TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 9, September
2014, pp. 64
7
9
~ 649
3
DOI: 10.115
9
1
/telkomni
ka.
v
12i9.601
2
6479
Re
cei
v
ed Ma
rch 2
3
, 2014;
Re
vised June
25, 2014; Accepte
d
Jul
y
1
0
, 2014
Structure and Mechan
ical Analysis of Single Cantilever
Piezoelectric E
n
ergy Harvester
Jie Zhao*
1
, Fenglin Yao
2
,
Shiqiao Gao
3
1
Departme
n
t of Computer En
g
i
ne
erin
g, T
a
iyu
an
Un
iversit
y
, 18 South D
a
ch
ang R
o
a
d
, Economic a
nd
T
e
chnolog
ica
l
Devel
opm
ent Z
one, T
a
i
y
u
an,
S
han
xi, 030
03
2,Chi
na, Ph./F
ax: (8
6)03
51-
8
378
00
6
2
Mechan
ical E
ngi
neer
in
g coll
ege, T
a
i
y
u
an U
n
iversit
y
of Sci
ence a
nd T
e
chnol
og
y, 66 W
a
l
i
u Street,
W
anba
ili
n Distr
ict, 03002
4, T
a
i
y
ua
n, Shan
xi, Chin
a, Ph./F
ax: (86) 0351-
69
981
15
3
School of Mec
hatron
i
cal E
ngi
neer
ing, Be
iji
n
g
Instit
ute of
T
e
chn
o
lo
g
y
, 5 South Z
hon
gg
ua
ncun Street,
Haid
ia
n District
,
10086, Be
iji
n
g
, Chin
a, Ph./Fax: (8
6) 010-
68
911
63
1
*Corres
p
o
ndi
n
g
author, em
ail
:
ty
d
x
c
o
mp
uter
@16
3
.com
1
，
jason
y
fl@
163.co
m
2
，
Shiqiao
g
a
o
@bit.e
du.cn
3
A
b
st
r
a
ct
Cantil
ever en
e
r
gy
harv
e
ster has
b
e
co
me the
ma
in struct
ure i
n
pi
e
z
o
e
l
ectric en
ergy
harvester.
T
here ar
e tw
o differe
nt methods t
o
b
u
il
d
their
mo
de
l. One is
lu
mpe
d
para
m
eter
mo
del; th
e oth
e
r
i
s
distrib
u
ted p
a
r
a
meter
mo
del.
By bui
ldi
ng t
heir
gover
n e
q
uatio
n a
nd so
l
v
ing th
e
m
, a
m
plitu
de-freq
ue
n
c
y
character
i
stics, pow
er
an
d
na
tural fre
q
u
ency
of the
mo
del
are obtai
ne
d. Co
mp
ariso
n
of
mod
e
l
fre
q
u
e
n
cy
and
a
m
pl
itud
e
are
ma
de
bet
w
een tw
o mod
e
ls. Prob
le
ms
,
scope
of ap
pl
icatio
n a
nd co
rrect met
hod f
o
r
ener
gy h
a
rvest
e
r are
als
o
g
i
v
en w
h
ic
h pr
ovi
des re
li
abl
e th
eoretic
al r
e
fere
nce a
n
d
make
s soli
d fou
n
d
a
tio
n
for energy h
a
rv
ester desi
gn.
Ke
y
w
ords
:
en
ergy harv
e
ster, lumpe
d
mass, Euler Ber
n
o
u
lli
bea
m, sin
g
le c
antil
ever
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
With the
ap
pl
ication
of
wireless
se
nsor
netwo
rks be
come m
o
re a
n
d
mo
re
wid
e
s
pread,
sup
p
ly energ
y
for these wirele
ss
sen
s
o
r
s p
r
ove
s
to be a signifi
ca
nt issu
e. At the sam
e
time
,
owin
g to the
appli
c
ation
of MEMS technolo
g
y,
the
power con
s
u
mmation
of wirel
e
ss sen
s
o
r
s
become m
o
re and m
o
re l
o
w. Mo
reove
r
, for the ambi
ent vibration l
i
es eve
r
ywh
e
re, the vibrati
on
can
supply
e
nergy
for th
e
low-po
we
r
consumption
wirel
e
ss
se
nsor via
en
ergy
ha
rveste
r
which
transfo
rm
s vi
bration
into
e
l
ectri
c
ity. It is a favo
rabl
e
approa
ch to
deal
with th
e
long
time
po
wer
sup
p
ly for th
e wi
rele
ss sensor [1]. T
herefo
r
e, n
u
m
ero
u
s
scho
lars have
re
sea
r
ched
en
ergy
harve
sting te
chn
o
logy. Th
e pie
z
oel
ectri
c
en
ergy
har
v
e
st
er
be
com
e
s f
o
cu
s f
o
r
simple
st
r
u
ct
ur
e
and hig
her e
n
e
rgy co
nversi
on efficien
cy [2].
