Int
ern
at
i
onal
Journ
al of
P
ower E
l
ectr
on
i
cs a
n
d
Drive
S
ystem
s
(
IJ
PEDS
)
Vo
l.
12
,
No.
2
,
Jun
2021
,
pp.
695
~
7
02
IS
S
N:
20
88
-
8694
,
DOI: 10
.11
591/
ij
peds
.
v12.i
2
.
pp
695
-
7
02
695
Journ
al h
om
e
page
:
http:
//
ij
pe
ds
.i
aescore.c
om
A study
on mod
eli
ng of a
piezoel
ectric mot
or
Youssef
Ba
ba
1
, Mos
taf
a
Bo
u
zi
2
1
Univer
sity
Hass
an
II
,
ENSA
M, 20670 Ca
sabl
an
ca
,
Morocc
o
2
Univer
sity
Hass
an
I, FS
T, 26000
Settat, Morocc
o
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
J
un
2
7
, 2
0
20
Re
vised
M
a
r
1
,
20
21
Accepte
d
M
ar
21
, 20
21
In
thi
s
pap
er,
a
study
on
modeli
ng
of
a
p
iezoe
l
e
ct
ri
c
tr
ansduc
er
type
rot
ary
tra
ve
li
ng
wave
ult
rasoni
c
mot
or
(US
M)
is
pre
s
ent
ed
.
First
a
m
at
he
ma
t
ic
a
l
mode
l
and
numerical
simul
at
ion
result
s
are
ac
h
ieved.
Th
e
mod
el
is
base
d
o
n
the
th
eor
y
of
p
i
ez
oe
le
c
tricit
y
an
d
physic
th
eor
y.
An
expe
ri
me
n
t
al
mod
el
is
worked
out
and
com
par
ed
to
t
he
nume
r
ic
a
l
m
odel
.
The
infl
u
enc
e
of
the
te
mp
era
tur
e
on
cha
ra
cteri
sti
cs
s
uch
as
th
e
rotat
iona
l
spee
d
of
t
he
m
otor
is
conside
red
.
Th
e
spee
d
of
th
e
US
M
is
me
asure
d
a
t
temper
at
ure
be
twee
n
17°C
and
50°C.
To
de
vel
op
sui
ta
bl
e
c
ontrol
str
at
eg
ie
s
for
the
driv
e,
a
f
uzz
y
mode
l
type
T
aka
gi
-
Su
geno
is
used.
The
unknown
par
amete
rs
of
the
outpu
t
me
mb
ership
fun
ct
ions
are
d
et
er
mi
ned
by
least
s
quar
e
method.
E
xper
imenta
l
dat
a
are
used
t
o
ex
amine
th
e
val
idity
of
th
e
fuz
zy
mod
el
.
C
ompa
rison
bet
wee
n
expe
ri
me
nt
al
and
c
al
c
ula
t
ed
d
at
a
of
th
e
fu
zz
y
mod
el
i
ndic
a
te
s
that
the
fu
zz
y
mode
l
ca
n
w
el
l
desc
ri
be
the
nonli
n
ea
r
cha
r
ac
t
eri
sti
cs
am
ong
t
h
e
fre
quenc
y
of
dr
i
ving
voltage
and
rotating
sp
ee
d.
Ke
yw
or
d
s
:
Fu
zz
y
lo
gic
M
odel
in
g
Tempe
ratu
re e
f
fect
Ultraso
nic m
ot
or
US
R6
0
This
is an
open
acc
ess arti
cl
e
un
der
the
CC
BY
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
Yousse
f
Ba
ba
Un
i
ver
sit
y Ha
s
san II
ENSA
M
, 2
0670 Casa
blanca
M
or
occo
Emai
l:
yous
se
f
.b
a
ba@gmail
.c
om
1.
INTROD
U
CTION
Ultraso
nic
m
ot
or
s
a
re
a
ty
pe
of
act
uat
or
wh
ic
h
hav
e
s
ome
excell
e
nt
performa
nces
su
c
h
as
high
ho
l
ding
tor
qu
e,
high
to
rqu
e
at
l
ow
s
pe
ed,
qu
ie
t
ope
rati
on,
simple
str
uctu
re,
c
ompact
siz
e
a
nd
no
el
ect
ro
ma
gn
et
i
c interfe
re
nce
[
1].
Du
e
t
o
su
c
h
c
har
act
erist
ic
s,
ul
traso
nic
m
oto
rs
ha
ve
bee
n
in
man
y
pr
act
ic
al
app
li
cat
ions,
su
c
h
as
i
n
rob
ots,
me
dical
instr
um
e
nts,
cameras,
ae
r
onautic
s,
M
E
MS
a
nd
ma
ny
ot
he
rs.
P
recise
t
he
or
et
ic
al
mode
li
ng
of
mo
ti
on
an
d
en
ergy
co
nversi
on
i
n
ultras
onic
m
otors
is
of
gr
eat
interest
bo
t
h
from
t
he
or
et
ic
al
a
nd
pract
ic
al
view
po
i
nts.
O
ne
of
t
he
m
os
t
interest
ing
t
ypes
of
m
otors
is
a
travell
in
g
w
ave
ultras
onic
mo
to
r.
Howe
ve
r,
it
is
diff
ic
ult
to
de
r
ive
co
mp
le
x
mathemat
ic
al
model
of
the
USM.
More
ov
er,
the
c
ontr
ol
char
act
e
risti
cs
of
t
he
mo
to
r
a
re
high
ly
no
nlinear
.
The
e
xact
val
ues
of
m
otor
par
a
mete
rs
a
re
diff
ic
ult
to
obta
in
an
d
th
ey
are
ti
me
-
va
r
yi
ng
du
e
t
o
increase
i
n
te
m
per
at
ur
e
a
nd
c
hanges
i
n
m
oto
r
dr
i
ve
op
e
rati
ng
c
onditi
ons
su
c
h
as
dri
vi
ng
fr
e
qu
e
nc
y,
a
nd
loa
d
tor
qu
e
.
S
eve
ral
at
te
mp
ts
ha
ve
been
ca
rr
ie
d
out
to
dev
el
op
t
he
model
of
th
e
m
otor.
The
e
qu
i
valent
ci
rcui
t
was
widely
us
e
d
on
ultraso
nic
m
otor
m
odel
ing
[1
]
-
[6
].
B
ut
thi
s
typ
e
of
m
ode
l
is
a
rude
modeli
ng
meth
od
of
t
he
dynamic
c
harac
te
risti
cs
of
ultraso
nic
m
oto
r
a
nd
ca
nnot
be
us
e
d
to
desig
n
m
otor’
s
c
o
nt
ro
ll
er
directl
y.
Don
gs
he
ng
Z
ha
ng,
S
hiyu
Wa
ng
a
nd
Jie
Xiud
[
7
]
de
velo
pe
d
an
a
nalytic
al
dynamic
m
od
el
of
ri
ng
ultra
so
nic
mo
to
r.
