Research Article
Synchronization investigation of vibration system of two co-rotating rotor with energy balance method
Advances in Mechanical Engineering 2016, Vol. 8(1) 1–19 Ó The Author(s) 2016 DOI: 10.1177/1687814015626023 aime.sagepub.com
Yongjun Hou1, Pan Fang1, Yanghai Nan2 and Mingjun Du1
Abstract The key technology of vibrating screens’ design is how to ascertain the synchronous state of vibrating systems. The occurrence of synchronization of a model, two co-rotating rotors fast excited by induction motors installed in a vibrating body, is treated in this article. The synchronization condition and stability criterion of the system are first derived by the energy balance method. The synchronization zones and stable phase difference are then estimated by the two corotating rotors operated in synchronous state. Moreover, the energy balance mechanism of the system is explained. The transfer of energy between the rotors by the vibrating body is presented as well. Finally, numerical studies of the dynamics are performed by Runge–Kutta method to verify the theoretical analysis. The results indicated that the synchronous state of the vibrating system is mainly determined by installation distance coefficient rl, installation angular b, symmetric coefficient d, and electromagnetic torque Tei, but little influenced by mass ratios h and rm. Keywords Synchronization, rotors, stability, energy balance
Date received: 24 March 2015; accepted: 22 November 2015 Academic Editor: Jia-Jang Wu
Introduction Vibration synchronization is a ubiquitous form of motion in nature. When the vibration motion of a system is influenced by the oscillations of (an)other vibration systems, this phenomenon is considered as an adjustment of rhythms of oscillating objects due to their internal weak couplings, which is called ‘‘synchronization.’’ Our surroundings are full of synchronization phenomenon, for example, violinists play in unison, insects in a population emit acoustic or light pulses with a common rate, birds in a flock simultaneously flap their wings, the heart of a rapidly galloping horse contracts once per locomotory cycle, and so on. Huygens1 described the notion of the synchronization by experiments that two pendulum clocks hanging from a suspended wooden bar. BVd Pol2 showed that the frequency of a generator can be entrained or
synchronized by a weak external signal of a slightly different frequency. J Rayleigh3 presented an interesting phenomenon of synchronization in acoustical systems. The theoretical explanation of synchronization in mechanical oscillators, pendulum clocks, the system of rotating elements, but also some electronic and quantum generator are given by II Blekhman.4–7 And L Cveticanin8 describes a method on balancing a flexible rotor with variable mass. A Pikovsky et al.9 published 1
School of Mechanical Engineering, Southwest Petroleum University, Chengdu, China 2 Department of Mechanical Engineering and Robotics, Universite´ Libre de Bruxelles, Brussel, Belgium Corresponding author: Pan Fang, School of Mechanical Engineering, Southwest Petroleum University, Chengdu 610500, China. Email:
[email protected]
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a monograph with respect to the explanation for synchronization regime, which consider synchronization as a universal concept in non-linear sciences. Nowadays, for synchronization study, the researchers mainly focus on physical, biological, chemical, mechanical, and social systems.10–12 The most representative is mechanical synchronization of multiple coupling pendulums and rotors in vibrating systems. For synchronization of multiple coupling pendulums, M Senator13 developed synchronization of two coupled, similarly sized and escapement-driven pendulum clocks. V Jovanovic and S Koshkin14 studied two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens with the Poincare´ method. J Pen˜a Ramirez et al.15 further perfected the theoretical investigation of synchronization of Huygens’ coupled clock combining Poincare´ method with the finite element method. P Koluda et al.16 studied a multiple self-excited double pendula hanging from a horizontal beam with the energy balanced method and explained how the energy is transferred between the pendula via the oscillating beam. The investigation of synchronization for rotor systems is a significant precondition to invent new vibration machines, including the self-synchronous vibration feeders, selfsynchronous vibration conveyors, synchronous rolling mills, and so on. Wen et al.17 developed the average method to study synchronization of the rotors fixed in a vibrating body, meanwhile, many synchronization machines are introduced by applying this synchronization theory. L Sperling et al.18 presented theoretical and numerical method to explore a two-plane automatic balancing device for equilibration of unbalance rigid-rotor. JM Balthazar et al.19,20 examined selfsynchronization of four non-ideal exciters in non-linear vibration system via computer simulations. Djanan et al.21 explored the synchronization condition for a system, three motors working on a plate, and the synchronous state depends on the physical characteristics of the motors and the plate. Zhao et al.22 and Zhang et al.23 proposed the average method of modified small parameters to investigate the synchronization of exciter systems, which immensely simplify the process of solving synchronization for the systems. Fang et al.24 applied Zhang’s method to investigate the selfsynchronization of two co-rotating rotors coupled with a pendulum rod in a far-resonant vibration system. For investigation of rotor dynamics, K Samantaray et al.25–27 considered that the non-ideal rotor and sommerfeld effect influence on the dynamics characteristics, which promotes the development of the dynamics theory of the rotors. In this article, we introduce energy balance method to study the synchronization of the rotor system. With this method, we can not only easily
understand the regime of the synchronization but also accurately compute the synchronous state of the system. This article is organized as follows. First, the dynamic model of the system is briefly described in section ‘‘The dynamic model of system.’’ Section ‘‘Energy balance of the vibration system’’ presents energy balance equation of the system in the synchronous state. Synchronization condition and stability criterion of the two rotors based on the energy balance method are derived in section ‘‘Synchronization condition and stability criterion.’’ In section ‘‘Numerical simulation discussions,’’ the synchronization zones for the system are estimated, and the theoretical approximate solutions for the stable phase difference are computed. In section ‘‘Sample validations,’’ computer simulations for the dynamics equations are performed with Runge–Kutta method to verify the correctness of the theoretical solutions. Finally, the results are summarized in section ‘‘Conclusion.’’
