Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 4
,
A
ugu
st
2016
, pp
. 15
41
~
1
550
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
4.9
955
ļ²
1
541
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Analysis
of
the Range of Accel
er
ation for an Accelerometer with
Extended Beams
Mar
g
arita Te
cpoyotl-T
o
rre
s
1
, Ram
o
n
Cabello-Ruiz
1
,
Jose Gerar
d
o Vera-Dimas
1
,
Alfons
o Torre
s-Jac
o
me
2
, Pe
dro V
a
rg
as
1
, S
v
et
la
na
K
o
shev
aya
1
1
Centro d
e
Inv
e
stigacion
en Ing
e
nieria
y
Cien
cias
Ap
licadas-(IIC
BA), Universid
a
d Au
tonoma del
Estado d
e
Morelos
2
Instituto N
acion
al d
e
Astrof
isi
c
a
,
Optic
a y El
ec
tro
n
ica
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 19, 2016
Rev
i
sed
Ap
r
15
, 20
16
Accepted Apr 29, 2016
The el
as
tic b
e
h
a
viour of a s
y
s
t
em
can be det
e
rm
ined b
y
an anal
ys
is
of
s
t
res
s
e
s
.
The
s
t
r
e
s
s
genera
ted
in
the
elem
ent
lo
ad
ed of
an
acc
el
er
om
eter is
of
inter
e
s
t
here
. I
n
thes
e devi
ces
, the s
u
s
p
ens
i
o
n
beam
s
are th
e elem
en
ts
subjected to greater stresses, as they
support th
e mass. The stress that th
ey
can support is limited b
y
the
elastic
limit of the material. Based on this
analy
s
is, th
e op
erating conditio
n
s to
prevent p
e
rmanent defor
m
ations are
determ
ined
. Th
e ana
l
y
s
is is f
o
cused on the
acc
ele
r
at
ion ap
plied
to the
acc
ele
r
om
eter b
ecaus
e
this
para
m
e
ter incr
eas
es
cons
iderab
l
y
the
s
t
res
s
e
s
in
the device. A relationship between nor
mal stress and gravity
applied is
obt
a
i
ne
d.
T
h
i
s
equa
t
i
on
i
s
use
d
in orde
r t
o
a
voi
d e
x
ce
e
d
i
n
g t
h
e el
a
s
t
i
c
l
i
mit,
during the ac
ce
lerom
e
ter op
erat
ion. This fa
ct d
e
term
ines the
a
cce
ler
a
tion
range supported
b
y
the d
e
vice.
In
the li
teratur
e
, studies about the
ph
y
s
ics
and
modelling of accelerometers ar
e perfo
rmed. H
o
wever,
about the specif
i
c
acceleration of operation which
they
ar
e subjected
,
informatio
n about its
determination is not provided. In this
paper, the ana
l
y
s
is is realiz
ed
considering a Conventional Capacitiv
e
Accelerom
e
t
e
r
(CC
A
)
and a
Capac
itiv
e Acc
e
l
erom
eter
with
Extend
ed Be
am
s (CAEB), part
i
c
ular
l
y
, on
the norm
a
l s
t
res
s
. W
h
en a range
of acce
ler
a
tion
values
ar
e appli
e
d, norm
a
l
s
t
res
s
occur whi
c
h m
u
s
t
not exc
eed th
e el
as
tic
li
m
it of the m
a
ter
i
al,
as
it was
mentioned befor
e
. Th
e Matlab
code used
to
c
a
lcu
l
at
e this r
e
l
a
tionship i
s
given
in Append
ix A.
Keyword:
Accelerom
eter
Grav
ity
MEMS
No
rm
al stress
Copyright Ā©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Marg
arita Tecp
o
y
o
tl-Torres,
C
e
nt
ro
de
I
n
ve
st
i
g
aci
on
en
I
n
geni
e
r
ia y Ciencias Ap
licad
as-(
II
CBA
),
Uni
v
ersi
dad
A
u
t
o
nom
a del
E
s
t
a
do
de
M
o
rel
o
s,
Av
. Uni
v
ersi
da
d 10
0
1
. 6
2
2
0
9
C
u
er
navaca
,
M
o
rel
o
s,
M
e
xi
co
.
Em
a
il: tecp
o
y
o
tl@u
aem
.
m
x
1.
INTRODUCTION
An accelerometer is a sensor de
vice that allows
the indirect
m
easurem
e
n
t of accelerat
ion (s
pee
d
vari
at
i
o
n wi
t
h
t
i
m
e
or rel
a
t
i
on bet
w
ee
n f
o
rc
e and m
a
ss) ac
co
rd
ing
to
on
e, two
,
or th
ree
d
i
rection
s
th
rou
gho
ut
a sensible axi
s
[1]. Accelerom
eters have expl
oited
their applications
in vari
ous
fields like m
onitori
ng
vibration, ine
r
t
i
al navi
gation,
an
d attitude c
o
ntrolling. The
m
o
st co
mm
on accelerom
eters are c
o
nve
ntionally
base
d on ca
pa
citive, piezore
sistive or
piez
oelectric be
ha
vior [2],[3]. To
ext
r
act the a
cceleration
value, the
sens
or
has
a
m
ovabl
e p
r
o
o
f
m
a
ss conne
ct
ed t
o
a
fi
xe
d
f
r
am
e t
h
ro
ug
h
spri
ng
st
r
u
ct
u
r
es.
Whe
n
t
h
er
e i
s
an
external acceleration, the
seis
mic
m
a
ss is displaced
from
its
rest
positi
on.
