Dr' S' K' Kudari,

1
Dr. S. K. Kudari, Professor, Department of
Mechanical Engineering, B V B College of Engg.
Tech., HUBLI email skkudari_at_bvb.edu
2
CHAPTER-8
Multi degree of freedom Systems
Exam TWO Questions 40 Marks

• Topics covered
• Eigen values and Eigen vectors 1-session
• Influence co-efficents
  1-session
• Approximate methods
  1-session (Covered in last session)
• Dunkerleys method
• Rayleighs method
• Numerical methods
• (i) Matrix iteration method
  2-session (Todays session)
• (ii) Stodolas method
  1-session
• (iii) Holzars method
  2-session

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
3
Multi degree of freedom Systems
Matrix iteration method
Using this method one can obtain natural
frequencies and modal vectors of a vibratory
system having multi-degree freedom It is
required to have ?1lt ?2lt.lt ?n
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
4
Multi degree of freedom Systems
Matrix iteration method
Eqns. of motion of a vibratory system (having n
DOF) in matrix form can be written as
Substitute
where
for principal modes of oscillations
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
5
Multi degree of freedom Systems
Matrix iteration method
Corresponding to rth mode
Where r is any value between 1 and n
Dynamic matrix
Converges to first natural frequency and first
modal vector
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
6
Multi degree of freedom Systems
Matrix iteration method
Inverse Dynamic matrix
Converges to last natural frequency and last
modal vector
In above Eqns by assuming trial modal vector and
iterating till the Eqn is satisfied, one can
estimate natural frequency of a system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
7
Multi degree of freedom Systems
Matrix iteration method
Obtain the natural frequencies and modal vectors
of the system by MATRIX ITERATION method
It is very much required to find that the method
converges to the correct solution. For this
reason, the present problem is selected, for
which already we have solution discussed in
session 5 (17/04/07) Refer the session 5
presentation for the solution using basic method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
8
Multi degree of freedom Systems
Matrix iteration method

• For matrix iteration method, it is required to
  have
• mass matrix and stiffness matrix of a system.
• They can be obtained by
• Eqns of motions
• (ii) Influence coefficents

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
9
Multi degree of freedom Systems
Matrix iteration method
Eqns of motion
Matrix form
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Multi degree of freedom Systems
Matrix iteration method
First natural frequency and modal vector
Obtain Dynamic matrix
Transpose of cofactors
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
11
Multi degree of freedom Systems
Matrix iteration method
For the given system, we have
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
12
Multi degree of freedom Systems
Matrix iteration method
Obtain Dynamic matrix
Use basic Eqn to obtain first frequency
Substitute
Assume trial vector
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Multi degree of freedom Systems
Matrix iteration method
First Iteration
As there is a difference in both the vectors the
solution is not converged
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Multi degree of freedom Systems
Matrix iteration method
Consider
as new trial vector
and iterate
Second Iteration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Multi degree of freedom Systems
Matrix iteration method
Fourth Iteration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Multi degree of freedom Systems
Matrix iteration method
Compare with basic Eqn.
Refer the session 5 (presentation) for the
solution using basic method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Multi degree of freedom Systems
Matrix iteration method
Last natural frequency and modal vector
Obtain inverse Dynamic matrix
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Multi degree of freedom Systems
Matrix iteration method
Use basic Eqn to obtain first frequency
Substitute
Assume trial vector
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Multi degree of freedom Systems
Matrix iteration method
First Iteration
As there is a difference in both the vectors the
solution is not converged
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Multi degree of freedom Systems
Matrix iteration method
Consider
as new trial vector
and iterate
Second Iteration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Multi degree of freedom Systems
Matrix iteration method
Fourth Iteration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Multi degree of freedom Systems
Matrix iteration method
Compare with basic Eqn.
Refer the session 5 presentation for the solution
using basic method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Multi degree of freedom Systems
Matrix iteration method
Find first natural frequency of the system shown
in the figure using matrix iteration method. Use
flexibility influence co-efficients
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
24
Multi degree of freedom Systems
Matrix iteration method
Obtain flexibility influence coefficents
2K
K
2m
K
m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
25
Multi degree of freedom Systems
Matrix iteration method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
26
Multi degree of freedom Systems
Matrix iteration method
First natural frequency and modal vector
Obtain Dynamic matrix
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
27
Multi degree of freedom Systems
Matrix iteration method
Dynamic matrix
Use basic Eqn to obtain first frequency
Substitute
Assume trial vector
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
28
Multi degree of freedom Systems
Matrix iteration method
First Iteration
As there is a difference in both the vectors the
solution is not converged
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
29
Multi degree of freedom Systems
Matrix iteration method
Third Iteration
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
30
Multi degree of freedom Systems
Matrix iteration method
Compare with basic Eqn.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
31
Multi degree of freedom Systems
Matrix iteration method
Practice problems
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
32
Summary
Multi degree of freedom Systems

• Numerical Method of obtaining fundamental
  natural frequency
• Matrix iteration method
• Convergence to first frequency
• Convergence to last frequency

Application of matrix iteration method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli