Biomedical Engineering Reference
In-Depth Information
Clearly, in analogy with the scalar one-dimensional case, the stretch of the spring
λ
is defined as
0
.
λ
=
(4.16)
To cause a stretch of the spring, a force must be applied to the end points A and B.
The forces applied on the end points are vectors represented by
F
A
and
F
B
. They
have equal magnitude but opposite direction:
F
B
=−
F
A
and are parallel to the orientation vector
a
:
F
B
=
Fa
. (4.17)
The scalar
F
represents the magnitude of the force vector
F
B
and for linearly
elastic springs this magnitude follows from the one-dimensional relation Eq. (
4.4
):
F
=
c
(
λ
−
1) .
(4.18)
Therefore, the force vector acting on point B is given by
F
B
=
c
(
λ
−
a
,
1)
(4.19)
while the force vector acting on point A is given by
F
A
=−
c
(
λ
−
a
.
1)
(4.20)
Example 4.1
Suppose a spring is mounted as depicted in Fig.
4.8
. The spring is fixed in space at
point A while it is free to translate in the vertical direction at point B. A Cartesian
coordinate system is attached to point A, as depicted in Fig.
4.8
. If point B is
moved in the vertical direction, the force on the spring at point B is computed
assuming linear elasticity according to Eq. (
4.19
). The length of the spring in the
e
y
e
y
y
e
x
e
x
A
B
A
B
0
(a)
(b)
Figure 4.8
Linear elastic spring, fixed at A and free to translate in the
e
y
direction at point B. (a) undeformed
configuration (b) deformed configuration.
