1.9. Supplement on Logarithmic Decrement Theory
1.
With the solution of the free vibration, the oscillation is when
B
2
is imaginary
Converting to a real term for oscillations,
Using the Euler formula, the exponentials can be written as sines and cosines
2.
The square bracket term is the oscillating behavior that does not decay.
The
amplitude decay causes the
y
oscillation to decay
The amplitude decay function is verified by generating
q(t)
from the
y(t)
data.
If
q(t)
appears as an undamped solution, then the amplitude decay solution is verified.
3.
Following the trigonometric functions for the oscillation, a peak amplitude is observed
at integer-multiple cycles,
b
2
t = m 2
B
,
where
m=[0,1,2,3,...]
.

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