The m
a
gnitude
of this
displace
m
e
nt
is p
r
o
p
o
r
ti
o
n
a
l
to
th
e m
a
g
n
itu
d
e
o
f
t
h
e acceler
atio
n
an
d
i
n
v
e
r
s
ely pr
opor
tio
n
a
l to
t
h
e stif
f
n
ess
of
th
e
sp
r
i
n
g
structures [4].
The accelerom
eters are c
o
nstituted
by st
ructural supports c
a
lled suspe
n
si
on
beam
s. It is im
portant to
realize an a
n
al
ysis of st
ress i
n
to t
h
em
, in order t
o
obt
ai
n
th
e ex
trem
e o
p
eratin
g cond
itio
n
s
.
In th
is
p
a
p
e
r, th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
15
41
ā
1
550
1
542
acceleration ra
nge i
n
which a
n
accelerom
ete
r
with e
x
tend
e
d
sus
p
e
n
sion
bea
m
s can ope
rate appropriate
l
y is
p
r
esen
ted. Fo
r
th
is reason
, th
e an
alysis o
f
norm
a
l
stre
ss in
th
e su
sp
en
sion
b
eam
s is realiz
ed
, using
th
e
m
o
d
e
l
o
f
a can
tilev
e
r b
eam
with
a
u
n
i
form
l
y
d
i
strib
u
t
ed
l
o
ad. Su
b
s
equ
e
n
tly, t
h
e in
fo
rm
atio
n o
b
t
ain
e
d
is v
a
lid
ated
usi
n
g An
sy
s.
1.
1.
Analytic
Rel
a
tion
Nor
m
al Stress-Acceleration
In ca
pacitive a
ccelerom
eters, the
struct
ural
ele
m
ents of support
are called beam
s. They
allow the
mass to be
s
u
s
p
ende
d a
n
d dis
p
laced. T
h
eir s
t
udy is
fund
amental to
unde
rs
tand t
h
e a
n
alysis of norm
a
l str
e
ss.
Due t
o
the a
p
plied loads
,
the
beam
s develop a sh
ear fo
rce and
a be
ndi
ng m
o
m
e
nt
t
h
at
,
i
n
ge
ne
ral
,
change from
point to point along th
e axis
of the beam
s [5].
The beam
s
can be classifi
ed accordi
ng t
o
their
conditions of support, as
fo
llows: a) simply supporte
d
beam
s. The reactions
ha
ppen in their e
n
ds,
b)
can
tilev
e
r.
On
e en
d
o
f
th
e
b
e
am is fix
e
d
to
prev
en
t ro
ta
tio
n
,
c) can
tilev
e
red
b
eam
s. On
e
o
r
b
o
t
h
end
s
o
f
th
em
st
ands
o
u
t
of t
h
e su
p
p
o
r
t
s
,
d)
const
a
nt
beam
s. A st
at
i
cally
indeterm
inate beam
that
spreads on three or
m
o
re
supports.
The loa
d
consi
s
ts on the a
ppli
e
d forces that a
c
t
on the beam
, whic
h can c
o
me from
the weight of the
beam
, besi
des
ot
he
r f
o
rces t
h
at
i
t
coul
d
s
u
p
p
o
r
t
.
T
h
e
r
e are
fi
ve
basi
c t
y
p
e
s o
f
l
o
a
d
s a
p
p
l
i
e
d i
n
beam
s, whi
c
h
are: a) without load.
The sa
m
e
b
eam
is co
n
s
id
ered
with
ou
t
weigh
t
(o
r at least v
e
ry sm
all
co
m
p
ared
with o
t
h
e
r
fo
rces t
h
at
co
u
l
d be a
ppl
i
e
d
)
,
b) c
once
n
t
r
at
e
d
l
o
a
d
.
A l
o
ad applied
on a relatively s
m
a
l
l area, c)
uni
form
ly
d
i
stribu
ted
lo
ad
s. Th
is lo
ad
is eq
u
a
lly d
i
stri
b
u
t
ed
on
a
po
rtio
n
of leng
th
o
f
b
eam
s, d
)
variab
le lo
ad
. Th
e lo
ad
varies its intensity form of one place
to other, e) torsion. This one is
generate
d when
a torsion is applied
on
any pa
rt
of t
h
e
beam
[6].
For t
h
e accele
r
om
eters used here
, the type of
b
eam
s are cantilevers a
nd t
h
e consi
d
ered l
o
ad is
uni
fo
rm
ly
di
st
ri
but
ed
. T
h
e
be
ndi
ng m
o
m
e
nt
M
, whe
r
e the
norm
al stress re
m
a
in
s b
e
low
th
e yield
streng
th
or
elastic l
i
m
it
Ļ
y
, is also
stu
d
i
ed
, b
ecau
s
e it serv
es as referen
ce p
a
ram
e
ter. Th
e stress in
can
tilev
e
rs m
u
st b
e
rem
a
in
ed
b
e
low th
e elastic l
i
m
i
t
so
th
ere will n
o
t
b
e
p
e
rm
an
en
t d
e
fo
rmatio
n
s
. Th
e
Hook
e's law can
b
e
ap
p
lied to
t
h
e stress
u
n
i
ax
ial calcu
latio
n
.
