a shock absorber modelE&P3 5.4.23. The natural frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdGPUY1 of the shock absorber is 80 cycles per minute (just 1.33 cycles per second, with a period slightly less than a second), but the actual frequency is \317\211 = 78 cycles/minute. Find the damping constant that corresponds to this situation. For the relaxing of the shock absorber, seconds are a better time unit. So we must convert time units to seconds and angle units to radians.
See http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/handouts/ep2_5-4-23.pdf. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==Using the textbook notation for the coefficients of the DE: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 are given that the mass is 100 kg:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSQxMDBGJ0ZARitGKw==If 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 (no shock absorber hooked up), then you get oscillating solutions with 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 , and we are told 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Changing the frequency units back to those assumed by the differential equation in radians per second: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So we can solve for the coefficient 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, (since by definition: 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) and then evaluate its ratio with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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Next we turn on the damping (connect the shock absorber) with this ratio of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GLzYlUSJtRidGMkY1RjJGNS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGQi8lKWJldmVsbGVkR1EmZmFsc2VGJy1GLzYjUSFGJy9GNlEnbm9ybWFsRic= in the deq and the result is oscillations with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== = 78 cycles/sec: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The deq in standard form is then (we have only to figure out what the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has to be)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSRkZXFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGKw==Without looking up any formulas for the natural frequency of the spring and the actual frequency of the combined damped spring system, recall that they are 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, with the characteristic time 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.To get the frequency formula, we just need the imaginary part (times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) of the solutions of the characteristic equation to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYlUSJJRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0YrRjc=: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Copying the second term in the plus sign solution and setting it equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmb21lZ2E7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYlUSJJRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0YrRjc= we get the damping constant which produces this frequency, choosing the positive solution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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 reciprocal of the absolute value of the real part is instead the characteristic time. The one percent level is reached after 4.6 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Initial Conditions. [for a graph to illustrate this]If we idealize the initial displacement is zero, and the shock is kicked over a very short time interval to get an initial velocity (we pick a unit value since everything will scale by this number due to lienarity and we are only interested in the time scale), but effectively zero force after that, we get the following IVP solution, setting that initial velocity to 1 to have something we can graph: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 characteristic time for the decaying exponential is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktRiM2Iy1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRidGMi1GIzYjLUZSNiRRKzEuODYxNTM5MDlGJ0YyLyUubGluZXRoaWNrbmVzc0dGVC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zobi8lKWJldmVsbGVkR0YxLUYsNiNRIUYnwhile the period of the sine function is comparable: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 use Maple tricks to pull out the exponential and sine factors without copying and pasting: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 amplitude decays to the 1 percent level after a time of 4.6 \317\204 or equivalently: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This is 4.6 characteristic times, we plot for 5 characteristic times: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 we use the frequency formulas of course this is a bit easierWe solved this without any formulas memorized about the various frequencies. However, if we look up the formula for the relationship between the two frequencies, it is:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==QyUvLCgqJkkibUc2IiIiIkknb21lZ2EwR0YnIiIjRigqJkYmRihJInJHRidGKkYoKiZJImNHRidGKEYsRihGKCIiIUYoLUkmc29sdmVHRic2JEkiJUdGJ0YsThe imaginary part coefficient, squared, is minus the square of the actual frequencyNiI=LyokSSZvbWVnYUc2IiIiIywmKiRJJ29tZWdhMEdGJUYmIiIiKiZJImNHRiVGJi1JIi9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiRJIm1HRiVGJiIiJUYqISIiQyUtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSIlR0YoSSJjR0YoIiIiPkYrJkYqNiNGLA==This is easily solved for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, given the two frequencies (unit conversion first into radians per second from cycles per minute) and mass (in kg).The quality factor for this system at this value of the damping is:Qyk+SSdvbWVnYTBHNiIkIis3L2V4JCkhIioiIiI+SSZvbWVnYUdGJSQiKys0OW8iKUYoRik+SSJtR0YlIiQrIkYpSSJjR0YlUsing bob's lowercase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= notation for the exponential rate factor, thenQyc+Jkkia0c2IjYjIiIhKiZJImNHRiYiIiJJIm1HRiYhIiJGKz4mSSR0YXVHRiZGJyokRiRGLUYrPkkiUUdGJiomSSdvbWVnYTBHRiZGK0YvRis=JSFHAnd you see roughly a bit less than 3 oscillations (a bit bigger than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for small values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) during 4 characteristic times of the actual system. Right.