The displacement of a linear damped oscillator is given, for weak damping by

Question

x=Ke^(-bΩt) sin⁡(βt+ ε)
Show x has a maximum at times given by βt = 2nπ + d where n is an integer and d is gives the position of the first maximum
Show also that the ratio of successive maxima is e^(2bΩπ/β).

Answer 1

maximum when sin(bt+e) = 1 or -1 but this seems only to look at the positive answer for max
that is when argument = pi/2
or pi/2 + n 2pi
first max when pi/2
so for first max
b t(1) + e = pi/2 = d
further maxes when
b t(n) + e = d + 2 n pi
so further max when b t/o = d + 2 n pi
x max n = k e (-d+2pin)o
xmax n+1 = k e^(-d +2 pi n +2 pi )o

xmax n+1/xmax n = e^( - 2pi o)