Dynamic Analysis and Experimental Study of Self ...paper.uscip.us/jvamc/jvamc.2014.1001.pdf ·...
Transcript of Dynamic Analysis and Experimental Study of Self ...paper.uscip.us/jvamc/jvamc.2014.1001.pdf ·...
Columbia International Publishing Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15 doi:10.7726/jvamc.2014.1001
Research Article
______________________________________________________________________________________ Corresponding e-mail: [email protected] 1* MCC Capital Engineering & Research Incorporation Limited, Beijing 100176, China 2 China ENFI Engineering Corporation, Beijing 100038, China 3 School of Mechanical Engineering and Automation, Northeastern University, Shenyang,
Liaoning 110004, China 1
Dynamic Analysis and Experimental Study of Self-synchronous Vibrating System
Driven by Three Motors
Degang Wang 1*, Ruijun Han 2, Hongliang Yao 3, and Bangchun Wen 3 Received 6 November 2013; Published online 22 March 2014 © The author(s) 2014. Published with open access at www.uscip.us
Abstract The self-synchronous theory of the vibrating system driven by three motors is studied in this paper. Mathematics model of electromechanical coupling of the vibrating system driven by three motors is established, and the self-synchronization and stability conditions of synchronous operation of the vibrating system are deduced by using Hamilton principle. Synchronous experimental table is used to study the characteristic of vibration synchronization. The synchronous experimental table can implement stable operation of vibration synchronization when the parameters of the synchronous experimental table meet the self-synchronization and stability conditions of synchronous operation deduced in the theoretical analysis. The experimental analysis validates the correctness of the theoretical analysis to the vibrating system, it provides theoretical basis for the design of the vibrating system driven by three or more motors.
Keywords: Vibrating system; Self-synchronization; Stability; Experiment
1. Introduction In vibration machine field, the mechanical system driven by two or more motors needs generally cooperate to complete uniform task, so that the synchronization characteristic of the machine is realized, and the technics purpose is reached. Self-synchronous vibrating machines with two or more motors are widely used in industry, such as molding machine, cooling machine and conveyor based on the theory of self-synchronous vibration. So it is important to apply vibration synchronization theory to mechanical engineering. Therefore, the further study should be done to investigate the dynamics characteristics of the synchronous vibrating system and to provide theoretical basis for the design of this sort of system by using the theory of electromechanical coupling. There are significant theory and engineering value for solving engineering application problem.
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
2
Scientific studies show that macroscopic dynamic behavior of complex system not only depends on motional characteristics of each subsystem, but has close relationship with interaction among subsystems. The change of motional characteristics and forms of complex system can be affected directly by interaction among the subsystem’s motion, it can result in new motional structure. It is important content of complex system science to study characteristics of interaction among the subsystem’s motion and the influence of the interaction among the subsystem’s motion to dynamic behavior of complex system. Zhang Yimin presented a generalized probabilistic perturbation finite element method and employed the method to solve the response analysis of multi-degree-of -freedom nonlinear vibrating systems with random parameters and vector-valued and matrix-valued functions (Zhang et al., 2003). Zhang Tianxia studied coupling effect in a synchronous vibrating system, and then he derived a differential equation with an analytical method of nonlinear vibration for the coupling motion of two eccentric rotors to describe mathematically the coupling parameters of the system. Then he studied the synchronization development course of two eccentric rotors in depth, and deducted the necessary coupling conditions to for synchronization state (Zhang et al., 2003). Zhao Chunyu analyzed the dynamic characteristic of a vibrating system with two-motor drives rotating in the reverse direction by using the dynamic theory and established the equations of frequency capture, he obtained the conditions of implementing frequency capture and the equations of calculating capturing frequency and the phase (Zhao et al., 2009). Wang Degang studied the dynamic coupling feature of a vibrating system driven by two motors in the same direction, converted the problem of synchronization into that of existence and stability of zero solution for the equation