Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 227-240, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.227

SYNCHRONIZATION AND STABILITY OF AN ELASTICALLY COUPLED
TRI-ROTOR VIBRATION SYSTEM

Yongjun Hou, Mingjun Du, Pan Fang, Liping Zhang

School of Mechanical Engineering, Southwest Petroleum University, Chengdu, China

e-mail: ckfangpan@126.com (Pan Fang – corresponding author)

Anewmechanism, an elastically coupled tri-rotor system, is proposed to implement synchro-
nization. It is composed of a rigid body, three induction motors, coupling unit and springs.
According to the Lagrange equation, the model of the system is established. The average
method of small parameters is applied to study the synchronization characteristics of the
system, therefore, the balance equation and stability criterion of the system can be obta-
ined. Obviously, many parameters affect the synchronous state of the rotors, especially the
spring stiffness, the stiffness of the coupling unit and the installation location of the system.
Finally, computer simulations are used to verify the correctness of theoretical analysis.

Keywords: tri-rotor, synchronization characteristics, stability, computer simulations

1. Introduction

The synchronization phenomenon is common in nature. General definitions of synchronization
were presented by Blekhman et al. (1997, 2002). The synchronization phenomenon is conside-
red as an adjustment of rhythms of oscillating objects due to their internal weak couplings.
Dutch scholar Huygens was first to discover the synchronization phenomenon, the synchronous
motion of a pendulum hanging on the common base in 1665 (Huygens, 1673). In 1960s, Blekh-
man proposed the synchronization theory of vibrating machines with two or multiple exciters
and successfully solved many engineering problems related to self-synchronization (Blekhman,
1998; Blekhman et al., 1997). Many fields, such as the modeling of nonlinear dynamics, co-
upling pendulums, mechanical rotors, have attracted attention of reserchers. In dynamics of
coupled pendulums and rotors, Blekhman proposed the Poincaré method for the synchroniza-
tion state and stability. Now it is amethodwidely used in engineering (Jovanovic andKoskhin,
2012). Based on Blekhman’s method, many scientists have developed other methods to analyze
synchronization of rotors (Blekhman, 1988). Koluda et al. (2014a,b) derived synchronization
conditions and explained observed types of synchronization for coupled double pendula. They
used an energy balance method to show how the energy is transferred between the pendula via
an oscillating beam. For synchronization of mechanical rotors, Zhao et al. (2010) and Zhang
et al. (2012) proposed an average method of modified small parameters, which was applied to
study of synchronous multiple unbalanced rotors (Zhang et al., 2013). Hou (2007) studied the
synchronism theory of threemotors using theHamilton principle. Balthazar (2004) andBaltha-
zar et al. (2005) described self-synchronization of two and four non-ideal rotating unbalanced
motors via numerical simulations. For synchronization and modeling of nonlinear dynamics, a
mechanism of interaction between two non-linear dissipative oscillators was presented by Rui
(2014). Twopendulums coupledwith aweak springwere proposed byBlekhman (1988). Kumon
et al. (2002) showed the synchronization phenomenon by designing the controller with applying
speed the Gradient Energy method. Fradkov and Andrievsky (2007) focused on the study of
phase relations between coupled oscillators.



228 Y. Hou et al.

However, for synchronization of three non-identical coupled exciters, the phase difference
of co-rotating motors stabilizes around 120◦ (Zhang et al., 2013). This results in a weakened
amplitude of the center ofmass. In order to improve vibration amplitude and screening efficiency
of the system, three rotors coupled with a weak spring are proposed in this paper. To explore
coupling characteristics of the system, synchronization conditions and the synchronous stability
criterion of the system are derived with the Poincaré method. Finally, computer simulations
are implemented to verify the results of theoretical analysis. It is demonstrated that the spring
stiffness, the coupling spring and the installation location plays a significant role in the vibration
system.

This paper is organized as follows. The analysis strategy and consideredmodel are described
in Section 2. In Section 3, the synchronization condition and the synchronization stability crite-
rion are obtained. In Section 4, the results of numerical simulations and results of the computer
simulations are presented, which validate correctness of the theoretical model of the vibration
system. Finally, the results are summarized in Section 5.

