Vietnam Journal of Mechanics, VAST, Vol.41, No. 1 (2019), pp. 79 – 87
DOI: https://doi.org/10.15625/0866-7136/12710
MECHANICAL RESPONSE OF OUTER FRAMES IN TUNING
FORK GYROSCOPE MODEL WITH CONNECTING
DIAMOND-SHAPED FRAME
Vu Van The1,∗, Tran Quang Dung1, Chu Duc Trinh2
1Le Quy Don Technical University, Hanoi, Vietnam
2Vietnam National University, Hanoi, Vietnam
∗E-mail: thevutb@gmail.com
Received: 24 July 2018 / Published online: 29 October 2018
Abstract. In tuning fork micro-g

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gyroscopes, two outer frames are connected by using the
linking elements. The driving vibrations of the two outer frames are required to be ex-
actly opposite to generate the opposite sensing modes perpendicular to driving direc-
tion. These opposite driving vibrations are provided by a mechanical structure named the
diamond-shaped frame. This paper presents mechanical responses of two outer frames in
a proposed model of tuning fork gyroscope when an external force with different types is
applied to them. The results show that the presence of a diamond-shaped frame guaran-
tees the absolute anti-phase mode for the driving vibrations of outer frames.
Keywords: tuning fork gyroscope; anti-phase mode; mechanical response.
1. INTRODUCTION
Gyroscopes are physical sensors that detect and measure the angle or angular ve-
locity of an object which relatively rotates in an inertial frame of reference. The name
“gyroscope” originated from a French Scientist, Lénon Foucault, combining the Greek
word “skopeein” meaning to see and the Greek word “gyros” meaning rotation, during
his experiments to measure the rotation of the Earth [1].
Micro-Electro-Mechanical Systems (MEMSs) are devices and systems integrated with
mechanical elements, sensors, actuators, and electronic circuits on a common silicon sub-
strate through micro-fabrication technology. A normal MEMS device consists of a central
unit that processes data, the microprocessor and several components that interact with
the outside such as micro-sensors and micro-actuators.
MEMS Vibratory Gyroscope (MVG) is a kind of micro-sensor used to detect and
determine the angular velocity or rotational angle of a body into which the MVG is inte-
grated. The operation of this micro-sensor is based on the Coriolis principle to transfer
energy from the primary vibration to a secondary one [1–3]. MVGs have been exten-
sively applied in automotive and aerospace industries and consumer electronics market
c 2019 Vietnam Academy of Science and Technology
80 Vu Van The, Tran Quang Dung, Chu Duc Trinh
given notable advantages including marked reduction in cost, size, and weight. Indeed,
previous researches into vibrational characteristics of the proof-mass in gyroscopes have
shown that the MVGs have many advantages over traditional gyroscopes for their small
size, low power consumption, low cost, batch fabrication and high performance [1–7].
The MEMS tuning fork gyroscope (TFG), which consists of two identical tines vibrat-
ing in opposite direction (anti-phase), is a widely used class of MVG. The advantage of
the tuning fork structure is the high resistance to the exciting phase deviation in operat-
ing [4–6]. However, the traditional MEMS tuning fork structure with the direct mechan-
ical coupling between two tines likely causes an in-phase vibratory mode [6]. Therefore,
it is necessary to design a novel TFG with a mechanism indirectly connecting two tines
to create an anti-phase state, where connecting mechanism plays an important role for
resisting phase deviation of two tines [7–9]. This mechanism is termed diamond-shaped
frame, and its detailed description can be found elsewhere [10].
2 This paper focuses onVu setting Van The, up Tran differential Quang Dung, equations and Chu Duc of Trinh motion and studying the
vibrations of two outer frames when they are connected indirectly by a diamond-shaped
frame.plays an These important outer role frames for resisting are expected phase deviation to vibrate of two with tines [7 the-9]. absolute This mechanism anti-phase is termed mode
todiamond create-shaped the anti-phase frame, and modeits detailed for description sensing vibrationscan be found ofelsewhere proof-masses [10]. in proposed TFG
model.This paper focuses on setting up differential equations of motion and studying the vibrations of
two outer frames when they are connected indirectly by a diamond-shaped frame. These outer frames
are expected to vibrate2. with CONFIGURATION the absolute anti-phase OF mode THE to PROPOSED create the anti- TFGphase mode for sensing
vibrations of proof-masses in proposed TFG model.