of frequency capture, and then obtain the condition of frequency capture and that of stable self-synchronous operation (Wang et al., 2010). These studies mostly analyzed motional rules of synchronous vibrating system driven by two motors, few studies analyzed system driven by three or more motors. To large vibrating machine, it is not fit to adopt two motors to drive vibrating system because of the large structure. It should adopt three or more motors to drive based on the structure need. There are no effective methods to analyze the self-synchronization and stability conditions of synchronous operation of the vibrating system driven by three or more motors. Because of complexity of dynamic model and restriction of mathematics method available, the research to self-synchronization and stability conditions of synchronous operation of the vibrating system is carried out mostly around approximate synchronous state. There are no further studies to nonlinear dynamic mechanism of synchronous system and synchronous stability problems. Blekhman (Blekhman, I.I., 1988) and Wen(Wen and Liu, 1983; Wen et al., 2003; Wen et al., 2005) studied self-synchronization problem by the approach of small parameter and averaging method. In this method, the average angular velocity of two motors is assumed to be constant and only phase difference between two exciters is considered to a small variable parameter. Zhao (Zhao et al., 2009) considered the difference of the two motors of a vibration system with two-motor drives rotating in the reverse direction, and then he analyzed the dynamic
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
3
characteristic by using dynamic theory and established the equations of frequency capture. He obtained the conditions of implementing frequency capture and the equations of calculating capturing frequency and the phase. He deduced the stable equations of implementing frequency capture in the system by using nonlinear dynamics theory and obtained the stable condition of synchronization by using Routh-Hurwitz Criterion. The method is very good to solve the problem of self-synchronous theory of the vibrating system driven by two motors. In addition, there are few scholars study self-synchronization problem driven by three or more motors by using experimental method, the theoretical analysis can not validate by experimental analysis. In this paper, the self-synchronous theory of the vibrating system driven by three motors is studied in depth, the difference of electromagnetic torque and the load torque of the three motors is considered in analysis process. By introducing three small variable parameters to average angular velocity of three exciters and their phase difference, the self-synchronization and stability conditions of synchronous operation of the vibrating system are gotten based on Hamilton principle of holonomic nonconservative dynamical system and stability of constrained system. It can indicate physical meaning of holonomic dynamical system by using Hamilton principle and stability of constrained system to solve the problem of self-synchronous theory of the vibrating system driven by three motors. The experimental study to the vibrating system of the synchronous experimental table verifies the correctness of the theoretical analysis.
2. Self-synchronous Theory of Vibrating System Driven by Three Motors
2.1 Mathematical Model of a Vibrating System Driven by Three Motors Dynamic model of a vibrating system driven by three motors in reverse direction is shown in Fig 1. O in the figure is the system centroid, O and O are its synthesized centroid. Oxy is fixed coordinates, yxO is moving coordinates. '''OO is the distance from synthesized centroid to
system centroid, 0
''' lOO . O1、O2、O3 are rotative centers of the three exciting motors, and they
are in one lines. O'O1=l1, O'O2=l2, O'O3=l3.
3O 2O 1O1
11rm
2
22rm
333rm
1
20
3
x
x
y y
O
xk2
1xk
2
1
yk2
1yk
2
1
OO
1l
2l3l
Fig 1. Dynamic model of a vibrating system driven by three motors in reverse direction
In working process, the first motor rotates in counter-clockwise direction, the second and third one rotate in clockwise direction. The three motors which supply by the same power drive three eccentric lumps each other. The three eccentric lumps excite the system to vibrate. The motions of the system are vibrations in horizontal direction x, in vertical direction y and in rocking
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
4
direction ( <<1) (Blekhman, I.I., 1988; Wen and Liu, 1983; Wen et al., 2003; Wen et al., 2005). With aid of the Langrange’s equations, in addition, choosing x, y, , φ1, φ2 and φ3 as variable parameters, the vibration equation can be established through the expression of kinetic energy and potential energy of the system. The kinetic energy expression of the vibrating system is shown in Equation (1).
