Biomedical Engineering Reference
In-Depth Information
where the rotation and translation of the rigid body element was introduced as
w
2
.t/
w
1
.t/
2l
w
2
.t/
C
w
1
.t/
2
V
1
.t/ D
;
2
.t/ D
:
(5.35)
The equivalent aerodynamic excitation forces F
1
.t/ and F
2
.t/ Fig.
5.4
baregiven
by the integrals of perturbation aerodynamic pressure
p.x;t/ along the vibrating
body surface
Z
L
0
p.x;t/
l C L
1
x
a
0
.x/ C V
1
.t/
.a.x/ .x L
1
/V
1
.t/ y
T
/
dx;
h
s
2l
F
1
.t/ D
(5.36)
Z
L
0
p.x;t/
l L
1
C x
a
0
.x/ C V
1
.t/
.a.x/ .x L
1
/V
1
.t/ y
T
/
dx;
h
s
2l
F
2
.t/ D
where h
s
is the width of the channel measured perpendicular to the direction of
airflow and parallel to the plane of symmetry, h
s
is identical with the width
of
the
rigi
d b
ody. Then by calculating the integrals (
5.36
) the aerodynamic forces F
1
.t/
and F
2
.t/ can be expressed as functions of the displacements V
1
.t/ and V
2
.t/.
After expressing the potential and kinetic energies of the system in a similar way
as in the article [
40
] and their substitution in the Lagrange equations (see, e.g., [
15
]),
the equations of motion are obtained in the form:
After substitution in the Lagrange equations, the equations of motion are
obtained in the form
M
V
C
B
V
C
K
V
C
F
D 0;
(5.37)
where the following displacement and excitation force vector were introduced
V
1
.t/
V
2
.t/
;
F
D
F
1
.t/
F
2
.t/
;
V
D
(5.38)
and where
M
,
B
,
K
are the structural mass, damping, and stiffness matrices:
lm
1
m
1
C
;
c
1
lc
1
c
2
lc
2
: (5.39)
m
3
2
M
D
B
D "
1
M
C "
2
K
;
K
D
m
3
2
lm
2
m
2
C
The damping matrix B represents a proportional model of structural damping (see,
e.g., [
15
]) and "
1
, "
2
are constants adjusted according to the desired damping ratios
for the two natural modes of vibration of the system. The structure of the matrices
M
reveals that a mass coupling caused by the mass m
3
is generally in the
system even if
F
D 0.
and
K
Search WWH ::
Custom Search