The proposed model consists of two identical tines as shown in Fig.1. Each tine is
deﬁned as a single gyroscope and includes a proof-mass (1) and an outer frame (2). The
conﬁguration and dynamic2. CONFIGURATION characteristics OF of THE each PROPOSED single gyroscope TFG are provided in [11].
This outerThe proposed frame is model connected consists to of the two proof-mass identical tines by as fourshown elastic in Fig. beams1. Each (3)tine andis defined suspended as a
onsingl substratee gyroscope thanks and includes to four a otherproof-mass elastic (1) and beams an outer (4). frame Each (2 of). The these configuration beams is and linked dynamic tothe
substratecharacteristics (not of be each presented single gyroscope in Fig.1 ) are by provided an anchor in [11] (5). This to allow outer the frame outer is connected frame and to the the
proof-massproof-mass by to four move elastic freely beam ins ( two3) and perpendicular suspended on substrate directions. thanks Two to four single other gyroscopes elastic beams are
connected(4). Each of through these beams a diamond-shaped is linked to the substrate frame (not to create be presented the proposed in Fig. 1) TFG.by an Thisanchor frame (5) to has
allow the outer frame and the proof-mass to move freely in two perpendicular directions. Two single
four rigid bars (7) with length L and the rectangular cross-section b × h, where h and
gyroscopes are connected through a diamond-shaped frame to create the proposed TFG. This frame
b hasare four the rigid thickness bars (7) and with width length ofL and each the bar, rectangular respectively. cross-section The b bars×h, where are connectedh and b are tothe the
connectorsthickness and by width elastic of stems each bar, (8) respectively. with the width The barss (s are< b connected). The conﬁguration to the connectors and by dynamics elastic
analysisstems (8) of with this the frame widthwas s (s< carriedb). The configuration out in our previousand dynamics research analysis [10 of]. this frame was carried
out in our previous research [10].
3 2 6 5 2
4
7
1 1
8
Fig. 1. 3D model of the proposed TFG with diamond-shaped frame
Fig. 1. 3D model of the proposed TFG with diamond-shaped frame
Fig. 2 describes a physical model of this TFG, where kx1, kx2, ky1, and ky2 are the equivalent
stiffness of the elastic beams; cx1, cx2, cy1, and cy2 are damping coefficients in x- and y-direction; mS1
and mS2 are values of the proof-masses; and mf1 and mf2 are masses of the outer frames. The points
(e.g. A, B, C, and D) are the nodes of the diamond-shaped frame.
ky/2
C
kx1 cy1/2 ky1/2 cy2/2 ky2/2 kx2
B
mS1 A mS2
cx1
cy1/2 k /2 c /2 k /2 cx2
y1 D y2 y2
ky/2
Fig. 2. Physical model of the TFG
2 Vu Van The, Tran Quang Dung, and Chu Duc Trinh
plays an important role for resisting phase deviation of two tines [7-9]. This mechanism is termed
diamond-shaped frame, and its detailed description can be found elsewhere [10].
This paper focuses on setting up differential equations of motion and studying the vibrations of
two outer frames when they are connected indirectly by a diamond-shaped frame. These outer frames
are expected to vibrate with the absolute anti-phase mode to create the anti-phase mode for sensing
vibrations of proof-masses in proposed TFG model.
2. CONFIGURATION OF THE PROPOSED TFG
The proposed model consists of two identical tines as shown in Fig. 1. Each tine is defined as a
single gyroscope and includes a proof-mass (1) and an outer frame (2). The configuration and dynamic
characteristics of each single gyroscope are provided in [11]. This outer frame is connected to the
proof-mass by four elastic beams (3) and suspended on substrate thanks to four other elastic beams
(4). Each of these beams is linked to the substrate (not be presented in Fig. 1) by an anchor (5) to
allow the outer frame and the proof-mass to move freely in two perpendicular directions. Two single
gyroscopes are connected through a diamond-shaped frame to create the proposed TFG. This frame
has four rigid bars (7) with length L and the rectangular cross-section b×h, where h and b are the
thickness and width of each bar, respectively. The bars are connected to the connectors by elastic
stems (8) with the width s (s<b). The configuration and dynamics analysis of this frame was carried
out in our previous research [10].
