# Nonlinear Response of Vibrational Conveyers with Nonideal Vibration Exciter: Superharmonic and Subharmonic Resonance

Authors: H. Bayıroğlu; G. F. Alışverişçi; G. Ünal
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Vibrational conveyers with a centrifugal vibration exciter transmit their load based on the jumping method. Common unbalanced-mass driver oscillates the trough. The motion is strictly related to the vibrational parameters. The transition over resonance of a vibratory system, excited by rotating unbalances, is important in terms of the maximum vibrational amplitude produced and the power demand on the drive for the crossover. The mechanical system is driven by the DC motor. In this study, the working ranges of oscillating shaking conveyers with nonideal vibration exciter have been analyzed analytically for superharmonic and subharmonic resonances by the method of multiple scales and numerically. The analytical results obtained in this study agree well with the numerical results.


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## Body

## 1. Introduction

The load-carrying element of a horizontal shaking conveyer performs, as a rule, linear (or sometimes circular or elliptical) symmetrical harmonic oscillations—with a sinusoidal variation of exciting force. In vertical shaking conveyers, the load-carrying element performs double harmonic oscillations: linear along the vertical axis and rotational around that axis (i.e., longitudinal and torsional oscillations). Conveyer drives with centrifugal vibration exciters may have (1) a single unbalanced-mass, (2) two equal unbalancing masses, (3) a pendulum-type unbalanced-mass, (4) four unbalanced-masses in two shafts, (5) four rotating unbalanced-masses for three principal modes of oscillation, that is, linear, elliptical, and circular. To induce strictly oriented linear oscillations of the load-carrying element, the conveyer drive should be arranged so that the line of excitation force passes through the inertial centre of the entire oscillating system. Nonideal drives find application in suspended and supported vibrational conveyers and feeders [1].By the characteristics and adjustment of the elastic support elements (oscillating system), we can distinguish between shaking conveyers with a resonant, subresonant, and superresonant system.A practical difficulty with unbalanced-mass exciters, observed as early as 1904 by A Sommerfeld, is that local instabilities may occur in operating speed of such devices.Rocard [2], Mazert [3], Panovko, and I. I. Gubanova [4] have studied the problem of the stability of the unbalanced-mass exciter.The first detailed study on the nonideal vibrating systems is presented by Kononenko. He obtained satisfactory results by the comparison of the experimental analysis and the approximated method [5].After this publication, the nonideal problem was presented by Evan-Ivanowski [6] or Nayfeh and Mook [7]. These authors showed that sometimes dynamical coupling between energy sources and structural response that had not to be ignored in reel engineering problems.Theorical studies and computations of Ganapathy and Parameswaran have indicated the beneficial effect of the “material load” during the starting and transition phase of an unbalanced-mass-driven vibrating conveyor [8].Bolla et al. analyzed through the multiple scales method a response of a simplified nonideal and nonlinear vibrating system [9].Götzendorfer in [10] presented a macromechanical model for the transport of granular matter on linear and horizontal conveyors subject to linear, circular or elliptic oscillations and compared it to experimental results [10].