Ass
u
m
i
ng a hom
ogene
ou
s m
a
t
e
ri
al
, and den
o
t
i
n
g f
o
r
E
t
o
t
h
e
m
odul
us o
f
el
ast
i
c
i
t
y
, on t
h
e
lo
ng
itu
d
i
n
a
l
d
i
rectio
n x, th
e stress is
g
i
v
e
n
by:
x
x
E
ļ„
ļ³
ļ½
(1)
whe
r
e
x
ļ„
is un
itary lon
g
itud
i
n
a
l
d
e
form
at
io
n
,
calcu
lated
from
,
m
x
c
y
ļ„
ļ„
ļ
ļ½
c
is the m
a
xim
u
m
distance to
the ne
utral s
u
rface, y is the
distance
of t
h
e ne
utral axis
to any
poi
nt of beam
s and
m
ļ„
is th
e m
a
x
i
m
u
m
ab
so
lu
te v
a
l
u
e o
f
th
e
u
n
itary
d
e
fo
rm
atio
n
.
Mu
ltip
lyin
g
both
m
e
m
b
ers o
f
(1) b
y
E
, the
norm
al stress
can be
obt
ai
ne
d:
m
x
c
y
ļ³
ļ³
ļ
ļ½
(
2
)
whe
r
e
m
ļ³
is th
e
max
i
m
u
m
ab
so
lu
te v
a
lu
e stress. Th
is resu
lt sh
o
w
s th
at, in
th
e elastic ran
g
e
, th
e no
rmal
stress ch
ang
e
s
lin
early with
t
h
e d
i
stan
ce to
t
h
e n
e
u
t
ral ax
is (Fig
ure
1
)
[7
].
Fi
gu
re
1.
Va
ri
at
i
on
of
t
h
e
n
o
r
m
al
st
ress [
7
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
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8
ļ²
Analysis of t
h
e
Range
of Ac
celera
tion for
an
Accelerometer
with Exte
nded
Beams (Margarita Tec
poy
otl
T.)
1
543
To calculate
m
ļ³
un
de
r t
h
e case
of
p
u
re
be
ndi
ng
, w
h
e
r
e t
h
e
neut
ral
axi
s
pa
sses f
o
r t
h
e ce
nt
r
o
i
d
sectio
n
,
t
h
e inertia m
o
m
e
n
t
I
, or the
second m
o
m
e
nt of t
h
e cross
sec
tion
with
resp
ect
to
th
e cen
t
ro
i
d
ax
is
per
p
e
ndi
c
u
l
a
rl
y
t
o
t
h
e
pl
a
n
e
of
t
h
e
par
M
[6], is also
con
s
id
ered
:
I
c
M
m
ļ
ļ½
ļ³
(
3
)
R
e
pl
aci
ng
m
ļ³
fr
o
m
(3) in
(
2
),
th
e n
o
rm
al stress
x
ļ³
to
an
y d
i
s
t
an
ce
y
o
f
t
h
e
neut
r
a
l
axi
s
i
s
obt
ai
ned:
I
y
M
x
ļ
ļ½
ļ³
(4)
In add
itio
n,
I
is
calculated by means
of (5) where
b
is th
e t
h
ick
n
e
ss and
h
i
s
t
h
e
wi
dt
h
of t
h
e
beam
.
12
3
bh
I
ļ½
(5)
If t
h
ere is no
t
ex
ist a l
o
ad co
n
c
en
trated
in th
e
free end
o
f
t
h
e can
tilever, t
h
e
u
n
i
q
u
e
lo
ad th
at it
expe
ri
ences
i
s
onl
y
t
h
e
p
r
o
d
u
ced
by
i
t
s
o
w
n
wei
g
ht
. T
h
e
n
t
h
e l
o
a
d
i
s
di
st
ri
b
u
t
e
d i
n
a
un
i
f
orm
way
,
as
Fi
gu
re
2
sh
ow
s [8
].
Fig
u
re 2
.
Can
tilev
e
r with
un
ifo
r
m
l
y
d
i
strib
u
t
ed
lo
ad
[7
]
To
calcu
late th
e
b
e
nd
ing
m
o
m
e
n
t
o
f
a can
tilev
e
r
with
u
n
i
form
l
y
d
i
stribu
ted
l
o
ad, th
e fo
llowing
equat
i
o
n i
s
use
d
:
2
2
1
wL
M
ļ½
(6)
whe
r
e
L
is th
e
len
g
t
h of t
h
e can
tilev
e
r and
w
is th
e l
o
ad d
i
st
ribu
ted
b
y
u
n
it
o
f
leng
th
:
L
g
m
w
ļ
ļ½
(
7
)
The t
y
pe
of
be
am
and l
o
ad c
onsi
d
ere
d
i
n
t
h
i
s
w
o
r
k
a
r
e s
h
o
w
e
d
i
n
Fi
g
u
r
e 2
.
R
e
pl
aci
n
g
(
7
) i
n
(6
)
,
fol
l
o
wi
n
g
e
x
pr
essi
on
i
s
obt
ai
ned:
L
g
m
M
ļ
ļ
ļ½
2
1
(
8
)
To
ob
tain
t
h
e
relatio
n
s
h
i
p
b
e
tween
n
o
rm
al
stress and
th
e
b
e
nd
ing
m
o
m
e
n
t
th
at will d
e
term
in
e th
e
rate of a
cceleration of the a
ccelerom
eter, (8) is re
placed in (4).
Finall
y, the desire
d
relations
hip
be
tween
n
o
rm
al stress an
d grav
ity, is ob
tain
ed
:
I
y
L
g
m
x
ļ
ļ
ļ
ļ
ļ½
2
ļ³
(
9
)
It is no
teworthy th
at in
the analyzed
literatu
re th
is relation
s
h
i
p
is no
t
d
e
v
e
l
o
p
e
d
or sh
own
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
15
41
ā
1
550
1
544
2.