The dynamic model of system The analyzed system is shown in Figure 1. It consists of a vibrating body, two unbalanced rotors, and some elements of spring and damper. The vibrating body of mass m can vibrate in x-, y-, and c-directions, and its movement directions are described by coordinates x, y, and c, respectively. The vibrating body is connected to the refuge by four linear springs symmetrically installed with stiffness coefficient kx/2 and damping coefficient fx/2 in x-direction, with stiffness coefficient ky/2 and damping coefficient fy/2 in ydirection, with stiffness coefficient kc and damping coefficient fc in c-direction, respectively. Unbalanced rotor i (for i = 1, 2) is modeled by a point mass mi attached at the end of a massless rod of length r. The rotation angle of rotor i is denoted by ui (for i = 1, 2). Moreover, Mi represents the electromagnetic torque of the induction motors, which provide the needed energy to compensate the energy dissipation to persistently keep the rotor rotation (the model of the motor introduced in Zhao et al.22). The mass center of the rigid vibrating body is point o. As illustrated in Figure 1(b), three reference frames of the system can be assigned as follows: the fixed frames oxy; the non-rotating moving frames o#x#y#, which undergoes the translation motion while remaining parallel to oxy; and the rotating frames o#x$y$, which dedicates the rotation motion around points o#. The three reference frames of the vibrating body coincide with each other in the static equilibrium state. According to Lagrange’s equations, the dynamic motion of the system can be given as follows16
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€ i þ fi u_ i þ mi r€y cos ui mi r€x sin ui J0i u € cosðui þ bi Þ þ ð1Þiþ1 mi lc þ ð1Þiþ1 mi rli c_ 2 sinðui þ bi Þ ¼ Mei
ði ¼ 1; 2Þ ð1Þ
2 X
mþ
! mi €x þ fx x_ þ kx x
i¼1
¼
2 X
mi r
u_ 2i
€ i sin ui cos ui þ u
ð2Þ
u1 ¼ u þ a1 ¼ u þ a u2 ¼ u þ a2 ¼ u a
Therefore, the average mechanical angular velocity of two rotors is u. Due to the periodical motion of this vibration system, the mechanical angular velocities of two rotors change periodically. If the least common multiple period of two motors is also supposed to be T, the average value of their average angular velocity could be considered as a constant,21 that is
i¼1
mþ
2 X
!
1 vm ¼ T
mi €y þ fy y_ þ ky y
¼
mi r
u_ 2i
€ i cos ui sin ui u
ð3Þ
i¼1
€ þ fc c_ þ kc c ¼ Jc
2 X i¼1
tþT ð
vðtÞdt
ð6Þ
t
i¼1 2 X
ð5Þ
ð1Þiþ1 mi rli
€ i cos ui þ ð1Þi b u_ 2i sin ui þ ð1Þi b u
ð4Þ
In equations (1)–(4), li is the distance between the rotating centers of the rotors and the mass center of the vibrating body, b is the angle between o$oi and x-direction, J is the moment inertia of the whole vibrating system about the mass center of the vibrating body, J0i is _ and the moment inertia of unbalanced rotor i, and ðÞ € represent dðÞ=dt and d 2 ðÞ=dt2 , respectively; ðÞ As shown in Figure 1, assuming the average phase and the phase difference of the two unbalanced rotors to be f and 2a,21 respectively, then we have
Assuming the instantaneous fluctuation coefficients of u and a_ to be e1 and e2 (e1 and e2 are the functions with respect to time t, meanwhile, e1 1 and e2 1), respectively, then u_ 1 and u_ 2 can be written as
u_ 1 ¼ u_ þ a_ 1 ¼ vm1 ¼ ð1 þ e1 þ e2 Þvm u_ 2 ¼ u_ þ a_ 2 ¼ vm2 ¼ ð1 þ e1 e2 Þvm
ð7Þ
Neglecting small parameters e1 and e2, with average angular velocity vm, the average phase can be approximately expressed as21
u¼
ðT 1 vm dt T
ð8Þ
0
According to equation (7), the rotation acceleration of the rotors can be expressed in the formulae
Figure 1. The model of the vibrating system: (a) the dynamic model of the vibrating system with two induction motors rotating in the same direction and (b) the reference frame system.
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€1 ¼ u €þa u € 1 ¼ ðe_1 þ e_2 Þvm €2 ¼ u €þa u € 2 ¼ ðe_2 e_2 Þvm
ð9Þ
In equation (9), considering e_1 1; e_2 1 in the synchronous state. According to equations (2)–(4), the steady response of the vibrating system, with neglecting the small parameters, can be expressed as follows
The rest four components of equation (11) describe the synchronous energy between the ith rotor and the vibrating body
WiSYS
ðT
ðT
¼ mi r€yu_ i cos ui dt mi r€xu_ i sin ui dt 0
0
ðT
8 > < x ¼ rrm mx ½cosðu þ a þ g x Þ þ h cosðu a þ gx Þ y ¼ rrm my sin u þ a þ gy þ h sin u a þ g y h hdrl1 i > rl1 : c ¼ rrm m sin u a þ b þ gc c le sin u þ a b þ g c le
€ u_ i cosðui þ bi Þdt þ ð1Þiþ1 mi lc
ð10Þ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where vnx ¼ kx =M , vny ¼ ky =M , vnc ¼ kc =J , pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi jnx ¼ fx = 4kx M , jnx ¼ fx = 4kx M , jnc ¼ fc = 4kc J , mx ¼ 1=ð1 v2nx =v2m Þ, my ¼ 1=ð1 v2ny =v2m Þ, mc ¼ pffiffiffiffiffiffiffiffiffiffi 1=ð1 v2nc =v2m Þ, rm ¼ m1 =M, h ¼ m2 =m1 , le ¼ J =M ,
2 þ ð1Þiþ1 mi rli u_ i c_ sinðui þ bi Þdt
d ¼ l2 =l1 , rl ¼ l1 =le , gj ¼ ½2jnj ðvnj =vm Þ= ½1 ðvnj =vm Þ2 ; ðj ¼ x;y; cÞ, and rl ¼ l2 =l1 . jnj is the damping ratio of the springs in the jdirection (considering jnj 0.07 in this article). Differentiating the formulas in equation (10) with _ and c € can be obtained. respect to time t, €x, €y, c,
Energy balance of the vibration system Multiplying equation (1) by angular velocity u_ i of the rotors and integrating it over the period T, the equation of energy balance can be expressed ðT
ðT
€ i u_ i dt þ fi u_ i u_ i dt þ mi r€yu_ i cos ui dt J0i u
0
0
ðT
The right-hand side of equation (11) depicts the output energy of the motors
WiDRIVE
0
0
ðT 2 þ ð1Þiþ1 mi rli u_ i c_ sinðui þ bi Þdt 0
ði ¼ 1; 2Þ
ð11Þ
0
The first component of equation (11) indicates the inertia energy that the induction motors act on the ith rotor. Because this term related to small parameters, therefore WiINERT ’ 0. The second component of equation (11) represents energy dissipated by the rotation joint of the ith rotor
WiDAMP
ðT
¼ fi u_ i u_ i dt 0
ð12Þ
ðT
¼ Mei u_ i dt
ð14Þ
0
Substituting equations (12)–(14) into equation (11), the energy balance of the ith rotor is obtained as WiDAMP þ WiSYS ¼ WiDRIVE
ð15Þ
Multiplying equation (2) by the velocity of the vibrating body in x-direction and integrating it over the period T, the equation of the energy balance of the vibrating body is given as
mþ
¼
2 X
!