## 2. The Governing Equations of the Motion

The equations of motion for the modified rocard system may be obtained by using Lagrange’s equation(2.1)ddt(∂T∂q̇i)+∂D∂q̇i-∂T∂qi+∂V∂qi=0,
where T, the kinetic energy, is
(2.2)T=12Mẏ2+12m[(ẏ+θ̇ecos⁡θ)2+(θ̇esin⁡θ)2]+12Imotθ̇2,
where e is the eccentricity of the mass m, m is the unbalanced-mass, M is the mass of the trough and the conveyed material on the trough of the conveyor, θ is the angle of rotation of the shafts carrying unbalanced-masses, Imot is the moment of inertia of the rotating parts in the motor, V, the potential energy, is
(2.3)V=12k1y2+14k2y4+mge(1+sin⁡θ),
where the constants k1 and k2 are the linear and nonlinear elastic coefficients, respectively, D, the Rayleigh dissipation function, is
(2.4)D=12cẏ2+12K(ωs-θ̇),
and qi is the generalized coordinate. Applying Lagrange’s equation for the two coordinates qi=y and qi=θ gives the differential equations of motion
(2.5)Mÿ+mÿ+cẏ+k1y+k2y3=me(θ̇2sin⁡θ-θ̈cos⁡θ),(Imot+me2︸Isys)θ̈+me(ÿcos⁡θ+gcos⁡θ)=12[∂K∂θ̇(θ̇-ωs)+K].
Equation (2.5) can be rewritten as follows:
(2.6)ÿ+ω02y=ε[-2μẏ-αy3-eθ̈cos⁡θ]+κ(θ̇2sin⁡θ),(2.7)θ̈=ε[-Icos⁡θ(ÿ+g)+E(θ̇)],
where
(2.8)I=(m+M)eIsys,κ=mem+M,2μ=cm,α=k2m,k1m+M=ω02,mm+M=ε,E(θ̇)=(m+M)L(θ̇)mIsys,L(θ̇)=12[∂K∂θ̇(θ̇-ωs)+K],
where c is the damping coefficient of the vibrating conveyor, g is acceleration due to gravity, Isys=(Im+IM+Imot) is the total moment of inertia of all the rotating parts in the system, and ωs is the synchronous angular speed of the induction motor [9]. K is the instantaneous drive torque available at the shafts. Note that E contains L(θ̇) that is the active torque generated by the electric circuit of the DC motor, shown in Figure 1.Figure 1
Vibrating model of the system.


## 3. Analytical Solution

Ideal system: if there is no coupling between motion of the rotor and vibrating system andθ̇=constant (θ=ωt, θ̈=0), (2.6) becomes
(3.1)ÿ+ω02y=ε[-2μẏ-αy3]+κω2sin⁡ωt.
On the right side of the equation, a function of time is present.Nonideal system:(3.2)θ̈=ε[-Icos⁡θ(ÿ+g)+E(θ̇)],ÿ+ωn2y=ε[-2μẏ-αy3-eθ̈cos⁡θ]+κ(θ̇2sin⁡θ),
where E(θ̇)=Mm(θ̇)-H(θ̇) is the difference between the torque generated by the motor and the resistance torque. Function E(θ̇)=u1-u2θ̇ is approximated by a straight line, where u1 is a control parameter that can be changed according to the voltage, and u2 is a constant parameter, characteristic for the model of the motor.We will obtain an approximate analytical solution to (3.2) by using the multiple scales method:
(3.3)θ(t,ε)≈θ0(T0,T1)+εθ1(T0,T1)+ε2θ2(T0,T1),y(t,ε)≈y0(T0,T1)+εy1(T0,T1)+ε2y2(T0,T1),Tn=εnt,n=0,1,…,T0=t,T1=εt
where the fast scale T0=t and slow scale T1=εt. The time derivatives transform according to
(3.4)ddt=dT0dt∂∂T0+dT1dt∂∂T1+⋯=D0+εD1+⋯d2dt2=D02+2εD0D1+⋯,
where Dn=∂/∂Tn, (n=0,1,…); then,
(3.5)θ̇=dθdt≈D0θ0+ε(D1θ0+D0θ1)+⋯,θ̈=d2θdt2≈D02θ0+ε(2D0D1θ0+D02θ1)+⋯,ẏ=dydt≈D0y0+ε(D1y0+D0y1)+⋯,ÿ=d2ydt2≈D02y0+ε(2D0D1y0+D02y1)+⋯.
Substituting (3.5) into (3.2), we will obtain
(3.6)D02θ0=ε[-2D0D1θ0-D02θ1-Icos⁡θ(D02y0+g)+E(θ̇)],D02y0+ε(2D0D1y0+D02y1)+ωn2(y0+εy1)=ε[-2μD0y0-α(y0)3-eD02θ0cos⁡θ]+κ[D0θ0+ε(D1θ0+D0θ1)]2sin⁡θ],
and equating coefficients of a like powers ε, we obtain(i)
forε0(3.7)D02θ0=0,D02y0+ω02y0=κ(D0θ0)2sin⁡θ,(ii)
forε1(3.8)D02θ1=-2D0D1θ0-Icos⁡θ(D02y0+g)+E(θ̇),(3.9)D02y1+ω02y1=-2D0D1y0-2μD0y0-α(y0)3-eD02θ0cos⁡θ+κ2D0θ0(D1θ0+D0θ1)sin⁡θ.
The solution of (3.7) can be written as
(3.10)θ0=BT0+σT1,(3.11)y0=Λsin⁡(T0Ω)+A(T1)eiω0T0+A̅(T1)e-iω0T0,
where
(3.12)cos⁡(θ0+εθ1)=cos⁡(θ0)+O(ε),sin⁡(θ0+εθ1)=sin⁡(θ0)+O(ε),Λ=-κΩ2(Ω2-ω02).


