Biomedical Engineering Reference
In-Depth Information
The simplified fluid-elastic system is schematically shown in Fig.
5.4
b including
a scheme of the equivalent mechanical part. The vocal fold can be approximated
by a two-degrees-of-freedom rigid body element with a defined shape a.x/,where
x is the axial coordinate. The element is supported by two discrete springs with
stiffnesses c
1
and c
2
and its vibration is described by its rotation and translation.
An equivalent three-mass system is used to formulate the equations of motion of
the element, based on three conditions of identical total mass, static moment and
moment of inertia of the rigid body. The vibrating rigid body with the center of
gravity T at the location (x
T
, y
T
) is replaced by three masses m
1
, m
2
,andm
3
joined together by a rigid massless rod of the total length L.
The distance between the positions of the masses m
1
and m
2
is denoted as 2l.
The distance between the location of the mass m
3
from the upstream end of t
he
rod
is de
n
oted by L
1
. The displacements of the masses m
1
and m
2
are denoted as
w
1
.t/
and
w
2
.t/,wheret is time. The length L should approximately correspond to the
anatomical data, the lengths l and L
1
can, however, be varied for the purpose of
tuning of the model.
Equations of Motion for the Equivalent Mechanical System of the
Vocal-Fold-Shaped Vibrating Element
The three masses m
1
, m
2
, m
3
of the equivalent mechanical system shown in
Fig.
5.4
b can be calculated from the following equations: equivalent mass of the
system:
m
1
C m
2
C m
3
D m;
(5.30)
equivalent static moment
L
2
C m
2
L
2
D me;
m
1
(5.31)
equivalent moment of inertia
m
1
L
2
2
C m
2
L
2
2
D I C me
2
;
(5.32)
which gives
mel
; m
3
D m
1
2
e
l
2l
2
I C me
2
1
I
l
2
:
m
1;2
D
(5.33)
The displacement of the rigid massless rod can be written as
w
.x;t/ D .x L
1
/V
1
.t/ C V
2
.t/;
(5.34)
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