Ba
se
d
on
t
his
m
odel
,
the
piezo
e
le
ct
ric
par
ame
tric
eff
ect
s
on
the
wa
ve
di
stortio
n
an
d
c
on
ta
ct
mecha
nics
a
re
examine
d.
I
n
[
8]
-
[10
]
dif
fer
e
ntial
ev
olu
ti
on
al
gorithm
is
us
ed
for
mo
t
or’s
model
ide
ntific
at
ion
.
The
al
go
rithm
reduces
t
he
t
ime
require
d
i
n
the
proces
s
of
ide
ntific
at
ion
.
M
a
ny
re
s
earche
rs
c
onsidere
d
mecha
nical
ch
aracte
risti
c
of
con
ta
ct
fr
ic
ti
onal
la
yer
a
nd
ca
lc
ulate
d
dis
pl
a
cement
a
nd
spe
ed
of
t
he
r
otor
usi
ng
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
2
,
J
une
2021
:
695
–
702
696
finite
el
ement method [
11]
-
[13
]
.
I
n
[
14]
-
[
1
6
]
dyna
mic
co
nta
ct
prob
le
ms b
e
tween
a
rigi
d
r
otor
an
d
de
for
mable
sta
tor
we
re
ap
pro
ximate
ly
s
olv
e
d.
Du
e
to
the
li
mit
at
ion
s
of
the
oret
ic
al
mo
delin
g,
c
ontr
ol
desig
n
usual
ly
adopts
ide
ntific
at
ion
meth
od
base
d
on
t
he
t
est
ed
data.
Ac
cordin
g
to
the
diff
e
re
nt
ide
ntific
at
ion
meth
ods,
the
forms
of
m
od
el
can
be
the
t
ran
s
fer
f
un
ct
i
on,
dif
fer
e
ntial
equ
at
io
n,
ne
ur
al
netw
ork,
a
nd
s
o
f
ort
h.
I
n
recent
year
s
,
f
uzz
y
modeli
ng
met
hod
base
d
on
fu
zz
y
reas
on
i
ng
is
gr
a
dual
ly
arisen
.
T
he
s
ame
as
in
the
ne
ur
al
netw
ork
m
odel
,
f
uzzy
model
i
s
al
so
ba
sed
on
exp
e
rime
ntal
data,
eas
y
to
s
how
the
no
nlin
ear
inf
ormat
io
n.
T
he
fu
zz
y met
hod i
s m
os
tl
y use
d
t
o real
iz
e sp
ee
d an
d posi
ti
on
c
on
t
ro
l
[1
7]
-
[
1
9
].
This
pa
per
pr
e
sents
both
a
nume
rical
m
od
e
l
an
d
a
fuzzy
modeli
ng
meth
od
b
ase
d
on
Takagi
-
Sugen
o
rea
soning
is
use
d
to
obta
in
the
m
od
el
of
ultr
aso
nic
m
otor.
The
Taka
gi
-
Sugeno
m
od
el
is
c
hose
n
t
o
ma
ke
t
he
pro
po
se
d
mod
el
easy
t
o
us
e
on
co
ntr
ol
design.
T
he
outp
ut
of
Ta
kag
i
-
S
uge
no
m
od
el
is
a
li
near
mathem
at
ic
al
expressi
on.
C
ontrari
wise,
the
M
am
da
ni
m
od
el
giv
e
s
a
f
uzz
y
outp
ut.
The
pro
po
se
d
w
ork
co
ns
i
der
s
the
eff
ect
of
te
m
pe
ratu
re
.
This
w
ork
will
be
ve
ry
us
e
f
ul
f
or
t
he
op
ti
mal
desig
n,
re
du
ci
ng
the
hea
t
loss
,
pr
ese
r
vin
g
the
desire
d
r
otar
y
sp
ee
d,
an
d
im
pro
veme
nt
of
pr
eci
se
co
ntr
ol
of
U
SM.
Thi
s
met
hod
is
ba
sed
on
f
uzz
y
m
od
el
structu
re
a
nd
exp
e
rime
ntal
measu
reme
nts.
The
data
c
onta
ins
the
dr
i
vin
g
f
reque
ncy,
the
te
m
per
at
ure
an
d
ro
ta
r
y
s
pee
d
of
ultraso
nic
m
oto
r
.
The
le
ast
s
qu
a
res
met
hod
is
us
ed
to
i
dent
ify
the
un
known
par
a
mete
rs
of
the
con
cl
us
io
ns
of
fu
zz
y ru
le
s
.
This
pap
e
r
is
orga
nized
as
i
n
Sect
io
n
2,
a
mathe
mati
cal
model
is
pr
ese
nted
.
Si
mu
la
ti
ons
are
achieve
d
on
MATL
AB.
I
n
Se
ct
ion
3,
ex
per
i
ments
are
perf
ormed
to
ve
rif
y
the
ef
fecti
ve
ness
of
the
a
na
lyti
cal
model.
T
he
e
ffec
t
of
te
mp
e
rat
ur
e
is
disc
us
se
d.
A
n
a
dap
ti
ve
f
uzzy
m
od
el
i
s
in
vestigat
ed
in
Sect
io
n
4,
ta
king
into acc
ount t
he
mo
t
or’s n
on
-
li
near
it
y.
Final
ly,
c
on
cl
us
io
ns o
f
this
w
ork
a
r
e presente
d
in
Sect
ion
5.
Table
1.
Desig
n
s
pecifica
t
io
ns an
d
sim
ulati
on
par
a
mete
rs
of USR
60
Item
s
Descripti
o
n
s
Valu
es
Sp
eed
Po
wer
Su
p
p
ly
f
requ
en
cy
Su
p
p
ly
vo
lta
g
e
Rated
sp
eed
Rated
po
wer
Load
torq
u
e
40
-
4
2
k
Hz
100V
rms
9
0
rpm
3w
0
-
0
.32
Nm
m
R
Stato
r
m
ass
Ro
to
r
m
ass
1
0
.1e
-
3
Kg
30e
-
3
Kg
,
Ro
to
r
d
am
p
in
g
Stato
r
d
am
p
in
g
Stiff
n
ess
Ap
p
lied
f
o
rce
Moment o
f
inertia
o
f
th
e r
o
to
r
5%
1
5
.4Ns
/m
5
.95
e+8
N/m
160N
7
.2e
-
6
Kg
m
2
Θ
Co
u
p
lin
g
f
acto
r
0
.22
6
3
N/V
2.
USM
MO
DE
LING
In
[
20
]
a
n
e
qu
i
valent
mec
han
ic
al
t
wo
-
m
od
e
a
ppr
ox
im
at
ion
is
giv
e
n,
r
ep
rese
nting
the
m
od
a
l
amplit
udes
1
,
an
d
2
,
of
the
sine
-
an
d
c
os
i
ne
-
m
od
e
of
the
vi
br
at
in
g
sy
ste
m
(stat
or
a
nd
c
eramic),
see
Fig
ure
1.