### 3.1. Subharmonic ResonancesΩ≈3ω0 (Table 1)

Table 1
Vibrational conveyer parameters in SI units.
εαμcIe(m)E1E0gω0k1k2mM0.050.010.0110.90.21.51.69.81120001005200Near resonance:(3.13)D0θ0=Ω,(3.14)Ω=3ω0+εσ.
The solution of (3.13) can be written as
(3.15)θ0=ΩT0=3ω0T0+σT1.
Taking
(3.16)A[T1,T2]=a[T1]2eiβ[T1],
where a and β are real, and substituting it into (3.11), we will obtain
(3.17)y0=12e-iT0ω0-iβ(T1)a(T1)+12eiT0ω0+iβ(T1)a(T1)+Λsin⁡(T0Ω),θ1=18(Ω3-Ωω02)2e-i(T0ω0+β(T1))×I((-4g+(-κΩ2+Λω02)sin⁡(T0Ω))-4Ω2ω02a(T1)×((1+e2i(T0ω0+β(T1)))(Ω2+ω02)cos⁡(T0Ω)-2i(-1+e2i(T0ω0+β(T1)))Ωω0sin⁡(T0Ω))-2ei(T0ω0+β(T1))(Ω2-ω02)cos⁡(T0Ω)(-4g+(-κΩ2+Λω02)sin⁡(T0Ω))),y1=(e-i(3T0Ω+4T0ω0+3β(T1))×(-8e3iβ(T1)Ω(Ω3+3Ω2ω0-Ωω02-3ω03)×(ieiT0(4Ω+3ω0)ω02(3αΛ3+8iΛμΩ+Iκ2Ω2-IκΛω02)(36Ω4-13Ω2ω02+ω04)+ieiT0(2Ω+3ω0)ω02(-3αΛ3+8iΛμΩ-Iκ2Ω2+IκΛω02)(36Ω4-13Ω2ω02+ω04)+4eiT0(Ω+3ω0)gIκω02(9Ω4-10Ω2ω02+ω04)+4eiT0(5Ω+3ω0)gIκω02×(9Ω4-10Ω2ω02+ω04)+8e3iT0(Ω+ω0)gIκ(36Ω6-49Ω4ω02+14Ω2ω04-ω06)-ie3iT0(2Ω+ω0)ω02(4Ω4-5Ω2ω02+ω04)(αΛ3+Iκ(κΩ2-Λω02))-ie3iT0ω0ω02(4Ω4-5Ω2ω02+ω04)(-αΛ3+Iκ(-κΩ2+Λω02)))+2e2iβ(T1)ω02(36Ω5+108Ω4ω02-39Ω2ω03+Ωω04+3ω05)×(eiT0(5Ω+2ω0)(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+ei(T0(Ω+4ω0)+2β(T1))(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+ei(T0(Ω+2ω0)(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0))+ei(5T0Ω+4T0ω0+2β(T1))×(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0)))a(T1)+24iei(T0(2Ω+ω0)+β(T1))α×ΛΩω02(-(-1-2e2i(T0ω0+β(T1))+2e2i(T0(Ω+ω0)+β(T1))+e2i(T0(Ω+2ω0)+2β(T1)))Ω+(-1+6e2i(T0ω0+β(T1))-6e2i(T0(Ω+ω0)+β(T1))+e2i(T0(Ω+2ω0)+2β(T1)))ω0)×(36Ω6-49Ω4ω02+14Ω2ω04-ω06)a(T1)2+e3iT0Ω(1+e6i(T0ω0+β(T1)))αΩ×(Ω2-ω02)2(36Ω5+108Ω4ω0-13Ω3ω02-39Ω2ω02+Ωω04+3ω05)a(T1)3)×(cos⁡(T0ω0)+isin⁡(T0ω0)))/(64Ω(6Ω2-5Ωω0+ω02)(6Ω2-5Ωω0+ω02)×(-Ω2ω0+ω03)2×(6Ω3+23Ω2ω0+16Ωω02+3ω03)).
Secular terms will be eliminated from the particular solution of (3.8) if we choose A to be a solution of
(3.18)-2εσ′+E(θ̇).In addition to the terms proportional toe(±iω0T0) or proportional to e[±i(Ω-2ω0)T0] there is another term that produces a secular term in (3.9). We express (Ω-2ω0)T0 in terms of ω0T0 according to
(3.19)(Ω-2ω0)T0=ω0T0+εσT0=ω0T0+σT1.


