Wind Energy Harvesting System - Energy Harvesting Autonomous Sensor Systems - page 71

dy
dx

is small; hence ( dy

dx ) 2

For small deflections, << 1, and therefore the radius

of curvature can be simplified to

d 2 y
dx 2

1
R =

(2.19)

Next, the bending moment at some distances x from the wall is equated to
the applied bending moment shown in Equation 2.15 . Using Equations 2.15 ,
2.16 , and 2.19 , the bending moment derived in Equation 2.15 can be described
in a simpler form as follows:

d 2 y
dx 2 =

−

Bending Moment = E y I F ( L x ) (2.20)

From the simplified bending moment equation, the beam curvature can be
found as

d 2 y
dx 2 =

F
E y I ( L

− x ) (2.21)

Integrate Equation 2.21 with respect to x once to derive the beam slope,
which is expressed as

Lx

x 2
2

dy
dx =

F
E y I

− + K (2.22)

Based on the boundary condition, which states that at x = 0, dy / dx = 0, so

0. Integrating the beam curvature equation a second time gives the beam
deflection

K =

Lx 2
2

x 3
6

F
E y I

K '

y = − + (2.23)

= =

0, so K '

=

There is a boundary condition that, at x 0, y 0. Hence, at

=

x L , the tip deflection is found to be

FL 3
3 E y I

y L = (2.24)

The relationship between the aerodynamic force F created by the wind
flow and the tip deflection of the piezoelectric wind energy harvester is es-
tablished. When a positive/negative net force is loaded on the tip of the
piezoelectric wind energy harvester mounted in cantilever form, the tip
downward/upward deflection of the harvester can be estimated using
Equation 2.24 . A summary of the theoretical and experimental deflections of
the piezoelectric wind energy harvester with respect to various wind speeds
is given in Table 2.1 .
Figure 2.32 exhibits the experimental setup used to measure the tip deflec-
tion of the piezoelectric wind energy harvester when wind flows across the