The
par
a
mete
r
des
cribes
the
m
od
al
mass
of
sta
tor
a
nd
ce
ramic
,
a
nd
represe
nt
s
the
str
uctu
ra
l
dam
ping.
and
ref
le
ct
the
eq
ui
valent
mecha
ni
cal
sti
ff
ness
of
the
sta
tor
a
nd
ceramic
,
res
pe
ct
ively.
1
and
2
are
dis
placem
ents
w
hich
a
re
pro
portion
al
t
o
the
el
ect
ric
currents
flo
wing
i
nto
t
he
cer
amic
la
yer
.
C
r
os
s
-
couplin
g
betw
een
the
tw
o
m
od
e
s
is
r
ep
rese
nted
by
the
s
mall
disturba
nc
es
1
,
2
.
The
deter
minati
on
of
t
he
model
par
a
mete
rs
ca
n
be
base
d
on
e
nerg
y
c
onside
rati
ons
or
detai
le
d
finite
el
ement
a
nalys
is
of
the
sta
to
r
[
21
]
-
[
24
]
.
Figure
1. M
ec
ha
nical
m
od
el
f
or the
(stat
or +
ceramic
) of
U
SM
1
,
t
w
2
,
t
w
1
y
1
y
2
y
2
y
C
c
C
c
S
c
S
c
S
1
d
S
2
d
1
e
C
c
2
e
C
c
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
A stu
dy on m
odel
ing of
a piez
oelect
ric
motor
(
Y
oussef B
aba
)
697
2.1.
Mathem
ati
cal
mo
d
el
An
ultraso
nic
mo
to
r
us
e
s
me
chan
ic
al
vi
br
at
ion
s
in
the
ultr
aso
nic
ra
nge
as
it
s
dr
ive
s
ourc
e.
A
volt
age
is app
li
ed
to
t
he
p
ie
zoelec
tric
ceramic
eleme
nts th
rou
gh
t
w
o
el
ect
r
od
es
to gene
rate al
te
rnat
ing
e
xp
a
ns
i
ons a
nd
con
t
racti
on
s
i
n
the
ce
ramic
body.
T
he
ma
gnit
ud
e
of
t
hes
e
os
ci
ll
at
ion
s
is
of
the
orde
r
of
1µm.
In
order
to
ob
ta
in a
highe
r
g
ai
n; the reso
nan
ce e
ff
ect
of the ceramics i
n
the
ultraso
ni
c range is u
se
d. The
s
pecifica
t
ion
s
of
the
U
SR6
0
-
obje
ct
of
t
his
stu
dy
-
a
re g
ive
n
in
Tab
le
1
.
T
he
st
at
or
o
f
USM
is
co
ns
ide
red
as a
piezoelec
tric
p
la
te
,
po
la
rized i
n
it
s
thick
ness dire
ct
ion
[1].
Wh
e
n
t
wo
sp
e
ci
fic
phase
vo
l
ta
ges
are
ap
plied
to
the
piez
oelect
ric
el
em
ents,
a
t
rav
el
li
ng
wa
ve
i
s
gen
e
rated
on
the
s
urface
of
t
he
sta
to
r.
T
he
travell
ing
wa
ve
pro
duces
vibrat
ion
with
a
n
el
li
ptic
locus
on
th
e
su
r
face
of
the
sta
tor.
T
his
sta
tor
vi
br
at
io
n
is
transfo
rme
d
into
a
r
otati
on
t
hro
ugh
f
rict
ion
con
ta
ct
bet
we
en
the
sta
tor
a
nd
r
otor.
The
ro
t
or
is
presse
d
agai
nst
the
sta
tor
by
a
disc
sprin
g
(
160N)
.
The
con
ta
ct
area
be
tween
sta
tor
a
nd
r
otor
m
ov
e
s
with
th
e
tra
velli
ng
w
ave.
T
he
r
oto
r
is
modele
d
by
a
rigid
strip
[
1].
I
n
deali
ng
wi
th
the
dynamics
of
t
he
r
otor,
t
wo
de
gr
ees
of
f
reedo
m
m
us
t
be
ta
ke
n
i
nto
acc
ou
nt
:
first
the
r
otati
on
of
t
he
ro
t
or
a
nd
seco
nd
the
mot
ion
in
directi
on,
as
sho
wn
by
[
25
].
T
he
dyna
mic
s
of
t
he
ve
rtic
al
ro
t
or
mo
ti
on
is
obta
ined
thr
ough the
forc
e eq
uili
br
iu
m
in
directi
on
.
̈
+
̇
=
−
(
1
)
with
is
the
ma
ss
of
th
e
r
otor,
an
d
is
an
a
ppr
opriat
e
da
mp
i
ng
te
r
m
of
t
he
ve
rtic
al
mo
ti
on.
The
ap
plie
d
axial
load
,
,
is
a
sy
ste
m
i
nput,
and
is
the
for
ce
at
the
co
nta
ct
area.
T
he
e
quat
ion
of
r
otati
on
al
mo
ti
on
is
cal
culat
ed by
.
̈
+
̇
=
−
(
2
)
w
he
re
is
the
app
li
ed
tor
que
(mec
han
ic
al
l
oad),
de
note
s
the
da
mp
i
ng
in
sp
i
nn
i
ng
di
recti
on
a
nd
represe
nts
the
mo
me
nt
of
i
ne
rtia
of
t
he
r
otor.
T
he
te
rm
is
the
f
rict
ion
al
pressu
re
at
the
con
ta
ct
ar
ea.
T
he
te
rm
is
small
, but
not ne
gligi
ble
. T
he d
yn
a
mics o
f
the
stat
or can
b
e
d
esc
r
ibed b
y
∙
̈
+
∙
̇
+
∙
=
∙
(
3
)
wh
e
re
th
e
pa
r
amet
er
desc
ribes
the
m
od
el
mass
vecto
r
of
sta
tor
a
nd
ceramic
,
re
flect
s
the
str
uctu
ral
dam
ping
vecto
r,
a
nd
re
pr
ese
nts
the
eq
uiv
al
ent
mec
ha
nical
sti
ffness
vect
or
of
t
he
sta
to
r
an
d
ce
ramic.
The
vecto
r
is
the
phase
volt
ages
,
and
re
presents
the
el
ect
rome
chan
ic
al
co
upli
ng
mat
rix
co
nsi
der
in
g
the
cr
oss
excit
at
ion
.
In
order
t
o
co
ns
i
der
the
i
nterac
ti
on
bet
ween
t
he
sta
tor/
ro
t
or
-
con
ta
ct
,
E
quat
ion
(3)
re
flect
ing
t
he
sta
tor’
s
dy
nam
ic
s is to e
xten
d b
y
a
nonline
ar
modal
force
ve
ct
or
yieldi
ng
.
∙
̈
+
∙
̇
+
∙
=
∙
+
(
4
)
Hen
ce
,
the
dy
namics
of
the
mo
to
r
is
c
omp
le
te
d.