ðT X 2 0
ðT
ðT
0
0
mi €xx_ dt þ fx x_ x_ dt þ kx x_xdt
i¼1
0
€ u_ i cosðui þ bi Þdt mi r€xu_ i sin ui dt þ ð1Þiþ1 mi lc
¼ Mei u_ i dt
0
0
ðT
ðT
0
ðT
ðT
ðT
ð13Þ
ð16Þ
€ i sin ui dt mi rx_ u_ 2i cos ui þ u
i¼1
The first component on the left-hand side of equation (16) represents the energy of inertia force in the xdirection during the period T. Owning to the periodic motion of vibrating body in the synchronous state, we have WxINERT ’ 0.The following component represents the dissipated energy in the spring damp in the xdirection
WxDAMP
ðT
¼ fx x_ x_ dt
ð17Þ
0
The last component on the left-hand side of equation (16) stands for the energy of the potential force in the x-direction. Due to the character of the potential force, we have WxPOT ’ 0. The right-hand side of equation (16) gives the resultant force of the two rotors acting on the vibrating body in x-direction
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WxSYS
¼
ðT X 2 0
mi rx_
u_ 2i
€ i sin ui dt cos ui þ u
ð18Þ
i¼1
WxDAMP
¼
WxSYS
ð19Þ
Then multiplying equation (3) by the velocity of the vibrating body in y-direction and integrating it over the period T, the equation of the energy balance of the vibrating body is presented as
mþ ðT X 2 0
0
0
€ i cos ui dt mi ry_ u_ 2i sin ui u
The first component on the left-hand side of equation (20) represents the energy of inertia force in the ydirection; therefore, we have WyINERT ’ 0.The second component represents the dissipated energy in the spring damp in the y-direction
The first component on the left-hand side of equation (24) represents the energy of inertia force in c-direction during the period T, and so we have WcINERT ¼ 0. The next component represents the dissipated energy in the spring in c-direction
WyDAMP ¼ fy y_ y_ dt
ð21Þ
0
The last component on the left-hand side of equation (20) stands for the energy of the potential force in the y-direction, and so we have WyPOT ¼ 0: The right-hand side of equation (19) gives the resultant force of the two rotors acting on the vibrating body in y-direction € i cos ui dt mi ry_ u_ 2i sin ui u
ð22Þ
WcSYS
¼
ðT X 2 i¼1
€ i cos ui þ ð1Þi b ð26Þgdt u Substituting equations (25) and (26) into equation (24), the energy balance of the vibrating body in y-direction can be expressed as WcDAMP ¼ WcSYS
ð27Þ
Adding together equations (19), (23), and (27), the energy balance of the vibrating body in x-, y-, and c-directions is yielded as
Substituting equations (21) and (22) into equation (20), the energy balance of the vibrating body in ydirection can be expressed as
Then multiplying equation (4) by the velocity of the vibrating body in c-direction and integrating it over the period T, the equation of the energy balance of the vibrating body is written as
ð25Þ
ð1Þiþ1 mi rli u_ 2i sin ui þ ð1Þi b
X
ð23Þ
_ ¼ fc c_ cdt
The last component on the left-hand side stands for the energy of the potential force in c-direction, so WcPOT ¼ 0. The right-hand side of equation (24) gives the resultant force of the rotors acting on the vibrating body in c-direction
i¼1
WyDAMP ¼ WySYS
ðT 0
0
ðT
0
€ i cos ui þ ð1Þi b gdt u
ð20Þ
ð24Þ
i¼1
WcDAMP
i¼1
WySYS ¼
0
0
ð1Þiþ1 mi rli u_ 2i sin ui þ ð1Þi b
ðT ðT mi y_ €ydt þ fy y_ y_ dt þ ky y_ ydt
ðT X 2
¼
0
ðT X 2
!
