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
• Dunkerleys method
• Rayleighs method
• Numerical methods
• (i) Matrix iteration method
  2-session
• (ii) Stodolas method
  1-session (Last session)
• (iii) Holzars method
  2-session (Todays session)

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
3
Multi degree of freedom Systems
Holzars method

• It is an iterative method, used to find the
  natural frequencies and modal vector of a
  vibratory system having multi-degree freedom.
• Undamped
• Damped
• Definite
• Semi-definite systems with linear or angular
  motions

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
4
Multi degree of freedom Systems
Holzars method
Principle
Consider a multi dof semi-definite torsional
system as shown in figure
Write Eqns. of motions
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
5
Multi degree of freedom Systems
Holzars method
Principle
Equations of motions
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
6
Multi degree of freedom Systems
Holzars method
Principle
Motion is harmonic
where i1,2,3,4
Substitute above Eqn. in Eqns. of motion
Add above equations
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
7
Multi degree of freedom Systems
Holzars method
Principle
Adding all equations
For n dof system
The above equation indicates that sum of inertia
torques (torsional systems) or inertia forces
(linear systems) is equal to zero for
semi-definite systems
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
8
Multi degree of freedom Systems
Holzars method
Principle
In this Eqn ? and ?i both are unknowns
Using this Eqn. one can obtain natural
frequencies and modal vectors by assuming a trial
frequency ? and amplitude ?1 so that the above
Eqn is satisfied.
Steps involved
1. Assume magnitude of a trial frequency ?
2. Assume amplitude of first disc/mass (for
simplicity assume ?11
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
9
Multi degree of freedom Systems
Holzars method
Steps involved
3. Calculate the amplitude of second disc/mass ?2
from first Eqn. of motion
4. Similarly calculate the amplitude of third
disc/mass ?3 from second Eqn. of motion
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Multi degree of freedom Systems
Holzars method
Steps involved
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
11
Multi degree of freedom Systems
Holzars method
Steps involved
The Eqn can be written as
5. Similarly calculate the amplitude of nth
disc/mass ?n from (n-1)th Eqn. of motion
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
12
Multi degree of freedom Systems
Holzars method
Steps involved
6. Substitute all computed ?i values in basic
constraint Eqn.
7. If the above Eqn. is satisfied, then assumed ?
is the natural frequency, if the eqn is not
satisfied, then assume another magnitude of ? and
follow the same steps
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Multi degree of freedom Systems
Holzars method
Prepare the following table, which facilitates
the calculations
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Multi degree of freedom Systems
Holzars method
Problem-1
For the system shown in the figure, obtain
natural frequencies using Holzars method
The given system is a semi-definite system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Multi degree of freedom Systems
Holzars method
Iteration-1
1
0.0625
0.0625
0.0625
1
1
1
1
0.121
0.0585
0.121
0.9375
0.816
1
?
0.051
0.172
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Multi degree of freedom Systems
Holzars method
Iteration-2
1
0. 25
0. 25
0. 25
1
1
1
1
0.44
0.19
0.44
0.75
0.31
1
?
0.07
0.51
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Multi degree of freedom Systems
Holzars method
Iteration-3
1
0. 56
0. 56
0. 56
1
1
1
1
0.80
0.24
0.80
0.44
-0.36
1
?
-0.20
0.60
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Multi degree of freedom Systems
Holzars method
Iteration-4
1.0
1
1.0
1.0
1
1
1
1
1.0
0.0
1.0
0.0
-1.0
1
?
-1.0
0.0
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Multi degree of freedom Systems
Holzars method
Iteration-5
1.56
1
1.56
1.56
1
1
1
1
0.69
-0.87
0.69
-0.56
-1.25
1
?
-1.95
-1.26
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Multi degree of freedom Systems
Holzars method
Iteration-6
2.25
1
2.25
2.25
1
1
1
1
-0.57
-2.82
-0.57
-1.25
-0.68
1
?
-1.53
-2.10
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Multi degree of freedom Systems
Holzars method
Iteration-7
3.06
1
3.06
3.06
1
1
1
1
-3.24
-6.30
-3.24
-2.06
1.18
1
?
3.60
0.36
?
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Multi degree of freedom Systems
Holzars method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Multi degree of freedom Systems
Holzars method
Iteration summary table
0.172
0.25
0.50
0.51
0.60
0.75
1.00
0.00
1.25
-1.26
-2.10
1. 50
0.36
1.75
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
24
Multi degree of freedom Systems
Holzars method
Analytical method of finding natural frequencies
As system is semi-definite, first natural
frequency is ZERO
second natural frequency of the system as
Third natural frequency is between 1.5 and 1.75.
One can get the value of ?3 by linear
interpolation
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
25
Multi degree of freedom Systems
Holzars method
Iteration summary plot
3
2
7
1
4
5
6
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
26
Multi degree of freedom Systems
Holzars method
From the plot
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
27
Multi degree of freedom Systems
Holzars method
If one need to obtain the modal vector, it is
required to iterate again with the magnitude of
?. For example, the third modal vector can be
obtained by iterating for ?1.71 rad/s
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
28
Multi degree of freedom Systems
Holzars method
Semi-definite linear system
For the system shown in the figure, obtain
natural frequencies using Holzars method
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
29
Multi degree of freedom Systems
Holzars method
Prepare the following table, which facilitates
the calculations
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
30
Multi degree of freedom Systems
Holzars method
Practice problems
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
31
Summary
Multi degree of freedom Systems
Numerical Method of obtaining fundamental
natural frequency Holzars method to find natural
frequencies and modal vectors of a semi-definite
system
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli