Biomedical Engineering Reference
In-Depth Information
affected by the prescribed displacement. In practice, the transformation can be
effectively carried out on the local level just prior to adding the local matrices to
the global assembled matrices.
2.3.6 Dynamic Equations
The equilibrium equations governing the linear dynamic response can be repre-
sented as in Eq. (
2.103
). The fundamental mathematical method used to solve
Eq. (
2.103
) is the separation of variables. In order to change the equilibrium
equations to the modal generalized displacements [
7
] it is proposed the following
transformation:
u
ð
t
Þ
¼U x
ð
t
Þ
ð
2
:
111
Þ
where U is a m
m square matrix containing m spatial vectors independent of the
time variable t, x
ð
t
Þ
is a time dependent vector and m ¼ 2N for the 2D case and
m ¼ 3N for the 3D case, being N the total number of nodes in the problem domain.
From Eq. (
2.111
) also follows that u
ð
t
Þ
¼U x
ð
t
Þ
and u
ð
t
Þ
¼U x
ð
t
Þ
. The com-
ponents of u(t) are called generalized displacements. For which the solution can be
presented in the form,
u
ð
t
Þ
¼/ sin x
ð
t
t
0
Þ
ð
Þ
ð
2
:
112
Þ
being / the vector of order m, t the time variable, the constant initial time is
defined by t
0
and x is the vibration frequency vector. Substituting Eqs. (
2.112
) into
(
2.103
) the generalized eigenproblem is obtained, from which / and x must be
determined,
K /
¼
x
2
M /
ð
2
:
113
Þ
Equation (
2.113
) yields the m eigensolutions,
2
K /
1
¼ x
1
M /
1
K /
2
¼ x
2
M /
2
:
K /
m
¼ x
m
M /
m
4
ð
2
:
114
Þ
The vector /
i
is called the ith mode shape vector and x
i
is the corresponding
frequency of vibration. Defining a matrix U whose columns are the eigenvectors /
i
,
U ¼ /
1
½
/
2
... /
m
ð
2
:
115
Þ
