Math 2250-1Mon Nov 1210.5, EP7.6Today we focus on the unit impulse ("delta function") Laplace transform entry, and the periodic function table entry, from those shown below. On Friday we discussed using the unit step function to turn functions on and off. We also briefly discussed convolution integrals, to find inverse Laplace transforms of products LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2JVEiR0YnRi9GMkY1LUkjbW9HRiQ2LVEiLkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0Y9 And, at the end of today's notes we'll see (EP7.6) why that convolution formula is so important in applications.
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for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEic0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPkYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEiTUYnRi9GMkY1Rjw=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 unit step 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 for turning components on and off at 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.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more complicated 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unit impulse/delta "function"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convolution integrals to invert Laplace transform productsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1Ri8vRjZRJ25vcm1hbEYnRi9GQEYvRkA= with period 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 for periodic forcing which is not sinusoidal.
Periodic functions: Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== be periodic with period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInRGJ0YvRjItSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRiw2JVEicEYnRi9GMkZBRkEtRj42LVEiPUYnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yNzc3Nzc4ZW1GJy9GVkZobkYrLUY2NiQtRiM2JEY6RkFGQS1GPjYtUSJ+RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWRmJvLUY+Ni1RKSZGb3JBbGw7RidGQUZDRkZGSEZKRkxGTkZQRmFvRmluRjpGQQ==. Notice that we can decompose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= into the sum of a function which equals LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtaUdGJDYlUSJwRicvJSdpdGFsaWNHRkAvRjVRJ2l0YWxpY0YnRjRGNC8lJW9wZW5HUSJbRidGNA== and is zero on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYlUSJwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYtUSgmaW5maW47RidGPkZAL0ZERkJGRUZHRklGS0ZNRk8vRlNGUUY+Rj4vJSVvcGVuR1EiW0YnRj4= , with a function that is zero on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtaUdGJDYlUSJwRicvJSdpdGFsaWNHRkAvRjVRJ2l0YWxpY0YnRjRGNC8lJW9wZW5HUSJbRidGNA== and equals LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYlUSJwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYtUSgmaW5maW47RidGPkZAL0ZERkJGRUZHRklGS0ZNRk8vRlNGUUY+Rj4vJSVvcGVuR1EiW0YnRj4=: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURitGNS1GNjYkLUYjNictSSNtbkdGJDYkUSIxRidGPS1GQDYtUSgmbWludXM7RidGPUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRl1vLUYsNiVRInVGJ0YvRjItRjY2JC1GIzYmRjpGaW4tRiw2JVEicEYnRi9GMkY9Rj1GPUY9LUZANi1RIitGJ0Y9RkNGRkZIRkpGTEZORlBGXG9GXm9GX29GYm9GK0Y1Rj0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUYjNistSSNtaUdGJDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRkk2JVEidEYnRkxGT0YvRi8tRlI2JC1GIzYnLUkjbW5HRiQ2JFEiMUYnRi8tRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZfby1GSTYlUSJ1RidGTEZPLUZSNiQtRiM2JkZWRltvLUZJNiVRInBGJ0ZMRk9GL0YvRi9GLy1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/Rl5vRmBvRmFvRmRvRkhGZG9GLy1GSTYjUSFGJ0Yv(since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZ1bWludXMwO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMRjk=periodic).If we take the Laplace transform of this identity use the definition of Laplace transform for the first term on the right and the translation theorem for the second term, we 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 .Exercise 1) Find the Laplace transform of the function whose graph is shown below. The function has period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv 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 )EP 7.6 impulse functions and the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy operator.Consider a force LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== acting on an object for only on a very short time interval 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, for example as when a bat hits a ball. This impulse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the force is defined to be the 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 it measures the net change in momentum of the object since by Newton's second 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEqJkltcGxpZXM7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkRGLw==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 .Since the impulse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= only depends on the integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, and since the exact form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is unlikely to be known in any case, the easiest model is to replace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with a constant force having the same total impulse, i.e. to 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where 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 is the unit impulse function given by 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 .Notice that 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 .Here's a graph of 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, for example: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the unit impulse function is a linear combination of unit step functions, we could solve differential equations with impulse functions so-constructed. As far as Laplace transform goes, it's even easier to take the limit as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZzcmFycjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= for the Laplace 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, and this effectively models impulses on very short time scales. 