i¼1
0
¼
2 X
ðT ðT _ _ _ _ € J ccdt þ fc ccdt þ kc ccdt
0
Substituting equations (17) and (18) into equation (16), the energy balance of the vibrating body can be expressed as
ðT
ðT
WiDAMP ¼
i¼x;y;c
X
WiSYS
ð28Þ
i¼x;y;c
Synchronization condition and stability criterion Condition of synchronization Adding equations (15)–(28), the energy balance of the whole system in following form can be expressed as
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Advances in Mechanical Engineering 2 X
X
WiSYS þ
i¼x;y;c
i¼1
¼
2 X
WiDAMP þ
2 X
WiDRIVE þ
WiDAMP
i¼1
X
ð29Þ
WiSYS
i¼x;y;c
i¼1
2 Y
During synchronous state, the energy supplied with the motors is dissipated by all of the dampers in the vibration system, we have X
manufacturing tolerances. Subtracting the energy balance of the two rotors according to equation (15), during the synchronous state, the difference equation of the energy of the system is given by
WiDAMP þ
i¼x;y;c
2 X
WiDAMP ¼
i¼1
2 X
WiDRIVE
ð30Þ
i¼1
with 2 X
WiSYS ¼
X
WiSYS
ð31Þ
i¼x;y;c
i¼1
Substituting equation (28) into equation (30) and considering equation (31), the energy balance of the 8 2 ÐT m rv € 2 ÐT P P > i my > > þ < i¼1 0 cos ui dt i¼1 0 2 ÐT m rv € 2 ÐT Q Q > y i m > > þ : cos u dt i i¼1 0 i¼1 0
2 ÐT P mi rvm€x þ sin ui dt i¼1 0 2 ÐT Q mi rvm€x þ sin ui dt i¼1 0
WiSYS þ
i¼1
2 Y
WiDAMP ¼
i¼1
2 Y
WiDRIVE
ð33Þ
i¼1
To ensure the implementation of the synchronous operation of the system, the system will adjust the phase difference between the rotors by itself. In the synchronous state, the phase angle difference (u12u2 = 2a) between the two rotors should be a constant and independent of the initial conditions, and instantaneous fluctuation coefficients e1 and e2 are equal to zero. If the parameters of the vibrating system satisfy the energy balance equation and the energy difference equation, that is, equations (32) and (33), the system can implement the synchronous operation. Substituting equations (7) and (10) into equations (32) and (33) with neglecting the small parameters, we have
iþ1 2 ÐT € P ð1Þ mi rvm ci þ cos ui þ ð1Þ b dt i¼1 0 iþ1 2 ÐT € Q ð1Þ mi rvm c þ cos ui þ ð1Þi b dt i¼1 0
2 iþ1 2 ÐT 2 ÐT P ð1Þ mi rvm li c_ þ P fi v2m dt ¼ Mei vm dt i sin ui þ ð1Þ b dt i¼1 0 i¼1 0 2 iþ1 2 ÐT 2 ÐT Q ð1Þ mi rvm li c_ þ Q fi v2m dt ¼ Mei vm dt sin ui þ ð1Þi b dt i¼1 0 i¼1 0
ð34Þ
Considering vm dt ¼ du, equation (34) can be rewritten as 8 2 2p 2 2p Ð mi r€y Ð P P > > > þ < i¼1 0 cos ui du i¼1 0 2 2p 2 2Ðp Q > > Q Ð mi r€y > þ : i¼1 0 cos ui du i¼1 0
2 2p Ð P mi r€x þ sin ui du i¼1 0 2 2Ðp Q mi r€x þ sin ui du i¼1 0
iþ1 2 2p Ð € P ð1Þ mi r c i þ cos ui þ ð1Þ b du i¼1 0 iþ1 2 2Ðp € Q ð1Þ mi r c i þ cos ui þ ð1Þ b du i¼1 0
2 iþ1 2 2p 2 2p Ð Ð P P ð1Þ mi rli c_ fi vm du ¼ Mei du þ i sin ui þ ð1Þ b du i¼1 0 i¼1 0 2 iþ1 2 2Ðp 2 2Ðp Q Q ð1Þ mi rli c_ fi vm du ¼ Mei du þ i sin ui þ ð1Þ b du i¼1 0 i¼1 0
ð35Þ _ and c € into equation (35) and Substituting €x, €y, c, integrating them over u = 2p, we have 1
2 2 2 m1 r vm ½ð1 1 2 2 m r 2 1 vm ½ð1
e1 þ M e2 þ h2 ÞTSþ þ 2TS cosð2a þ uS Þ þ ðf1 þ f2 Þvm ¼ M e1 M e2 h2 ÞTS þ 2TC sinð2a þ uC Þ þ ðf1 f2 Þvm ¼ M
whole system during the synchronous state can be rewritten as 2 X i¼1
WiSYS þ
2 X i¼1
WiDAMP ¼
2 X
WiDRIVE
ð32Þ
with
TS1
i¼1
Equation (32) describes the energy balance of the whole system within the synchronous state in period T. According to this equation, it is known that the energy produced by the motors is balanced by synchronization energy and the dissipated energy of the motors. In actual engineering applications, the same type motors hold the different electrical characteristic as
ð36Þ
TS2
TSþ ¼ TS1 þ TS2 ; TS ¼ TS1 TS2
¼ rm my sin g y þ mx sin g x þ rl2 mc sin g c ;
¼ h2 rm my sin g y þ mx sin g x þ mc d2 rl2 sin g c TS ¼ hrm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2S þ b2S ;
TC ¼ hrm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2C þ b2C
aS ¼ my sin g y þ mx sin gx dmc rl2 sin g c cosð2bÞ; bS ¼ dmc rl2 sin g c sinð2bÞ
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aC ¼ my cos g y mx cos g x þ dmc rl2 cos g c cosð2bÞ; bC ¼ dmc rl2 cos g c sinð2bÞ
S arctan b aS 0 aS ;
; uS ¼ > S : p þ arctan b aS \0 aS ; 8
> < arctan baCC ; aC 0
uC ¼ > : p þ arctan baC ; aC \0 C
2a ¼ arcsin
ei is average electromagnetic torque of the where M motors in period T. The first formula of equation (36) is the equation of torque balance of the system in the synchronous state, which serves to find the approximation value of synchronous angular velocity vm. Moreover, the second formula of equation (36) is the balance equation of the torque difference of the system in the synchronous state, which serves to determine the approximation value of stable phase difference 2a. Therefore, the system adjusts the phase difference between the rotors to satisfy the second formula of equation (36). Specifying TSYS as the synchronization torque of the vibration system, we obtain TSYS ¼ m1 r