Cantileve
rs i
s
the
mo
st
simple,
effect
ive and g
e
n
e
rally u
s
e
d
geomet
ry. Study an
d
analyze the stru
cture and
mech
ani
cal
relation
ship
is the ba
si
s of the re
search of en
ergy
harve
ster. G
enerally, there are
two m
e
ch
ani
cal
mo
dels fo
r e
nergy harve
ster.
One i
s
lum
p
ed
para
m
eter m
odel (S
prin
g
vibrator
mo
del), the
oth
e
r i
s
di
stribu
ted pa
ramet
e
r mo
del
(Euler
Bernoulli model) [3].
2. Lumped Parameter Mo
del for Single Can
t
ilev
e
r
Energ
y
Har
v
este
r
Non
coupl
ed
lumped
pa
ra
meter
mod
e
l
is lu
mped
pa
ramete
r m
o
d
e
l. It is
a
con
v
enient
way to model
. Acquire
d the parameter
of the mec
h
a
n
ical p
a
rt of the ha
rveste
r, the mech
ani
cal
equilib
rium
equatio
n an
d electri
c
al
balan
ce e
q
u
a
tion ca
n b
e
build up
by piezo
e
le
ctric
con
s
titutive relation.And t
he tra
n
sfo
r
mi
ng relation
sh
ip is b
u
ild u
p
. The
simpl
i
fied model
can
interp
ret som
e
feature of the ene
rgy ha
rveste
r more accurately.
2.1. Model Structur
e
The sket
ch o
f
the single
cantilever b
e
a
m
energy
ha
rveste
r is
sh
own in
Figu
re 1. It is
con
s
tituted
of pola
r
ized
pie
z
oel
ectri
c
plat
e alo
ng th
e d
epth of th
e la
yer an
d
an
elastic layer.
T
h
e
whol
e stru
ctu
r
e is cl
amp
e
d
end of the beam an
d fo
rm into a ca
ntilever struct
ure. The e
n
e
r
gy
harve
ster vibrates
harmoni
cally in
the
a
m
bient vib
r
ati
on. The
am
pli
t
ude of it i
s
A
,
and f
r
equ
en
cy
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 54
79 – 649
3
6480
is
. And the cantilever vibration belon
gs to
bending vibration m
ode
. Moreover, a
lumped ma
ss
is fixed on the free end of
the beam tha
t
adjust
the reso
nate freq
uen
cy and in
cre
a
se the o
u
tput
power. T
h
e
p
i
ezo
e
le
ctric p
l
ate is
covere
d with
ele
c
tro
de o
n
th
e u
p
per an
d lo
we
r
surfa
c
e.
Th
e
electrode
is conne
cted
wit
h
loa
d
circuit
and fo
rm
a
cl
ose
ci
rcuit. T
he imp
eda
nce of lo
ad
circuit is
r
e
pr
es
e
n
t
ed
w
i
th
L
Z
.
L
Z
m
L
Figure 1. Model of Single Cantileve
r En
ergy
Harve
s
te
r wit
h
Lumpe
d Ma
ss o
n
the End
Figure 2. Equivalent System of Single
Cantileve
r En
ergy Ha
rve
s
ter
Single cantil
ever e
nergy harve
ster
can be
sim
p
li
fied as
sin
g
l
e
deg
ree
of
freed
om
system if m
a
ss of the
bea
m is
reg
a
rdl
e
ss and
lump
ed ma
ss a
n
d
beam
vibrat
e in the
verti
c
al
dire
ction. Th
e
n
, the ene
rgy
harve
ster m
odel
can
b
e
repla
c
ed by th
e sp
ring
-ma
s
s sy
stem sho
w
n
in Figu
re 2. T
he sprin
g
-m
a
ss
syste
m
is
more
se
nsitiv
e to ambie
n
t vibration a
nd
gene
rate fo
rced
vibration.
In the mod
e
l
above m
ent
ioned, m
a
in
mech
ani
cal
compon
ents a
r
e in
ertia m
a
ss
and
sup
port
sp
rin
g
. Mass i
s
conne
ct to the
ba
se th
rou
g
h
the
sp
ring.
The
st
iffness of th
e en
ergy
harve
ster
can
be expre
s
se
d
by the sp
ring stiffness
k
.The syste
m
m
a
ss ca
n be repla
c
ed by th
e
lumped
mass. With the a
c
t of vibrate a
c
celeration,
the ha
rveste
r
will vibrate.
Displa
ceme
nt of
the base is repre
s
e
n
ted with
1
z
, relative displa
cem
ent of the
mass to the base is represente
d
with
0
z
, and the
n
ma
ss di
spl
a
cem
ent
relat
i
ve to the fra
m
e is re
presented
with
1
0
1
0
z
z
z
. (Only
the relative d
i
spla
cem
ent can p
r
od
uce deform
a
ti
on in spri
ng.) Re
serve
d
ene
rg
y is in form of
elasti
c p
o
tent
ial en
ergy i
n
t
he p
r
o
c
e
s
s of
tr
an
sform of
system. T
h
e
output
a
nd di
ssi
pate ene
rgy
is refle
c
t on the dampi
ng. Dampi
ng is
repre
s
e
n
ted with
c
.
