Nonlinear response of vibrational conveyers with non-ideal vibration exciter: primary resonance

Nonlinear Dyn (2012) 69:1611–1619DOI 10.1007/s11071-012-0372-8

O R I G I NA L PA P E R

Nonlinear response of vibrational conveyers with non-idealvibration exciter: primary resonance

G.F. Alısverisçi · H. Bayıroglu · G. Ünal

Received: 17 October 2011 / Accepted: 15 February 2012 / Published online: 10 March 2012© Springer Science+Business Media B.V. 2012

Abstract Vibrational conveyers with a centrifugal vi-bration exciter transmit their load based on the jump-ing method. The trough is oscillated by a commonunbalanced-mass driver. This vibration causes the loadto move forward and upward. The motion is substan-tially related to the vibrational parameters. The tran-sition of the vibratory system for over resonance ex-cited by rotating unbalances is important in terms ofthe maximum vibrational amplitude and the power re-quirement from the drive for the cross-over. The me-chanical system depends on the motion of the DC mo-tor. In this study, the working ranges of oscillatingshaking conveyers with a non-ideal vibration exciterhave been analyzed analytically for primary resonanceby the method of multiple scales with reconstitution,and numerically. The analytical results obtained in thisstudy have been compared with the numerical results,and have been found to be well matched.

Keywords Centrifugal vibration exciter · Vibrationalconveyers · Non-ideal vibrations · Method of multiplescales with reconstitution · Primary resonance

G.F. Alısverisçi (�) · H. BayırogluYıldız Technical University, Istanbul, Turkeye-mail: [email protected]

H. Bayıroglue-mail: [email protected]

G. ÜnalYeditepe University, Istanbul, Turkeye-mail: [email protected]

1 Introduction

The load-carrying element of a horizontal shakingconveyer performs, as a rule, linear (or sometimescircular or elliptical) symmetrical harmonic oscilla-tions with harmonic exciting force. In vertical shakingconveyers, the load-carrying element performs dou-ble harmonic oscillations: linear along the verticalaxis and rotational around that axis (i.e. longitudi-nal and torsional oscillations). Conveyer drives withcentrifugal vibration exciters may have (1) a singleunbalanced mass, (2) two equal unbalancing masses,(3) a pendulum-type unbalanced mass, (4) four unbal-anced masses in two shafts, (5) four rotating unbal-anced masses for three principal modes of oscillation,i.e. linear, elliptical, and circular. To induce strictly ori-ented linear oscillations of the load-carrying element,the conveyer drive should be arranged so that the lineof excitation force passes through the inertial centreof the entire oscillating system [16]. Non-ideal drivesfind applications in suspended and supported vibra-tional conveyers and feeders [2].

By the characteristics and adjustment of the elas-tic support elements (oscillating system), it is distin-guished between shaking conveyers with a resonant,sub-resonant, and super-resonant system.

A practical difficulty with unbalanced mass ex-citers, observed as early as 1904 by Sommerfeld,is that local instabilities may occur in an operatingspeed of such devices. Rocard [17], Mazert [18], andPanovko and Gubanova [19] have studied the prob-

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1612 G.F. Alısverisçi et al.

lem of the stability of the unbalanced mass exciter.The first detailed study on the non-ideal vibrating sys-tems is presented by Konokenko. He obtained sat-isfactory results by the comparison of the experi-mental analysis and approximated method [9]. Af-ter this publication, the non-ideal problem was pre-sented by Evan-Ivanowski [20] and Nayfeh and Mook[14]. These authors showed that sometimes dynamiccoupling between energy sources and structural re-sponse should not be ignored in real engineering prob-lems. Ganapathy and Parameswaran’s theoretical stud-ies and computations have indicated the beneficial ef-fect of the “material load” during the starting and tran-sition phases of an unbalanced mass-driven vibrat-ing conveyor (1986). Bolla et al. analyzed throughthe multiple scales method the response of a simpli-fied non-ideal and non-linear vibrating system (2007).Götzendorfer presented a macro-mechanical modelfor the transport of granular matter on linear, hori-zontal conveyors subject to linear, circular, or ellip-tic oscillations and compared it to experimental re-sults [7]. An overview of the main properties of non-ideal vibrating systems was presented by Balthazaret al. [3].