v2m TC
ð37Þ
Then assigning TEXC as the excessive torque of the rotors, we have TEXC ¼ TRES1 TRES2
TRES1 ¼ Te1 f1 vm 12 m1 r2 v2m0 TS1 TRES2 ¼ Te2 f2 vm 12 m1 r2 v2m0 TS2
ð39Þ
Rewriting the second formula of equation (36), we can obtain
8 >
> m sinð2a þ r Þ m sin ð 2a r Þ þ m sr sin 2a r 2b cos ð Da Da Þ y x c 1 2 > > y x c l > > = < h i 2 ¼0 > þ my sinð2a þ ry Þ mx sinð2a rx Þ þ mc srl sinð2a rc 2bÞ sinðDa1 Da2 Þ > > > > > ; : þmy sin 2a þ ry þ mx sinð2a rx Þ mc srl2 sin 2a rc 2b 1 J02 D€ a2 þ f2 Da_ 2 þ hm1 r2 v2m rm 2 9 8h i 2 > > þ m m sin 2a þ r sin ð 2a þ r Þ m sr sin 2a þ r 2b cos ð Da Da Þ y x c 1 2 > > y x c l2 > > = < h i 2 þ my sin 2a þ ry þ mx sinð2a þ rx Þ mc srl2 sin 2a þ rc 2b sinðDa1 Da2 Þ > ¼ 0 > > > > > ; : 2 my sin 2a þ ry mx sinð2a þ rx Þ þ mc srl2 sin 2a þ rc 2b where TRES1 and TRES2 represent the residual torques of rotors 1 and 2, respectively, which can be written as
ð40Þ
ð42Þ
From equation (42), it should be noted that ½Da1 ; Da2 T ¼ ½0; 0T is an equilibrium solution when the two rotors rotate synchronously. Moreover,
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linearizing equation (36) around ½Da1 ; Da2 T ¼ ½0; 0T , we have h 8 i < J01 D€ a1 þ f1 Da_ 1 þ 12 hm1 r2 v2m rm my sin 2a þ ry mx sinð2a rx Þ þ mc srl2 sin 2a rc 2b ðDa1 Da2 Þ ¼ 0 h i : J02 D€ a2 þ f2 Da_ 2 þ 12 hm1 r2 v2m rm my sin 2a þ ry þ mx sinð2a þ rx Þ mc srl2 sin 2a þ rc 2b ðDa1 Da2 Þ ¼ 0
ð43Þ
According to equation (36), equation (43) can be rewritten as (
D€ a1 þ mf11r2 Da_ 1 þ h2 v2m ½TC cosð2a þ uC Þ þ TS sinð2a þ uS ÞðDa1 Da2 Þ ¼ 0 D€ a2 þ hmf21 r2 Da_ 2 þ h2 v2m ½TC cosð2a þ uC Þ þ TS sinð2a þ uS ÞðDa1 Da2 Þ ¼ 0
Actually, the damping coefficients in the two motor shafts are very small, and so f1 ’ f2 =h. Therefore, the disturbance differential equation can be obtained with the subtraction of the two formulas in equation (44) D€ aþ
f1 Da_ þ hv2m ½TC cosð2a þ uC ÞDa ¼ 0 m1 r 2
ð45Þ
a2 , Da_ ¼ Da_ 1 Da_ 2 , and where D€ a ¼ D€ a1 D€ a, Da, _ and Da denote Da ¼ Da1 Da2 . In addition, D€ the acceleration, velocity, and displacement of the phase difference when the disturbance phase angular acts on the rotors, respectively. According to the stability theory, the coefficient of the third item of equation (45) must be positive as the first two terms are obviously greater than zero. In this case, the synchronous state of the system can be stable. As h.0 and v2m .0, the criterion of the synchronous stability of the system can be written as TC cosð2a þ uC Þ.0
ð46Þ
In the weak damper system, it can be seen that cosð2a þ uC Þ.0 as TC .0 in equation (46). So 2a þ uC 2 ðp=2; p=2Þ, and the stable phase difference is determined by uc.
Numerical simulation discussions In the numerical simulation discussions, the following system parameters are considered. We assume the parameters of the two motors are identical (i.e. rated power = 0.7 kW, rated voltage = 220 V, rated frequency = 50 Hz, pole pairs = 2, stator resistance = 0.56 O, rotor resistance = 0.54 O, stator inductance = 0.1 H, rotor inductance = 0.12 H, mutual inductance = 0.13 H, and the damping coefficient of shafting = 0.04 N m/ (rad s)). The structural parameters of the system are as follows: m1 = 4 kg, m2 = 1–4 kg, m = 80 kg, r = 0.04 m, le = 0.3 m 0 \ l1 \ 2.1 m, l2 l1 (rl 2 ½0; 7 according to Djanan et al.21), kx= ky = 63 kN/m, kc = 5 kN m/rad, j = 8 kg/m2, fx= fy = 0.38 kN s/m, fc= 0.034 kN s/m.
ð44Þ
Synchronization zones of the system Equation (41) has given the synchronous condition of the system, on which the zones of synchronization of the rotors can be ascertained. Substituting the abovementioned parameters into equation (41), the synchronization zones can be depicted, as shown in Figure 2. These figures are divided into the white, black, and gray zone, respectively. If the value of the system parameters locate in the gray zone, the value of the stable phase difference belongs to the interval of ½p=2; 3p=2. If the value of the parameters locate in the black zone, the phase difference belongs to the interval of ½p=2; p=2 (Huygens1 describes that the motion, as the existence of 2a 2 ðp=2; p=2Þ, is called as synphase synchronization; and the motion, as the existence of 2a 2 ðp=2; 3p=2Þ, is called as anti-phase synchronization). If the parameters locate in the white zone, not satisfying the synchronization condition, the two rotors cannot implement the synchronous operation. It should be noted that the gray zone (i.e. 2a 2 ½p=2; 3p=2) gradually is decreased with increasing value of parameter d as well. Figure 3 describes the synchronization zones for the variation of parameter b when d= 1.0. With increasing the value of parameter b, the white and black zones are gradually shrined. Therefore, it can be concluded that the large value of parameter b is in favor of the implementation of the anti-phase synchronization.