### 3.2. Superharmonic ResonancesΩ≈(1/3)ω0 (Table 2)

Table 2
Vibrational conveyer parameters in SI units.
εαμcIe(m)E1E0gω0k1k2mM0.050.010.0110.90.21.51.69.81102000001005200We consider(3.20)3Ω=ω0+εσ.In addition to the terms proportional toe(±iω0T0) in (3.9), there is another term that produces a secular term in (3.9). This is -αΛ3e(±3iΩT0). T0 eliminate the secular terms, we express 3ΩT0 in terms of ω0T0 according to
(3.21)3ΩT0=(ω0+εσ)T0=ω0T0+σεT0=ω0T0+σT1.
Eliminating secular terms in equations, taking (3.16), and separating real and imaginary parts, the system of equations solved it is as follows [11]:
(3.22)y0=12e-iT0ω0-iβ(T1)a(T1)+12eiT0ω0+iβ(T1)a(T1)+Λsin⁡(T0Ω),θ1=18(Ω3-Ωω02)2e-i(T0ω0+β(T1))×I(-4Ω2ω02a(T1)×((1+e2i(T0ω0+β(T1)))(Ω2+ω02)cos⁡(T0Ω)-2i(-1+e2i(T0ω0+β(T1)))Ωω0sin⁡(T0Ω))-2ei(T0ω0+β(T1))(Ω2-ω02)cos⁡(T0Ω)(-4g+(-κΩ2+Λω02)sin⁡(T0Ω))),y1=(e-i(2T0Ω+5T0ω0+3β(T1))×(-8e3iβ(T1)Ω(Ω4-10Ω2ω02+9ω04)×(4eiT0(4Ω+5ω0)gIκω02(Ω2-ω02)-4e5iT0ω0gIκω02(-Ω2+ω02)+ieiT0(3Ω+5ω0)ω02(4Ω2-ω02)(3αΛ3+8iΛμΩ+Iκ2Ω2-IκΛω02)+eiT0(Ω+5ω0)ω02(-4Ω2+ω02)(3iαΛ3+8ΛμΩ+iIκ2Ω2-iIκΛω02)+8eiT0(2Ω+5ω0)gIκ(4Ω4-5Ω4ω02+ω04))+2e2iβ(T1)ω02(4Ω4-37Ω2ω02+9ω0)×(e4iT0(Ω+ω0)(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+e2i(3T0ω0+β(T1))(Ω+ω0)2×(2IκΩω02+3αΛ2(-Ω+ω0)+e4iT0ω0(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0))+e2i(2T0Ω+3T0ω0+β(T1))(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0)))a(T1)-24ieiβ(T1)αΛΩω02(4Ω4-5Ω2ω02+ω04)×(2ei(3T0Ω+5T0ω0+2β(T1))(Ω2-9ω02)-2ei(T0(Ω+5ω0)+2β(T1))(Ω2-9ω02)-eiT0(Ω+3ω0)(Ω2-4Ωω0+3ω02)+ei(3T0Ω+7T0ω0+4β(T1))(Ω2-4Ωω0+3ω02)+e3iT0(Ω+ω0)(Ω2+4Ωω0+3ω02)-ei(T0(Ω+7ω0)+4β(T1))×(Ω2+4Ωω0+3ω02))a(T1)2+(e2iT0(Ω+ω0)+e2i(T0(Ω+4ω0)+3β(T1)))αΩ(Ω2-ω02)2×(4Ω4-37Ω2ω02+9ω04)a(T1)3))/(64(-Ω2ω0+ω03)2(4Ω5-37Ω3ω02+9Ωω04)).


