This document provides an overview of numerical analysis methods for multi-degree of freedom mechanical vibration systems. It covers Rayleigh's method, Dunkerley's method, Rayleigh-Ritz method, critical speeds of shafts, whirling of uniform shafts, shafts with discs with and without damping, and multi-disc shafts. The document also provides examples of applying the Rayleigh's method and Dunkerley's formula to calculate vibration frequencies of beams and shafts.

1. .
COURSE NAME: MECHANICAL VIBRATIONS
Prepared By:
MD ATEEQUE KHAN
(Assistant Professor)
Mechanical Engineering Department
JIT,Barabanki,U.P. INDIA
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2. Table of Contents
Unit-5
1. Multi Degree Freedom system: Numerical Analysis by Rayleigh’s method,
2. Multi Degree Freedom system: Numerical Analysis by, Dunkerely’s,
method
3. Multi Degree Freedom system: Numerical Analysis by Holzer’s method
4. Multi Degree Freedom system: Numerical Analysis Stodola methods,
5. Multi Degree Freedom system: Numerical Analysis Rayleigh-Ritz method
6. Critical speed of shafts
7. Whirling of uniform shaft
8. Shaft with one disc with damping
9. Shaft with one disc without damping
10.Multi-disc shafts.
11.Secondary critical speed.
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3. Unit-5: Mechanical Vibrations
Numerical Analysis by Rayleigh’s method
Rayleigh’s Method
• Rayleigh method gives a fast and rather accurate computation of the fundamental frequency
of the system.
• It applies for both discrete and continuous systems.
Consider a uniform beam of length L and of mass per unit length. The maximum Potential/
elastic-strain energy: According to rayleigh Method:
Kinetic Energy:

dx
dx
yd
EI
MdEP
L
L
2
0
2
2
0
2
1
2
1
..








 
  dxyEK
L
n
2
0
2
1
..  







































L
L
n
dxy
dx
dx
yd
EI
0
2
0
2
2
2
2


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4. Unit-5: Mechanical Vibrations
Dunkerely’s, method
Let W1, W2 ,….Wn be the concentrated loads on the shaft due to masses m1, m2,….mn and D1,
D2,…… D3 are the static deflections of the shaft under each load. Also let the shaft carry a
uniformly distributed mass of m per unit length over its whole span and static deflection at the
mid span due to the load of this mass be .
= Frequency of transverse vibration of the whole system.
= Frequency with distributed load acting alone.
= Frequency of transverse vibration when each of W1, W2 ,….Wn ....act alone
According to Dunkerley's formula:
s
2n1n
ne
n
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5. Unit-5: Mechanical Vibrations
Rayleigh-Ritz method
The Rayleigh-Ritz method
• This is considered as an extension of Rayleigh’s method. A closer approximation to the
natural mode can be obtained by superposing a number of assumed functions than using by a
single assume functions as in Rayleigh’s method.
•
• It gives the more accurate result than the previous method.
•
• In the case of transverse vibration of beams, if n functions are chosen for approximating the
deflection , can be written as
• Where, are linear independent functions of the spatial coordinate x which satisfy the
boundary condition of the problem, and are the coefficient to be found.
•
• As the Rayleigh quotients have stationary value near the natural mode by differentient by
differentiating the Rayleigh quotient with respect to these coefficients will yield a set of
homogeneous algebraic equations, which can be solved to obtain the frequencies.
•
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6. Unit-5: Mechanical Vibrations
Critical speed of shafts
Whirling is defined as the rotation of the plane made by the bent shaft and the line of the centre of
the bearing. It occurs due to a number of factors, some of which may include (i) eccentricity,
(ii) unbalanced mass, (iii) gyroscopic forces, (iv) fluid friction in bearing, viscous damping.
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7. Unit-5: Mechanical Vibrations
Whirling of uniform shaft
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8. Unit-5: Mechanical Vibrations
Shaft with one disc with damping
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9. Unit-5: Mechanical Vibrations
Multi-disc shafts
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