Piezoelectric polymer wind generators
by Hadi Darejeh
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Physics
Montana State University
© Copyright by Hadi Darejeh (1983)
Abstract:
Small wind generators based on the piezoelectric effect in poly(vinylidene fluoride), or PVF2 for short
were designed, built and tested. The design was based on developing a voltage across a bimorph made
of two PVF2 sheets glued back-to-back and coated with electrodes. Suitable means of setting these
bimorphs into oscillation in the wind were developed. One of these designs (oscillating leaf) is based
on forcing the blades into oscillation at 60 Hz and feeding the output directly into the ac line. The other
two designs had the blades as parts of rotors which were forced to rotate by the wind. For these designs
the power was brought out through the rotor bearings and could be fed into the line by means of a
rectifier and synchronous inverter. The poled PVF2 is very expensive, but reducing the cost of the
poling process could make PVF2 wind generators practical for commercial use.
PIEZOELECTRIC POLYMER WIND GENERATORS
by
Hadi Darej eh
A thesis submitted in partial fulfillment
of the requirements for the degree
Master of Science
ii
.
Phy sics
MONTANA STATE UNIVERSITY
Bozeman, Montana
D e c embe r 19 83
Main
lib .
cop* ^
APPROVAL
of a thesis submitted by
Hadi Darejeh
This thesis has been read by each member of the thesis
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College of Graduate Studies.
f) ^
Date
. io,
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it
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7OTl, Graduate
____
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Approved for the Major Department
Date
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Approved for the College
Date
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epartment
of Graduate Studies
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D ean
iii
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University,
I
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quotations
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may
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in the opinion of either,
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ission.
iv
TABLE OF CONTENTS
Page
LIST OF FIGURES ..............................
ABSTRACT. ...................................................
v
vii
1.
INTRODUCTION........
I
2.
T H E O R Y ...............................
5
3.
P R O C E D U R E ......
24
4.
CONCLUSION.................
55
R E F E R E N C E S ................. ...............................
58
AP P E N D I C E S......
60
Appendix I - Figures, Graphs and D a t a s ....... .
Appendix II - Figures and Plots . . . . . .............
Appendix III - Energy Analysis Calculation......;
\
61
68
74
V
LIST OF FIGURES
Page
Figure I:
Figure 2:
Figure 3:
Figure 4:
Electromechanical coupling in piezo­
e l e c t r i c s ................................
5
An applied force produces electric polari­
zation in piezoelectric solids............
6
Dimensional changes of a piezoelectric
in an external field........
7
solid
(a) Piezoelectric material.
(b) An electric
field induces dimensional expansion.
(c ) Reverse polarity produces a contraction.
Figure 5:
(a) t g + tg
Figure 6:
Schematic representation
electric f i l m
Figure 7:
Blade
(b) all trans.....................
8
10
ofmaking piezo­
11
tip sh a p e............ ............. .
13
Figure 8 : Geometry of deflection a ngle............
16
Figure 9:
18
Cantilever b l a d e ..............
Figure 10:
Cantilever torque
........... .
analysis. ............... .
. 18
Figure
11: Frequency vs.. amplitude r e s ponse.........
21
Figure
12: Circuit to measure Y, d ^ j .......
24
Figure
13: Geometry of blade r o o t ...................
28
Figure
14: Capacitive blade g e nerator.........
32
Figure
15: Blade as current g e n e r a t o r.................
33
Figure
16: Blade as voltage
33
Figure
17: RLC representation of generator............
34
Figure
18: Impedance matching circuit for the
generator, power o u t p u t ..... ...............
35
19: Power v ^ . load resistance
37
Figure
g enerator.................
curves..;......,.
vi
LIST OF FIGURES
(continued)
Page
Figure 20:
Savonius rotor design for oscillating
b l a d e s ........................................
3g
Top view cross section of Savonius rotor
with PVF2 b i m o r p h ...........................
40
Plot of power v s . load resistance of
S-rotor generator.... ......................
43.
Figure 23:
Cross
section of oscillator le a f ...........
44
Figure 24:
Front view of an oscillator leaf. ..........
44
Figure 25:
Output power vs., load resistance for
oscillating l e a f .........
45
Circuit representation of three generators
in p a r a llel........ ..............
51
Figure 27:
Equivalent
52
Figure 28:
Y and d ^j measurement, apparatus..........
62
Figure 29:
Stress-strain c u r v e ............
63
Figure 21:
Figure 22:
Figure 26:
circuit of figure 2 6 ............
..........
Figure 30:
Load v^s . voltage drop graph for d g ^ ..... .
64
Figure 31:
Load vs., voltage drop graph for d ^ 2 ......
65
Figure 32:
Circuit
diagram for determining Q and f r ..
66
Figure 33:
Plot of frequency .vs. amplitude for Q and
fr determination.........................
57
Figure 34:
Plot of
a v_s . w .............................
69
Figure 35:
Lateral
leaf rotor construction figure...
70
vii
ABSTRACT
S m a l l w i n d g e n e r a t o r s b a s e d on the p i e z o e l e c t r i c
effect in p o ly(vinylidene fluoride), or PVF2 for short were
designed,
b u i l t and tested.
The design was based on
d e v e l o p i n g a v o l t a g e across a b i m o r p h m a d e of two P V F 2
sheets glued b a c k - t o - b a c k and coated w i t h electrodes.
Suitable means of setting these bimorphs into oscillation
in the w i n d w e r e d e v e l o p e d .
One of t h e s e d e s i g n s
( o s c i l l a t i n g leaf) is b a s e d on forcing the blades into
o s c i l l a t i o n at 60 Hz and f e e d i n g the output d i r e c t l y into
the ac line.
The other two designs had the blades as parts
of rotors w h i c h w e r e forc e d to rotate by the wind.
For
these designs the p o w e r was b r o u g h t out t h r o u g h the rotor
b e a r i n g s and co u l d be fed into the line by means of a
rectifier and synchronous inverter.
The poled PVF2 is very
expensive, but r e d u c i n g the cost of the p o l i n g process
could m a k e P V F 2 w i n d g e n e r a t o r s p r a c t i c a l for c o m m e r c i a l
use.
I
CHAPTER I
INTRODUCTION
When Coulomb
between
two
stated the well-known law of the force
charges,
it was
thought
that
electricity
co u l d be p r o d u c e d by pressure. First Hauy and then A., C.
Becquerel
crystals
conducted
showed
experiments
electrical
in
effects
which
when
particular
pressure
was
applied.
Credit s h o u l d also be given to the b r o t h e r s Pierre
and J a c q u e s
crystals
Curie
when
for
the
discovery
compressed
produce positive and negative
of their surfaces,
the pressure,
these
and which
in
in 1880
particular
that
some
directions
charges on certain portions
charges being p r o p o r t i o n a l
then vanish when
the pressure
to
is
removed.
This wasn't just a lucky discovery, b e c a u s e Pierre
Curie's previous
of them
study of pyroelectric phenomena
to look for electricity
from
pressure.
led both
They also
look e d for a p a r t i c u l a r d i r e c t i o n of a p p l y i n g the force
and
studied which
So the n ame
pressure,
One
groups
of
piezoelectric,
crystals
exhibit
which means
the effect.
e l e c t r i c i t y by
was given to this class of material.
of
these
piezoelectric
materials
is
a
2
piezoelectric
polymer,
namely
p o ly(vinyIidene
a b b r e v i a t e d P V D F or P V F 2 . The
piezoelectric
PVF2 was first discovered by Kawai,
in
1969.
As
electric
the
name
forces are acting.
is p r o p o r t i o n a l
force
implies,
polarization
is
of
effect
a Japanese
this
in
scientist,
phenomenon
the p o l y m e r
(solid)
is
an
on w h i c h
For reasonable forces the polarization
to the
reversed
in
applied
sign
force.
the
direction. An inverse effect
strain)
fluoride),
If the
external
polarization
changes
is a d i m e n s i o n a l change (a
caused by applying an electric
field.
P V F 2 and some of its c o p o l y m e r s have b e e n sh o w n to
be ferroelectric,
while
other piezoelectric
polymers
are
less likely to be. PVF2 , whose molecular chain formula is
(CH^-CF^)^,
appears
to be
the
strongest
of piezoelectric
these
polymers
m a c r 0 sc0 p i c a 11y
and pyroelectric polymers.
In
polar,
order
there
electrical
array
oriented
of
has
poling.
reorientation
an
to mak e
of
the
to
be
Mechanical
original
crystallites
in the
mechanical
direction
of
extension
spherulitic
which
the
extension
now
and
causes
structure
a
into
has
its
molecules
force.
Now
the
final
step consists of evaporating electrodes on the sample and
connecting
them
to
a high
voltage
field of 0.5 megavolt per centimeter.
permanent polar film.
source,
applying
a
This step creates a
3
Wind
generator
main aspect
detail.
application
of this project,
Today
electricity
wind
by
of P V F 2 , w h i c h
is explored herein
power
can
be
conventional
used
to
is the
in great
provide
windmills,
but
a
?
piezoelectric
technique
wind
to produce
generator
this
is
quite
a different
electricity.
