Forced Oscillations and Resonance: Definition, Types, Solved Examples

Forced Oscillations and Resonance: Would you believe if you were told that a group of soldiers once broke a bridge just by marching over it? It is indeed true! The soldiers, while crossing a bridge, are asked to break their ordered march and walk randomly. Wonder why does that happen? This is because the natural frequency of the parade of soldiers might be similar to the natural frequency of the bridge causing it to break and thus causing harm to the people in its surroundings.

Consider another example-Have you ever used a radio? These are installed in our cars and music systems. But how do we select a particular channel on a radio? By shifting the knob on the radio or by entering a given frequency, we force the radio to work at a certain frequency that we wish to hear. Also, have you ever seen an opera singer sing? They are capable of breaking glasses just by the power of their voices! How do they do that? The opera singers force the frequency of their voice to match with the natural frequency of the glass, and when that happens, the glass breaks. Let us read further to understand the phenomena of forced oscillations and resonance.

What are Oscillations?

oscillations are the vibrations that take place in the absence of an external damping force. The frequency of these oscillations solely depends on the frequency of the source of vibrations, and the value of this frequency remains constant. In the case of the oscillations, the amplitude remains constant. All oscillations eventually die down because of the presence of the damping forces.

Natural Frequency:

The natural frequency of a system is the frequency at which it oscillates in the absence of any driving force or damping. It is also known as Eigen frequency. Thus, any given system oscillates at its natural frequency when no external forces are acting on it. It is represented by \(\omega\).
A simple pendulum or a spring block system starts oscillating when displaced from its equilibrium position and released. Such a system oscillates at its natural frequency, and these oscillations are called oscillations.

What are the Forced Oscillations?

The oscillations that take place under the effect of an external driving force are called forced oscillations. These oscillations are driven by a periodic force that lies outside the oscillating system. The frequency of the forced oscillations is affected by the frequency of the source of vibration and the driving force, and their amplitude can increase, decrease or remain constant. These are also known as driven oscillations.

Driven Frequency

The frequency of the external agency at which the forced oscillations of a system take place is known as driven frequency. The frequency of the forced oscillations can be varied by changing the value of the driven frequency. It is represented by \(\omega_d\).
While enjoying park swings, kids periodically apply the force by pressing their feet on the ground or by asking their friends to push them from behind to maintain their oscillations. Such oscillations are forced oscillations and the frequency at which the frequency of the external force can determine the swing moves.

Expression for Forced Oscillations

Consider that an external force, varying periodically with time, is applied to a damped oscillator. Let the external force be \(F(t)\), and its amplitude be \(F_0\), then, this force can be represented as:
\(F\left( t \right) = {F_0}\cos {\omega _d}t\) …………(1)
If \(a(t)\) be the acceleration of the particle in time \(t\). If \(m\) be the mass of the body undergoing oscillations, then the force acting on the particle can be given as,
\(F\left( t \right) = m.a\left( t \right)\)
The equation of the damping force acting on the oscillator can be given as,
\({F_d} = – bv\left( t \right)\)
Where \(b\) is the damping constant and \(v(t)\) is the velocity of the oscillator in time \(t\).
The equation of the restoring force can be given as:
\(F = – kx\left( t \right)\)
Where \(k\) is the force constant and \(x(t)\) is the displacement of the oscillator in time \(t\).
We can write the particle’s equation of motion under the combined effect of a linear restoring force, damping force, and time-dependent force. It can be given as:
\(m.a\left( t \right) = – bv\left( t \right) – kx\left( t \right) + {F_0}\cos {\omega _d}t\) …………(2)
Substitute, \(v\left( t \right) = \frac{{dx}}{{dt}}\) and \(a\left( t \right) = \frac{{{d^2}x}}{{d{t^2}}}\) in the above equation. We get:
\(m\frac{{{d^2}x}}{{d{t^2}}} = – b\frac{{dx}}{{dt}} – kx + {F_0}\cos {\omega _d}t\) ………….(3)
The oscillator is initially oscillating at its natural frequency \(\omega\). Without any external agency, the oscillations of the oscillator will eventually die out. But when an external force of frequency \(\omega_d\) is applied to the oscillator, then it will oscillate with the frequency of the applied external force. Thus, the expression for the displacement of the oscillator after its natural oscillations dies out:
\(x\left( t \right) = A\cos \left( {{\omega _d}t + \phi } \right)\) …………(4)
Here, \(t\) is the time measured from the moment periodic force is applied, \(A\) is the amplitude of the displacement, and it is a function of \(\omega _d\) (driven frequency) and \(\omega\) (natural frequency). From the analysis, it can be given that:
\(A = \frac{{{F_o}}}{{{{\left[ {{m^2}{{\left( {{\omega ^2} – \omega _d^2} \right)}^2} + \omega _d^2{b^2}} \right]}^{1/2}}}}\) ………….(5)
\(\tan \phi = \frac{{ – {v_o}}}{{{\omega _d}{x_o}}}\) ………….(6)
Here,
\(m\): is the mass of the particle
\(x_0\): is the displacement at \(t = 0\) (the moment at which force is applied)
\(v_0\): is the velocity at \(t = 0\)
From equation (5), the amplitude of a forced oscillator varies with the angular frequency of the driving force. Based on the values of \(\omega_d\) and \(\omega\) we can find the different behaviour of the oscillator.

