Experimental and theoretical investigations of unbalance flexible rotor in active magnetic bearings considering backup bearings contacts

Abstract

This paper describes the experimental verification of an unbalance flexible rotor model in active magnetic bearings. The dynamic modeling takes into account the gyroscopic moments of the disk, geometric coupling of the magnetic actuators, and contact forces of the backup bearings. The Rung–Kutta method is used to integrate the equations of motion. The nonlinear dynamic response is analyzed using bifurcation plots, disc center trajectories, Fast Fourier Transforms, Poincaré maps, and maximum Lyapunov exponent. The analysis is carried out for different values of rotating speed and unbalance eccentricity. In the unbalance test, a concentrated mass of 5 gr is attached in three radial position of the disk. The numerical simulations and experiments prove that a variety of nonlinear dynamical phenomena such as period-3, -4, -8, periodic, quasi-periodic, and chaotic motions occur in the system. The results indicate that the response of the rotor returns back to a regular motion by increasing the rotational speed. Also, by increasing the unbalance eccentricity, the first irregular motion initiates at much higher rotational speeds. Therefore, sufficient attention should be paid to these factors in design of a flexible rotor system equipped with both active magnetic bearings and backup bearings in order to ensure system reliability.

Keywords

Nonlinear dynamics active magnetic bearings system elastic shaft backup bearing experimental observations

Introduction

Active magnetic bearings (AMB) have merits such as low loss, low noise, absent lubrication, non-friction, and adjustable stiffness and damping. So it has been increasingly used in the rotary machines, especially for high rotating speeds.¹ In high speed rotating machinery, the major cause of vibration problems is rotor unbalance. Furthermore, because of the strong nonlinearities in the AMB systems, nonperiodic vibration can occur in rotor.^2-4 So, how to increase range of periodic motion and avoid the appearance of nonperiodic motion becomes the major execution of this paper.

Wu et al.⁵ investigated nonlinear dynamic of a rigid rotor—AMB with 16-pole legs and time varying stiffness under primary and sub-harmonic resonances. They found that the time varying stiffness can be controlled to achieve the stable state motions.

Inayat-Hussain^6,7 considered the misalignment and geometric coupling effects on the bifurcation behavior of a flexible rotor AMB system. The numerical results revealed the occurrence of sub-synchronous vibrations of period-2, -3, -6, 9, -17, quasi-periodicity, and chaos that cause the rotor’s position to move from one basin of attraction to another.

A rotor–AMB system requires backup bearings to prevent excessive displacements that would otherwise result in rotor–stator contact.⁸ More recently, Inayat-Hussain⁹ also analyzed nonlinear dynamics of a magnetically supported rigid rotor in backup bearings. He indicated that the design parameters of backup bearing have almost negligible effect on the rotor’s response for the large unbalance parameter. Also, the occurrence of non-synchronous response reduces with decreasing the stiffness of backup bearing for rotors with small level of unbalance.

Liu et al.¹⁰ used analytical and numerical methods to recognize multi-attractor coexisting in a magnetically supported rigid rotor with backup bearings. Also, they reveal occurrence of a Hopf bifurcation in the system by using center manifold and normal form.

Modeling of the flexible rotor–AMB system with backup bearings contacts is mentioned in the following:

Jang and Chen¹¹ simulated the response of a flexible rotor supported by magnetic and backup bearings. The results showed the occurrence of sub-synchronous vibrations of periods-2, -4, and -8, quasi-periodicity, and chaos.

Xie et al.¹² proposed model of a flexible rotor–AMB system supported by backup bearings. They found that the most favorable rotor behavior can be achieved by auxiliary bearing designs with low clearance, low support stiffness, and high support damping.