Solutions of stable phase difference between the two rotors In this section, the theoretical solutions of the stable phase difference will be quantitatively discussed considering the following features of the system: identical rotors symmetrically placed, non-identical rotors asymmetrically placed, non-identical rotors symmetrically placed, and turning off the power source. From equation (40), the main parameters influen e2 , f1 , f2 , e1 , M cing on the stable phase difference are M TS1 , TS2 , and TC . However, TS1 , TS2 , and TC are
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Figure 2. The synchronization zones for the variation of parameter d when b = 0°: (a) d ¼ 0, (b) d ¼ 0:2, (c) d ¼ 0:4, (d) d ¼ 0:6, (e) d ¼ 0:8, and (f) d ¼ 1:0.
functions of dimensionless parameters rl , s, h, mx , my , and mc . In the far-resonant vibration system, the value of parameters mx , my , and mc range from 1.01 to 1.07.16 However, parameters h and rm have little influence on the phase difference according to equations (36) and (40). Therefore, the dimensionless parameters b, rl, and d will be concerned. When the two identical motors drive two non-identical rotors, we have
1 TEXE ¼ m1 r2 v2m TS 2
ð47Þ
We assume here that Te1 f1 vm ðTe2 f2 vm Þ ’ 0 just for the convenient discussions. Therefore, equation (40) can be simplified in the form
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2a ¼ arcsin
TS uC 2TC
ð48Þ
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Figure 3. The synchronization zones for the variation of parameter b when d= 1.0: (a) b ¼ 158 , (b) b ¼ 308 , and (c) b ¼ 458 .
According to equation (48), the stable phase difference between the rotors can be computed with numerical method. Figure 4 shows the approximate value of the phase difference in the power-supplying state. It can be seen that the value of parameters b, rl, and d determine the synchronous state of the system. Especially, it is noted that the phase difference is close to 180° for b = 90°. However, for d6¼ 0, the synchronous behavior of the system will approach to the in-synchronous state with increasing the value of parameter rl and decreasing the value of parameter b. The phenomenon implies that the farther installation distance and smaller installation angular of the rotors are beneficial to the anti-phase synchronization of the system. Otherwise, the synphase synchronization will be conducted. Then, the synchronous state of the system will be discussed, when turning off the power source of the second motor. Therefore, parameters Te2 ¼ 0 and f2 vm0 f1 vm0 ’ 0 can be assumed, and equation (40) can be simplified in the form 2a ¼ arcsin
e1 TS M uC TSYS
ð49Þ
According to Djanan et al.21 and Zhao et al.,22 when the induction motor operates with the steady velocity ei of the motor can be vm , the electromagnetic torque M simplified as 2 2 ei ¼ n2p Lmi US0 vs np vm M 2 Lsi vs Rri
ð50Þ
where Lmi is the mutual inductance of the ith induction motor, Lsi is the stator inductance of the ith induction motor, np is the number of pole pairs of the induction motor, vs is the angular velocity of synchronous electric, Rri is the rotor resistance of the ith induction motor, and Us0 is the amplitude of the stator voltage vector. Substituting the given parameters of the motors into equation (50), the electromagnetic torque of the second motor can be defined. In addition, according to equation (49), the synchronous state of the system in the power-cutting state can be estimated. Figure 4 shows the stable phase difference in the power-cutting state. The phase difference in the power-cutting state is larger than the power-supplying state owing to the existence e1 . For example, the of the electromagnetic torque M
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Figure 4. Approximate theoretical solutions for synchronous state in the power-supplying state. In this figure, coordinate X represents the value of the installation angular b of the rotors. Coordinate Y represents the value of the installation displacement rl of the rotors; and coordinate Z represents the value of stable phase difference 2a: (a) d ¼ 1:0, (b) d ¼ 0:8, (c) d ¼ 0:6, (d) d ¼ 0:4, (e) d ¼ 0:2, and (f) d ¼ 0.
value of the stable phase difference for rl = 2, b = 0°, and d= 1 in the power-supplying state is equal to 0° (see Figure 4(a)); however, the value increases to 41° in the power-cutting state (see Figure 5(a)).
machine SI units and three-phase programmable voltage source in MATLAB. The Runge–Kutta algorithm with adaptive step size control is used to compute the dynamics equations (1)–(4).
Sample validations
Simulations for rl = 2, b = 0°, d= 1, rm = 0.02, and h=1
Further analyses have been performed by MATLAB to verify the correctness of the above theoretical solutions for synchronous state of the system, and the models of the induction motors are established with asynchronous
Before entering into the power-cutting state, the synchronous implementation of the system must go through two operation steps. First, the two rotors
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Figure 5. Approximate theoretical solutions for the synchronous state in the power-cutting state. In this figure, coordinate X represents the value of the installation angular b of the rotors; coordinate Y represents the value of the installation displacement rl of the rotors; and coordinate Z represents the value of stable phase difference 2a: (a) d ¼ 1:0, (b) d ¼ 0:8, (c) d ¼ 0:6, (d) d ¼ 0:4, (e) d ¼ 0:2 and (f) d ¼ 0.
connected with the induction motors are continuously supplied with the power sources; after a while, the system achieves the synchronous state, which can be called as the power-supplying stage. Second, the power source of the second motor is cut off; after a while, the system achieves another synchronous state, which can be called as the power-cutting stage.
A synchronous example for the power-cutting state is performed with MATLAB. The following parameters, rl = 2, b = 0°, d= 1, rm = 0.02, and h = 1, are considered (in this case, the value of the parameters in equations (1)–(4) are M = 80 kg, m1 =m2 = 4 kg, r = 004 m, l0 = 0.3 m, and l1 = l2 = 0.6 m). In the simulation model, the two rotors are supplied with the
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Figure 6. Simulation results for rl = 3, b = 45°, d= 1, rm = 0.02, and h = 1: (a) Rotor velocity, (b) Rotor torque, (c) Phase difference, (d) Displacement in x-axis, (e) Displacement in y-axis and (f) Displacement in c-axis.