3
1
22
0
2
33333
2
333333
2
22222
2
222222
2
11111
2
111111
2
00
2
000
2
1
2
1
coscossinsin2
1
coscossinsin2
1
coscossinsin2
1
πcosπsin2
1
i
iiJJ
rlyrlxm
rlyrlxm
rlyrlxm
lylxmT
(1)
where x, y and are the displacements of the vibrating body in the directions x, y and ,
respectively, , , yx are speeds of the vibrating body in the directions x, y and , respectively,
m0 is mass of the vibrating body (not including the three eccentric lumps), mi (i=1, 2, 3) are masses of the three eccentric lumps, respectively, ri(i=1, 2, 3) are eccentricities of the three eccentric lumps, respectively,
i (i=1, 2, 3) are angular displacements of the three eccentric lumps, respectively,
i (i=1, 2, 3) are angular velocities of the three eccentric lumps, respectively,
J0 is the moment of inertia of the vibrating body (not including the three eccentric lumps),
iJ (i=1, 2, 3) are the moments of inertia of the three eccentric lumps encircled respective
rotative centers. The potential energy expression of the vibrating system is shown in Equation (2).
222
2
1
2
1
2
1kykxkV
yx (2)
where kx, ky, and k are the stiffness of spring in the directions x, y, , respectively. The expression of energy dissipate function of the vibrating system is shown in Equation (3).
2
33
2
22
2
11
222 )(2
1)(
2
1)(
2
1
2
1
2
1
2
1 ffffyfxfD
yx (3)
where fx, fy and f are the damping coefficient in the directions x, y, , respectively,
if (i=1, 2, 3) are the damping coefficient of the three rotors, respectively.
Assuming x, y, and
i are generalized coordinates
iq , then we can get the expression of
generalized force which is shown in Equation (4).
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
5
3f3e
2f2e
1f1e
332211)()()(
TT
TT
TT
ffff
yf
xf
Q
y
x
i
(4)
where
iT
e(i=1, 2, 3) are the electromagnetic torque of the three motors, respectively,
iT
f(i=1, 2, 3) are the load torque of the three motors, respectively.
The Langrange’s equation is shown in Equation (5).
i
iiii
D
q
V
q
T
q
T
t
d
d (5)
Applying the expressions of kinetic energy, potential energy, energy dissipate function and generalized force into Equation (5), we can derive the rotation differential equations of the vibrating system in three directions and that of the three eccentric lumps which are shown in Equation (6).
)cossin(
)cossin()cossin(
2
2
22222
3
2
333331
2
11111
rm
rmrmxkxfxMxx
)sincos(
)sincos()sincos(
2
2
22222
3
2
333331
2
11111
rm
rmrmykyfyMyy
)]sin()cos([
)]sin()cos([
)]sin()cos([
)()()(
22
2
2222222
33
2
3333333
11
2
1111111
223311
rlm
rlm
rlm
fffkfJ
)]sin()cos([
]cossin[)(
11
2
1111111
1111111f1e110
rlm
yxrmfTTJ
3,2)]sin()cos([
]cossin[)(
2
fe0
irlm
yxrmfTTJ
iiiiiiiii
iiiiiiiiii
(6)
where M is mass of the vibrating system (including the three exciting motors and the three
eccentric lumps), J is the moment of inertia of the vibrating system surround O (including the three exciting
motors and the three eccentric lumps),
3
1
22
000
i
iilmlmJJ ,
x , y and are accelerations of the vibrating body in the directions x, y and ,
respectively,
i (i=1, 2, 3) are angular accelerations of the three eccentric lumps.
Assuming that the phase of eccentric rotor 1 leads that of eccentric rotor 2 by
12 and that of
eccentric rotor 2 leads that of eccentric rotor 3 by23
, i.e. 1221
, 2332
. Assuming
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
6
that the average phase of the three eccentric rotors is when the vibrating system operates at
stable state, and then we have:
t
23123
122
121
2
1
2
1
2
1
(7)
where is the average angular velocity of the three motors when the vibrating system operates at stable state. Assuming the instantaneous variation coefficients of ,
12 and
23 are
1 ,
2 and
3 (
1 ,
2 and
3 are the functions of time t),
323
212
1)1(
(8)
From Equations (7) and (8), we can obtain the angular velocities of the three motors:
)2
11(
)2
11(
)2
11(
32133
2122
2111
(9)
The angular accelerations of the three motors can be obtained by Equation (9).
