Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system

International Journal of Non–Linear Mechanics 87 (2016) 109–124

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Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system

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Nasser Abdul-Fadeel Abdul-Hameed Saeed , M. Kamel Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt

A R T I C L E I N F O

A BS T RAC T

Keywords: Primary resonance Bifurcation Multi-jump phenomenon Localized vibrations Non-localized vibrations Nonlinear PD-controller

This paper investigates the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The system frequency (angular speed) -response curve is plotted at different values of the rotor eccentricity. The analysis illustrated that the system has a high oscillation amplitude and exhibits some nonlinear behaviors before control. A Proportional-Derivative (PD)-controller is integrated into the system via two pairs of electromagnetic magnetic poles. The nonlinearity due to the electromagnetic coupling is considered in the system model. A second-order approximate solution is obtained by utilizing multiple scales perturbation method. The bifurcation analyses of the controlled system are conducted. The results showed the high efficiency of the controller to mitigate the nonlinear vibrations of the considered system. Numerical simulations are carried out to validate the accuracy of the analytical results. The numerical results confirmed the excellent agreement with the analytical solutions. Then, the optimal working conditions of the system are concluded. Finally, a comparative study with previously published work is reported.

1. Introduction Rotating machinery has an important role in modern industry due to their numerous applications such as automobile engines, turbomachinery, large-scale manufacturing, compressors, aerospace, generators, and home appliances, etc. The existence of vibrations in rotating machinery is an unavoidable phenomenon, which ultimately can lead to machine failures and dangerous accidents. Such undesired motions are often caused by the dynamic interaction between the stator and rotating parts and the mass unbalance that occurs if a mismatch between the principal axis of the moment of inertia of the rotating shaft and its axis of rotation exists. Generally, nonlinear vibration is inherent phenomenon and has quite negative results in terms of reliability, durability and safe operation of machines. Therefore, vibration analysis and control have received major attention in the last few years. Great efforts have been made to mitigate vibrations in rotating machines. Passive control elements such as squeeze film damper bearings and active control such as magnetic bearings have been integrated into such systems to suppress and control their vibrations. Active magnetic bearings (AMB) system generates magnetic forces through magnetic fields to act against the undesired oscillations in the rotating parts. The main advantage of AMB system is no-contact between the bearing and the rotating parts and thus prevents the mechanical wear and the need for lubrication during the machine operation. Generating electromag-

⁎

netic force of a controllable magnitude and direction via controlling the dynamics of the AMB system is the main principle. The sensors measure the rotor-required states (displacement, velocity, or acceleration) from their reference position. A controller drives the control action according to the control algorithm and the measured values. A power amplifier converts the control action to control current that produces a controllable magnetic force in a way such that the rotor does not oscillate away from its reference position. The control algorithm is responsible for the system stability and the vibration level as well as the damping and the stiffness of the controlled system. Many researchers studied different control methodology to mitigate or to suppress the nonlinear vibrations in the rotating machines. Ji et al. [1– 3] investigated two different nonlinear model simulating the vibration behaviors of a rotor system supported by active magnetic bearing. The authors studied the rotor system supported via eight poles baring at primary and super-harmonic resonance with 1:1 internal resonance [1,2]. They observed the existence of nonlinear phenomena that did not appear in the linear model such as static bifurcation, jump phenomenon, sensitivity to the initial conditions, and existence of more than one stable solution. In additions, the authors showed that the rotor unbalance and the control gain have great influences on the system response. In Ref. [3], they studied four poles magnetic bearing supported rotor system. They derived two independent second-order differential equations that describe the motions in x and y directions.

Corresponding author. E-mail addresses: [email protected]fia.edu.eg, [email protected] (N.A.-F. Abdul-Hameed Saeed).

http://dx.doi.org/10.1016/j.ijnonlinmec.2016.10.003 Received 13 March 2016; Received in revised form 3 October 2016; Accepted 4 October 2016 Available online 05 October 2016 0020-7462/ © 2016 Elsevier Ltd. All rights reserved.

International Journal of Non–Linear Mechanics 87 (2016) 109–124

N.A.-F. Abdul-Hameed Saeed, M. Kamel

vertical oscillation modes, respectively. λ Quadratic nonlinearity coefficients of the rotor system. γ , α1, α2, α3 Cubic nonlinearity stiffness coefficients of the rotor system. p, d Proportional and derivative gains of PD-controller, respectively. Ω The rotor angular speed. f The rotor eccentricity.

Nomenclature

u,̈ u,̇ u v ,̈ v,̇ v

μ1, μ2 ω1, ω 2

Acceleration, velocity and displacement of the horizontal direction. Acceleration, velocity and displacement of the vertical direction. Linear damping coefficients of both the horizontal and the vertical oscillation modes, respectively. Linear natural frequencies of both the horizontal and the

approximate analyses illustrated the high efficiency of the controller to mitigate the system vibrations, and the nonlinear behaviors such as sensitivity to initial conditions, multi-valued solution, and multi-jump phenomenon have been eliminated even at extremely large rotor eccentricity.

The bifurcation analysis is presented in the vertical direction only at autonomous and non-autonomous cases. The author concluded that the system offers extremely complicated nonlinear behaviors in both autonomous and non-autonomous cases. The autonomous equation reveals the existence of saddle node, Hopf bifurcations, and saddle connection, and the non-autonomous equation confirms the existence of transversal intersection of the homoclinic orbit. In Refs [4–6], Zhang et al. investigated the nonlinear vibrations of active magnetic bearing supported rotor system having a time-varying stiffness. The existence of multi-pulse, and Shilnikov type chaotic motions are the main observation of the nonlinear phenomena. In [7–11], the authors studied active magnetic bearing supported rotor system with timevarying stiffness at different resonance cases and different excitation forms by utilizing the perturbation methods. They investigated the shape of the chaotic motions in both the vertical and horizontal directions. The nonlinear vibrations of an active magnetic bearing system subjected to different excitation forms are studied in [12–15]. Eissa et al. [16,17], investigated the nonlinear vibrations and control of active magnetic bearing supported rotor system. They applied the saturation controller to suppress the system vibration near the primary resonance. The analysis illustrated that the saturation control strategy has the ability to suppress the system vibrations. Vlajic et al. [18], discussed the torsional vibrations of a Jeffcott-rotor system. The authors assumed that the rotor system exposed to continuous contact with the stator. The forward and backward whirling motions are investigated numerically and analytically. The authors found that, at forward whirling, a Hopf instability could excite the torsional vibrations in the reduced-order model. In addition, the torsional vibrations are subject to a centrifugal stiffening effect during both backward and forward whirling. Ishida et al. [19] studied the vibration mitigation in a vertically supported Jeffcott rotor system using a linear dynamic absorber. They performed a theoretical study to investigate the effects of nonlinearity on the vibration characteristics of the system. The authors concluded that the absorber fails to reduce the system vibrations if its parameters are optimized according to the linearized system model, but the control method exhibits high efficiency if the absorber parameters are tuned according to the nonlinear system model. Then, they implemented the dynamic absorber practically via two pairs of electromagnetic poles to generate push-pull control force in two perpendicular directions. Within this paper, a PD-controller is proposed to control the nonlinear vibrations of a horizontally supported Jeffcott-rotor system [20]. The controller is integrated into the system via four electromagnetic poles. The two pairs of the electromagnets are fixed on a fixed frame and located at two perpendicular directions. The nonlinearities due to the restoring force, electromagnetic force (control force), and the rotor weight are considered in the system model. Equations of the motion of the modified system (controlled system) are derived. Then, the multiple scales perturbation technique is sought to analyze the system dynamics. The system frequency (angular speed)-response curve, and the force (eccentricity)-response curve are plotted before and after control. Influences of the control parameters on the vibration amplitude are explored. Numerical confirmations are performed to validate the accuracy of the approximate results. The numerical and the

2. System model and perturbation analysis After adding four magnetic poles and integrating a PD-controller to the Jeffcott-rotor system, the governing equations of the horizontal and vertical oscillation modes, respectively are given as follows:

u ̈ + μ1 u ̇ + ω12 u + 2λvu + γv 2u = f Ω2 cos(Ωt ) − ( pu + du ̇ + α1 u3 + α2 u2u ̇ + α3 uu 2̇ ) (1.a)

v ̈ + μ2 v ̇ +

ω22 v

+

λ (u 2

+3

v 2)

+

γu2v

=

f Ω2

sin(Ωt )

− ( pv + dv ̇ + α1 v 3 + α2 v 2v ̇ + α3 vv 2̇ )