1. Small Damping: When driving frequency is far from the natural frequency.
Here, \(\omega_d b\) will be much less than \(m\left( {{\omega ^2} – \omega _d^2} \right)\) and hence it can be ignored. Thus, the amplitude of the forced oscillation becomes:
\(A = \frac{{{F_0}}}{{m\left( {{\omega ^2} – \omega _d^2} \right)}}\)
For different amounts of damping present in the system, the dependence of the displacement amplitude of an oscillator on the angular frequency of the driving force can be graphically represented as:

As can be seen from the curves, the oscillation’s amplitude is maximum when \(\frac{{{\omega _d}}}{\omega } = 1\) and the smaller is the value of damping, the taller and narrower is the resonance peak.
In the case of an ideal scenario, when the system’s damping is zero, the value of amplitude tends to infinity when the value of driving frequency equals the system’s natural frequency. Such a case never really happens in real life.
This can be visualised while taking swings; when the timing of the push exactly matches the time period of the swing, its amplitude will become large, but it still won’t be infinity because, in real life, damping forces are always present.

2. Large Damping: When driving frequency is close to the natural frequency.
Here, \(m\left( {{\omega ^2} – \omega _d^2} \right)\) will be much less than \(\omega_d b\) and hence it can be ignored. Thus, for any value of damping constant \(b\), the amplitude of the forced oscillation becomes:
\(A = \frac{{{F_0}}}{{{\omega _d}b}}\)
Thus, the maximum possible amplitude for a given value of driving frequency depends on the value of this frequency and damping constant.

Resonance

Resonance can be defined as the phenomenon of the increase in the oscillation amplitude when the value of the driving force frequency is close to the natural frequency of the oscillator.

We can observe the effect of resonance several times in our day-to-day life. While taking swings, we can attain maximum heights only when the push force provided by our feet or someone from behind is in sync with the swing’s natural frequency. The swings are now said to be in resonance.

A mechanical structure like a building, bridge, or aircraft can have several possible values of natural frequencies. We know that an external periodic force is required to set a system into forced oscillations. When the forced frequency becomes close to one of the system’s natural frequencies, its amplitude will increase due to resonance and will lead to damage.
This is the reason why:

1. Soldiers are suggested to step out of order while crossing a bridge.
2. An earthquake doesn’t lead to uniform destruction to all the buildings in the region, even when they are built with the same material and have the same design. The natural frequencies of a building vary with its height, size parameter, and nature of the raw material used for constructing it. Buildings having natural frequencies closer to the frequency of seismic waves face much more destruction.

Summary

oscillations are the vibrations that take place in the absence of an external damping force. The frequency of these oscillations solely depends on the source of vibrations, and the value of frequency remains constant. The natural frequency of a system is the frequency at which it oscillates in the absence of any driving force or damping. It is also known as Eigen frequency.

The oscillations that take place under the effect of an external driving force are called forced oscillations. These oscillations are driven by a periodic force that lies outside the oscillating system.
The frequency of the external agency at which the forced oscillations of a system take place is known as driven frequency. The frequency of the forced oscillations can be varied by changing the value of the driven frequency.

Small Damping: When driving frequency is far from the natural frequency. Here, \(\omega_d b\) will be much less than \(m\left( {{\omega ^2} – \omega _d^2} \right)\) and hence it can be ignored.
Large Damping: When driving frequency is close to the natural frequency. Here, \(m\left( {{\omega ^2} – \omega _d^2} \right)\) will be much less than \(\omega_d b\) and hence it can be ignored.
Resonance can be defined as the phenomenon of the increase in the oscillation amplitude when the value of the driving force frequency is close to the natural frequency of the oscillator.

Frequently Asked Questions

Q.1. Write the expression for the amplitude of forced oscillator when driving frequency is far from natural frequency.
Ans: The amplitude of forced oscillator when driving frequency is far from natural frequency:
\(A = \frac{{{F_o}}}{{m\left( {{\omega ^2} – \omega _d^2} \right)}}\).

Q.2. What is the natural frequency of a system?
Ans: The natural frequency of a system is the frequency at which it oscillates in the absence of any driving force or damping.

Q.3. Define forced oscillations.
Ans: The oscillations that take place under the effect of an external driving force are called forced oscillations.

Q.4. Give the expression for the amplitude of forced oscillator when driving frequency is close to the natural frequency.
Ans: The amplitude of forced oscillator when driving frequency is close to the natural frequency:
\(A = \frac{{{F_0}}}{{{\omega _d}b}}\).

Q.5. What happens to the amplitude of the oscillations when the driving frequency becomes close to the natural frequency of an oscillator.
Ans: The system is said to be in resonance and the amplitude of oscillation increases.

Now you are provided with all the necessary information on forced oscillations and resonance and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.