# Rub-Impact Force Induces Periodic, Quasiperiodic, and Chaotic Motions of a Controlled Asymmetric Rotor System

Authors: N. A. Saeed; Emad Mahrous Awwad; Ali Maarouf; Hassan M. H. Farh; Fahd A. Alturki; Jan Awrejcewicz
Journal: Shock and Vibration
 (2021)

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## Abstract

This article aims to explore the oscillatory characteristics of a controlled asymmetric rotor system when subjected to rub and impact forces between the rotor and stator. Four electromagnetic poles are used to control the whirling motion of the rotor system through a linear proportional-derivative control law. The equations of motion that govern the whole system dynamics are derived including the rub and impact forces. The derived mathematical model is analyzed in two basic steps. Firstly, the obtained model is treated as a weakly nonlinear system using perturbation analysis to obtain the slow-flow modulating equations when neglecting the rub and impact forces. Depending on the obtained slow-flow equations, different response curves are plotted to explore the system’s periodic vibrations and determine the conditions at which the system can exhibit rub and impact force. Secondly, the whole system model including the rub and impact forces is investigated by using the bifurcation diagrams, Poincare map, frequency spectrums, and temporal oscillations. The obtained results revealed that the applied control law could mitigate the system whirling oscillations and prevent the rub and impact forces if the control gains are tuned properly. However, the system can perform period-n, quasiperiodic, or chaotic motion depending on the shaft spinning speed if the controller fails to eliminate the contact between the rotor and stator.


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## Body

## 1. Introduction

The rotating machinery has many engineering applications and is considered as the backbone of a huge number of industries. Therefore, ensuring safe operation of such machines without any catastrophic failure under different operational conditions is the vital role of the scientists and engineers. One of the main reasons that causes failure of rotating machines is nonlinear vibrations. There are many sources for the rotating shaft oscillations such as the shaft’s eccentricity, the improper coupling in the case of multirotor systems, the development of the shaft cracks due to the working conditions, the existence of wears, the occurrences of the rub-impact force between the rotating parts and housing, the shaft’s asymmetry, and so on. Accordingly, vibration analysis and control of the rotors are the main research subjects of many researchers for a long time. Iwatsubo et al. [1] investigated free and forced vibrations of an asymmetric rotor when supported by asymmetric bearings. The authors applied the asymptotic analysis to find an approximate solution for the system mathematical model, where the obtained results showed that the asymmetry of shaft stiffness coefficients and/or the oil whip of the oil film bearings can destabilize the system motion. Park [2] and Hsieh et al. [3] explored the oscillatory behaviours of the two-pole rotor system. They reported that the system may oscillate with high vibration amplitudes when the shaft angular speed is close to twice the shaft critical speed. Ishida and Liu [4] utilized a discontinuous-spring characteristic to stabilize the unstable motion of an asymmetric shaft system, asymmetric rotor system, and rotor system partially filled with liquid. Shahgholi and Khadem [5] investigated the nonlinear vibrations of an asymmetric rotating shaft system in both the primary and parametric resonance cases. The authors considered that both the moment of inertial and bending stiffness are the main sources of the shaft asymmetry. Han and Chu [6, 7] and Saeed et al. [8–10] studied the dynamical behaviours of an asymmetric rotor system with a transverse crack. The main conclusion is that the increase of the crack size can stabilize the system’s unstable motion. Xiang et al. [11] studied the parametric instability of an asymmetric rotor system having a transverse crack when subjected to both rub-impact and oil-film forces. They investigated the system model using the frequency spectrum, Poincare map, and bifurcation diagram. It is concluded that the crack depth and the stator stiffness have an important role in both the system instability and oscillation amplitudes. Przybylowicz et al. [12] studied the influence of the kinematic excitation on the asymmetric rotor numerically. They showed that the proper design of both the system stiffness coefficient and damping parameter can eliminate the system’s chaotic motion. Saeed [13] investigated the nonlinear dynamics of an asymmetric rotor supported vertically. The obtained results showed that the asymmetry of the shaft stiffness coefficients can induce a backward whirling motion. In addition, the authors reported that the asymmetric system may have four stable periodic solutions at the same angular speed. Saeed [14] explored the dynamical behaviours of an asymmetric horizontally suspended rotor system. The author obtained an analytical solution for the studied system applying perturbation analysis. He showed that the asymmetry of the linear stiffness coefficient can suppress the system’s backward whirling motion. Felix et al. [15] performed numerical simulations for the oscillatory behaviours of both the one and two-disk rotor system when excited by a nonideal motor.On the other hand, many researchers introduced different passive and active control techniques to suppress or mitigate the undesired vibrations of the rotating machines. For example, Tehrani et al. [16, 17] employed both the nonlinear energy sinks and vibration absorber as passive control methods to mitigate the vibrations of the rotating machine. Ishida and Inoue [18] applied a nonlinear active vibration absorber to control the lateral vibrations of a vertically suspended rotating shaft. Saeed et al. [19–22] applied different active control strategies utilizing the active magnetic bearings as an actuator. Saeed and Kamel [19] employed a nonlinear combination of both the position and velocity controllers to control the vibrations of the nonlinear Jeffcott rotor model supported horizontally via four electromagnetic poles. Saeed et al. [20] proposed a combination of both the nonlinear and linear proportional-derivative controllers to mitigate the nonlinear vibrations of the asymmetric rotor system that is supported vertically. Saeed et al. [21, 22] introduced six electromagnetic poles to control the rotating disk vibrations utilizing PD controller. They explored two control configurations that are the Cartesian and radial control schemes. It is reported that the Cartesian control scheme has high efficiency than the radial control technique in mitigating the system vibrations. However, they concluded that the radial control strategy has high robustness against the system instability. Piccirillo et al. [23] investigated the impact force elimination and chaos control utilizing magnetorheological fluid. The authors applied the damped passive control technique to suppress the system’s chaotic oscillations, while a fuzzy logic controller is proposed to control the system’s lateral vibrations. Singh et al. [24] explored the unique nonlinear features of the permanent magnet synchronous motor (PMSM) system. The authors reported that the PMSM system has many nonlinear properties such as multistability occurrence, the existence of the hidden attractors, the existence of two stable node-foci, and one saddle index-1. Recently, Saeed et al. [25] investigated the rub-impact force of an asymmetric nonlinear rotor system supported vertically. The authors integrated both linear position-velocity and cubic-velocity controllers simultaneously to control the system vibrations. The rub and impact forces between the pole legs and rotating shaft have been included in the derived model. The obtained results illustrated that the rub and impact forces can induce unbounded oscillation for the uncontrolled system. In addition, it is found that the proper tuning of the control parameters can prevent the rub and impact forces between the rotor and the legs of the poles.Based on the above discussion, the dynamical behaviours of the asymmetric horizontally suspended rotor system with rub-impact force have not been investigated before as summarized in Table1. Therefore, this work is dedicated to investigating the dynamical characteristics of an asymmetric nonlinear rotor system that is supported horizontally. The studied system is governed by two coupled second-order nonlinear differential equations having both cubic and quadratic nonlinearities which is excited by external and multiparametric forces. The linear proportional-derivative control algorithm is applied via four electromagnetic poles to control the nonlinear vibrations of the considered system. The rub and impact forces between stator and rotor are considered. The equations of motion of the controlled system are derived as a discontinuous nonlinear dynamical system. Due to the discontinuity of the derived model, the equations of motion have been analyzed in two stages. Firstly, the system is investigated analytically applying asymptotic methods when the rub and impact forces are neglected. The different response curves (i.e., spinning speed and eccentricity response curves) are obtained to explore the system’s periodic oscillations and to determine the conditions at which the system may be subject to rub and/or impact forces. Secondly, utilizing the orbit plot, frequency spectrum, Poincare map, and bifurcation diagram, the whole system model is investigated including the rub and impact forces. The obtained results revealed that the studied system exhibits periodic oscillations only as long as the rub and impact forces do not occur. However, the occurrence of rub and impact forces is resulting in period-n, quasiperiodic, or chaotic oscillation for the rotor system. Moreover, the analytical and numerical simulations approved that the proposed control strategy can eliminate the nonlinear behaviours of the studied system as well as prevent the rub and impact forces when the control gains are designed properly.Table 1
Categorization of available literature on the asymmetric rotating shaft.
CategorySubclassificationReferencesDynamical analysis of the rotating shafts(i) Symmetric rotating shaft system(i) References [8–10, 15, 26, 27](ii) Asymmetric rotating shaft system(ii) References [1–7, 12–14]Vibration control of the rotating shafts(i) Symmetric rotating shaft with passive control(i) References [16–18](ii) Symmetric rotating shaft with active control(ii) References [19–22](iii) Asymmetric rotating shaft system with active control(iii) References [20, 28]Active vibration control of asymmetric rotating shafts including the rub-impact forces(i) Nonlinear vertically supported rotor(i) Reference [25](ii) Nonlinear horizontally supported rotor(ii) This work