2. Model description

2.1. Strategy

The equations ofmotion for the considered rotation system are as follows (Fang et al., 2015)

Jsϕs = µΦs(ϕs, ẍ) s =1, . . . ,k

ẍ+2ωxξxẋ+ω
2
xx =

k
∑

j=1

Fj(ωt,α1, . . . ,αk)+µFk+1(ωt,α1, . . . ,αk)
(2.1)

where µΦs = Tms −Tfs, µ is the small parameter, Js is the rotational inertia of s-th induction
motor, Tms is the driving torque of the inductionmotors, Tfs is themechanical damping torque
of the induction motors, ξx and ωx are the damping coefficient and the natural frequency of
the system in the x-direction, ω and ϕs are mechanical velocity and rotation angles of the s-th
unbalanced rotor.

In the synchronous state, the velocity of the rotors is assumed as ω. Steady forced vibrations
with T =2π/ω are determined by

x = x(ωt,α1, . . . ,αk) (2.2)

Considering that the rotors are uniformly rotating with an initial phase α1, . . . ,αk, then the
phase angle of rotors should satisfy the synchronous solutions from the second formula Eq. (2.2)

ϕs = ϕ
0
s = ωt+αs (2.3)

Theabove-mentioned basic equationmaybe satisfiedwith suchvalues of constantsα1, . . . ,αk

Ps(α1, . . . ,αk)= µ〈Φs(ϕs(ϕs, ẍ))〉=0 (2.4)

Here, the angle brackets 〈∗〉 show the average value for one period by the variable t, and the
symbol ∗ represents a function related to time t

〈∗〉=
1

T

T
∫

0

∗ dt (2.5)



Synchronization and stability of an elastically coupling tri-rotors... 229

If a certain set of constants α1, . . . ,αk is satisfied byEq. (2.4), all the roots χ of the algebraic
equation
∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂(P1−Pk)

∂α1
−χ

∂(P1−Pk)

∂α2
. . .

∂(P1−Pk)

∂αk−1
∂(P2−Pk)

∂α1

∂(P2−Pk)

∂α2
−χ .. .

∂(P2−Pk)

∂αk−1
. . . . . . . . . . . .

∂(Pk−1−Pk)

∂α1

∂(Pk−1−Pk)

∂α2
. . .

∂(Pk−1−Pk)

∂αk−1
−χ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

=0 (2.6)

would have negative real parts, then unique constant values α1,α2, . . . ,αk are determinedwhen
the parameter µ is sufficiently small.Meanwhile, there exists an asymptotic periodic solution to
Eq. (2.1).Only a single root have apositive real part, and the corresponding solution is unstable.
For zero or imaginary roots, additional analysis would further be explored (Blekhman, 1998).

2.2. Kinematic equation of the vibrating system

The model of the vibration system is shown in Fig. 1. The system is mainly composed of
three induction motors, coupling unit, crossbeam, screen frame, motor seat. And two motors
rotate in the same direction connected with the coupling unit, which consists of a connecting
rod, chutes, coupling springs and slide blocks. The chute, linked to the end of the connecting rod
by welding, should bemutually parallel. The slide blocks and the coupling springs are installed
in the chutes. Besides, the stiffness of the connecting rod is bigger than the coupling springs,
and the connecting rodhas smaller density. The cross-section area of the connecting rod changes
with stiffness of the simplified spring.

Fig. 1. The model of an elastically coupled tri-rotor system

Figure 2 describes the dynamical model of the considered model. The exciters mi
(i = 1,2,3) are installed in the screen frame. The rigid vibro-body m0 is supported on an
elastic foundation by some stronger stiffness springs kx,ky,kψ in x-, y-, ψ-directions. The foun-
dation is characterized by damping constants Cx,Cy,Cψ. The elastic coupling unit is simplified
as a linear spring k, and the distance between the point of connecting of the springs and themo-
tors axles is assumed to be a. As illustrated in Fig. 2b, themass centers of the rigid vibro-body



230 Y. Hou et al.

is the point o. Three reference coordinate system of the vibration system is designed as follows:
the non-rotating moving frame o′x′y′ is always parallel to the fixed coordinate frame oxy in
the x- and y-directions, and the moving frame o′x′′y′′ swings around the point o′. The exciters
also rotate around their own spin axes, which are denoted by ϕi (i =1,2,3). M is mass of the
system, and the installation angle of the motor is expressed by βi (i = 1,2,3). The responses
x,y and the angular rotation ψ are considered as independent coordinates.