3 2 6 5 2
4
7
1 1
8
Mechanical response of outer frames in tuning fork gyroscope model with connecting diamond-shaped frame 81
Fig.2 describesFig. 1. a3D physical model of model the proposed of this TFG TFG, with where diamond-shapedkx1, kx2, kframey1, and ky2 are the
equivalent stiffness of the elastic beams; cx1, cx2, cy1, and cy2 are damping coefﬁcients
Fig. 2 describes a physical model of this TFG, where kx1, kx2, ky1, and ky2 are the equivalent
in x- and y-direction; mS1 and mS2 are values of the proof-masses; and m f 1 and m f 2 are
stiffness ofmasses the elastic of the beams; outer frames.cx1, cx2 The, cy1, points and cy (e.g.2 are A, damping B, C, and coefficients D) are the nodesin x- and of the y-direction; diamond-mS1
and mS2 areshaped values frame. of the proof-masses; and mf1 and mf2 are masses of the outer frames. The points
(e.g. A, B, C, and D) are the nodes of the diamond-shaped frame.
ky/2
C
kx1 cy1/2 ky1/2 cy2/2 ky2/2 kx2
mS1 A B mS2
cx1
cy1/2 k /2 c /2 k /2 cx2
y1 D y2 y2
ky/2
Fig. 22.. Physical model of of the the TFG TFG
3. DIFFERENTIAL EQUATIONS OF MOTION
The four beams of the diamond-shaped frame are assumed to be absolutely rigid.
The displacement at the end of the beams (A, B, C, and D) is carried out by the elasticity
of stems with the smaller section. When the diamond-shaped frame links two single
gyroscopes to create tuning fork structure, points A and B only displace in x-direction
and points C and D only do in y-direction. The displacement of point A is x1, while
point C displaces y1 from the initial position. Points B and D are the same displacements
with A and C except for the direction of motion (Fig. 3(a)). These displacements depend
mutually and have a relation as follows
q
2 2
y1 = L − (L1 − x1) − L2 ,
q (1)
2 2
y2 = L − (L1 − x2) − L2 ,
where L1 = Lcosa0, L2 = Lsina0, and a0 is the angle to deﬁne initial position of rigid bars
of the diamond-shaped frame.
Thence the elastic forces are deﬁned by the followed expressions
FDy = kyyD/2 = kyy1/2, (2)
FCy = kyyC/2 = kyy2/2. (3)
MechanicalMechanical response response of of outer outer frames frames in in TFG TFG model model with with connecting connecting diamond-shaped diamond-shaped frame frame 3 3
3.3. DIFFERENTIAL DIFFERENTIAL EQUATIONS EQUATIONS OF OF MOTION MOTION
TheThe four four beams beams of of the the diamond-shaped diamond-shaped frame frame are are assumed assumed to to be be absolutely absolutely rigid. rigid. The The
displacementdisplacement at at the the end end of of the the beams beams (A, (A, B, B, C, C, and and D) D) is iscarried carried out out by by the the elasticity elasticity of of stems stems with with the the
smallersmaller section. section. When When the the diamond-shaped diamond-shaped frame frame links links two two single single gyroscopes gyroscopes to to create create tuning tuning fork fork
structure,structure, points points A A and and B B only only displace displace in in x -directionx-direction and and points points C C and and D D only only do do in in y -direction.y-direction. The The
displacementdisplacement of of point point A Ais is x 1x, 1while, while point point C C displaces displaces y 1yfrom1 from the the initial initial position. position. Points Points B B and and D D are are
thethe same same displacements displacements with with A A and and C C except except for for the the direction direction of of motion motion (Fi (Fig.g. 3a). 3a). These These
displacementsdisplacements depend depend mutually mutually and and have have a arelation relation as as follows: follows:
y22 L22 () L x L
y1 1 L () L1 1 x 1 1 L 2 2
(1)(1)
y22 L22 () L x L
y2 2 L () L1 1 x 2 2 L 2 2
82 Vu Van The, Tran Quang Dung, Chu Duc Trinh
wherewhere L 1L1= =L cosLcosα0α, 0,L 2L2= =L sinLsinα0α, 0, and andα0α0isis the the angle angle to to define define initial initial position position of of rigid rigid bars bars of of the the
diamond-shapeddiamond-shaped frame. frame.