2.2. Diffe
ren
t
ial Equation of Spring Vibrator M
odel
If displaceme
nt of the measu
r
ed b
a
se is
1
z
(velocity is
dt
dz
1
,ac
c
e
leration is
2
1
2
d
t
z
d
) that
use
d
as in
put
,
2
01
2
d
t
z
d
of the mass ca
n be u
s
e
d
as outp
u
t. So, differential equatio
n o
f
the mass
c
an be written as
follow [4].
2
2
01
2
01
2
01
2
2
dt
z
d
z
dt
dz
dt
z
d
n
(1)
m
c
2
/
is dampin
g
coefficient,
m
k
n
/
is natural frequ
ency of the system.
The am
bient
vibration
ca
n be rega
rde
d
as
sy
nthe
sis of many v
i
bration
s
of
different
freque
nci
e
s.
The e
nerg
y
harveste
r
natural
fre
quen
cy take
s the m
a
in
freque
ncy
of the
environ
ment i
n
to acco
unt d
u
ring
de
sign
pro
c
e
ss. So,
singl
e freq
ue
ncy vibratio
n
can
be referred
as re
se
arch e
m
pha
sis. Th
e
ambient
vibration ca
n be
written a
s
follow.
t
A
t
z
sin
)
(
1
(2)
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TELKOM
NIKA
ISSN:
2302-4
046
Structu
r
e and
Mecha
n
ical Analysis of Si
ngle
Cantilever Piezoele
c
tric Energy… (Jie Zha
o
)
6481
A
is amplitu
d
e
of vibration,
is freq
uen
cy
. The accel
e
ration of the
vibration
can
be
obtain by differentiate o
n
the displa
cem
ent function.
t
A
dt
z
d
sin
2
1
2
(3)
Subs
titute (3) into (1), then:
t
A
z
dt
dz
dt
z
d
n
sin
2
2
01
2
01
2
01
2
(4)
The
solutio
n
of differe
ntial Equation
(4) incl
ude
s t
w
o pa
rts. T
h
e
first
part
is
the fre
e
vibration whi
c
h do
esn’t take am
bient vibration ex
ci
ta
tion into con
s
iderin
g. i.e. the right
side
of
the differe
ntial eq
uation
equal
s
ze
ro
that ma
ke
a hom
oge
ne
ous second
-orde
r
differe
ntial
equatio
n. If the attenu
atio
n vibration i
s
think
a
bout,
gene
ral
solu
tion of the e
quation
ca
n
be
written as
follow:
)
sin(
)
(
)
sin
cos
(
2
2
2
2
2
2
1
t
c
m
k
mA
t
C
t
C
e
z
t
(5)
)
(
arctan
2
m
k
c
(6)
Equation expl
ain that the lumped p
a
ra
meter sy
stem
that make u
p
of sprin
g
, mass and
dampin
g
vibrate und
er fo
rced vib
r
ation.
The first ite
m
)
sin
cos
(
2
1
t
C
t
C
e
t
, when
t
,i
t
trend
s to ze
ro and
kno
w
n
as tran
sie
n
t terms. It
expresse
s that the amp
litud
e of the vibration
grad
ually attenuate
s
. The
second item
is kno
w
n
a
s
stea
dy item. And it expre
s
ses that
the
amplitude a
n
d
the cycle a
r
e invariant wit
h
the time.
Therefore,
when the
ambi
ent is
re
son
a
n
t exci
tation,
the sprin
g
-m
a
s
s vibratio
n system is
a stable p
e
ri
odic vibration
.
If
c
and
)
(
2
m
k
are too small, the vibration at
tenuate
s
mild
ly. If
the am
bient
freq
uen
cy i
s
clo
s
e
to n
a
tural
freq
ue
ncy
2
2
4
2
1
m
c
m
k
f
, amplitu
de m
a
y be
tremen
dou
s l
a
rge, this
situ
ation is called
as re
son
a
n
c
e.
Thus, the rela
tive vibration of spri
n
g
-m
ass syste
m
is gi
ven as bel
ow.
)
sin(
01
t
A
z
(7)
Whe
r
e,
2
2
2
2
)]
/
(
2
[
]
)
/
(
1
[
)
/
(
n
n
n
A
A
(8)
2
)
/
(
1
)
/
(
2
arctan
n
n
(9)
The displa
ce
ment
)
(
1
0
t
z
of
t
h
e mass
m
relative to the fra
m
e is harmo
nic vibratio
n
obviously i.e.
)
sin(
)
(
1
0
t
A
t
z
,but the phase an
gle is differ
.
is d
a
mping ratio
and
km
c
2
.
n
is the natu
r
al
freque
ncy of
the syste
m
a
nd
m
k
n
/
.The difference of ampli
t
ude
and ph
ase is
depe
nd on
and
n
.