## 3.1. Subharmonic ResonancesΩ≈3ω0 (Table 1)

Table 1
Vibrational conveyer parameters in SI units.
εαμcIe(m)E1E0gω0k1k2mM0.050.010.0110.90.21.51.69.81120001005200Near resonance:(3.13)D0θ0=Ω,(3.14)Ω=3ω0+εσ.
The solution of (3.13) can be written as
(3.15)θ0=ΩT0=3ω0T0+σT1.
Taking
(3.16)A[T1,T2]=a[T1]2eiβ[T1],
where a and β are real, and substituting it into (3.11), we will obtain
(3.17)y0=12e-iT0ω0-iβ(T1)a(T1)+12eiT0ω0+iβ(T1)a(T1)+Λsin⁡(T0Ω),θ1=18(Ω3-Ωω02)2e-i(T0ω0+β(T1))×I((-4g+(-κΩ2+Λω02)sin⁡(T0Ω))-4Ω2ω02a(T1)×((1+e2i(T0ω0+β(T1)))(Ω2+ω02)cos⁡(T0Ω)-2i(-1+e2i(T0ω0+β(T1)))Ωω0sin⁡(T0Ω))-2ei(T0ω0+β(T1))(Ω2-ω02)cos⁡(T0Ω)(-4g+(-κΩ2+Λω02)sin⁡(T0Ω))),y1=(e-i(3T0Ω+4T0ω0+3β(T1))×(-8e3iβ(T1)Ω(Ω3+3Ω2ω0-Ωω02-3ω03)×(ieiT0(4Ω+3ω0)ω02(3αΛ3+8iΛμΩ+Iκ2Ω2-IκΛω02)(36Ω4-13Ω2ω02+ω04)+ieiT0(2Ω+3ω0)ω02(-3αΛ3+8iΛμΩ-Iκ2Ω2+IκΛω02)(36Ω4-13Ω2ω02+ω04)+4eiT0(Ω+3ω0)gIκω02(9Ω4-10Ω2ω02+ω04)+4eiT0(5Ω+3ω0)gIκω02×(9Ω4-10Ω2ω02+ω04)+8e3iT0(Ω+ω0)gIκ(36Ω6-49Ω4ω02+14Ω2ω04-ω06)-ie3iT0(2Ω+ω0)ω02(4Ω4-5Ω2ω02+ω04)(αΛ3+Iκ(κΩ2-Λω02))-ie3iT0ω0ω02(4Ω4-5Ω2ω02+ω04)(-αΛ3+Iκ(-κΩ2+Λω02)))+2e2iβ(T1)ω02(36Ω5+108Ω4ω02-39Ω2ω03+Ωω04+3ω05)×(eiT0(5Ω+2ω0)(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+ei(T0(Ω+4ω0)+2β(T1))(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+ei(T0(Ω+2ω0)(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0))+ei(5T0Ω+4T0ω0+2β(T1))×(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0)))a(T1)+24iei(T0(2Ω+ω0)+β(T1))α×ΛΩω02(-(-1-2e2i(T0ω0+β(T1))+2e2i(T0(Ω+ω0)+β(T1))+e2i(T0(Ω+2ω0)+2β(T1)))Ω+(-1+6e2i(T0ω0+β(T1))-6e2i(T0(Ω+ω0)+β(T1))+e2i(T0(Ω+2ω0)+2β(T1)))ω0)×(36Ω6-49Ω4ω02+14Ω2ω04-ω06)a(T1)2+e3iT0Ω(1+e6i(T0ω0+β(T1)))αΩ×(Ω2-ω02)2(36Ω5+108Ω4ω0-13Ω3ω02-39Ω2ω02+Ωω04+3ω05)a(T1)3)×(cos⁡(T0ω0)+isin⁡(T0ω0)))/(64Ω(6Ω2-5Ωω0+ω02)(6Ω2-5Ωω0+ω02)×(-Ω2ω0+ω03)2×(6Ω3+23Ω2ω0+16Ωω02+3ω03)).
Secular terms will be eliminated from the particular solution of (3.8) if we choose A to be a solution of
(3.18)-2εσ′+E(θ̇).In addition to the terms proportional toe(±iω0T0) or proportional to e[±i(Ω-2ω0)T0] there is another term that produces a secular term in (3.9). We express (Ω-2ω0)T0 in terms of ω0T0 according to
(3.19)(Ω-2ω0)T0=ω0T0+εσT0=ω0T0+σT1.


