72
4 Multi-Degree of Freedom (MDOF) Systems
by transposition and right multiplication with i . Since the left sides of the
above equations coincide, we have the equality of the corresponding right sides,
i.e., ω 2
i T
j m i = ω 2
j T
j m i or
ω 2
i − ω 2
j
T
j m i = 0, which implies
T
j m i = 0 since ω i = ω j by assumption. The above equations also show that
T
j k i = 0, i = j , and that T
i k i = ω 2
i T
i m i .
2. The matrix form of orthogonality conditions in Eq. 4.9 is
T m = diag{ ˜
m i } and
T k = diag{ ˜
k i },
(4.10)
where the columns of the (n, n)-matrix = [ 1 2 · · · n ] are the system modal
shapes. Note that the MATLAB output u in Eq. 4.8 is the matrix .
4.3.3 Approximate Calculation of Modal Frequencies
Consider an n-DOF structure with modal frequencies ω 1 < ω 2 < · · · < ω n and
modal shapes 1 , 2 , . . . , n . We have established the following relationship (see
Eq. 4.9):
ω
2
i =
T
i k i
T
i m i
=
˜
k i
˜
m i
(4.11)
between modal shapes, modal frequencies, and mass/stiffness matrices. This relationship can be used to estimate modal frequencies from guesses of modal shapes
based on the following two properties.
1. If u is a possible deformed shape, i.e., it satisfies geometrical boundary conditions, then
ω
2
1 ≤
u T k u
u T m u
≤ ω
2
n .
(4.12)
2. If u = i + ε v is a possible deformed shape and ε is a small parameter, then
u T k u
u T m u
= ω
2
i + O(ε).
(4.13)
Proof Since u is a possible deformation, it is a vector in the linear space spanned
by modal shapes, i.e., u =
n
i=1 b i i , where {b i } are coefficients that determine
the precise form of u (see Appendix C). This representation of u inserted in Eq. 4.11
gives
ω
2
=
u T k u
u T m u
=
n
i,j =1 b i b j T
i k j
n
i,j =1 b i b j T
i m j
=
n
i=1 b 2
i
˜
k i
n
i=1 b 2
i ˜
m i
(by orthogonality and Eq. 4.11)
4 Multi-Degree of Freedom (MDOF) Systems
by transposition and right multiplication with i . Since the left sides of the
above equations coincide, we have the equality of the corresponding right sides,
i.e., ω 2
i T
j m i = ω 2
j T
j m i or
ω 2
i − ω 2
j
T
j m i = 0, which implies
T
j m i = 0 since ω i = ω j by assumption. The above equations also show that
T
j k i = 0, i = j , and that T
i k i = ω 2
i T
i m i .
2. The matrix form of orthogonality conditions in Eq. 4.9 is
T m = diag{ ˜
m i } and
T k = diag{ ˜
k i },
(4.10)
where the columns of the (n, n)-matrix = [ 1 2 · · · n ] are the system modal
shapes. Note that the MATLAB output u in Eq. 4.8 is the matrix .
4.3.3 Approximate Calculation of Modal Frequencies
Consider an n-DOF structure with modal frequencies ω 1 < ω 2 < · · · < ω n and
modal shapes 1 , 2 , . . . , n . We have established the following relationship (see
Eq. 4.9):
ω
2
i =
T
i k i
T
i m i
=
˜
k i
˜
m i
(4.11)
between modal shapes, modal frequencies, and mass/stiffness matrices. This relationship can be used to estimate modal frequencies from guesses of modal shapes
based on the following two properties.
1. If u is a possible deformed shape, i.e., it satisfies geometrical boundary conditions, then
ω
2
1 ≤
u T k u
u T m u
≤ ω
2
n .
(4.12)
2. If u = i + ε v is a possible deformed shape and ε is a small parameter, then
u T k u
u T m u
= ω
2
i + O(ε).
(4.13)
Proof Since u is a possible deformation, it is a vector in the linear space spanned
by modal shapes, i.e., u =
n
i=1 b i i , where {b i } are coefficients that determine
the precise form of u (see Appendix C). This representation of u inserted in Eq. 4.11
gives
ω
2
=
u T k u
u T m u
=
n
i,j =1 b i b j T
i k j
n
i,j =1 b i b j T
i m j
=
n
i=1 b 2
i
˜
k i
n
i=1 b 2
i ˜
m i
(by orthogonality and Eq. 4.11)