Despite of significant efforts on rotor–AMB systems, a model that considers simultaneously effects of flexibility of shaft, gyroscopic moments of disk, geometric coupling of magnetic actuators, and backup bearings contact forces has not been addressed. Therefore, in this paper the nonlinear dynamics of a flexible rotor–AMB system in backup bearings is studied, where the influences of rotating speed and unbalance eccentricity on the bifurcation behavior of the system are considered. Also, the results are demonstrated by experiments on a test rig. This is a continuation of work presented in Ref. 13, where the mathematical modeling of system was discussed and the influences of rotating speed and backup bearings stiffness were analyzed.

In the next sections, the formulated model is used and nonlinear dynamic equations of motion are solved numerically by Runge-Kutta method. Dynamical system theory was used as a conceptual framework and comparisons are made between numerical and experimental results for different values of rotating speed and unbalance eccentricity. So, an overview of nonlinear dynamic behavior of the system is obtained, which can be consequently used to suppress non periodic vibration.

Materials and methods

A flexible rotor is supported at each end (points A and D) by an AMB equipped with a backup bearing (Figure 1). Internal damping, axial motion, and torsional vibrations have been ignored. The mass of flexible shaft and its journals are concentrated at the end-points (m₁ and m₃). The two end masses m₁ and m₃ are allowed only planar displacements in (x₁, y₁) and (x₃, y₃), respectively. The disk d is allowed both planar displacements in (x₂, y₂) and angular displacements in (Ɵ_x, Ɵ_y). So, the model of a flexible rotor–AMB system supported by backup bearings is described by eight degrees-of-freedom nonlinear ordinary differential equations. In derivation of the equations of motion, the following assumptions hold:

The flexible rotor–AMB system is symmetric.

The external damping at disk due to aerodynamics is viscous.

Flux density does not change through the core and gap. So, there are no eddy currents in the iron core.

The magnetic flux leakage and the saturation of the core material are negligible.

Figure 1.

Schematic of a flexible rotor–AMB with backup bearings.

The total kinetic energy of the flexible rotor–AMB system is characterized as the sum of the kinetic energies for disk d and masses m₁ and m₃. For the disk, kinetic energy can be expressed as the sum of its mass center’s translational kinetic energy and its rotational kinetic energy about the mass center. For the flexible shaft in bending, the potential energy is derived by integrating the strain energy over the length of the flexible shaft. The dissipation energy is due to the viscous damping force acting on the disk. The potential energy V, kinetic energy T, and dissipation energy D.E. are, respectively, written as follows¹³

\begin{array}{l} T = T_{m_{1}} + T_{m_{3}} + T_{d} \\ = \frac{1}{2} [m_{1} ({\dot{x}}_{1}^{2} + {\dot{y}}_{1}^{2}) + m_{2} ({\dot{x}}_{2}^{2} + {\dot{y}}_{2}^{2}) + m_{3} ({\dot{x}}_{3}^{2} + {\dot{y}}_{3}^{2}) + I_{T} ({\dot{θ}}_{x}^{2} + {\dot{θ}}_{y}^{2}) + I_{P} (Ω^{2} - 2 Ω {\dot{θ}}_{y} θ_{x})] \end{array}

(1)

\begin{array}{l} V = \frac{1}{2} \int_{0}^{α_{1} L} E I [{(x^{″} (z))}^{2} + {(y^{″} (z))}^{2}] d z + \frac{1}{2} \int_{α_{1} L}^{L} E I [{(x^{″} (z))}^{2} + {(y^{″} (z))}^{2}] d z \\ = \frac{3 E I}{2 α_{1}^{3} L^{3}} [x_{1}^{2} - 2 x_{1} x_{2} + 2 α_{1} x_{1} L θ_{y} + y_{1}^{2} - 2 y_{1} y_{2} - 2 α_{1} y_{1} L θ_{x}] \\ + \frac{3 E I}{2 L^{3}} [\frac{1 + 3 α_{1}^{2} - 3 α_{1}}{α_{1}^{3} {(1 - α_{1})}^{3}} (x_{2}^{2} + y_{2}^{2}) + \frac{L^{2}}{α_{1} (1 - α_{1})} (θ_{x}^{2} + θ_{y}^{2}) + \frac{2 (1 - 2 α_{1}) L}{α_{1}^{2} {(1 - α_{1})}^{2}} (y_{2} θ_{x} - x_{2} θ_{y})] \\ + \frac{3 E I}{2 {(1 - α_{1})}^{3} L^{3}} [x_{3}^{2} - 2 x_{2} x_{3} - 2 (1 - α_{1}) x_{3} L θ_{y} + y_{3}^{2} - 2 y_{2} y_{3} + 2 (1 - α_{1}) y_{3} L θ_{x}] \end{array}

