Biomedical Engineering Reference
In-Depth Information
F
u
B
u
−
F
u
A
x
0,B
x
0,A
x
x
0
(a) Displacement
vector
u
(b) Small extension of the spring
Figure 4.14
Displacement vector and spring extension.
If, as before, the reference position of a certain point is denoted by
x
0
, and the
current position by
u
of this point is defined as the
difference between the current and initial position of the point, see Fig.
4.14
(a):
x
, then the displacement vector
u
=
x
−
x
0
.
(4.21)
If a spring, as in Fig.
4.14
(b), is loaded by a force vector
F
, the end points A and
B will displace by the displacement vectors
u
A
and
u
B
, respectively. The current
position of the end points can be written as
x
A
=
x
0,A
+
u
A
x
B
=
x
0,B
+
u
B
.
(4.22)
The current length of the spring
may also be expressed in terms of the end-point
displacements:
=|
(
x
0,B
+
u
B
)
−
(
x
0,A
+
u
A
)
|
.
(4.23)
Define
R
as the end-to-end vector from point A to point B in the initial, unloaded,
configuration, hence
R
=
x
0,B
−
x
0,A
(4.24)
and let
δ
denote the difference in displacement of point B and point A:
δ
=
u
B
−
u
A
.
(4.25)
The current length
can be expressed in terms of the vectors
R
and
δ
:
=|
R
+
δ
|
.
(4.26)
