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VIBRATIONS
Any motion that exactly repeats itself after a certain interval of time is a periodic motion . A periodic motion which is having to and fro motion is called vibration.
TYPE OF VIBRATION
BASIC ELEMENT OF VIBRATING SYSTEM
FREE LONGITUDINAL VIBRATION
Different methods for finding natural frequency of a vibrating system
(i) Equilibrium method
Let, D = static deflection
S = stiffness of the spring
Inertia force = m (upwards) = ma, where a =
Spring force = sx (upwards)
This is the equation of a simple harmonic motion and is analogous to
where A and B are the constants of integration and their values depend upon the manner in which the vibration starts.
Time period
(ii) Energy Method
• In a conservative system (system with no damping) the total mechanical energy i.e. the sum of the kinetic and the potential energies remains constant.
We know
where v =
or ma + sx = 0
(iii) Rayleigh's Method
Let the motion be simple harmonic
therefore, x = Xsinw_{n}t
where X = maximum displacement from the mean position of the extreme position
or KE at mean position = PE at extreme position
If we consider mass of spring is 'm_{1}' then
DAMPED LONGITUDINAL VIBRATION
Assumption
• The damping force is proportional to the velocity of vibration at lower values of speed and proportional to the square of velocity at higher speeds.
Let, s = stiffness of the spring
c = damping coefficient (damping force per unit velocity)
wn = frequency of natural undamped vibrations
x = displacement of mass from mean position at time 't'
v = = velocity of the mass at time 't'
f = = acceleration of the mass at time 't'.
As the sum of the inertial force and the external forces on a body in any direction is
to be zero
Damping factor
Points to Remember z = 1, the damping is critical z > 1, the system is overdamped z < 1, the system is underdamped 
Logarithmic Decrement (d) : The ratio of two successive oscillations is constant in an underdamped system. Natural logarithm of this ratio is called logarithmic decrement.
FORCED VIBRATION
FORCED DAMPED VIBRATIONS
If the mass is subjected to an oscillating force F = F_{0} sinwt, the forces acting on the mass at any instant will be
F = F_{0} sinwt
=m
Thus the equation of motion will be
TRANSVERSE VIBRATION
Natural frequency of shaft and beams under different type of load and end condition, shows transverse vibration.
Where,
for cantilevers supporting a concentrated mass at free end.
for simply supported beams
for beams fixed at both ends
Note : A shaft supported in long bearings is assumed to have both end fixed while one in short bearing is considered to be simply supported.
(ii) Uniformly loaded Shaft (Simply supported)
Then, taking the smallest value of f_{n}
Note:
(iii)Shaft Carrying several loads
A. Dunkerley's Method
f_{n} = frequency of transverse vibration of the whole system
f_{ns} = frequency with the distributed load acting alone.
f_{n1}, f_{n2}, f_{n3} ....... = frequencies of transverse vibrations when each of w_{1}, w_{2}, w_{3}
.... alone.
Then, according do Dunkerley's empirical formula
where
Similarly,
B. Energy Method
Maximum P.E.
Maximum K.E.
ω = circular frequency of vibration
WHIRLING OF SHAFT
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