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 .In Laplace land we can use L'Hopital's rule (in the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy) to take the limit as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZzcmFycjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI=: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.The result in time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= space is not really a function but we call it the "delta function" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUSJhRidGPUY/RjJGMkYy anyways, and visualize it as a function that is zero everywhere except at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk=, and that it is infinite at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyRjk= in such a way that its integral over any open interval containing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= equals one. As explained in EP7.6, the delta "function" can be thought of in a rigorous way as an operator, not as a function. It can also be thought of as the derivative of the unit step function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInRGJ0YvRjItSSNtb0dGJDYtUSgmbWludXM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRiw2JVEiYUYnRi9GMkZBRkFGQQ==, and this is consistent with the Laplace table entries for derivatives of functions. In any case, this leads to the very useful Laplace transform table entry
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 unit impulse 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 for impulse forcing
Exercise 2) Revisit the swing from Friday and solve the IVP below for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. In this case the parent is providing an impulse each time the child passes through equilibrium position after completing a cycle.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRjpGPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0Y5RjlGOS1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORmduRlVGOQ== .Impulse forcing of mechanical systems EP 7.6LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEld2l0aEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiMtRiw2JlEmcGxvdHNGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy1JI21vR0YkNi5RIjpGJ0YvRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVw==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impulse solution: five equal impulses to get same final amplitude of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NjA7RicvJSVzaXplR1EjMTBGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi9GNQ== meters - Exercise 2:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1GLDYmUSJ0RidGL0YyRjUtRjk2LlEoJnNyYXJyO0YnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRRlktSSNtbkdGJDYlUSMuMkYnRi9GPC1GOTYuUScmc2RvdDtGJ0YvRjxGPkZBRkNGRUZHRklGS0ZYRlotRiw2JlEjUGlGJ0YvL0YzRkBGPEZpbi1GLDYmUSRzdW1GJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JS1GIzYwLUYsNiZRKkhlYXZpc2lkZUYnRi9GX29GPC1GZG82JS1GIzYsRlItRjk2LlEoJm1pbnVzO0YnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSwwLjIyMjIyMjJlbUYnL0ZRRmNwLUYsNiZRImtGJ0YvRjJGNUZpbi1GZm42JVEiMkYnRi9GPEZpbkZcb0YvLyUrZXhlY3V0YWJsZUdGQEY8Ri9GPEZpbi1GLDYmUSRzaW5GJ0YvRl9vRjxGW3AtRjk2LlEiLEYnRi9GPEY+L0ZCRjRGQ0ZFRkdGSUZLRlgvRlFRLDAuMzMzMzMzM2VtRidGZXAtRjk2LlEiPUYnRi9GPEY+RkFGQ0ZFRkdGSUZLRk1GUC1GZm42JVEiMEYnRi9GPC1GOTYuUSMuLkYnRi9GPEY+RkFGQ0ZFRkdGSUZLRmJwRlotRmZuNiVRIjRGJ0YvRjxGL0ZbcUY8Ri9GPC1GOTYuUSI6RidGL0Y8Rj5GQUZDRkVGR0ZJRktGTUZQRi9GW3FGPA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlElcGxvdEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNjctRiw2JlEiZkYnRi9GMkY1LUY5NiUtRiM2Ji1GLDYmUSJ0RidGL0YyRjVGLy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YvRkotSSNtb0dGJDYuUSIsRidGL0ZKLyUmZmVuY2VHRkkvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGRC1GTTYuUSI9RidGL0ZKRlAvRlNGSUZURlZGWEZaRmZuL0ZpblEsMC4yNzc3Nzc4ZW1GJy9GXG9GY28tSSNtbkdGJDYlUSIwRidGL0ZKLUZNNi5RIy4uRidGL0ZKRlBGYW9GVEZWRlhGWkZmbi9GaW5RLDAuMjIyMjIyMmVtRicvRlxvRmpuLUZmbzYlUSMyMEYnRi9GSi1GTTYuUScmc2RvdDtGJ0YvRkpGUEZhb0ZURlZGWEZaRmZuRmhuRl5wLUYsNiZRI1BpRidGLy9GM0ZJRkpGTC1GLDYmUSZjb2xvckYnRi9GMkY1Rl5vLUYsNiZRJmJsYWNrRidGL0YyRjVGTC1GLDYmUSZ0aXRsZUYnRi9GMkY1Rl5vLUYsNiZROGBsYXp5fnBhcmVudH5vbn5Nb25kYXlgRidGL0YyRjVGL0ZHRkpGL0ZKLUZNNi5RIjtGJ0YvRkpGUEZSRlRGVkZYRlpGZm5GaG5GZG9GL0ZHRko=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Or, an impulse at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Ri9GPA== and another one at 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 .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Convolutions and solutions to non-homogeneous physical oscillation problems (EP7.6 p. 499-501) Consider a mechanical or electrical forced oscillation problem for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, and the particular solution that begins at rest: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRjpGPg==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 .Then in Laplace land, this equation is equivalent 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 .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 is called the weight function for the physical system, and because of the convolution table entry