C C
y C' C
' y2 2
C
FFy
y F
A B B FFx AB Fx x
A A'A ' B'B ' x A B
x x
x1 1 x2 2
Fy
Fy
DD
y1y1
D'D ' DD
a)(a)a) (b)b)b)
Fig.Fig. 3. 3. Schema Schema of of deformation deformation (a) (a) and and elastic elastic forces forces (b) (b) of of diamond-shaped diamond-shaped frame frame
Fig. 3. Schema of deformation (a) and elastic forces (b) of diamond-shaped frame
ThenceThence the the elastic elastic forces forces are are defined defined by by the the followed followed expressions: expressions:
FFDy k k y y y D/ 2/ 2 k k y y y /1 2 / 2 (2)(2)
The elastic force applied to theDy outer y D frames y 1 in x-direction is determined as the fol-
lowed expression F k y/ 2 k y / 2 (3)
FCyCy k y y y C C/ 2 k y y y2 /2 2 (3)
1
The elastic force applied to F the= outer(F frames+ F in) cotgx-directiona, is determined as the followed (4)
The elastic force applied to thex outer2 framesCy inDy x-direction is determined as the followed
4 expression:expression: Vu Van The, Tran Quang Dung, and Chu Duc Trinh
with a is an angular rotation of a rigid beam when diamond-shaped frame operated.
11 y
The Eq. (4) describes the relationFFF between( elastic )cotg forces in -direction and the corre-(4)
FFFxx( Cy Cy Dy Dy )cotg (4)
dampingsponding forces force with in dampingx-direction. coefficients cx21 2and cx2 known as the slide air damping between the
masseswithwith and αIn αis theis thisan an angularsubstrate, angular issue, rotation bothrotation and F outerofx of isa arigidelastic rigid frame beam beamforce and when when mentioned proof-mass diamond diamond above.-shaped-shaped vibrate frame frame in operated. operated. the driving direction. In
essence, they are considered as one element with total mass m and m , respectively (m =
TheThe Eq. Eq. (4) (4) describes describes the the relation relation between between elastic elastic forces forces in in y -directiony-direction1 and 2and the the corresponding correspondingi
FL1 k x 1 x 1; F L 2 k x 2 x 2 (5)
forcemforcef i + in inm x -direction.Six-direction.). The component forces applying to the masses m1 and m2 are shown in Fig.4
after splitting them. F c x; F c x (6)
InIn this this issue, issue, both both outer outer frame frame and andC1 proof-mass proof-mass x 1 1 C 2 vibrate vibrate x 2 2 in in the the driving driving direction. direction. In In essence, essence, they they
areare considered considered as as one one element element with with total total mass mass m m1 and1 and m m2, 2respectively, respectively (m (mi =i =m mfi +fi +m mSi).Si ).The The component component
F F2
forcesforces applying applying to to the the masses massesmm1 and1 and m m2 are2 are shown shown1 in in Fig. Fig. 4 4after after splitting splitting them. them.
x1(t) x2(t)
In Fig. 4 F and F are external forces applied to the outer frames; F and F are elastic
In Fig. 4 F1 1and F2 2are external forces applied to the outer frames; FL1Land1 FL2L2are elastic
F FL2
forces of elastic beamsL1 with the stiffness coefficientsF F kx1 and kx2 respectively; F and F are
forces of elastic beams with the stiffness coefficientsx x kx1 and kx2 respectively; FC1C1and FC 2C 2are
m1 m2
F FC2
C1
Fig. 4. The forces applied to the outer frames
Fig. 4. The forces applied to the outer frames
By using the second Newton law, the motion differential equations for the system are obtained
as follows:
~ ~ ~ ~
In Fig.4, F1 and F2 are external forces applied to the outer frames; FL1and FL2 are
m1 x 1 FL 1 F C 1 F x F 1 ~
elastic forces of elastic beams with the stiffness coefﬁcients kx1 and kx2 respectively;(7F)C 1
~
and FC2 are damping forces withm2 x damping 2 FL 2 F coefﬁcients C 2 F x F 2 cx1 and cx2 known as the slide air
damping between the masses and the substrate, and ~F is elastic force mentioned above.