A
n
i
nteg
rati
on
of
this
modeli
ng
ap
pr
oach
base
d
on
cl
os
ed
form sol
ution
s
for
the
contac
t
mecha
nism i
nto
the
sim
ulati
on
m
odel
is fe
as
ible.
2.2.
Simul
at
i
on
re
sults
The
im
pleme
ntati
on
is
perf
ormed
on
M
A
TLAB
a
nd
t
he
par
a
mete
rs
of
USM
a
re
s
cheduled
in
Table
1.
T
he
t
empe
rature
of
the
mo
t
or
is
suppose
d
to
be
20°C.
T
he
s
pee
d
of
t
he
m
otor
ha
s
it
s
ma
xim
um
at
the
mecha
nical
res
onant
f
re
quency.
It
is
du
e
to
th
e
fact
t
hat
the
re
volvi
ng
sp
ee
d
of
t
he
m
otor
is
propo
rtion
a
l
to
t
he
vibrat
io
n
f
or
ce
of
piezoelec
tric
el
e
ments.
S
o,
a
ny
de
viati
on
fro
m
t
his
fr
e
quen
cy
de
gr
a
des
th
e
m
oto
r
performa
nce.
Howe
ver, this
eff
ect
see
ms
m
or
e
ser
i
ou
s
for
fr
e
qu
e
nc
y decr
ements.
To
a
vo
i
d
th
ese
phen
om
e
na
,
s
peed
of
USM
was
c
on
t
ro
ll
ed
in
40
-
42
kH
z
fr
e
qu
e
nc
y
rang
e.
Fig
ure
2
sh
ows
s
peed
-
f
r
equ
e
nc
y
c
ha
ra
ct
erist
ic
of
USM
unde
r
dif
fere
nt
loa
d
to
rque
s.
S
pee
d
-
fr
e
qu
ency
c
har
act
e
risti
c
of
USM
is
not
li
ne
ar.
R
otar
y
spe
ed
dec
reases
as
the
loa
d
t
orqu
e
inc
reases
.
As
menti
one
d,
va
riat
ion
s
in
dr
i
vin
g
fr
e
qu
e
nc
y
cha
ng
e
s
sp
ee
d
of
USM.
Fi
g
ure
3
re
pr
ese
nts
sp
ee
d
-
to
r
qu
e
c
har
act
erist
ic
of
USM
for
dr
iving
fr
e
qu
e
nc
y
=
4
0
kH
z
. Wh
en
the
loa
d
t
orqu
e
ex
cee
ds 0
.
32Nm, the
m
odel
b
ec
ome
s i
mpreci
se.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
2
,
J
une
2021
:
695
–
702
698
Figure
2. S
pee
d
-
fr
e
quenc
y
c
ha
racteri
sti
c
s of
USM
Figure
3. S
pee
d
-
t
orqu
e
ch
a
ra
ct
erist
ic
U
S
M
3.
EXPERI
MEN
TAL D
ATA
A
ND R
ES
ULT
S
To
obta
in
the
t
empe
rature
a
nd
t
he
pe
rfo
rma
nce
cha
racteri
s
ti
cs
of
t
he
m
oto
r
,
a
n
e
xp
e
rim
ent
set
up
is
est
ablished
.
Th
e
m
oto
r
is
al
im
ented
by
t
he
t
wo
-
phase
vo
lt
a
ges
with
90°
phase
sh
ift.
T
he
te
ens
y
de
velo
pm
e
nt
bo
a
r
d
r
unni
ng
at
72
M
Hz
an
d
the
dr
ive
r
c
ircuit
are
util
iz
ed
to
gen
e
rate
,
am
plifie
r,
a
nd
filt
er
t
he
a
pp
li
e
d
vo
lt
age
s
of
t
he US
M
.
A
te
m
per
at
ur
e
sens
or
(L
M3
5)
is
us
ed
to
m
easur
e
the
te
m
per
at
ur
e
of
the
sta
tor
i
n
the
te
mp
e
rature
range
of
17°C
to
50°C.
A
hi
gh
est
sp
ee
d
of
the
USM
ca
n
be
obta
ine
d
w
he
n
th
e
sta
t
or
is
e
xcite
d
by
the
resona
nt
f
re
quency.
T
he
dri
vi
ng
f
reque
nc
y
i
s
meas
ur
e
d
by
os
ci
ll
os
co
pe.
A
ca
pacit
ive
e
ncode
r
(
A
M
T
102
with
1024
pulse
s
pe
r
re
vo
l
ution)
is
us
ed
t
o
meas
ure
the
r
otati
ng
sp
ee
d
of
U
SM.
The
de
vel
opm
ent
boar
d
is
use
d
to
measu
re
da
ta
f
rom
both
t
he
posit
ion
a
nd
te
mp
e
ratur
e
sen
s
or
s
.
T
he
ex
pe
rimental
be
nc
h
a
nd
it
s
blo
c
k
di
agr
a
m
are s
how
n
in
F
ig
ure
4.
A
c
omparis
on
betwee
n
sim
ul
at
ion
an
d
e
xp
erimental
re
su
l
ts
for
s
peed
-
f
r
equ
e
nc
y
c
har
a
ct
erist
ic
of
USM
unde
r
di
ff
e
ren
t
l
oad
to
rques,
is
giv
e
n
in
Fig
ure
5.
The
fig
ur
e
s
hows
that
t
he
s
imulat
ion
res
ul
ts
are
com
patible
wi
th
the
e
xp
e
rimental
val
ues.
Howe
ver,
f
or
fr
e
qu
e
ncies
a
bove
41.1
kH
z
we
noti
ce
a
sli
gh
t
dev
ia
ti
on
du
e
t
o
th
e
in
crease
in
mo
t
or
te
m
pe
ratur
e
.
T
he
m
at
la
b
model
is
no
t
ti
me
dep
e
ndent.
Ne
ver
t
he
le
ss,
durin
g
mo
t
or
op
e
rati
on
a
nd
as
ti
me
pas
s
es,
the
te
m
perat
ur
e
of
t
he
c
eramic
el
eme
nt
s
increas
es,
wh
ic
h
decr
ease
s the
USM s
pee
d.
In
order
t
o
me
asur
e
t
he
te
mpe
ratur
e
of
t
he
USM,
the
sta
t
or
is
excit
e
d
by
the
res
ona
nt
fr
e
qu
e
nc
y.
Fig
ure
6
s
how
s
the
s
peed
of
U
SM
versus
ti
me.
It
is
e
vi
den
t
that
sp
ee
d
dec
reases
ac
cordin
g
to
the
ti
me
dep
e
ndent
te
m
per
at
ur
e
rises.
Fig
ure
7
sho
w
s
the
meas
ur
e
d
te
mpe
ratu
re
ver
s
us
ti
me.
A
fter
13
min
utes,
th
e
mo
to
r
te
m
per
a
ture
e
xceed
s
50°C.
T
he
te
m
per
at
ur
e
-
ti
me
curve
is
al
most
li
near
.