Fig. 2. Simplified model: (a) dynamic model of three rotors coupled with a weak spring, (b) the
reference frame system

The expressions for the kinetic energy of the system can be written as follows

T =
1

2
m0
{

[ẋ− ℓ0ψ̇ sin(β0+ψ+π)]
2+[ẏ + ℓ0ψ̇cos(β0+ψ+π)]

2
}

+
1

2

3
∑

i=1

Jiϕ̇
2
i

+
1

2

2
∑

i=1

mi
{

[ẋ− ℓiψ̇ sin(βi +ψ)+riϕ̇i sinϕi]
2

+[ẏ+ ℓiψ̇cos(βi +ψ)+riϕ̇icosϕi]
2
}

+
1

2
m3
{

[ẋ− ℓ3ψ̇ sin(β3+ψ)−r3ϕ̇3 sinϕ3]
2

+[ẏ+ ℓ3ψ̇cos(β3+ψ)+r3ϕ̇3cosϕ3]
2
}

+
1

2
J0ψ̇

2

(2.7)

Moreover, considering that the distance of the co-rotating induction motors is r, and assu-
ming that the ratio (a/r ≪ 1) is infinitesimally small, the elongation of the coupled spring can
be obtained

∆ℓ = ℓ− ℓ0 ≈ a(cosϕ1− cosϕ2) (2.8)

And the potential energy of the system can be written as

V =
1

2
kxx
2+
1

2
kyy
2+
1

2
kψψ

2+
1

2
∆ℓ2 (2.9)

In addition, the viscous dissipation function of the vibration system can be expressed as

D =
1

2
Cxẋ

2+
1

2
Cyẏ

2+
1

2
Cψψ̇

2+
1

2
C1ϕ̇

2
1+
1

2
C2ϕ̇

2
2+
1

2
C3ϕ̇

2
3 (2.10)



Synchronization and stability of an elastically coupling tri-rotors... 231

The dynamics equation of the system can be obtained according to Lagrange’s equation

d

dt

∂(T −V )

∂q̇i
−
∂(T −V )

∂qi
+
∂D

∂qi
= Qi (2.11)

If q = [x,y,ψ,ϕ1,ϕ2,ϕ3]
T is chosen as the generalized coordinates, the generalized forces are

Qx = Qy = Qψ = 0, Qϕi = Tmi − Tfi. It can be seen that mi ≪ m0 and ψ ≪ 1 in the
system, and the inertia coupling from asymmetry of the system can be neglected. Considering
M =

∑3
i=1mi +m0, m1 = m2, r1 = r2, the kinetic equation of the vibration system is derived

as

Mẍ+Cxẋ+kxx = m3r3(ϕ̈3 sinϕ3+ ϕ̇
2
3cosϕ3)−

2
∑

i=1

miri(ϕ̈i sinϕi + ϕ̇
2
i cosϕi)

Mÿ +Cyẏ+kyy = m3r3(ϕ̇
2
3 sinϕ3− ϕ̈3cosϕ3)+

2
∑

i=1

miri(ϕ̇
2
i sinϕi − ϕ̈icosϕi)

Jψ̈+Cψψ̇+kψψ =
2
∑

i=1

miℓiri[ϕ̇
2
i sin(ϕi +βi +ψ)− ϕ̈icos(ϕi +βi +ψ)]

+m3r3ℓ3[ϕ̇
2
3 sin(ϕ3−β3−ψ)− ϕ̈3cos(ϕ3−β3−ψ)]

+C3(ϕ̇3− ψ̇)−C1(ϕ̇1+ ψ̇)−C2(ϕ̇2+ ψ̇)

Jo1ϕ̈1 = Tm1−Tf1−C1(ϕ̇1+ ψ̇)−m1r1[ẍsinϕ1+ ÿcosϕ1]