If a p i e z o e l e c t r i c p o l y m e r is set into o s c i l l a t i o n
by m e a n s
voltage
of wind,
and
since
the
strain
the polymer
o u tput w i l l be alternating.
these
polymers
three
different
lateral
leaf
in the p o l y m e r
rotor,
of
the voltage
Suitable means
of setting
are
developed
generators.
Savonius
rotor,
T hey
thr o u g h
are
and Procedure
The
reason
for
named
and oscillating leaf.
Their e n g i n e e r i n g as p e c t s are dealt w i t h m a i n l y
Theory
a
is oscillating
into o s c i l l a t i o n
types
creates
in the
sections.
the design of the lateral
leaf
rotor
is to have as large a rotor as p o s s i b l e c o n s i s t e n t w i t h
the r e q u i r e m e n t of 60 Hz blade o s c i l l a t i o n f r e q u e n c y so
that its 60 Hz p o w e r output
can be fed into the utility
line.
The
Savonius
(S— c r o s s — section)
vertical axis rotor
has a low tip speed ratio,
but its design shape is very
s u i table
blades.
for
oscillating
This
has
a flexible
blade root m a d e of P V F 2 a t t a c h e d to a central rod w h i c h
holds
the
entire assembly together.
4
For bot h
of these
de s i g n s
the output
c urrent
from
the rotor is taken out through the rotor bearings and fed
into
a resistor
measuring
bimorph
as
a test
the v o l t a g e
load,
w ith
an o s c i l l o s c o p e
c r e a t e d due to the
s t r a i n in the
blade.
The o s c i l l a t i n g leaf g e n e r a t o r sets a P V F 2 b i m o r p h
blade
into
oscillating
an ac voltage.
bending
motion,
thereby
creating
A cantilever mounted thin spring steel bar
has a PVFg bl a d e m o u n t e d on the free end. That puts
the
entire
the
system
into oscillation
if bar
same resonant frequency. The blade
and blade have
consists of two sheets
of the piezoelectric polymer PVFg glued back-to-back onto
an inert p l a s t i c c entral layer. The b e n d i n g a l t e r n a t e l y
causes one PVFg sheet to be in t e n s i o n and the other one
in c o m p r e s s i o n ,
producing
a net
electric
field
in the
same d i r e c t i o n in both sheets for one half cycle, while
both fields reverse
in the next half
cycle.
5
CHAPTER 2
THEORY
Piezoelectric
crystals
do
not
have
a
center
of
s y m m e t r y . If they have net dipole m o m e n t s in their unit
cells piezoelectrics
are also pyroelectrics.
all
unit
piezoelectric
ther e f o r e
not
all
Piezoelectricity
subjected
to
is
produced
mechanical
field is applied,
have
piezoelectrics
s u r f a c e s of the crystal
Some
cells
when
net
are
However,
dipoles
and
pyroelectrics,
a suitable
stress.
not
crystal
Conversely,
if
is
the
are e l e c t r o d e d and an electric
deformation of the crystal will
of the electromechanical
coupling
effects
are
result.
shown
in figure I.
Q
Figure
trics .
I.
l
_ +„
Electromechanical
coupling
in
piezoelec­
6
As
the
name
implies,
of
solid
polarization
(figure
2).
For
polarization
applied
a
is
on
the
the
forces
forces,
proportional
If
phenomenon
which
reasonable
stress.
direction,
thi s
to
e x t ernal
polarization
the
are
the
a
acting
amount
magnitude
force
is
is
of
of
the
cha n g e d
changes
in
direction
accordingly.
The
relationship
(for a simple
b e t w e e n P and F can be e x p r e s s e d
geometric figure) as P = constant
P
F
F
Figure
2.
An
polarization
in
piezoelectric
is s h o w n here
any angle
The
is,
applied
force
produces
solids.
This
col I inear w i t h F. In general,
electric
polarization
P can make
with F.
inverse
piezoelectric
a dimensional
crystal
* F.
when
it
change
is
placed
(a
effect
strain)
in
an
exists
is
also.
produced
e l e ctric
field.
That
in
a
The
inverse piezoelectric effect however is a linear function
of
the
first
fields).
power
Therefore
of the applied field
the
magnitude
of
(for appreciable
the
piezoelectric
stra i n S x i n a s p e c i m e n p l a c e d in an elect r i c field F for
7
a s i m p l e shape s p e c i m e n is given by the r e l a t i o n s h i p S j, =
const a n t * E.
The
figure
geometry
3.
With
represented
then
the
the
above
initial
I and
strain
sens i t i v e
field
by
of
is
the
relation
length
leng t h
S x = A I/I.
(no
field
in the
The
is
shown
present)
field by
inverse
in
I+A I,
effect
is
to the d i r e c t i o n of the app l i e d field. If the
is reversed,
the strain is a contraction.
M
/2
of
a piezoelectric
E
Figure
solid
3.
Dimensional
in an external
field.
A d e s c r i p t i o n of wha t
applying
This
a field
figure
sound wave.
also
cha n g e s
to the
shows
truly happens
polymer
the
is
r e ason
in the case of
sh o w n
for
in figure
generation
4.
of a
8
+ +
+ +
99
AA
UU
Figure
4.
(a)
Piezoelectric
electric field induces dimensional
polarity
produces
material.
expansion,
(b ) An
(c) Reverse
a contraction.
The f igure of m e r i t for p i e z o e l e c t r i c s K is called
the coefficient of electromechanical
coupling,
or
o
Electrical Energy out
Mechanical Energy out
g* = _________________— ----- = ----------------------Mechanical Energy in
Elect Energy in
K is also
related
to the
dielectric
constant
e in
the following manner.
Bfree ^
where
the
* ~ eClamped
Sfree
clamped
frequency
is
dielectric
where
own inertia.
dielectric
constant
constant
the device
at
is
low
frequency
measured
at
is effectively clamped by
and
high
its
Fiom the m e c h a n i c a l point of v i e w K is also
related to Young's Modulus Y , such that:
Y open ckt *
The value
of Y is m e a s u r e d w i t h closed c i r c u i t and wit h
electroded
sho r t e d
^ = ^closed ckt
the
s u r faces
value
of
connected
Y
is
less
to
each
than
other.
that
for
When
an open
9
circuit.
Let's c o n s i d e r that there is a single crystal w i t h
the
dipoles
4.
In
of the unit cells
applying
an
aligned as shown
electric
field
as
in figure
e x h i bited,
the
crystal l e n g t h e n s b e c a u s e the ions are a t t r a c t e d to the
pole
plates.
expands
If
an AC v o l t a g e
and contracts
into the
is applied,
in oscillation,
surrounding medium,
the
crystal
sending out
a wave
whether air or water.
If we
apply a m e c h a n i c a l force, the charges s h o w n above build
up
on
the
surface,
creating
exponentialIy when the force
responsive
fluoride),
A few
would
piezoelectric
PVF2 , with
words
pifobab Iy
piezoelectricity
have
on
many
To
constituents
helical
shapes
polarization.
of
monomers
good
should
crystallization
linked
These
obtain
in
repeating
are
is
be
so
These
produce
(CHg-CF^)^.
of PVFg
understanding
Macromolecular
unit
cells,
have
big
chemical
polymers,
they
or make
internal
macromolecules
chains
during
polar
that
the
called
chemically
the macromolecules
which
off
of the most
composition
piezoelectric
not
dies
p o l y ( v i n y lidene
formula
chemical
this material.
that
polymerization.
polymers
useful
elementary
monomers,
groups.
the
which
is removed. One
the chemical
be
of
a vol t a g e
their
prevent
them
have
gathering
should
also
of
be
chemically stable and not cross-linked into infusible and
10
insoluble
solids. B e c a u s e
fluorocarbons
are
the
piezoelectric
polymer
of the above
best
considerations,
monomers
crystals.
A
to
yield
schematic
representation of the two most common crystalline chains
is shown in figure 5.
H
Figure
indicate
planes
5. (a) t g+ t g-
projections
of
(b) (all
the -cF2
trans).
dipole
The arrows
directions
on
defined by the carbon backbone.
The tg+tg- configuration has its dipole moments both
parallel
other
one
molecular
The
obtained,
figure 6 .
and p e r p e n d i c u l a r to the chain axis, wh i l e the
has
its
dipole
moments
perpendicular
to the
axis.
process
mentioned
by
which
piezoelectric
films
on page 2 , is shown schematically
are
in
11
Melt Modified PVF0
a-phase
&
Mechanical Extension
Polar 3-phase
Random dipole direction
Electrical
Poling
(1) stretch direction
(2) transverse direction
(3) electric field direction
. I t * t f t t ? I f~TT
Dipole normal to film
Figure
electric
6 . Schematic
representation
of mating
film.
We have built
two kinds
of rotary w i n d g e n e r a t o r s
w h i c h use P V F 2 p o l y m e r to produce power.
lateral
are
used
piezo
leaf
rotor
in these
and
a Savonius
rotors
are
rotor.
called
The y e m p l o y a
The blades
bimo r p h s .
that
This
is
12
because
two layers o f .PVF2 are glued t o g e t h e r b a c k - t o -
back
p i e z o e I e c t r i c a I Iy
to
sheets w h i c h add w h i l e
sheet
induce
voltages
the b i m o r p h
is bent
in
the
two
so that one
is in tension and the other is in compression.