## 2. Mathematical Formulation

The dynamical model of the horizontally suspended rotor system shown in Figure1(a) is given as follows [26–29]:(1a)mu¨1+c1u˙1+FU1=medω2cosωt+γ,(1b)mu¨2+c2u˙2+FU2=medω2sinωt+γ−mg,where m represents the disk mass, c1 and c2 denote the bearings damping in U1 and U2 directions, FU1 and FU2 are the restoring forces of the shaft in U1 and U2 directions, ed is the displacement between the geometric centre Gu1,u2 of the rotor and its centroid is eu1e,u2e as in Figure 1(b), ω represents the rotor angular speed, mg is the disk weight, and γ is the angle between oς1⟶ axis and the eccentricity direction Ge⟶. It is considered that the rotor system performs a planer vibration in the stationary U1−U2 frame, while ς1−ς2 is a rotational frame that rotates with angular speed ω. When the rotating shaft is displaced by small radial displacement s=u12+u22 as shown in Figure 1(b), an exerted nonlinear restoring force is developed in U1 and U2 directions that can be expressed as follows [26–29]:(2a)Fu1=k1s+k2s3cosψ=k1u1+k2u12+u22u1,(2b)Fu2=k1s+k2s3sinψ=k1u2+k2u12+u22u2,where k1 is linear stiffness coefficient and k2 is the nonlinear stiffness coefficient.Figure 1
Horizontally supported asymmetric nonlinear rotor system: (a) Jeffcott rotor model, (b) symmetric disk, (c) asymmetric disk, (d) nondeflected rotating disk within the stator, and (e) deflected rotating disk within the stator.
(a)(b)(c)(d)(e)Substituting equations (2a) and (2b) into equations (1a) and (1b) yields(3a)mu¨1+c1u˙1+k1u1+k2u12+u22u1=medω2cosωt+γ,(3b)mu¨2+c2u˙2+k1u2+k2u12+u22u2=medω2sinωt+γ−mg.Figure1(a) shows that the disk weight mg causes a small static deflection us for the rotating shaft in U2 direction. Accordingly, at static equilibrium, we have ω=u¨1=u˙1=u1=u¨2=u˙2=0.0 and u2=us. Substituting these conditions into equations (3a) and (3b) yields(4)k1us+k2us3=−mg.To eliminate the constantmg from equations (3a) and (3b), a new coordinate system V1−V2 is introduced such that U1=V1 and U2=us+V2. So, we have u¨1=v¨1,u˙1=v˙1,u1=v1,u¨2=v¨2,u˙2=v˙2,u2=us+v2. Inserting the new variables into equations (3a) and (3b) considering equation (4) yields the following nonlinear dynamical system:(5a)mv¨1+c1v˙1+k1+k2us2v1+2k2usv1v2+k2v12+v22v1=medω2cosωt+γ,(5b)mv¨2+c2v˙2+k1+3k2us2v2+k2usv12+3v22+k2v12+v22v2=medω2sinωt+γ.For the asymmetric rotating machinery, the restoring force inς1 and ς2 directions may be not symmetric due to the winding slots as in the case of 2-pole rotor generators [29] (see Figure 1(c)). Therefore, the equations of motions for such asymmetric machines can be obtained via modifying equations (5a) and (5b) as follows:(6a)mv¨1+c1v˙1+k1+k2us2v1+2k2usv1v2+k2v12+v22v1+ΔFV1=medω2cosωt+γ,(6b)mv¨2+c2v˙2+k1+3k2us2v2+k2usv12+3v22+k2v12+v22v2+ΔFV2=medω2sinωt+γ,where ΔFV1 and ΔFV2 represent the asymmetry of the restoring forces in V1 and V2 directions. It is considered that the difference between the restoring force in ς1 direction and in ς2 is ΔF, where ΔF is given as follows [28]:(7)ΔF=Δk1ς1+Δk2ς13=Δk1v1cosωt+v2sinωt+Δk2v1cosωt+v2sinωt3,where Δk1ς1 and Δk2ς13 are the linear and nonlinear extra restoring forces in ς1 direction, respectively. By resolving ΔF in V1 and V2 directions, we can express their components ΔFV1 and ΔFV2 as follows:(8a)ΔFV1=ΔFcosωt=12Δk1v1+38Δk2v13+38Δk2v1v22+12Δk1v1cos2ωt+12Δk1v2sin2ωt+12Δk2v13cos2ωt+34Δk2v12v2sin2ωt+14Δk2v23sin2ωt+18Δk2v13cos4ωt−18Δk2v23sin4ωt+38Δk2v12v2sin4ωt−38Δk2v1v22cos4ωt,(8b)ΔFV2=ΔFsinωt=12Δk1v2+38Δk2v23+38Δk2v12v2−12Δk1v2cos2ωt+12Δk1v1sin2ωt−12Δk2v23cos2ωt+34Δk2v1v22sin2ωt+14Δk2v13sin2ωt+18Δk2v23cos4ωt+18Δk2v13sin4ωt−38Δk2v1v22sin4ωt−38Δk2v12v2cos4ωt.Inserting equations (8a) and (8b) into equations (6a) and (6b), we have(9a)mv¨1+c1v˙1+k1+12Δk1+k2us2v1+2k2usv1v2+k2+38Δk2v12+v22v1=medω2cosωt+γ−12Δk1v1cos2ωt−12Δk1v2sin2ωt−12Δk2v13cos2ωt−34Δk2v12v2sin2ωt−14Δk2v23sin2ωt−18Δk2v13cos4ωt+18Δk2v23sin4ωt−38Δk2v12v2sin4ωt+38Δk2v1v22cos4ωt+F11+F12,(9b)mv¨2+c2v˙2+k1+12Δk1+3k2us2v2+k2usv12+3v22+k2+38Δk2v12+v22v2=medω2sinωt+γ+12Δk1v2cos2ωt−12Δk1v1sin2ωt+12Δk2v23cos2ωt−34Δk2v1v22sin2ωt−14Δk2v13sin2ωt−18Δk2v23cos4ωt−18Δk2v13sin4ωt+38Δk2v1v22sin4ωt+38Δk2v12v2cos4ωt+F21+F22.the above nonlinear dynamical system governs the lateral vibrationsV1 and V2of the asymmetric horizontally supported rotating shaft. The forces F11 and F21have been added to equations (9a) and (9b) as control forces generated via electromagnetic poles as shown in Figure 1(d) to control the system vibrations in V1 and V2 directions, while the forces F12 and F22 are induced in V1 and V2 directions due to the rub and impact forces (FN and FT) that occur between the rotor and stator when the applied control forces (F11 and F21) fail to prevent the contact of the rotating shaft and the stator as illustrated in Figure 1(e). Depending on the proposed electromagnetic poles (see Figures 1(d) and 1(e)), the forces F11 and F21 are given as follows [30]:(10a)F11=Fv1,iv1=ξI0+iv12h0+v12−I0−iv12h0−v12,(10b)F21=Fv2,iv2=ξI0+iv22h0+v22−I0−iv22h0−v22,where pole 1 and pole 3 apply force F11, while pole 2 and pole 4 apply force F21, ξ denotes the magnetic force constant, iv1 and iv2 are the control currents, I0 is the magnetizing current, and h0 denotes the clearance between the stator and the rotor (i.e., h0 is the static air gap as shown in Figure 1(d)). Expanding equations (10a) and (10b) in terms of iv1,iv2,v1, and v2 up to the third-order approximation using Maclaurin series yields(11)F11=4ξh05−h03I0iv1+h02I02v1+2I02v13−3h0I0iv1v12+h02iv12v1,(12)F21=4ξh05−h03I0iv2+h02I02v2+2I02v23−3h0I0iv2v22+h02iv22v2,where iv1 and iv2 are selected to be a function of both the disk displacements (v1 and v2) and disk velocities (v˙1 and v˙2) as follows:(13)iv1=kpv1+kdv˙1,iv2=kpv2+kdv˙2,where kp and kd are the proportional and derivative control gains, respectively. The considered system may be subject to rub and impact forces as illustrated in Figure 1(e). The rub and/or impact forces occur when the radial displacement (rt=v1t2+v2t2) exceeds the air-gap size (i.e., when t≥h0). The impact force FN and rub force FT develop at the stator-rotor interface as depicted in Figure 1(e). Accordingly, the rub can be expressed as follows [25]:(14)FN=kcr−h0Hr−h0,FT=μdFN=μdkcr−h0Hr−h0,where kc denotes the impact stiffness coefficient, μd represents the dynamic frictional coefficient between the rotating disk and the pole leg, and H denotes the Heaviside function. Resolving the forces FN and FT into their components in V1 and V2 directions and considering cosωt=v1/r,sinωt=v2/r, we get(15)F21=FTsinωt−FNcosωt=kcrr−h0μdv2−v1Hr−h0,(16)F22=−FTcosωt−FNsinωt=−kcrr−h0μdv1+v2Hr−h0.Inserting equations (11), (12), (15), and (16) into equations (9a) and (9b) taking into account equation (13) with introducing the dimensionless variables τ=ωnt,z1=v1/h0,z2=v2/h0,R=r/h0=z12+z22, we get the following dimensionless equations:(17a)z¨1+μ1z˙1+ω12z1+2ηz1z2+βz12+z22z1=fΩ2cosΩτ+γ−12δ1+δ2z12z1cos2Ωτ−12δ1+12δ23z12+z22z2sin2Ωτ+18δ23z1z22−z13cos4Ωτ+18δ2z23−3z12z2sin4Ωτ−α1z1+α2z˙1+α3z13+α4z12z˙1+α5z1z˙12+δ3RR−1μdz2−z1HR−1,(17b)z¨2+μ2z˙2+ω22z2+ηz12+3z22+βz12+z22z2=fΩ2sinΩτ+γ+12δ1+δ2z22z2cos2Ωτ−12δ1+12δ23z22+z12z1sin2Ωτ+18δ23z12z2−z23cos4Ωτ+18δ23z1z22−z13sin4Ωτ−α1z2+α2z˙2+α3z23+α4z22z˙2+α5z2z˙22−δ3RR−1μdz1+z2HR−1,where μ1=c1/k1m,μ2=c2/k1m,ωn2=k1/m=ξI02/mh0,Ω=ω/ωn,f=ed/h0, λ=us/h0,ρ=h02k2/k1,η=λρ,δ1=Δk1/k1,δ2=h02Δk2/k1,δ3=kc/k1,ω1=1+λ2ρ+1/2δ1, ω2=1+3λ2ρ+1/2δ1,β=ρ+3/8δ2,p=h0/I0kp,d=h0ωn/I0kd, α1=p−1,α2=d,α3=−2+3p−p2,α4=3d−2pd,α5=−d2.Equations (17a) and (17b) govern the whole system dynamics. As the studied system is governed by a discontinuous dynamical model, equations (17a) and (17b) are analyzed in two steps. In the first step, the rub-impact force is neglected via setting δ3=kc/k1=0.0. The corresponding continuous dynamical system is investigated analytically using the perturbation method to obtain the amplitude-phase modulation equations as given in Section 3. Based on the obtained amplitude-phase modulation equations (when δ3=0.0), the conditions at which the whole system can exhibit rub-impact force can be predicted via plotting the different response curves. In the second step of the analysis, the dynamical behaviours of the whole system (i.e., equations (17a) and (17b) when δ3≠0.0) are simulated using the bifurcation diagrams, Poincare map, frequency spectrum, and whirling orbits.