+m1r1ℓ1[ψ̇
2 sin(ϕ1+β1+ψ)− ψ̈cos(ϕ1+β1+ψ)]−ka

2(cosϕ2− sinϕ1)sinϕ1

Jo2ϕ̈2 = Tm2−Tf2−C2(ϕ̇2+ ψ̇)−m2r2[ẍsinϕ2+ ÿcosϕ2]

+m2r2ℓ2[ψ̇
2 sin(ϕ2+β2+ψ)− ψ̈cos(ϕ2+β2+ψ)]−ka

2(cosϕ1− sinϕ2)sinϕ2

J03ϕ̈3 = Tm3−Tf3−C3(ϕ̇3− ψ̇)+m3r3[ẍsinϕ3− ÿcosϕ3]

+m3r3ℓ3[−ψ̇
2 sin(ϕ3−β3−ψ)− ψ̈cos(ϕ3−β3−ψ)]

(2.12)

3. Criterion of synchronization and stability of synchronous states

3.1. Method description

According to the Poincaré method (i.e., based on fundamental Eq. (2.1)), introducing the
small parameter µ into Eq. (2.12), the influence of the small parameter can be ignored, then a
new form of Eq. (2.12) is given

Mẍ+kxx = m3r3(ϕ̈3 sinϕ3+ ϕ̇
2
3cosϕ3)−

2
∑

i=1

miri(ϕ̈i sinϕi +ϕ
2
i cosϕi)

Mÿ +kyy = m3r3(ϕ̇
2
3 sinϕ3− ϕ̈3cosϕ3)+

2
∑

i=1

miri(ϕ̇
2
i sinϕi − ϕ̈icosϕi)

Jψ̈+kψψ =
2
∑

i=1

miℓiri[ϕ̇
2
i sin(ϕi +βi +ψ)− ϕ̈icos(ϕi +βi +ψ)]

+m3r3ℓ3[ϕ̇
2
3 sin(ϕ3−β3−ψ)− ϕ̈3cos(ϕ3−β3−ψ)]+C3ϕ̇3−C1ϕ̇1−C2ϕ̇2

Jo1φ̈1 = µφ1 Jo2φ̈2 = µφ2 J03φ̈3 = µφ3

(3.1)



232 Y. Hou et al.

where

µφ1 = Tm1−Tf1−m1r1[ẍsinϕ1+ ÿcosϕ1]

+m1r1ℓ1[ψ̇
2 sin(ϕ1+β1+ψ)− ψ̈cos(ϕ1+β1+ψ)]−ka

2(cosϕ2− sinϕ1)sinϕ1

µφ2 = Tm2−Tf2−m2r2[ẍsinϕ2+ ÿcosϕ2]

+m2r2ℓ2[ψ̇
2 sin(ϕ2+β2+ψ)− ψ̈cos(ϕ2+β2+ψ)]−ka

2(cosϕ1− sinϕ2)sinϕ2

µφ3 = Tm3−Tf3+m3r3[ẍsinϕ3− ÿcosϕ3]

+m3r3ℓ3[−ψ̇
2 sin(ϕ3−β3−ψ)− ψ̈cos(ϕ3−β3−ψ)]

(3.2)

Solving Eq. (3.1), the steady responses in the x-, y- and ψ-directions are obtained

x = a3cosϕ3−a1cosϕ1−a2cosϕ2

y = b1 sinϕ1+ b2 sinϕ2+ b3 sinϕ3

ψ = c1 sin(ϕ1+β1)+c2 sin(ϕ2+β2)+ c3 sin(ϕ3+β3)

(3.3)

where

αi =
miriϕ̇

2
i

kx −Mϕ̇
2
i

bi =
miriϕ̇

2
i

ky −Mϕ̇
2
i

ci =
miriϕ̇

2
i ℓi

kψ −Jϕ̇
2
i

i =1,2,3 (3.4)

Here, introducing the following dimensionless parameters, the standard mass m is defined, and
the natural frequencies are denoted by ωx,ωy,ωϕ in the x-, y-, ψ-direction, respectively.

