44
N. Herisanu et al.
ε 0 = u x +
1
2
W
2
x −
1
8
W
4
x −
1
2
u x W
2
x
(2)
ε 1 = z
W xx −
3
2
W
2
x W xx
(3)
where u x = ∂u/∂x. It follows that the strain potential energy of the beam is
U =
E A
2
l
0
u x +
1
2
W
2
x −
1
8
W
4
x −
1
2
u x W
2
x
2
dx +
E I
2
l
0
W
2
x −
3
2
W
2
x W xx
2
dx
(4)
where A is the area of the beam of length dx and I =
l
0 z
2 dA.
The virtual work is given by the applied force P 0 and by the damping, such that
t 2
t 1
δW dt = −
t 2
t 1
P 0 δu(l, t)dt −
t 2
t 1
l
0
cW t δW dxdt
(5)
where c is the damping coefficient.
If it is considered that no interaction occurs between transverse and longitudinal
vibrations and therefore the longitudinal inertia can be neglected, then the governing
equation of hinged–hinged beam subjected to axial constant force is
mW tt + cW t + E I
27
2
W
2
x W
3
xx − 3W
3
xx − 3W
2
x W xxxx +
9
4
W
4
x W xxxx + W xxxx
+ P 0
W xx +
3
2
W
2
x W xx
= 0
( 6 )
Supposing the transverse deflection w in the form
w(x, t) = X (x)T (t)
(7)
where T is the amplitude of the fundamental transverse mode and X is the first
eigenmode of the hinged–hinged Euler–Bernoulli beam, of the form X(x) = sinπx/l,
and applying the Galerkin method, from (6) we obtain
m ¨
T + c ˙
T +
E I π
4
l 4 −
P 0 π
2
l 2
T −
P 0 π
4
2l 4 +
7E I π
6
8l 6
T
3
−
27E I π
8
20l 8 T
5
= 0
(8)
Making the transformations