## 3. Amplitude-Phase Modulating Equations Whenδ3=0.0

Within this section, a second-order approximate solution to equations (17a) and (17b) when δ3=0.0can be sought using the multiple scales perturbation method as follows [31]:(18a)z1τ=z11T0,T1,T2+εz12T0,T1,T2+ε2z13T0,T1,T2+Oε3,(18b)z2τ=z21T0,T1,T2+εz22T0,T1,T2+ε2z23T0,T1,T2+Oε3,where ε is a book-keeping perturbation parameter and Tj=εjτ,j=0,1,2. In terms of T0,T1, and T2, the derivatives d/dτ can be expressed as(19)ddτ=D0+εD1+ε2D2,d2dτ2=D02+2εD0D1+ε2D12+2D2D0,Dj=∂∂Tj,j=0,1,2.The system parameters are scaled such that(20)μ1=ε2μ^1,μ2=ε2μ^2,β=ε2β^,η=εη^,f=εf^,δ1=ε2δ^1,δ2=ε2δ^2,αn=ε2α^n,n=1,2,…,5.Substituting equations (18a) to (20) into equations (17a) and (17b) and then equating coefficients of the same power of ε, we getOε0:(21a)D02+ω12z11=0,(21b)D02+ω22z21=0.Oε1:(22a)D02+ω12z12=−2D0D1z11−2η^z21z11+f^Ω2cosΩT0+γ,(22b)D02+ω22z22=−2D0D1z21−η^z112−3η^z212+f^Ω2sinΩT0+γ.Oε2:(23a)D02+ω12z13=−2D0D1z12−D12+2D2D0z11−μ^1D0z11−2η^z21z12−2η^z22z11−β^z212z11−β^z113−12δ^1z11cos2ΩT0−12δ^1z21sin2ΩT0−12δ^2z113cos2ΩT0−14δ^23z112z21+z213sin2ΩT0+18δ^23z11z212−z113cos4ΩT0+18δ^2z213−3z112z21sin4ΩT0−α^1z11−α^2D0z11−α^3z113−α^4z112D0z11−α^5z11D0z112,(23b)D02+ω22z23=−2D0D1z22−D12+2D2D0z21−μ^2D0z21−2η^z11z12−6η^z21z22−β^z21z112−β^z213−12δ^1z11sin2ΩT0+12δ^1z21cos2ΩT0+12δ^2z213cos2ΩT0−14δ^23z11z212+z113sin2ΩT0+18δ^23z112z21−z213cos4ΩT0+18δ^23z11z212−z113sin4ΩT0−α^1z21−α^2D0z21−α^3z213−α^4z212D0z21−α^5z21D0z212.The solution of equations (21a) and (21b) can be expressed as(24a)z11T0,T1,T2=χ1T1,T2eiω1T0+χ¯1T1,T2e−iω1T0,(24b)z21T0,T1,T2=χ2T1,T2eiω2T0+χ¯2T1,T2e−iω2T0,where χ1T1,T2 and χ2T1,T2 are unknown functions of the time scales T1 and T2. Inserting equations (24a) and (24b) into equations (22a) and (22b), we have(25a)D02+ω12z12=−2iω1D1χ1eiω1T0−2η^χ1χ2eiω1+ω2T0−2η^χ1χ¯2eiω1−ω2T0+f^Ω22eiΩT0+γ+cc,(25b)D02+ω22z22=−2iω2D1χ2eiω2T0−η^χ12e2iω1T0+χ1χ¯1−3η^χ22e2iω2T0+χ2χ¯2−if^Ω22eiΩT0+γ+cc,where cc denotes the complex conjugate term. The primary resonance and internal resonance cases (i.e., Ω=ω1=ω2) are investigated simultaneously in this work, where the detuning parameters σ and σ1 are utilized to describe the closeness of Ωandω2 to ω1 as(26)Ω=ω1+σ=ω1+εσ^,ω2=ω1+σ1=ω1+εσ^1.Substituting equation (26) into equations (25a) and (25b) yields(27a)D02+ω12z12=2−2iω1D1χ1+f^Ω22eiσ^T1+γeiω1T0−2η^χ1χ2eiω1+ω2T0−2η^χ1χ¯2eiω1−ω2T0+cc,(27b)D02+ω22z22=−2iω2D1χ2+f^Ω22ieiσ^−σ^1T1+γeiω2T0−η^χ12e2iω1T0+χ1χ¯1−3η^χ22e2iω2T0+χ2χ¯2+cc.The solvability conditions of equations (27a) and (27b) are(28)2iω1D1χ1=f^Ω22eiσ^T1+γ,2iω2D1χ2=−if^Ω22eiσ^−σ^1T1+γ.According to equation (28), the solution of equations (27a) and (27b) can be expressed as follows:(29a)z12T0,T1,T2=2η^ω22ω1+ω2χ1χ2eiω1+ω2T0+2η^ω2ω2−2ω1χ1χ¯2eiω1−ω2T0+cc,(29b)z22T0,T1,T2=η^4ω12−ω22χ12e2iω1T0+η^ω22χ22e2iω2T0−η^χ1χ¯1ω22−3δ^χ2χ¯2ω22+cc.Substituting equations (24a), (24b), (29a), and (29b) into the right-hand side of equations (23a) and (23b), one finds the following solvability conditions of equations (23a) and (23b):(30a)2iω1D2χ1=−D12χ1−iω1μ^1+α^1+iω1α^2χ1+4η^2ω22−2η^24ω12−ω22−3β^−3α^3−iω1α^4−ω12α^5χ12χ¯1+12η^2ω22−4η^2ω2ω2−2ω1−4η^2ω22ω1+ω2−2β^χ1χ2χ¯2−2η^2ω22+4η^2ω2ω2−2ω1+β^χ¯1χ22e2iσ^1T1−14δ^1χ¯1e2iσ^T1+i4δ^1χ¯2ei2σ^−σ^1T1−34δ^2χ1χ¯12e2iσ^T1−14δ^2χ13e−2iσ^T1+3i8δ^2χ¯12χ2ei2σ^−σ^1T1−3i8δ^2χ12χ2ei−2σ^−σ^1T1+3i4δ^2χ1χ¯1χ¯2ei2σ^−σ^1T1+3i8δ^2χ2χ¯22ei2σ^−σ^1T1−i8δ^2χ23ei−2σ^−3σ^1T1+316δ^2χ¯1χ¯22ei4σ^−2σ^1T1−116δ^2χ¯13e4iσ^T1−i16δ^2χ¯23ei4σ^−3σ^1T1+3i16δ^2χ¯12χ¯2ei4σ^−σ^1T1,(30b)2iω2D2χ2=−D12χ2−iω2μ^2+α^1+iω2α^2χ2+30η^2ω22−3β^−3α^3−iω2α^4−ω22α^5χ22χ¯2−4η^2ω22ω1+ω2+4η^2ω2ω2−2ω1−12η^2ω22+2β^χ1χ¯1χ2−4η^2ω2ω2−2ω1+6η^24ω12−ω22+β^χ12χ¯2e−2iσ^1T1+14δ^1χ¯2ei2σ^−2σ^1T1+i4δ^1χ¯1ei2σ^−2σ^1T1+34δ^2χ2χ¯22ei2σ^−2σ^1T1+14δ^2χ23ei2σ^1−2σ^T1+3i8δ^2χ1χ¯22ei2σ^−3σ^1T1−3i8δ^2χ1χ22eiσ^1−2σ^T1+3i4δ^2χ¯1χ2χ¯2ei2σ^−σ^1T1+3i8δ^2χ1χ¯12ei2σ^−σ^1T1−i8δ^2χ13ei−2σ^−σ^1T1+316δ^2χ¯12χ¯2ei4σ^−2σ^1T1−116δ^2χ¯23ei4σ^−4σ^1T1−3i16δ^2χ¯1χ¯22ei4σ^−3σ^1T1+i16δ^2χ¯13ei4σ^−σ^1T1.Utilizing the multiple scales reconstitution method [32], we can combine the solvability conditions of equations (28), (30a), and (30b) as(31a)2iω1ddtχ1=εf^Ω22eiσ^T1+γ−ε2σ^f^Ω24ω1eiσ^T1+γ−ε2iω1μ^1+α^1+iω1α^2χ1+ε24η^2ω22−2η^24ω12−ω22−3β^−3α^3−iω1α^4−ω12α^5χ12χ¯1+ε212η^2ω22−4η^2ω2ω2−2ω1−4η^2ω22ω1+ω2−2β^χ1χ2χ¯2−ε22η^2ω22+4η^2ω2ω2−2ω1+β^χ¯1χ22e2iσ^1T1−14ε2δ^1χ¯1e2iσ^T1+i4ε2δ^1χ¯2ei2σ^−σ^1T1−34ε2δ^2χ1χ¯12e2iσ^T1−14ε2δ^2χ13e−2iσ^T1+3i8ε2δ^2χ¯12χ2ei2σ^+σ^1T1−3i8ε2δ^2χ12χ2ei−2σ^+σ^1T1+3i4ε2δ^2χ1χ¯1χ¯2ei2σ^−σ^1T1+3i8ε2δ^2χ2χ¯22ei2σ^−σ^1T1−i8ε2δ^2χ23ei−2σ^+3σ^1T1+316ε2δ^2χ¯1χ¯22ei4σ^−2σ^1T1−116ε2δ^2χ¯13e4iσ^T1−i16ε2δ^2χ¯23ei4σ^−3σ^1T1+3i16ε2δ^2χ¯12χ¯2ei4σ^−σ^1T1,(31b)2iω2ddtχ2=−iεf^Ω22eiσ^−σ^1T1+γ−iε2σ^1−σ^f^Ω24ω2eiσ^−σ^1T1+γ−ε2iω2μ^2+α^1+iω2α^2χ2+ε230η^2ω22−3β^−3α^3−iω2α^4−ω22α^5χ22A¯2−ε24η^2ω22ω1+ω2+4η^2ω2ω2−2ω1−12η^2ω22+2β^χ1χ¯1χ2−ε24η^2ω2ω2−2ω1+6η^24ω12−ω22+β^χ12χ¯2e−2iσ^1T1+14ε2δ^1χ¯2ei2σ^−2σ^1T1+i4ε2δ^1χ¯1ei2σ^−σ^1T1+34ε2δ^2χ2χ¯22ei2σ^−2σ^1T1+14ε2δ^2χ23ei2σ^1−2σ^T1+3i8ε2δ^2χ1χ¯22ei2σ^−3σ^1T1−3i8ε2δ^2χ1χ22eiσ^1−2σ^T1+3i4ε2δ^2χ¯1χ2χ¯2ei2σ^−σ^1T1+3i8ε2δ^2χ1χ¯12ei2σ^−σ^1T1−i8ε2δ^2χ13ei−2σ^−σ^1T1+316ε2δ^2χ¯12χ¯2ei4σ^−2σ^1T1−116ε2δ^2χ¯23ei4σ^−4σ^1T1−3i16ε2δ^2χ¯1χ¯22ei4σ^−3σ^1T1+i16ε2δ^2χ¯13ei4σ^−σ^1T1.To