Savonius RojtjDjr
The
blade
Savonius
roots
Procedure
made
rotor
of
a
has
PVFg
two
blades
bimorph
with
as
f l e xible
shown
in
the
sec t i o n in figure 20. The change in d i r e c t i o n
of the w i n d force on each blade as the rotor turns will
give
the
desired
oscillation
of
the blades.
The Savonius
rotor is a non— synchronized ac generator.
It is d e s i r e d to have a c o n s t a n t bend radius of the
bimorph
so that
all portions
in generating power.
of it are equally
effective
Calculations show that a blade with
exponentially decreasing wid t h would attain a constant
b end
radius
whole blade.
if
uniform
Now,
this
so it must be truncated.
wind
pressure
shape would have
exists
over
infinite
the
length.
13
M
Figure 7.
Blade
tip shape
The following calculation
particular
shape.
truncating
"cap" ,
supports the idea behind a
Using
a
it
desired
is
parabolic
torque per unit width as x varies
to
shape
get
(figure
a
for
a
constant
7) which
is:
(x'-x)y'dx'
Eq. I
N is the
We
also w a n t
torque,
with
y is the w i d t h
this
shape
and p is constant.
a constant
Then integrating the second term
slope
at x=a.
on the right results
in:
N(X)
Y(X)
Eq. 2
The cap portion acts as a point
acting
at X q = (2b+3a)/5.
can be
replaced
gravity of
the
by
cap.
source F = 2 P y o (b-a) / 3
So for any cap shape,
a force
PA
acting
at the
the cap
center
of
14
The
theoretical
developed
by
the
value
bimorph
at
of
a
the
voltage
constant
that
is
frequency
is
o b t a i n e d u s i n g the f u n d a m e n t a l p i e z o e l e c t r i c e q u a t i o n s
and the e l e c t r i c d i s p l a c e m e n t
and r e a s o n i n g b e h i n d these
Computational
Procedure
equation. The
equations
are
derivation
stated
in the
section.
The cur r e n t that is p r o d u c e d in the b i m o r p h is equal
to the time rate of change of the charge.
equal
to the free
The charge
is
surface charge times the surface area
A:
V
2 Et
dQ
. dcr,
= .. =S . =S A —— *
R
R
dt
dt
I
The
direction
electric
(I)
displacement
Eq. 3
equation for a stretch in
is:
D 3 = 8 oE 3 + P 3
Eq. 4
and the piezoelectric
stress and strain equations
are:
P 3 = d31 Sl+(8r"1)coE 3
Eq. 5
and
Eq. 6
8I " d 31E3+Sl /Y
For a free surface
charge
density:
D3 = Of
Eq. 7
Substituting equation 5 into 3 gives
dQ
dt
.
ds
dt
.
the result:
dE
dt
Eq. 8
w h e r e 8. is the strain, E is the e l e c t r i c field.
stress
D
is
the
electric
displacement,
P
S is the
is
the
15
polarization,
and Y is Young's modulus.
2Et
Substituting — — f or
R
2Et
ds
Ad -- +
R
dt
dQ
—— in equation 8 gives the equation:
dt
.
dE
As -Eq. 9
dt
A s o l u t i o n of the for m
is tried for b oth E and S and
the following recursion relation is obtained:
2t
E(^— - j (i)As )E=JAdtoS
Substitution
for
S
Eq . 10
from
equation
5
into
the
above
equation yields:
21
o
E (—
-j toAs + j (odzYA) = j AdtoY8
Using E
of
, we eventually obtain the recursion relation
the following form:
2 tAYd
AYdz- A s r s0 + 2 J t/toR
5
This
calculation
sheet
is
bent,
for
the
6
sh o w s
shape
the
achieved
circle with radius r. The diagram
By
sector,
taking
we
the
differential
get dL = 0dr.
fact
is
that
while
a portion
the
of
a
i s .shown in figure 8.
of
the
equation
If the d i s p l a c e m e n t
for
a
is also a
sector with radius L then the equation is y = L0/2.
Figure
8.
Geometry
of deflection angle.
Now using the above e q u a t i o n s and the r e l a t i o n s h i p
dr = t / 2 , we get dL = yt/L. Since
the
the strain
strain in terms of L and t is
To find
the
theoretical
is 6 = dL/L,
= yt/L^.
value
for voltage,
we use
equation 11 to get V/ 6 , and knowing 6 , we get V.
Lal^ral Leaf Rotor
The
reason
achieving
as
requirement
power
basic
this
large
a rotor
be
directly
of this design
A p p e n d i x 1 1 is to have
pitch
similar
machine,
desi g n
as
is the d e s i r a b i l i t y of
possible
and
but
to
into
which
for
the
is shown
two b l a d e s
that
mounted
fed
power
the
that both have
a high-speed
on a v e r t i c a l
and the wind force
line.
The
in figure 35 in
small
horizontal
rotation
p i t c h (bending) of the blade m u s t be o p p osite
revolution,
also
of a 60 Hz blade oscillation frequency so the
could
idea
for
itself achieves
axis.
The
each half
this pitch
.17
change
the
as it acts on the entire
wind
velocity
accordingly,
increases,
tending
the wind velocity
to keep
blade
the
pitch
the rotor
increases.
at all
times.
wil l
speed
As
increase
the
same as
The bending torque
tries to
b r i n g the blade b a c k to its free position. This b e n d i n g
torque
is quite
produce
electrical
advantageous
having
large
to
compared
power
get
equal
excite
In order
one
analysis.
blade,
blade
so
the blade
The
to ana l y z e
has
to
exact
do
(resonant)
the blade
it b e c o m e s
at its
natural
frequency be
in the lateral
cantilever
is quite
desi g n
is
calculation made
The
later
improved
using
a more
in the computational procedure
cantilever
blade
has
a
leaf
beam
complicated but
an approximate approach is sufficient for a trial
This
by
frequency.
piezoelectric
calculation
required to
deflection
This requires that the rotational
to the blade natural
rotor,
the
large
the wind pressure
frequency.
in
to the torque
t hickness
of
design.
detailed
section.
2 Z Q and
w i d t h w that is d i v i d e d into two parts. The root length
X 2 bends w i t h a u n i f o r m radius
thickness 2Z C, and length X q -X^
straight
shape.
shown in figure
The
9.
total
and the cap of w i d t h W ,
is assumed
blade
length
to maintain a
is
then X q , as
I8
Cantilever blade
Figure
To compute
torque N along
the
root
the bending of
the root
is neglected,
the blade
is considered and the
so this
torque
torque of the c a n t i l e v e r blade
figure
10. Cantilever
torque
a uniform
inertia
of
is balanced by the
elements
10 .
Figure
root,
analysis
dz as s h o w n in
19
Now,
the torque is given by the equation:
N = J
ZdF = / _ f ° ZWS(Z)dZ =
ZWY5(Z)dZ
zo
J Zo
w h e r e S is the stress,
strain.
Now
Eq. 12
Y is Young's m o d u l u s and 8 is the
the strain 6 = ZZR1 and the deflection angle
Q = X 1 Z R 1 ; U s i n g the f o r m u l a for root d e flection, Y 1 = 9
X 1 ZZ = X 1 2 Z Z R 1 , we
Therefore
N =
If
the
the
8 = Z Y 1ZZX12 .
torque,
,
f_l° Z 0ZWYy1Xj2Z2dZ
centrifugal
constant
get R 1 = X 1 2 Z Z Y 1 and
blades
are
force
on
= 4WYy11Z 0Z Sx1
not
each
and would affect
connected
blade
at
can
E q . 13
the
be
tips,
the
considered
the blade's radial position but
not its r e s o n a n t f requency, just as gra v i t y affects the
e q u i l i b r i u m p o s i t i o n but not the natural f r e q u e n c y of a
mass hanging on a spring. The blade resonant frequency is
then
determined
from
a
tors i o n a l
analog
of
Newton's
se cond law:
N = 4 WYy1 Z 0ZSx12 = Ia=mx02W2#/3
and
N = 4 PcZ cton y i WXo/3xl
Eq. 14
“5 - lz^ e cz C1O*!
Eq. 15
then
Now
angular
it is d e s i r e d to get m r
a>n , where
(or is the
frequency
so
exists
of
the
rotor
there
a
synchronization between the blade's oscillation frequency
20
and the rotational
frequency.
More generally,
»r+»n=»o
where (Uq is the blade
oscillation at zero rotor speed.
f 0 is 60 Hz at r e s o n a n c e w h e r e
If
= w r , it is i m p o r t a n t
to have PcZ fi in accord with equation 15.
There is a new f a ctor here that needs to be def i n e d
for
a resonant
blade,
and
that
is
c a lled
the
qua l i t y
fact o r or Q. If one obt a i n s a plot of f r e q u e n c y of this
rotor
v.s^
(while
electrically
amplitude
following
shows
the
amplitude
oscillation
then
one
of a shaker)
obtains
the
■■
plot.
