Answer to Question #252144 in Mechanics | Relativity for Tithi

Question

Discuss analytically the theory of forced vibrations in the presence of damping force

Answer 1

A tuning fork is an example of a damped harmonic oscillator. Indeed we hear the note because some of the energy of oscillation is converted into sound. After it is struck the intensity of the sound, which is proportional to the energy of the tuning fork, steadily decreases. However, the frequency of the note does not change. The ends of the tuning fork make thousands of oscillations before the sound disappears and so we can reasonably assume that the degree of damping is small. We may suspect, therefore, that the frequency of oscillation would not be very different if there were no damping. Thus we infer that the displacement x of an end of the tuning fork is described by a relationship of the form

$x = (\text{amplitude that reduces with t}) × \cos{ ωt}$

where the angular frequency ω is about but not necessarily the same as would be obtained if there were no damping. We shall assume that the amplitude of oscillation decays exponentially with time. The displacement of an end of the tuning fork will therefore vary according to

$x =A_0\cdot \exp{(-\beta t)}\cdot \cos{ ωt}$

where A₀ is the initial value of the amplitude and β is a measure of the degree of damping. The minus sign indicates that the amplitude reduces with time. As we shall see, this expression correctly describes the motion of a damped harmonic oscillator when the degree of damping is small and so the assumptions we have made above are reasonable.

Clearly the rate at which the oscillator loses energy will depend on the degree of damping and this is described by the quality of the oscillator. At first sight, damping in an oscillator may be thought undesirable. However, there are many examples where a controlled amount of damping is used to quench unwanted oscillations. Damping is added to the suspension system of a car to stop it from bouncing up and down long after it has passed over a bump in the road. Additional damping was installed on London’s Millennium Bridge shortly after it opened because it suffered from undesirable oscillations.

Reference:

• King, G. C. (2013). Vibrations and waves. John Wiley & Sons.