analyze equations (31a) and (31b), the functions χ1T1,T2 and χ2T1,T2 can be expressed in polar form as follows:(32)χj=12ajτeiθjτ⇒ddtχj=12a˙jτeiθjτ+i2ajτθ˙jτeiθjτ,j=1,2,where a1 and a2 denote the rotating disk vibration amplitudes in V1 and V2 directions, respectively, and θ1 and θ2 are the motion phase angles. Substituting equation (32) into equations (31a) and (31b), one can derive the following amplitude-phase modulation equations:(33a)da1dτ=1−σ2ω1fΩ22ω1sinϕ1+γ−12μ1+α2a1−18α4a13−18ω12η2ω22+4η2ω2ω2−2ω1+βa1a22sin2ϕ1−2ϕ2−18ω1δ1a1sin2ϕ1+18ω1δ1a2cosϕ1+ϕ2−332ω1δ2a13sin2ϕ1+132ω1δ2a13sin2ϕ1+364ω1δ2a12a2cos3ϕ1−ϕ2−364ω1δ2a12a2cosϕ1+ϕ2+332ω1δ2a12a2cosϕ1+ϕ2+364ω1δ2a23cosϕ1+ϕ2−164ω1δ2a23cosϕ1−3ϕ2+3128ω1δ2a1a22sin2ϕ1+2ϕ2−1128ω1δ2a13sin4ϕ1−1128ω1δ2a23cosϕ1+3ϕ2+3128ω1δ2a12a2cos3ϕ1+ϕ2,(33b)da2dτ=σ2ω2−σ12ω2−1fΩ22ω2cosϕ2+γ−12μ2+α2a2−18α4a23−18ω24η2ω2ω2−2ω1+6η24ω12−ω22+βa12a2sin−2ϕ1+2ϕ2+18ω2δ1a2sin2ϕ2+18ω2δ1a1cosϕ1+ϕ2+332ω2δ2a23sin2ϕ2−132ω2δ2a23sin2ϕ2+364ω2δ2a1a22cos3ϕ2−ϕ1−364ω2δ2a1a22cosϕ1+ϕ2+332ω2δ2a1a22cosϕ1+ϕ2+364ω2δ2a13cosϕ1+ϕ2−164ω2δ2a13cosϕ2−3ϕ1+3128ω2δ2a12a2sin2ϕ1+2ϕ2−1128ω2δ2a23sin4ϕ2−3128ω2δ2a1a22cosϕ1+3ϕ2+1128ω2δ2a13cos3ϕ1+ϕ2,(33c)dϕ1dτ=σ+1−σ2ω1fΩ22ω1a1cosϕ1+γ−α12ω1+18ω14η2ω22−2η24ω12−ω22−3β−3α3−ω12α5a12+18ω112η2ω22−4η2ω2ω2−2ω1−4η2ω22ω1+ω2−2βa22−18ω12η2ω22+4η2ω2ω2−2ω1+βa22cos2ϕ1−2ϕ2−18ω1δ1cos2ϕ1−18ω1a1δ1a2sinϕ1+ϕ2−332ω1δ2a12cos2ϕ1−132ω1δ2a12cos2ϕ1−364ω1δ2a1a2sin3ϕ1−ϕ2−364ω1δ2a1a2sinϕ1+ϕ2−332ω1δ2a1a2sinϕ1+ϕ2−364ω1a1δ2a23sinϕ1+ϕ2+164ω1a1δ2a23sinϕ1−3ϕ2+3128ω1δ2a22cos2ϕ1+2ϕ2−1128ω1δ2a12cos4ϕ1+1128ω1a1δ2a23sinϕ1+3ϕ2−3128ω1δ2a1a2sin3ϕ1+ϕ2,(33d)dϕ2dτ=σ−σ1−σ2ω2−σ12ω2−1fΩ22ω2a2sinϕ2+γ−α12ω2+18ω230η2ω22−3β−3α3−ω22α5a22−18ω24η2ω22ω1+ω2+4η2ω2ω2−2ω1−12η2ω22+2βa12−18ω24η2ω2ω2−2ω1+6η24ω12−ω22+βa12cos−2ϕ1+2ϕ2+18ω2δ1cos2ϕ2−18ω2a2δ1a1sinϕ1+ϕ2+332ω2δ2a22cos2ϕ2+132ω2δ2a22cos2ϕ2−364ω2δ2a1a2sin3ϕ2−ϕ1−364ω2δ2a1a2sinϕ1+ϕ2−332ω2δ2a1a2sinϕ1+ϕ2−364ω2a2δ2a13sinϕ1+ϕ2+164ω2a2δ2a13sinϕ2−3ϕ1+3128ω2δ2a12cos2ϕ1+2ϕ2−1128ω2δ2a2cos4ϕ2+3128ω2δ2a1a2sinϕ1+3ϕ2−1128ω2a2δ2a13sin3ϕ2−ϕ1,where ϕ1=στ−θ1 and ϕ2=σ−σ1τ−θ2. Now, by inserting equation (32) into equations (24a) and (24b) and then inserting the obtained equations into equations (18a) and (18b), one can derive a periodic solution for the system equations of motion (i.e., equations (17a) and (17b) when δ3=0.0) as follows:(34a)z1τ=a1τcosΩτ−ϕ1τ,(34b)z2τ=a2τcosΩτ−ϕ2τ.The oscillation amplitudes (a1τ & a2τ) and the phase angles (ϕ1τ&ϕ2τ) are governed by the autonomous dynamical system given by equations (33a)–(33d). Setting da1/dτ=da2/dτ=dϕ1/dτ=dϕ2/dτ=0 in equations (33a)–(33d), we have(35a)1−σ2ω1fΩ22ω1sinϕ1+γ−12μ1+α2a1−18α4a13−18ω12η2ω22+4η2ω2ω2−2ω1+β×a1a22sin2ϕ1−2ϕ2−18ω1δ1a1sin2ϕ1+18ω1δ1a2cosϕ1+ϕ2−332ω1δ2a13sin2ϕ1+132ω1δ2a13sin2ϕ1+364ω1δ2a12a2cos3ϕ1−ϕ2−364ω1δ2a12a2cosϕ1+ϕ2+332ω1δ2a12a2cosϕ1+ϕ2+364ω1δ2a23cosϕ1+ϕ2−164ω1δ2a23cosϕ1−3ϕ2+3128ω1δ2a1a22sin2ϕ1+2ϕ2−1128ω1δ2a13sin4ϕ1−1128ω1δ2a23cosϕ1+3ϕ2+3128ω1δ2a12a2cos3ϕ1+ϕ2=0,(35b)σ2ω2−σ12ω2−1fΩ22ω2cosϕ2+γ−12μ2+α2a2−18α4a23−18ω24η2ω2ω2−2ω1+6η24ω12−ω22+β×a12a2sin−2ϕ1+2ϕ2+18ω2δ1a2sin2ϕ2+18ω2δ1a1cosϕ1+ϕ2+332ω2δ2a23sin2ϕ2−132ω2δ2a23sin2ϕ2+364ω2δ2a1a22cos3ϕ2−ϕ1−364ω2δ2a1a22cosϕ1+ϕ2+332ω2δ2a1a22cosϕ1+ϕ2+364ω2δ2a13cosϕ1+ϕ2−164ω2δ2a13cosϕ2−3ϕ1+3128ω2δ2a12a2sin2ϕ1+2ϕ2−1128ω2δ2a23sin4ϕ2−3128ω2δ2a1a22cosϕ1+3ϕ2+1128ω2δ2a13cos3ϕ1+ϕ2=0,(35c)σ+1−σ2ω1fΩ22ω1a1cosϕ1+γ−α12ω1+18ω14η2ω22−2η24ω12−ω22−3β−3α3−ω12α5a12+18ω112η2ω22−4η2ω2ω2−2ω1−4η2ω22ω1+ω2−2βa22−18ω12η2ω22+4η2ω2ω2−2ω1+βa22cos2ϕ1−2ϕ2−18ω1δ1cos2ϕ1−18ω1a1δ1a2sinϕ1+ϕ2−332ω1δ2a12cos2ϕ1−132ω1δ2a12cos2ϕ1−364ω1δ2a1a2sin3ϕ1−ϕ2−364ω1δ2a1a2sinϕ1+ϕ2−332ω1δ2a1a2sinϕ1+ϕ2−364ω1a1δ2a23sinϕ1+ϕ2+164ω1a1δ2a23sinϕ1−3ϕ2+3128ω1δ2a22cos2ϕ1+2ϕ2−1128ω1δ2a12cos4ϕ1+1128ω1a1δ2a23sinϕ1+3ϕ2−3128ω1δ2a1a2sin3ϕ1+ϕ2=0,(35d)σ−σ1−σ2ω2−σ12ω2−1fΩ22ω2a2sinϕ2+γ−α12ω2+18ω230η2ω22−3β−3α3−ω22α5a22−18ω24η2ω22ω1+ω2+4η2ω2ω2−2ω1−12η2ω22+2βa12−18ω24η2ω2ω2−2ω1+6η24ω12−ω22+βa12cos−2ϕ1+2ϕ2+18ω2δ1cos2ϕ2−18ω2a2δ1a1sinϕ1+ϕ2+332ω2δ2a22cos2ϕ2+132ω2δ2a22cos2ϕ2−364ω2δ2a1a2sin3ϕ2−ϕ1−364ω2δ2a1a2sinϕ1+ϕ2−332ω2δ2a1a2sinϕ1+ϕ2−364ω2a2δ2a13sinϕ1+ϕ2+164ω2a2δ2a13sinϕ2−3ϕ1+3128ω2δ2a12cos2ϕ1+2ϕ2−1128ω2δ2a2cos4ϕ2+3128ω2δ2a1a2sinϕ1+3ϕ2−1128ω2a2δ2a13sin3ϕ1+ϕ2=0.The nonlinear system of algebraic equations given by equations (35a)–(35d) governs the steady-state vibration amplitudes (a1 & a2) of the system given by equations (17a) and (17b) when δ3=0.0. By solving equations (35a)–(35d) numerically in terms of the parameters σ,f,δ1, and δ2, one can obtain the different response curves as in Section 4.