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 54
79 – 649
3
6482
Becau
s
e
of relative di
spl
a
cem
ent of the ine
r
tia m
a
ss is
01
z
, according to
Ne
wton’
s
se
con
d
la
w, the deformati
on force th
at inerti
a syste
m
pro
d
u
c
ed unde
r
the
ex
citation can be
written a
s
bel
ow.
)
sin(
2
01
t
A
m
z
m
F
(10)
2.3. Amplitude-frequ
enc
y
Characteri
s
tic of the Spring-v
i
brator Model
Acco
rdi
ng to the re
sult
that m
entioned p
r
evio
usly, amplit
ude-f
r
eq
uen
cy
)
(
x
A
cha
r
a
c
teri
stic and pha
se
-f
reque
ncy characteri
stic
)
(
are
sho
w
n a
s
bel
ow.
2
2
2
2
1
01
)]
/
(
2
[
]
)
/
(
1
[
)
/
(
)
(
n
n
n
x
Z
Z
A
(11)
2
)
/
(
1
)
/
(
2
arctan
)
(
n
n
(12)
Amplitude-f
r
e
quen
cy cu
rve
s
and
pha
se
-freque
ncy
cu
rves a
r
e
sho
w
n in Fig
u
re
3 and
Figure 4 acco
rding to the t
w
o eq
uation t
hat mention p
r
eviou
s
ly.
0
2468
1
0
0
1
2
3
4
5
z
01
/z
1
ω
/
ω
n
0
2468
1
0
-2
0
0
20
40
60
80
10
0
12
0
14
0
16
0
18
0
20
0
φ
(
ω
)
ω
/
ω
n
Figure 3. Amplitude un
der
Different Da
mping
Ratio
Figure 4. Phase Angle u
n
d
e
r Different
Dampi
ng Ratio
As ca
n be
se
en from
Figu
re 3, the ampli
t
ude of
the
cantilever i
s
la
rge
r
5 time
s than the
ambient vibra
t
ion whe
n
1
.
0
and
n
.In other wo
rds,
cantileve
r model i
s
mo
re se
nsitive to
the ambie
n
t environ
ment
and am
plify the amplitu
d
e
.
On the oth
e
r ha
nd, the
amplitude i
s
th
e
bigge
st wh
en
vibration is reso
nan
ce, an
d sma
lle
r da
mping can ge
t bigger am
pli
t
ude. More
over,
it can be
co
nclu
ded that
dampin
g
is
a criti
c
al
p
a
rameter fo
r cantilever e
n
e
r
gy ha
rveste
r. It
affects not on
ly the amplitude of
the ene
rgy harve
ster
but also the
p
hase that sho
w
n in Figu
re
4.
Phase
will
affect both
ele
c
tri
c
current
and volt
of t
he en
ergy h
a
rveste
r
circuit. Gene
rally
, the
dampin
g
is b
e
twee
n 0.01
and 0.05.
2.4. Po
w
e
r o
f
Spring-v
i
br
ator Mod
e
l
As sho
w
n in
Equation (7), the amplitude
of the vibrator is
A
Z
01
.
2
2
2
2
)
2
(
)
1
(
'
A
A
(13)
Whe
r
e
n
, the powe
r
of it is shown as b
e
lo
w.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Structu
r
e and
Mecha
n
ical Analysis of Si
ngle
Cantilever Piezoele
c
tric Energy… (Jie Zha
o
)
6483
2
2
2
3
2
3
)
2
(
)
1
(
A
m
P
e
(14)
The output p
o
we
r will be
max when th
e freque
ncy
of excitation equal to the reso
nan
ce
freque
ncy. i.e.
1
4
2
3
A
m
P
e
(15)
The affect factors
can be
concl
ude
d that as belo
w
[1].
(1) Bigg
er m
a
ss of ene
rg
y harveste
r
can harve
st
more p
o
wer.
So it is important to
increa
se th
e
mass of
the
energy ha
rve
s
ter a
s
po
ssi
b
le.
And add
mass
o
n
the
end of
the be
am
can n
o
t only decrea
s
e the
natural frequ
ency but al
so
incre
a
se out
put power.
(2)
High
er freque
ncy of the ene
rgy h
a
rveste
r can
harve
st hig
her p
o
wer.
Ho
wever,
highe
r frequ
e
n
cy m
a
kes smaller amplit
ude. So
it is
i
m
porta
nt
to consi
der
the a
m
plitude as well
as the fre
que
ncy is con
c
erned.
(
3
) Sma
ller
da
mp
in
g mak
e
s
b
i
gg
er
p
o
we
r
.
Bu
t it is no
t p
r
e
tty as
sma
ll as
p
o
ss
ib
le
. T
h
e
output of ene
rgy harve
ster
is dep
endin
g
upon the p
o
wer that dampi
ng co
nsumm
a
tion.
(4) Incre
a
se
of dam
ping
ratio is hel
p t
o
fr
e
quen
cy sen
s
itivity
of
energy ha
rve
s
ter that
near the
re
so
nan
ce freq
ue
ncy.