(2)

D . E . = \frac{1}{2} C ({\dot{x}}_{2}^{2} + {\dot{y}}_{2}^{2})

(3)

The generalized forces appear in the flexible rotor–AMB model as the applied forces from AMBs, radial unbalance at disk and rotor/backup bearings contact. The generalized forces

Q_{q_{i}}

can be represented as

Q_{x_{1}} = F_{M_{x_{1}}} + F_{C_{x_{1}}}

(4)

Q_{y_{1}} = F_{M_{y_{1}}} + F_{C_{y_{1}}}

(5)

Q_{x_{2}} = m_{2} u Ω^{2} cos (Ω t)

(6)

Q_{y_{2}} = m_{2} u Ω^{2} sin (Ω t)

(7)

Q_{θ_{x}} = 0

(8)

Q_{θ_{y}} = 0

(9)

Q_{x_{3}} = F_{M_{x_{3}}} + F_{C_{x_{3}}}

(10)

Q_{y_{3}} = F_{M_{y_{3}}} + F_{C_{y_{3}}}

(11)

Each AMB has a stator with four poles in the horizontal and vertical direction (Figure 2). According to electromagnetic theory, the electromagnetic forces exerted on the rotor (at points A and D) can be expressed as follows¹

F_{M_{x_{1}}} = \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{x_{1}}}{g_{0} - x_{1}})}^{2} - {(\frac{I_{0} - i_{x_{1}}}{g_{0} + x_{1}})}^{2}] + α x_{1} \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{y_{1}}}{g_{0} - y_{1}})}^{2} + {(\frac{I_{0} - i_{y_{1}}}{g_{0} + y_{1}})}^{2}]

(12)

F_{M_{y_{1}}} = \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{y_{1}}}{g_{0} - y_{1}})}^{2} - {(\frac{I_{0} - i_{y_{1}}}{g_{0} + y_{1}})}^{2}] + α y_{1} \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{x_{1}}}{g_{0} - x_{1}})}^{2} + {(\frac{I_{0} - i_{x_{1}}}{g_{0} + x_{1}})}^{2}]

(13)

F_{M_{x_{3}}} = \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{x_{3}}}{g_{0} - x_{3}})}^{2} - {(\frac{I_{0} - i_{x_{3}}}{g_{0} + x_{3}})}^{2}] + α x_{3} \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{y_{3}}}{g_{0} - y_{3}})}^{2} + {(\frac{I_{0} - i_{y_{3}}}{g_{0} + y_{3}})}^{2}]

(14)

F_{M_{y_{3}}} = \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{y_{3}}}{g_{0} - y_{3}})}^{2} - {(\frac{I_{0} - i_{y_{3}}}{g_{0} + y_{3}})}^{2}] + α y_{3} \frac{A_{p} μ_{0} N^{2}}{4} [{(\frac{I_{0} + i_{x_{3}}}{g_{0} - x_{3}})}^{2} + {(\frac{I_{0} - i_{x_{3}}}{g_{0} + x_{3}})}^{2}]

(15)