Figure
amplitude
of
f
vibrated by means
11
shows
of a sample
bandwidth
the
r e s ponse
of
the p e a k - t o - p e a k
vs., the applied frequency.
at
half
power
i
It
corresponding
of oscillation I / (2) of its m a x i m u m value,
also
to
that
is :
'
where
R.F.
is the
obvious from
resonant
the plot.
Q
frequency
and f 2 and f ^ are
21
FREQ
Figure 11.
( H 2)
Frequency v.s. amplitude response
22
For a driving force F q
Mt,
cos
the displacement
is Q
times as large as the displacement for a static force F q.
Therefore,
a resonantly driven blade
and produce
more
piezoelectric
power
can be much
than
thicker
a nonresonant
blade.
Bneraz Analysis
The q u a l i t y f a c t o r Q is also a m e a s u r e of the e n e r g y
s t ored in a r e s o n a n t
per
radian.
The
system
fraction
d i v i d e d by the e n e r g y lost
of
energy
lost
per
cycle
is
2jt/Q. This e n e r g y must be s u p p l i e d by the w i n d s t r e a m to
retain
a
constant
mechanical
amplitude
of
oscillation.
The
e n e r g y in the P V F 2 b l a d e root is c o m p u t e d as
follows:
1st: Energy density = S5/2 = Y5^/2 and 8 = S q Z Z Z o
2nd:
For
integral
two b l a d e s
this
mechanical
energy
is
the
of the energy density:
^stored
Now the p o w e r loss is
the
kinetic
rotor's
From
e n ergy
of
projected area
the
relations
E s tored w/Q. The p o w e r due to
the
windstream
incident
on
the
is
given
in the
Computational
section, we get for the output power:
Procedure
23
P out = V 02 o>C/4 where v 0= g Y 6 0Z 0, and C = e W X 1/2Z0
for one blade. Here V
is the p e a k open c i r c u i t vol t a g e
and C is the blade cap a c i t a n c e . This f o r m u l a a s s u m e s an
impedance-matched
resistive
load.
The
efficiency,
P 0 U t ZPin' a n ^ i nternal losses are c o m p u t e d and s h o w n in
the Procedure
section.
CHAPTER
3
PROCEDURE AND TESTS
In order
rotors
to o b t a i n
the
best
we measured mechanical
various manufacturers
quality.
In
doing
PVF 2 p o l y m e r
properties of samples
so as to choose a polymer
so,
we
achieved
power output as well as mechanical
The first
for
b e tter
our
from
of higher
e l e c trical
responses.
series of tests wer e p e r f o r m e d to obtain
Young's m o d u l u s ,
y i e l d stra i n and p i e z o e l e c t r i c
strain
coefficients.
The experimental set-up is shown in figure
%
2^ of A p p e n d i x I. In order to make the m e a s u r e m e n t s of
the above constants,
Figure 12.
the following circuit was designed.
Circuit to measure Y,
d
ij
‘
25
To o b t a i n Young's M o d u l u s ,
equation
we start w i t h the basic
of:
„
stress
S
Y — ------ — —
strain
8
F
S =—
A
where
and s t r a i n
F
=—
wt
mg
wt
is 8
Now,
t e n s i o n in the wire,
m is the load m ass
creating
g is a c c e l e r a t i o n of gravity,
the l e ngth of the sample,
I is
Al is the change of length,
w
is width of the sample and t is the thickness.
The
fundamental
displacement
piezoelectric
piezoelectric
equations
are
used
strain constant.
to
and
electric
calculate
the
They are the following:
P = S ’d + e0 (er-l) -E,
and
D = e0E+P,
where
P is the
stress
D
Eq. I
is
S (stress
the
Eq. 2
polarization vector
to
the
applied
tensor), E is the electric field vector,
electric
dielectric
due
tensor,
displacement
and
d is
the
vector,
er
piezoelectric
is
the
strain
tensor.
Now
substitution of P from
2 results
equation I into equation
in the relation
S*d = D - e •E, where e = e0 er
In
stress
our
application
is I and the
the
direction
Eq. 3
of
the
d e s i r e d f i e l d is p r o d u c e d
applied
in the 3
26
direction.
This
piezoelectric
simplifies
equation 3,
because
only one
strain coefficient dg^ is needed:
d31 Sl = D 3- 8 33E 3
The
the
stress
e l e ctric
Eq* 4
is the force per area,
field
in
the
mg
so S1 = —
.
-L wt
dielectric
is
the
Also
voltage
divided
V
T
by the thickness. Eg
The electric displacement
to
the
surface
divided
total
free
surface
charge
by
the
charge
capacitance
charge
in the dielectric is equal
density,
density is also equal
surface
is also
area,
equal
o =
Dg = <y.
to the
or Dg
to the voltage
The
total
free
charge
=
divided by
The
the
C.
Now s u b s t i t u t i o n of the above values in e q u a t i o n 4
results
d
,
i
in
=
VtC
;
mg I
A plot
PVFg
Vw
V d , - ,
8 8 — — or
=
o r mg
mg
_
I - eo erw
of mg vs.
—
V (voltage)
is shown in the Appendix I,
that line
----------------------------
is proportional
developed
figure 30.
across
the
The slope of
to dg ^.
The complete apparatus
figure for this experiment is
presented in figure 28 of Appendix I.
27
Test Procedure for Determining Natural Frequency
and Quality Factor Q
Since
on
Q,
the energy loss
a
careful
necessary.
higher
in our
measurement
The e n e r g y loss
Q
is,
system
the
less
of
this
lossy
the
b i m o r p h was
frequency
and
system.
quality
depends
quantity
is d e t e r m i n e d by Q,
physics of Q is mentioned in the Theory
Natural
greatly
is
so the
The
basic
section.
factor
d e t e r m i n e d by e l e c t r i c a l
Q
for
PVF2
excita t i o n .
The
c i r c u i t that was used is s h o w n in figure 32 of Ap p e n d i x
I.
The v a r i a b l e
various
frequencies
corresponding
frequencies
amplified
the
frequency
through
shaker.
and c o n s t a n t
amplitude
was
oscillator
as
meas u r e d .
the
of
and
at
Then the
change
oscillator
amplifier
The rotor natural
operated
a m p l itude.
result
The
power
was
output
then
fed
in
was
into
frequency was measured by
m o u n t i n g the ends of the central rod on the shaker so the
blades
were
and then the
means
of
perpendicular
amplitude
The
measurements
of
the
direction
of o s c i l l a t i o n was
a stroboscope.
resonant frequency
to
The
method
is e x p l a i n e d
natural
leaf were a little different.
of
of
vibration
m e a s u r e d by
finding
in the T h e o r y
frequency
First
of
Q and
section.
the
oscillator
the blade
itself was
m o u n t e d on a strip of rigid a l u m i n u m and then v i b r a t i o n
applied
so
the
natural
frequency
of
the
blade
was
28
measured.
and
the
Then the blade was mounted on the spring steel
length
was
adjusted
so that
the
entire
outfit
could o s c i l l a t e w i t h the nat u r a l f r e q u e n c y of the blade
measured previously.
This
the
leng t h was used in the win d
tunnel.
Sometime
second
measured
if it was a p p r e ciable.
occurs at the same length of the
Blade
This
some
thickness
at all times.
f by
The
of
vibration
assumes
spring
the
the wind
following
was
This second m ode also
steel.
calculation for lateral
computation
frequency
mode
rotor
force.
diagram
is
leaf rotor:
turning
with
The wind speed is u
shows variables
and
constants for this computation
The calculation for maximum kinetic energy follows:
r (6 ,t ) = r (6) c o suit
where
r(0) = ro (l-cos0)
v (0 ,t ) = -air (0) s indit
so
KE max
Eq.
5
29
KEmax = PZoro2woR“2
Let KEfflax = PEfflax to get the thickness h.
Then after a lot of derivation and substitution
.
/(5/3-jt/2) p n^r +
E’ - 6
where p is density of PVF2 , u wind velocity,
ratio,
rt tip speed
and <i>6 is the line frequency of 377 rad/s.
R U0 = u r t
then R = --- 1 . Our p a r t i c u l a r
cm.
(I)q
Now
the
second part
d e sign
requires
of the c a l c u l a t i o n
R = 2.65
is to find
the parameter a in the following blade shape equation:
W = M sin ad
Eq. 7
w h e r e M is blade height, a is a constant,
shape angle.
To do so,
force
and
balance
They a r e :
we use
torque
the g e o m e t r i c
balance
dT = -pTdd + «4
and d is blade
equation
for
equation,
the blade.
Eq . 8
77 w
A
A0 C
dc = pc d@ +
-dd
Pd J
Eq • 9
dT + dc + dFc=0
E q . 10
O
Il
.^
O
+
(-T + — — + BW)p +
ap
A
dN = kRcdd
E q . 12
w h e r e T is tension,
c i s shear force.
centrifugal
and
boundary
force,
conditions
E q . 11
B
is
N is torque, F c i s
a constant.
and s u b s t i t u t i o n s
With
of one
proper
equation
30
into another,
are
the
simplified forms
of the equations
used
stated as:
9T
I dN
dt$ + R 96$
0 =5> T ----R
N = DW for constant curvature, so T -----W
R
where
D
is
differential
constant.