)2
1(
)2
1(
)2
1(
3213
212
211
(10)
For the operating speed of the induction motor is slightly lower than synchronous speed, the influence of
1 ,
2 and
3 can be ignored when the system operate in a stable state. And then
the prior three equations of Equation (6) can be rewritten to Equation (11).
)sin()sin(
)sin()()()(
sinsinsin
coscoscos
22
2
222233
2
3333
11
2
1111223311
2
2
2223
2
3331
2
111
2
2
2223
2
3331
2
111
rlmrlm
rlmfffkfJ
rmrmrmykyfyM
rmrmrmxkxfxM
yy
xx
(11)
The effect of the damping to the amplitude of the vibrating system can be neglected because the damping constant is very small ( 07.0 ) (Wen and Liu, 1983; Wen et al., 2003). So the
response of x, y and in Equation (11) can be expressed approximately as
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
7
)sin(cos
12'
x
x
x
m
Ax
)sin(cos
22'
y
y
y
m
By
)sin(cos
32
J
C (12)
where
)arctan(
x
x
xm
f
, )arctan(
y
y
ym
f
, )arctan(
J
f
,
2x
x
kMm ,
2
y
y
kMm ,
2
kJJ ,
)(cos)]sin(sin)cos([12312
22
3
2
12312312
22
2
2
1221aAaAAAAA ,
)(sin)]cos(sin)cos([12312
22
3
2
12312312
22
2
2
1221bBbBBBBB ,
)(sin
)]cos()(sin)]cos([[
1312312
22
3
2
131231232112
22
2
2
211221
cC
cCCCCC
,
211215.0 aa ,
211225.0 bb ,
1211235.0 cc ,
2
11111rmBA ,
2
22222rmBA ,
2
33333rmBA ,
2
11111rlmC , 2
22222rlmC , 2
33333rlmC ,
122
1221
1sin
cosarctan
A
AAa
,
)sin(sin)cos(
)cos(arctan
12312312
22
2
2
1221
123123
2
aAAAA
aAa
,
1221
122
1cos
sinarctan
BB
Bb
,
)cos(sin)cos(
)sin(arctan
12312312
22
2
2
1221
123123
2
bBBBB
bBb
,
)cos(
)sin(arctan
211221
21122
1
CC
Cc ,
)cos()(sin)]cos([
)sin(arctan
131231232112
22
2
2
211221
13123123
2
cCCCC
cCc
.
2.2 Self-synchronization Conditions of the Vibrating System The overall kinetic energy of the vibrating system includes motional kinetic energy of the vibrating body, rotatory kinetic energy of the vibrating body surround its centroid, rotatory kinetic energy of the three eccentric lumps. The expression of the overall kinetic energy is shown in Equation (13).
3,2,12
1
2
1
2
1 222 iTJyMxMTi
(13)
where )3,2,1( iTi
is rotatory kinetic energy of the three eccentric lumps, respectively.
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
8
Rotatory kinetic energy of the eccentric lumps can be seen constant when motor operates in a stable state. The expression of the potential energy is shown in Equation (2). By using Hamilton principle (Bian and Du, 1998; Qiu, 1998), we can get the Hamilton action over a single period when the vibrating system operates in a stable state.
i
y
y
x
x TJ
C
m
B
m
AtVTH π2
2
cosπ
2
cosπ
2
cosπ)(d)(
2
22
2
22
2
22π2
0
(14)
The elastic force and damping force of the vibrating system is far less than the inertia force and exciting force of the vibrating body. If the influence of elastic force and damping force is neglected, the dynamical system is affected by driving moment
iM
g(i=1, 2, 3) and friction torque
iM
f(i=1,
2, 3), besides potential force, i.e. gravitation. So the system is a holonomic nonconservative dynamical system. By using Hamilton principle of holonomic nonconservative dynamical system, we can obtain
0)(dδδπ2
0
2
1
tqQHi
i
i (15)
where i
Q is generalized force of the system, i
q is generalized coordinates.