Non-ideal vibrating systems have two importantproperties: the jump phenomena and the increase inpower required by the energy source operating near theresonance. Often the operating frequency lies aboveresonance, and hence passage through the resonancespeed becomes necessary, while starting and stopping.In many cases, linear analysis is insufficient to de-scribe the behavior of the physical system accurately.One of the main reasons for modeling a physical sys-tem as a non-linear one is that totally unexpectedphenomena sometimes occur in non-linear systems–phenomena that are not predicted or even hinted at bylinear theory.

In this work, the vibrating model of the system ex-cited by the rotation of two equal unbalanced massesas shown in Fig. 1 is analyzed [5–7, 16]. We consider amathematical model with spring of cubic non-linearityand a DC motor of quadratic non-linearity [8, 9].

2 The governing equations of motion

The equations of motion may be obtained by using La-grange’s equation

d

dt

(∂T

∂qi

)+ ∂D

∂qi

− ∂T

∂qi

+ ∂V

∂qi

= 0. (1)

Fig. 1 Vibrating model of the system

Here, T is the kinetic energy, V is the potential energy,and D is the Rayleigh dissipation function. They aregiven by

T = 1

2My2 + 1

2m

[(y + θ e cos θ)2 + (θe sin θ)2]

+ 1

2Imotθ

2, (2)

V = 1

2k1y

2 + 1

4k2y

4 + mge(1 + sin θ), (3)

D = 1

2cy2 + 1

2K(ωs − θ ), (4)

where the constants k1 and k2 are the linear and non-linear elastic co-efficients, respectively, c is the damp-ing co-efficient, g is acceleration due to gravity, Isys =(Im + IM + Imot) is the total moment of inertia of theall the rotating parts in the system, m is the unbalancedmass, M is the mass of the trough and the conveyedmaterial on the trough of the conveyor, e is the eccen-tricity of the unbalanced mass, K is the instantaneousdrive torque available at the shafts carrying the unbal-anced masses, ωs is synchronous angular speed of theinduction motor, θ is the angle of the rotation of theshafts carrying unbalanced mass, and qi is the gener-alized coordinate. Lagrange’s equation of motion forthe coordinates q1 = y and q2 = θ can be written as

Isysθ + me cos θ(y + g) = L(θ),

L(θ) = 1

2

[∂K

∂θ(θ − ωs) + K

],

(5)

(m + M)y + cy + k1y + k2y3

= me(θ2 sin θ − θ cos θ

). (6)

Nonlinear response of vibrational conveyers with non-ideal vibration exciter: primary resonance 1613

In terms of non-dimensional variables (5) and (6) readas

θ ′′ = ε[E

(θ ′) − I cos θ

(ρ′′ + G

)], (7)

ρ′′ + ρ = ε[−2μρ′ − αρ3 + θ ′2 sin θ − θ ′′ cos θ

], (8)

where y = eρ, τ = ωnt (the non-dimensional time),ω2

n = k1m+M

, ε = mm+M

,

I = (m + M)e2

Isys, G = g

eω2n

, 2μ = c

mωn

,

α = k2e2

mω2n

, E(θ ′) = m + M

mIsysω2n

L(ωnθ

′).Note that in (7) and (8), ε is a small parameter of thisproblem. If we take ε = 0 , the equation describes aclassical harmonic vibration with frequency 1 and arotation with constant angular velocity dθ

dt. Note that

E contains L(ωnθ′) that is the active torque generated

by the electric circuit of the DC motor, as it is seen inFig. 1.