These
result
in
the
following
equation
D 92W
(— + B) W +
R
R 94% = "
E q . 13
A solution of the above is
W = Msinad + ncosad
.
From boundary condition n = 0 ,
W = Msinad
In
finding
E q . 14
a,
there
are
tremendous
engineering mechanics involved,
results
calculus
so here we just
and
state the
of those calculations:
By the means
of the previous
equation,
a is shown to
be
“
_ /l + BR .
V
D
°
/I + 3R4a)2p0
E q . 15
YZo
The plot of a vs w is shown in figure 34
in Appendix II.
Calculation of elastic energy of the PVF2 blade
shows
YMZ
—
Ra
(1-cosad-)
°
E q . 16
31
Centrifugal
energy is
a(cosad0-cos2d0 )
Dc
p^<i)^M( 2ZQ )Rr
(I-Cosod0 )
(cosad0-cosd0 )
+ 2a
It can e a s i l y be
shown
that
a depends
on 4 in the
following manner.
3a^-21a^+36+(— a^+7a^-18)cosad0
for
+2a^(a^-l)cos^d0— 4a^(a^-4)cosdo=0
E q . 17
d o = 90° =$>
in the
a = 1.6.
Now
all
design consideration are found,
obtained through
some
the p a r a m e t e r s
so a particular
shape
is
tedious calculation.
Electrical Analysis of Lateral Leaf Rotor
The
lateral
leaf
rotor
c o m p u t a t i o n s and analysis,
Appendix
II.
In m a k i n g
is b a s e d
on
the
following
and is s h o w n in f i gure 35 in
this analysis,
we c o n s i d e r e d a
b i m o r p h w i t h o u t e r layers of p i e z o e l e c t r i c p o l y m e r and
center
layer
stiffness,
of
just
plastic
with
strong enough
negligible
to make
density
the blade
and
act as a
beam.
It is c o n s i d e r e d that this b i m o r p h is f e e d i n g a line
whose voltage
is lower
circuit is as follows:
than that
of the bimorph.
So the
32
E=EqCOS (Wt+<f>)
Figure
14.
By
Capacitive
adding
blade
the voltages
Kirchhoff voltage
law,
generator.
around
the
loop according to
I is found.
I0 = JwC(E0 Z d - V 0 ), so
I = w C (-E0 (sin wt cosd+cosw t sind)+V q sin w t )
Average
power = <P> = VI */2 = - W C V 0E0 sin (d )/2
M a x i m u m p o w e r is w h e n i = -90°, but I is then not in
phase
w i t h V.
To have
I in phase
w ith V as des i r e d by
the utility company,
V
-E.cosd+V
°
o
o =O or cosd = —E0
1 V
<P> = - (l/2)wCV0E 0 sin[cos'1 — °]
E q . 18
Then <P> = (l/2)wCV0 V e ^ - V q ^
Now
if the blade
consisting
is c o n s i d e r e d
of very
together, we have
as a current
generator,
thin layers of t hickness
d z stacked
the following
eqivalent
circuit:
33
T y
Figure 15.
Blade as current
Therefore
Iq = A ^
Cn = 8A/dz.
generator
[dxn+ (e-I)E q ]
It is known that Iq varies with n, so Q n
will be an additional
variable.
After using the fact that V=ZI
V = tdx + ( 6 - l ) E ] Z 0 /e
The
blade
following
considered
equivalent
------- ©
Figure
16.
as
a vol t a g e
generator
has
circuit:
— K—
—
Blade as voltage
I^— (5)— —
generator
A Thevinen and Norton analysis of the blade results
V.v
J
Nor
ws A
in
the
34
In t e r m s of strain the i m p e d a n c e incr e a s e d by a factor of
8e
Using
the
6e
stress
equation,
the
impedance
is again of the
form
- i±P
(1)8A
Again calculation
and stress
result
if
equation
for a blade
energy by using strain
of t hickness
w ItoeV, V e 02- 1
the
capacitive,
blade
circuit
is
inductive
L
C
.
|
6
17.
as
it is represented by the following
R
Figure
2Z 0 w o u l d
in:
<Wm >
Now
of m e c h a n i c a l
RLC representation of generator
well
figure.
as
35
so if to is close to <i>0 , d e f i n e d as (LC)
*^^ ,
E
j 2 ---0 QR+R
wo
where
a) L
R
The
radian
Now
inductor
is
is I
energy
2 and the
energy
R/ 2o).
the e f f i c i e n c y of blade for the parallel
the form
loss per
case is of
eff = I/Q c Ic (17Q eIc + 1ZQm ech}
AE , /radian
where 1/Q.
'elc = e _
------The
data
that
shows
this design is included
To m e a s u r e
generator,
power
the power
in Appendix
output
and voltage
II.
of the lateral
the following circuit
output of
leaf
rotor
is used:
GRD
Figure
power
18.
output
Impedance
matching
circuit
for the
generator
36
R
EICO.
is
a variable
The
resistor
oscilloscope
box
model
is Model
T922 made
vs,.
resistance
1100
made
by
by Tektronix.
Rprobe is 10M ohmThe
power
plotted for
m/s
output
the
the wind velocities
is s h o w n in figure 19.
of 9.6,
of
11.6,
the
box
14.3 and 22
The m a x i m u m p o w e r output of
this generator is 750 pw at a wind velocity of 22 m/s.
V
2
Now the power is
where Vp_p is peak-to-peak voltage
of the generator
(Ri ) (R probe ^
and R = -------;
-----Rprobe+ Rl
For
resistance
the
case
of
of the box.
R probe
>>
R1,
R
is
just
the
Resistance (Cl
Load
/00
300
P o w e r
Figure 19.
In
Power vs.
summary
500
TOO
f
load resistance curves
from
both
the
Theory
and
Procedure
sections the lateral leaf rotor power design requirements
are
A.
Develop an oscillating blade
B.
Find the mechanical
C.
Determine
energy needed to bend blade
the fraction of energy going to stored
mechanical energy
D.
E.
Determine
the
electrical
energy
Calculate
fraction
internal
energy
of
energy
going
lost and friction
to
F.
Determine
how
the
above
to electromechanical
G.
Determine
H.
Decide
statements
are
related
coupling constant.
the role of dielectric constant
on series or parallel blade
connections
Savoni us Rotor
The basic physics of this design is explained
Theory
section.
rotor.
In addition to drag on
shaft power,
This
rotor
is
basically
the vane
in the
a d rag-type
producing
rotary
that drag c r e a t e s d o w n w i n d forces on the
tower.
The S - r o t o r w h i c h
slows
air d o w n on one
other.
The d i a g r a m
is subject
side w h i l e
below
shows
to the M a g n u s
s p e eding
effect
it up on the
the basic shape
of our
S a v o n i u s rotor w h i c h has a fairly high tip speed ratio.
almost
I.
Figure 20
Savonius
rotor
design
for oscillating blades
In
order
to
get
a high
RPM
from
a S-rotor,
two
considerations ought to be taken into account:
1.
Maximize
the
difference
in drag
between upwind and downwind vanes,
2.
Minimize
the
These
wind
moving
vane.
two
proper
desi g n of shapes
and
force
against
requirements
like
coefficients
cone,
the
can be
wedges,
upwind
met
by
a
extra vane,
etc.
.
The
S-rotor
oscillating
P V F 2 is a little d i f f e r e n t
rotary
application
of
than the r e g u l a r S - r o t o r for
S-rotor was built by the wind generator
the f o l l o w i n g m anner.
thin-walled
s ide
of
the
The g e n e r a t o r frame
tubes.
approximately one half
one
for
shaft power.
This
two
blade
Each
tube
has
a
lab in
c o n s i s t s of
diameter
of
the length of the active region of
bimorph.
The
t o g e t h e r by two n y l o n bolts.
tubes
These
are
connected
tubes have a brass
rod with a cone-shaped tip connected at each end.
The brass
rods
are electrically
insulated
from
each
other and serve as b e a r i n g s w h i c h also carry current to
the
external
for
the
circuit.
S-rotor
is
The
two
bimorph
layers
of
that
was
constructed
PVFj
glued
together
back-to-back so the piezoelectricalIy induced voltages in
the two sheets add w h e n the b i m o r p h is bent so that one
sheet is in tension and the other is in compression.
The
40
two
sheets
were
connected with
epoxy glue.
The hardener
was p l a c e d on one surface and the resin was p l a c e d on the
other
surface.
completely
off
The
and
hardener
the
sheets
and
were
resin
w ere
cleaned
then placed together.
A top v i e w of the S-r o t o r wit h the b i m o r p h m o u n t e d on its
center
rod
is shown in figure 21.
wind
back stop tube
rotation
PVF0 bimorph blade
>■
Figure 21.
end cap
Top view
cross
section of Savonius
rotor with
P V F2 bimorph.
The
electrical
measurements
were
recorded
by
an
o s c i l l o s c o p e w i t h a v a r i a b l e r e s i s t o r box c o n n e c t e d
in
41
parallel
to
resistance
the
leads.
of 4.12M ohm.
area
should
power
outp u t
v±.
the
load
is 95.2p w at a load r e s i s t a n c e
The deviation of experimental
the t h e o r e t i c a l
area
output
is shown in figure 22.