η1 =
m1
m

η2 =
m2
m

η3 =
m3
m

rm =
m
M

re =
m

J

ωx =

√

kx
M

ωy =

√

ky
M

ωψ =

√

kψ
J

σ =
re
rm

ρ =
r3
r1

λ1 =
ω2m

ω2m −ω
2
x

λ2 =
ω2m

ω2m −ω
2
y

λ3 =
ω2m

ω2m −ω
2
ψ

(3.5)

Consequently, basic Eq. (3.3) will be written as

x = rmλ1(η1r1cosϕ1+η2r2cosϕ2−η3r3cosϕ3)

y =−rmλ2(η1r1 sinϕ1+η2r2 sinϕ2+η3r3 sinϕ3)

ψ =−reλ3[η1r1 sin(ϕ1+β1)+η2r2 sin(ϕ2+β2)+η3r3 sin(ϕ3−β3)]

(3.6)

3.2. Synchronization criterion

Theoretical derivation of the synchronization condition is discussed in this Section. Assume
that αi, ϕi are the initial phase and phase angle of the unbalanced rotor i, respectively. The
solution mentioned above is corresponding with Eq. (2.3)

ϕ1 =ωt+α1 ϕ2 = ωt+α2 ϕ3 = ωt+α3 (3.7)



Synchronization and stability of an elastically coupling tri-rotors... 233

According to Eq. (2.4), substituting Eq. (3.7) into Eq.(3.2), Pi can be calculated

P1 = 〈µφ1〉= Tm1−Tf1−
1

2
m1rmr1ω

2[η2r2(λ1+λ2)sin(α2−α1)

+η3r3(λ2−λ1)sin(α3−α1)]−
1

2
m1r1ℓ1reλ3ω

2[η2r2 sin(α2−α1+β2−β1)

+η3r3 sin(α3−α1−β3−β1)]+
1

2
ka2[sin(α2−α1)+1]= 0

P2 = 〈µφ2〉= Tm2−Tf2−
1

2
m2rmr2ω

2[−η1r1(λ1+λ2)sin(α2−α1)

+η3r3(λ1−λ2)sin(α2−α3)]−
1

2
m2r2ℓ2reλ3ω

2[η1r1 sin(α1−α2+β1−β2)

+η3r3 sin(α3−α2−β3−β2)]+
1

2
ka2[−sin(α2−α1)+1]= 0

P3 = 〈µφ3〉= Tm3−Tf3+
1

2
m3rmr3ω

2[η1r1(λ1−λ2)sin(α1−α3)

+η2r2(λ1−λ2)sin(α2−α3)]

−
1

2
m3r3ℓ3reλ3ω

2[η1r1 sin(α1−α3+β1+β3)+η2r2 sin(α2−α3+β2+β3)] = 0

(3.8)

When the angular velocity of the tri-rotors is near to the synchronous velocity ωm, the excessive
torque Zs(ω) of the rotors is equal to zero in the synchronization state

Zi(ω)= Tmi −Tfi =0 i =1,2,3 (3.9)

The balance equation of synchronization of the vibrating system can be expressed as

µ1[sin(α3−α1)+sin(α3−α2)]+µ2 sin(α2−α1+β2−β1)

+µ3 sin(α3−α1−β3−β1)+µ4 sin(α3−α2−β3−β2)−µ7 =0

µ5[sin(α1−α3)+sin(α2−α3)]+µ6[sin(α3−α1−β1−β3)

+sin(α3−α2−β2−β3)] = 0

(3.10)

where

µ1 = η1η3ρ(λ2−λ1) µ2 = η1η2σλ3(ℓ1− ℓ2) µ3 = η1η3ℓ1σρλ3

µ4 = η2η3ℓ2σρλ3 µ5 =λ1−λ2 µ6 = ℓ3σλ3

µ7 =
2ka2

m0r
2
1rmω

2

(3.11)

3.3. Stability criterion of synchronization states

Introduce now new parameters A, B, C, D, i.e:

A =
∂(P1−P3)

∂α1
B =

∂(P2−P3)

∂α2
C =

∂(P2−P3)

∂α1
D =

∂(P2−P3)

∂α2
(3.12)

According to Eq.(2.6), the stability criterion of synchronization of the system can be expressed
as

A+B < 0 (3.13)



