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pjcappaert
 3 years ago
Why is it said that critical damping leads to the fastest decay? If a system with amplitude 1 is slightly underdamped it crosses the x axis but it's absolute value is less than that of the system with critical damping up to f(x) ~ 10^25.
For the system: x''+cx'+x=0
critical damping (c=2)
x=e^(x)+x*e^(x)
underdamped (c=1.9, wd=sqrt(41.9^2)/2)
x=e^(x*b/2)*cos(wd*x)+(b/2/wd)*sin(wd*x)
=e^(x*1.9/2)*(cos(.39^.5/2*x)+(1.9/.39^.5)*sin(.39^.5/2*x))
pjcappaert
 3 years ago
Why is it said that critical damping leads to the fastest decay? If a system with amplitude 1 is slightly underdamped it crosses the x axis but it's absolute value is less than that of the system with critical damping up to f(x) ~ 10^25. For the system: x''+cx'+x=0 critical damping (c=2) x=e^(x)+x*e^(x) underdamped (c=1.9, wd=sqrt(41.9^2)/2) x=e^(x*b/2)*cos(wd*x)+(b/2/wd)*sin(wd*x) =e^(x*1.9/2)*(cos(.39^.5/2*x)+(1.9/.39^.5)*sin(.39^.5/2*x))

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pjcappaert
 3 years ago
Best ResponseYou've already chosen the best response.0I looked in a textbook and saw that the wording in the 18.03 lecture notes is not ideal. In the textbook the wording was "fastest decay without oscillation." I agree with this statement.
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