The m a x i m u m
bimorph
The
results
is due to the use of the actual
in the c a l c u l a t i o n s .
be
used
as
results from
the
bimorph
A smaller effective
was
not
uniformly
strained as assumed.
Experiment
Theory
LOAD
F i g u r e 22.
generator.
RESISTANCE(MD)
Plot of p o w e r vs load r e s i s t a n c e of S-rotor
42
The
d e sign
parameters
of
the
found for a r b i t r a r y P V F 2 t h i c k n e s s
Numerical
Savonius
rotor
are
t and blade w i d t h w.
results here are for t=100 microns and w = l O cm,
and a w i n d
speed for this
wind velocity
d e s i g n is V q = IO m/sec.
is the required value
This
to bend the blade
its m i n i m u m radius as l i m i t e d by the bac k stop tube.
steps
to
The
in the Savonius rotor analysis are:
1.
Calculation of allowable
strain S max
2.
Calculation of allowable
curvature Rffl^n
3.
Calculation
of
allowable
torque
Tq
needed
to
required
to
obtain Rmin
4.
Calculation
of
blade
length
L
achieve T 0 at d e s i g n w i n d speed V 0
5.
Computation of blade oscillation frequency fQ at
design wind speed V q
6.
Computation
of
blade
open
ci r c u i t
voltage
V fl
when bent to radius R min
7.
Calculation of blade electrical
8.
Measurements
impedance Z.
of power per cm^ of PVF2 volume
at
design wind speed V q
9.
and
calculation
of
output
intercepted wind area
of V q.
power
per
m
2
of
Oscillating Flag Generator
This is the n e w e s t idea in a p p l i c a t i o n of PVF 2 i n a
43
wind generator.
a
The rotor is eliminated,
c a n t i l e v e r - m onn t.e d
nat u r a l
frequency.
blade
The
which
first
and replaced by
oscillates
idea
was
to
at
its
de v e l o p
a
desi g n like a tree leaf w h i c h o s c i l l a t e s b a c k and forth
in the wind.
which
shows
together.
desi g n
The
idea
comes
from
oscillation of the entire
The
of
second
combination of these
the
oscillator
construction involves
leaf
a flexible
a road
sign,
sign and its mount
two
ideas
generator.
piece
led, to the
I
The
basic
of plasticj cut
in
the shape of a tree leaf m o u n t e d on a thin sprinig steel
strip
that
is
set
into
oscillation
by
the
wind: force.
Our first d e sign w i t h P V F 0 gl u e d b a c k - t o - b a c k onto the
I
neu t r a l layer didn't o s c i l l a t e by itself due |to the
change
in the
The g l uing
. flexible
mechanical
properties
of the
s t i f f e n e d the entire blade,
composite.
creating
leaf w h i c h could not oscillate.
a less
This p r o b l e m
!
was overcome in two stages:
A.
First, by c o v e r i n g only o n e - f o u r t h of the blade
w i t h P VFg
glued b a c k - t o - b a c k w i t h
double
sided
scotch
tape.
B.
Second,
of a c y l i n d e r
length,
pull
blade
the
by designing a^vortex maker in the shape
that had
its
diameter
half
of
the blade
so it could deflect w i n d off its side to easily
blade
and the
tip
in and
spring steel
then
out.
to oscillate
This
helped
the
with much bigger
44
amplitude
A
and greater bending at the blade root.
sample
model
construction are
of
this
generator
show n in figures
lea f
and
its
23 and 24.
neutral layer
Figure 23.
Cross
Again
the
section of oscillator leaf
power
output
was
measured
by
an
oscilloscope with variable resistor connected in parallel
to
the
lead s .
resistance
The
of 3.2 M ohms.
for a very small
6 c m ^ on each
the
plot
maximum
of
R v.s.
was
The m a x i m u m
area of PVF^•
side
power
obtained
p o w e r was
for
10 pW
The area of PVF2 used was
of the
neutral
power
output
layer.
is
In figure
shown
for
25
th r e e
45
different
velocities.
The
blade
25.5
change
was
Hz.
as result of change
The
resonant
frequency
in the resonant
of
this
frequency
in wind velocity is about + 1Hz.
v* = 2.85 m / s
NA,=4.6 m / S
v.-7 / m / s
P o w e r (U W )
Figure
25.
oscillating
Output
leaf
power
v si.
loa d
resistance
for
46
The g e n e r a t o r as r e c e n t l y made c o n s i s t e d of a thin,
steel leaf spring mounted as a cantilever
in) w i t h
a blade
a t t a c h e d near
(thickness 0.02
its tip.
The blade was
m a d e of p l a s t i c w h i c h was 230 m i c r o n thick.
T w o layers
of PVFg polymer with thickness of 28 micron were attached
on either
side,
near
the blade's
b l a d e s e x t e n d o u t w a r d from
like
two
blowing
wings,
the
in
entire
a T
central
portion.
the tip of the steel
configuration.
assembly
lies
With
With wind blowing perpendicular to this plane,
bends
and
bot h
assembly
goes
starts,
it
position.
blade
tips
bend
into o s c i l l a t i o n .
does
so
at
this
The back and forth
backward
When
the
bent-back
oscillation
spring
no
in a v e r t i c a l
The
win d
plane.
the spring
before
the
oscillation
equilibrium
is relative
to
I
that equilibrium position.
The blade
r e s p e c t to their e q u i l i b r i u m
tips oscillate with
condition.
the spring do not o s c i l l a t e in phase.
r e l a t i o n s h i p of this o s c i l l a t i o n ,
were
taken by m e a n s
shapes have
The
reason
and
To find the phase
a series of p i c tures
of a stroboscope.
different
The blade
D i f f e r e n t blade
phase relationships.
for o s c i l l a t i o n and cycle
of o p e r a t i o n
are as f o i l o w s :
I)
The
spring
blowing
blades
begins
ag a i n s t
are
in
at
its
its
back)
their
rearward
position
neutral
(wind
but
is
the
curved
47
I
configuration.
Now as the blade
the d i r e c t i o n of the wind,
rearward
straightening
pr o v i d e
the m a x i m u m
w i n d by
the
equilibrium
2)
time
blade
sec t i o n
spring has
tip goes
so
as
toward
to
the
g o t t e n to its
position.
The spring keeps moving forward from its neutral
to its
most
continues
forward
position
while
the
blade
f o r w a r d to its e q u i l i b r i u m p o s i t i o n
(both motions
3)
the
the blade
the
cross
goes forward in
are relative
to the
spring).
The spring c o m e s b a c k to its neutral position,
while
the b l a d e m o v e s
spring,
t hus
smallest
forward relative
providing
area,
at
the
the
time
wind
when
to the
with
the
the
spring
motion toward the wind is greatest.
4)
The
spring
most
continues
position
blade
tips
going rearward
finishing
go
the
rearward
to
to
cycle
its
while
their
rear
the
neutral
configuration.
If small
amplitude of oscillation is considered,
blade gets more energy from the wind while
than
it
position.
gives
This
up
while
is because
coming
back
to
the
going forward
its
neutral
the area which the wind pushes
on is larger during the forward motion.
As the amplitude
of oscillation gets larger, the wind velocity against the
48
blade
becomes
smaller
during
the
forward
motion
faster during the rearward part of the cycle.
to d e c r e a s e
and
This tends
the net e n e r g y input per Cycle,
but this is
c o m p e n s a t e d by g r e a t e r b e n d i n g of the blade p r o d u c e d by
the
greater
i n c rease
wind
in
velocity
area
difference
difference
resulting
between
the
from
the
forward
and
rearward motion,
At
last
the blade
bends
so far at the e x t r e m e s of
its m o t i o n that m ore b e n d i n g h a r d l y
differential
input
and
amplitude
any more.
los s
per
For
cycle
this
i ncreases
amplitude
becomes
the
the area
the
Same
energy
and
the
of oscillation stabilizes.
Line Synchronization Calculation
Td
synchronize
operation,
considered
the
PVF2 blade
1 0 60
Hz
cycle
the c o u p l i n g of P V F 2 to the ac line s h o u l d be
in d e t a i l .
An
estimate
is p r e s e n t e d
of
a
s im pl e c r i t e r i o n for f r e q u e n c y locking.
It is t h o u g h t
that
frequency
the
oscillator
will
chOdse
O s c i l l a t i o n which gives m a x i m u m
blade
is
resonance.
Oscillating
Now
with
x 6 > the
the
amplitude.
a given
amplitude
of
of
C o n s i d e r the
amplitude
a forced
x ^ at
damped
h a r m o n i c o s c i l l a t o r d r i v e n at f r e q u e n c y m is as f o l l o w s :
*6 = F o /mr(“ o2" 6,2)2 + to4/Q2 ]l/2
E q . 19
If it is a s s u m e d t h a t o s c i l l a t i o n t a k e s p l a c e at the
49
frequency
Wq , which
undamped system,
is
the
resonant
frequency
of
then Eq. 19 becomes
Xq (re s ) = F 0 ZCmto2ZQ')
where
Q'
includes
electrical
the
both
output
Eq. 20
the
mechanical
at resonant
losses
frequency toQ from
and
the
the PVF2
blade.