3 Approximate analytical solution

When there is no coupling between motion of the rotorand vibrating system it becomes an ideal system andθ = ωτ . In this case, (8) takes the form of

ρ′′ + ω2nρ = ε

[−2μρ′ − αρ3 + ω2 sinωτ]. (9)

Equations of motion for the non-ideal system read

θ ′′ = ε[E


(ρ′′ + G

)],

ρ′′ + ω2nρ = ε

[−2μρ′ − αρ3 + θ ′2 sin θ − θ ′′ cos θ],

(10)

where E(θ ′) = Mm(θ ′) − H(θ ′) is the difference be-tween the torque generated by the motor and the resis-tance torque. The function E(θ ′) = u1 − u2θ

′ is ap-proximated by a straight line. Here, u1 is a controlparameter and it can be varied according to the volt-age u2. The latter is a constant parameter and it is acharacteristic for the model of the motor.

We will seek the approximate analytical solu-tions to (7) and (8) by using the method of multiplescales [12, 14].

θ(τ, ε) ≈ θ0(T0, T1, T2) + εθ1(T0, T1, T2)

+ ε2θ2(T0, T1, T2),

ρ(τ, ε) ≈ ρ0(T0, T1, T2)

+ ερ1(T0, T1, T2) + ε2ρ2(T0, T1, T2),

(11)

Tn = εnτ, n = 0,1,2, . . . , T0 = τ,

T1 = ετ, T2 = ε2τ, Dn = ∂∂τn

,(12)

d

dτ= dT0

dτ

∂

∂T0+ dT1

dτ

∂

∂T1+ dT2

dτ

∂

∂T2· · ·

= D0 + εD1 + ε2D2 + · · · , (13)

d2

dτ 2= D2

0 + 2εD0D1

+ ε2(D21 + 2D0D2

) + · · · . (14)

Substitution of (11)–(14) into (7) and (8) leads to

D20θ0 + ε

(2D0D1θ0 + D2

0θ1) + ε2(D2

0θ2 + 2D0D1θ1

+ D21θ0 + 2D0D2θ0

)= ε

[E


(D2

0ρ0

+ ε(2D0D1ρ0 + D2

0ρ1) + G

)], (15)

D20ρ0 + ε

(2D0D1ρ0 + D2

0ρ1)

+ ε2(D20ρ2 + 2D0D1ρ1 + D2

1ρ0

+ 2D0D2ρ0) + (

ρ0 + ερ1 + ε2ρ2)

= ε{−2μ

[D0ρ0 + ε(D1ρ0 + D0ρ1)

]− α

(ρ3

0 + 3ερ20ρ1

)+ [

(D0θ0)2 + 2D0θ0(D0θ1 + D1θ0)

]ε sin θ

− [D2

0θ0 + ε(2D0D1θ0 + D20θ1)

]cos θ

}. (16)

We now equate the co-efficients of like powers of ε toobtain:

D20θ0 = 0, (17)

D20θ1 = −2D0D1θ0 + E

(θ ′)

− I cos θ(D2

0ρ0 + G), (18)

D20θ2 = −2D0D1θ1 − D2

1θ0 − 2D0D2θ0

− I cos θ(2D0D1ρ0 + D2

0ρ1), (19)

D20ρ0 + ρ0 = 0, (20)

D20ρ1 + ρ1 = −2D0D1ρ0 − 2μD0ρ0 − αρ3

0

+ (D0θ0)2 sin θ − (

D20θ0

)cos θ, (21)

D20ρ2 + ρ2 = −2D0D1ρ1 − D2

1ρ0 − 2D0D2ρ0

− 2μ[(D1ρ0 + D0ρ1)

] − α(3ρ2

0ρ1)

+ [2D0θ0(D0θ1 + D1θ0)

]sin θ

− [(2D0D1θ0 + D2

0θ1)]

cos θ, (22)


θ1 = O(1), τ → ∞, (23)

cos(θ0 + εθ1) = cos(θ0) + O(ε)

= cos(T0 + σT1) + O(ε), (24)

ω = θ ′ ∼= D0θ0 = 1 + εσ (T1, T2) → θ0

= T0 + εσ (T1, T2)T0

= T0 + σ(T1, T2)T1. (25)

