# Linear Vibration Theory

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In this chapter, the theory related to vibration, crack and the Linear Elastic Fracture Mechanics (LEFM) are presented. Then the attention is given to the mathematical formulation of a cracked uniform cantilever beam. The presence of crack reduces the local stiffness matrix which alters the dynamic response of the system.
2. Theory
2.1 Concept of vibration
Vibration definition
Most of the human activities like hearing, seeing, breathing etc are due to vibration. Hearing involves vibration of the eardrum, seeing is associated with vibration of light waves,breathing is based on the periodic motion of lungs. For the safe design, construction and operation of variety of machines and structures, an understanding of the vibratory behavior of structural
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a) Longitudinal or Rectilinear vibration: When the particles of the body move parallel to the axis of the body, it is called longitudinal vibration
b) Lateral or Transverse vibration: When the particles of the body move perpendicular to the axis of the body is called transverse vibration.
c) torsional vibration ()
3. Linear and Non-linear vibration:If the components of vibratory system( spring, mass, behave in linear way, then the vibration is known to be Linear vibration. Principle of superposition is valid here.
When the components vibrate in non-linear way, then it is called non-linear vibration.
2.2 History and Importance of vibration
The equation of motion for the transverse vibration of thin beams was derived by Daniel Bernoulli in 1735. The first solutions of the equation for various support conditions were given by Euler in 1744, which is known to be Euler-Bernoulli or Thin beam theory. Rayleigh included the effect of inertia and presented a beam theory. The improved theory by including the effect of rotary inertia and shear deformation known as Timoshenko or thick beam theory was presented by Stephen Timoshenko in
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2.3 Theory of crack
Crack can be defined as a line on the surface of a structure along which splitting takes place without breaking apart. Due to the presence of crack, the vibration response of the structure is affected. This property is utilized for detection of the location and depth of crack in the member.
Classification of cracks based upon their geometries:
1. Transverse cracks:
These are most common and very serious because the presence of this type of cracks reduces the cross sectional area that weakens the beam.
2. Longitudinal cracks:
These cracks act parallel to the beam axis. To the right angles of crack direction, when tensile loading is applied, they are very dangerous..
3. Open cracks:
These cracks remain open. More precisely they are known as notches. These are very easily adaptable for the lab environment. So, mostly for the experimental purpose, open cracks are preferred.
4. Breathing crack :
These cracks depend on the nature of the stresses acting on the material, if tensile stresses act , the cracks open. If the stresses are reversed, the cracks close. When cracks are open, its more critical. Due to the breathing of crack, the vibration response of the beam is