14.10 Forced oscillations and resonance

When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency w, and the oscillations are called free oscillations. All free oscillations eventually die out because of the ever-present damping forces. However, an external agency can maintain these oscillations. These are called forced or driven oscillations.

We consider the case when the external force is itself periodic, with a frequency \(\omega_{\mathrm{d}}\) called the driven frequency. The most important fact of forced periodic oscillations is that the system oscillates not with its natural frequency \(\omega\), but at the frequency \(\omega_{\mathrm{d}}\) of the external agency; the free oscillations die out due to damping. The most familiar example of forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations.

Suppose an external force \(F(t)\) of amplitude \(F_0\) that varies periodically with time is applied to a damped oscillator. Such a force can be represented as,
\(
F(t)=F_o \cos \omega_d t \dots(14.36)
\)
The motion of a particle under the combined action of a linear restoring force, damping force, and a time-dependent driving force represented by Eq. (14.36) is given by,
\(
m a(t)=-k x(t)-b v(t)+F_o \cos \omega_d t \quad(14.37 a)
\)
Substituting \(\mathrm{d}^2 x / \mathrm{d} t^2\) for acceleration in Eq. (14.37a) and rearranging it, we get
\(
m \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2}+b \frac{\mathrm{d} x}{\mathrm{~d} t}+k x=F_o \cos \omega_d t \dots(14.37b)
\)
This is the equation of an oscillator of mass \(m\) on which a periodic force of (angular) frequency \(\omega_d\) is applied. The oscillator, initially, oscillates with its natural frequency \(\omega\). When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force. Its displacement, after the natural oscillations die out, is given by
\(
x(t)=A \cos \left(\omega_d t+\phi\right) \dots(14.38)
\)
where \(t\) is the time measured from the moment when we apply the periodic force.

The amplitude \(A\) is a function of the forced frequency \(\omega_d\) and the natural frequency \(\omega\). Analysis shows that it is given by
\(
A=\frac{F_o}{\left\{m^2\left(\omega^2-\omega_d^2\right)^2+\omega_d^2 b^2\right\}^{1 / 2}} \dots(14.39a)
\)
and \(\tan \phi=\frac{-v_o}{\omega_d x_o} \dots(14.39b)\)
where \(m\) is the mass of the particle and \(v_0\) and \(x_0\) are the velocity and the displacement of the particle at time \(t=0\), which is the moment when we apply the periodic force. Equation (14.39) shows that the amplitude of the forced oscillator depends on the (angular) frequency of the driving force. We can see a different behaviour of the oscillator when \(\omega_d\) is far from \(\omega\) and when it is close to \(\omega\). We consider these two cases.

Case 1: Small Damping, Driving Frequency far from Natural Frequency

In this case, \(\omega_d b\) will be much smaller than \(m\left(\omega^2-\omega_d^2\right)\), and we can neglect that term. Then Eq. (14.39) reduces to
\(
A=\frac{F_o}{m\left(\omega^2-\omega_d^2\right)} \dots(14.40)
\)
Fig. 14.21 shows the dependence of the displacement amplitude of an oscillator on the angular frequency of the driving force for different amounts of damping present in the system. It may be noted that in all cases the amplitude is the greatest when \(\omega_d / \omega=1\). The curves in this figure show that the smaller the damping, the taller and narrower is the resonance peak.

If we go on changing the driving frequency, the amplitude tends to infinity when it equals the natural frequency. But this is the ideal case of zero damping, a case which never arises in a real system as the damping is never perfectly zero.

Case-2: Driving Frequency Close to Natural Frequency

If \(\omega_d\) is very close to \(\omega, m\left(\omega^2-\omega_d^2\right)\) would be much less than \(\omega_d b\), for any reasonable value of \(b\), then Eq. (14.39) reduces to
\(
A=\frac{F_o}{\omega_d b} \dots(14.41)
\)
This makes it clear that the maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance.