Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

Mechanical Engineering SSC JE (Technical)

Mechanical Engineering : Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

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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

  •  Free Vibrations : Vibrations in which there are no friction and external force after the initial release of the body, are known as free or natural vibrations.
  •  Forced Vibrations : When a repeated force continuously acts on a system, the vibrations are said to be forced vibration. The frequency of the vibrations depends upon the applied force and is independent of there own natural frequency of vibrators.
  •  Damped Vibrations : When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped vibration.
  •  Undamped Vibrations (Hypothetical) :When there is no friction or resistance present in system to counter act vibration then body execute undamped vibration.
  •  Longitudinal Vibrations : If the shaft is elongated and shortened so that the same moves up and down resulting in tensile and compressive stresses in the shaft, the vibrations are said to be longitudinal.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Transverse Vibrations : When the shaft is bent alternately and tensile and compressive stresses due to bending result, the vibrations are said to be transverse vibration.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Torsional Vibrations :When the shaft is twisted and untwisted alternately and torsional shear stresses are induced the vibrations are known as torsional vibrations.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

BASIC ELEMENT OF VIBRATING SYSTEM

  •   Inertial element : These are represented by lumped masses for rectilinear motion and by lumped moment of inertia for angular motion.
  •   Restoring Elements : Massless linear or torsional springs represent the restoring elements for rectilinear and torsional motions respectively.
  •  Damping Elements : Massless dampers of rigid elements may be considered for energy dissipation in a system.

FREE LONGITUDINAL VIBRATION
Different methods for finding natural frequency of a vibrating system
(i) Equilibrium method
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  It is based on the principle that whenever a vibratory system is in equilibrium the algebraic sum of forces and moments acting on it is zero.
  •  This is in accordance with D' Alembert's principle that the sum of the inertial forces and the external forces on a body in equilibrium must be zero.

Let, D = static deflection
S = stiffness of the spring
Inertia force = m Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev (upwards) = ma, where a = Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev 
Spring force = sx (upwards)
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
This is the equation of a simple harmonic motion and is analogous to
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  The solution of which is given by

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
where A and B are the constants of integration and their values depend upon the manner in which the vibration starts.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Linear frequency

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Time period
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
(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.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
We know
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev where v = Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
or ma + sx = 0
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
(iii) Rayleigh's Method

  •  In this method, the maximum kinetic energy at the mean position is made equal to the maximum potential energy (or strain energy) of the extreme position.

Let the motion be simple harmonic
therefore, x = Xsinwnt
where X = maximum displacement from the mean position of the extreme position
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
or KE at mean position = PE at extreme position
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Inertial effect of mass of spring

If we consider mass of spring is 'm1' then
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
DAMPED LONGITUDINAL VIBRATION 

  •  When an elastic body is set in vibratory motion, the vibrations die out after some  time due to the internal molecular friction of the mass of the body and the friction of  the medium in which it vibrates. The diminishing of vibrations with time is called  damping.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
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 = Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev = velocity of the mass at time 't'
f = Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev = 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
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Damping factor
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 

Points to Remember
z = 1, the damping is critical
z > 1, the system is over-damped
z < 1, the system is under-damped

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Logarithmic Decrement (d) : The ratio of two successive oscillations is constant in an underdamped system. Natural logarithm of this ratio is called logarithmic decrement.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
FORCED VIBRATION

  •  Step-input force

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Harmonic force

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
FORCED DAMPED VIBRATIONS 

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

If the mass is subjected to an oscillating force F = F0 sinwt, the forces acting on the mass at any instant will be

  •  Impressed oscillating force (downwards)

F = F0 sinwt

  •   Inertial force (upward)

=mChapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Damping force (upward) = cChapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
  •  Spring force (restoring force) (upwards)

Thus the equation of motion will be
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
TRANSVERSE VIBRATION
Natural frequency of shaft and beams under different type of load and end condition, shows transverse vibration.

  •  In case of shafts and beams of negligible mass carrying a concentrated mass, the  force is proportional to the deflection of the mass from the equilibrium position.

Where,
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev  for cantilevers supporting a concentrated mass at free end.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev  for simply supported beams
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 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) 

  •   A simply supported shaft carrying a uniformly distributed mass has maximum  deflection at the mid-span.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Then, taking the smallest value of fn
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Note:

  •  This is the lowest frequency of transverse vibrations and is called the fundamental frequency. It has a node at each end.
  •  The next higher frequency is four times the fundamental frequency. It has three nodes.
  •   The next higher frequency is nine times the fundamental. It has four nodes.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev

  •  Thus a simply supported shaft will have an infinite number of frequencies under uniformly distributed load.
  •   Similarly, the cases of cantilevers and shafts fixed at both ends can be considered.

(iii)Shaft Carrying several loads
A. Dunkerley's Method

  •  It is semi-empirical. This gives approximate results but is simple.

fn = frequency of transverse vibration of the whole system
fns = frequency with the distributed load acting alone.
fn1, fn2, fn3 ....... = frequencies of transverse vibrations when each of w1, w2, w3
.... alone.
Then, according do Dunkerley's empirical formula
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
where  
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
 Similarly, 
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
B. Energy Method

  •  Which gives accurate results but involves heavy calculations if there are many  loads.

Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Maximum P.E.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
Maximum K.E.
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
ω = circular frequency of vibration
Chapter 7 Vibrations - Theory of Machine, Mechanical Engineering Mechanical Engineering Notes | EduRev
WHIRLING OF SHAFT

  • When a rotor is mounted on a shaft, its centre of mass does not usually coincide  with the centre line of the shaft. Therefore, when the shaft rotates, it is subjected to  a centrifugal force which makes the shaft bend in the direction of eccentricity of the  centre of mass. Thus further increases the eccentricity, and hence centrifugal force.  The bending of shaft depends upon the eccentricity of the centre of mass of the  rotor as well as upon the speed at which the shaft rotates.
  •  Critical or whirling or whipping speed is the speed at which the shaft tends to  vibrate violently in the transverse direction . 
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