modeling, simulation and technology of ... - Scientific Bulletin

22
Georgeta Ionascu, Lucian Bogatu, Adriana Sandu, Elena Manea, Ileana Cernica
z
x
z G
t 1
(Si) t 2
(SiO 2
) t 3
(polymer)
Fig. 4. Succession of layers that compose the microcantilevers;
the centroidal axis of the composite beam is marked
In this case, the bending stiffness EI and density ρ are replaced in
equations 1 and 2 with the composite bending stiffness EI and composite
density ρ :
N
EI = ∑ E i I i
i=
1
N
∑ ρiti
i=
1
ρ = N
∑ ti
i=
1
where: N – the number of layers in the composite cantilever, E i I i - the bending
stiffness of the individual layers, ρ i – the density of the individual layers, t i – the
thickness of the individual layers.
The centroid coordinate (the location z G of the neutral plane) of the
composite beam is found by:
∑
∑
z G =
Ei
Ai
zi
Ei
Ai
(5)
where
t i
= + ∑ − 1
z i
i
t
2 k=
1 k
(6)
A i = bech
ti
(7)
b1 l1
+ b2l
b 2
ech =
l1
+ l2
(8)
b ech – is an equivalent width of a microbeam of constant width and mass
equal to that one of the microcantilever of variable width (obviously, b ech = b for
the microcantilever of constant width).
A i – is the cross-sectional area of each individual layer.
The individual moments of inertia I i of each layer are computed with the
following equation:
(3)
(4)