We choose the detuning parameter asσ to study thecase of primary resonance. Approximate solution of(20) has the form

ρ0 = A(T1, T2)eiT0 + A(T1, T2)e

−iT0 . (26)

Substitution of (25) and (26) into (21) yields,

D20ρ1 + ρ1

= −[2iD1A(T1, T2) + 2iμA(T1, T2)

]eiT0

+ [2iD1A(T1, T2) + 2iμA(T1, T2)

]e−iT0

− α[A3(T1, T2)e

3iT0 + A3(T1, T2)e−3iT0

+ 3A2(T1, T2)A(T1, T2)eiT0

+ 3A2(T1, T2)A(T1, T2)e−iT0

]− i

2

[ei(T0+σT1) − e−i(T0+σT1)

], (27)

D20ρ1 + ρ1 = −α

[A3(T1, T2)e

3iT0

+ A3(T1, T2)e−3iT0

]. (28)

We now let

A(T1, T2) = a(T1, T2)

2eiβ(T1,T2), (29)

where a and β are real numbers. Equation (26) takesthe form

ρ0 = a[T1, T2] cos[T0 + β[T1, T2]

]. (30)

Solution to (28) reads

ρ1 = 1

64e−3i(T0+β[T1,T2])(1 + e6i(T0+β[T1,T2]))

× αa[T1, T2]3. (31)

The secular terms of (27) can be eliminated by setting

−[2iD1A(T1, T2) + 2iμA(T1, T2)

]eiT0

+ [2iD1A(T1, T2) + 2iμA(T1, T2)

]e−iT0

− α[3A2(T1, T2)A(T1, T2)e

iT0

+ 3A2(T1, T2)A(T1, T2)e−iT0

]− i

2

[ei(T0+σT1) − e−i(T0+σT1)

] = 0, (32)

−[2D1A(T1, T2) + 2μA(T1, T2)

]eiT0

− 3αA2(T1, T2)A(T1, T2)eiT0 − i

2ei(T0+σT1)

= 0,(33)[

2iD1A(T1, T2) + 2iμA(T1, T2)]e−iT0

+ 3αA2(T1, T2)A(T1, T2)e−iT0 + i

2e−i(T0+σT1)

= 0.

The solvability conditions can be obtained as

−2iD1A(T1, T2) − 2iμA(T1, T2)

− 3αA2(T1, T2)A(T1, T2) − i

2

(ei(σT1)

) = 0,

2iD1A(T1, T2) + 2iμA(T1, T2)

+ 3αA2(T1, T2)A(T1, T2) + i

2e−i(σT1) = 0.

(34)

Substituting of (29) into (34) yields{−[

iD1a(T1, T2) + a(T1, T2)D1β(T1, T2)

+ ia(T1, T2)μ] − 3

8αa3(T1, T2)

}eiβ(T1,T2)

− i

2eiσT1 = 0. (35)

In terms of trigonometric functions, it can berewritten as

−iD1a(T1, T2) + a(T1, T2)D1β(T1, T2) + ia(T1, T2)μ

− 3

8αa3(T1, T2) + i

2

[cos(β − σT1)

+ i sin(β − σT1)] = 0. (36)

Upon separating real and imaginary parts in (36),we obtain

−D1a(T1, T2) + a(T1, T2)μ

+ 1

2cos(β − σT1) = 0, (37)

a(T1, T2)D1β(T1, T2) − 3

8a3(T1, T2)

+1

2sin(β − σT1) = 0. (38)

Then

D1a(T1, T2) = +[a(T1, T2)μ + 1

2cos(β − σT1)

],

(39)

−D1β(T1, T2) = −3

8a2(T1, T2) + 1

2asin(β − σT1).

(40)


Upon invoking (25), the general solution to (17) canbe obtained as

D1θ0 = σ. (41)

Substitution of (25), (26), (29), and (33) into (18)yields

D20θ1 = −2εσ ′ + E


[−a

2ei(T0+β) + G

].