R
E
SEARC
H M
ETHOD
2.
1.
Theoretic
al
analysis
and si
mulati
on
of norm
al
stres
s i
n
a conve
n
ti
onal
accelerom
e
ter
at
1
g
The accelerometers are devic
e
s use
d
to m
e
a
s
ure accel
e
r
ation a
n
d vi
brati
o
n. The
s
e de
vi
ces conve
r
t
the acceleration of the
gravit
y or
of t
h
e movem
e
nt into
an electrical
analogical signal,
proportiona
l to the
fo
rce a
pplie
d t
o
the
sy
stem
[9]
.
Fig
u
re
3
shows the m
a
in ele
m
ents of a C
C
A.
Fi
gu
re
3.
M
a
i
n
el
em
ent
s
of
t
h
e C
C
A
The calculation of
norm
al stress
x
ļ³
i
s
m
a
de
fr
om
(9). The
corres
p
on
di
n
g
val
u
es are s
h
o
w
n i
n
Tabl
e 1.
Tabl
e
1. C
a
l
c
ul
at
ed pa
ram
e
t
e
r val
u
e
s
Para
m
e
ter Value
Lo
ad
,
w
0.
014
N/m
Bending m
o
m
e
nt,
M
2.
81x1
0
-8
NĀ·
m
I
n
e
r
t
i
a
mo
me
n
t
,
I
3.
25x1
0
-2
0
m
4
Norm
al
stress,
x
ļ³
10.
8 M
P
a
In orde
r t
o
c
o
m
p
are the approxim
at
io
n
to calcu
late
th
e no
rm
al
stress, giv
e
n
b
y
(9),
the
sim
u
lat
i
o
n
was i
m
pl
em
ent
e
d.
A
p
pl
y
i
ng
1
g
(9.81 m
/
s2
) to
CCA.
A
2
.
59 MPa
n
o
rm
al stress
v
a
lu
e was ob
tain
ed
.
As it can
be obse
rve
d
, t
h
e the
o
retical value a
ppea
r
s
to be larger
t
h
an
the one obtained by
sim
u
lation.
T
h
is
is because
th
e calcu
latio
n is p
e
rform
ed
fo
r on
ly o
n
e
susp
en
sion
b
eam. To
ob
tain
th
e n
o
r
m
a
l stress
v
a
lu
e it is n
ecessary
to
d
i
v
i
d
e
th
e to
tal v
a
lu
e b
e
t
w
een
t
h
e number
of
beam
s.
The res
u
lt is 2.
7 MPa for eac
h one of the s
u
spe
n
sion
beam
s.
The differe
n
ce pro
duc
es an error
of 4%.
Tab
l
e
2
sh
ows th
e
p
r
op
erties o
f
s
ilico
n
used
in th
eo
retical calcu
latio
n
s
an
d b
y
sim
u
latio
n
,
i
n
th
e
devel
opm
ent
o
f
t
h
i
s
w
o
r
k
.
Tab
l
e 2
.
Silico
n
p
r
op
erties u
s
ed
[10
]
Pr
oper
t
y Value
Desnsity
(
Ļ
),
in kg/m
3
2330
YoungĀ“
s M
odulus
(
E
),
in GPa
131
PoissonĀ“
s r
a
tio,
dim
e
nsionless
0.
33
Fi
gu
re
4 s
h
ows
t
h
e c
o
n
s
i
d
ere
d
di
m
e
nsi
ons
f
o
r t
h
e
C
C
A
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Analysis of t
h
e
Range
of Ac
celera
tion for
an
Accelerometer
with Exte
nded
Beams (Margarita Tec
poy
otl
T.)
1
545
Fi
gu
re
4.
Di
m
e
nsi
o
ns
of
t
h
e
C
C
A
In Fi
g
u
r
e 5 an
d 6, t
h
e n
o
r
m
a
l st
ress gene
rat
e
d i
n
t
h
e sus
p
en
si
on beam
s and a zoom
i
n
at
one
of t
h
em
are s
h
own,
res
p
ectively.
Fi
gu
re
5.
N
o
r
m
al
st
ress ge
ne
rat
e
d i
n
t
h
e
s
u
s
p
en
si
o
n
beam
s of a CC
A
Figure
6. Zoom
in at one
of the
fol
d
ed bea
m
2.
2.
T
h
eoreti
c
al
a
n
al
ysi
s
a
nd si
mul
a
ti
on
of norm
al stres
s i
n
a CAEB
at
1
g
Once t
h
e calculations
have been
m
a
de fo
r
a C
C
A
sub
j
e
c
t
e
d at
1
g
, t
h
e case o
f
th
e C
A
EB will b
e
analysed. Figure
7 s
h
ows
the
dim
e
nsions
of this accelerometer, ge
nerat
e
d
fr
om
a cha
nge
in the
ge
ometry of
t
h
e con
v
e
n
t
i
o
n
a
l
m
a
ss, i
n
t
e
nd
ed t
o
ext
e
nd t
h
e l
e
ngt
h
of t
h
e
sus
p
ensi
o
n
bea
m
s, wi
t
hout
ex
cessi
vel
y
red
u
ce t
h
e
v
a
lu
e
of th
e mass, as it is
requ
ired
b
y
equ
a
tio
n of sen
s
itiv
ity, g
i
v
e
n
b
y
(10):
k
g
m
x
ļ
ļ½
(
1
0)
whe
r
e
x
is th
e
sen
s
itiv
ity d
i
splace
m
e
n
t
an
d
k
is th
e stiffn
ess con
s
tan
t
.