In this system,
12 and
23 are generalized coordinates, so the expression of generalized force
1Q and
2Q can be shown in Equation (16).
)()(
)(2
1)(
2
1)(
2
1)(
3f3g
3
1 23
fg2
3f3g2f2g1f1g
3
1 12
fg1
MMMMQ
MMMMMMMMQ
i
i
ii
i
i
ii
(16)
Considering the independence of phase difference
12 and
23 , applying Equations (14) and (16)
into Equation (15), we have
0)]sin()sin([
)sinsin()sinsin(
2
3223332231122211
2
23321221
2
23321221
WlElElElE
EEEEEEEEyx
(17)
where
111rmE ,
222rmE ,
333rmE ,
x
x
xm
2cos
, y
y
ym
2cos
,
m
2cos,
)()()(3f3g2f2g1f1g
MMMMMMW .
Equation (17) can be rewritten as
0)sin()sin(223
2
4
2
3112
2
2
2
1 WHHHH (18)
where )]cos([
1221
2
211
llEEH
yx,
)sin(1221
2
22112
lllElEH ,
)]cos([3232
2
323
llEEH
yx,
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
9
)sin(32
2
33224
lElEH .
Form Equation (18), we can obtain
2
2
2
1
223
2
4
2
3
112
)sin()sin(
HH
HHW
(19)
If the phase difference
12 and
23 can be stable over a single period /π2T , the three
motors of the vibrating system operate at the same rotational speed, the system is in a vibratory synchronization state. At this time, the phase difference
12 keep stability over a single period,
thus, Equation (19) must have a solution of 12
. Consequently, we have
1)sin(
2
2
2
1
223
2
4
2
3
HH
HHW (20)
We can get Inequation (21) easily
2
2
2
1
2
4
2
3
2
2
2
1
223
2
4
2
3)sin(
HH
HHW
HH
HHW
(21)
In addition, we define the ratio in Inequation (21) as 1
D :
2
2
2
1
2
4
2
3
1
HH
HHWD
(22)
If we order
11D (23)
then Inequation (20) will come into existence without fail. Inequation (23) is one of the synchronization conditions of the vibrating system. By the same method, we can get
12
4
2
3
2
2
2
1
2
HH
HHWD (24)
From what have been discussed above, it can be seen that Inequation (23) in company with Inequation (24), i.e. 1
1D and 1
2D , compose self-synchronization conditions of the
vibrating system driven by three motors in reverse direction. 2.3 Stability Conditions of Synchronous Operation The stability conditions of the vibrating system driven by three motors is analyzed based on extreme value theory of Hamilton action, stability criterion of system with function of many variables, extreme value theory of function. From stability of constrained system, we can know that there is minimum value in Hamilton action of true motion. Consequently, we can get stability conditions of synchronous operation of the vibrating system driven by three motors.
0
0
2
2312
2
2
23
2
2
12
2
2
12
2
HHH
H
(25)
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
10
Applying Equation (15) into (25), we have
0)cos()]sin([)]cos([
)cos()]sin([)]cos([
112
2
2121
2
21212
22312
2
3131
2
31313
llllE
llllE
yx
yx
(26)
0)cos()cos()]sin([)]cos([
)]sin([)]cos([
)cos()]sin([)]cos([
)]sin([)]cos([
)cos()cos()]sin([)]cos([
)]sin([)]cos([
323112
2
3232
2
3232
2
2121
2
21212
22312
2
3232
2
3232
2
3131
2
31313
11222312
2
3131
2
3131
2
2121
2
21211
llll
llllE
llll
llllE
llll
llllE
yx
yx
yx
yx
yx
yx
(27)
From Equations (26) and (27), we can obtain
)cos()cos(112222231233 PEPE (28)
)cos()cos()cos(])cos([3231122122231213112213
PPEPEPEP (29)
where 2
3232
2
32321)]sin([)]cos([
llllP
yx,
2
2121
2
21212)]sin([)]cos([
llllP
yx,
2
3131
2
31313)]sin([)]cos([
llllP
yx.
