2.5 Several Special Cases of the Euler Equation and Their Integrals
131
Fig. 2.6 The force diagram
when the sewing machine
needlepoint is piercing a
fabric
O
x
y
2x 1
y
1
f d N
dN
dP
dQ
α
The elastic force dQ can be expressed by the fabric tension σ and the rotation
elemental surface dA
dQ = σ d A cos α =
σ cos α2πxdx
sin α
= 2πσ cot αdx
(3)
Substituting Eq. (3) into Eq. (2), and putting y
= cot α, then integrating, the
functional of the resistance that the needlepoint pierces the fabric is
P =
x 1
0
2πσ y
1 + f y
y − f
xdx
(4)
At the needlepoint piercing the fabric, if the deformation of the fabric is not large,
then it can be approximate thought that the tension is directly proportional to the
strain of the fabric, namely σ = Eε, where, E is the elastic modulus of the fabric,
and ε = x/x 1 , substituting them into Eq. (4), there is
P =
2πE
x 1
x 1
0
y
(1 + f y
)x
2
y − f
dx
(5)
The boundary conditions are y(0) = 0, y(x 1 ) = y 1 , x 1 =
d 0
2
, y 1 = L 0 .
Let y = f x + z, there is y
= f + z
, then Eq. (5) can be written as