Howev
e
r, th
e i
n
crem
en
t on
bea
m
s len
g
t
h imp
lies greater b
e
n
d
i
n
g
m
o
m
e
n
t
(6),
wh
ich also im
p
lies an
increase
of normal stress (9).
The
n
,
the
normal stress limits
the beam
s extensi
on a
n
d t
h
e acceleration value
.
Th
e an
alysis i
n
th
is case is
si
m
ilar to
th
e
o
n
e
realized
in section 2.1.
In Ta
ble
3, the
calculated
val
u
es a
r
e
prese
n
t
e
d
.
Tabl
e
3. C
a
l
c
ul
at
ed pa
ram
e
t
e
r val
u
e
s
Para
m
e
ter Value
Lo
ad
,
w
0.
005
N/m
Bending m
o
m
e
nt,
M
5.
43x1
0
-8
NĀ·
m
I
n
e
r
t
i
a
mo
me
n
t
,
I
3.
25x1
0
-2
0
m
4
Norm
al
stress,
x
ļ³
20.
8 M
P
a
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
15
41
ā
1
550
1
546
Fi
gu
re
7 s
h
ows
t
h
e
di
m
e
nsi
o
n
s
o
f
a C
A
EB
.
Fi
gu
re
7.
Di
m
e
nsi
o
ns
of
a
n
ac
cel
erom
et
er wi
t
h
ext
e
nde
d
be
am
s
Fi
gu
re 8 s
h
o
w
s t
h
e no
rm
al stress ge
nerat
e
d i
n
t
h
e sus
p
ensi
on
beam
s, and i
n
Fi
gure
9 a zoom
i
n
at
one
o
f
t
h
e s
u
s
p
ensi
o
n
beam
i
s
sh
ow
n.
Fi
gu
re
8.
N
o
r
m
al
st
ress ge
ne
rat
e
d i
n
t
h
e
s
u
s
p
en
si
o
n
beam
s of a C
A
EB
Figure
9. Zoom
in at one
of the
fol
d
ed bea
m
Fro
m
Fig
u
r
e
9
,
th
e si
m
u
latio
n
resu
lts p
r
ov
ide a n
o
r
m
a
l stre
ss v
a
lu
e of 5.03
MPa. In
a similar fash
ion
to
th
e pro
cedure fo
llo
wed
i
n
th
e case of th
e CCA, it is n
e
cessary to
d
i
v
i
de th
is to
tal v
a
lu
e b
y
th
e nu
mb
er
of
sus
p
ensi
o
n
bea
m
s. 5.2 M
P
a n
o
rm
al
st
ress va
l
u
e was
obt
ai
n
e
d,
whi
c
h i
s
ve
ry
cl
ose t
o
5.
0
3
M
P
a o
b
t
a
i
n
e
d
f
r
om
th
e si
m
u
latio
n
,
th
rowing
an
erro
r
o
f
3%. As it is
sh
own, th
e ch
an
ge in
geo
m
etry o
f
th
e
m
a
ss d
e
ter
m
i
n
es a
m
o
re precise analytical calculation.
T
h
e incre
m
ent in the arm length im
plie
s an increase
in the norm
al stress,
because a
n
inc
r
eased be
nding m
o
m
e
nt occurs.
3.
R
E
SU
LTS AN
D ANA
LY
SIS
In t
h
is section, the theoretical and
sim
u
lated
resu
lts of
the accelerom
eters
ar
e prese
n
ted, perform
i
ng
a swee
p in the
range
of the a
pplied a
ccelera
tion.
As it
is shown,
from
sim
u
la
tion res
u
lts of
Sections 2.1 a
nd
2.
2, t
h
e a
n
al
y
t
i
cal
appr
oac
h
i
s
u
s
ef
ul
f
o
r
st
ress
cal
cul
a
t
i
on.
Si
nce
t
h
e n
o
rm
al
st
ress ge
ne
rat
e
d
i
n
t
h
e
suspensi
on be
a
m
s does not
exceed the e
l
astic l
i
m
it of silicon (250
MPa) when
1
g
is ap
p
lied
t
o
th
e
accelerom
eters shown pre
v
iously (section 2.1
and
2.2),
we
procee
d to
re
alize an a
n
al
ysis with large
r
value
s
of
g
. Fi
gu
re
10
sh
ow
s t
h
e
n
o
r
m
a
l
st
ress ge
ne
rat
e
d i
n
t
h
e
s
u
s
p
en
si
o
n
beam
s o
f
C
C
A
,
w
h
e
n
9
7
g'
s
are a
p
plied.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Analysis of t
h
e
Range
of Ac
celera
tion for
an
Accelerometer
with Exte
nded
Beams (Margarita Tec
poy
otl
T.)
1
547
The
val
u
e
o
f
t
h
e
no
rm
al
st
ress ge
nerat
e
d i
n
the s
u
spe
n
si
on
beam
s is 251 MPa at 97
g'
s
(
F
igur
e 10).