## 4. Results and Discussion

Based on the introduced dimensionless variables that are given in equations (17a) and (17b) (i.e., z1=v1/h0,z2=v2/h0) and the periodic solution that is given by equations (34a) and (34b), it can be deduced that the actual solution of the physical system (i.e., v1t and v2t) is also periodic because h0 is a constant representing the air-gap size. Accordingly, the rotor can oscillate without physical contact with the pole legs if z1=v1/h0<1 and z2=v2/h0<1. This means that z1=a1τcosΩτ−ϕ1τ<1 and z2=a2τcosΩτ−ϕ2τ<1 depending on equations (34a) and (34b). Therefore, the sufficient conditions for the oscillation of the rotating disk without any physical contact with the pole leg are that a1<1 and a2<1. Hence, equations (35a)–(35d) govern the steady-state vibration amplitudes in V1 and V2 as long as a1<1 and a2<1; otherwise, the impact and rub forces occur that require the investigation equations (17a) and (17b) when δ3≠0. Because the system model given by equations (17a) and (17b) is a strong discontinuous dynamical system when δ3≠0, it can be studied numerically using the temporal oscillations, frequency spectrum, Poincare map, and bifurcation diagrams when the rub and impact forces occur (i.e., when a1≥1 and/or a2≥1). Based on this explanation, the oscillatory behaviours of the system model given by equations (17a) and (17b) can be investigated via solving the algebraic equations (i.e., equations (35a)–(35d)) as long as a1<1 and a2<1. But, when the rub and impact occur (i.e., a1≥1 and/or a2≥1), the system model is analyzed numerically via solving equations (17a) and (17b). It is worth mentioning that the numerical simulations in the whole article are performed utilizing MATLAB ODE45 solver to obtain the system the bifurcation diagrams, temporal oscillations, orbit plot, and frequency spectrum.The dynamical characteristics of the considered system are analyzed utilizing the dimensionless system parameters [26–28]: μ1=0.015,μ2=0.025,Ω=ω1+σ,f=0.025,λ=1,ρ=0.05,η=λρ,δ1=0.1,δ2=0.1,δ3=5.0, ω1=1+λ2ρ+1/2δ1,ω2=1+3λ2ρ+1/2δ1, β=ρ+3/8δ2,p=1,d=0.025,α1=p−1,α2=d,α3=−2+3p−p2,α4=3d−2pd,α5=−d2. The rest of Section 4 is organized in a way such that Section 4.1 is assigned to explore the effects of both the linear asymmetric stiffness coefficient δ1 and the nonlinear asymmetric stiffness coefficient δ2 on dynamical behaviours of the rotating disk when the controller is turned off, while the effects of the different control parameters on the system dynamics are explored in Section 4.2.


### 4.1. Rub and Impact Forces of the System without Control

The effects of the stiffness asymmetry on both the lateral vibrations and the rub-impact occurrence are discussed when the controller is turned off (i.e.,αi=0,i=1,2,…,5). Referring to equation (26), the parameter σ describes the closeness of the rotor speed Ω to its natural frequency ω1, while σ1 is constant such that σ1=ω2−ω1=1+3λ2ρ+1/2δ1−1+λ2ρ+1/2δ1. Therefore, σ is utilized as a bifurcation control parameter in the whole article to investigate the system dynamics when Ω is close to ω1 (i.e., studying the primary resonance Ω⟶ω1). Accordingly, the amplitude of oscillations (i.e., a1 and a2) in V1 and V2 directions at resonance condition (i.e., σ≅0) can be predicted via solving equations (35a)–(35d) numerically utilizing σ as a bifurcation control parameter (see Figures 2 and 3). In addition, the occurrence of the rub and impact forces between the rotor and stator can be predicted if a1≥1 and/or a2≥1 as explained above.Figure 2
Spinning-speed response curve of the rotor system before control atδ1=0.05,δ2=0.0.
(a)(b)Figure 3
Spinning-speed response curve of the rotor system before control atδ1=0.1,δ2=0.0.
(a)(b)In Figure2, the system oscillation amplitudes a1 and a2 are plotted versus σ via solving equations (35a)–(35d) numerically when δ1=0.05 and δ2=0.0, while Figure 3 shows the system lateral vibration amplitudes a1 and a2 when δ1=0.1 and δ2=0.0. It is clear from Figures 2 and 3 that the increase of δ1 from δ1=0.05 to δ1=0.1 increases the system oscillation amplitudes and widens the spinning-speed interval at which the system may perform rub and impact forces. In other words, Figure 2 shows that the system will oscillate with vibration amplitude higher than unity (i.e., a1>1 and/or a2>1) either in the horizontal or the vertical direction if 0.0086<σ<0.075, while Figure 3 confirms that the system can oscillate with oscillation amplitude higher than unity (i.e., a1>1 and/or a2>1) if 0.0<σ<0.084. Accordingly, the system performs rub and/or impact forces as long as 0.0086<σ<0.075 if δ1=0.05 and δ2=0.0, but when δ1=0.1 and δ2=0.0, the system will suffer from rub and/or impact forces as long as 0.0<σ<0.084.To explore the system motion when the rub and impact occurs between the rotor and stator as explained above (i.e., whena1≥1 and/or a2≥1), the bifurcation diagram of the whole system model (i.e., equations (17a) and (17b) when the impact stiffness coefficient δ3=0.5) is obtained as shown in Figure 4, where Figure 4(a) shows the system bifurcation diagram according to Figure 2 (i.e., δ1=0.05 and δ2=0.0) and Figure 4(b) depicts the system bifurcation diagram according to Figure 3 (i.e., δ1=0.1 and δ2=0.0). It is worth mentioning that the bifurcation diagrams in this article are obtained via plotting the Poincare return map of the steady-state radial displacement Rτ=z1τ2+z2τ2 versus the targeted system parameter. Figure 4 shows that the system can perform periodic, quasiperiodic, or chaotic motion depending on both the magnitude of the linear asymmetric stiffness coefficient δ1 and the disk spinning speed Ω=ω1+σ when the rub and impact forces occurred. Although Figure 2 illustrates that the system suffers from rub and impact forces at σ=0.05, Figure 4(a) confirms that the system performs periodic motion at σ=0.05 depending on the obtained Poincare map. Accordingly, the radial temporal oscillation Rτ, orbit plot (i.e., z1τ versus z2τ), and frequency spectrum for the asymmetric rotating disk are simulated in Figure 5 when σ=0.05 according to the bifurcation diagram given in Figure 4(a). By examining Figure 5, it can be confirmed that the system exhibits periodic motion as is clear from Figure 5(a). At the same time, the system orbit plot in Figure 5(b) illustrates the occurrence of rub and impact forces.Figure 4
Bifurcation diagram of the system before control: (a) bifurcation diagram according to Figure2 (i.e., when δ1=0.05,δ2=0.0); (b) bifurcation diagram according to Figure 3 (i.e., when δ1=0.1,δ2=0.0).
(a)(b)Figure 5
Uncontrolled system time response according to Figure4(a) (i.e., at δ1=0.05,δ2=0.0) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)By comparing Figures3 and 4(b) at σ=0.05, it can be revealed that the system suffers from rub and impact forces as it is clear from Figure 3 (i.e., when a1≥1 and/or a2≥1). At the same time, Figure 4(b) illustrates that the system exhibits chaotic motion at σ=0.05. Accordingly, the radial temporal oscillation Rτ, orbit plot (i.e., z1τ versus z2τ), and frequency spectrum for the asymmetric rotating disk are simulated in Figure 6 when σ=0.05 according to the bifurcation diagram given in Figure 4(b). The system exhibits a chaotic motion as shown in Figures 6(a) and 6(c). At the same time, the system orbit plot in Figure 6(b) illustrates the occurrence of rub and impact forces between the rotating disk and the magnetic poles legs.Figure 6
Uncontrolled system time response according to Figure4(b) (i.e., at δ1=0.1,δ2=0.0) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)The existence of both linear and nonlinear asymmetric stiffness (i.e.,δ1=δ2≠0) on the lateral vibrations of the considered system is investigated in Figures 7 and 8, where Figure 7 illustrates the system spinning-speed response curves when δ1=δ2=0.05, while Figure 8 shows the spinning-speed response curves when δ1=δ2=0.1. Comparing Figures 7 and 8 with Figures 2 and 3, we can report that the asymmetry in both the linear and nonlinear stiffness coefficients may increase the system vibration amplitudes that result in widening the spinning-speed interval at which the rotor can perform rub and impact forces. It is clear from Figure 2 (i.e., δ1=0.05,δ2=0) that system can exhibit rub and impact forces as long as 0.0086<σ<0.075, while Figure 7 (i.e., δ1=δ2=0.05) illustrates that the system will perform rub and impact as long as 0.01<σ<0.085. Also, Figure 3 (i.e., δ1=0.1,δ2=0) shows that the system can exhibit rub and impact as long as 0.0<σ<0.084, while Figure 8 (i.e., δ1=δ2=0.1) confirms that the rotor will exhibits rub and impact as long as 0.0<σ<0.1.Figure 7
Spinning-speed response curve of the rotor system before control atδ1=δ2=0.05.
(a)(b)Figure 8
Spinning-speed response curve of the rotor system before control atδ1=δ2=0.1.
(a)(b)According to Figures7 and 8, the rotor bifurcation diagrams are illustrated as shown in Figure 9 via solving equations (17a) and (17b) numerically when δ3=5.0. Figure 9(a) depicts the system bifurcation diagram according to Figure 7 (i.e., when δ1=δ2=0.05), while Figure 9(b) shows the bifurcation diagram according to Figure 8 (i.e., when δ1=δ2=0.1). It is clear from Figure 9 that the system can perform either periodic or nonperiodic motion as long as the rub and impact forces between the rotor and stator occur (i.e., when 0.01<σ<0.085 regarding Figure 9(a) and 0.0<σ<0.1 regarding Figure 9(b)), but the system oscillates periodically when the rum/impact force disappeared. Numerical simulations for the system temporal oscillation and the corresponding whirling orbit according to Figure 9(b) when σ=0.025 and 0.05 are illustrated in Figures 10 and 11, respectively. Figure 10 confirms that the system performs periodic motion when σ=0.025, while Figure 11 exhibits chaotic motion when σ=0.05.Figure 9
Bifurcation diagram of the uncontrolled system: (a) bifurcation diagram according to Figure7 (i.e., when δ1=δ2=0.05); (b) bifurcation diagram according to Figure 8 (i.e., when δ1=δ2=0.1).
(a)(b)Figure 10
Uncontrolled system time response according to Figure9(a) (i.e., at δ1=δ2=0.05) when σ=0.025: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 11
Uncontrolled system time response according to Figure9(b) (i.e., at δ1=δ2=0.1) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)