(1.b)

where −(pu + du )̇ , −(pv + dv )̇ are the linear control forces, and −(α1 u3 + α2 u2u ̇ + α3 uu 2̇ ),−(α1 v 3 + α2 v 2v ̇ + α3 vv 2̇ ) are the nonlinear control forces due to the electromagnetic coupling. The derivation of Eqs. (1) is given in Appendix A. 2.1. Multiple scale perturbation analysis By utilizing the multiple scales perturbation technique (MSTP) [21,22], we can obtain a second-order approximate solution to Eqs. (1) by seeking a solution in the following form:

u (t , ε ) = εu1 (T0, T1, T2 ) + ε 2u2 (T0, T1, T2 ) + ε 3u3 (T0, T1, T2 ) + O (ε 4 )

(2.a)

v (t , ε ) = εv1 (T0, T1, T2 ) + ε 2v2 (T0, T1, T2 ) + ε 3v3 (T0, T1, T2 ) + O (ε 4 )

(2.b)

where ε is the perturbation parameter, T0=t , T1=εt and T2=ε 2t are three time scales. According to the chain rule, the time derivatives can be written in terms of T0, T1, and T2 as follows:

d = D0 + εD1 + ε 2D2 , dt ∂ , j = 0, 1, 2 Dj = ∂Tj

d2 = D02 + 2εD0 D1 + ε 2 (D12 + 2D2 D0 ), dt 2 (3)

A new scaling for the system parameters according to their values is considered such that:

μ1 = ε 2μˆ1,

μ2 = ε 2μˆ 2 ,

p = ε 2pˆ ,

d = ε 2dˆ,

f = ε 2fˆ

(4)

By substituting Eqs. (2)–(4) into Eqs. (1), then equating the coefficients of like powers of ε , we obtain the following six differential equations. O (ε ):

(D02 + ω12 ) u1 = 0

(5.a)

(D02 + ω22 ) v1 = 0

(5.b)

O 110

(ε 2 ):

International Journal of Non–Linear Mechanics 87 (2016) 109–124

N.A.-F. Abdul-Hameed Saeed, M. Kamel

(D02 + ω12 ) u2 = −2D0 D1 u1 − 2λv1 u1 + fˆ Ω2 cos(ΩT0 )

(6.a)

(D02 + ω22 ) v2 = −2D0 D1 v1 − λu12 − 3 λv12 + fˆ Ω2 sin(ΩT0 )

(6.b)

v2 (T0, T1, T2 ) =

(13.b)

ε 3):

O(

Inserting Eqs. (8) and (13) into Eqs. (7), we get

ˆ 0 u1 ˆ 1 − dD (D02 + ω12 ) u3 = −2D0 D1 u2 − (D12 + 2D2 D0 ) u1 − μˆ1 D0 u1 − pu − 2λv1 u2 − 2λv2 u1 −

γv12 u1

−

α1 u13

−

⎛ 4λi (ω1 + ω 2 ) ⎞ (D02 + ω12 ) u3 = −⎜ D1 AB⎟ ei (ω1+ ω2) T0 ⎝ ω 2 (2ω1 + ω 2 ) ⎠ ⎛ 4λi (ω1 − ω 2 ) ⎞ +⎜ D1 A B⎟ e−i (ω1− ω2) T0 ⎝ ω 2 (ω 2 − 2ω1) ⎠ ⎞ ⎛ 4λ 2 2λ 2 −⎜ + + γ ⎟ A B2e−i (ω1−2ω2) T0 ω22 ⎠ ⎝ ω 2 (2ω1 − ω 2 ) 2 2 ⎞ ⎛ 4λ 2λ − ⎜γ + + ⎟ AB2ei (ω1+2ω2) T0 ⎝ ω 2 (2ω1 + ω 2 ) ω22 ⎠ ⎞ ⎛ 2λ 2 −⎜ + α1 + iα2 ω1 − α3 ω12⎟ A3 e3iω1T0 ⎠ ⎝ 4ω12 − ω22

α2 u12 D0 u1

− α3 u1 (D0 u1)2

(7.a)

ˆ 0 v1 ˆ 1 − dD (D02 + ω22 ) v3 = −2D0 D1 v2 − (D12 + 2D2 D0 ) v1 − μˆ 2 D0 v1 − pv − 2λu1 u2 − 6λv1 v2 − γv1 u12 − α1 v13 − α2 v12 D0 v1 − α3 v1 (D0 v1)2

(7.b)

The solution of Eqs. (5), can be expressed in the form

u1 (T0, T1, T2 ) = A (T1, T2 ) eiω1T0 + cc

(8.a)

v1 (T0, T1, T2 ) = B (T1, T2 ) eiω2 T0 + cc

(8.b)

⎛ ˆ 1) A + ⎜ −2iω1 D2 A − D12 A − (iμˆ1 ω1 + pˆ + idω ⎝

where cc denotes the complex conjugate of the preceding terms. The coefficients A (T1, T2 ) and B (T1, T2 ) are unknown functions of T1 and T2 up to this step of the analysis. They will be obtained at the next solution step when the secular and the small-divisor terms are eliminated. By inserting Eqs. (8) into (6), we get

4λ2 4λ2 12 λ2 + − + 2γ ) ABB ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎞ 2λ2 4λ2 −( − 2 + 3α1 + iα2 ω1 + α3 ω12 ) A2 A ⎟ eiω1T0 2 2 ω2 4ω1 − ω2 ⎠

−(

(D02 + ω12 ) u2 = −2iω1 eiω1T0 D1 A − 2λABei (ω1+ ω2) T0 − 2λAB ei (ω1− ω2) T0 fˆ Ω2 iΩT0 + e + cc (9.a) 2 (D02 + ω22 ) v2 = −2iω 2 eiω2 T0 D1 B − λ (A2 e 2iω1T0 + AA ) fˆ Ω2 iΩT0 − 3 λ (B2e 2iω2 T0 + BB ) + e + cc 2i

⎞ ⎛ 4λiω1 ⎞ ⎛ 4λi D1 A2 ⎟ e 2iω1T0 − ⎜ D1 B2⎟ e 2iω2 T0 (D02 + ω22 ) v3 = −⎜ ⎠ ⎝ ω2 ⎠ ⎝ 4ω12 − ω22 2 2 ⎞ ⎛ 4λ 6λ − ⎜γ + + ⎟ A2 B ei (2ω1− ω2) T0 ⎝ ω 2 (ω 2 − 2ω1) 4ω12 − ω22 ⎠ ⎞ ⎛ 4λ 2 6λ 2 −⎜ + + γ ⎟ A2 Bei (2ω1+ ω2) T0 4ω12 − ω22 ⎠ ⎝ ω 2 (2ω1 + ω 2 ) ⎞ ⎛ 6λ 2 − ⎜ 2 + α1 + iα2 ω 2 − α3 ω22⎟ B3e3iω2 T0 + (−2iω 2 D2 B ⎠ ⎝ ω2

(9.b)

1. Primary resonance: Ω ≅ ω1,Ω ≅ ω 2 2. Internal resonance: ω1≅ω 2 3. Simultaneous resonance: Ω ≅ ω1≅ω 2

ˆ 2) B − ( − D12 B − (iμˆ 2 ω 2 + pˆ + idω

The simultaneous resonance case (Ω ≅ ω1≅ω 2 ) is studied in this article. So, the closeness of the considered resonance case can be described quantitatively by introducing the two detuning parameters σ1 and σ2 according to

Ω = ω1 + σ1 = ω1 + εσˆ1,

+

The conditions for a bounded solution of Eqs. (14) are

ˆ 1) A 2iω1 D2 A = −D12 A − (iμˆ1 ω1 + pˆ + idω ⎞ ⎛ 4λ2 4λ2 12 λ2 −⎜ + − + 2γ ⎟ ABB ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎠ 2 2 ⎞ ⎛ 2λ 4λ −⎜ − 2 + 3α1 + iα2 ω1 + α3 ω12⎟ A2 A ω2 ⎠ ⎝ 4ω12 − ω22 2 2 ⎞ ⎛ 4λ 2λ −⎜ + + γ ⎟ A B2e 2iσˆ 2 T1 ω22 ⎠ ⎝ ω 2 (ω 2 − 2ω1)

(11.a)