Since
line
the r e s o n a n t f r e q u e n c y is d i f f e r e n t from the
frequency
impedance,
If
and
the
the electrical
resistance
(voltage
to,
R
is
E2 ZR,
generated by
the
blade
line
power
wh e r e
has
basically
dissipated
e is
the
in the
rms
zero
source
source
emf
the bimorph).
locks
into
the
line
frequency,
then
e q u a t i o n 19 applies.
This w a y Q differs fro m Q' in that
the electrical
output
is smaller because
supplying
line.
the
in w h i c h
both
expressions for X q in equations 19 and 20 are equal,
Q is
sufficiently
larger
For
a value
of
the generator is
than Q' so that
Mq
the other
term
which
when AtoZto = (k2 Z4Q)
where
appears only in equation 19 is compensated.
This
Ato is
M0
equality
the
and
to,
takes
magnitude
and
k2
of
is
place
the
the
frequency
difference
electromechanical
c o n s t a n t w h i c h is about 0.01 for P V F 2 20 for the b l a d e s we have m e a s u r e d ,
that
there
can
be
about
5%
between
c o u pling
Since Q is about
this r e l a t i o n says
deviation
of
the
resonant
f r e q u e n c y from the line f r e q u e n c y before the blade gets
50
out
of
synchronism
supports
the
maximum
idea
with
the
line
that the blade
power output.
frequency.
This
is to be d e s i g n e d for
In our measurements
the frequency
s t a b i l i t y as w i n d v e l o c i t y ch a n g e s over a w i d e range is
about 1% so obtaining synchronization with the line seems
to be a possibility.
Computational
The basic
Procedure for Line Synchronization
assumptions
that one. needs
to make
are as
follows:
1.
its
The blade o s c i l l a t e s at the f r e q u e n c y at w h i c h
amplitude
is
oscillation remains
at resonance
the blade
2.
the
remains
3.
at
the
the
the
natural
The
same
with
largest.
amplitude
same regardless
line
((i)j = 377
of whether
rad/S)
xQ
of
it is
or resonant
at
frequency Mq.
dri v i n g
ia and
same
The
force
(i>0
as
amplitude
long
as
the
F q of the w i n d is
blade
amplitude
at x 0 (g)j ) and x0 (M 0 )•
This x Q is given by the s t a n d a r d d r i v e n d a m p e d
harmonic oscillator expression
Xr. = F./m[ ((U2 -to2 ) 2 + a>4 /Q2 ] 1 ^ 2
Now for m = (i)Q the above r e l a t i o n is
xo = p OtV
If
Q
is
not
m w O2
dependent
on w then
resu l t in the m a x i m u m x 0.
this
value
M = M 0 will
B a s i c a l l y Q is l i m i t e d by both
51
mechanical
and electrical
that Q ffl is independent
As earlier
a voltage
losses.
It
is also
of to.
in this section,
generator with
series
we consider the blade as
capacitive
a lot of blades are connected in parallel
impedance.
three
and make
generators
it mor e resistive.
connections
If
then connecting
an i n d u c t o r in series w i t h these blades will
impedance
considered
reduce
the
Figure 26 shows
in parallel.
■)(----V W
Figure
generators
26.
Circuit
in parallel
representation
of
three
52
Now
ba s e d
on the above c o n d i t i o n s ,
figure 26 reduces
I
Figure 27.
The
to the circuit
the c ircuit
in figure
in
27.
—
Equivalent
line
has
circuit
a vol t a g e
of figure 26.
of
V
and
zero
impedance.
Since the v o l t a g e s are a c then we have
E=Eo Sinwt and V = V 0 Sinw^t
In the first
the p o w e r
output
also the power
case,
it is assumed that WjfeWj therefore
is s i m p l y
E ^ m g Z R eq = E q 2 / 2 R eq w h i c h
is
loss.
This is supported by the following derivation:
Heat loss in R
After
eS
is
substitution
of
relation and integrating
E 2
TC2
- V
- W Equation 21
and
cas e
the
is
when
synchronized.
E( t ) a n d
it,
V (t ) i n t o
the heat
the
above
loss is given by
Eq. 21
-
g l
to
W=W j
Now
T (E-V)'
(e-V)Idt =(T
JO
Req
V2
exhibits
generator
f^
JO
the heat
the
so
load
both
the heat
contribution
from
separately.
line
loss
and
The
the line
second
generator
calculation
in the
are
load
53
is coupled such that
"'I, - f
eq
The real power into the line is,
<iv> = 4 - ^ «
v QR eq
Addition of power
losses results in the power output
of generator e or,
® - T eO fe o-vo >/6
Q Calculation for Synchronized and Unsynchronized Cases
It
is k n o w n
that
the
mechanical
energy
stored
in
blade is
Wm = T Y S 2At
where
A is area and t is thickness of blade.
The voltage £ of the generator as seen previously is
B = -Y— —
e
F r o m the d e f i n i t i o n of Q it is obv i o u s that
Q
-
Eq. 23
' "Re,=
"Re, =A
N o w by m a n i p u l a t i n g e q u a t i o n s 22 and 23 it can be sh o w n
that :
Eq. 24
” R1 Q T .
54
Similarly
Q_ =
E q . 25
t'Qelc(Eo-Vo>
Using equations 19, 22, 23,
and forth results
and 25,
substitution back
in:
E q . 26
<r
-Wm
iW ) " 2
After a lot of algebra and a binomial
expansion,
equation
26 can be w r i t t e n as
k zQ„
(<o0- “ l)
+ italic’
Eq. 27
U s i n g n u m e r i c a l v a l u e s from our first design such as k'
= 0.01, Q elc = 20 and Q m = 20, we o b t a i n
(O0-O)I
= 0.056
The
above
value
is b a s e d
on
assumptions
s t a t e d above.
E v e n t u a l l y a b e t t e r c a l c u l a t i o n of the s y n c h r o n i z a t i o n
condition must be made
55
CHAPTER 4
CONCLUSION
The
there
Savonins
is
rotor
a minor
is one
problem
of the most
with
this
efficient
design.
The
but
top
bearing does not make constant contact with the top mount
where
the
lead
is
connected,
b e c o m e s a very no i s y sine wave.
that
has
a spring
the
output
wave
form
We d e v e l o p e d a top m o u n t
an old re l a y
which
contact with the bearing at all times.
This
signal mode s t l y .
from
so
adjusts
its
improved the
It is p o s s i b l e to de v e l o p an i m p r o v e d
bearing design so the contact becomes permanent.
The
tip
speed
Savonius
rotor
ratio
rotor.
could
directly
for
lateral
this
be
leaf
rotor
is quite
An
generator
low
compared
intermediate
practical
to the line.
is e f f i c i e n t
if
it
size
can
goal
of
to
that
but
its
of
the
of lateral
be
leaf
synchronized
The calculation of synchronization
limits
the v a l u e s
that they have to be fairly high for
The
also,
improvement
of
g e n e r a t o r is of m a j o r impor t a n c e .
of Q and m r such
its operation.
the
oscillating
leaf
This is b e c a u s e
this
56
new generator oscillates
with
at frequencies
(tip s p e e d ) / ( w i n d speed) ratios
from 25
to 55 Hz
from 0.7 to 2.
To
get 60 Hz o s c i l l a t i o n s y n c h r o n i z e d w i t h the a c line,
is
desirable
oscillation
2
k .
to
and
maximize
the
the
mechanical
electromechanical
Q
coupling
of
it
the
constant
The m e a s u r e d v a lues of Q are in the 13 to 15 range,
and improvements
in Q are both possible and desirable.
To increase Q of the b i m o r p h one has to exp l o r e the
following
alternatives:
1.
Blade
2.
G l uing
from
shape
design.
technique
if
inert
different
PVF 2 »
3.
New calculation for a better geometry of blade.
4.
A better vortex maker
of
to increase
locations
may be
device.
compactness.
uniaxial
amplitude
generator
in remote
cost-effective
application
testing of PVF2 bimorphs
under
of
extreme
and wind conditions has not yet been done.
A major
present
the first
But
temperature
is no need
the
oscillation.
Operation .as. a line-independent
this
layer , is
advantage
Since
of the PV F2 wind generator
the PVF2 will
for gear boxes
cost
of
a 100
"Kynar"
almost proportional
create
or separate
c m 2 piece
P V F 2 film
is
of
the power,
there
generators.
HO
$75.00
to its thickness.
is its
and
Using
micron
the
The
thick
cost
is
the value of
57
0.212
w a t t s / cm^
power
output
computed
in
the
a n a l y s i s s e c t i o n in A p p e n d i x III, this b e c o m e s
watt.
for solar cell cost in 1985 .
exp e n s i v e ,
Clearly
wha t
almost
three
the
amount
meantime
of
generator
times
is n e e d e d to be done
the poling process which makes
In
$321 per
This is 2 00 times as e x p e n s i v e as the $1.60 / watt
DOE goal
very
energy-
money
concept
it
to
seems
the
price
of
gold.
is a b r e a k t h r o u g h in
the polymer piezoelectric.
w ise
develop
The p o l e d PVFg is
to
the
spend
a ver y
piezoelectric
m o dest
wind
so it will be ready when the prices of
PVFg polymer drop sharply.
preliminary wind tunnel
These
tests
better piezoelectric properties
design computations
and
show that a combination of
and lower cost are needed
b e f o r e these PVFg w i n d g e n e r a t o r s will be e c o n o m i c a l l y
practical.
