## Types of Vibration

Understanding the several types of vibration is necessary to comprehend vibratory motion, which can be desirable or undesirable depending on the situation. There are three different types of vibration.

- Free or Natural Vibration
- Forced Vibration
- Damped Vibration

## Free or Natural Vibration

It is considered free or natural vibration when no external force operates on the body after it has experienced an initial displacement. Free vibration occurs when a mechanical system is started in motion with an initial input and allowed to vibrate freely.

The mechanical system vibrates at one or more natural frequencies before becoming stationary. Free or natural frequency refers to the frequency of free or natural vibration. The amplitude appears to be diminishing over time.

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### Free Vibration Definition

“Free or natural vibrations are elastic vibrations in which there are no friction or external forces after the initial release of the body.”

### Free Vibration Examples

Here are some examples of free vibration.

- Pulling a child back on a swing and letting it go
- Tapping a tuning fork and letting it ring

### Types of Free Vibration

There are mainly three types of free vibration that an object may experience.

- Longitudinal Vibration
- Transverse Vibration
- Torsional Vibration

### Longitudinal Vibrations

The vibrations are known as longitudinal vibrations when the particles of the shaft or disc travel parallel to the shaft's axis, as depicted in Figure. In this situation, the shaft is alternately lengthened and shortened, causing tensile and compressive stresses to be created in the shaft.

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### Transverse Vibrations

The vibrations are known as transverse vibrations when the particles of the shaft or disc move approximately perpendicular to the shaft axis, as shown in Figure. In this scenario, the shaft is straight and bent, causing bending stresses in the shaft.

### Torsional Vibrations

Torsional vibrations occur when the particles of the shaft or disc move in a circle around the shaft's axis, as seen in the figure. The shaft is twisted and untwisted alternatively in this scenario, causing torsional shear stresses.

## Forced Vibrations

Forced vibration occurs when a mechanical system is subjected to a time-varying disturbance (load, displacement, velocity, or acceleration). A periodic and steady-state input, a transient input, or a random input can all be used as disturbances. The frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion for linear systems, with the size of the response being dependent on the particular mechanical system.

### Forced Vibration Examples

Here are some examples of forced vibration.

- A harmonic or non-harmonic disturbance can be used as the periodic input.
- A washing machine shaking due to an imbalance
- Transportation vibration induced by an engine or uneven road
- Building vibration during an earthquake.

## Damped Vibration

Vibrations are considered to be damped when the energy of a vibrating system is progressively absorbed by friction and other resistances. The vibrations progressively decrease in frequency or intensity or stop altogether, and the system returns to balance.

### Damped Vibration Examples

The following are examples of damped vibration

- The vehicle suspension
- Which is dampened by the shock absorber

## Methods to Find the Natural Frequency

Any of the methods listed below can be used to determine a vibratory system's natural frequency.

- Equilibrium Method
- Energy Method
- Rayleigh's Method

### Equilibrium Method

It is founded on the principle that when a vibratory system is in equilibrium, the algebraic total of forces and moments acting on it is zero, which corresponds to D'Alembert's Principle that the sum of inertia forces and external forces acting on a body in equilibrium must be zero.

Let,

- Δ = static deflection
- k = Stiffness of the spring

Inertial force = ma (upwards, a = acceleration)

Spring force = kx (upwards)

So the equation becomes

ma + kx = 0

⇒ω_{n} = √(k/m)

Linear frequency f_{n} = (1/2π)√(k/m)

Time period T = 1/f_{n} = 2π√(m/k)

### Energy Method

The total mechanical energy, the sum of the kinetic and potential energies, remains constant in a conservative system (system with no damping).

d/dt (K.E+ P.E.) = 0

### Rayleigh's Method

In this manner, the maximum kinetic energy at the mean position is equivalent to the maximum potential energy (or strain energy) at the extreme position. At any given time, the displacement of the mass 'm' from the mean position is given by

a+ω_{n}^{2} x = 0

x = A sinω_{n} t + B Cosωn t

Let A = X cos φ; B = X Sin φ

x = X sin(ωω_{n} t +φ)

Velocity, V = Xω_{n} Sin [π/2 + (ω_{n} t +φ)]

Acceleration, f = Xω_{n}^{2} Sin[ π + (ω_{n} t +φ)]

These relationships indicate that

- The velocity vector leads the displacement vector by π/2.
- The acceleration vector leads the displacement vector by π.

Consider, 'm' = man of the spring wire per unit length

l = total length of the spring wire m_{1} = m'l

KE of the spring = 1/3 * KE with the same mass as the spring and the same velocity as the free end.

f_{n} = (1/2π) √ (s/(m+(m_{1}/m)))

f_{n} = (1/2π) √g/Δ

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