⎛ fˆ Ω2 i (σˆ1− σˆ 2) T1⎞ iω2 T0 ⎟e (D02 + ω22 ) v2 = ⎜ −2iω 2 D1 B + e − λ (A2 e 2iω1T0 + AA ) 2i ⎠ ⎝ (11.b)

fˆ Ω2 i (σˆ1− σˆ 2) T1 e 2

⎞ ⎛ 4λ 2 4λ 2 12λ2 −⎜ + − + 2γ ⎟ AA B ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎠ ⎞ ⎛ 30 λ2 − ⎜ − 2 + 3α1 + iα2 ω 2 + α3 ω22⎟ B2B ⎠ ⎝ ω2 2 2 ⎞ ⎛ 4λ 6λ + γ ⎟ A2 B e−2iσˆ 2 T1 −⎜ + ⎝ ω 2 (ω 2 − 2ω1) 4ω12 − ω22 ⎠ (15.b)

(12)

Therefore, the solutions of Eq. (11) under the conditions (12) are

u2 (T0, T1, T2 ) =

(15.a)

ˆ 2) B 2iω 2 D2 B = −D12 B − (iμˆ 2 ω 2 + pˆ + idω

The conditions for Eq. (11) so as not produce secular terms are

2iω 2 D1 B = −i

30 λ2 + 3α1 + iα2 ω 2 + α3 ω22 ) B2B ) eiω2 T0 + cc ω22 (14.b)

⎛ fˆ Ω2 iσˆ1T1⎞ iω1T0 (D02 + ω12 ) u2 = ⎜ −2iω1 D1 A + e ⎟e − 2λABei (ω1+ ω2) T0 2 ⎠ ⎝

fˆ Ω2 iσˆ1T1 e , 2

12λ2 4λ 2 − + 2γ ) AA B ω 2 (ω 2 − 2ω1) ω22

− (−

Inserting Eqs. (10) into the secular and small-divisor terms of Eqs. (9), we get

− 3 λ (B2e 2iω2 T0 + BB ) + cc

4λ 2 ω 2 (2ω1 + ω 2 )

(10)

ω 2 = ω1 + σ2 = ω1 + εσˆ2

− 2λAB ei (ω1− ω2) T0 + cc

(14.a)

+ cc

Before we proceed to the next solution step, we must define all possible resonance cases at this approximation order, which are

2iω1 D1 A =

λ λ λAA 3 λBB A2 e 2iω1T0 + 2 B2e 2iω2 T0 − − + cc 4ω12 − ω22 ω2 ω22 ω22

2λ 2λ ABei (ω1+ ω2) T0 + AB ei (ω1− ω2) T0 + cc ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1)

After eliminating D12 A, D12 B from Eqs. (15) with the aid of Eq. (12),

(13.a) 111

International Journal of Non–Linear Mechanics 87 (2016) 109–124

N.A.-F. Abdul-Hameed Saeed, M. Kamel

the reconstitution method [23] is applied to combine the resulting equation and Eq. (12) using the first relation of Eqs. (3) into

φ⋅ 2 = σ1 − σ2 −

⎞ 1 ⎛ 4λ2 4λ2 12λ2 + − + 2γ ⎟ a12 ⎜ 8ω 2 ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎠ 2 ⎞ ⎛ ⎛ 1 30 λ 1 4λ2 − + ⎜ − 2 + 3α1 + α3 ω22⎟ a 22 − ⎜γ + 8ω 2 ⎝ ω2 8ω 2 ⎝ ω 2 (ω 2 − 2ω1) ⎠

fˆ Ω2 iσˆ1T1 σˆ fˆ Ω2 iσˆ1T1 ⋅ ˆ 1) A 2iω1 A = ε − ε2 1 − ε 2 (iμˆ1 ω1 + pˆ + idω e e 2 4ω1 ⎞ ⎛ 4λ2 4λ2 12 λ2 − ε2 ⎜ + − + 2γ ⎟ ABB ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎠ 2 2 ⎞ ⎛ 2λ 4λ − ε2 ⎜ − 2 + 3α1 + iα2 ω1 + α3 ω12⎟ A2 A ω2 ⎠ ⎝ 4ω12 − ω22 ⎞ ⎛ 4λ2 2λ2 − ε2 ⎜ + + γ ⎟ A B2e 2iσˆ 2 T1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠ ifˆ Ω2 i (σˆ1− σˆ 2) T1 i (σˆ − σˆ2 ) fˆ Ω2 i (σˆ1− σˆ 2) T1 ⋅ 2iω 2 B = −ε e + ε2 1 e 2 4ω 2 ⎛ 4λ 2 ˆ 2) B − ε2 ⎜ − ε 2 (iμˆ 2 ω 2 + pˆ + idω ⎝ ω 2 (2ω1 + ω 2 ) ⎞ 4λ 2 12λ2 + − + 2γ ⎟ AA B ω 2 (ω 2 − 2ω1) ω22 ⎠ ⎞ ⎛ 30 λ2 − ε 2 ⎜ − 2 + 3α1 + iα2 ω 2 + α3 ω22⎟ B2B ⎠ ⎝ ω2 2 2 ⎞ ⎛ 4λ 6λ + γ ⎟ A2 B e−2iσˆ 2 T1 − ε2 ⎜ + ⎝ ω 2 (ω 2 − 2ω1) 4ω12 − ω22 ⎠ d

−

⎞ 6λ2 ⎟ a12 cos(2φ2 − 2φ1) 4ω12 − ω22 ⎠

2.2. Equilibrium solutions At steady state oscillations, the variations of the amplitudes and phases are zero (a1̇ =a2̇ =φ1̇ =φ2̇ =0 ). Substituting this condition into Eqs. (18), we obtain the following four algebraic equations. 1 ⎛ ⎜1 2ω1 ⎝

(16.b)

+

d

1 iδ1 1 1 aˆ1 e ⇒ Ȧ = aˆ1̇ eiδ1 + iaˆ1 δ1̇ eiδ1 , 2 2 2

1 1 1 B = aˆ2 eiδ2 ⇒ B ̇ = aˆ2̇ eiδ2 + iaˆ2 δ 2̇ eiδ2 , 2 2 2

a1 = εaˆ1

2λ2 ω22

σ1 + (17.a)

(17.b)

−

where a1 and a2 , represent the steady state oscillation amplitudes at both the horizontal and the vertical vibration modes, respectively, and δ1, δ 2 , are the phases of the two oscillation modes. By inserting Eqs. (17) into (16) and separating the real and imaginary parts with restoring each scaled parameter back to its original form (i.e. μ μ p f d σ σ a a μˆ1= 21 , μˆ 2 = 22 , pˆ1= 21 , dˆ2= 22 , fˆ = 2 , σˆ1= ε1 , σˆ2= ε2 , T1=εt , aˆ1= ε1 , aˆ2= ε2 ) ε ε ε ε ε in the resulting equations, we obtain the following four first order autonomous differential equations.

1 ⎛ σ ⎞ 1 1 a⋅1 = ⎜1 − 1 ⎟ f Ω2 sin(φ1) − (μ1 + d ) a1 − α2 a13 2ω1 ⎝ 2ω1 ⎠ 2 8 1 ⎛ 4λ2 2λ2 ⎞ − + ⎟ a1 a 22 sin(2φ1 − 2φ2 ) ⎜γ + 8ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠

φ⋅ 1 = σ1 +

1

1 ⎛ 4λ2 ⎜ 8ω1 ⎝ ω2 (ω2 − 2ω1)

1

sin(φ1) − 2 (μ1 + d ) a1 − 8 α2 a13 −

⎞ + γ ⎟ a1 a 22 sin(2φ1 − 2φ2 ) = 0 ⎠

1 ⎛ ⎜1 2ω1a1 ⎝ 1 ⎛ ⎜ 2γ 8ω1 ⎝

−

σ1 ⎞ ⎟ f Ω2 2ω1 ⎠

cos(φ1) −

4λ2 ω2 (2ω1 + ω2)

+

1 ⎛ 2λ2 ⎜ 8ω1 ⎝ 4ω12 − ω22

−

4λ2 ω22

+

1 p 2ω1

4λ2 12 λ2 ⎞ − 2 ⎟ a 22 ω2 (ω2 − 2ω1) ω2 ⎠

⎞ + 3α1 + α3 ω12⎟ a12 − ⎠

1 ⎛ ⎜γ 8ω1 ⎝

+

4λ2 2λ2 ⎞ + 2⎟ ω2 (ω2 − 2ω1) ω2 ⎠

(19.b)

⎞ 1 1 − 1⎟ f Ω2 cos(φ2 ) − 2 (μ2 + d ) a2 − 8 α2 a 23 − ⎠ ⎞ 6λ 2 + 2 2 ⎟ a12 a2 sin(2φ2 − 2φ1) = 0 4ω1 − ω2 ⎠ 1 ⎛ σ1 − σ 2 ⎜ 2ω2 ⎝ 2ω2

