Forced Oscillations And Resonance

As the name suggests, forced oscillations are body oscillations that do not occur naturally. Forced oscillations can also be referred to as the vibrations of a system or a body that oscillates under the effect of an external periodic force. Also, the new frequency of the oscillating body is different from its natural frequency. Forced oscillations are the converse of free oscillations in which the body can do oscillations independently, for example, free moving pendulum. Moreover, when talking about resonance, it is a frequency of special cases wherein the natural frequency of the body is equal to the driven or forced frequency of the body. 

Free Oscillations 

Let us consider a case of a simple pendulum oscillating with some frequency. Once the pendulum is displaced from its mean position, it executes to and fro motion. It freely vibrates with its own natural frequency. Such oscillations are said to be free oscillations. 

• Hence, free oscillations can be defined as oscillations possessed by a body that, once set into oscillations, is free to vibrate with its own natural frequency.
• The factors on which the natural frequency of free oscillation depends are the body’s structure and the nature of the body. 
• Also, the vibrating body’s frequency, amplitude, and energy remain constant for a body to perform free oscillations. 

Damped Oscillations

When a body like a simple pendulum is allowed to oscillate freely, we notice that the pendulum comes back to its original position after some time. This happens due to certain factors like air resistance, etc., which resist the pendulum’s motion. As a result, there is an effect on the amplitude of the body vibrating. The amplitude of the vibrations decreases with time. The body is then performing damped vibrations. 

Forced or Driven Oscillations

• Forced or driven oscillations are those vibrations wherein the body oscillates under the effect of an external periodic force, having a frequency different from the natural frequency of the oscillating body. 
• In the case of forced vibrations, the body vibrates with the frequency due to external force. 
• Also, the amplitude for such oscillations is usually tiny. 
• The expression for external driving force is given by: 

F(t) = F0cos⍵dt

Remember that the motion of the body or the vibrations caused in a body are due to the combined action of: 

• Restoring force, that is F= -kx 
• Damping force, that is F= -bv, where b is damping constant 
• Driving force, that is F(t)

By combining all three, we get, 

ma=-kx-bv+F0cos⍵dt

 Solving the above equation will give us the solution as; 

  x=x0sin(⍵dt+ ɸ)

The expression for natural frequency would be;

  ⍵0  =k/m = natural frequency

Resonance

In a nutshell, resonance is the process where an oscillator gets vibrations by driving frequency far and near-natural frequency. Moreover, Resonance is the special case of forced vibration. The external force and natural force’s frequencies are entirely the same in this case. The vibrations are termed resonant vibrations whenever the body oscillates with such properties. Also, an instance of resonance increases the amplitude of the body considerably since it gets affected. 

Examples of resonance are as follows:

1. When a giant sound system is turned on around you or your house – One can always observe the trembling occurring within the home’s walls and furnishings whenever a loud sound system is turned on around it. This occurs due to the resonance frequencies of the furnishings resonating with the frequency of a music’s noise, forcing the furnishings vibration.
2. Pendulum – The swing and the pendulum both function along with an identical premise. The pendulum will swing backward and forward when we force it. The movement of the pendulum can grow if you keep pressing at the same time frame. The pendulum’s momentum can be further increased by giving it regular pushes. It is resonance frequency, the number of times at which the pendulum will move forward and backward per second.
3. Microwave oven – We can see that food certainly gets heated in the oven. It has taken place since the resonance. Microwaves emit electromagnetic radiation at a specified wavelength and frequency. And water and fat molecules also have resonance frequencies. At specific frequencies, the molecules absorb the radiation and heat up.

Conclusion

From the above, we have gone through various aspects of forced oscillations and resonance; we have learned that forced oscillations are the converse of free oscillations. The vibrations of forced oscillations can only be observed when there is a certain external force acting on the body. Moreover, resonance is the phenomenon where the body has elevated amplitude and has (nearly) equal driving and natural frequency.