power source at first 2.5 s, and the power source on motor 2 is cut off at 2.5 s. It is noted that the value of the system parameters located in the black zone in Figure 2(f); therefore, the phase difference should belong to interval ½p=2; p=2, that is, synphase synchronization. Figure 6(a)–(c) represents the dynamics response of the system in the powercutting state. First, the synchronous rotational velocity of the rotors stabilizes at vicinity of 152 rad/s in the power-supplying stage. The electromagnetic torques of the induction motors are equal to 3.3 and 3.6 N m, respectively. Moreover, the phase difference approaches 1.2°, that is, synphase synchronization, which is corresponding to Figure 2(f). Comparing simulation results with Figure 4(a), it can be seen that the value of the stable phase difference is in agreement with the results
obtained by the theoretical solutions (i.e. the phase difference in Figure 4(a) is equal to 0°, and here, the phase difference is equal to 1.2°). When entering into the power-cutting state, the rotational velocity of the rotors is decreased to 147 rad/s. The phase difference is increased to 38° from 1.2°, which is according with theoretical computation in Figure 5(a) (i.e. the stable phase difference in Figure 5(a) is equal to 41.8°, and here the phase difference is equal to 38°). The electromagnetic torque of rotor 1 becomes zero when the power source of induction motor 2 is cut off at 2.5 s, and the electromagnetic torque of motor 2 is increased to 6.85 N m for balancing the resistances and frictions of the system. Figure 6(d)–(f) describes the responses of the system in x-, y-, and c-directions. In the starting stage, the
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displacements in x-, y-, and c-directions are by far larger than other stages as the exciting frequency go through the resonant region. In the power-supplying state, the steady displacements in x- and y-directions are smaller than that of the power-supplying state. In the power-supplying state, the part energy supplied by the two induction motors is dissipated by their friction dampers (W1DAMP and W2DAMP ), and the other part is balanced by the synchronization energy (W1SYS and W2SYS ) of the vibrating body. Moreover, the synchronization energy excites the vibrating body (WxSYS and WySYS ) and is dissipated by the dampers (WxDAMP and WyDAMP ). Owing to the small displacement in c-direction, the energy in the direction WcSYS can be neglected. Then, the energy balance equation of the vibration system during the power-cutting state can be expressed as W1DAMP þ W1SYS ¼ W1DRIVE ;
W2DAMP þ W2SYS ¼ W2DRIVE ;
W1DAMP þ W1SYS þ W2DAMP þ W2SYS ¼ W1DRIVE þ W2DRIVE ; W1SYS þ W2SYS ¼ WxSYS þ WySYS ; WxSYS þ WySYS ¼ WxDAMP þ WyDAMP
ð51Þ
In the synchronous state, the energy balance of the power-supplying system is depicted in Figure 7(a). However, the electromagnetic torque of the rotor is equal to zero in the power-cutting stage (see in Figure 6(b)). It is noted that the displacement in c-direction cannot be neglected. In this case, the synchronous energy (W1SYS and W2SYS ) and the dissipated energy (W1DAMP and W2DAMP ) are supplied by the first motor through the oscillation of the vibrating body. Moreover, the synchronous energy are transferred to synchronization energy (WxSYS , WySYS , and WcSYS ) of the vibrating body, which are balanced by the dissipated energy of the vibrating body (WxDAMP , WyDAMP , and WcDAMP ). Therefore, the energy balance equation of the system in the power-cutting state can be obtained as W1DAMP þ W1SYS ¼ W1DRIVE ; W1SYS ¼ WxSYS þ WySYS þ WcSYS ; WxSYS þ WySYS þ WcSYS ¼ WxDAMP þ WyDAMP þ WcDAMP þ W2DAMP ð52Þ In the synchronous state, the energy balance of the power-cutting system is depicted in Figure 7(b).
Simulations for rl = 2, b = 0°, d= 0, rm = 0.02, and h = 0.25 The synchronous state, considering the asymmetrical installation of the non-identical rotors, will be discussed in this section. According to Figure 4(f), it can be that the phase difference stabilizes at 2180° when the second rotors installed in the mass center of the system.
Figure 7. Energy balance of the vibrating system in the synchronous state: (a) The power-suppling state, (b) The power cutting state, (c) The asymmetrical installation of the nonidentical rotors and (d) The symmetrical installation of the nonidentical rotors.
The dimensionless parameters rl = 2, b = 0°, d= 0, rm = 0.02, and h = 0.25 are assumed to verify the theoretical solutions (in this case, the parameters in equations
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Figure 8. Simulation results for rl = 2, b = 0°, d= 0, rm = 0.02, and h = 0.25: (a) Rotor velocity, (b) Rotor torque, (c) Phase difference, (d) Displacement in x-axis, (e) Displacement in y-axis and (f) Displacement in c-axis.
(1)–(4) are M = 80 kg, m1 = 4 kg, m2 = 1 kg, r = 004 m, l0 = 0.3 m, l1 = 0.6 m, and l2 = 0 m). From Figure 2(a), it can be seen that the parameters are located in the gray zone; therefore, the phase difference belongs to interval ½p=2; 3p=2. The simulation results by MATLAB for the case are shown in Figure 8. As becomes clear from them, the synchronous velocity of the system stabilizes at vicinity of 153 rad/s. The electromagnetic torques of the induction motors are equal to 3.1 and 3.5 N m, respectively. And the stable phase difference approaches 2186°. Comparing simulation results with Figure 4(f), it follows that the phase difference is close to the value obtained by theoretical solutions (i.e. the stable phase difference in Figure 4(f) is equal to 2180°, and here the stable phase difference is equal to 2186°). In the case, when the two rotors rotate in the synchronous state, the part energy supplied by the two induction motors is dissipated by their frictions and
dampers (W1DAMP and W2DAMP ), and the other part is balanced by the vibrating body (synchronization energy W1SYS and W2SYS ). Moreover, the synchronization energy excites the vibrating body (WxSYS , WySYS , and WcSYS ), which is dissipated by the damper of the vibrating body (WxDAMP , WyDAMP , and WcDAMP ). So the energy balance equation of the system in the state can be written by W1DAMP þ W1SYS ¼ W1DRIVE ; W2DAMP þ W2SYS ¼ W2DRIVE ; W1DAMP þ W1SYS þ W2DAMP þ W2SYS ¼ W1DRIVE þ W2DRIVE ; W1SYS þ W2SYS ¼ WxSYS þ WySYS þ WcSYS ; WxSYS þ WySYS þ WcSYS ¼ WxDAMP þ WyDAMP þ WcDAMP ð53Þ In the synchronous state, the energy balance of the system, considering the asymmetrical installation of the non-identical rotors, is depicted in Figure 7(c).
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Figure 9. Simulation results for rl = 1, b = 0°, d = 1, rm = 0.05, and h = 1: (a) Rotor velocity, (b) Rotor torque, (c) Phase difference, (d) Displacement in x-axis, (e) Displacement in y-axis and (f) Displacement in c-axis.