234 Y. Hou et al.

InsertingEq. (3.8) andEq. (3.12) intoEq. (3.13), the stability criterion of synchronization states
can be simplified as

2µ8cos(α2−α1)+2µ1[cos(α3−ϕ1)+cos(α2−α3)]+µ9cos(α2−α1+β2−β1)

+µ10cos(α1−α3+β1+β3)+µ11cos(α2−α3+β2+β3)−µ7cos(α2−α1) < 0
(3.14)

where

µ8 = η1η2(λ1+λ2) µ9 = η1η2(ℓ1+ ℓ2)σλ3

µ10 = η1η3ρσλ3(ℓ1+ ℓ3) µ11 = η1η2ρσλ3(ℓ2+ ℓ3)
(3.15)

Therefore, only the systemparameters satisfy balance equations (3.10) and the stability criterion
of synchronization (3.14) can be implemented in the considered case.

4. Numerical verification

In the above Sections, the differential equations, balanced equations and the stability criterion
of synchronization have been derived. The theoretical and simulation results are presented in
this Section, where the correctness of the theory is to be verified.

4.1. Analysis of numerical results

Some examples are used to prove the correctness of the results of the above theoretical de-
rivation. Based on Eq. (4.1), the stiffness coefficients kx, ky and kϕ are separately transformed
into frequency ration ηx, ηy, ηϕ. Balance equations (3.10) are nonlinear equations related to
the system parameters, including the supporting spring stiffness, stiffness of elastic spring k,
installation location, etc., which seriously influence the stability of self-synchronization of the
system.When the system parameters are simultaneously satisfied, balance equation (3.10), sta-
bility criterion (3.14) and the stable phase difference can be estimated by applying a numerical
method. In order to simplify calculations, we assume ηx = ηy = ηϕ, i.e.,

ηx =
ω

ωx
ηy =

ω

ωy
ηϕ =

ω

ωϕ
(4.1)

Studying synchronization of the vibration system, the parameters are shown inTable 1, and
the dimensionless values are shown in Table 2 according to Eq. (3.5).

Table 1.Parameter values of the system

Unbalanced rotor for i =1,2,3 mi =3kg, ri =0.02m, ωm =156∼ 157rad/s,
ci =0.01N·s/m

Vibro-platform M =100kg J =10kg·m2, fx =1000N/(m/s),
fy =1000N/(m/s), fz =1000N/(m/s),
kx =1 ·10

4 ∼ 3.65 ·105N/m,
ky =1 ·10

4 ∼ 3.65 ·105N/m,
kψ =1 ·10

3 ∼ 3.65 ·104N/m

Other parameters l1 =0.8m, l2 =0.73, 0.41m, l3 =0.8m,
β1 =2π/3, 5π/6, β2 =2π/5, 5π/12, β3 = π/3, π/6

The spring k =0∼ 1.4 ·105N/m, a =0.01m

Equations (3.10) and (3.14) describe the approximate analytical solution for the stable phase
difference. Based on the parameters in Table 2, we can acquire an approximate value of ϕ1−ϕ2
and ϕ1 − ϕ3 considering different parameters k,ηx,ηy,ηϕ when three motors are installed in



Synchronization and stability of an elastically coupling tri-rotors... 235

Table 2.Parameter values according to dimensionless Eq. (3.5)

η1 =1, η2 =1, η3 =1, rm =0.02, re =0.18, σ =8.82, ρ =1,
nx =1∼ 19, ny =1∼ 19, nϕ =1∼ 15.7

different positions. The analytical results are shown in Fig. 3 and Fig. 4. They indicate that
the parameters ηx, ηy, ηϕ, have little influence on the value of the phase difference when the
above-mentioned balance equation and the stability criterion equation are satisfied. But the
parameter k directly affects the phase difference. Figure 3 shows numerical results for positional
parameters (i.e, l1 = 0.8m, l2 = 0.73m, l3 = 0.8m, β1 = 2π/3, β2 = 2π/5, β3 = π/3) for
different frequency ratios. When k = 0N/m (there is no coupling unit in co-rotating motors),
the phase difference ϕ1−ϕ2 of the co-rotating motors is stabilized in the vicinity of 3rad, and
the phase difference ϕ1−ϕ3 of the counter-rotatingmotors is near 1rad.When k ­ 30000N/m,
the phase difference of the co-rotating motors is close to 0rad and the stable difference ϕ1−ϕ3
is near 1rad. Meanwhile, the vibration amplitude improves when the stiffness of the coupling
spring k exceeds the maximum value kmax = 140000N/m (Fig. 3a), and kmax = 120000N/m
(Fig. 3b,c,d), which means that the synchronous motion is unstable. The numerical results for
l1 = 0.8m, l2 = 0.41m, l3 = 0.8m, β1 = 5π/6, β2 = 5π/12, β3 = π/6 are displayed in Fig. 4.
Similar conclusions are also obtained.