(42)

Then the trigonometric form becomes

D20θ1 = −2εσ ′ + E

(θ ′) + I

a

4

[cos(2T0 + σT1 + β)

+ i sin(2T0 + σT1 + β) + cos(β − σT1)

+ i sin(β − σT1)] − IG cos(T0 + σT1). (43)

Elimination of the terms that lead to secular termsfrom (43) can be achieved by letting

−2εσ ′ + E(θ ′) + I

a

4

[cos(β − σT1)

+ i sin(β − σT1)] = 0, (44)

D20θ1 = I

a

4

[cos(2T0 + σT1 + β)

+ i sin(2T0 + σT1 + β)]

− IG cos(T0 + σT1), (45)

−2i(D1A + μA) − 3αA2A − i

2eiσT1 = 0, (46)

θ1 = GI cos[T0 + T1σ ] − 1

8Ia[T1, T2]

× cos[2T0 + T1σ + β[T1, T2]

]. (47)

Substitution of (30), (31), and (47) into (22) leadsto

D20ρ2 + ρ2

= −2

{−3i

64αD1A[T1, T2]3e−3i(T0+β[T1,T2])

+ 9i2

64α(D1β[T1, T2]

)A[T1, T2]3e−3i(T0+β[T1,T2])

+ αD1A[T1, T2]3 3i

64e3i(T0+β[T1,T2])

+ α(D1β[T1, T2]

)A[T1, T2]3 9i2

64e3i(T0+β[T1,T2])

}

− {A′′(T1, T2)e

iT0 + A′′(T1, T2)e−iT0

}

− 2

{iD2a(T1, T2)

2eiβ(T1,T2)eiT0

+ i2(D2β(T1, T2))a(T1, T2)

2eiβ(T1,T2)eiT0

− iD2a(T1, T2)

2eiβ(T1,T2)e−iT0

− i2(D2β(T1, T2))a(T1, T2)

2eiβ(T1,T2)e−iT0

}

− 2μ

[({A′(T1, T2)e

iT0 + A′(T1, T2)e−iT0

}

+{−3i

64αA[T1, T2]3e−3i(T0+β[T1,T2])

+ αA[T1, T2]3 3i

64e3i(T0+β[T1,T2])

})]

− α(3ρ2

0ρ1) +

[2{1 + εσ }

({−GI sin[T0 + T1σ ]

+ 2

8IA[T1, T2] sin

[2T0 + T1σ + β[T1, T2]

]}

+ {σ })]

sin θ −[(

2{εσ ′}

+{−GI cos[T0 + T1σ ] + 4

8IA[T1, T2]

× cos[2T0 + T1σ + β[T1, T2]

]})]cos θ. (48)

Elimination of the terms that lead to secular termsfrom (48) can be achieved by letting

−{A′′(T1, T2)e

iT0 + A′′(T1, T2)e−iT0

}

− 2

{iD2a(T1, T2)

2eiβ(T1,T2)eiT0

+ i2(D2β(T1, T2))a(T1, T2)

2eiβ(T1,T2)eiT0

− iD2a(T1, T2)

2eiβ(T1,T2)e−iT0

− i2(D2β(T1, T2))a(T1, T2)

2eiβ(T1,T2)e−iT0

}

− 2μ

[({A′(T1, T2)e

iT0 + A′(T1, T2)e−iT0

}

+{−3i

64αA[T1, T2]3e−3i(T0+β[T1,T2])

+ αA[T1, T2]3 3i

64e3i(T0+β[T1,T2])

})]

− α(3ρ2

0ρ1) +

[2{1 + εσ }

({−GI sin[T0 + T1σ ]

+ 2

8IA[T1, T2] sin

[2T0 + T1σ

+ β[T1, T2]]} + {σ }

)]sin θ −

[(2{εσ ′}


+{−GI cos[T0 + T1σ ]

+ 4

8IA[T1, T2] cos

[2T0 + T1σ

+ β[T1, T2]]})]

cos θ = 0, (49)

ρ2 = 1

6144e−i(5T0+2T1σ+5β[T1,T2])

