6.1 Basic Concepts and Formulae
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Both the exponential terms in (6.50) are negative and they correspond to exponential decrease. The motion is not oscillatory. The general solution is of the form
x = e
−bt
(Ae
Rt
+ Be
−Rt
)
(6.56)
Fig. 6.4 Overdamped motion
Case 3: Critical damping
b = ω, R = 0
Fig. 6.5 Criticallydamped
motion
The exponentials in the square bracket may be expanded to terms linear in Rt. The
solution is of the form
x = x 0 e
−bt
(1 + bt)
(6.57)
The motion is not oscillatory and is said to be critically damped. It is a transition
case and the motion is just aperiodic or non-oscillatory. There is an initial rise in
the displacement due to the factor (1 + bt) but subsequently the exponential term
dominates.
Energy and Amplitude of a Damped Oscillator
E(t) = E 0 e
−t/t c
(6.58)