I
n
te
rms
of
it
s
ef
f
ect
on
resona
nt
fr
e
quency,
howe
ve
r,
te
mp
e
ratu
re
c
hange
is
a
m
ore
imp
ort
ant
fac
tor
t
ha
n
dime
nsi
on
al
preci
sio
n.
Th
e
eff
ect
of
te
m
pe
ratur
e
is
sho
wn
i
n
Fi
g
ure
8
.
F
r
om
this
fi
gure,
it
is
co
nc
lud
e
d
that
th
e
resona
nt
f
re
qu
e
nc
y
-
te
mp
erat
ur
e
c
ha
racteri
sti
c
is
li
near
f
r
om
20°
C
to
50°C,
a
nd
it
can
be
ap
pr
ox
imat
e
d
by
a
li
near
eq
uatio
n
[
26]
-
[27
]
. In
[
1] the
auth
or
s
af
firme
d
that t
his c
harac
te
risti
c is li
near
fr
om
-
10°C
to 55°C
.
Figure
4. Ex
pe
rimental
ben
c
h an
d
it
s
blo
c
k d
ia
gr
am
20
40
60
80
1
0
0
1
2
0
1
4
0
4
0
.
0
0
4
0
.
5
0
4
1
.
0
0
4
1
.
5
0
4
2
.
0
0
Sp
eed
(r
p
m
)
Drr
iv
e
Frequ
en
cy
(kHz
)
Sim
u
latio
n
(0.0
Nm
)
Sim
u
latio
n
(0.1
Nm
)
Sim
u
latio
n
(0.2
Nm
)
Sim
u
latio
n
(0.3
Nm
)
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
A stu
dy on m
odel
ing of
a piez
oelect
ric
motor
(
Y
oussef B
aba
)
699
Figure
5. Com
par
is
on b
et
wee
n
sim
ulati
on a
nd expe
riment
resu
lt
s
Figure
6. S
pee
d
-
ti
me c
ha
racteri
sti
c o
f
USM
Figure
7. Tem
per
at
ur
e
of the
USM
ver
s
us
ti
me
Figure
8. Tem
per
at
ur
e
-
de
pe
ndence
of
res
onant freq
ue
ncy
4.
ADAPTI
VE F
UZZY
MO
DE
L
The
ultras
onic
mo
to
r
has
nonl
inear
sp
ee
d
c
ha
racteri
sti
cs
w
hich
va
r
y
with
dr
i
ve
operati
ng
co
ndit
ion
s
su
c
h
as
te
m
pe
r
at
ur
e
rise
a
nd
app
li
ed
loa
d
to
rque
c
ha
ng
es
.
Fu
rt
hermo
re,
the
m
otor
has
no
mat
hemati
c
model
for
c
on
tr
ol
by
far
beca
us
e
th
e
dr
i
ve
pr
inci
pl
e
is
diff
e
ren
t
from
t
he
ge
ne
r
al
el
ect
ro
ma
gnet
ic
typ
e
m
otors.
In
this
sect
io
n,
a
fu
zz
y
model
is
pro
pose
d
to
s
olv
e
these
pr
oble
ms
from
th
e
point
of
util
it
y.
T
he
in
pu
t
of
the
fu
zz
y
m
od
el
is
the
fr
e
qu
e
nc
y
,
and
the
outp
ut
is
the
sp
eed
.
The
f
reque
nc
y
is
def
ine
d
in
the
range
f
r
om
40
to
42
kH
z
.
Lo
ok
i
ng
at
th
e
in
pu
t
-
outp
ut
data
sho
wn
in
Fi
g
ure
5,
w
e
can
see
t
hat
the
sp
ee
d
-
fr
e
qu
e
nc
y
char
act
e
risti
cs
change
as
the
fr
e
qu
e
nc
y
dec
r
eases.
F
ive
f
uz
zy
set
s
:
“
very
_
small
”,
“s
mall
”,
“m
edi
um
”,
“big”,
and “ve
r
y
_
big
”
are
def
i
ned
as
sh
ow
n
in
Fi
gur
e 9
.
W
e
h
a
ve a
model
with t
he
five
impli
cat
ion
s
.
very_
sm
all
ℎ
1
=
1
∙
+
1
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
2
,
J
une
2021
:
695
–
702
700
sm
all
ℎ
2
=
2
∙
+
2
(
6
)
ℎ
3
=
3
∙
+
3
(
7
)
ℎ
4
=
4
∙
+
4
(
8
)
very_
bi
g
ℎ
5
=
5
∙
+
5
(
9
)
wh
e
re
(i =
1…5)
is the
r
otary
sp
ee
d.
T
he five
mem
ber
s
hip f
un
ct
io
ns o
f
Fi
g
ure
9 are
giv
e
n b
y
.
1
(
)
=
(
−
(
−
40000
)
2
)
(
10
)
2
(
)
=
(
−
(
−
40500
)
2
)
(
11
)
3
(
)
=
(
−
(
−
41000
)
2
)
(
12
)
4
(
)
=
(
−
(
−
41500
)
2
)
(
13
)
5
(
)
=
(
−
(
−
42000
)
2
)
(
14
)
with
=
6
,
65
−
6
. T
hen the
mo
t
or sp
ee
d
is
obtai
ned by
de
fu
zzi
ficat
io
n.
=
∑
∙
5
1
=
(
15
)
with
=
∑
5
1
⁄
,
=
[
,
]
,
and
=
[
,
1
]
.
I
n
this
pa
per
t
he
vect
or
is
cal
culat
ed
by
the
le
ast
sq
ua
res
alg
or
it
hm
.
+
1
=
+
+
1
+
1
(
+
1
−
+
1
)
+
1
=
−
+
1
+
1
1
+
+
1
+
1
=
1
…
999
0
=
0
,
0
=
1000
∙
Figure
9. M
e
m
ber
s
hip f
unct
io
ns
of fuzz
y
set
s
Fig
ure
10
s
hows
th
e
e
xp
e
rimental
m
odel
(no
l
oad)
a
nd
the
f
uzz
y
m
od
el
of
the
s
pee
d
-
fr
e
quenc
y
char
act
e
risti
c.
Su
c
h
cha
racter
ist
ic
ind
ic
at
es
the
simi
la
riti
es
betwee
n
the
e
xperime
ntal
nonl
inear
m
od
el
a
nd
t
he
fu
zz
y
m
od
el
.
Th
is
m
od
el
ca
n
the
n
be
us
e
d
to
pe
rfo
rm
a
dire
ct
co
ntr
ol
desig
n.
With
the
ada
ptive
fu
zz
y
appr
oach,
mo
t
or’s
nonlinea
ri
ti
es
are
maske
d
by
loc
al
li
ne
arizat
ion
us
i
ng
the
Taka
gi
-
Sugeno
reas
onin
g
.
The
same c
on
cl
us
io
ns
a
re
valua
ble
for
l
oad b
el
ow 0
.
3Nm.