× 512e3iT0+5iβ[T1,T2](1 + e4i(T0+T1σ))GI

+ 3e2iT1σ(1 + 6e2i(T0+β[T1,T2])

+ 6eθi(T0+β[T1,T2])

+ e10i(T0+β[T1,T2]))α2a[T1, T2]5

+ 216ie2i(T0+T1σ+β[T1,T2])

× (−1 + e6i(T0+β[T1,T2]αa[T1,T2]2)D1a[T1, T2]+ 72ie2i(T0+T1σ+β[T1,T2])αa[T1, T2]3

× ((−1 + e6i(T0+β[T1,T2]))μ× +3i

(1 + e6i(T0+β[T1,T2]))D1β[T1, T2]

)). (50)

We now substitute (25), (26), (30), and (47) into (19).Then we eliminate the secular terms in the resultingequation. Finally, we solve for θ2 to obtain

θ2 = − 1

2048e−i(4T0+T1σ+3β[T1,T2])

× I(9(1 + 4e2i(T0+T1σ) + 4e6i(T0+β[T1,T2])

+ e2i(4T0+T1σ+3β[T1,T2]))αa[T1, T2]3

+ 128e2i(T0+β[T1,T2])(16ei(T0+β[T1,T2])

× (1 + e2i(T0+T1σ)

)Gσ

− i(−1 + e2i(2T0+T1σ+β[T1,T2]))D1a[T1, T2]

)− 128e2i(T0+β[T1,T2])

× (1 + e2i(2T0+T1σ+β[T1,T2]))a[T1, T2]

× (σ − D1β[T1, T2]

)). (51)

4 Stability analysis

Upon using the method of reconstitution [11] and sec-ular terms of (27) and (48), at first order, we obtain asolvability condition of the form

D1A = g1(A, A, T1), (52a)

D1A = 1

4

(−eiT1σ − 4μA(T1, T2)

+ 6iαA(T1, T2)2A(T1, T2)

), (52b)

where A is a complex-valued function representing theamplitude and phase of the response, Dn = ∂/∂Tn, andTn = εnt . Then, at second order, we obtain a solvabil-ity condition of the form

2iωD2A = −D21A − 2μD1A

+ g2(A, A, T1, T2), (53a)

D2A = 1

16

(−8eiT1σ σ + i(3α2A(T1, T2)

3A2(T1, T2))

+ 16μD1A(T1, T2) + 8D21A(T1, T2)

). (53b)

Instead of seeking solutions to (52a), (52b) and (53a),(53b) as partial differential equations, we combinethem into a unique single ordinary differential equa-tion in what we refer to as the method of reconstitution[11, 13]. To this end, we note that

dA

dt(t; ε) = εD1A + ε2D2A + · · · . (54)

Then we differentiate (54) once with respect to T1

to obtain

D21A = ∂g1

∂Ag1 + ∂g1

∂Ag1 + ∂g1

∂T1, (55a)

D21A = 1

4

(−ieiT1σ σ − 4μD1A(T1, T2)

+ 12iαA(T1, T2)A(T1, T2)D1A(T1, T2)

+ 6iαA2(T1, T2)D1A(T1, T2)). (55b)

Finally, we substitute (52a), (52b), (53a), (53b), and(55a), (55b), into (54) to obtain the complex-valuedmodulation equation

dA

dt(t; ε) = εg1 − iε2

2ω

(−2μg1 − ∂g1

∂Ag1

− ∂g1

∂Ag1 − ∂g1

∂T1+ g2

)+ O

(ε3), (56a)

dA

dt(t; ε) = 1

4ε(−eiT1σ − 4μA(t) + 6iαA2(t)A(t)

)

+ 1

16ε2(−8eiT1σ σ + i

(−2ieiT1σ σ

+ 3α2A3(t)A2(t) + 4μ(−eiT1σ

− 4μA(t) + 6iαA2(t)A(t))))

. (56b)