2.5. Natu
ral Freque
nc
y
Solv
ing of Spring-v
i
brator
L
x
c
z
Figure 5. Defl
ection
Curve
of Cantilever
2.5.1. Lumped Mass is Ta
ken into Acc
ount
Whe
n
singl
e
cantilever b
eam simplifie
d as sp
ring
-vibrator mo
d
e
l, elastic el
ement is
cantileve
r structure. The
mass of ca
ntilever hav
e significa
nt pro
portion of th
e system. So it
cannot be i
g
nored. Otherwise, the cal
c
ulated
frequency
will be obvious
ly high. Generall
y,
Rayleig
h
method is u
s
ed
to calculate the natur
al freque
ncy of single ca
ntilever pie
z
oel
ect
r
ic
energy harve
ster [5].
3
)
'
140
33
(
420
'
)
140
/
33
(
L
m
m
YI
m
m
k
c
n
(16)
'
m
is lumped
mass on the
free end,
m
is mass of the
cantilever.
B
e
ca
use of the assume
d
bendi
ng
cu
rve is differe
nt from th
e real v
i
bration
cu
re,
the calculate
d
natu
r
al f
r
eq
uen
cy is sli
g
h
t
ly
highe
r than th
e accurate.
2.5.2. Non L
u
mped Mas
s
Condition
Acco
rdi
ng to the derivation
as above, if
t
he mass on the end doe
sn’t take into account,
the natu
r
al
freque
ncy i
s
a
s
Eq
uation
(16). T
h
e
eq
u
i
valent ma
ss is
m
140
33
. Compare to the
con
d
ition of lumped m
a
ss
cantileve
r, na
tural fr
eq
uen
cy of the cantilever is obvio
usly lower.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 54
79 – 649
3
6484
3
11
140
)
140
/
33
(
mL
YI
m
k
c
n
(17)
3. Euler Bernoulli Model of Single Cantilev
e
r
Although
lum
ped
pa
ramet
e
r m
odel
ha
s give
n p
r
eli
m
inary
sol
u
tion of
the
problem
by
simplification.
But it is co
nfined in
sin
g
le
freed
om vibration. It lacks of detailed
d
e
formatio
n a
nd
vibration of th
e cantileve
r, su
ch a
s
vibra
t
ion mode,
a
c
curate strain
distrib
u
tion a
nd ele
c
tri
c
affect
etc. Fa
cts
sh
own th
at for the tran
sve
r
se vib
r
ation
cantileve
r, h
a
rmo
n
ic
excit
a
tion of lum
ped
para
m
eter m
odel m
a
y lea
d
great e
rro
r.
The e
r
ror
dep
end
s on
spe
c
ific ratio
of en
d ma
ss and
the
mass of
cant
ilever. As
sh
own i
n
Fig
u
re 6, for t
r
an
sverse
vibratio
n of sl
end
er
beam,
sup
p
o
s
e
prin
cipal
axis
of inertia
of e
v
ery cro
s
s se
ction
s
a
r
e i
n
t
he
same
pla
n
e XOZ. External loa
d
i
s
al
so
in the plan
e. The be
am wil
l
vibrate in th
e pl
ane. T
h
e
n
main defo
r
mation of the
beam i
s
ben
d
i
ng.
If length is
great 5 time
s t
han the
heig
h
t of beam
s, she
a
r deform
a
tions and
th
e
cro
ss se
cti
o
n
spin
ning a
r
o
und their p
r
i
n
cip
a
l axis o
f
inertia can
be ignore
d
.
In this case, the beam
is
equivalent to
Euler Bernoul
li beam [6].
3.1. Laminate Struc
t
ure a
nd Part
ial Equation of Single Cantilever
R
c
z
Figure 6. Euler Bernoulli Cantilever Model
The e
nd fixe
d cantilever i
s
ma
de
up t
w
o l
a
yers. T
he le
ngth of
it is
L
, width i
s
b
,
t
h
ick
n
e
ss ar
e
p
t
(PZT laye
r) and
m
t
(elastic layer), th
e l
u
mped
ma
ss is
'
m
(
L
x
),the en
d i
s
fixed(
0
x
).Top la
yer is pie
z
oel
ectri
c
layer a
nd the bottom layer is ela
s
tic layer. Two layers a
r
e
smooth
co
ntinuou
s an
d h
a
ve no relati
ve sliding.
It is suppo
se
d
that the layers
are unifo
rm.
Elastic m
odel
of PZT layer is
p
Y
, bendin
g
moment of i
n
ertia i
s
p
I
, thic
kness
is
p
t
, and
cross-
se
ct
ion a
r
e
a
i
s
p
A
(
p
bt
). Elasti
c
model
of ela
s
tic layer i
s
m
Y
, bendin
g
mom
ent of ine
r
tia
is
m
I
,
t
h
ick
n
e
ss i
s
m
t
, and cro
s
s-se
ction area is
m
A
(
m
bt
)
.