To sum up, Inequation (28) in company with Inequation (29) compose stability conditions of synchronous operation of the vibrating system driven by three motors.
3. Experimental Results and Discussion The synchronous experimental table is shown in Fig 2. The experimental table includes three exciting motors (i.e. exciter), vibrating body and supporting equipment (springs and bottom saddles). The experimental table is excited to vibrate by the three motors. We can change the exciting frequency of the system by means of changing the input frequency of the motors. B&K vibration measurement and analysis system is used to collect and analyze the sensor signals, as shown in Fig 3.
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
11
Fig 2. Synchronous experimental table
Fig 3. B&K vibration measurement and analysis system
The parameters of the vibrating system of the experimental table are regulated based on Table 1. When the input frequency of the motors is regulated to 45 Hz, the experimental results are shown in Fig 4.
Table 1 Parameters of the vibrating system.
Content Parameter Mass of body of the vibrating body, M (kg) 157 Moment of inertia of about its centriod, J (kg·m2) 17 Mass of eccentric lump, m0 (kg) 3.5 Spring constant in the direction of x, kx (N/m) 77600 Spring constant in the direction of y, ky (N/m) 30000 Spring constant in the direction of ψ, kψ (Nm/rad) 3000 Damping constant in the direction of x, fx (N/(m/s)) 270(approx.) Damping constant in the direction of y, fy (N/(m/s)) 270(approx.) Damping constant in the direction of ,f (N/(m/s)) 220(approx.)
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
12
Fig 4 (a) and (b) are acceleration of the vibrating system in horizontal and vertical direction, Fig 4 (c) and (d) are displacement of the vibrating system in horizontal and vertical direction. Fig 4 (e), (f) and (g) are rotational speed of the three motors, and Fig 4 (h) and (i) are rotational speed difference between motor 1 and 2 and that between motor 1 and 3. Fig 4 (j) and (k) are phase difference between motor 1 and 2 and that between motor 1 and 3.
Acc
eler
atio
n i
n x
dir
ecti
on (
m/s
2)
Time (s)
Time (s)Acc
eler
atio
n i
n y
dir
ecti
on
(m
/s2)
(a) Acceleration in horizontal direction (b) Acceleration in vertical direction
Time (s)
Dis
pla
cem
ent
in x
dir
ecti
on
(m
m)
Time (s)
Dis
pla
cem
ent
in y
dir
ecti
on (
mm
)
(c) Displacement in horizontal direction (d) Displacement in vertical direction
Time (s)
Rota
tional
spee
d o
f m
oto
r 1 (
r/m
in)
Time (s)
Rota
tional
spee
d o
f m
oto
r 2 (
r/m
in)
(e) Rotational speed of motor 1 (f) Rotational speed of motor 2
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
13
Time (s)
Rota
tional
spee
d o
f m
oto
r 3 (
r/m
in)
Time (s)
Ro
tati
on
al s
pee
d d
iffe
ren
ce (
r/m
in)
(g) Rotational speed of motor 3 (h) Rotational speed difference between
motor 1 and 2
Time (s)
Ro
tati
on
al s
pee
d d
iffe
ren
ce (
r/m
in)
Time (s)
Ph
ase
dif
fere
nce
(d
egre
e)
(i) Rotational speed difference between (j) Phase difference between motor 1 and 2
motor 1 and 3
Time (s)
Ph
ase
dif
fere
nce
(d
egre
e)
(k) Phase difference between motor 1 and 3
Fig 4. Experimental results of the vibrating system As is shown in Fig 4, we can see that there are great fluctuations in the acceleration and displacement of the vibrating system when the system starts to operate. After operating a few
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
14
seconds, the system is in a relatively stable synchronous operation state. To account for the reason of this phenomenon, we can get the answer from what have been discussed above in the theoretical analysis. This phenomenon results from that the phase difference between the motors does not reach a stable state at initial stage after the system’s operating. At this time, the vibrating system has not reach the stability conditions of synchronous operation. Because the structure parameters of the vibrating system meet the self-synchronization and stability conditions of synchronous operation, consequently, we can see from Fig 4 that the phase difference between the motors reach a stable state and meet the self-synchronization and stability conditions of synchronous operation. Therefore, there are great fluctuations in the motion of the vibrating system at initial stage, but it reach periodic stable operation soon, and the vibrating system is in a stable synchronous operation state. From Fig 4 we can see that the stable synchronous speed is 896 r/min, phase difference between motor 1 and 2 is 8.4°, phase difference between motor 1 and 3 is 5.8°. To sum up, the self-synchronization and stability conditions of synchronous operation of the vibrating system which have been deduced in the theoretical analysis are validated by experimental results.
