Environmental Engineering Reference
In-Depth Information
φ
R
R
dz
z
t
Compression
Neutral Plane (NA)
M
t
M
No Stress:
Neutral Axis (NA)
a
Tension
(a)
(b)
FIGURE 2.30
Section of the beam subjected to pure bending.
deflects downward/upward, respectively. The relationship between the net
pressure created by the wind flow v and the beam deflection y is investigated
and discussed. Once this relationship is established and the wind speed flow-
ing above and below the wind energy harvester is known, the amount of
vibration being generated on the piezoelectric material can then be calcu-
lated.
In this research, the vibration of the piezoelectric wind energy harvester
is examined based on a simple beam theory studied in structural mechanics
to understand the beam deflection behaviour. When a beam is subjected to a
bending moment M of small deflections as shown in Figure 2.30 , the outside
of the bend is stretched, the inside is compressed, and in between them, there
is the neutral axis or neutral plane, which does not experience any tensile
stress at all [76]. The bending process of a section of the beam exhibited in
Figure 2.30 is subjected to pure bending. For a homogeneous and symmetrical
material, the neutral axis should be located at the geometrical centre.
Considering an element of the beam as shown in Figure 2.30a for bending
analysis, the radius of curvature of the neutral axis NA is R , and the element
of the beam includes an angle
at the centre of curvature. An incremental
change in the distance z from the NA has the length of ( R
±
z )
along the
NA, so the extension/compression of the incremental distance is
±
z
, and the
±
/
±
/
strain is
R . Now consider a symmetrical beam with thickness
t and width w pointing out of the page as shown in Figure 2.30b . When
the symmetrical beam bends, the rectangle shown in Figure 2.30b would be
rotated, and the bending moment needed can be calculated by considering
the tensile forces involved. Since the strain
z
R
=
z
and stress
on the element dz
R and E y ( z )
caused by the bending are
,respectively, the force F required
to achieve the moment M can be expressed as
±
z
/
E y ( z ) ±
adz
z
R
F
=
( stress )( area )
=
(2.13)
 
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