The real-valued modulation equations can be obtainedby substituting

A(t) = 1

2a(t)eiβ(t), (57)


into (56b) and by separating the real and imaginaryparts. They read as

a = εA1(a, γ ) + ε2A2(a, γ ) + O(ε3), (58a)

a = −1

2εμa(t) − 3

16αε3μa3(t)

− 1

8ε(2 + 3εσ ) cos

[εtσ − β(t)

]

+ 1

4ε2μ sin

[εtσ − β(t)

] − a′(t)2

, (58b)

aγ = εθ1(a, γ ) + ε2θ2(a, γ ) + O(ε3), (59a)

aγ = −1

2ε2μ2a(t) + 3

16αεa3(t) + 3

512α2ε2a(t)5

− 1

4ε2μ cos

[εtσ − β(t)

]

− 1

8ε(2 + 3εσ ) sin

[εtσ − β(t)

]

− 1

2a(t)β ′(t). (59b)

Equations (58a), (58b), (59a), (59b) can be trans-formed into an autonomous system by setting

γ = εtσ − β, (60)

a′ = −εμa − 3

8αε2μa3 − 1

4ε(2 + 3εσ ) cos[γ ]

+ 1

2ε2μ sin[γ ],

γ ′ = ε2μ2 − 3

8αεa2 − 3

256α2ε2a4

+ 1

2aε2μ cos[γ ] + 1

4aε(2 + 3εσ ) sin[γ ] + εσ,

f1 = a′, f2 = γ ′, F =[

∂af1 ∂γ f1

∂af2 ∂γ f2

].

(61)

We now perform the stability analysis of the mod-ulation equations in the neighborhood of the equi-librium points, using (61), where F, Jacobian matrixof (61). Stability of the approximate solutions dependson the value of the eigenvalues of the Jacobian ma-trix F [2, 4]. The solutions are unstable if the real partof the eigenvalues is positives [1]. Figure 2 shows thefrequency-response curves for primary resonance ofthe unbalanced vibratory conveyor. These curves showthat the non-linearity bends the frequency-responsecurves. The bending of the frequency-response curvesleads to multi-valued amplitudes, and hence to jumpphenomenon. In contrast to linear systems, for in-stance the mass non-linear spring system exhibits noresonance (Fig. 2).

Fig. 2 Frequency-responsecurve, (a) effect of thenon-linearity, (b) withstability, – stable, · · ·unstable(σ = 0.2272269–0.774372,a = 3.3–10.1), (c) effect ofdamping parameters,(d) effect of thenon-linearity co-efficients


Table 1 Vibrational Conveyer parameters in SI units

ε α μ c I e(m) E1 EO g ωn k1 k2 m M

0.05 0.02 0.05 1 0.9 0.2 1.5 1.6 9.81 1 20 000 100 1 200

Fig. 3(a) Displacement-timeresponse, (b) powerspectrum, (c) Poincaré map(before stabilization),(d) Poincaré map (afterstabilization)

5 Numerical results

The numerical calculations of the vibrating system areperformed with the help of the software Mathemat-ica [10, 13]. Figure 3 shows the power spectrum [15],phase portrait and Poincaré map for the primary reso-nance. In Fig. 3, the curves obtained numerically andanalytically by solving the governing equations of themotion are plotted. Comparing the solutions it can beconcluded that the difference between the numericalsolution and approximate analytical solution is neg-ligible. The main characteristics values used in thisstudy are given in the (Table 1).

6 Conclusions

A non-ideal vibrating system has been analyzed, nu-merically and analytically for primary resonance usingthe method of multiple scales with reconstitution. The

effect of damping, detuning, cubic term, and magni-tude of random excitation are analyzed. The transitionof the vibratory system for over resonance excited byrotating unbalances is important in terms of the maxi-mum vibrational amplitude and the power requirementfrom the drive for the cross-over. The maximum am-plitude of vibration is important in determining thestructural safety of the vibrating members. The dottedcurve region in the amplitude-frequency plot is unsta-ble (see Fig. 1); the extend of unstableness depends ona number of factors such as the magnitude of damp-ing, non-linearity of spring, and the rate of chanceof the exciting frequency (Fig. 2). The results of thenumerical simulations, obtained from the analyticalequations, showed that the important dynamic charac-teristics of the system such as damping, non-linearityand the amplitude excitations effects, and presenteda periodic behavior for these situations. The bend-ing of response is due to the nonlinearity is responsi-ble for a jump phenomenon. The jump phenomenon


