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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


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


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).


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

Qq

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


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


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.


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,


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)


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.


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.)


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


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


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


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basis for the design of the self-synchronous vibrating system driven by several motors.

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http://dx.doi.org/10.3901/JME.2009.09.024

http://dx.doi.org/10.3901/JME.2009.09.024

dynamic analysis and experimental study of self ...paper.uscip.us/jvamc/jvamc.2014.1001.pdf ·...

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