Equations (7) are expanded as: x
F = k x ; F = k x , (5)
L1 x1 11 L2 x2 2
m1 x 1 cx 1 x 1 k x 1 x 1 k y ( y 1 y 2 )cotg F 1
F = c 4x˙ ; F = cx x˙ . (6)
C1 x1 1 C2 2 2 (8)
1
m x c x k x k( y y )cotg F
2 2x 2 2 x 2 24 y 1 2 2
Adding some equations describing the geometric relations between the displacements x1, x2 and
y1, y2:
LLLL1cos 0 ; 2 sin 0
22
y1 L () L 1 x 1 L 2
y L22 () L x L (9)
2 1 2 2
y1
arc tg
x1
the differential equations of motion for the system with displacements in x-direction become:
1
mxcxkxkLLx 2 ( ) 2 LLx 2 ( ) 2 2 L cotg F
111111x x4 y 11 12 2 1
1
mxcxkx kLLx2 ( ) 2 LLx 2 ( ) 2 2 L cotg F
222222x x4 y 11 12 2 2
LL10 cos (10)
LL sin
20
L22() L x L
atan1 1 2
x
1
Mechanical response of outer frames in tuning fork gyroscope model with connecting diamond-shaped frame 83
By using the second Newton law, the motion differential equations for the system
are obtained as follows
¨ ~ ~ ~ ~
m1~x1 = FL1 + FC1 + Fx + F1,
¨ ~ ~ ~ ~ (7)
m2~x2 = FL2 + FC2 + Fx + F2.
Eqs. (7) are expanded as
1
m x¨ + c x˙ + k x + k (y + y )cotga = F ,
1 1 x1 1 x1 1 4 y 1 2 1
(8)
1
m x¨ + c x˙ + k x + k (y + y )cotga = F .
2 2 x2 2 x2 2 4 y 1 2 2
Adding some equations describing the geometric relations between the displacements
x1, x2 and y1, y2
L1 = L cos a0; L2 = L sin a0,
q
2 2
y1 = L − (L1 − x1) − L2,
q
2 2 (9)
y2 = L − (L1 − x2) − L2,
y
a = arctg 1 .
x1
The differential equations of motion for the system with displacements in x-direction
become
1 q q
m x¨ + c x˙ + k x + k L2 − (L − x )2 + L2 − (L − x )2 − 2L cotga = F ,
1 1 x1 1 x1 1 4 y 1 1 1 2 2 1
1 q q
m x¨ + c x˙ + k x + k L2 − (L − x )2 + L2 − (L − x )2 − 2L cotga = F ,
2 2 x2 2 x2 2 4 y 1 1 1 2 2 2
L1 = L cos a0,
L2 = L sin a0,
p !
L2 − (L − x )2 − L
a = atan 1 1 2 .
x1
(10)
4. VIBRATIONAL CHARACTERISTICS OF TWO OUTER FRAMES
The result of the vibratory problem is received by using Matlab software with suit-
able initial parameters. In this paper, the research cases consist of: free vibration; forced
vibration with harmonic impulse form. The characteristic parameters of the TFG system
are shown in Tab.1.
4.1. Free vibration
In order to demonstrate the ability to vibrate in x-direction, a study on free vibration
of the system is carried out ﬁrstly. When the system has no exciting force and the initial
−5 −5
parameters are: x1 = 2.5 × 10 m, x2 = 2.5 × 10 m, the free vibration of two element
masses is showed in Fig.5.
The results show that the outer frames are able to oppositely vibrate in the driving di-
rection. Their amplitude seems like a constant in a short time. However, the air damping
Mechanical response of outer frames in TFG model with connecting diamond-shaped frame 5
4. VIBRATIONAL CHARACTERISTICS OF TWO OUTER FRAMES
The result of the vibratory problem is received by using Matlab software with suitable initial
parameters.
In this paper, the research cases consist of: free vibration; forced vibration with harmonic
impulse form.
The characteristic parameters of the TFG system are shown in Table 1.