Fig. 3. Stable phase difference with theoretical analysis for l1 =0.8m, l2 =0.73m, l3 =0.8m,
β1 =2π/3, β2 =2π/5, β3 = π/3; (a) ηx = ηy = ηϕ =1.76, (b) ηx = ηy = ηϕ =3.51,

(c) ηx = ηy = ηϕ =5.23, (d) ηx = ηy = ηϕ =7.85;−·− shows there is no stable phase difference

The above analysis implies that these parameters play an important role in the synchronous
state, whichmainly include the stiffness coefficient k, frequency ratios and installation location
of three induction motors. Besides, the coupled spring connecting the co-rotation rotors is also
compliant with the condition and stability of synchronization. By selecting a large value of k,
the vibration amplitude and the screening efficiency of the system can be improved.

4.2. Buckling analysis of the connecting rod

The two chutes are connected by the connecting rod. During the process of self-
-synchronization, the elasticity coupling between the two induction motors can be achieved



236 Y. Hou et al.

Fig. 4. Stable phase difference with theoretical analysis for l1 =0.8m, l2 =0.41m, l3 =0.8m,
β1 =5π/6, β2 =5π/12, β3 = π/6; (a) ηx = ηy = ηϕ =2.22, (b) ηx = ηy = ηϕ =4.96,

(c) ηx = ηϕ =5.55, (d) ηx = ηy = ηϕ =7.02,−·− shows there is no stable phase difference

by the springs in the chutes. Owing to the connecting rod with smaller density and the strong
stiffness, the centrifugal inertia force of the rod is small, in which case a deflection of the elastic
rod can be ignored. For example, the location parameters of vibrationmotors are identical with
the parameters in Fig. 3. According to theoretical analysis (Fig. 3), the stiffness of the simplified
spring has the maximum value kmax =120000N/m. Assume k =80000N/m in this case.

The phase difference of co-rotating motors is expressed as α, which satisfies

α = |ϕ1−ϕ2| (4.2)

The range of the phase difference α is obtained as 0¬ α ¬ π.When α =0◦, the deformation of
the simplified spring is equal to 0; when α =180◦, the deformation of the simplified spring has
the maximum value, xmax = 2a = 0.02m (the simplified spring is in stretched state or under
compression), 0¬ x ¬ xmax.

The force in the connecting rod satisfies

F = kx ¬ Fmax = kxmax =80000
N

m
·0.02m=1600N (4.3)

Assuming that thematerial of the rod is 2024(LY12), then the yield strength and the density
are [σ] = 325MPa, ρ = 2770kg·m−3, E = 72GPa = 7.2 · 104N/mm2. Then the cross-section
area A of the connecting rod can be determined

A ­
Fmax
[σ]
= 4.92mm2 (4.4)

Themodel of the connecting rod in buckling analysis is shown in Fig. 5. The applied load is
expressed as F (µ =1). The inertia moment of circular section can be expressed as

I =
1

32
πd4 (4.5)



Synchronization and stability of an elastically coupling tri-rotors... 237

Fig. 5. Themodel of the connecting rod in buckling analysis

The cross-sectional area A of the connecting rod

A =
πd2

4
(4.6)

The critical load of the rod can be computed by Euler’s formula

Fcr =
π2EI

(µL)2
(4.7)

If the buckling of the rod is not achieved, the statical criterion for elastic stability satisfies

F < Fcr (4.8)

Based on the critical condition F = Fmax, from equation (4.5)-(4.8), A can be calculated

A >

√

2Fmax(µL)2

Eπ
=35.7mm2 (4.9)

Therefore, the cross-section area the connecting rod A can be obtained

A > 35.7mm2 (4.10)

If A =36mm2, so the mass can be calculated

m = ρLS =2770
kg

m2
·0.3m ·36mm2 =0.03kg≪ M =100kg (4.11)

Therefore, the mass is too small to be neglected.
According to the national design standard, the size of the coupling springs can be finally

established.