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convolution integrals to invert Laplace transform products
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=the solution is given 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 .This idea generalizes to much more complicated mechanical and circuit systems, and is how engineers experiment mathematically with how proposed configurations will respond to various input forcing functions, once they figure out the weight function for their system.Exercise 3. Let's play the resonance game, with non-sinusoidal forcing functions. We'll stick with our earlier swing, but consider various forcing periodic functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=that we haven't thought about before.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVRjpGPg==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a) Find the weight function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==.b) Write down the solution formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== as a convolution integral.JSFHc) Play the resonance game on the following pages ... We worked out that the solution to our DE IVP will be 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 the unforced system has a natural angular frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnRjpGOi8lL3N1YnNjcmlwdHNoaWZ0R0ZCLUkjbW9HRiQ2LVEiPUYnRjovJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZZLUZANiRRIjFGJ0Y6RjpGK0Y6 , we expect resonance when the forcing function has the corresponding period of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRldGPg==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. We will discover that there is the possibility for resonance if the period of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a multiple of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIi5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlctRkM2LVEifkYnRj5GRkZJRktGTUZPRlFGU0ZVRlhGPg== (Also, forcing at the natural period doesn't guarantee resonance...it depends what function you force with.)Example 1) A square wave forcing function with amplitude 1 and period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1. Let's talk about how we came up with the formula (which works until 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 ) .LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=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1) What's your vote? Is this square wave going to induce resonance, i.e. a response with linearly growing amplitude?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 2) A triangle wave forcing function, same 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2) Resonance?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Example 3) Forcing not at the natural period, e.g. with a square wave having period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk= .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3) Resonance?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Example 4) Forcing not at the natural period, e.g. with a particular wave having period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjZGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVy1GLDYlUScmIzk2MDtGJy9GMEY9RjlGOQ== .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) Resonance?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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Hey, what happened???? How do we need to modify our thinking if we force a system with something which is not sinusoidal, in terms of worrying about resonance? In the case that this was modeling a swing (pendulum), how is it getting pushed?Precise Answer: It turns out that any periodic function with period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a (possibly infinite) superposition of a constant function with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnY29zaW5lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk=and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElc2luZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= functions of periods 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 Equivalently, these functions in the superposition are 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 with \317\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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8= This is the theory of Fourier series, which you will study in other courses, e.g. Math 3150, Partial Differential Equations. If the given periodic forcing function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=has non-zero terms in this superposition for which 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(the natural angular frequency) (equivalently LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRIlBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GLzYlUSJuRidGMkY1RjJGNS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQi8lKWJldmVsbGVkR1EmZmFsc2VGJy9GNlEnbm9ybWFsRic=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), there will be resonance; otherwise, no resonance. We could already have understood some of this in Chapter 5, for exampleExercise 4) The natural period of the following DE is (still) 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 . Which of the two DE's below will have solutions that exhibit resonance? Use Chapter 5 superposition ideas. Compare your answer with how the period of the forcing function compares to the natural period of the system.a) 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)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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=