## 3.2. Superharmonic ResonancesΩ≈(1/3)ω0 (Table 2)

Table 2
Vibrational conveyer parameters in SI units.
εαμcIe(m)E1E0gω0k1k2mM0.050.010.0110.90.21.51.69.81102000001005200We consider(3.20)3Ω=ω0+εσ.In addition to the terms proportional toe(±iω0T0) in (3.9), there is another term that produces a secular term in (3.9). This is -αΛ3e(±3iΩT0). T0 eliminate the secular terms, we express 3ΩT0 in terms of ω0T0 according to
(3.21)3ΩT0=(ω0+εσ)T0=ω0T0+σεT0=ω0T0+σT1.
Eliminating secular terms in equations, taking (3.16), and separating real and imaginary parts, the system of equations solved it is as follows [11]:
(3.22)y0=12e-iT0ω0-iβ(T1)a(T1)+12eiT0ω0+iβ(T1)a(T1)+Λsin⁡(T0Ω),θ1=18(Ω3-Ωω02)2e-i(T0ω0+β(T1))×I(-4Ω2ω02a(T1)×((1+e2i(T0ω0+β(T1)))(Ω2+ω02)cos⁡(T0Ω)-2i(-1+e2i(T0ω0+β(T1)))Ωω0sin⁡(T0Ω))-2ei(T0ω0+β(T1))(Ω2-ω02)cos⁡(T0Ω)(-4g+(-κΩ2+Λω02)sin⁡(T0Ω))),y1=(e-i(2T0Ω+5T0ω0+3β(T1))×(-8e3iβ(T1)Ω(Ω4-10Ω2ω02+9ω04)×(4eiT0(4Ω+5ω0)gIκω02(Ω2-ω02)-4e5iT0ω0gIκω02(-Ω2+ω02)+ieiT0(3Ω+5ω0)ω02(4Ω2-ω02)(3αΛ3+8iΛμΩ+Iκ2Ω2-IκΛω02)+eiT0(Ω+5ω0)ω02(-4Ω2+ω02)(3iαΛ3+8ΛμΩ+iIκ2Ω2-iIκΛω02)+8eiT0(2Ω+5ω0)gIκ(4Ω4-5Ω4ω02+ω04))+2e2iβ(T1)ω02(4Ω4-37Ω2ω02+9ω0)×(e4iT0(Ω+ω0)(Ω+ω0)2(2IκΩω02+3αΛ2(-Ω+ω0))+e2i(3T0ω0+β(T1))(Ω+ω0)2×(2IκΩω02+3αΛ2(-Ω+ω0)+e4iT0ω0(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0))+e2i(2T0Ω+3T0ω0+β(T1))(Ω-ω0)2(2IκΩω02-3αΛ2(Ω+ω0)))a(T1)-24ieiβ(T1)αΛΩω02(4Ω4-5Ω2ω02+ω04)×(2ei(3T0Ω+5T0ω0+2β(T1))(Ω2-9ω02)-2ei(T0(Ω+5ω0)+2β(T1))(Ω2-9ω02)-eiT0(Ω+3ω0)(Ω2-4Ωω0+3ω02)+ei(3T0Ω+7T0ω0+4β(T1))(Ω2-4Ωω0+3ω02)+e3iT0(Ω+ω0)(Ω2+4Ωω0+3ω02)-ei(T0(Ω+7ω0)+4β(T1))×(Ω2+4Ωω0+3ω02))a(T1)2+(e2iT0(Ω+ω0)+e2i(T0(Ω+4ω0)+3β(T1)))αΩ(Ω2-ω02)2×(4Ω4-37Ω2ω02+9ω04)a(T1)3))/(64(-Ω2ω0+ω03)2(4Ω5-37Ω3ω02+9Ωω04)).


