What is the Basic Concept of Vibration

What is the basic concept of vibration? All bodies having mass and elasticity are capable of vibration. When external force is applied on the body, the internal forces are set up in the body which tends to bring the body in the original position. The internal forces which are set up are the elastic forces which tend to bring the body in the equilibrium position. Consider an example of swinging of pendulum. At extreme position whole of the kinetic energy of the ball is converted into elastic energy which tends to bring the ball in the equilibrium/mean position. At mean position whole of the, elastic energy is converted into kinetic energy and body continues to move in opposite direction because of it. Now the whole of kinetic energy is converted into elastic energy and this elastic energy again brings the ball to the equilibrium position. In this way, vibratory motion is repeated indefinitely and exchange of energy takes place. This motion which repeats itself after certain interval of time is called vibration.

(i) Periodic Motion (ii) Time period (iii) Frequency (iv) Amplitude (v) Natural frequency (vi) Fundamental mode of vibration (vii) Degree of freedom (viii) Simple Harmonic Motion (S.H.M.) (ix) Resonance (x) Damping (xi) Phase Difference (xi,) Spring stiffness (i) Periodic motion: A motion which repeats itself after certain interval of time iscalled periodic motion. (ii) Time Period : It is time taken to complete One cycle .(iii) Frequency: No’s of cycles in one sec. Units = H (iv) Amplitude: Maximum displacement of a vibrating body from mean position is called Amplitude. (v) Natural frequency: When there is no external force applied on the system and it is given a slight displacement the body vibrates. These vibrations are called free vibrations and frequency of free vibration is called Natural frequency.

(vi) Fundamental mode of vibration: Fundamental mode of vibr!sternis a mode (vii) Degree of freedom:

• The minimum no’s of co-ordinates required to specify motion of a system at any instant is called degree of freedom. (viii) Simple Harmonic Motion (S.H.M..) : The motion of a body “to” and “fro” about a fixed point is called S.H.M. S.FLM.. is a periodic motion and it is function of “Sine” or “Cosine”. Acceleration of S.H.M. is proportional to displacement from the mean position and is directed towards the centre.

In S.H.M. acceleration is directly proportional to the displacement from the mean position and is directed towards the centre.

(zx) Resonance : When the frequency of external force is equal to the natural frequency of a vibrating body, the amplitude of vibration becomes excessively large. This is known as “Resonance” . At resonance there are chances of machine part or structure to fail due to excessively large amplitude. It is thus important to find natural freuqencies of the system in order to avoid resonance. (x) Damping: It is resistance provided to the vibrating body and vibrations related to it are called damped vibration. (xi) Phase difference : Suppose there are two vectors

(xii) Spring stiffness : It is defined as unit deflection. Units : N/m. What are the various parts of a vibrating system? Various parts of the mechanical system (vibratory system) are : — (A) Spring (B) Damper (C) Mass

Damping force c ± acting upwards Accelerating force m i acting downwards Spring force kx acting upwards

According to this method the sum of forces and moments acting on the system is zero if no external force is applied on the system. Consider fig. I

Classify different types of vibrations. Types of Vibrations I. Free and Forced To and fro motion of the system when disturbed initially without any external force acting on it are called free vibrations. e.g. pendulum. To and fro motion of the system under the influence of external force is called forced vibrations. e.g. Bell, Earthquake. II. Linear and Non-linear vibrations Linear vibrations: The linear vibrations are those which obey law of superimposition. If a1 and a2 are the solutions of a differential equation, then a1 + a2 should also be the solution.

Non-linear vibrations: When amplitude of vibrations tends towards large value, then vibrations become non-linear in nature. They do not obey law of superimposition. III. Damped and Undamped vibrations Damped vibrations are those in which amplitude of vibration decreases with time. These vibrations are real and are also called transient vibrations.

Undamped vibrations are those in which amplitude of vibration remains constant. In ideal system there would be no damping and so amplitude of vibration is steady. 1V. Deterministic and Random vibrations (Non-Deterministic). Deterministic vibrations are those whose external excitation are known or can be determined whereas Random vibrations are those whose external excitation cannot be determined. e.g. Earthquake

V. Longitudinal, Transverse and Torsional vibrations

What do you mean by undamped free vibrations? If the body vibrates with internal forces and no external force is included, it is Further during vibrations if there is no loss of energy due to friction or resistance, it is known as undamped free vibration. Consider the relation for the frequency of spring mass system in vertical position.

What is D’Alembert’s Principle? D’Alembert’s principle states that if the resultant force acting on a body along with the inertia force is zero, then the body will be in static equilibrium. Inertia force acting on the body is given by

Assuming that the resultant force acting on body is F, then the body will be in static equilibrium if

Consider fig. 2.2., the spring force of the body Kx is acting upwards and acceleration of the body i is acting in downward direction. The accelerating force is acting downward so inertia force is acting upwards, so the body is M static equilibrium under the action of the two forces Kx and mi. Mathematically it can be written as