Hardware Reference
In-Depth Information
By summing equations (9.29) we have
n
1
k
i
L
d
=
L
0
−
F
.
i
=1
Thus, force
F
can be expressed as
F
=
K
p
(
L
0
−
L
d
)
,
(9.30)
where
1
i
=1
K
p
=
.
(9.31)
1
k
i
Substituting expression (9.30) into Equation (9.29) we finally achieve
L
d
)
K
p
k
i
∀
ix
i
=
x
i
0
−
(
L
0
−
.
(9.32)
Equation (9.32) allows us to compute how each spring has to be compressed in order
to have a desired total length
L
d
.
For a set of elastic tasks, Equation (9.32) can be translated as follows:
U
d
)
E
i
E
0
∀
iU
i
=
U
i
0
−
(
U
0
−
.
(9.33)
where
E
i
=1
/k
i
and
E
0
=
i
=1
E
i
.
INTRODUCING LENGTH CONSTRAINTS
If each spring has a length constraint, in the sense that its length cannot be less than a
minimum value
x
i
min
, then the problem of finding the values
x
i
requires an iterative
solution. In fact, if during compression one or more springs reach their minimum
length, the additional compression force will only deform the remaining springs. Such
a situation is depicted in Figure 9.28.
Thus, at each instant, the set Γ can be divided into two subsets: a set Γ
f
of fixed springs
having minimum length, and a set Γ
v
of variable springs that can still be compressed.
Applying Equations (9.32) to the set Γ
v
of variable springs, we have
L
d
+
L
f
)
K
v
k
i
∀
S
i
∈
Γ
v
x
i
=
x
i
0
−
(
L
v
0
−
(9.34)
where
L
v
0
=
S
i
∈
Γ
v
x
i
0
(9.35)
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