Question

Differentiate between free and forced vibrations.

Answer 1

Hint: We can oscillate a body in two ways, by providing an initial displacement and leaving it to oscillate by itself. Another way is to constantly apply an external force to maintain the oscillations.

Complete step by step answer:
We know that vibration occurs when an external force or displacement is applied on a body. After the application of force or displacement, if the body is set to vibrate freely with its frequency obtained from the initial displacement called the natural frequency, then the vibrations executed are known as the free vibrations. For example, consider a spring of spring constant k, which is displaced along the x-axis by a quantity x and is allowed to vibrate freely. The equation of motion of this vibration can be written as,

$F=-kx$

The minus sign indicates that the force is acting opposite to the direction of displacement.

So, when a time-varying external force is acting on a body, suppose a spring of spring constant k, having a resistance ‘r’ which is executing vibrations, then the body is said to be executing forced vibrations. The disturbance can be a periodic and steady-state input, a transient input, or a random input. The periodic input can be a harmonic or a non-harmonic disturbance. When the external force frequency is equal to the natural frequency of vibrations, resonance occurs, and the amplitude of vibrations will be high.

The equation of a periodic external force acting on a body of stiffness k and having a resistance ‘r’ executing forced vibrations along the x-axis is given by,

$m\left( \dfrac{{{d}^{2}}x}{d{{t}^{2}}} \right)+r\left( \dfrac{dx}{dt} \right)+kx={{F}_{0}}\sin \left( \omega t \right)$

Where

${{F}_{0}}$ is the amplitude of the external periodic force.
$\omega $ is the angular frequency of the external periodic force.
$x$ is the displacement of the body from its mean position.
$\dfrac{dx}{dt}$ is the velocity of the body.
$\dfrac{{{d}^{2}}x}{d{{t}^{2}}}$ is the acceleration of the body.

Also, note that the resistance in regular oscillatory motion and forced oscillatory motion is proportional to the velocity of the body.

The main difference between the free and forced vibrations is that in the case of free oscillation there is no external force acting on the body executing harmonic motion, while in the case of forced harmonic motion there is an external force acting on the body.

Another difference is that the body when executing a free oscillation oscillate with its own natural oscillation. For a body executing forced vibrations, the frequency of the oscillation is the frequency of the external force acting on the body.

The amplitude of a body executing free oscillation reduces due to damping, while the amplitude of a forced oscillation does not reduce with time.

Note: When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change in frequency or intensity or cease, and the system rests in its equilibrium position.
Types of damped oscillations are underdamped, critically damped and overdamped vibrations.