It indicates tha
t
the elastic limit
has been e
x
ceede
d
,
which m
a
y
indicate th
at this accelerom
eter, fabri
cated in
silico
n
,
can
operate p
r
o
p
e
rly u
n
til 9
6
g'
s
app
r
ox
im
a
t
ely, w
h
er
e th
e nor
m
a
l str
e
ss v
a
lu
e is o
f
24
9.149
MPa. In
Fi
gu
re
11
, t
h
e
no
rm
al
st
resses val
u
es
o
b
t
a
i
n
ed t
h
e
o
ret
i
cal
l
y
and
by
si
m
u
lat
i
on are
sh
o
w
n,
fr
om
1
g
t
o
96
g's
for CCA. To
en
sure t
h
e in
tegrity o
f
t
h
e
d
e
v
i
ce we
sugg
est
as op
erating
li
mit u
p
to
92
g's.
Fi
gu
re
1
0
.
N
o
r
m
al
st
ress ge
ne
rat
e
d i
n
s
u
s
p
en
si
on
beam
s
of the c
o
nve
ntional acc
elerom
eter,
whe
n
97
g'
s
are
ap
p
lied
Fi
gu
re
1
1
. C
a
l
c
ul
at
ed a
n
d si
m
u
l
a
t
e
d n
o
rm
al
stress
gene
rate
d in s
u
spe
n
si
on
beam
s of a C
C
A
Fi
gu
re
12
s
h
o
w
s t
h
e
n
o
r
m
a
l st
ress
ge
nerat
e
d i
n
t
h
e s
u
s
p
ensi
o
n
beam
s of t
h
e C
A
EB
,
sh
ow
n i
n
sect
i
on 2.
2,
wi
t
h
50
g's
app
lied
.
Fi
gu
re
1
2
.
N
o
r
m
al
st
ress ge
ne
rat
e
d i
n
s
u
s
p
en
si
on
beam
s of
C
A
EB
,
wi
t
h
5
0
g'
s
app
lied
Figure
12 shows
how the
elastic li
mit of the
silicon is e
x
ceede
d
. 251
MPa are
obtai
ned at 50
g'
s
.
Fig
u
re
1
3
shows th
e no
rm
al stress
o
b
t
ain
e
d th
eoretically an
d b
y
sim
u
lati
o
n
, fro
m
1
g
up
to
50
g'
s
. At
49
g's
th
e valu
e
o
f
the no
rm
al stress is 246
.8
4 MPa. Ag
ain, t
o
ensu
re d
e
v
i
ce i
n
teg
r
ity op
erating
lim
i
t
u
n
til 47
g's
is
suggeste
d.
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
15
41
ā
1
550
1
548
Fi
gu
re
1
3
. C
a
l
c
ul
at
ed a
n
d si
m
u
l
a
t
e
d n
o
rm
al
st
ress
gene
rat
e
d i
n
s
u
spe
n
si
o
n
beam
s of a C
A
EB
4.
CO
NCL
USI
O
N
Th
e id
en
tificatio
n
of th
e typ
e
s o
f
supp
ort o
f
th
e su
spe
n
sion beam
s and of lo
ad
are im
p
o
r
tan
t
in
th
e
th
eoretical an
alysis o
f
n
o
rm
al
stress. Th
ey hav
e
b
e
en
id
en
t
i
fied
as can
tilev
e
r an
d
un
iformly
d
i
strib
u
t
ed lo
ad
,
respectively.
T
h
e e
r
ror
bet
w
e
e
n sim
u
lated and calculated
re
su
lts of
n
o
rmal stress for
b
o
th
cases is
sm
a
ll, 4
%
for CCA and
3% fo
r CAEB.
Th
is
fact is du
e to
th
e v
a
riab
les u
s
ed
i
n
th
e calcu
l
atio
n
o
f
th
e lo
ad.
Th
e ex
ten
s
ion o
f
th
e leng
th o
f
th
e b
eam
s p
r
oduces considera
b
ly great
er stresses.
At 1
g
, norm
al
st
resses o
f
2
.
5
M
P
a and
5.
2
M
P
a i
s
gene
rat
e
d f
o
r cases
o
f
C
C
A
an
d o
f
C
A
EB
, re
spect
i
v
el
y
.
Whi
l
e
, t
h
e
up
p
e
r
limit values are reache
d
at
accelerations
of 96 a
n
d 47
gĀ“
s
, pr
o
duci
ng
si
m
u
l
a
t
e
d no
rm
al
st
resses nea
r
t
o
t
h
e
y
i
el
d st
ress
of
24
9 M
P
a
,
fo
r t
h
e C
C
A
an
d C
A
EB
,
res
p
ect
i
v
el
y
.
It can
be
observed a
n
inve
rse
relationshi
p
between
the a
pplied norm
al for
ce and accele
r
ation. That
mean
s, CCA su
ppo
r
t
s a
w
i
der
r
a
ng
e
of ac
celeration tha
n
CAEB. T
h
er
ef
or
e
,
CA
E
B
is
r
e
co
mme
n
d
e
d fo
r
syste
m
s requiring l
o
w le
vels
of acceleration.
Knowle
dge
of the
relationshi
p
betwee
n norm
al stress and
acceleration is
fundam
ental in the
calculation of the
ra
nge of accelerat
i
o
n supported
by the accele
r
om
eter.
Th
e
p
a
ram
e
ters th
at in
fl
u
e
nce th
is relation
s
h
i
p
ar
e o
f
phy
si
cal
an
d geom
et
ri
c nature
. The
calculation
pr
oce
d
u
r
e i
s
gi
ven
i
n
t
h
i
s
wo
r
k
a
n
d
s
u
m
m
a
rized i
n
(
9
)
.