0
0.5
1
40
40.5
41
41.5
42
Dr
ive
Fr
equ
ency
(
kH
z)
m
1
:
very
_
sm
all
m
2
:
sm
all
m
3
:
medi
u
m
m
4
:
big
m
5
:
very
_
b
ig
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
A stu
dy on m
odel
ing of
a piez
oelect
ric
motor
(
Y
oussef B
aba
)
701
Figure
10. T
he
sp
ee
d
-
fr
e
quen
cy
c
har
act
e
risti
c
5.
CO
NC
L
US
I
O
N
In
this
pap
e
r,
a
stu
dy
on
modeli
ng
of
USM
is
prese
nted.
A
nume
rical
model
of
the
m
ot
or
is
achieve
d
.
T
he
simulat
ion
res
ults
are
co
mp
a
red
with
the
e
xperime
ntal
one
s
wh
ic
h
gi
ve
good
a
gr
ee
ment
of
the
model.
T
he
e
ffec
t
of
the
te
m
pe
ratur
e
on
t
he
char
act
e
risti
cs
of
t
he
USM
is
discusse
d.
F
ur
t
hermo
re,
t
he
e
f
fects
of
dri
ving
fr
e
qu
e
nc
y
a
nd
l
oa
d
to
rque
a
re
identifie
d.
Ba
s
ed
on
t
he
e
xperimental
data,
a
f
uzz
y
m
od
el
is
achieve
d.
T
he
model
can
wel
l
simulat
e
the
nonlinea
r
relat
ion
s
hip
a
mon
g
the
dri
ving
frequ
e
nc
y
a
nd
r
otar
y
sp
ee
d.
Re
s
ults
of
the
ex
per
im
ents
a
nd
meas
ur
e
ments
sho
w
ed
t
he
validat
ion
of
t
he
fuzzy
m
od
el
in
g.
T
he
pa
per
sh
ows
that
the
nonlinea
r
c
ha
r
act
erist
ic
s
of
U
SM
are
fit
to
be
de
scri
bed.
Further
more,
the
pr
ese
nted
meth
od
is
gen
e
ral,
a
nd
c
an
be
e
xte
nde
d
to
oth
e
r
trav
el
li
ng
wa
ve
ty
pe
m
otors.
To
make
pr
eci
se
con
t
ro
l
of
USM,
t
he
pro
po
se
d
f
uzz
y
model
ca
n
be
us
e
d.
A
f
uz
zy
PID
re
gu
l
at
or
is
fea
sibl
e
to
bala
nce
t
he
var
ia
ti
ons
in
t
he
op
e
rati
ng
c
ondi
ti
on
s
a
nd
pres
erv
e
the
desire
d
ro
ta
r
y
sp
ee
d.
As
a
f
uture
w
ork
,
f
uzzy
lo
gic
will
be
co
mbi
ned
t
o
sli
din
g
mode
t
echn
i
qu
e
to
a
chieve
a
r
obust
con
t
ro
l
sc
he
me.
T
he
mai
n
a
dv
a
ntage
of
f
uzz
y
sli
ding
m
ode
con
t
ro
l i
s t
hat t
he
c
on
t
ro
l
method ac
hieve
s a
sy
m
pto
ti
c sta
bili
ty o
f
the cl
ose
d
-
l
oop
s
ys
te
m
.
REFERE
NCE
S
[1]
T.
Sashid
a and T
.
Kenjo
,
An
int
ro
duct
ion
to
u
lt
r
asonic
mot
ors,
Oxf
ord Univ
ersity P
ress
,
1993.
[2]
Hong
Zha
ng
,
Hui
Z
hao
,
We
i
ji
a
S
hi
and
Yan
li
n
L
i
,
"
A
novel
h
al
f
-
bridge
drive
ci
rc
uit
for
u
lt
rasoni
c
mot
o
r
b
ase
d
on
FPGA,
"
2017
IEE
E
Tr
anspor
tat
i
on
Elec
tri
fi
ca
ti
o
n
Confe
ren
ce an
d
Ex
p
o
,
Asia
-
Pa
ci
fic
,
Harbi
n
,
20
17
,
pp
.
1
-
6.
[3]
Hame
d
Moj
al
l
ali
,
Rou
zbe
h
Ami
ni
,
Roo
zbe
h
Izadi
Z
am
an
aba
d
i
and
Ali
A.
Ja
la
l
i
,
"
Sys
te
m
at
i
c
mode
li
ng
for
f
re
e
stat
ors
of
rot
ary
pi
ez
oe
lectr
i
c
u
l
tra
sonic
mo
tors
,
"
IEEE/A
SM
E
Tr
ansacti
ons
on
Me
chat
roni
cs
,
v
ol.
12,
no.
2
,
pp
.
219
-
223,
2007
,
DOI
:
10.
1109/TM
ECH.
2007.
89
2829
[4]
Jian
-
Guo
Zha
ng
,
Zhi
-
Li
Long,
W
en
-
Ju
Ma,
Guan
g
-
Hao
Hu
and
Yang
-
Min
L
i
,
"
Elec
tro
me
ch
anica
l
dynam
i
cs
model
of
ult
rasoni
c
trans
duce
r
in
u
lt
r
asonic
m
ac
h
ini
ng
base
d
on
equi
va
le
nt
ci
r
cui
t
appr
oac
h
,
"
S
ensors
19,
no.
6
,
pp
.
1
-
6
,
2019
,
DOI
:
10
.
3
390/s19061405
[5]
Rola
nd
Rynd
zionek,
Łuka
sz
Sienki
ewi
cz
,
Michał
Michn
a
a
nd
Fili
p
Kut
t
,
"
Design
and
expe
ri
me
nts
of
a
pie
zo
el
e
ct
ri
c
m
otor
using
thr
ee
rot
ating
mo
de
a
ct
ua
tors
,
"
Sensors
19,
no.
23
,
pp.
1
-
16
,
2019
,
DOI
:
10.
3390/s19235
184
[6]
Přemysl
Janů
,
J
osef
Baj
er
,
Pav
e
l
Dyčka
and
Ra
dek
Bystřic
ký
,
"
Prec
ise
exp
erim
ent
a
l
determ
ina
t
ion
of
elec
tri
c
al
equi
va
le
nt
ci
r
cuit
par
am
e
te
rs
for
ult
rasoni
c
pi
ez
o
el
e
ct
ri
c
c
erami
c
tra
nsduce
rs
fro
m
the
ir
measured
cha
ra
cteri
sti
cs,
"
Ultrasonics
,
vol
.
112
,
106341,
20
21
,
DOI
:
10
.
101
6/j
.
ul
tra
s.2020
.
1
06341
[7]
Dongs
heng
Zhang
,
Shiyu
Wa
ng
and
Ji
e
Xiud
,
"
Piez
oelec
t
ric
p
ara
m
et
ri
c
eff
e
cts
on
wave
vibr
at
ion
and
conta
ct
me
ch
ani
cs
of
tra
ve
li
ng
wa
ve
ult
r
asonic
mot
or,
"
Ultr
asonics
,
vol.