Table 1. The parameters of the TFG
Parameter Value Unit
-7
Mass of the left single gyroscope: m1 2.65×10 kg
-7
Mass of the right single gyroscope: m2 2.65×10 kg
Driving stiffness of the left single gyroscope: kx1 25.2 N/m
Driving stiffness of the right single gyroscope: kx2 25.2 N/m
-5
84Damping coefficient in Vu left Van drive The, Trandirection: Quang Dung,cx1 Chu Duc2×10 Trinh kg/s
-5
Damping coefficient in right drive direction: cx2 2×10 kg/s
Stiffness of diamond-shapedTable 1frame. The in parameters y-direction: of k they TFG10 N/m
Length of a rigid bar: L 10-4 m
0
Initial angle of rigidParameter bar: α0 60 Valuedegree Unit
−7
Mass of the left single gyroscope: m1 2.65×10 kg
−7
4.1. FreeMass vibration of the right single gyroscope: m2 2.65×10 kg
InDriving order to stiffness demonstrate of the the left ability single to gyroscope: vibrate in x-kdirectionx1 , a study 25.2 on free vibration N/m of the
system Drivingis carried stiffness out firstly. of the When right the single systemgyroscope: has no excitingkx2 force and the25.2 initial parameters N/m are:
-5 -5 −5
x1 = 2.5×10Dampingm, x2 coefﬁcient= 2.5×10 m, in the left free drive vibration direction: of twoc elementx1 masses is2 showed×10 in Fig. 5. kg/s
c × −5
TheDamping results show coefﬁcient that the in outer right frames drive direction:are able to oppositelyx2 vibrate2 in10 the driving direction.kg/s
Their amplitudeStiffness se ofems diamond-shaped like a constant in frame a short in time.y-direction: However,ky the air damping10 causes decreasing N/m
−4
slowly Length in amplitude of a rigid to time. bar: TwoL elements vibrate around equilibrium position10 (±5×10-5 m)m with
-5 ◦
decreasedInitial amplitude angle after of rigid every bar: perioda0 (initial amplitude 2.5×10 m). These vibr60 ations are symmetricdegree
through the centre of the diamond-shaped frame.
Fig. 5. The free vibration of two outer frames
Fig. 5. The free vibration of two outer frames
causes decreasing slowly in amplitude to time. Two elements vibrate around equilibrium
position (±5 × 10−5 m) with decreased amplitude after every period (initial amplitude
2.5 × 10−5 m). These vibrations are symmetric through the centre of the diamond-shaped
frame.
4.2. Force vibration
When applying two external forces to the outer frames with a constant value (3 ×
10−4 N) and their direction from the centre of the diamond-shaped frame to each outer
frame, these outer frames vibrate from their equilibrium position (±5 × 10−5 m) and
outward from the centre (Fig. 6(a)). While vibrations of them are anti-phase mode when
applied forces with opposite direction toward to the centre (Fig. 6(b)). The amplitudes in
both cases of external forces are the same value (23 mm).
According to the results in Fig.6, the vibrational amplitude appears as constant
through some continuous periods. These vibrations still guarantee anti-phase mode and
symmetry through the centre of the diamond-shaped frame.
6 6 Vu VuVan Van The, The, Tran Tran Quang Quang Dung, Dung, and andChu ChuDuc DucTrinh Trinh
4.2.4.2. Force Force vibration vibration
-4
6When When applying applying two two external externalVu force Van forces The, tos t hetoTran touterhe Quang outer frames Dung, frames with and withChu a constant Duc a constant Trinh value value (3×10 (3×10N) and-4N) theirand their
directiondirection from from the the ce ntrecentre of theof the diamond diamond-shaped-shaped frame frame to each to each outer outer frame frame, the,se the outerse outer frames frames vibrate vibrate
-5
fromfrom their4.2. their Forceequilibrium equilibrium vibration position position (±5×10 (±5×10m)-5 m)and and outward outward from from the centrethe centre (Fig .(Fig 6a).. 6Whilea). While vibrations vibrations of of
them are anti-phaseMechanical mode response when of outerapplied frames forces in tuning with fork opposite gyroscope modeldirection with connecting toward diamond-shapedto the centre frame(Fig. 6b). 85
them are anti-phase mode when applied forces with opposite direction toward to the-4 centre (Fig. 6b).
The amplitudesWhen in applyingboth cases two of external external force forcess to tarehe outerthe same frames value with (23 a constant μm). value (3×10 N) and their
Thedirection amplitudes from in the both centre cases of theof externaldiamond -forcesshaped are frame the to same each value outer frame(23 μ,m). the se outer frames vibrate
from their equilibrium position (±5×10-5m) and outward from the centre (Fig. 6a). While vibrations of
them are anti-phase mode when applied forces with opposite direction toward to the centre (Fig. 6b).
