1
Questions and Answers on Natural Frequencies and
Mode Shapes
Background
The free response of an undamped, discrete system with N DOF s is given by:
x
t
( )
=
f
j
( )
c
j
cos
w
j
t
+
s
j
sin
j
t
[ ]
j
=
1
N
(1)
where
j
and
j
( )
(j = 1, 2, , N)
are the
natural frequencies
and
mode shapes
for the
system, respectively, that are found from the solution of the eigenvalue problem :

2
M
[ ]
+
K
[ ]
[ ]
=
0
(2)
Question
:
How do I find the natural
frequencies?
Answer
:
The natural frequencies are found from the solution of
the characteristic equation determined from:
det

2
M
[ ]
+
K
[ ]
[ ]
=
0
which is an Nth order polynomial in
2
:
a
N
2
N
+
a
N

1
2
N

1
( )
+
...
+
a
1
2
+
a
0
=
0
For N > 2, we need to use a numerical solver to find
these roots.
Question
:
How do I find the mode
shapes?
Answer
:
The mode shapes are found from the solution of the
algebraic equations of:

j
2
M
[ ] +
K
[ ]
[ ]
j
( )
=
0
That is, for a given root
j
of the characteristic
equation, we solve for the components
i
j
( )
(i =
1,2, ,N)
of the corresponding vector
j
( )
.
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Question
:
What does the notation
f
i
j
( )
mean?
Answer
:
i
j
( )
represents the ith component of the jth mode
shape
j
( )
; that is, the jth mode shape is made up the
N components of:
j
( )
=
1
j
( )
2
j
( )
M
N
j
( )
Question
:
Why do we need to choose
a value for one of the
components when solving
for a mode shape?
Answer
:
Since the eigenvalue problem (1) is a set of
homogeneous
equations, then if
is a solution, any
scalar multiple of
(say
af
) is also a solution.
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 Fall '09

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