl0aW1lfihzKUYkUTFkaXNwbGFjZW1lbnR+KG0pRiQvSTBsYWJlbGRpcmVjdGlvbnNHRiQ3JFEraG9yaXpvbnRhbEYkUSl2ZXJ0aWNhbEYk%;As the plot shows, the system overshoots to approximately 0.52 m before reaching a steady-state value of 0.5 m at approximately 6 seconds.
<Text-field style="Heading 2" layout="Heading 2">Unit impulse</Text-field>The position as a function of time for a unit impulse response can be calculated by using an inverse Laplace transform:JSFHLSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSokLCgqJEkic0dGKCIiIyIiIkYvRjBGMEYxISIiRi9JInRHRig= = print(); # input placeholderJSFHPlotting the response plot for a 10 second simulation can be seen below:JSFHLUk0SW1wdWxzZVJlc3BvbnNlUGxvdEc2IjYnSSN0ZkdGJC9JK3BhcmFtZXRlcnNHRiQ8JS9JImNHRiQiIiMvSSJrR0YkRiwvSSJtR0YkIiIiL0kmdGl0bGVHRiRRQUltcHVsc2V+cmVzcG9uc2V+d2l0aH5jPX4yfk5+cy9tRiQvSSdsYWJlbHNHRiQ3JFEpdGltZX4ocylGJFExZGlzcGxhY2VtZW50fihtKUYkL0kwbGFiZWxkaXJlY3Rpb25zR0YkNyRRK2hvcml6b250YWxGJFEpdmVydGljYWxGJA==%;As the plot shows, the system peaks at 0.32 m and reaches steady-state value of 0 meters at approximately 6.5 seconds.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Part 3</Text-field>The first case has parameter values m=1 kg, k=2 N/m, and c=0 N s/m. The transfer function for the system is:JSFHLUklZXZhbEclKnByb3RlY3RlZEc2JEkldGZlcUc2IjwlL0kiY0dGJyIiIS9JImtHRiciIiMvSSJtR0YnIiIiprint(); # input placeholderJSFH
<Text-field style="Heading 2" layout="Heading 2">Unit step input</Text-field>The position as a function of time for a unit step response can be calculated by using an inverse Laplace transform:JSFHLSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSomSSJzR0YoISIiLCYqJEYtIiIjIiIiRjFGMkYuRi1JInRHRig= = print(); # input placeholderJSFHPlotting the response plot for a 10 second simulation can be seen below:JSFHLUktUmVzcG9uc2VQbG90RzYiNihJI3RmR0YkLUklU3RlcEdGJEYkL0krcGFyYW1ldGVyc0dGJDwlL0kiY0dGJCIiIS9JImtHRiQiIiMvSSJtR0YkIiIiL0kmdGl0bGVHRiRRPlN0ZXB+cmVzcG9uc2V+d2l0aH5jPX4wfk5+cy9tRiQvSSdsYWJlbHNHRiQ3JFEpdGltZX4ocylGJFExZGlzcGxhY2VtZW50fihtKUYkL0kwbGFiZWxkaXJlY3Rpb25zR0YkNyRRK2hvcml6b250YWxGJFEpdmVydGljYWxGJA==%;
<Text-field style="Heading 2" layout="Heading 2">Unit impulse</Text-field>The position as a function of time for a unit impulse response can be calculated by using an inverse Laplace transform:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK2ludmxhcGxhY2VHRig2JSokLCYqJEkic0dGKCIiIyIiIkYwRjEhIiJGL0kidEdGKA== = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Plotting the response plot for a 10 second simulation can be seen below:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUk0SW1wdWxzZVJlc3BvbnNlUGxvdEc2IjYnSSN0ZkdGJC9JK3BhcmFtZXRlcnNHRiQ8JS9JImNHRiQiIiEvSSJrR0YkIiIjL0kibUdGJCIiIi9JJnRpdGxlR0YkUUFJbXB1bHNlfnJlc3BvbnNlfndpdGh+Yz1+MH5OfnMvbUYkL0knbGFiZWxzR0YkNyRRKXRpbWV+KHMpRiRRMWRpc3BsYWNlbWVudH4obSlGJC9JMGxhYmVsZGlyZWN0aW9uc0dGJDckUStob3Jpem9udGFsRiRRKXZlcnRpY2FsRiQ=%;JSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Rough work</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==QyQtSSV3aXRoRzYiNiNJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiJSFHQyQ+SSN0Zkc2Ii1JMVRyYW5zZmVyRnVuY3Rpb25HRiU2IyokLCgqJkkibUdGJSIiIkkic0dGJSIiI0YtKiZJImNHRiVGLUYuRi1GLUkia0dGJUYtISIiRjM=JSFHQyQ+SSNrMUc2IjcnIiIiIiIlIiIqIiM7IiNEISIiprint(); # input placeholderQyQ+SSNjMUc2IjcnJCIiJiEiIiIiIiQiIzpGKSIiIyQiI0RGKUYpQyQ+SSRjb2xHNiI3J0kkcmVkR0YlSSdvcmFuZ2VHRiVJJmdyZWVuR0YlSSVibHVlR0YlSSd2aW9sZXRHRiUhIiI=print(); # input placeholderJSFHJSFHJSFHLSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKGRpc3BsYXlHRig2Iy1JJHNlcUdGJjYkLUktUmVzcG9uc2VQbG90R0YoNiZJI3RmR0YoLUklU3RlcEdGKEYoL0kmY29sb3JHRigmSSRjb2xHRig2I0kiaUdGKC9JK3BhcmFtZXRlcnNHRig8JS9JImNHRigmSSNjMUdGKEY5L0kia0dGKCZJI2sxR0YoNiMiIiIvSSJtR0YoRkcvRjo7RkciIiY=%;LSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKGRpc3BsYXlHRig2Iy1JJHNlcUdGJjYkLUktUmVzcG9uc2VQbG90R0YoNiZJI3RmR0YoLUklU3RlcEdGKEYoL0kmY29sb3JHRigmSSRjb2xHRig2I0kiaUdGKC9JK3BhcmFtZXRlcnNHRig8JS9JImNHRigmSSNjMUdGKDYjIiIiL0kia0dGKCZJI2sxR0YoRjkvSSJtR0YoRkMvRjo7RkMiIiY=%;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%;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%;Unit 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LSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNfRihJK2ludmxhcGxhY2VHRig2JSomSSJzR0YoISIiLCgqJEYuIiIjIiIiRi5GMkYyRjNGL0YuSSJ0R0YoJSFHKiQsKCokSSJzRzYiIiIjIiIiRiVGJ0YnRighIiI=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LSZJKWludHRyYW5zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNfRihJK2ludmxhcGxhY2VHRig2JSokLCgqJEkic0dGKCIiIyIiIkYwRjFGMUYyISIiRjBJInRHRig=JSFHJSFH