1 ⎛ ⎜γ 8ω 2 ⎝

+

4λ 2 ω2 (ω2 − 2ω1)

(19.c)

⎞ 1 ⎛σ − σ σ1 − σ2 − 2ω a ⎜ 12ω 2 − 1⎟ f Ω2 sin(φ2 ) − 2 2⎝ 2 ⎠ (18.a)

+

−

a⋅2 =

σ1 ⎞ ⎟ f Ω2 2ω1 ⎠

a 22 cos(2φ1 − 2φ2 ) = 0

1 ⎛ 1 σ ⎞ p ⎜1 − 1 ⎟ f Ω2 cos(φ1) − 2ω1 a1 ⎝ 2ω1 ⎠ 2ω1

⎞ 1 ⎛ 4λ2 4λ2 12 λ2 + − + 2γ ⎟ a 22 ⎜ 8ω1 ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω22 ⎠ 2 2 ⎞ ⎛ 1 2λ 4λ − − 2 + 3α1 + α3 ω12⎟ a12 ⎜ 8ω1 ⎝ 4ω12 − ω22 ω2 ⎠ 2 2 ⎞ ⎛ 1 4λ 2λ − + + γ ⎟ a 22 cos(2φ1 − 2φ2 ) ⎜ 8ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠

−

(19.a)

− a2 = εaˆ2

(18.d)

where φ1=σ1 t − δ1, φ2=σ1 t − σ2 t − δ 2 . Eqs. (18) forms the amplitudephase modulating equations of the controlled system, and based on these equations the oscillatory behavior of the system can be studied in terms of the controller parameters p, d as a function of the detuning parameter σ1.

(16.a)

̇ B , and to analyze Eqs. (16), we rewrite A (T1, T2 ) and where Ȧ = dt A, B= dt B (T1, T2 ) in the polar form as follows:

A=

⎞ 1 ⎛ σ1 − σ2 1 − 1⎟ f Ω2 sin(φ2 ) − p ⎜ ⎠ 2ω 2 a2 ⎝ 2ω 2 2ω 2

1 1 ⎛ p − 8ω ⎜ 2γ + 2ω 2 2⎝

4λ 2 ω2 (2ω1 + ω2)

4λ 2 ω2 (ω2 − 2ω1) 1 ⎛ 30 λ2 ⎜− ω 2 8ω 2 ⎝ 2

−

12λ2 ⎞ 2 ⎟a ω22 ⎠ 1

+

⎞ 2 6λ 2 ⎟a 4ω12 − ω22 ⎠ 1

−

⎞ + 3α1 + α3 ω22⎟ a 22 − ⎠

1 ⎛ ⎜γ 8ω 2 ⎝

+

4λ 2 ω2 (ω2 − 2ω1)

cos(2φ2 − 2φ1) = 0 (19.d)

The solution of Eqs. (19) gives the steady-state solution of Eqs. (1) in terms of the system and the controller parameters. Since there is no closed-form solution for Eqs. (19), we resorted to the numerical techniques (Newton–Raphson algorithm is employed to solve the above four equations simultaneously).

(18.b)

⎞ 1 ⎛ σ1 − σ2 1 1 − 1⎟ f Ω2 cos(φ2 ) − (μ2 + d ) a2 − α2 a 23 ⎜ ⎠ 2ω 2 ⎝ 2ω 2 2 8 ⎞ 1 ⎛ 4λ2 6λ2 − + ⎟ a12 a2 sin(2φ2 − 2φ1) ⎜γ + 2 2 8ω 2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠

2.3. Stability analysis The stability of the steady state oscillations was determined by exploring the eigenvalues of the Jacobian matrix of the right-hand side of Eqs. (18). Then, Lyapunov first method for stability is utilized. To

(18.c) 112

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Fig. 1. Effect of varying the eccentricity ( f ) on the uncontrolled Jeffcott-rotor frequency-response curve: (left column) horizontal oscillations, and (right column) vertical oscillations.

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derive the stability criteria, we need to check the behavior of small deviation from the steady state solutions a10 , a20 , φ10 , and φ20 (i.e. linearization about the equilibrium point). Thus, we assume that

a1 = a11 + a10 , a2 = a21 + a20 , a1̇ = a11 ̇ , a2̇ = a21 ̇ , φ1̇ = φ11 ̇ ,

φ1 = φ11 + φ10 , φ2̇ = φ21 ̇

Δ0 = 8ρ3 − 3ρ12 ,

Δ2 = ρ22 − 3ρ1 ρ3 + 12ρ4 ,

Δ3 = 64ρ4 − 16ρ22 + 16ρ12 ρ2 − 16ρ1 ρ3 − 3ρ14 .

φ2 = φ21 + φ20 ⎫ ⎬ ⎭

By combining the conditions (24) with each one of the following possible cases, one can determine the nature of the roots of Eq. (23) as follows:

(20) where a10 , a20 , φ10 , and φ20 are the solutions of (19), and a11, a21, φ11,φ21 are perturbations which are assumed to be small compared to a10 , a20 , φ10 , and φ20 . By substituting Eqs. (20) into (18) and expanding for small a11, a21, φ11, and φ21 with keeping the linear terms only, we get

⎡ a11 ̇ ⎤ ⎛ R11 ⎢ φ̇ ⎥ ⎜ R 11 ⎢ ⎥ = ⎜ 21 ̇ ⎥ ⎜ R31 ⎢ a21 ⎜ ⎢⎣ φ21 ̇ ⎥⎦ ⎝ R 41

R12 R22 R32 R 42

R13 R23 R33 R 43

R14 ⎞ ⎡ a11 ⎤ R24 ⎟⎟ ⎢ φ11⎥ R34 ⎟ ⎢⎢ a21 ⎥⎥ ⎟ R 44 ⎠ ⎣ φ21⎦

a) If ∆ < 0 then the equation has two complex conjugate roots and two different real roots. b) If ∆ > 0, then either the four roots of the equation are all complex conjugate or all real according to the following two cases: i. If ∆0 <0 and ∆3 <0 then all four roots are real and different. ii. If ∆0 >0 or if ∆3 >0 then the roots are two pairs of complex conjugate. c) If ∆ = 0 then the equation has a multiple root according to the following four cases: i. If ∆0 <0 and ∆3 <0 and ∆2 ≠0 , there are two real simple roots and a real double roo ii. If ∆3 >0 or (∆0 >0 and (∆3≠0 or ∆1≠0 )), there are two complex conjugate roots and two real equal roots. iii. If ∆2 =0 and ∆3≠0 , there are three real equal roots and one real different root. iv. If ∆3=0 , then: 1. If ∆0 <0 , there are two real double roots. 2. If ∆0 >0 and ∆1=0 , there are two complex conjugate double roots. ρ 3. If ∆2 =0 , all four roots are equal to− 41 .

(21)

where the coefficients Rjk :j = 1,2,3,4, k = 1,2,3,4 are given in the Appendix B. The above square matrix is the Jacobian matrix. Thus, the stability of the steady state oscillations depends on the matrix eigenvalues. One can obtain the following eigenvalue equation:

R11 − γ R21 R31 R 41

R12 R22 − γ R32 R 42

R13 R23 R33 − γ R 43

R14 R24 R34 R 44 − γ

=0 (22)

Expanding the above determinant, yields

γ4

+ ρ1 γ 3 + ρ2 γ 2 + ρ3 γ + ρ4 = 0

(23) 3. Result and discussion

where γ denotes the eigenvalues of the Jacobian matrix and ρ1 , ρ2 , ρ3 , and ρ4 are the coefficient of Eq. (23) and given in Appendix B. By applying Routh-Hurwitz criterion to investigate the stability of the equilibrium solutions, we find the necessary and sufficient conditions for the system to be stable are:

ρ1 > 0,

ρ1 ρ2 − ρ3 > 0,

ρ3 (ρ1 ρ2 − ρ3) − ρ12 ρ4 > 0,

Δ1 = ρ13 + 8ρ32 − 4ρ1 ρ2 ,

ρ4 > 0

In this section, the vibrations in both the horizontal and vertical directions of the Jeffcott-rotor system are explored. The analysis is introduced by adopting the following value of the system parameters unless otherwise mentioned: p = 0.3, d = 0.05, μ1=0.015, μ2 =0.025, λ = γ = η = 0.05, α1=γ + p (1 − p ), α2=d (1 − 2p ), α3=−d 2, . In the fol-