### 4.2. Rub and Impact Forces of the System with Control

The efficiency of the proposed control strategy in suppressing rotor lateral vibrations and eliminating the rub-impact forces is discussed within this subsection. The system spinning-speed response curves at three different values of the asymmetric stiffness coefficients (δ1 and δ2) are illustrated in Figure 12 when the proportional gain p=1.0 and the derivative control gain d=0.025. Figures 12(a) and 12(b) show the system response curves at δ1=δ2=0.05,075, while Figures 12(c) and 12(d) illustrate the controlled system response curves at δ1=δ2=0.1. Figures 12(a) and 12(b) prove that the applied controller has mitigated the system oscillation amplitudes below unity (i.e., a1<1 and a2<1) when δ1=δ2=0.05, which ultimately prevents the rub and impact forces. Accordingly, it can be emphasized that the asymmetric system with asymmetric coefficients δ1=δ2=0.05 that has strong nonlinear characteristics as illustrated in Figure 7 can be controlled very well to respond as a linear system utilizing the applied controller with proportional control gain p=1.0 and derivative control gain d=0.025. However, Figures 12(a) and 12(b) illustrate that the increase of δ1 and δ2 from δ1=δ2=0.05 to δ1=δ2=0.075 has increased the controlled system oscillation amplitudes a1 and a2 to become higher than unity at the spinning-speed interval 0.02<σ<0.078, which means that the occurrence of rub and impact forces when the controller is turned on with control gains p=1.0 and d=0.025. Also, Figures 12(c) and 12(d) confirm that the increase of δ1 and δ2 to become δ1=δ2=0.15 makes the controlled system (i.e., p=1.0 and d=0.025) respond as a strong nonlinear one again with high oscillation amplitudes where the rub and impact occur at the spinning-speed interval −0.003<σ<0.11.Figure 12
Controlled system spinning-speed response curves when the proportional control gainp=1.0 and derivative control gain d=0.025 at three different values of the linear and nonlinear asymmetric stiffness coefficient (i.e., δ1=δ2=0.05,0.075,0.15): (a, b) spinning-speed response curves when δ1=δ2=0.05,0.1 and (c, d) spinning-speed response curves when δ1=δ2=0.15.
(a)(b)(c)(d)Depending on the obtained spinning-speed response curves in Figure12, the system bifurcation diagrams are simulated as shown in Figure 13, where Figure 13(a) depicts the system bifurcation diagram according to Figures 12(a) and 12(b) when δ1=δ2=0.075, and Figure 13(b) shows the rotating shaft bifurcation diagram according to Figures 12(c) and 12(d) when δ1=δ2=0.15. Generally, Figure 13 illustrates that the asymmetric system can perform periodic, quasiperiodic, chaotic, or n-period motion depending on both the angular speed Ω=ω1+σ and the asymmetric stiffness coefficients (δ1 and δ2) even if the controller was turned on.Figure 13
Bifurcation diagram of the controlled system (i.e.,p=1.0d=0.025): (a) bifurcation diagram according to Figures 12(a) and 12(b) when δ1=δ2=0.075; (b) bifurcation diagram according to Figures 12(c) and 12(d) (i.e., when δ1=δ2=0.15).
(a)(b)Based on Figure13, numerical simulations for the rotor radial displacement and whirling orbit are illustrated in Figures 14 to 16, where Figure 14 shows the system temporal oscillation according to Figure 13 at σ=0.04 and Figures 15 and 16 illustrate the system temporal oscillation according to Figure 14 at σ=0.06,0.085. It is clear from Figure 14 that the system can oscillate with quasiperiodic motion, while Figures 15 and 16 confirm the oscillation of the system with chaotic and 10-period motions depending on the disk spinning-speed Ω=ω1+σ.Figure 14
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(a) when δ1=δ2=0.075 at σ=0.04: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 15
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(b) when δ1=δ2=0.15 at σ=0.06: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 16
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(b) when δ1=δ2=0.15 at σ=0.085: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure12 shows the failure of the applied controller in suppressing the rub and impact forces with the control gains p=1.0 and d=0.025 when δ1 = δ2>0.05, and the effect of increasing the derivative control gain from d=0.025 to d=0.1 is illustrated in Figure 17 when δ1 = δ2=0.15 at f=0.025,0.05. Figure 17 confirms that the applied control law can suppress the system oscillation amplitudes in a way such that a1<1 and a2<1 even at the large asymmetric stiffness coefficients (i.e., δ1=δ2=0.15). In addition, the system responds as a linear system without rub and impact forces occurrence. To determine the boundaries of the asymmetric stiffness at which the system will start to perform rub and impact forces, the system oscillation amplitudes a1 and a2 are plotted versus δ1=δ2 when p=1.0, d=0.1, and f=0.025 as shown in Figure 18. Figures 18(a) and 18(b) show δ1=δ2 response curve when σ=0, and Figures 18(c) and 18(d) depict δ1=δ2 response curve when σ=0.05. Figures 18(a) and 18(b) confirm that the controlled system can operate safely without rub-impact as long as δ1=δ2<0.29, while Figures 18(c) and 18(d) illustrate that the controlled rotor can work safely without rub-impact as long as δ1=δ2<0.1977.Figure 17
Controlled system spinning-speed response curves when the proportional control gainp=1.0 and derivative control gain d=0.1 at two different values of the disk eccentricity f when δ1=δ2=0.15.
(a)(b)Figure 18
Controlled systemδ1=δ2 response curves when the proportional control gain p=1.0 and derivative control gain d=0.1 at two different values of the detuning parameter σ: (a, b) δ1=δ2 response curves when σ=0.0 and (c, d) δ1=δ2 response curves when σ=0.05.
(a)(b)(c)(d)The controlled system bifurcation diagram utilizingδ1=δ2 as the bifurcation control parameter is obtained in Figure 19 depending on Figure 18. Figure 19(a) shows the bifurcation diagram according to Figures 18(a) and 18(b) (i.e., when σ=0), while Figure 19(b) shows the system bifurcation diagram according to Figures 18(c) and 18(d) (i.e., when σ=0.05). Based on the bifurcation diagrams shown in Figure 19, it can be deduced that the controlled system can oscillate with periodic, quasiperiodic, chaotic, or period-n motion depending on the magnitudes of δ1,δ2, and σ. Numerical simulation for the system temporal oscillation and whirling motion according to Figure 19(b) (i.e., p=1.0,d=0.1,σ=0.05,f=0.025,δ3=5) at three different values of δ1=δ2 is illustrated in Figures 20–22. Figure 20 shows the system period-2 motion when δ1=δ2=0.3, and Figure 21 confirms the system period-5 motion when δ1=δ2=0.45, while Figure 22 depicts the system quasiperiodic motion when δ1=δ2=0.5.Figure 19
Bifurcation diagram of the controlled system according to Figure18 (i.e., p=1.0,d=0.1): (a) bifurcation diagram according to Figures 18(a) and 18(b) when σ=0.0; (b) bifurcation diagram according to Figures 18(c) and 18(d) (i.e., when σ=0.05).
(a)(b)Figure 20
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.3: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 21
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.45: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 22
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.5: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)The influence of increasingf on the dynamical behaviours and the nature of motion (i.e., p=1.0 and d=0.1) is illustrated in Figures 23 and 24, utilizing f as the main bifurcation control parameter. Figures 23(a) and 23(b) illustrate the system eccentricity response curves when δ1=δ2=0.15,σ=0, while Figures 23(c) and 23(d) depict the eccentricity response curves when δ1=δ2=0.15,σ=0.05. By investigating the figure, we can deduce that the controlled rotor may be subject to rub and impact forces if f>0.12 at σ=0, but the system will exhibit rub and impact force at σ=0.05 if the disk eccentricity f>0.064. According to Figure 23, the system bifurcation diagrams are depicted in Figure 24 employing the disk eccentricity f as the bifurcation parameter. Figure 24(a) shows the bifurcation diagram according to Figures 23(a) and 23(b) (i.e., when p=1.0,d=0.1,δ1=δ2=0.15,σ=0.0), while Figure 24(b) depicts the bifurcation diagram according to Figures 23(c) and 23(d) (i.e., when p=1.0,d=0.1,δ1=δ2=0.15.σ=0.05). Comparing Figures 23 and 24, we can deduce that the controlled system can exhibit periodic oscillation only as long as the rub and impact forces between the rotor and stator do not occur (i.e., as long as f>0.12 as in Figure 24(a) or as long as f>0.064 as in Figure 24(b)), but the system can exhibit periodic, period-n, quasiperiodic, or chaotic motion when the rub and impact forces occur.Figure 23
Controlled system eccentricity response curves when the proportional control gainp=1.0, derivative control gain d=0.1, and δ1=δ2=0.15 at two different values of the detuning parameter σ: (a, b) eccentricity response curves when σ=0.0 and (c, d) eccentricity response curves when σ=0.05.
(a)(b)(c)(d)Figure 24
Bifurcation diagram of the controlled system according to Figure23 (i.e., p=1.0,d=0.1,δ1=δ2=0.15): (a) bifurcation diagram according to Figures 23(a) and 23(b) when σ=0.0; (b) bifurcation diagram according to Figures 23(c) and 23(d) (i.e., when σ=0.05).
(a)(b)


