Q7.18 Damping (T)
The figure shows the displacement (x, in meters) as a function of time (t, in seconds) for an oscillating object. Describe the behaviour in words. Estimate:
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Solution
The figure shows damped simple harmonic motion. One can think of the system oscillating in a conventional SHM fashion, but with an amplitude that is slowly decaying with time.- The time taken for one cycle can be estimated from the figure as T=1s so the SHM frequency is 1 Hz.
- The figure suggests (what we saw in S6.6)
that the amplitude decays exponentially with time:
xm(t)=xm(0)e-λtwhere λ depends on the strength of the damping. An eyeball estimate suggests that the amplitude decays from xm=1 to xm=0.25 in time t=7 s. ThusThe time for the amplitude to decay to 1% of its initial value is the solution to
![\[ -t \lambda= \ln \left[ \frac{x_m(t)}{x_m(0)} \right] \hspace{0.25cm}\mbox{\rm implying}\hspace{0.25cm} \lambda = - \ln [0.25]/7 = 0.2 \]](mastermathpng-0.png)
![\[ 0.01 = e^{-\lambda t} \hspace{0.25cm}\mbox{\rm or}\hspace{0.25cm} t = - \frac{\ln(0.01)}{\lambda} = \frac{4.6}{0.2}= 23\,s \]](mastermathpng-1.png)
The total energy associated with SHM is proportional to the square of the amplitude. Thus
![\[ \frac{E(t)}{E(0)} = \frac{x_m^2(t)}{x_m^2(0)} = \left [ e^{-\lambda t} \right] ^2 = e^{-2\lambda t} \]](mastermathpng-2.png)
[If you got that last step wrong then review the properties of exponentials!]
The time for the energy to decay to 1% of its initial value is thus the solution to
![\[ 0.01 = e^{-2\lambda t} \hspace{0.25cm}\mbox{\rm giving}\hspace{0.25cm} t=\frac{4.6}{0.4}= 11.5\,s \]](mastermathpng-3.png)