(24)

ω1= 1 + η , ω 2= 1 + 3η , σ1=0, σ2=ω 2 −ω1,f = 0.025 lowing bifurcation diagrams, dashed lines refer to unstable solutions, while the solid lines refer to stable solutions. In Fig. 1, various frequency-response curves are plotted at different values of the shaft eccentricity f for the system before control (i.e. p = d = 0 ). It can be seen that as the eccentricity increases, the frequency-response curves bend away from the linear curves to the right resulting in multivalued solutions and the jump phenomenon at both the horizontal and vertical oscillation modes. In addition, the figure shows that the oscillation

Moreover, to determine the nature of the roots of Eq. (23), we defined the following discriminants ∆,∆0 , ∆1, ∆2 ,and∆3 (see [24,25]) where:

Δ = 256ρ43 − 192ρ1 ρ3 ρ42 − 128ρ22 ρ42 + 144ρ2 ρ32 ρ4 − 27ρ34 + 144ρ12 ρ2 ρ42 − 6ρ12 ρ32 ρ4 − 80ρ1 ρ22 ρ3 ρ4 + 18ρ1 ρ2 ρ33 + 16ρ24 ρ4 − 4ρ23 ρ32 − 27ρ14 ρ42 + 18ρ13 ρ2 ρ3 ρ4 − 4ρ13 ρ33 − 4ρ12 ρ23 ρ4 + ρ12 ρ22 ρ32 .

Fig. 2. Influences of varying the proportional gain p on the controlled system frequency-response curve: (a) horizontal oscillations, and (b) vertical oscillations.

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peak amplitude of the two oscillation modes and shift them to the right. Also the figure shows that the peak amplitude of the two oscillation modes decreases and shifts to the left at the negative values of p . Fig. 3 shows the proportional gain p -response curve at different values of σ1. The figure shows that the peak amplitudes of the two oscillation modes are a monotonic increasing function in the proportional gain. According to Figs. 2 and 3, the best value of the proportional gain p is to be negative value within the stable range (i.e.−1 < p < 0 ). The influence of the derivative gain d on the system frequency-response curve when p = 0 is shown in Figs. 4. It is noticed that increasing the value of d , decreases the oscillation peak amplitude. Fig. 5 shows the d -response curve at three different values of the detuning parameter σ1. It is clear from the figure that the two oscillation amplitudes are a monotonic decreasing function in d . Figs. 6 and 7 illustrate the controlled system frequency-response curve at various values of the rotor eccentricity f . Where Fig. 6 shows the system frequency-response curves when p = −0.3, and Fig. 7 shows the system frequency-response curves when p = −0.3. By comparing the two figures, it can be noticed that the system peak amplitudes in Fig. 6 are about the half of the system peak amplitudes in Fig. 7. Therefore, choosing the proportional gain to be negative is the optimal choice, but it must be selected within the stable range that reported in Figs. 3. Fig. 8 shows the system force (eccentricity)-response curve before and after control at different values of detuning parameter σ1. The figure shows that the relation between the system amplitude and the rotor eccentricity (before control) is a nonlinear relation that exhibits

amplitudes are a monotonic increasing function of the rotor eccentricity. Figs. 1g and h show the system frequency-response curve at f = 0.045. Arrows indicate the oscillation associated with each mode. Whereas σ1 is increased gradually from a negative value (i.e. rotor speed lower than the critical speed), the system has small amplitude nonlinear oscillations for both the horizontal and the vertical modes until σ1 reaching the point A, then they increased at an observable rate from A to B until they reach the point C. At this point C, there are saddle-node bifurcations. Subsequent to this bifurcation point, the two modes amplitude jump down from the point C to D. Then they oscillate with small amplitudes when σ1 cross D toward E. As σ1 is decreased gradually from the point E (i.e. rotor speed higher than the critical speed), the system motion will follow the E-D-F path with a small oscillation amplitude for both a1 and a2 until σ1 reaching the point G. At this point G, there are saddle-node bifurcations. Subsequent to this bifurcation, the two modes amplitude jump up from the point G to H. Then, the horizontal mode (a1) oscillates with a moderate amplitude while the vertical mode (a2 ) oscillates with a large amplitude along the H-I path until σ1 reaching the point I. At this point I, there is another saddlenode bifurcation point. Subsequent to this point, the two modes amplitude jump from the point I to J. The horizontal mode jumps up, while the vertical mode jumps down from I to J, then the two modes oscillation amplitude decreases along the J-K-A path. Fig. 2 shows the system frequency-response curve of the controlled system at different values of the proportional gain p . It can be noted from the figure that increasing the proportional gain p , increases the

Fig. 3. The proportional gain p -response curve at different values of σ1: (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 4. Influence of varying the derivative gain d on controlled system frequency-response curve when p = 0 : (a) horizontal oscillations, and (b) vertical oscillations.

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Fig. 5. The derivative gain d -response curve at different values of σ1: (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 6. The controlled system frequency-response curve at different values of the eccentricity ( f ) when p = −0.3: (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 7. The controlled system frequency-response curve at different values of the eccentricity ( f ) when p = 0.3: (a) horizontal oscillations, and (b) vertical oscillations.

Runge-Kutta algorithm, and the obtained results are compared with the analytical solutions that given by applying MSPT in the above section. The numerical results are marked as small circles when sweeping the bifurcation parameter from the small values to large values, and as big-dots when sweeping the bifurcation parameter from the large values to small values. In Fig. 9 the system frequency (angular speed)-response curve has been plotted before control ( p = d = 0.0 ) and after control ( p = −0.3, d = 0.05) when the rotor eccentricity f = 0.025. Before control the figure shows that the system exhibits a

extremely large system amplitudes for a slight increasing in the rotor eccentricity. After control, the relation becomes linear with an extremely small slope, which produces system amplitudes less than that before control. In addition, all the jump phenomena have been eliminated after control even in the case of extremely large eccentricity. 4. Numerical simulations We have integrated the system original Eqs. (1) numerically using 116

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Fig. 8. Effect of varying the detuning parameter σ1 on Jeffcott rotor eccentricity-response curve before and after control: horizontal oscillations (left column), and vertical oscillations (right column).

behaviors have been eliminated, and the maximum oscillation amplitude becomes extremely small compared to that before control. The figure also confirms the good agreement between the numerical

large oscillation amplitudes, multi-valued solutions, jump phenomenon, and the sensitivity to initial conditions when the rotor speed near the resonant speed. After control, the figure shows that all nonlinear

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18. The analytical results that obtained in Figs. 6 and 7 have been verified numerically in Fig. 14. The system frequency-response curves at two different values of the rotor eccentricity (i.e. f = 0.025, f = 0.055) are plotted in Fig. 14 when the proportional gain p = ± 0.3. Then, numerical validations for the obtained curves are performed. The resulting figures confirm that the oscillation peak in the case of the negative value of the proportional gain ( p = −0.3) is about the half of the oscillation peak in the case of the positive value of the proportional gain ( p = 0.3). This confirms the preference of the proportional gain to be negative (i.e. the best control method is a positive position feedback controller with a negative velocity feedback controller). In Figs. 15 and 16, the force-response curves have been validated numerically before and after control at two different values of detuning parameter σ1. The jump phenomenon is reported numerically in Fig. 16 before control, and has been eliminated after control.

simulations and the approximate analytical solution. Fig. 10 shows the system time histories, according to point p1 (i.e. σ1=0.05) that marked on Fig. 9 at the system initial conditions u (0)=2, v (0)=0.5, u ̇ (0)=v ̇ (0)=0 , while Fig. 11 shows the system time histories according to point p2 at the system initial conditions u (0)=v (0)=u ̇ (0)=v ̇ (0)=0 . Based on Fig. 10 and 11, the Jeffcott- rotor system is sensitive to the initial conditions. The Eqs. (18(a)–(d)) describe the modulating amplitudes a1, a2 and the modified phases φ1 and φ2 for the considered resonant case. The numerical solutions of the Eqs. (18(a)–(d)) at chosen values of the system parameters are plotted in Figs. 11–13. The dashed lines show the modulation amplitudes (a1, a2 ) of the two coordinates u and v . However, the continuous lines represent the time histories that were acquired by solving the original Eqs. (1) numerically. The solution that presented in the Fig. 11 was obtained according to point p2 that marked on Fig. 9 (i.e. σ1=0.05, a1 (0)=a2 (0)≅0, φ1 (0)=φ2 (0)=0 ). The simulation results illustrate that Eqs. (18(a)–(d)) describe with high accuracy not only the steady-state modulating amplitudes, but also the transient modulating amplitudes of the Jeffcott-rotor system. Fig. 12 shows the controlled system time-histories, according to point p3 (i.e. σ1=0.05, p = −0.3, d = 0.05) that marked on Fig. 9 at the system initial conditions u (0)=v (0)=u ̇ (0)=v ̇ (0)=0 , and Fig. 13 shows the controlled system time-histories, according to point p3 at the system initial conditions u (0)=2, v (0)=0.5, u ̇ (0)=v ̇ (0)=0 . Based on Figs. 12 and 13, all nonlinear behaviors of the controlled Jeffcott-rotor system such as sensitivity to initial condition, jump phenomenon, and high oscillation amplitudes have been eliminated. In addition, Figs. 10–13 show that the controller effectiveness Ea (Ea = the system steady state amplitude before control / system steady state amplitude after control) is about

