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
  (Covered in last session)
• Influence co-efficents
  1-session (Todays session)
• Approximate methods
  1-session
• Dunkerleys method
• Rayleighs method
• 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
Influence co-efficents
Influence co-efficients. It is the influence of
unit displacement at one point on the forces at
various points of a multi-DOF system. OR It is
the influence of unit Force at one point on the
displacements at various points of a multi-DOF
system.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
4
Multi degree of freedom Systems
Influence co-efficents
The equations of motion of a multi-degree
freedom system can be written in terms of
influence co-efficients. A set of influence
co-efficents can be associated with each of
matrices involved in the equations of motion.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
5
Multi degree of freedom Systems
Influence co-efficents
For a simple linear spring the force necessary to
cause unit elongation is referred as stiffness of
spring. For a multi-DOF system one can express
the relationship between displacement at a point
and forces acting at various other points of the
system by using influence co-efficents referred
as stiffness influence coefficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
6
Multi degree of freedom Systems
Influence co-efficents
The equations of motion of a multi-degree
freedom system can be written in terms of inverse
of stiffness matrix referred as flexibility
influence co-efficients. matrix of flexibility
influence co-efficients The elements
corresponds to inverse mass matrix are referred
as flexibility mass/inertia co-efficients.
matrix of flexibility mass/inertia co-efficients

Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
7
Multi degree of freedom Systems
Influence co-efficents
The flexibility influence co-efficients are
popular as these coefficents give elements of
inverse of stiffness matrix The flexibility
mass/inertia co-efficients give elements of
inverse of mass matrix
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
8
Multi degree of freedom Systems
Stiffness influence co-efficents.
For a simple linear spring the force necessary to
cause unit elongation is referred as stiffness of
spring. For a multi-DOF system one can express
the relationship between displacement at a point
and forces acting at various other points of the
system by using influence co-efficents referred
as stiffness influence coefficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
9
Multi degree of freedom Systems
Stiffness influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
10
Multi degree of freedom Systems
Stiffness influence co-efficents.
Obtain of the Stiffness influence co-efficents of
the system shown in the figure
Stiffness is force / unit deflection
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
11
Multi degree of freedom Systems
Stiffness influence co-efficents.
K1x1 K1
k11
m1
K2(x2-x1) -K2
k21
m2
K3(x3-x2) 0
k31
m3
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
12
Multi degree of freedom Systems
Stiffness influence co-efficents.
K1x1 0
K1
k12
m1
k12
x10
K2(x2-x1) K2
K2
k22
m2
k22
m2
x21 Unit
K3(x3-x2) -K3
K3
k32
k32
m3
m3
x30
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
13
Multi degree of freedom Systems
Stiffness influence co-efficents.
K1x1 0
K1
k13
m1
k13
x10
K2(x2-x1) 0
K2
k23
m2
k23
m2
x20
K3(x3-x2) K3
K3
k33
k33
m3
m3
x31 Unit
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
14
Multi degree of freedom Systems
Stiffness influence co-efficents.
Maxwells reciprocal theorem
K matrix can be obtained without writing Eqns. of
motion
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
15
Multi degree of freedom Systems
Eqns. of motion
Matrix form
Symmetric about this axis
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
16
Multi degree of freedom Systems
Flexibility influence co-efficents.
Matrix of Flexibility influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
17
Multi degree of freedom Systems
Flexibility influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
18
Multi degree of freedom Systems
Flexibility influence co-efficents.
Obtain of the Flexibility influence co-efficents
of the system shown in the figure
Flexibility is deflection / unit force
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
19
Multi degree of freedom Systems
Flexibility influence co-efficents.
K1 ?11
F11
m1
K2(?21 - ?11)
F20
m2
K3(?31 - ?21)
F30
m3
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
20
Multi degree of freedom Systems
Flexibility influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
21
Multi degree of freedom Systems
Flexibility influence co-efficents.
K1 ?12
F10
m1
K2(?22 - ?12)
F21
m2
K3(?32 - ?22)
F30
m3
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
22
Multi degree of freedom Systems
Flexibility influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
23
Multi degree of freedom Systems
Flexibility influence co-efficents.
K1 ?13
F10
m1
K2(?23 - ?13)
F20
m2
K3(?33 - ?23)
F31
m3
Force equilibrium diagram
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
24
Multi degree of freedom Systems
Flexibility influence co-efficents.
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
25
Multi degree of freedom Systems
Flexibility influence co-efficents.
By applying unit force on mass 1
By applying unit force on mass 2
By applying unit force on mass 3
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
26
Multi degree of freedom Systems
Flexibility influence co-efficents.
For simplification, let us consider
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
27
Multi degree of freedom Systems
Flexibility influence co-efficents.
Very important
Try to verify
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
28
Multi degree of freedom Systems
Flexibility influence co-efficents.
Obtain of the Flexibility influence co-efficents
of the system shown in the figure
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
29
Multi degree of freedom Systems
Flexibility influence co-efficents.
?
T
l
?11
F1unit
m
l
m
l
m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
30
Multi degree of freedom Systems
Flexibility influence co-efficents.
?
l
?11
m
l
T
F1unit
m
?22
l
m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
31
Multi degree of freedom Systems
Flexibility influence co-efficents.
?
l
?11
m
l
m
?22
l
T
F1unit
m
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
32
Multi degree of freedom Systems
Flexibility influence co-efficents.
By applying unit force on mass 1
By applying unit force on mass 2
By applying unit force on mass 3
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli
33
Summary
Stiffness influence coefficents
Flexible influence coefficents
There importance
Solved some Problems
Dr. S. K. Kudari, Professor, BVB College of Engg.
Tech., Hubli