## 4.1. Rub and Impact Forces of the System without Control

The effects of the stiffness asymmetry on both the lateral vibrations and the rub-impact occurrence are discussed when the controller is turned off (i.e.,αi=0,i=1,2,…,5). Referring to equation (26), the parameter σ describes the closeness of the rotor speed Ω to its natural frequency ω1, while σ1 is constant such that σ1=ω2−ω1=1+3λ2ρ+1/2δ1−1+λ2ρ+1/2δ1. Therefore, σ is utilized as a bifurcation control parameter in the whole article to investigate the system dynamics when Ω is close to ω1 (i.e., studying the primary resonance Ω⟶ω1). Accordingly, the amplitude of oscillations (i.e., a1 and a2) in V1 and V2 directions at resonance condition (i.e., σ≅0) can be predicted via solving equations (35a)–(35d) numerically utilizing σ as a bifurcation control parameter (see Figures 2 and 3). In addition, the occurrence of the rub and impact forces between the rotor and stator can be predicted if a1≥1 and/or a2≥1 as explained above.Figure 2
Spinning-speed response curve of the rotor system before control atδ1=0.05,δ2=0.0.
(a)(b)Figure 3
Spinning-speed response curve of the rotor system before control atδ1=0.1,δ2=0.0.
(a)(b)In Figure2, the system oscillation amplitudes a1 and a2 are plotted versus σ via solving equations (35a)–(35d) numerically when δ1=0.05 and δ2=0.0, while Figure 3 shows the system lateral vibration amplitudes a1 and a2 when δ1=0.1 and δ2=0.0. It is clear from Figures 2 and 3 that the increase of δ1 from δ1=0.05 to δ1=0.1 increases the system oscillation amplitudes and widens the spinning-speed interval at which the system may perform rub and impact forces. In other words, Figure 2 shows that the system will oscillate with vibration amplitude higher than unity (i.e., a1>1 and/or a2>1) either in the horizontal or the vertical direction if 0.0086<σ<0.075, while Figure 3 confirms that the system can oscillate with oscillation amplitude higher than unity (i.e., a1>1 and/or a2>1) if 0.0<σ<0.084. Accordingly, the system performs rub and/or impact forces as long as 0.0086<σ<0.075 if δ1=0.05 and δ2=0.0, but when δ1=0.1 and δ2=0.0, the system will suffer from rub and/or impact forces as long as 0.0<σ<0.084.To explore the system motion when the rub and impact occurs between the rotor and stator as explained above (i.e., whena1≥1 and/or a2≥1), the bifurcation diagram of the whole system model (i.e., equations (17a) and (17b) when the impact stiffness coefficient δ3=0.5) is obtained as shown in Figure 4, where Figure 4(a) shows the system bifurcation diagram according to Figure 2 (i.e., δ1=0.05 and δ2=0.0) and Figure 4(b) depicts the system bifurcation diagram according to Figure 3 (i.e., δ1=0.1 and δ2=0.0). It is worth mentioning that the bifurcation diagrams in this article are obtained via plotting the Poincare return map of the steady-state radial displacement Rτ=z1τ2+z2τ2 versus the targeted system parameter. Figure 4 shows that the system can perform periodic, quasiperiodic, or chaotic motion depending on both the magnitude of the linear asymmetric stiffness coefficient δ1 and the disk spinning speed Ω=ω1+σ when the rub and impact forces occurred. Although Figure 2 illustrates that the system suffers from rub and impact forces at σ=0.05, Figure 4(a) confirms that the system performs periodic motion at σ=0.05 depending on the obtained Poincare map. Accordingly, the radial temporal oscillation Rτ, orbit plot (i.e., z1τ versus z2τ), and frequency spectrum for the asymmetric rotating disk are simulated in Figure 5 when σ=0.05 according to the bifurcation diagram given in Figure 4(a). By examining Figure 5, it can be confirmed that the system exhibits periodic motion as is clear from Figure 5(a). At the same time, the system orbit plot in Figure 5(b) illustrates the occurrence of rub and impact forces.Figure 4
Bifurcation diagram of the system before control: (a) bifurcation diagram according to Figure2 (i.e., when δ1=0.05,δ2=0.0); (b) bifurcation diagram according to Figure 3 (i.e., when δ1=0.1,δ2=0.0).
(a)(b)Figure 5
Uncontrolled system time response according to Figure4(a) (i.e., at δ1=0.05,δ2=0.0) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)By comparing Figures3 and 4(b) at σ=0.05, it can be revealed that the system suffers from rub and impact forces as it is clear from Figure 3 (i.e., when a1≥1 and/or a2≥1). At the same time, Figure 4(b) illustrates that the system exhibits chaotic motion at σ=0.05. Accordingly, the radial temporal oscillation Rτ, orbit plot (i.e., z1τ versus z2τ), and frequency spectrum for the asymmetric rotating disk are simulated in Figure 6 when σ=0.05 according to the bifurcation diagram given in Figure 4(b). The system exhibits a chaotic motion as shown in Figures 6(a) and 6(c). At the same time, the system orbit plot in Figure 6(b) illustrates the occurrence of rub and impact forces between the rotating disk and the magnetic poles legs.Figure 6
Uncontrolled system time response according to Figure4(b) (i.e., at δ1=0.1,δ2=0.0) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)The existence of both linear and nonlinear asymmetric stiffness (i.e.,δ1=δ2≠0) on the lateral vibrations of the considered system is investigated in Figures 7 and 8, where Figure 7 illustrates the system spinning-speed response curves when δ1=δ2=0.05, while Figure 8 shows the spinning-speed response curves when δ1=δ2=0.1. Comparing Figures 7 and 8 with Figures 2 and 3, we can report that the asymmetry in both the linear and nonlinear stiffness coefficients may increase the system vibration amplitudes that result in widening the spinning-speed interval at which the rotor can perform rub and impact forces. It is clear from Figure 2 (i.e., δ1=0.05,δ2=0) that system can exhibit rub and impact forces as long as 0.0086<σ<0.075, while Figure 7 (i.e., δ1=δ2=0.05) illustrates that the system will perform rub and impact as long as 0.01<σ<0.085. Also, Figure 3 (i.e., δ1=0.1,δ2=0) shows that the system can exhibit rub and impact as long as 0.0<σ<0.084, while Figure 8 (i.e., δ1=δ2=0.1) confirms that the rotor will exhibits rub and impact as long as 0.0<σ<0.1.Figure 7
Spinning-speed response curve of the rotor system before control atδ1=δ2=0.05.
(a)(b)Figure 8
Spinning-speed response curve of the rotor system before control atδ1=δ2=0.1.
(a)(b)According to Figures7 and 8, the rotor bifurcation diagrams are illustrated as shown in Figure 9 via solving equations (17a) and (17b) numerically when δ3=5.0. Figure 9(a) depicts the system bifurcation diagram according to Figure 7 (i.e., when δ1=δ2=0.05), while Figure 9(b) shows the bifurcation diagram according to Figure 8 (i.e., when δ1=δ2=0.1). It is clear from Figure 9 that the system can perform either periodic or nonperiodic motion as long as the rub and impact forces between the rotor and stator occur (i.e., when 0.01<σ<0.085 regarding Figure 9(a) and 0.0<σ<0.1 regarding Figure 9(b)), but the system oscillates periodically when the rum/impact force disappeared. Numerical simulations for the system temporal oscillation and the corresponding whirling orbit according to Figure 9(b) when σ=0.025 and 0.05 are illustrated in Figures 10 and 11, respectively. Figure 10 confirms that the system performs periodic motion when σ=0.025, while Figure 11 exhibits chaotic motion when σ=0.05.Figure 9
Bifurcation diagram of the uncontrolled system: (a) bifurcation diagram according to Figure7 (i.e., when δ1=δ2=0.05); (b) bifurcation diagram according to Figure 8 (i.e., when δ1=δ2=0.1).
(a)(b)Figure 10
Uncontrolled system time response according to Figure9(a) (i.e., at δ1=δ2=0.05) when σ=0.025: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 11
Uncontrolled system time response according to Figure9(b) (i.e., at δ1=δ2=0.1) when σ=0.05: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)