Subscr
ipt
p
si
gnifies PZT l
a
yer, ). Subscript
m
signifie
s
ela
s
tic laye
r. Curvatu
r
e i
s
R
C
/
1
, and the dimen
s
ionl
ess co
uple
effect of
piezoele
c
tri
c
effect
is
2
/
1
0
2
31
)
/
(
p
p
Y
d
k
that sup
p
o
s
e to
be le
ss th
an
L
/
1
.
31
d
is polarizatio
n
cou
p
ling coef
ficient
in
z
dire
ction when
subje
c
ted to st
ress/
s
train in
x
dire
ction.
0
is
vacuum
permittivity,
p
is relative pe
rmittivity of piezo
e
le
ctric
material. Be
n
d
ing
stiffness of the b
eam
about the n
e
u
tral axis
c
z
is
sho
w
n a
s
b
e
l
o
w in Fi
gure
7. (
p
D
is b
endin
g
stiffness of
unit width.
[7])
]
)
12
)
[(
]
)
12
)
[(
1
2
2
1
2
2
N
i
i
N
i
i
i
N
i
i
N
i
i
i
p
t
z
z
Y
bt
t
z
z
Y
A
b
D
YI
(18)
If the beam i
s
mad
e
up o
f
two layers, the equatio
n
can
be si
mpl
i
fied as b
e
lo
w. The
uppe
r is pie
z
oele
c
tric m
a
terial an
d the lowe
r is ela
s
ti
c materi
al.
)
(
12
)
3
2
2
(
2
2
2
4
2
4
2
m
m
p
p
m
P
m
p
m
P
m
p
m
m
p
p
p
t
Y
t
Y
t
t
t
t
t
t
Y
Y
t
Y
t
Y
D
(19)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Structu
r
e and
Mecha
n
ical Analysis of Si
ngle
Cantilever Piezoele
c
tric Energy… (Jie Zha
o
)
6485
For laminated layer,
c
z
N
i
i
N
i
i
i
Yt
z
Yt
1
1
,
i
z
is center
axis of
the co
ordi
nat
e of the
i
th layer.
i
t
is the
t
h
ick
n
e
ss of
i
th layer that is sho
w
n in Fig
u
re 7.
Figure 7.
Model of Euler Berno
u
lli
)
(
2
2
2
1
1
p
p
m
m
m
m
p
p
N
i
i
i
N
i
i
i
i
c
t
Y
t
Y
t
Y
t
Y
t
Y
z
t
Y
z
(20)
m
m
p
p
N
i
i
i
i
bt
bt
t
b
m
1
(21)
m
is the mass o
f
unit length, for the co
nven
ience of
derivation, it can be con
c
lud
ed as belo
w
by
prin
ciple of virtual wo
rk [5]
.
0
)
,
(
)
,
(
)
,
(
2
2
4
4
t
x
F
t
t
x
w
m
x
t
x
w
YI
(
L
x
0
)
(22)
)
(
)
,
(
)
,
(
t
F
x
t
x
z
b
t
x
F
m
(23)
The first item
in equatio
n (2
3) is d
a
mpin
g
force, the se
con
d
item
)
(
t
F
is prod
uced force
that cau
s
ed b
y
vibration along
z
.
)
,
(
t
x
w
is the deflection when
time is
t
.
3.2. Free Vibration Soluti
on tha
t
w
i
th
out Lumpe
d Mass on the
Beam End
For th
e
ene
rg
y harve
ster i
s
ca
ntilever st
ructure
wi
th
si
ngle-end
fixed, t
he vib
r
atio
n of it i
s
force
d
vibrati
on und
er b
a
s
e ex
citation.
In this ca
se,
the forced b
a
se
of singl
e
-
end
ca
nnot
be
rega
rd
ed a
s
fixed. Base excitation (small deflect
io
n con
d
ition
s
) is take
n ba
se excitatio
n
that
prop
osed
by
Erturk an
d In
man [3]. If the cro
s
s secti
on of
ca
ntilever is unifo
rm and
have
no
lumped
ma
ss on the
en
d, the tra
n
sl
ation
a
l motion
is
)
(
t
g
, tiny rotation
o
f
the ro
ot is
)
(
t
h
whi
c
h
is sh
own in Figure 8.
Figure 8. Fre
e
Vibration of
Single Cantil
ever Co
nsi
d
e
r
the Rotation
of Root
As de
scriptio
n of Timo
sh
enko [8], ab
solute
displa
ceme
nt
)
,
(
t
x
w
is sum of
ba
se
displ
a
cement
)
,
(
t
x
w
b
and tran
sverse di
spla
cem
ent
)
,
(
t
x
w
rel
[9].
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 54
79 – 649
3
6486
)
,
(
)
,
(
)
,
(
t
x
w
t
x
w
t
x
w
rel
b
(24)
)
,
(
t
x
w
b
rep
r
e
s
ent
s the di
spla
cem
ent of the fixed end,
)
,
(
t
x
w
rel
is the
transve
rse di
spla
cem
ent t
o
the fixed end.
