A vibratory system basically consists of 3 elements, namely the mass, the spring and damper. In a vibrating body, there is exchange of energy from one form to another. energy is stored by mass in the form of kinetic energy $(\frac{1}{2} mx^2)$ , in the spring in the form of potential energy $(\frac{1}{2} kx^2)$ and dissipated in the damper in the form of heat energy which opposes the motion of the system, energy enters the system with the application of external force known as excitation. the excitation disturbs the mass from its mean position and mass goes up and down from the mean position. the kinetic energy is converted into potential energy and potential energy into kinetic energy, this sequence goes on repeating and the system continues to vibrate. at the same time damping force cx acts on the mass and opposes its motion. thus, some energy is dissipated in each cycle of vibration due to dumping, the free vibrations die out and the system remains at its static equilibrium position, a basic vibratory system is shown in figure.

The equation of motion of such a vibratory system can be written as:

mx + cx + kx = 0

where, cx = damping force,

kx = spring force,

mx = inertia force.

VIBRATION TERMINOLOGY :

1] A disturbance :

Nothing can vibrate on its own because of Newton's first law of motion, initial displacement may be intentionally given or be the result of accident or the disturbance itself may be repeatedly given after an interval of time.

2] Restoring force/ Torque :

The displaced body does not remain in its new position but it comes back to a mean position by an agency, called the resorting force or restoring torque, which may be provided by elasticity of system or gravity.

3] Inertia:

It is the property of system due to which the body remains in continuous motion or at rest unless an external force is not acting on it.

4] Damping:

The resistant to the motion is called damping. damping is present due to external forces which oppose the motion of system. these forces may be due to air resistance or any other fluid resistance due to its viscous nature. damping force is proportional to the velocity of the system.

5] Periodic motion :

The motion which repeats itself after equal time interval is known as periodic motion.

6] Time period (T) :

It is the time taken to complete one cycle an its given by relation.

$T = \frac{2 \pi }{ω}$

where w is the angular velocity or circular frequency.

7] Frequency :

It is defined as the number of cycles completed per unit time and it is given by relation:

$f = \frac{1}{T} = \frac{1}{2 \pi } = \frac{ω}{2 \pi } $ or ω= $2 \pi f$

8] Simple harmonic motion (S.H.M.) :

A periodic motion of a system or body in which acceleration is always directed towards the mean position and it is proportional to its distance from the mean position.

Consider the simplest equation of a vibrational motion.

X = A sin ωt

where x is the displacement measured from a mean position at time 't'

Velocity at time t, $\frac{dx}{dt} = ω$ A cos ωt

and acceleration at time t, $\frac{d^2x}{dt^2} = - ω^2$ S sin ωt = $- ω^2 x$

such a vibration where acceleration is proportional to the displacement and is directed towards the mean position (minus sign gives this direction) is called S.H.M.

X = A cos ωt

is another example of a S.H.M.

There are 2 equations very frequently used in vibration analysis. the reason is that a complicated vibrational motion, which is a periodic function, may be analysed into its Fourier components which are terms containing sines and cosines.

If $x = a_1 sin ωt + a_2 sin2ωt + . . . . . .,$

$A_1$ sin ωt is called the first harmonic

$A_2$ sin2 ωt is called the second harmonic.

9] Amplitude :

It is the maximum displacement of body or system from the mean position.

10] Free or natural vibration :

The vibrations of system because of its own elastic properties are known as free vibrations, in this case, there is no any external force or torque acting on the system.

11] Forced vibration :

Vibration of a system which is executed under the external exciting force is known as forced vibration.

12] Natural frequency :

The frequency at which the system vibrations under free vibration condition is known as natural frequency of system. It is denoted by $ω_n$

13] Resonance :

When the vibrations of system takes place under external force then the condition at which frequency of vibration under external force is same as that of natural frequency of the system is known as resonance.

14] Degree of freedom :

It is the minimum number of co ordinates required to locate the system, the number of degrees of freedom of a system equals the minimum number o independent co-ordinates necessary to define the configuration of the system. these co ordinates necessary to describe the system constitutes a set of " generalized co ordinates " these generalized co ordinates are usually denoted by $q_1 q_2, q_3. . . . . q_n$ and can represent non Cartesian co ordinates as well as Cartesian co ordinates. eg. particular directional vibration, D.O.F = Number of masses by elastic members.

Examples :

1] Single degree of freedom systems :

2] Two degrees of freedom systems

a] $X_1$ and $X_2$ are required to define the configuration.

b] $\theta_1$ and $\theta_2$ are required to describe the configuration.

3] Three degrees of freedom systems.

4] Infinite degrees of freedom system or continuous systems :

A cantilever has infinite mass points. so deflection points is to be defined to define the system configuration.

Vibrations of beams, plates, rings, etc are studied by treating them as finite lumped masses and springs, the larger the number into which the continuous systems are divided, the greater the accuracy of the analysis achieved.

5] Static Equilibrium position (SEP) :

The position in which system is in dynamic equilibrium, having zero retardation is called SEP. At SEP, kinetic energy is maximum and potential energy is zero.