5. Conclusions Within this article, a nonlinear PD-controller is proposed to eliminate or to control the nonlinear oscillations of a horizontally supported Jeffcott rotor system. The control law was applied via two pairs of electromagnetic poles. Where, the sensors measure the rotor required states (displacement, velocity, or acceleration) from their reference position. A controller drives the control action according to the control algorithm and the measured values. A power amplifier converts the control action to control current that produces a controllable magnetic force in a way such that the rotor does not oscillate away from its reference position. The rotor weight, the nonlinearity of

Fig. 9. The system frequency-response before and after control when p = −0.3: (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 10. Uncontrolled Jeffcott rotor time-histories according to the point p1 that marked on Fig. 9 (i.e.σ1=0.05) at the system initial conditions u (0)=2, v (0)=0.5, u ̇ (0)=v ̇ (0)=0 : (a) the horizontal oscillations, (b) the vertical oscillations, and (c) orbit plot.

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Fig. 11. Uncontrolled Jeffcott rotor time-histories according to the point p2 that marked on Fig. 9 (i.e. σ1=0.05) at the system initial conditions u (0)=v (0)=u ̇ (0)=v ̇ (0)=0 : (a) the horizontal oscillations, (b) the vertical oscillations, and (c) orbit plot.

Fig. 12. Controlled Jeffcott rotor time-histories according to the point p3 that marked on Fig. 9 (i.e. σ1=0.05) at the system initial conditions u (0)=v (0)=u ̇ (0)=v ̇ (0)=0 : (a) the horizontal oscillations, (b) the vertical oscillations, and (c) orbit plot.

Fig. 13. Controlled Jeffcott rotor time-histories according to the point p3 that marked on Fig. 9 (i.e. σ1=0.05) at the system initial conditions u (0)=2, v (0)=0.5, u ̇ (0)=v ̇ (0)=0 : (a) the horizontal oscillations, (b) the vertical oscillations, and (c) orbit plot.

the rotor restoring force, and the nonlinearity due to electromagnetic forces are considered in the system model. MSPT is employed to derive a second-order approximate solution to the system governing equations. The stability analysis was introduced to determine the stable and unstable boundaries of the control parameters. Based on the above discussion, we may conclude the following.

4.

5.

1. At small values of the rotor eccentricity ( f < 0.01), uncontrolled system responds with a small and localized oscillation in the horizontal mode or in the vertical mode according to the rotor speed. 2. If the rotor eccentricity in the range of (0.01 < f < 0.02 ), uncontrolled system responds with a large and localized oscillation in one of the two oscillation modes according to the rotor speed, and a single jump occurs when the rotor system has to be accelerated or decelerated through the resonant speed. 3. At the large rotor eccentricity ( f > 0.02 ), nonlinear interactions

6. 7.

119

between the two oscillation modes occurs (non-localized oscillation). In addition, more than one jump phenomenon appears when the rotor system has to be accelerated or decelerated through the resonant speed. The integration of the controller to the system has eliminated the nonlinear phenomena of the Jeffcott-rotor system, and the frequency response curve is turned to a linear form. The relation between the response amplitude and the rotor eccentricity becomes linear with an extremely small slope that make the vibration amplitude less than that before control. In addition, all the jump phenomena have been eliminated even in the case of extremely large eccentricity. The PD-controller mitigates the oscillation amplitudes with effectiveness EA around 18. The increasing of the proportional gain beyond zero, increases the peak amplitude of the two oscillation modes, and shifts them to the right.

International Journal of Non–Linear Mechanics 87 (2016) 109–124

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Fig. 14. The controlled Jeffcott rotor frequency-response curve for two different values of the eccentricity f when p = ± 0.3: (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 15. Jeffcott rotor eccentricity-response curve before and after control when σ1=0 : (a) horizontal oscillations, and (b) vertical oscillations.

Fig. 16. Jeffcott rotor eccentricity-response curve before and after control when σ1=0.03: (a) horizontal oscillations, and (b) vertical oscillations.

8. The decreasing of the proportional gain beyond zero decreases the peak amplitude of the two oscillation modes and shifts them to the left. 9. Increasing the derivative gain beyond zero eliminates the nonlinear behavior of the two oscillation modes via increasing the system linear damping coefficients μ1, μ2 .

10. The peak amplitude of the two oscillation modes is a monotonic increasing function in the proportional gain. 11. The optimal value of the proportional gain must be negative value within the stable range (−1 ≪ p < 0 ). 12. The best control method is a positive position feedback and a negative velocity feedback controller. 120

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[20]. The controller is integrated into the system via four electromagnetic poles. The two pairs of the electromagnets are fixed on a fixed frame and located at two perpendicular directions. The nonlinearities due to the restoring force, electromagnetic force (control force), and the rotor weight are considered in the system model. The equations of the motions of the modified system (controlled system) are derived. Then, the multiple scales perturbation technique is sought to analyze the system dynamics. The system frequency-response curves is plotted before and after control. Influences of the control parameters on the vibration amplitude are explored. The analyses illustrated the capability of the controller to mitigate the vibration amplitudes of the system to about 6%, and nonlinear behavior such as sensitivity to initial conditions, multi-valued solution, and multi-jump phenomenon have been eliminated even at extremely large rotor eccentricity.

6. Comparison with previously published work In comparison with previous work [19], Ishida et al studied the vibration control of symmetric two-degree-of-freedom nonlinear system simulating a vertically supported Jeffcott-rotor. Due to the bearing clearance, the authors considered the restoring force is a symmetric cubic nonlinear function. A passive vibration control is applied using dynamic vibration absorber to mitigate the system vibrations. A theoretical study to explore the effects of nonlinearity on the vibration behaviors of the system is performed. The authors concluded that the absorber fails to reduce the system vibrations if its parameters are optimized according to the linearized system model, but the control method gives high efficiency to mitigate the system vibration if the absorber parameters are tuned according to the nonlinear model. Then, they implemented the dynamic absorber practically via non-contact electromagnetic damper consisting of two-pairs of electromagnetic poles. Each pair of electromagnet poles is fixed on a movable frame. Moreover, the two frames are designed to move in two perpendicular directions to generate push-pull control force. In Ref. [20], the authors discussed the nonlinear oscillations of a horizontally supported Jeffcott-rotor system at the primary resonance case. The rotor weight and the nonlinearities due to bearing-clearance are considered in the system model. The authors derived two coupled and asymmetrical differential equations having cubic and quadratic nonlinearities to simulate the system vibrations. The multiple scales perturbation method is utilized to obtain the slow-flow modulating equations. Then, the system frequency (angular speed) response-curve is plotted at different values of the rotor eccentricity. The authors concluded that at small rotor eccentricity there are almost localized response amplitudes, and at large eccentricity the response amplitude curve becomes more complex and illustrates localized and non-localized nonlinear motions. Within this paper, a PD-controller is proposed to control the nonlinear vibrations of a horizontally supported Jeffcott-rotor system

Fig. A. 3. PD-controller connected to vertical oscillation modes of the Jeffcott-rotor system.

Fig. A. 1. Jeffcott-rotor system.

Fig. A. 2. Jeffcott rotor system with magnetic bearing.

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Acknowledgement

T. Inoue, and Prof. Y. Ishida, as we modified their mathematical model that used in Ref. [20] to be the basic model for our article.