)
(
)
(
)
(
)
(
)
,
(
2
1
t
h
x
t
g
x
t
x
w
b
(25)
)
(
1
x
and
)
(
2
x
are tran
slate affect fu
nction a
nd rotate
affect fun
c
tion. For
ca
n
t
ilever whi
c
h
has
no lumpe
d
m
a
ss on the en
d, the partial
equatio
n of it
can b
e
sh
own as bel
ow.
)
(
)]
,
(
[
)
,
(
)
,
(
2
2
4
4
t
F
t
x
w
x
b
t
t
x
w
m
x
t
x
w
YI
b
m
rel
rel
(26)
If
)
(
t
F
=0 i
n
eq
uat
ion, the eq
ua
tion turn i
n
to
corre
s
p
ondin
g
homo
gen
e
ous
equ
ation
and
can
be
solved by variable
s
se
para
t
ion. Gene
ral
solutio
n
of
de
flection is
su
perp
o
sitio
n
of every prin
cip
a
l
vibration mo
d
e
and can be
written a
s
bel
ow.
1
)
(
)
(
)
,
(
r
r
r
rel
t
x
t
x
w
(27)
After derivation,
)
(
)]}
(
)
(
2
)
(
)[
(
{
2
2
2
2
2
1
t
f
t
dt
t
d
dt
t
d
x
r
r
r
r
r
r
r
r
(28)
Whe
r
e
，
m
b
m
r
r
2
，
m
YIk
r
r
4
2
，
m
t
F
t
f
)
(
)
(
If
)
(
x
r
is reg
u
lar
mode, the vibration mo
de satisfies:
0
)
(
)
(
4
4
4
x
k
dx
x
d
r
r
r
(29)
From
)
(
)]}
(
)
(
2
)
(
)[
(
{
2
2
2
2
2
1
t
f
t
dt
t
d
dt
t
d
x
r
r
r
r
r
r
r
n
.
Ac
c
o
rding to formula
)
(
)]}
(
)
(
2
)
(
)[
(
{
2
2
2
2
2
1
t
f
t
dt
t
d
dt
t
d
x
r
r
r
r
r
r
r
n
, it can
be obtaine
d as:
)
(
)
(
)
(
2
)
(
2
2
2
2
2
t
N
t
dt
t
d
dt
t
d
r
r
r
r
r
r
r
(30)
Modal excitati
on functio
n
of
r
th is
:
)]
(
)
(
[
)
(
t
a
t
a
m
t
N
r
w
w
r
r
(31)
dx
x
L
x
r
w
r
0
)
(
and
dx
x
L
x
r
r
0
)
(
are
r
th integration
con
s
tant.
2
2
)
(
)
(
dt
t
g
d
t
a
w
2
2
)
(
)
(
dt
t
h
d
t
a
are tran
slate
and rotate vel
o
city re
spe
c
ti
vely. If bas
e tran
slate a
nd
rotate are
arbi
trary fun
c
tion,
it
can b
e
obtain
ed by Duh
a
m
e
l integratio
n.
d
t
e
t
N
t
rd
t
t
r
rd
r
r
r
)
(
sin
)
(
1
)
(
)
(
0
(32)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Structu
r
e and
Mecha
n
ical Analysis of Si
ngle
Cantilever Piezoele
c
tric Energy… (Jie Zha
o
)
6487
rd
is dampin
g
freque
ncy of
r
th mode.
r
r
rd
1
,and
r
is dampi
ng ratio of
r
th mode.
1
)
(
0
)
(
sin
)
(
)
(
)
,
(
r
rd
t
t
r
rd
r
rel
d
t
e
t
N
x
t
x
w
r
r
(33)
3.3. Forced
Vibration So
lution tha
t
w
i
thout Lump
e
d Mass on
the Beam En
d
Back to
non
-homog
ene
ou
s e
quatio
n, fo
cu
s o
n
the
lo
we
st o
r
de
r m
ode
(
1
r
) that
clo
s
ely
relate
s to ene
rgy harve
ster
[10].
t
i
t
i
e
Z
dt
e
Z
d
m
t
F
t
f
1
1
2
1
0
2
0
2
)
(
)
(
)
(
(34)
Acco
rdi
ng to Equation (28), the following
equation
can
be got.
t
i
e
Z
t
t
t
x
1
2
1
0
1
2
1
1
1
1
1
1
)]
(
)
(
2
)
(
)[
(
(35)
That is,
)
sinh(
)
sin(
)]
sin(
)
cosh(
)
sinh(
)
[cos(
2
1
)
(
1
1
1
1
1
1
1
L
k
L
k
L
k
L
k
L
k
L
k
mL
L
(36)
Acco
rdi
ng to Equation (30)
, the resp
onse of vibration mode is
cha
n
ged a
s
belo
w
.
t
i
w
r
r
r
r
r
r
r
e
Z
m
t
d
t
t
d
d
t
t
d
0
2
2
2
2
2
2
)
(
)
(
2
)
(
(37)
By way of Duhamel integ
r
a
t
ion,
t
i
r
r
r
w
r
r
e
Z
i
m
t
0
2
2
2
2
)
(
(38)
Substitute the
integration
consta
nt of
r
th mode,
m
L
L
k
dx
x
r
L
x
r
w
r
r
0
r
2
)
(
(39)
t
i
r
r
r
r
r
e
Z
i
L
k
mL
t
0
2
2
2
r
)
2
(
r
2
)
(
(40)
Substitute the
above into
1
)
(
)
(
)
,
(
n
r
r
rel
t
x
t
x
w
, the re
sp
on
se
of cantileve
r
unde
r force
d
vibration can
be obtain
ed.