58
REFERENCES
59
REFERENCES
Bikales, N.M.
Mechanical Properties
Wiley and Sons New; York, 1971.
Cady, W.G.
Piezoelectricity,
Inc., 19 4 6 .
Ka w a i , H.
of P o l y m e r s ,
McGraw-Hill
Book
John
Company,
Jpn. J. App I. Phy s . (?,. 9 7 5 , 1.969.
Key s e r , M a t e r i a l s Sc ie nc e in E n g i n e e r i n g ,
3rd
Charles E. Merrill Publishing Company, 1974.
Ed.,
Loving er, A.J., Sc ie nce 220, 4602, 1983.
Park, J.
The Wind Power Book,
California, 1981.
Cheshire Books,
Palo Alto,
Schmidt, V.H., Klakken, M., and Darej eh, H.
Experimental
Study and Electromechanical Analysis of Oscillation
of Piezoelectric Polymer Bimorphs as Wind Generator
Elements, to appear in Proceedings of Wind Workshop
VI
(Minneapolis, June 1983).
Sch mid t, V.H.
P r o c e e d i n g s of the 1981 W i n d and Solar
Energy Technology Conference (University of MissouriC o l u m b i a Press), p. 97.
Yoler, Y.A.
A Stud y of P i e z o e l e c t r i c E l e m e n t s for the
Measurement
of T r a n s i e n t Forces,
U n i t e d States
Department of Commerce Office of Technical Services,
PB 111702, February I, 1955.
APPENDICES
61
APPENDIX I
Measur eme nts , equipment
diagrams and test plots.
62
VISES
SAMPLE
CARRIAGE
I""'""!
T 'T
BALL BEARINGS
WEIGHT
HOLDER
Figure 28
Y and d _
measurement apparatus.
63
Y,
= I.AxlO9 N
3m
Slope - I.633x10® kg
~2
m
Yield strength at 0.3%
T
m
Y-
I.6x1O9 ^
2
m
Figure 29.
S tress-strain curve.
64
S ize I by 2 cm
Thickness 30 micron
Slope ~ 0.038 ^
gm
d„. = 4xl0'12 C
31
N
VOLTAGE (
m
Sample made by 3M
2 0 0
load
Figure 30.
(
gm )
Load vs. voltage drop graph for d
31
65
Sample made by 3M
SIze I by 2 cm
Thickness 30 micron
Slope ~ 0.0504
d
- 5.36x10-12 c
200
load
Figure
300
cg m
400
)
Load vs. voltage drop graph for d
66
Cable
U
Shaker
I
Power Amp
Oscillator
CFJ-
Figure
32.
Circuit diagram for determining Q and f
r'
67
Shape : Tree
Width = 13.5
Height = 4.5
Blade thickness
leaf
cm
cm
= 350 micron
io6
a m p litu d e
(
E
Q.
5—
I
a
60
- H
25
30
FREQ ( H Z J
Figure
33.
Plot of frequency vs. amplitude for Q
and fr determination.
68
APPENDIX II
Construction figures of lateral leaf rotor, design
parameter plot of a and power data.
69
IOOOO^
IOOO-
Figure 34.
Plot of a vs. a).
Also see Procedure Section for a (to)
70
CONNECTOR T0P ELECTRODE
WIRE
1/4“ BRASS SCREW
STEEL
SQUARE
PHENOLIC
,SHAFT
NYLON
"FISH IN G
LINE /
BIMORPH
NYLON
SCREWS
1/4
PLEXIGLAS
DISK
CONNECTOR
W IRE
BOTTOM
ELECTRODE
(CROT
SO
SPSV
EI
CE
TW
ION)
Figure 35.
SI
(CRO
SD
SESV
EI
CE
TW
ION)
Lateral leaf rotor construction figure.
71
Power Data for Lateral Leaf Rotor
RPM
Tip
Velocity
(m/s)
Voltage
(P-P)
(V)
Load
R(Q)
Power
(fiW)
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
750
750
7 50
750
750
750
750
750
750
750
750
750
750
2 .1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2.1
2 .1
2.1
2.1
5
12
17
25
30
35
40
42
44
50
45
45
45
IOOK
15 0K
220K
3 3 OK
4 7 OK
680K
IM
I .SM
2.2M
3 .3M
5.7M
6 .SM
10M
31 .3
120
164
237
2 40
- 225
200
147
HO
126
79
62
50
11.6
11.6
11.6
1 1 .6
11.6
11.6
11.6
11.6
11.6
11.6
11.6
11.6
1 1 .6
770
770
770
770
770
770
770
770
770
770
77 0
770
770
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
15
20
25
30
35
43
52
55
60
- 65
60
60
60
IOOK
I 5 OK
220K
. 3 3 OK
47 OK
6 SOK
IM
I .SM
2 .2M
3 .SM
4.7M
6.SM
10M
281
333
355
341
326
340
338
2 89
250
212 .8
150.3
111
90
15.7
15.7
15.7
15.7
15.7
15.7
15.7
15.7
15 .7
770
770
7 70
770
770
770
770
770
770
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
2.14
25
30
35
40
60
70
65
65
65
1 5 OK
220K
3 3 OK
470K
IM
2.2M
4.7M
6.SM
10M
521
511
464
425
450
339
165
130
105.6
Wind
Velocity
v w(»/«>
72
Wind
Velocity
Vw (m/s)
RPM
Tip
Velocity
(m/s)
22
22
22
22
22
22
22
22
920
920
92.0
920
920
920
920
920
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
Voltage
(P-P)
(V)
30
35 "
40
. 55
60
65
60
60
Load
R(Q).
Power
(pW)
15 0K
220K
3 3 OK
IM
2.2M
4 . 7M
6 .SM
IOM
750
696
606
587
250
165
73
Power Data for Oscillating Leaf
Load
R(Q)
Power
(pW)
Wind
Velocity
(m/s)
Freq.
(Hz)
V o l tage
(P-P)
(V)
1.6
1.6
1.6
1.6
1.6
1.6
I .6
1.6
1.6
1.6
1.6
1.6
25
25
25
25
25
25
25
25
25
25
25
25
I
1.5
2.3
3
4
5 .5
8
10
12
15
17.5
17.5
1 5 OK
220K
3 3 OK
47 OK
680K
0.91M
I .SN
I .SM
2 .SM
3 .2M
4.55M
SM
0.83
1.28
2
2.4
2.9
4.2
6.1
6.9
7.2
8.9
9.4
8
2.85
2.85
2.85
2.85
2.85
2.85
2.85
2.85
2.85
2.85
2.85
2.85
25
25
25
25
25
25
25
25
25
25
25
25
1.3
1.7
2 .4
3.5
4.5
6
8
11
13
16
17.5
19
I 5 OK
220K
3 3 OK
470K
680K
0.91M
I .SM
I .SM
2 .SM
3.2M
4.5 5 M
SM
I .4
1.6
2.2
3.3
3.7
4.95
6.2
8.4
8.5
10
9.5
9
7 .1
7.1
7.1
7.1
7.1
7 .1
7.1
7 .1
7.1
7.1
7.1
7 .1
25
25
25
25
25
25
25
25
25
25
25
25
2
2 .4
3
4
5
6
8
11
13
15
17
18
I 5 OK
220K
3 3 OK
47 OK
680K
0.91M
I .3M
I .SM
2 .SM
3 .2M
4.55M
SM
3.3
3.3
3 .4
4.3
4.6
4.95
6.2
8.4
8.5
8.8
8.9
8.1
74
APPENDIX III
Energy analysis calculation.
75
Energy Analysis
8
= -S-=
0.0294
Rmin = t/5max = 3 *33 mm
for t = 100 micron
T 0 = 2YWt3 /3Rmin
T q = 8x10~^ N-m
Blade
impedance Z = — j/wc
w h e r e , w = 2itf = 2jr (60)
C=
80 erWL/2t
S q = Constant
Er = 12
W = 10 cm
L = 0.52 cm
.then
IZ |. = 9.61x10** ohms
V q = 920 volts
output power =
mc
PVF2 Volume = 10 cm x 0.52 cm x 0.02 cm
Power/Volume = 0.212 W/cm3
The output p o w e r is 0.022 w a t t s for a blade of 5.2x10^
PVFj area facing the wind, but only 45% of the blade area is
composed
meter
blades,
of PVFj-
area
the
facing
output
Accordingly
if a sy stem
the
packed
wi n d
power
is
would
be
fairly good compared with 100 watts/m
of one
densely
19.1
watts.
square
wi t h
such
This
is
available from large
conventional wind generators at design wind speed
MONTANA STATE UNIVERSITY LIBRARIES
stks N378.D245@Theses
Piezoelectric polymer wind generators /
RL
3 1762 00184023 8
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