We would like to thank Prof. H. Yabuno, prof. T. Kashimura, prof. system applies nonlinear magnetic forces FMu in u direction and FMv in v direction. Therefore, the modified mathematical model of the rotormagnetic bearing system can be described as follows: Appendix A

mu ̈ + cx u ̇ + (k1 + k2 yst2) u + 2k2 yst uv + k2 (u2 + v 2 ) u = med ω 2 cos(ωt ) − FMu mv ̈ + cy v ̇ + (k1 +

The whirling motion of a horizontally supported Jeffcott-rotor system as shown in Fig. A.1 is supposed to happen on the x– y plane. The distance between the two support points of the rotor system is denoted as l . The disc has mass m . It is assumed the gravity center e (xe , ye) departs a bit from the geometric center G , and the eccentricity is given by Ge=ed . Assume the rotor angular velocity to be ω . Due to the clearance of the bearing, the rotor restoring force F is assumed a symmetric cubic nonlinear function with respect to the vertical deflection r of the shaft and can be expressed as follows [26]:

Fx = k1 x + k2 (x 2 + y 2) x

Fy = k1 y + k2

(x 2

+

. Then, the equations of motion in the x- and y- directions are expressed as follows: (A.3)

my ̈ + cy y ̇ + k1 y + k2 (x 2 + y 2) y = med ω 2 sin(ωt ) − mg

(A.4)

(A.5)

where yst is the static displacement of the geometric center G due to the disc weight mg . Accordingly, it is possible to express the motion of the geometrical center G in terms of the deviations (u , v ) from the static equilibrium by substituting x = 0 + u , y = yst +v into Eqs. (A.3) and (A.4), with considering Eq. (A.5), we get mu ̈ + cx u ̇ + (k1 + k2 yst2) u + 2k2 yst uv + k2 (u2 + v 2 ) u = med ω 2 cos(ωt ) mv ̈ + cy v ̇ + (k1 +

3k2 yst2) v

+ k2 yst

(u 2

+3

v 2)

+ k2

(u 2

+

v 2) v

= med

ω2

δ g03

(A.10)

2 ⎧ ⎛ I − i v ⎞2 ⎫ ⎪⎛ I + i ⎞ ⎪ v ⎟⎬ ⎟ −⎜ 0 ⎜ 0 FMv = F (v, iv ) = B0 ⎨ ⎪ ⎝ g0 − v ⎠ ⎪ ⎭ ⎩ ⎝ g0 + v ⎠

(A.11)

⎛ ⎞ 2I 2 I2 3I 1 FMu = δ ⎜⎜I0 iu − 0 u − 30 u3 + 20 iu u2 − iu2 u⎟⎟ g0 g0 g0 g0 ⎝ ⎠

(A.12)

⎛ ⎞ 2I 2 I2 3I 1 FMv = δ ⎜⎜I0 iv − 0 v − 30 v 3 + 20 iv v 2 − iv2 v⎟⎟ g g g g ⎝ ⎠ 0 0 0 0

(A.13)

4B0 g02

. In this research, a PD-controller is proposed to produce a

control current that is proportional to the vibration displacement and velocity of the rotor. Therefore, the control law can be expressed as:

iu = kp u + kd u ,̇

(A.7)

iv = k p v + kd v ̇

(A.14)

where Fig. A.3 shows the schematic connection of the PD-controller to the vertical direction of the Jeffcott-rotor system (v oscillation mode). Inserting Eq. (A.14) into Eqs. (A.12) and (A.13), then by substitution of the resulting equations into Eqs. (A.8) and (A.9), we get

By adding an additional magnetic bearing locating around the disc as shown in Fig. A.2, where the center of the stator of the bearing is assumed to be the origin of coordinates u and v . The magnetic bearing

mu ̈ + cx u ̇ + (k1 + k2 yst2) u + 2k2 yst uv + k2 (u2 + v 2 ) u +

2 ⎧ ⎛ I − iu ⎞ 2 ⎫ ⎪⎛ I + i ⎞ ⎪ u ⎟⎬ ⎟ −⎜ 0 ⎜ 0 FMu = F (u , iu ) = B0 ⎨ ⎪ ⎪ ⎝ g0 − u ⎠ ⎭ ⎩ ⎝ g0 + u ⎠

whereδ =

(A.6)

sin(ωt )

(A.9)

where B0 is the magnetic force constant, I0 is the bias current,iu and iv are the control currents at u and v directions, respectively, g0 is the nominal air gap between the magnetic poles and the rotor disc. The nonlinear magnetic force can be approximated by expanding Eqs. (A.10) and (A.11) utilizing Taylor series up to the third order about the equilibrium point u = v = iu=iv=0 as follows:

where g is the acceleration of gravity. At static equilibrium, we have x= ̈ x= ̇ x = ω = 0, y = yst . Substituting this condition into Eqs. (A.3) ̈ y= ̇ y= and (A.4), yields

k1 yst + k2 yst3 = −mg

+ k 2 yst (u2 + 3 v 2) + k 2 (u2 + v 2) v = med ω2 sin(ωt ) − FMv

It is supposed that the magnetic bearing system consists of four poles system, each direction has two identical and opposite poles. The coupling between the two forces FMu and FMv is neglected. Therefore, the total force generated in both u and v directions can be expressed as [27]:

(A.2)

mx ̈ + cx x ̇ + k1 x + k2 (x 2 + y 2) x = med ω 2 cos(ωt )

(A.8)

.

(A.1)

y 2) y

3k 2 yst2) v

[(g03 I0 kp − g02 I02 ) u + g03 I0 kd u ̇

+ (3g0 I0 kp − 2I02 − g02 kp2 ) u3 + (3g0 I0 kd − 2g02 kp kd ) u2u ̇ − g02 kd2 uu 2̇ ] = med ω 2 cos(ωt ) mv ̈ + cy v ̇ + (k1 + 3k2 yst2) v + k2 yst (u2 + 3 v 2 ) + k2 (u2 + v 2 ) v +

δ g03

(A.15)

[(g03 I0 kp − g02 I02 ) v + g03 I0 kd v ̇

+ (3g0 I0 kp − 2I02 − g02 kp2 ) v 3 + (3g0 I0 kd − 2g02 kp kd ) v 2v ̇ − g02 kd2 vv 2̇ ] = med ω 2 sin(ωt )

(A.16)

Introducing the dimensionless parameters t = ς −1t *,u = g0 u*,v = g0 v* into Eqs. (A.15), (A.16) and omitting the asterisk for brevity, we get the following dimensionless equations of motion

u ̈ + μ1 u ̇ + ω12 u + 2λvu + γv 2u = f Ω2 cos(Ωt ) − ( pu + du ̇ + α1 u3 + α2 u2u ̇ + α3 uu 2̇ )

(A.17)

v ̈ + μ2 v ̇ + ω22 v + λ (u2 + 3 v 2 ) + γu2v = f Ω2 sin(Ωt ) − ( pv + dv ̇ + α1 v 3 + α2 v 2v ̇ + α3 vv 2̇ ) where p=

ς=

g0 kp

ςg k −1, d = I0 d , I0 0 δI02 mg0

=

k1 m

.

e

f = gd ,Ω = 0

ω , ς

c

cy

μ1= mςx , μ2 = mς , λ=

k 2 g0 yst mς 2

, γ=

(A.18) k 2 g02 mς 2

, η= 122

k 2 yst2 mς 2

, ω12=1 + η, ω22=1 + 3η, α1=γ +p (1 − p ), α2=d (1 − 2p ), α3=−d 2,

International Journal of Non–Linear Mechanics 87 (2016) 109–124

N.A.-F. Abdul-Hameed Saeed, M. Kamel

Appendix B

1 3 a ⎛ 4λ 2 2λ 2 ⎞ 2 R11 = − (μ1 + d ) − α2 a10 − 20 ⎜γ + + ⎟ sin(4φ10 − 4φ20 ) 2 2 4ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠

R12 =

1 ⎛ σ ⎞ a a ⎛ 4λ2 2λ2 ⎞ + ⎟ cos(4φ10 − 4φ20 ) ⎜1 − 1 ⎟ f Ω2 cos(2φ10 ) − 10 20 ⎜γ + 2ω1 ⎝ 2ω1 ⎠ ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠ a10 ⎛ 4λ 2 2λ 2 ⎞ + ⎟ sin(4φ10 − 4φ20 ) ⎜γ + 4ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠

R13 = −

R14 =

2 ⎛ 2a10 a 20 4λ 2 2λ 2 ⎞ + ⎟ cos(4φ10 − 4φ20 ) ⎜γ + ω1 ⎝ ω 2 (ω 2 − 2ω1) ω22 ⎠

R21 = −

⎞ f Ω2 ⎛ 4λ2 σ ⎞ a ⎛ 2λ2 − 2 + 3α1 + α3 ω12⎟ ⎜1 − 1 ⎟ cos(2φ10 ) − 10 ⎜ 2 ⎝ 2 2 2ω1 ⎠ 2ω1 ⎝ 4ω1 − ω2 8ω1 a10 ω2 ⎠