4.3. Simulation results for nx = ny = nϕ =5.23, k =60000N/m, l1 =0.8m, l2 =0.73m,
l3 =0.8m

Simulation results for the dimensionless parameters in Table 3 are shown in Fig. 6. Here,
kx = ky = 9.0 · 10

4N/m,kψ = 9.0 · 10
3N/rad, l1 = 0.8m, l2 = 0.73m, l3 = 0.8m, and the

other parameters are identical with those in Table 1. From Fig. 6a to Fig. 6f, it can be seen
that the self-synchronization of the system is implemented. The three induction motors cannot



238 Y. Hou et al.

Table 3.Dimensionless parameter values

η1 =1, η2 =1, η3 =1, rm =0.02, re =0.20, σ =8.70, ρ =1,
nx =8.54, ny =8.54, nϕ =8.54

Fig. 6. Results of computer simulations. (a),(b),(c) Displacement responses of the vibrating body in the
x-, y-, ψ-directions, respectively; (d) rotational velocities of the vibration system; (e) electromagnetic

torques of the tri-rotors; (f) phase difference of unbalanced rotors

simultaneously start at the same angular velocity owing to the difference coupling characteri-
stics when three exciters are switched on at the same time. Eventually, the value of angular
velocity is identical, see Fig. 6d. The average angular velocity of the three unbalanced rotors is
155.7rad/s at about 2s, which is always defined as the synchronous velocity. In addition, the
coupling torques (Fig. 6e), keeping the vibration system working in a steady synchronization
state, are approximately 0.4N/m.The phase difference ϕ1−ϕ2 of the co-rotatingmotors is near
0.205rad. The phase difference ϕ1−ϕ3 of the counter-rotating motors stabilized in the vicinity
of 88.73rad (88.73rad= (28π+0.77)rad,Fig. 6f), agreeswith the approximate theoretical value
0.83rad. The stable difference ϕ1−ϕ2 is equal to 0.28rad, Fig. 3c. It can be seen that the ideal
phase synchronization is achieved by two co-rotation rotors coupledwith aweak spring, and the
excitation forces of the system are improved. The displacement response of the vibrating body
is displayed in the x-, y- and ψ-directions, respectively, Fig.6a,b,c. The computer simulation
results further proved the validity of theoretical analysis.



Synchronization and stability of an elastically coupling tri-rotors... 239

5. Conclusions

Based on the theoretical research andnumerical analysis, the following conclusions are obtained.

In this paper, a newvibrationmechanism, an elastically coupled tri-rotor system, is proposed
to implement synchronization.Theaveragemethodof small parameters is used to study synchro-
nization characteristics of the system.Thedynamical equations are converted into dimensionless
equations, and the synchronized state have been investigated. When the values of the system
parameters satisfy the balance equations and the stability criterion of synchronization, the vi-
bration system can operate in a steady state. The study indicates that many factors, such as
the spring stiffness, stiffness of the elastic unit and the installation location, influence stability
of the system. Finally, computer simulations have been preformed to verify the correctness of
the approximate solution from computations for the vibration system. Besides, it can be found
that the spring connecting the co-rotation rotors makes the phase difference stabilized in the
vicinity of 0rad, and the vibration amplitude of the system is improved in contrast to the former
one. In this case, the screening efficiency of the system can be improved aswell.Moreover, when
stiffness of the coupling spring exceeds the maximum value, the vibration system locates in an
unstable state. In short, a new balanced elliptical vibrating screen is proposed having a bright
future in applications.

Acknowledgment

This study has been supported by science and technology support plan of Sichuan province

(2016RZ0059) and the National Natural Science Foundation of China (Grant No. 51074132).

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Manuscript received January 16, 2016; accepted for print July 14, 2016