81,
pp.
118
-
1
26,
2017
,
D
OI
:
10.
1016/
j
.
ult
r
as.
2017.
05.
013
[8]
J.
Shi
and
W.
H
uang,
"
Improv
ed
DEA
for
mot
or’
s
mode
l
ide
nt
ifi
c
at
ion
,
"
COMPE
L
-
The
Inte
rnat
i
onal
Journal
for
Co
mputati
on
an
d
Mathe
m
at
ic
s
i
n
E
l
ec
tri
cal
and
El
e
ct
ronic
Eng
i
nee
ring
,
vol
.
38
no.
6
,
pp
.
1846
-
1854,
2019
,
DO
I
:
10.
1108/COMP
EL
-
05
-
2019
-
018
5
[9]
S.
Jingzhuo
and
W.
Huang
,
"
Model
ref
ere
n
ce
a
dapt
iv
e
itera
ti
ve
le
arn
ing
spee
d
cont
rol
for
ult
r
a
sonic
mot
or
,
"
in
IEE
E
Acce
ss
,
vo
l.
8
,
pp
.
181815
-
181824,
2020
,
D
OI
:
10.
1109/AC
CESS
.
2020.
302
9106
[10]
S.
Lu
and
S.
Jingzhuo,
"
Nonlin
e
ar
Ham
m
erstein
mode
l
of
u
lt
ras
onic
mo
tor
for
p
ositi
on
con
trol
u
sing
diffe
r
ent
i
al
evol
uti
on
a
lgori
t
hm,
"
Ultrasonic
s
,
v
ol
.
94,
pp.
20
-
27
,
2019
,
DOI
:
10.
1016/j.ult
r
as.
2018.
12.
012
[11]
Jun
Li
u,
Zi
-
Ji
e
Niu,
Hua
Zhu
a
nd
Chun
-
Sheng
Zha
o
,
"
Design
a
nd
expe
r
im
en
t
o
f
a
la
r
ge
-
ape
r
tur
e
hol
low
tra
v
eli
ng
wave
ul
tra
sonic
mot
or
with
low
spee
d
and
high
t
orque
,
"
Appl
i
ed
Sci
en
ce
s
,
vol.
9
,
no.
19
,
pp.
1
-
16
,
sep
2019
,
DOI
:
10.
3390/a
pp919
3979
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
12
, N
o.
2
,
J
une
2021
:
695
–
702
702
[12]
Hongpeng
Y
u,
Qiquan
Quan
,
X
inqi
T
ia
n
and
He
L
i
,
"
Optim
i
zat
ion
and
an
al
ysis
of
a
u
-
shape
d
l
i
nea
r
p
iezoe
l
ectri
c
ult
rasoni
c
mot
o
r
using
longitudi
nal
tr
ansduc
er
s,
"
Sensors
(Base
l)
,
vo
l.
18
,
no
.
3
,
pp.
1
-
12
,
Mar
.
2018
,
DOI
:
10.
3390/s18030809
[13]
Dawei
An,
We
i
qing
Huang
,
W
ei
quan
L
iu,
Jin
rui
Xi
ao,
Xiao
c
hu
Li
u
and
Zh
ongwei
Li
ang
,
"
Meshing
d
rive
me
ch
ani
sm
o
f
d
ouble
tra
v
el
ing
wave
s
for
ro
ta
ry
piezoe
le
c
tri
c
m
otors
,
"
Math
emati
cs
,
vol
.
9
,
no
.
4
,
pp
.
1
-
19
,
2021
,
DOI
:
10.
3390/math9040445
[14]
Xiang
Li,
Zhi
we
i
Ch
en
and
Zhi
y
uan
Yao
,
"
Cont
ac
t
an
al
ysis
and
per
for
ma
n
ce
ev
al
ua
ti
on
of
stan
ding
-
wave
li
n
ea
r
ult
rasoni
c
mo
tor
s
via
a
physics
-
b
ase
d
con
ta
c
t
mo
del
,
"
Smar
t
Ma
t
erial
s
and
Struc
t
ures
,
vol.
28,
no
.
1,
2019
,
DOI
:
10.
1088/1361
-
6
65X/aa
f11a
[15]
We
i
ji
a
Shi
,
Hui
Zha
o
,
Ji
e
Ma
a
nd
Yu
Yao
,
"
Dea
d
-
zon
e
com
pe
nsati
on
of
an
ul
tra
sonic
mot
or
u
sing
an
ada
p
ti
v
e
dit
her
,
"
IEEE
Tr
ansacti
ons
on
Industrial
El
e
ct
roni
cs
,
vol.
65
,
no.
5,
pp.
3730
-
3
739,
2018
,
DOI
:
10.
1109/T
I
E.
20
17.
2760854
[16]
T.
Mashimo
and
K.
T
era
shim
a,
"
Expe
ri
me
nt
al
ve
rifi
c
at
ion
of
e
ll
i
pti
c
al
motion
m
odel
in
tr
ave
l
ing
wave
u
lt
rasoni
c
mot
ors
,
"
I
EEE/
ASME
Tr
ansacti
ons
on
Me
c
hatronic
s
,
vol.
20,
no
.
6
,
pp.
2699
-
27
07
,
2015
,
DOI
:
10.
1109/T
MEC
H.2015.
2392126
[17]
Shi
Jingzhuo
,
Huang
W
enwe
n
and
Zhou
Yin
g
,
"
T
–
S
fuz
zy
cont
rol
of
tr
avelli
ng
-
wav
e
u
lt
ra
sonic
mot
or
,
"
J
o
f
Control
Aut
om
El
e
ct
r Sy
st
,
vo
l.
31,
pp
.
319
-
328
,
2020
,
DOI
:
10.
1
007/s40313
-
019
-
00538
-
y
[18]
Yunlai
Shi
,
Jun
Zha
ng
,
Yuyang
Li
n
and
We
nbo
Wu
,
"
Improv
ement
of
low
-
spee
d
pre
ci
sion
cont
r
ol
of
a
but
te
rfly
-
shape
d
l
ine
ar
ult
r
asonic
mot
or
,
"
I
EEE
Acce
ss
,
vo
l.
8
,
pp
.
135131
-
135137,
2020
,
DOI
:
10.
1109/ACCESS
.
2020.
3007773
[19]
Jingwen
Le
ng
,
Peng
Pan
,
Zhi
k
e
Xu
,
Ta
o
Wa
ng
and
Long
Jin
,
"
Design
of
low
spee
d
cont
rol
le
r
f
or
tra
v
el
ing
wav
e
ult
rasoni
c
mo
tor
,
"
2019
22nd
In
te
rnational
Con
fe
renc
e
on
E
le
c
tric
al
Mac
hin
es
and
S
yste
ms
(I
CEMS)
,
Harbi
n
,
China
,
pp.
1
-
5,
2
019
,
DOI
:
10
.
