unit-4

1. Mechanical Department
COURSE NAME: MECHANICAL VIBRATIONS
Prepared By:
MD ATEEQUE KHAN
(Assistant Professor)
Mechanical Engineering Department
JIT,Barabanki,U.P. INDIA
12/31/2016 1NME-013 MA KHAN

2. Table of Contents
Unit-4
1. Multi-degree Freedom system
2. Exact Analysis
3. Undamped free
4. forced vibrations of multi-degree freedom systems
5. influence coefficients
6. Reciprocal theorem
7. Torsional vibration of multi-degree rotor system
8. Vibration of gear system
9. Principal coordinates
10.Continuous systems- Longitudinal vibrations of bars.
11.Torsional vibrations of circular shafts.
12/31/2016 1NME-013 MA KHAN

3. Unit-4: Mechanical Vibrations
Multi-degree Freedom system
Flexibility matrix:
For a three degree-of-freedom system, the displacement as forces are related by flexibility matrix as :
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4. Unit-4: Mechanical Vibrations
Stiffness Matrix
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5. Unit-4: Mechanical Vibrations
Example:
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6. Unit-4: Mechanical Vibrations
FBD
Solution Matrix
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7. Unit-4: Mechanical Vibrations
Reciprocal theorem
Reciprocity theorem : States that in a linear system
Proof : Consider a linear system and now applying force the work done force X
displacement=
However due to application of force i undergoes further displacement, and the
additional work done by becomes
So Total Work done
Now if one reverses the order application of forces,workdone will be:
work done in the two cases must be equal, so we find:
jiij aa 
if
2
1
  iiiiiii afaff
2
2
1
2
1

jf jij fa
if ijij ffa
jiij aa 12/31/2016 1NME-013 MA KHAN

8. Unit-4: Mechanical Vibrations
Torsional vibration of multi-degree rotor system
1. When there are several numbers of discs in the rotor system it
becomes a multi degree of freedom (MDOF) system.
2. When the mass of the shaft itself may be significant then the analysis
described in previous section (i.e. single or two-disc rotor systems) is
inadequate to model such systems, however, they could be extended to
allow for more number of lumped masses (i.e., rigid discs) but
resulting mathematics becomes cumbersome.
3. Alternative methods for analysis of MDOF systems are
(a) Transfer matrix method and
(b) Finite element method
12/31/2016 1NME-013 MA KHAN

9. Unit-4: Mechanical Vibrations
Example:
1k 4k3k2k 1nk
1PI
3PI
2PI 1PnI
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10. Unit-4: Mechanical Vibrations
Vibration of gear system
Geared system: In some machine the shaft may not be continuous from one end of the machine to the other, but
may have a gearbox installed at one or more locations. So shafts will be having different angular velocities
as shown in figure.
Real system
Equivalent System
  T
e e eT
ek
12/31/2016 1NME-013 MA KHAN

11. Unit-4: Mechanical Vibrations
Continuous systems- Longitudinal vibrations of bars
Introduction to Continuous systems:
In the previous modules we have studied about discrete mass system, which are modeled
as single, two or multi-degrees of freedom systems. In these cases the system has a
definite number of lumped masses, stiffness elements and damping elements. For
example the cantilever beam with a tip mass as shown in Figure 10.1 is modeled as a
single degree of freedom system with a spring and a mass. The stiffness k of the system
was calculated using the following equation.
W
m
k
n 
g
W
m 12/31/2016 1NME-013 MA KHAN

12. Unit-4: Mechanical Vibrations
Torsional vibrations of circular shafts
tk
PI
t
12/31/2016 1NME-013 MA KHAN