occurs in the motion of the system near resonance.A periodic solution in the case of the angular velocityfor above the resonance (after stabilization) is illus-trated in (Fig. 3). Numerical results (Fig. 3 xy) havebeen compared with the analytical ones (Fig. 3 wz)and good correspondence has been observed betweenthem.

Acknowledgements We gratefully acknowledge the effortsof anonymous referees.

References

1. Awrejcewicz, J., Lamarque, C.H.: Bifurcation and Chaos inNonsmooth Mechanical Systems. World Scientific, RiverEdge (2003) (electronic resource)

2. Balthazar, J.M., Mook, D.T., Weber, H.I., Brasıl,R.M.L.R.F., Fenili, A., Belato, D., Felix, J.L.P.: Anoverview on non-ideal vibrations. Mechanica 38, 613–621(2003)

3. Balthazar, J.M., Brasıl, R.M.L.R.F., Weber, H.I., Fenili, A.,Belato, D., Felix, J.L.P., Garzelli, F.J.: A Review of NewVibration Issues due to Non-Ideal Energy Sources. CRCPress, Boca Raton (2004)

4. Bolla, M.R., Balthazar, J.M., Felix, J.L.P., Mook, D.T.: Onan approximate analytical solution to a nonlinear vibratingproblem, excited by a nonideal motor. Nonlinear Dyn. 50,841–847 (2007)

5. Ganapathy, S., Parameswaran, M.A.: Transition over res-onance and power requirements of an unbalanced massdriven vibratory system. Mech. Mach. Theory 21, 73–85(1986)

6. Ganapathy, S., Parameswaran, M.A.: Effect of materialloading on the starting and transition over resonance of avibratory conveyor. Mech. Mach. Theory 22(2), 169–176(1987)

7. Götzendorfer, A.: Vibrated granular matter: transport, flu-idization, and patterns. Ph.D. University Bayreuth (2007)

8. Hallanger, L.W.: The dynamic stability of an unbalancedmass exciter. Thesis California Institute of TechnologyPasadena, California (1967)

9. Kononenko, V.O.: Vibrating Problems with a LimitedPower Supply. Iliffe, London (1969)

10. Lynch, S.: Dynamical Systems with Applications UsingMathematica. Springer, Boston, Basel, Berlin (2007)

11. Nayfeh, A.H.: Resolving controversies in the application ofthe method of multiple scales and the generalized methodof averaging. Nonlinear Dyn. 40, 61–102 (2005)

12. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dy-namics: Analytical, Computation and Experimental Meth-ods. Wiley, New York (2004)

13. Nayfeh, A.H., Chin, C.M.: Perturbation Methods withMathematica. Dynamics Press, Blacksburg (1999)

14. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley,New York (1979)

15. Schmidt, G., Tondl, A.: Non-Linear Vibrations. CambridgeUniversity Press, Cambridge (1986)

16. Spivakovasky, A.O., Dyachkov, V.K.: Conveying Ma-chines, vols. (I, II). Mir, Moscow (1985)

17. Rocard, Y.: General Dynamics of Vibrations. Ungar, NewYork (1960), trans. 3rd French edn

18. Mazert, R.: Mécanique Vibratoire. C. Béranger, Paris(1955)

19. Panovko, Y.G., Gubanova, I.I.: Stability and Oscillations ofElastic Systems. Consultans Bureau, New York (1965)

20. Evan-Ivanowski, R.M.: Resonance Oscillations in Mechan-ical Systems. Elsevier, Amsterdam (1976)

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Nonlinear response of vibrational conveyers with non-ideal vibration exciter: primary resonance