Hardware Reference
In-Depth Information
x
x
x
x
10
20
30
40
k
k
2
k
k
1
3
4
(a)
L
L
max
L
0
x
x
x
x
1
2
3
4
k
k
k
k
(b)
1
2
3
4
F
L
d
L
L
0
max
Figure 9.27
A linear spring system: (a) the total length is
L
0
when springs are uncom-
pressed; (b) the total length is
L
d
<L
0
when springs are compressed by a force
F
.
For the sake of clarity, we first solve the problem for a spring system without length
constraints (i.e.,
x
i
min
=0), then we show how the solution can be modified by
introducing length constraints, and finally we show how the solution can be adapted
to the case of a task set.
SPRINGS WITH NO LENGTH CONSTRAINTS
Consider a set Γ of
n
springs with nominal length
x
i
0
and rigidity coefficient
k
i
po-
sitioned one after the other, as depicted in Figure 9.27. Let
L
0
be the total length of
the array; that is, the sum of the nominal lengths:
L
0
=
i
=1
x
i
0
. If the array is
compressed so that its total length is equal to a desired length
L
d
(0
<L
d
<L
0
), the
first problem we want to solve is to find the new length
x
i
of each spring, assuming
that for all
i
, 0
<x
i
<x
i
0
(i.e.,
x
i
min
=0).
Being
L
d
the total length of the compressed array of springs, we have
n
L
d
=
x
i
.
(9.28)
i
=1
If
F
is the force that keeps a spring in its compressed state, then for the equilibrium of
the system, it must be
∀
iF
=
k
i
(
x
i
0
−
x
i
)
,
from which we derive
F
k
i
∀
ix
i
=
x
i
0
−
.
(9.29)
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