1
2
2
r
r
0
2
)
2
(
)]}
sinh(
)
[sin(
r
)]
cosh(
)
{[cos(
r
2
)
,
(
r
n
n
r
r
r
r
r
r
t
i
rel
i
L
k
x
k
x
k
x
k
x
k
e
Z
t
L
w
(41)
3.4. Model Take th
e Lum
ped Mas
s
into Acco
unt
The
cantileve
r that previou
s
ly cal
c
ul
ated
is
unifo
rm
cross sectio
n a
nd ha
s n
o
lu
mped
mass. Nevertheless, for decrea
s
e n
a
t
ural fr
equ
en
cy and dimi
nish dim
e
n
s
i
on of energ
y
harvester, a lumped
mass will
be
attached
on the end of
be
am. The vibration m
ode
and
eigenvalu
e
are not appli
c
a
b
le for this m
odel [9]. If
the lumped ma
ss is
'
m
as sho
w
n in Figure 9.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 9, September 20
14: 54
79 – 649
3
6488
Figure 9.
Single Ca
ntilever with Lumpe
d
Mass
Acco
rdi
ng to
Equation
(26
)
and (28), fo
r singl
e
ca
ntilever with l
u
m
ped ma
ss, the partial
equatio
n ca
n be written a
s
belo
w
.
2
2
2
2
4
4
)
,
(
)]
(
'
[
)
,
(
)
,
(
t
t
x
w
L
x
m
m
t
t
x
w
m
x
t
x
w
YI
b
rel
rel
(42)
The corre
s
po
nding Eig
env
alue
s will
ch
ange
with re
spe
c
t to the
cha
nge of o
r
t
hogo
nal
con
d
ition. Th
e vibration m
ode fun
c
tion
will ch
ange a
s
belo
w
.
)]
)
(
sinh
-
)
(
(s
s
)
(
cos
)
(
cosh
[
)
(
r
r
x
k
x
k
in
x
k
x
k
C
x
r
r
r
r
r
(43)
)]
(
sinh
)
(
[sin
'
)]
(
cos
)
(
mL[cosh
)]
(
cos
)
(
[cosh
'
)]
(
sinh
-
)
(
mL[s
s
r
L
k
L
k
Lm
k
L
k
L
k
L
k
L
k
Lm
k
L
k
L
k
in
r
r
r
r
r
r
r
r
r
r
(44)
If
L
k
r
, eigenvalu
e
of the vib
r
a
t
ion mod
e
ca
n sati
sfy the
equatio
n a
s
belo
w
. It is a
transce
nde
ntal equatio
n a
nd ca
n be sol
v
ed only by numeri
c
al met
hod
s.
0
)
h
cos
cos
1
(
'
)
c
s
sin
cosh
(
'
)
cosh
s
s
(cos
'
cos
cosh
1
4
2
4
3
3
L
m
I
m
os
inh
mL
I
m
in
inh
mL
m
t
t
(45
)
t
I
is
Mome
nt
of ine
r
tia to t
he
cent
roid,
all fre
quen
cie
s
of
every
m
ode
ca
n b
e
get by
solve
s
the ab
ove equatio
n.
Similarly, accordin
g to equ
ation,
)
(
)
(
)
(
2
)
(
2
2
2
2
2
t
N
t
dt
t
d
dt
t
d
r
r
r
r
r
r
r
(46)
dx
t
t
x
w
L
x
m
m
x
t
N
b
L
r
r
2
2
0
)
,
(
)]
(
'
[
)
(
)
(
(47)
The re
sp
on
se
of the lumped mass on th
e
end of the cantilever is
sh
own a
s
bel
ow.
1
)
(
0
)
(
sin
)
(
)
(
)
,
(
r
rd
t
t
r
rd
n
rel
d
t
e
t
N
x
t
x
w
r
r
(48)
4
.
D
i
f
f
e
r
e
n
ce o
f
T
w
o M
o
de
ls
Spring
-vibrator mod
e
l is
suppo
se
d to b
e
si
ngl
e deg
ree free
dom
system; actual
ly, it is
aco
n
tinuo
us
syste. If the singl
e deg
re
e freed
om m
odel is
used,
the error i
s
inevitable. Th
us,
compare them will correct
the system theoreti
c
ally.
Evaluation Warning : The document was created with Spire.PDF for Python.