## 4.2. Rub and Impact Forces of the System with Control

The efficiency of the proposed control strategy in suppressing rotor lateral vibrations and eliminating the rub-impact forces is discussed within this subsection. The system spinning-speed response curves at three different values of the asymmetric stiffness coefficients (δ1 and δ2) are illustrated in Figure 12 when the proportional gain p=1.0 and the derivative control gain d=0.025. Figures 12(a) and 12(b) show the system response curves at δ1=δ2=0.05,075, while Figures 12(c) and 12(d) illustrate the controlled system response curves at δ1=δ2=0.1. Figures 12(a) and 12(b) prove that the applied controller has mitigated the system oscillation amplitudes below unity (i.e., a1<1 and a2<1) when δ1=δ2=0.05, which ultimately prevents the rub and impact forces. Accordingly, it can be emphasized that the asymmetric system with asymmetric coefficients δ1=δ2=0.05 that has strong nonlinear characteristics as illustrated in Figure 7 can be controlled very well to respond as a linear system utilizing the applied controller with proportional control gain p=1.0 and derivative control gain d=0.025. However, Figures 12(a) and 12(b) illustrate that the increase of δ1 and δ2 from δ1=δ2=0.05 to δ1=δ2=0.075 has increased the controlled system oscillation amplitudes a1 and a2 to become higher than unity at the spinning-speed interval 0.02<σ<0.078, which means that the occurrence of rub and impact forces when the controller is turned on with control gains p=1.0 and d=0.025. Also, Figures 12(c) and 12(d) confirm that the increase of δ1 and δ2 to become δ1=δ2=0.15 makes the controlled system (i.e., p=1.0 and d=0.025) respond as a strong nonlinear one again with high oscillation amplitudes where the rub and impact occur at the spinning-speed interval −0.003<σ<0.11.Figure 12
Controlled system spinning-speed response curves when the proportional control gainp=1.0 and derivative control gain d=0.025 at three different values of the linear and nonlinear asymmetric stiffness coefficient (i.e., δ1=δ2=0.05,0.075,0.15): (a, b) spinning-speed response curves when δ1=δ2=0.05,0.1 and (c, d) spinning-speed response curves when δ1=δ2=0.15.
(a)(b)(c)(d)Depending on the obtained spinning-speed response curves in Figure12, the system bifurcation diagrams are simulated as shown in Figure 13, where Figure 13(a) depicts the system bifurcation diagram according to Figures 12(a) and 12(b) when δ1=δ2=0.075, and Figure 13(b) shows the rotating shaft bifurcation diagram according to Figures 12(c) and 12(d) when δ1=δ2=0.15. Generally, Figure 13 illustrates that the asymmetric system can perform periodic, quasiperiodic, chaotic, or n-period motion depending on both the angular speed Ω=ω1+σ and the asymmetric stiffness coefficients (δ1 and δ2) even if the controller was turned on.Figure 13
Bifurcation diagram of the controlled system (i.e.,p=1.0d=0.025): (a) bifurcation diagram according to Figures 12(a) and 12(b) when δ1=δ2=0.075; (b) bifurcation diagram according to Figures 12(c) and 12(d) (i.e., when δ1=δ2=0.15).
(a)(b)Based on Figure13, numerical simulations for the rotor radial displacement and whirling orbit are illustrated in Figures 14 to 16, where Figure 14 shows the system temporal oscillation according to Figure 13 at σ=0.04 and Figures 15 and 16 illustrate the system temporal oscillation according to Figure 14 at σ=0.06,0.085. It is clear from Figure 14 that the system can oscillate with quasiperiodic motion, while Figures 15 and 16 confirm the oscillation of the system with chaotic and 10-period motions depending on the disk spinning-speed Ω=ω1+σ.Figure 14
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(a) when δ1=δ2=0.075 at σ=0.04: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 15
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(b) when δ1=δ2=0.15 at σ=0.06: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 16
Controlled system time response (i.e.,p=1.0,d=0.025) according to Figure 13(b) when δ1=δ2=0.15 at σ=0.085: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure12 shows the failure of the applied controller in suppressing the rub and impact forces with the control gains p=1.0 and d=0.025 when δ1 = δ2>0.05, and the effect of increasing the derivative control gain from d=0.025 to d=0.1 is illustrated in Figure 17 when δ1 = δ2=0.15 at f=0.025,0.05. Figure 17 confirms that the applied control law can suppress the system oscillation amplitudes in a way such that a1<1 and a2<1 even at the large asymmetric stiffness coefficients (i.e., δ1=δ2=0.15). In addition, the system responds as a linear system without rub and impact forces occurrence. To determine the boundaries of the asymmetric stiffness at which the system will start to perform rub and impact forces, the system oscillation amplitudes a1 and a2 are plotted versus δ1=δ2 when p=1.0, d=0.1, and f=0.025 as shown in Figure 18. Figures 18(a) and 18(b) show δ1=δ2 response curve when σ=0, and Figures 18(c) and 18(d) depict δ1=δ2 response curve when σ=0.05. Figures 18(a) and 18(b) confirm that the controlled system can operate safely without rub-impact as long as δ1=δ2<0.29, while Figures 18(c) and 18(d) illustrate that the controlled rotor can work safely without rub-impact as long as δ1=δ2<0.1977.Figure 17
Controlled system spinning-speed response curves when the proportional control gainp=1.0 and derivative control gain d=0.1 at two different values of the disk eccentricity f when δ1=δ2=0.15.
(a)(b)Figure 18
Controlled systemδ1=δ2 response curves when the proportional control gain p=1.0 and derivative control gain d=0.1 at two different values of the detuning parameter σ: (a, b) δ1=δ2 response curves when σ=0.0 and (c, d) δ1=δ2 response curves when σ=0.05.
(a)(b)(c)(d)The controlled system bifurcation diagram utilizingδ1=δ2 as the bifurcation control parameter is obtained in Figure 19 depending on Figure 18. Figure 19(a) shows the bifurcation diagram according to Figures 18(a) and 18(b) (i.e., when σ=0), while Figure 19(b) shows the system bifurcation diagram according to Figures 18(c) and 18(d) (i.e., when σ=0.05). Based on the bifurcation diagrams shown in Figure 19, it can be deduced that the controlled system can oscillate with periodic, quasiperiodic, chaotic, or period-n motion depending on the magnitudes of δ1,δ2, and σ. Numerical simulation for the system temporal oscillation and whirling motion according to Figure 19(b) (i.e., p=1.0,d=0.1,σ=0.05,f=0.025,δ3=5) at three different values of δ1=δ2 is illustrated in Figures 20–22. Figure 20 shows the system period-2 motion when δ1=δ2=0.3, and Figure 21 confirms the system period-5 motion when δ1=δ2=0.45, while Figure 22 depicts the system quasiperiodic motion when δ1=δ2=0.5.Figure 19
Bifurcation diagram of the controlled system according to Figure18 (i.e., p=1.0,d=0.1): (a) bifurcation diagram according to Figures 18(a) and 18(b) when σ=0.0; (b) bifurcation diagram according to Figures 18(c) and 18(d) (i.e., when σ=0.05).
(a)(b)Figure 20
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.3: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 21
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.45: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)Figure 22
Controlled system time response (i.e.,p=1.0,d=0.1) according to Figure 19(b) when δ1=δ2=0.5: (a) the temporal oscillation of the radial displacement Rτ=z12+z22, (b) whirling orbit z1τ versus z2τ, and (c) frequency spectrum.
(a)(b)(c)The influence of increasingf on the dynamical behaviours and the nature of motion (i.e., p=1.0 and d=0.1) is illustrated in Figures 23 and 24, utilizing f as the main bifurcation control parameter. Figures 23(a) and 23(b) illustrate the system eccentricity response curves when δ1=δ2=0.15,σ=0, while Figures 23(c) and 23(d) depict the eccentricity response curves when δ1=δ2=0.15,σ=0.05. By investigating the figure, we can deduce that the controlled rotor may be subject to rub and impact forces if f>0.12 at σ=0, but the system will exhibit rub and impact force at σ=0.05 if the disk eccentricity f>0.064. According to Figure 23, the system bifurcation diagrams are depicted in Figure 24 employing the disk eccentricity f as the bifurcation parameter. Figure 24(a) shows the bifurcation diagram according to Figures 23(a) and 23(b) (i.e., when p=1.0,d=0.1,δ1=δ2=0.15,σ=0.0), while Figure 24(b) depicts the bifurcation diagram according to Figures 23(c) and 23(d) (i.e., when p=1.0,d=0.1,δ1=δ2=0.15.σ=0.05). Comparing Figures 23 and 24, we can deduce that the controlled system can exhibit periodic oscillation only as long as the rub and impact forces between the rotor and stator do not occur (i.e., as long as f>0.12 as in Figure 24(a) or as long as f>0.064 as in Figure 24(b)), but the system can exhibit periodic, period-n, quasiperiodic, or chaotic motion when the rub and impact forces occur.Figure 23
Controlled system eccentricity response curves when the proportional control gainp=1.0, derivative control gain d=0.1, and δ1=δ2=0.15 at two different values of the detuning parameter σ: (a, b) eccentricity response curves when σ=0.0 and (c, d) eccentricity response curves when σ=0.05.
(a)(b)(c)(d)Figure 24
Bifurcation diagram of the controlled system according to Figure23 (i.e., p=1.0,d=0.1,δ1=δ2=0.15): (a) bifurcation diagram according to Figures 23(a) and 23(b) when σ=0.0; (b) bifurcation diagram according to Figures 23(c) and 23(d) (i.e., when σ=0.05).
(a)(b)


## 5. Conclusions

The nonlinear oscillation control of an asymmetric horizontally suspended rotor is discussed in this article. The rotor is modeled as a two-degree-of-freedom discontinuous dynamical system with asymmetric stiffness. A proportional-derivative control law is proposed to mitigate the lateral oscillations of the considered rotor system. It is suggested to couple the proposed controller to the rotor system via poles’ active magnetic bearings fixed on the stator housing. The whole system model (i.e., the rotor system and the proposed controller) is derived considering the rub and impact forces between the rotor and stator housing. The obtained system model is investigated in two steps. Firstly, the rub and impact forces are neglected to analyze the periodic response and determine the conditions at which the system can exhibit rub and impact between the rotor and stator applying the asymptotic analysis. Secondly, the nature of the system motions is explored via plotting the different bifurcation diagrams, temporal oscillations, whirling orbits, and frequency spectrum of the system model including the rub and impact forces. According to the introduced analysis and discussion, some conclusions can be summarized as follows:(a)
The rotor oscillation amplitudes are monotonic increasing functions of both the disk eccentricity and the asymmetric stiffness coefficients.(b)
The uncontrolled system exhibits many nonlinear phenomena such as the sensitivity to the initial conditions, jump phenomenon, and the coexistence of multiple whirling orbits.(c)
The rotating shaft system exhibits periodic motions only as long as the rub and impact forces do not occur between the rotor and the stator housing.(d)
The occurrence of rub and impact forces results in the oscillation of the system with period-n, quasiperiodic, or chaotic motion depending on the disk spinning speed.(e)
It is possible to force the nonlinear rotor system to respond as a linear system without rub and impact forces via the proper design of the control parameters (p and d).



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*Source: 1800022-2021-08-12.xml*