The amplitudes in both cases of external forces are the same value (23 μm).
a) Outward from the centre b) Toward to the centre
(a)a) O Outwardutward from from the the centre centre b) Toward(b) Towardto the centre to the centre
Fig. 6. Vibration of outer frames with constant external force
a) OutwardFig. 6from. Vibration the centre of outer frames with constantb) T externaloward to forcethe centre
According to theFig. results 6. Vibration in Fig. of6, outerthe vibrationa frames withl amplitude constant appears external as force constant through some
According to the resultsFig in . Fig.6. Vibration 6, the ofvibrationa outer framesl amplitude with constant appears external as force constant through some
continuouscontinuous periods periods. These. These vibrations vibrations still still guarantee guarantee anti anti-phase-phase mode mode and andsymmetry symmetry through through the centre the centre
of the diamondAccording-shaped to the frame. results in Fig. 6, the vibrational amplitude appears as constant through some
of the continuousdiamondThe- relationshaped periods frame.. betweenThese vibrations the vibratory still guarantee amplitude anti-phase and mode the and force symmetry value through in case the of centre changing
theofThe theThe value relation diamond relation of kbetween-yshaped betweenis shown frame.the the vibratory in vibratory Fig.7 .amplitude The amplitude displacements and andthe theforc force of value outere value in framescase in case of inchanging of driving changing the direction value the value
−
of kshouldy is shown be in smaller Fig. 7. thanThe displacements 3 × 10 5 m, henceof outer the frames value in ofdriving exciting direction force should should be besmaller smaller than
of ky is -shown5 The in relation Fig. 7. betweenThe displacements the vibratory of amplitude outer frames and the in drivingforce value direction in -4case should of changing be smaller the value than
3×10-5 m, hence− the4 value of exciting force should be smaller than 4×10-4 N (Fig. 7).
3×10thanof m, ky henceis 4 ×shown10 the in valueN Fig. (Fig. of7. Theexciting7). displacements force should of outer be smaller frames thanin driving 4×10 direction N (Fig. should 7). be smaller than
3×10-5 m, hence the value of exciting force should be smaller than 4×10-4 N (Fig. 7).
Fig.Fig. 7. 7. Relation Relation betweenbetween vvibrationalibrational amplitude amplitude and and value value of external of external force force
Fig.Fig. 7. Relation 7. Relation between between vibrationalvibrational amplitude amplitude and and value value of external of external force force
TheThe exciting exciting harmonic harmonic force force appliedapplied toto outer outer frames frames should should ha veha ave form a form as follows: as follows:
The exciting harmonic force applied to outer frames should have a form as follows:
F Fsin2 ft ; F F sin(2 ft ) (11)
The exciting harmonicF force11 F 0 appliedsin2 ft ; toF 2 2 outer 0 F 0 sin(2 frames ft should ) have a form as follows(11)
F1 F 0sin2 ft ; F 2 F 0 sin(2 ft ) (11)
Function (11) is defined by the force value F0 and the exciting frequency f. These parameters
Function (11) is definedF1 by= Fthe0 sin force 2p fvalue t; F F2 0= andF0 sinthe (exciting2p f t + pfrequency). f. These parameters (11)
are aredeterminedFunction determined (11) by by isanalyz analyzdefinedinging bythe the the amplitudeamplitude force value -- frequencyfrequency F0 and response theresponse exciting of theof frequency thesystem system. T of. . reduce TheseTo reduce theparameters time the oftime of
arethe determined the calculation calculationFunction by andanalyz and (11 guarantee ) guaranteeing is deﬁned the amplitude the the by eff efficiency, the - forcefrequency the the value exciting exciting responseF0 frequencyand frequency of the the exciting ofsystem the of the system. frequencyT systemo reduce is assess is the fassessed. Thesetime fromed of pa-from
1400 Hz to 1800 Hz (Fig. 8a). According to the result shown in Fig. 8a, the exciting frequency should
the1400 calculationrameters Hz to 1800 areand determinedHz guarantee (Fig. 8 a the). byAccording eff analyzingiciency, to the the the exciting result amplitude shown frequency in - frequencyFig. of 8 thea, the system response exciting is assess frequency of theed system.from should
1400 ToHz reduceto 1800 theHz ( timeFig. 8 ofa). the According calculation to the and result guarantee shown in Fig. the efﬁciency,8a, the exciting the excitingfrequency frequency should
of the system is assessed from 1400 Hz to 1800 Hz (Fig. 8(a)). According to the result
shown in Fig. 8(a), the exciting frequency should be 1590 Hz. With this frequency, the
86 Vu Van The, Tran Quang Dung, Chu Duc Trinh
MechanicalMechanical response response of outer of outer frames frames in TFG in modelTFG model with connecting with connecting diamond diamond-shaped-shaped frame frame7 7
amplitude of driving vibration depends on the value of F0. To match with the conﬁgu-
ration of the diamond-shaped frame presented in [10] (i.e. less than 30 mm), the value of
be 1590 Hz. With this frequency, the amplitu

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