## 4. Stability Analysis

We analyzed in this system the stability(a,γ) in the equilibrium point, using (4.1), where F is the Jacobian matrix of (4.2), and γsub=3β[T1]-T1σ[T1], γsup=[β[T1]-T1σ[T1]]. Stability of the approximate solutions depends on the value of the eigenvalues of the Jacobian matrix F [12]. The solutions are unstable if the real part of the eigenvalues is positives Figures 2(a1) and 2(a2). Figure 2 shows the frequency-response curves for the subharmonic and superharmonic resonance of the unbalanced vibratory conveyor:
(4.1)asub′=-μa-38ω0αΛa2cos⁡[γ],γsub′=-3(2IκΩ2ω02+3αΛ2(-Ω2+ω02))4(Ω2-ω02)ω0+98ω0αa2-98ω0αΛasin⁡[γ]-σ,asub′=-μa-38ω0(αΛ3+Iκ(κΩ2-Λω02))cos⁡[γ],γsub′=-(2IκΩ2ω02+3αΛ2(-Ω2+ω02))4(Ω2-ω02)ω0+3αa28ω0+(αΛ3+Iκ(κΩ2-Λω02))sin⁡[γ]8ω0a-σ,(4.2)F={∂af1,∂γf1},{∂af2,∂γf2},f1=a′,f2=γ′.Figure 2
Subharmonic resonance and superharmonic resonance. (a) Frequency-response curves with stability,— stable, ⋯ unstable {(a1):a=2.78–10, σ=0.43258–2.62092 and (a2): a=2.73–7.21, σ=0.6782–0.986448}, (b) effect of detuning parameter, and (c) effect of damping parameter.


## 5. Numerical Results

The numerical calculations of the vibrating system are performed with the help of the software Mathematica [13, 14]. We analyze the subharmonic resonance Ω≈3ω0 and superharmonic resonance 3Ω≈ω0. Figure 3 shows the power spectrum, phase portrait and the Poincaré map for superharmonic resonance and Figure 3 shows the power spectrum for superharmonic resonance.Subharmonic resonance: (a) power spectrum and superharmonic resonance, (b) power spectrum, (c) phase portrait, and (d) Poincaré sections.
(a)(b)(c)(d)


## 6. Conclusions

The vibrating system is analyzed, analytically, and numerically for superharmonic and subharmonic resonance by the method of multiple scales. Very often in the motion of the system near resonance the jump phenomenon occurs. The frequency-response curves of the subharmonic resonance consist of two branches; the left one is stable and the right one is unstable (saddle node bifurcation). The frequency-response curves of the superharmonic resonance consist of three branches; the left one is stable, the middle one is unstable, and the right one is stable (pitchfork bifurcation). The stable motions of the oscillator are shown with one peaks in the power spectrum for superharmonic resonance and with two peaks in the power spectrum for subharmonic resonance. Both analytical and numerical results that we have obtained are in good agreement. The system studied here exhibits chaotic behaviour in case of strong nonlinearity. This will be reported in the forthcoming paper. Furthermore, control methods in the passage through resonance await for future publication.



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*Source: 717543-2012-03-25.xml*