R22 = −

⎞ a2 ⎛ f Ω2 ⎛ σ ⎞ 4λ 2 2λ 2 + 2 + γ ⎟ sin(4φ10 − 4φ20 ) ⎜1 − 1 ⎟ sin(2φ10 ) + 20 ⎜ 4ω1 a10 ⎝ 2ω1 ⎠ ω1 ⎝ ω 2 (ω 2 − 2ω1) ω2 ⎠

R23 = −

⎞ ⎞ a20 ⎛ 4λ 2 4λ 2 12 λ2 a ⎛ 4λ2 2λ2 + − 2 + 2γ ⎟ − 20 ⎜ + 2 + γ ⎟ cos(4φ10 − 4φ20 ) ⎜ 2ω1 ⎝ ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω2 2ω1 ⎝ ω 2 (ω 2 − 2ω1) ω2 ⎠ ⎠

R24 = −

2 ⎛ ⎞ a 20 4λ 2 2λ 2 + 2 + γ ⎟ sin(4φ10 − 2φ20 ) ⎜ ω1 ⎝ ω 2 (ω 2 − 2ω1) ω2 ⎠

R31 = −

⎞ a10 a20 ⎛ 4λ2 6λ2 + ⎟ sin(4φ20 − 4φ10 ) ⎜γ + 2 2 ω2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠

R32 =

2 ⎞ 2a10 a20 ⎛ 4λ 2 6λ2 + ⎟ cos(4φ20 − 4φ10 ) ⎜γ + 2 2 ω2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠

⎞ a2 ⎛ 1 3 4λ2 6λ2 2 R33 = − (μ2 + d ) − α2 a 20 − 10 ⎜γ + + ⎟ sin(4φ20 − 4φ10 ) 2 2 2 2 2ω 2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠

R34 = −

⎞ ⎞ 2a 2 a20 ⎛ f Ω2 ⎛ σ1 − σ2 4λ 2 6λ2 − 1⎟ sin(2φ20 ) − 10 + ⎟ cos(4φ20 − 4φ10 ) ⎜ ⎜γ + 2 2 ⎠ 2ω 2 ⎝ 2ω 2 ω2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠

⎞ ⎞ a10 ⎛ 4λ2 4λ2 12λ2 a ⎛ 4λ2 6λ2 + − 2 + 2γ ⎟ − 10 ⎜γ + + ⎟ cos(4φ20 − 4φ10 ) R 42 ⎜ 2 2 ⎝ ⎝ 2ω 2 ω 2 (2ω1 + ω 2 ) ω 2 (ω 2 − 2ω1) ω2 2ω 2 ω 2 (ω 2 − 2ω1) 4ω1 − ω2 ⎠ ⎠ ⎞ a2 ⎛ 4λ 2 6λ2 = 10 ⎜γ + + ⎟ sin(4φ20 − 4φ10 ) ω2 ⎝ ω 2 (ω 2 − 2ω1) 4ω12 − ω22 ⎠

R 41 = −

R 43 =

⎞ ⎞ f Ω2 ⎛ σ1 − σ2 a ⎛ 30 λ2 − 1⎟ sin(2φ20 ) − 20 ⎜ − 2 + 3α1 + α3 ω22⎟ ⎜ 2 ⎝ ⎠ 2ω 2 2ω 2 ⎝ ω 2 8ω 2 a 20 ⎠

R 44 = −

⎞ ⎞ a2 ⎛ f Ω2 ⎛ σ1 − σ2 4λ2 6λ2 − 1⎟ cos(2φ20 ) + 10 ⎜γ + + ⎟ sin(4φ20 − 4φ10 ) ⎜ 2 ⎠ 4ω 2 a20 ⎝ 2ω 2 ω2 ⎝ ω 2 (ω 2 − 2ω1) 4ω1 − ω22 ⎠

ρ1 = −R 44 − R33 − R22 − R11 ρ2 = R11 R 44 + R11 R33 + R11 R22 + R22 R 44 + R22 R33 + R33 R 44 − R21 R12 − R31 R13 − R 41 R14 − R34 R 43 − R32 R23 − R 42 R24 ρ3 = R11 R32 R23 − R31 R12 R23 − R11 R22 R 44 − R 42 R23 R34 + R31 R22 R13 + R32 R23 R 44 − R11 R22 R33 − R32 R 43 R24 − R21 R32 R13 + R11 R 42 R24 + R31 R13 R 44 + R21 R12 R 44 − R21 R 42 R14 + R 42 R24 R33 + R21 R12 R33 − R11 R33 R 44 − R22 R33 R 44 − R 41 R12 R24 − R31 R 43 R14 + R 41 R22 R14 + R22 R34 R 44 + R 41 R14 R33 + R11 R34 R 43 − R 41 R13 R34 123

International Journal of Non–Linear Mechanics 87 (2016) 109–124

N.A.-F. Abdul-Hameed Saeed, M. Kamel

ρ4 = −R 41 R12 R23 R34 − R21 R12 R33 R 44 + R11 R22 R33 R 44 − R11 R 42 R24 R33 + R21 R32 R13 R 44 + R11 R32 R 43 R24 − R 41 R32 R13 R24 + R 41 R32 R23 R14 − R11 R22 R34 R 43 + R21 R12 R34 R 43 − R31 R12 R 43 R24 − R21 R 42 R13 R34 − R21 R32 R 43 R14 + R31 R12 R14 R 44 + R31 R 42 R13 R24 − R31 R 42 R23 R14 + R21 R 42 R14 R33 + R11 R 42 R23 R34 − R11 R32 R23 R 44 + R31 R22 R 43 R14 − R31 R22 R13 R 44 + R 41 R12 R24 R33 − R 41 R22 R14 R33 + R 41 R22 R13 R34 [13] M. Eissa, U.H. Hegazy, Y.A. Amer, Dynamic behavior of an AMB supported rotor subject to harmonic excitation, Appl. Math. Model. 32 (2008) 1370–1380. [14] M. Kamel, H.S. Bauomy, Nonlinear study of a rotor-AMB system under simultaneous primary-internal resonance, Appl. Math. Model. 34 (2010) 2763–2777. [15] M. Kamel, H.S. Bauomy, Nonlinear behavior of a rotor-AMB system under multiparametric excitations, Meccanica 45 (2010) 7–22. [16] N.A. Saeed, M. Eissa, W.A. El-Ganini, Nonlinear oscillations of rotor active magnetic bearings system, Nonlinear Dyn. 74 (2013) 1–20. [17] M. Eissa, N.A. Saeed, W.A. El-Ganini, Saturation-based active controller for vibration suppression of a four-degree-of-freedom rotor–AMB system, Nonlinear Dyn. 76 (2014) 743–764. [18] N. Vlajic, X. Liu, H. Karki, B. Balachandran, Torsional oscillations of a rotor with continuous stator contact, Int. J. Mech. Sci. 83 (2014) 65–75. [19] Y. Ishida, T. Inoue, Vibration Suppression of nonlinear rotor systems using a dynamic damper, J. Vib. Control 13 (8) (2007) 1127–1143. [20] H. Yabuno, T. Kashimura, T. Inoue, Y. Ishida, Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor, Nonlinear Dyn. 66 (2011) 377–387. [21] A. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley, New York, 1995. [22] A. Nayfeh, Perturbation Methods, Wiley, New York, 1973. [23] A. Nayfeh, Resolving controversies in the application of the method of multiple scales and the generalized method of averaging, Nonlinear Dyn. 40 (2005) 61–102. [24] E.L. Rees, Graphical discussion of the roots of a quartic equation, Am. Math. Mon. 29 (2) (1922) 51–55. [25] D. Lazard, Quantifier elimination: optimal solution for two classical examples, J. Symb. Comput. 5 (1988) 261–266. [26] T. Yamamoto, Y. Ishida, Linear and Nonlinear Rotor Dynamics, Wiley-VCH, Weinheim, Germany, 2012. [27] H. Bleuler, M. Cole, P. Keogh, R. Larsonneur, E. Maslen, R. Nordmann, Y. Okada, G. Schweitzer, A. Traxler, Magnetic Bearings Theory, Design, and Application to Rotating Machinery, Springer, Berlin Heidelberg, 2009.

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