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

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

Exam TWO Questions 40 Marks
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
3
Multi degree of freedom Systems
Eigen values and Eigen vectors
In vibration problems Eigen values are the
natural frequencies and the and Eigen vectors are
modal vectors. This method is a basic tool, which
can be used to analyse any vibratory problems.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Multi degree of freedom Systems
Eigen values and Eigen vectors
Obtain natural frequencies and modal vectors of
the system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Multi degree of freedom Systems
Eigen values and Eigen vectors
Governing equations
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Multi degree of freedom Systems
Eigen values and Eigen vectors
Eqns. of motion
Matrix form
Symmetric about this axis
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Multi degree of freedom Systems
Eigen values and Eigen vectors
Matrix form
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
8
Multi degree of freedom Systems
Eigen values and Eigen vectors
Dynamic matrix
Substitute
For harmonic analysis with frequency ?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
9
Multi degree of freedom Systems
Eigen values and Eigen vectors
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Multi degree of freedom Systems
Eigen values and Eigen vectors
For simplification, let us consider
The equations in matrix form changes to
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
11
Multi degree of freedom Systems
Eigen values and Eigen vectors
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
12
Multi degree of freedom Systems
Eigen values and Eigen vectors
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Multi degree of freedom Systems
Eigen values and Eigen vectors
Eigen values can be obtained by
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Multi degree of freedom Systems
Eigen values and Eigen vectors
Solve the above Eqn. to get three values of ?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Multi degree of freedom Systems
Eigen values and Eigen vectors
For obtaining Eigen vectors consider the equation
Eigen or modal vector
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Multi degree of freedom Systems
Eigen values and Eigen vectors
First modal vector,
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Multi degree of freedom Systems
Eigen values and Eigen vectors
First modal vector,
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Multi degree of freedom Systems
Eigen values and Eigen vectors
First modal vector
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Multi degree of freedom Systems
Eigen values and Eigen vectors
Second modal vector,
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Multi degree of freedom Systems
Eigen values and Eigen vectors
Second modal vector,
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Multi degree of freedom Systems
Eigen values and Eigen vectors
Second modal vector
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Multi degree of freedom Systems
Eigen values and Eigen vectors
Third modal vector,
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Multi degree of freedom Systems
Eigen values and Eigen vectors
Third modal vector,
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Multi degree of freedom Systems
Eigen values and Eigen vectors
Third modal vector
In terms of amplitude ratios
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
25
Multi degree of freedom Systems
Eigen values and Eigen vectors
Modal vectors
Modal matrix
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
26
Multi degree of freedom Systems
Eigen values and Eigen vectors
Mode Shapes
Modal matrix
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
27
Multi degree of freedom Systems
Eigen values and Eigen vectors
Practice problems
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
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Summary
Multi degree of freedom Systems
Method of obtaining Eigen values and Eigen
vectors, Which are natural frequencies and modal
vectors
Mode shapes of a 3DOF system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli