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(4-1.9^2)/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))

Question

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(4-1.9^2)/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))

anonymous · Answer

I 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.