APPE
NDI
X
A
In
t
h
i
s
a
ppe
n
d
i
x
,
we s
h
ow
t
h
e
co
de
gene
rat
e
d e
n
M
A
TL
A
B
.
clear
all
close
all
clc
datax=[];
dataE=[];
wb=25*10^-6
E=131*10^9;
h=25*10^-6;
g=9.81;
pi=3.141592;
lb=input(
'introduce the beam length '
);
k1=(wb/lb)^3;
%
k=E*h*k1;
m=input (
'introduce the mass of the system '
);
f=(1/(2*pi))*((4*k)/m)^0.5
%frequency
I=(wb*h*h*h)/(12);
%inertia moment
c=wb/2;
%maximum distance to the neutral surface
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
ļ²
Analysis of t
h
e
Range
of Ac
celera
tion for
an
Accelerometer
with Exte
nded
Beams (Margarita Tec
poy
otl
T.)
1
549
for
i=1:100
p=g*i;
x=(m*p)/(4*k);
force=m*p;
w=force/lb;
%load
M=(w*lb*lb)/2;
%bending moment
Normal stress=((M*c)/I)/4;
datax=[datax;x];
dataE=[dataE;Esfuerzo];
end
save(
'esfuerzo.txt'
,
'datax'
,
'-ascii'
);
save(
'desplazamiento.txt'
,
'dataE'
,
'-ascii'
);
ACKNOWLE
DGE
M
ENTS
R Cabello a
nd P
Vargas
express
e
d their s
i
ncere t
h
anks
to CONA
CYT fo
r th
e scho
l
a
rsh
i
p
with
g
r
an
ts r
e
f
e
r
e
n
c
e
37
656
6
/
2
4857
6
and
48
439
2/2
7
3
928
, r
e
sp
ectiv
ely.
REFERE
NC
ES
[1]
Albarbar A
.
, āPerformance
evalu
a
tion
of MEMS
acc
elerometer
Measurement,
ā vo
l. 42
, pp
. 790ā5
,
2009.
[2]
Macdonald G. A
., āA review of low cost
accelero
m
eters for vehicle d
y
n
a
mics Sens,ā
Actuators A
, v
o
l. 21, pp
. 303ā
307, 1990
.
[3]
Yazdi N
.
,
et al.
,
ā
M
icrom
achined
iner
tia
l sensors,
ā
Proc
. I
E
E
E
, vo
l. 86
, pp
. 1640ā1
659, 1998
.
[4]
B. S
h
afaa
t,
et a
l
.
, āMonolith
ic
Tri-Axes Nick
el
-Based Accelero
m
e
ter Design V
e
rifi
ed Through
Finite
Elem
en
t
Analy
s
is
,
ā
Arab J
Sci Eng
, vol. 3
8
, pp
. 2103ā211
3, 2013
.
[5]
H. Jones R. C., ā
M
ecani
c
a
de
m
a
teri
ales,ā
Ed. Pe
arson, 2006
.
[6]
F
itzger
a
ld R
.
W
.
, ā
M
ecan
ic
a de
m
a
teria
l
es
,
ā
Ed
.
Alfaom
ega,
200
7.
[7]
Be
e
r
F. P.,
et
al
.
,
ā
M
ecani
c
a
de m
a
ter
i
al
es,ā
Ed
. M
c
Graw-Hill,
200
9.
[8]
G. Cruz C. T., ā
D
iseƱo de s
e
ns
ores basados
en la tecnologia
d
e
m
i
crosistemas,ā Thesis, 2011
.
[9]
Manzanar
es A.,
āEstudio de modelos matematico
s de acelerometr
o
s come
rciales,ā Thesis, 2008.
[10]
http://www.
ans
y
s.
com/
BIOGRAP
HI
ES OF
AUTH
ORS
M
a
rgarit
a T
ecpo
y
ot
l Torr
es
rec
e
i
v
ed the M
a
them
ati
c
ian d
e
gre
e
fr
om
the Univers
i
t
y
o
f
P
u
ebla
,
Mexico, in 1991
. From this Uni
v
ersity
, she was
also graduated
as Electronic
En
gineer in 1993.
She rec
e
iv
ed th
e
M.Sc.
and Ph
.D. degr
ees
in
Ele
c
t
ronics from
Na
t
i
onal
Institu
te o
f
Astroph
y
s
i
c
s,
Optics and Electronics, INAOE, MĆ©xico, in 1997
and 1999, respectiv
ely
.
Dr. Tecpo
y
otl works,
since 1999, at CIICAp of Auton
o
mous
University
of Morelos,
Mexico, wher
e she is curren
tly
titul
a
r professor. She has been vi
siting resear
ch sc
ien
tist in Universit
y
of Bristol (
2001), UK. Sh
e
led the Winner team of Boot Camp, UAEM Pot
e
ntial obtaining
support b
y
TEC
H
BA to go t
o
SILICON VAL
LEY in May
of
2014. In th
e sam
e
y
e
ar, she was
co-founder of
I
NNTECVER. In
2015 she won
the third place in the Ro
y
a
l Academ
y
of Engin
eeringĀ“
s Leaders
in Innovation
Fellowships final pitch session,
in UK. Her m
a
in research in
ter
e
st includ
es MEMS, Antenna
design, Microw
ave devices, entrepren
e
urship
and innovatio
n; and also, d
e
velopment of
educational prog
rams. She has currently
four pa
t
e
nts tit
les from
IMPI, tw
o cop
y
rights, and one
trademark
.
She h
o
lds the status
of
Nation
a
l Resear
cher (SNI), in
Mexico
since 1999
.
