Damping is a condition in which external force operates in such a manner that it impedes the motion of oscillatory body. As a matter of fact, all real time harmonic motions that we consider to be simple are actually damped SHM. We consider them SHM only as an approximation. The motion of a block hanging from a spring, for example, is not SHM as air works to oppose the motion – at every instant. As a consequence, the span (amplitude) of the motion keeps decreasing every cycle. Diminishing amplitude is the characterizing feature of damped oscillation. A typical displacement – time plot looks as shown here.

Figure 2: The amplitude of oscillation diminishes with time.

Damped oscillation

The real time situation may be a bit complex to describe damped oscillation mathematically. Here, we consider one simplified situation in which air resistance can be considered to be proportional to the velocity of the oscillatory body. In such case, net force on the oscillating body is the resultant of restoring and damping force (with a negative sign) :

F net = - k x - b v F net = - k x - b v

m a + k x + b v = 0 m a + k x + b v = 0

In terms of displacement derivatives :

⇒ m ⅆ 2 x ⅆ t 2 + b ⅆ x ⅆ t + k x = 0 ⇒ m ⅆ 2 x ⅆ t 2 + b ⅆ x ⅆ t + k x = 0

For small damping constant “b”, the solution of this differential equation yields :

x = A 0 e - b t / 2 m sin ω b t + φ x = A 0 e - b t / 2 m sin ω b t + φ

The amplitude of the oscillation is a decreasing function in time, which tends to become zero :

A = A 0 e - b t 2 m A = A 0 e - b t 2 m

A damped oscillation thus dies down as the case with most of the oscillatory systems, which are not provided with external energy to compensate energy dissipated due to damping. On the other hand, the angular frequency of damped oscillation depends on additional factor of proportionality constant “b” :

ω b = { k m − b 2 m 2 } ω b = { k m − b 2 m 2 }

Clearly, angular frequency of damped oscillation is lesser than corresponding angular frequency if damping is absent i.e. b = 0,

ω b = k m = ω o ω b = k m = ω o

It is clear from the discussion so far that most of artificial oscillation system tends to cease as damping is part of the natural set up. There can be various sources of damping force, but friction is one common source. There can be air resistance or resistance at the fixed hinge from which oscillating part is hung.

It is imperative that we supply appropriate energy (force) to compensate for the loss of energy due to damping. To meet this requirement, the oscillating system is subjected to oscillatory external force. The external force imparted is itself oscillatory and is, therefore, described by harmonic trigonometric function. Considering presence of damping, the force equation is :

F net = - k x - b v + F 0 sin ω e t F net = - k x - b v + F 0 sin ω e t

m a + k x + b v − F 0 sin ω e t = 0 m a + k x + b v − F 0 sin ω e t = 0

In terms of displacement derivatives :

⇒ m ⅆ 2 x ⅆ t 2 + b ⅆ x ⅆ t + k x − F 0 sin ω e t = 0 ⇒ m ⅆ 2 x ⅆ t 2 + b ⅆ x ⅆ t + k x − F 0 sin ω e t = 0

The solution of this differential equation yields :

x = A sin ω e t + φ x = A sin ω e t + φ

As is evident from the expression, the system oscillates with the same frequency as that of external force. The amplitude of the oscillation is described in terms of frequency of external force ( ω e ω e ) and natural frequency ( ω o ω o ) as :

A = F m { ω e 2 − ω o 2 + b ω e m 2 } A = F m { ω e 2 − ω o 2 + b ω e m 2 }

We can see that this expression is a constant for given set up. It means we can sustain a constant amplitude of the oscillation by applying external oscillatory force - even if damping force is present.

The resonance is an interesting feature of oscillation. This phenomenon attracts interest as it makes possible to achieve extra-ordinary result (material failure of large structure) with small force! Resonance also explains why earthquake causes differentiating result to different structures – most devastating where resonance occurs!

The condition for maximum amplitude is obtained by differentiating amplitude function with respect to applied frequency “ ω e ω e ” and setting the resulting expression equal to zero. This gives the resonance frequency as :

ω R = ω o 2 − b 2 4 m 2 ω R = ω o 2 − b 2 4 m 2

If damping is absent, then the amplitude function is maximum when applied frequency is equal to natural frequency. The amplitude is infinite in such case. However, damping is always present in actuality and as such resonance amplitude is finite. For small value of damping constant, the resonance frequency is close to natural frequency and resonance is sharply defined. As the damping increases, resonance amplitude is reduced. The plots of amplitude – frequency curves outline the features of resonance amplitude as shown in the figure. Note that resonance becomes flatter as damping force increases.

Figure 3: The amplitude is maximum at resonance.

Resonance

Resonance underlines certain fundamental aspects of oscillatory system. First, it demonstrates that oscillation is a storing mechanism of energy. Second, it demonstrates that the energy can be supplemented in a constructive manner to increase the energy of the oscillating system to an extra-ordinary level. We can experience both these aspects easily by observing motion of swings in a nearby park. Ask a very strong adult to apply the most of his/her power to swing in one go and compare the result with that of a child who produces large swings with no such power. What is the difference? The child begins with small swing and synchronizes subsequent jerk with the oscillation. Each time the energy of the system is increased by the small amount associated with each jerk. The key, here, is the synchronization (timing of external force) and periodicity with which the jerk is applied to the swing.

Resonance is not specific to mechanical oscillatory system. The concept is equally applicable to electrical system and waves in general. Tunning of a particular radio station at a specific frequency is one of the most striking applications of this phenomenon.

We should also realize that many vibrating systems like atomic arrangements, vibrating strings etc have multiple natural frequencies. This means that these systems can be subjected to resonance at more than one frequency.