11
09/ICE
MS
.
2019.
8921667
[20]
J.
Wall
asc
hek
,
"
Piez
oe
le
c
tri
c
ul
t
rasonic
mot
ors
,
"
Journal
o
f
In
telli
gen
t
Ma
te
rial
Syste
ms
and
St
ructures
,
vol
.
6
,
pp.
71
-
83
,
1995
,
DOI
:
10.
1016
/B
978
-
0
-
12
-
803581
-
8.
03979
-
5
[21]
Anis
Maisara
h
Mohd
As
ry,
Fara
hiya
h
Mus
ta
fa
,
Maiz
u
l
Ishak
,
a
nd
Aznizam
Ah
ma
d
,
"
Pow
er
ge
ner
ation
by
usin
g
pie
zo
el
e
ct
ri
c
trans
duce
r
wi
th
b
e
nding
m
ec
h
ani
s
m
support
,
"
Int
e
rnational
Journal
of
Pow
er
E
lectronic
s
and
Dr
i
ve
Syste
ms
(IJ
PE
D
S)
,
vol
.
10
,
no
.
1
,
pp
.
562
-
567
,
2
019
,
DOI
:
10
.
11
591/i
jpe
ds
.
v10.
i
1.
pp562
-
567
[22]
A.
Augu
stai
ti
s
a
nd
V.
Jurèna
s,
"
Dynami
cs
of
tru
nk
type
robot
wi
th
spherical
p
ie
z
oel
e
ct
ri
c
a
ct
ua
to
rs,
"
Inte
rnationa
l
Journal
of
Robotics
and
Aut
omati
on
(IJ
R
A)
,
vol
.
9
,
no
.
2
,
pp
.
113
-
122,
2020
,
DOI
:
10.
11591/ijra.
v9
i2.
pp113
-
122
[23]
Jianm
in
Qiu
,
Yi
ng
Yang
,
Xin
Hong
,
Piotr
V
asiljev
,
Dali
us
Ma
z
ei
ka
and
Serg
ejus
Borodina
s
,
"
A
disc
-
type
h
ig
h
spee
d
rot
ary
u
lt
r
asonic
mot
or
wit
h
interna
l
con
tac
t
t
ee
th
,
"
App
l. Sci
.
11
,
no
.
5
,
2021
,
DOI
:
10
.
3390/
a
pp11052386
[24]
Yingxia
ng
Li
u
,
Shengjun
Shi
,
Ji
peng
Yan
,
Weishan
Ch
en
and
Dongmei
Xu
,
"
A
novel
piezoe
l
ect
ric
actua
t
or
wi
th
two
oper
a
ti
ng
m
odes
,
"
Journal
o
f
Intelli
g
ent
Mat
erial
Syst
ems
and
Struct
ures
,
vol
.
29
,
no.
6
,
pp
.
1
157
-
1164
,
2018
,
DOI
:
10.
1177/1
045389X17730921
[25]
N.
W
.
Hagood
a
nd
A.
J.
M
c
Far
l
and,
"
Modeli
ng
of
a
pi
ez
o
el
e
ct
ri
c
ro
ta
ry
ult
r
asonic
mot
or
,
"
I
EE
E
Tr
ansacti
ons
on
Ultrasonics,
F
erroele
c
tric
s,
and
Freque
nc
y
Con
t
rol
,
vol
.
42
,
no
.
2,
pp
.
210
-
224
,
1995
,
DOI
:
10
.
1
109/58.
365235
[26]
He
Li
,
W
ei
shan
Chen
,
Xinqi
Tia
n
and
Junkao
Liu
,
"
An
expe
r
im
e
nt
study
on
t
em
p
era
tur
e
c
h
aract
e
r
isti
cs
of
a
l
ine
ar
ult
rasoni
c
mo
t
or
using
lon
git
udinal
tra
ns
duce
rs,
"
Ul
trasonics
,
vol
.
95,
pp
.
6
-
12
,
2019
,
DOI
:
10.
1016/j.ult
r
as.
2019.
03.
003
[27]
Li
fen
g
Zhou
,
Zh
iyua
n
Yao
,
Xiao
niu
Li
and
Shich
ao
Da
i
"
Mode
li
n
g
and
ver
ifica
t
io
n
of
the
r
mal
-
mecha
ni
ca
l
-
elec
t
r
ic
coupl
ing
dyn
am
i
cs
of
a
V
-
shap
e
li
ne
ar
ultrasonic
mot
or,
"
S
ensors
and
Ac
tua
tors
A
:
Phy
si
cal
,
vol.
298
,
2019
,
DOI
:
10.
1016/j.sna.
20
19.
111580
BIOGR
AP
HI
ES OF
A
UTH
ORS
You
ss
ef
Bab
a
was
born
in
El
Kela
a
des
Sragh
na
,
Morocc
o
,
in
1982.
He
re
cei
ved
the
B.
S.
degr
ee
in
E
le
c
trica
l
Engi
n
ee
r
ing
from
th
e
Na
ti
on
al
Schoo
l
of
Ap
pli
ed
Scie
n
ce
s
o
f
Marra
k
esh,
Morocc
o,
in
20
07.
He
obt
ai
ne
d
the
PhD
degr
ee
in
“
E
le
c
trica
l
Eng
ineeri
ng
”
at
H
assan
1
Univer
sity,
in
2
018.
He
is
cur
r
ent
ly
a
n
assista
nt
prof
essor
a
t
the
depa
r
tment
of
E
le
c
trica
l
Engi
ne
eri
ng,
École
Nati
on
ale
S
upér
ie
ur
e
d
'
A
rts
e
t
M
ét
i
ers
(
ENSA
M)
Casabl
an
ca
,
Hass
an
II
unive
rsity
,
Moro
cc
o.
His re
s
ea
rc
h
intere
st
includ
es
mot
or
con
trol
.
Mostafa
B
ouzi
i
s
a
Profess
or
in
El
e
ct
ri
ca
l
Engi
n
ee
ring
Dep
art
m
e
nt
at
Sci
ence
an
d
t
ec
hnology
fac
ul
ty,
Hass
an
1
Univer
sity.
He
re
ce
iv
ed
th
e
E
ngine
er
ing
degr
ee
in
Elec
t
rical
Engi
ne
eri
ng
,
from
Eng
ine
e
r
school
of
Li
l
le
in
Franc
e,
obtai
nd
the
PhD
de
gre
e
in
pho
tovo
lt
aic
sys
te
m
cont
rol
a
t
univ
ersit
y
Hass
an1
and
got
his
ha
bil
itati
on
to
su
per
vise
rese
a
rch
at
Hass
an1
unive
rsity
.
His
cur
ren
t
rese
arc
h
ac
t
ivi
t
ie
s
are
i
n
the
f
ields
of
el
e
ct
ri
ca
l
sys
tems,
ren
ewa
bl
e
ene
rgy conversi
on
sys
te
ms a
nd
nonli
ne
ar
con
tro
l
sys
te
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.