4. Conclusion and Discussion From Fig 4 we can see that the vibrating system driven by three motors can operate in a stable synchronous state when the parameters of the vibrating system of the experimental table are regulated based on the self-synchronization and stability conditions of synchronous operation which are deduced in the above theoretical analysis. When the three motors operate respectively at initial stage, the rotational speed difference and phase difference is not stable and the vibrating system can not operate in a stable state. After operating a few seconds, the rotational speed difference and phase difference get stable under interaction of the dynamic effect, and then the vibrating system become stable synchronous state. The experimental results demonstrate that the vibrating system realizes speed synchronization and phase synchronization after operating a few seconds when adopting proper parameters in the synchronous experimental table based on the self-synchronization conditions and stability conditions. Thus, it can be concluded that the experimental results of the synchronous experimental table verify the correctness of the theoretical analysis. Self-synchronous vibrating system driven by three motors is analyzed based on dynamic theory and experimental study. The self-synchronization and stability conditions of synchronous operation of the vibrating system are deduced by using Hamilton principle. In the design of vibrating system, if we can regulate the parameters of the vibrating system based on the self-synchronization and stability conditions of synchronous operation, the vibrating system can realize stable self-synchronous operation. On account of restrict of the mathematic method, the self-synchronization and stability conditions of synchronous operation are not very accurate because the influence of elastic force and damping force is neglected in dynamic analysis. The accuracy of the dynamic analysis should be improved when the influence of elastic force and damping force can be calculated by some mathematic method. The dynamic analysis of the self-synchronous vibrating system has significant theory and engineering value for solving engineering application problem, it can provide a good theoretical
Degang Wang, Ruijun Han, Hongliang Yao, and Bangchun Wen / Journal of Vibration Analysis, Measurement, and Control (2014) Vol. 2 No. 1 pp. 1-15
15
basis for the design of the self-synchronous vibrating system driven by several motors.
References Bian, Y. and Du, G.., 1998. The basis of analytical mechanics and multi-body dynamics, The press of
mechanical industry, Beijing. Blekhman, I.I., 1988. Synchronization in science and technology, ASME Press, New York. Qiu, B, 1998. Analytical mechanic., The press of Chinese railway, Beijing. Wang, D, Zhao, Q, Zhao C., et al., 2010. Self-synchronous feature of a vibrating system driven by two motors
in the same direction. Journal of Vibration Measurement & Diagnosis 30, 217-222. Wen, B., Li, Y., Zhang, Y., et al., 2005. Vibration utilization engineering, Science Press, Beijing. Wen, B., Liu, F., 1983. The theory of vibrating machine and its applications. The press of mechanical
industry, Beijing. Wen, B., Zhao, C., Su, D, et al., 2003. Vibrated synchronization and controlled synchronization of mechanical
system, Science Press, Beijing. Zhang, T., E, X., Wen, B., 2003. Coupling effect in a synchronous vibration system. Journal of Northeastern
University (Natural Science) 24, 839-942. Zhang, Y., Liu, Q., Wen, B., 2003. Probability perturbation finite element method for response analysis of
multi-degree-of-freedom nonlinear vibration systems with random parameters. Chinese Journal of Computational Mechanics, 20, 8-11.
Zhao, C., Liu, K.,, Ye, X., et al., 2009. Self-synchronization theory of a vibrating system with two-motor drives rotating in opposite directions. Journal of Mechanical Engineering 45, 24-30.
http://dx.doi.org/10.3901/JME.2009.09.024