Simulations for rl = 1, b = 0°, d= 1, rm = 0.05, and h=1 In the section, we will consider the synchronous state when the two identical rotors are symmetrically installed more closely than section ‘‘Simulations for rl = 2, b = 0°, d= 1, rm = 0.02, and h = 1.’’ The values of the dimensionless parameters rl = 1, b = 0°, d = 1, rm = 0.05, and h = 1 are supposed (in this case, the value of the parameters in equations (1)–(4) are M = 80 kg, m1 = m2 = 4 kg, r = 004 m, l0 = 0.3 m, and l1 =l2 = 0.3 m). From the theoretical computations in Figure 2(f), it is known that the value of the parameters of the system are located in the gray zone; therefore, the value of the stable phase difference belongs to interval ½p=2; 3p=2. From Figure 4(a), it can be seen that the stable phase difference is equal to 2180°. Figure 9
depicts the simulation results with MATLAB. The synchronous velocity of the rotors stabilizes at vicinity of 153.5 rad/s; the electromagnetic torques of the induction motors are equal to 3.1 and 3.4 N m, respectively; the stable phase difference between the two rotors approaches 2178.6° (i.e. anti-phase synchronization state), which is according to the result of the above theoretical computation in Figure 4(a). The displacement amplitude in x- and y-directions is very small. Therefore, the energy transition (WxSYS and WySYS ) in x- and y-directions can be neglected. The part energy supplied by the two induction motors is dissipated by their friction dampers (W1DAMP and W2DAMP ) and the other part is balanced by the vibrating body (synchronization energy W1SYS and W2SYS ) in the synchronous state. Nevertheless, the synchronization energy excites the vibrating body in c-direction (WcSYS ), which
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Figure 10. Simulation results for rl = 3, b = 45°, d = 1, rm = 0.02, and h = 1: (a) Rotor velocity, (b) Rotor torque, (c) Phase difference, (d) Displacement in x-axis, (e) Displacement in y-axis and (f) Displacement in c-axis.
is dissipated by the damper (WcDAMP ) of the vibrating body. Now, the energy balance equation of the system can be written by W1DAMP þ W1SYS ¼ W1DRIVE ; W2DAMP þ W2SYS ¼ W2DRIVE ; W1DAMP þ W1SYS þ W2DAMP þ W2SYS ¼ W1DRIVE þ W2DRIVE ; W1SYS þ W2SYS ¼ WcSYS ; WcSYS ¼ WcDAMP
ð54Þ
In the synchronous state, the energy balance of the system, considering the symmetrical installation of the non-identical rotors, is depicted in Figure 7(d).
Simulations for rl = 3, b = 45°, d = 1, rm = 0.02, and h=1 According to Figure 4(a), it is indicated that the synphase synchronization of the two rotors can be
pffiffiffi implemented when rl . 2 and b = 0°, but the synchronous state of the system gradually approaches to the anti-phase synchronization with increasing the value of parameter b. The dimensionless parameters rl = 3, b = 45°, d = 1, and h = 1 are defined to verify the theoretical computation results (in this case, the parameters in equations (1)–(4) are M = 80 kg, m1 = m2 = 4 kg, r = 004 m, l0 = 0.3 m, and l1 =l2 = 0.9 m). From Figure 3(c), it can be seen that the system parameters are located in the gray zone, and so the phase difference belongs to interval ½p=2; 3p=2. Moreover, according to Figure 4(a), it can be found that the phase difference is equal to 298.22°. The simulation results with MATLAB are shown in Figure 10. As becomes clear from them, the synchronous velocity of the rotors stabilizes at vicinity of 155 rad/s. The electromagnetic torque of the induction motors is identical.
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The phase difference between the rotors closely stabilizes at 2104°, which is according to the result of the theoretical computation in Figure 4(a). The displacement amplitudes in x-, y-, and c-directions are lager in this synchronous state. In the case, the part energy supplied by the two induction motors is dissipated by their friction dampers (W1DAMP and W2DAMP ), and the other part is balanced by the vibrating body (synchronization energy W1SYS , W2SYS ) when the two rotors rotate in the synchronous state. Moreover, the synchronous energy excites the vibrating body (WxSYS , WySYS , and WcSYS ), which is dissipated by the damper of the vibrating body (WxDAMP , WyDAMP , and WcDAMP ). So the energy balance equation of the system is identical with equation (54), that is W1DAMP þ W1SYS ¼ W1DRIVE ; W2DAMP þ W2SYS ¼ W2DRIVE ; W1DAMP þ W1SYS þ W2DAMP þ W2SYS ¼ W1DRIVE þ W2DRIVE ; W1SYS þ W2SYS ¼ WxSYS þ WySYS þ WcSYS ; WxSYS þ WySYS þ WcSYS ¼ WxDAMP þ WyDAMP þ WcDAMP ð55Þ Therefore, the energy balance of the system in the synchronous state is identical with Figure 7(c).
Conclusion To design new vibrating screens, the synchronization phenomenon for two co-rotating rotors, interacting via the vibrating body in the far-resonance vibration system, has been investigated with the energy balance method. According to this method, the synchronous zone, synchronous state, synchronous stability, and the energy balance of the system can be determined. It is observed that the dynamic characteristics of the vibrating body are related to the synchronous state of the system. When designing the vibrating screens, the synchronous state can be chosen to acquire the desired dynamics characteristics of the vibrating screens through this method. And the synchronous state of the system is mainly determined by installation distance coefficient rl, installation angular b, symmetric coefficient d, and electromagnetic torque Tei, while little influenced by the mass ratio h and rm. On the other hand, these theoretical investigations can be used to evaluate and discriminate whether a self-synchronous machine used in industries is able to achieve vibratory synchronization or not, as well as to supervise whether the design of a self-synchronous vibrating machine has the capacity of achieving vibratory synchronization.
Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study supported by the National Natural Science Foundation of China (Grant No. 51074132) and the Key Project of Talent Engineering of Sichuan, China (Grant No. 2016RZ0059).
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