Ramon Cabello
Ruiz receiv
e
d th
e degree of Me
chanical Eng
i
neer
(2010) from the Autonomous
Unive
r
sity
of M
o
re
los (UAEM), Me
xic
o
. And he
studied
the M
a
ster
y
of
Engin
eering
in th
e
Center for Applied Resear
ch in
Engineering an
d
Applied Scien
c
es (CIICAp), Mexico
, in 2012.
Ramon Cabello
is stud
y
i
ng
a Ph
D in Eng
i
neerin
g
in CIICAp Sciences, moreover
,
th
at has giv
e
n
classes at
the S
c
hool of Ch
emical Sc
iences
and
Engineer
ing at
the UAEM.
Evaluation Warning : The document was created with Spire.PDF for Python.
ļ²
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
15
41
ā
1
550
1
550
JosƩ Gerardo V
e
ra
is graduated from the
Tec
hnological Institute of
Morelia as
Electron
i
c
Engine
er. Curre
ntl
y
, he is organ
i
zing m
e
m
b
er of
ROPEC editions VII and VIII.
He receiv
e
d th
e
EGRETEC 200
9 award as Young Graduate
d from the Graduat
e
d Association of the
Techno
logical I
n
stitute o
f
Morel
i
a. He got his
M
a
ster degr
ee wi
t
h
honors in the
Research C
e
nt
er
of Engin
eering
and Applied Sciences (CIICAp)
belong
ing to
the Autonomous University
of
More
los Sta
t
e
(UAEM).
He
a
l
so got the
PhD
de
gree in Eng
i
n
eering and Applied Scien
ce in
Ele
c
tri
cal
are
a
.
Gerardo Vera
is
part of
the d
e
v
e
lopm
ent of th
e
curricu
lum
of t
h
e M
a
s
t
er in
Commercialization of Innovative Knowledge and
the
Bach
elor on
Technolog
y
,
bo
th of CIICAp-
UAEM. He was winner of the
BootCamp 2013 organi
zed b
y
TechBA, also he is part of the
winner team of
a scholarship
to
be par
t
of
the pr
ogram FULL I
MMERSION in TechBA Silicon
Valle
y.
P
e
dro Vargas
ChablƩ R
e
c
e
ived
the B. S
c
. Degr
ee b
y
th
e Auton
o
m
ous
Univers
ity J
u
ar
ez from
Tabasco in 200
8. From 2009
to 2012, he was
Technical Specialist A
ssesme
n
t of Lighting
Conditions and
Non-Ionizing
Radiation,
NO
M-025-STPS-20
08 and NOM-013-STPS-199
3
res
p
ect
ivel
y,
in
Environm
enta
l Techno
log
y
S
.
A
of C.V. In 2014 he Receiv
e
d M
.
S
c
. Degree a
t
the Autonomous University
of
Morelos State (
UAEM). He is
a PhD student
at the Research
Center on Eng
i
neering
and Applied Science (
C
IICAp) of the UAEM. His current research
inter
e
s
t
ar
e F
E
A
,
m
i
crogripper
,
m
i
croa
ctua
tors
an
d VLS
I
.
S
v
etlan
a
Kos
h
eva
y
a re
ce
ived
t
h
e Diplom
a of
M
a
s
t
er from
F
acult
y of R
a
diop
h
y
s
i
cs
, P
h
y
s
i
cal
Electroni
cs Dept., Kiev Universi
t
y
, in 1964, the
Ph. D. in Radioph
y
s
ics from
Kiev Institute of
Radioproblems, Kiev University in 1969, and th
e diploma of Doctor of Science, from Kiev
Universit
y
, in 1
986. Dra. Koshevay
a worked
as Engineer (196
4-1968), in
Kiev Institut
e
of
Radioproblems, Ukraine. She was
Junior Senior
Research Scien
tist Research Scientist (1968
-
1970) in Kiev Institute of Rad
i
oproblem
s and Seni
or Research
Scientist in Ins
titut
e
"Orion"
(1979-1972), Kiev, Ukrain
e. In
Faculty
of R
a
dioph
y
s
ics of Kiev National Univ
ersity
, she w
a
s
Senior Resear
ch
Scientist (1972
-1974), Princip
a
l Lecturer (197
4-1980), Associate Professor
(1980-1987) and Full Professor (
1987 -1995). She was Titular Researcher "Cā (19
95- 1998), in
INAOE, Puebla, Mexico. Since
1998, she is Titu
lar Resear
cher
āCā at CIICAp, Autonomo
u
s
Univers
i
t
y
of S
t
at
e M
o
relos
(
UAEM
)
, Cuern
a
vac
a
, M
e
xico
.
Her res
e
arch
i
n
teres
t
s
in
clud
e
rem
o
te sensing
s
y
stem
for sei
s
m
and volca
no activ
it
y
,
phot
onics and subm
illim
eter wav
e
integr
ated techn
i
que, nonlin
ear
radiolo
cation,
and solitonics in nonlinear ph
y
s
ics. She has 7
books (in Russia), two chapters in books published in
English, one book with student in Spanish,
15 cert
i
fic
a
te of
paten
t
s, 196 Pap
e
rs in intern
at
io
nal journ
a
ls and
143 Articl
e
s in Proceedings of
S
y
mposiums . She is member of
Mexican Acad
emia
of Scien
c
e,
member of National S
y
s
t
em of
Res
earch
ers
(S
NI) and
M
e
m
b
er o
f
W
HO IS
W
H
O
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