4.4 Variational Problems of Functionals with Functions of Several Variables
293
w
2
x + w
2
y = 0 (on Γ 2 )
(9)
The other boundary conditions are
w = 0 (on Γ 1 )
(10)
w = d (on Γ 2 )
(11)
By the functional (6), the Ostrogradsky equation is given
N
∂
2 w
∂ x 2 +
∂
2 w
∂ y 2
+ q = 0 (in D)
(12)
In this way, the discussed problem becomes the probelm using the boundary
condition (9) ~ the boundary condition (11) to solve the Ostrogradsky Eq. (12).
For the circular membrane, see Fig. 4.6, rectangular coordinates can be converted
into polar coordinates, according to the expression (1.4.7) and the Eq. (1.4.12), there
are
∂w
∂ x
2
+
∂w
∂ y
2
=
∂w
∂r
2
+
1
r 2
∂w
∂θ
2
(13)
R
R 0
d
R
R 0
w(r )
Fig. 4.6 Contacting area caused by a circular membrane the under uniform load