Equations (12–15) include coupling effect between the electromagnets. To control bearing forces, the controller generates commands that determine the current in the coils. A common strategy is the proportional-derivative (PD) control

i_{x_{1}} = - \bar{P} x_{1} - \bar{D} {\dot{x}}_{1}

(16)

i_{y_{1}} = - \bar{P} y_{1} - \bar{D} {\dot{y}}_{1}

(17)

i_{x_{3}} = - \bar{P} x_{3} - \bar{D} {\dot{x}}_{3}

(18)

i_{y_{3}} = - \bar{P} y_{3} - \bar{D} {\dot{y}}_{3}

(19)

An elastic contact model between the flexible shaft and the backup bearings is used. Also the Coulomb type of frictional relationship is assumed in the analysis (Figure 3). The normal force

f_{n_{A}}

and tangential force

f_{t_{A}}

can be expressed as

{\begin{matrix} f_{n_{A}} = k_{a} e_{A} \\ f_{t_{A}} = μ f_{n_{A}} \end{matrix}

(20)

where

e_{A} = (\sqrt{x_{1}^{2} + y_{1}^{2}} - g_{1}) H (\sqrt{x_{1}^{2} + y_{1}^{2}} - g_{1})

(21)

It should be noted that the value of Heaviside step function H(x) is zero for x≤0 and one for x>0. The components of the contact forces in the x and y directions are

F_{C_{x_{1}}} = - f_{n_{A}} cos ψ_{A} + f_{t_{A}} sin ψ_{A}

(22)

F_{C_{y_{1}}} = - f_{n_{A}} sin ψ_{A} - f_{t_{A}} cos ψ_{A}

(23)

where

ψ_{A} = \tan^{- 1} (\frac{y_{1}}{x_{1}})

(24)

Similarly, the contact forces applied to the flexible shaft at point D can be obtained. The Lagrange’s equation is expressed as¹⁴

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - (\frac{\partial T}{\partial q_{i}}) + (\frac{\partial V}{\partial q_{i}}) + (\frac{\partial D . E .}{\partial {\dot{q}}_{i}}) = Q_{q_{i}} i = 1, 2

(25)

The non-dimensional parameters are introduced as

\begin{array}{l} x_{i} = \frac{x_{i}}{g_{0}} y_{i} = \frac{y_{i}}{g_{0}} θ_{x} = \frac{L θ_{x}}{g_{0}} θ_{y} = \frac{L θ_{y}}{g_{0}} \\ τ = ϖ_{n} t \frac{d}{d t} = ϖ_{n} \frac{d}{d τ} n_{1} = \frac{g_{1}}{g_{0}} n_{2} = \frac{L}{r_{1}} \\ λ_{1} = \frac{ϖ_{n}}{ϖ_{1}} λ_{2} = \frac{ϖ_{n}}{ϖ_{2}} α_{2} = \frac{m_{1}}{m_{2}} ζ = \frac{c}{2 m_{2} ϖ_{1}} \\ U = \frac{u}{g_{0}} S = \frac{Ω}{ϖ_{n}} P = \frac{g_{0} \bar{P}}{I_{0}} D = \frac{g_{0} ϖ_{n} \bar{D}}{I_{0}} \end{array}

By applying the Lagrange’s equation, eight coupled nonlinear differential equations of motion can be obtained as

\begin{array}{l} {\ddot{x}}_{1} + \frac{η_{1}}{α_{2} λ_{1}^{2}} (x_{1} - x_{2}) + \frac{η_{2}}{α_{2} λ_{1}^{2}} θ_{y} \\ = \frac{1}{4 (P - 1)} [{(\frac{1 - P x_{1} - D {\dot{x}}_{1}}{1 - x_{1}})}^{2} - {(\frac{1 + P x_{1} + D {\dot{x}}_{1}}{1 + x_{1}})}^{2}] \\ + \frac{α x_{1}}{4 (P - 1)} [{(\frac{1 - P y_{1} - D {\dot{y}}_{1}}{1 - y_{1}})}^{2} + {(\frac{1 + P y_{1} + D {\dot{y}}_{1}}{1 + y_{1}})}^{2}] \\ - \frac{(\sqrt{x_{1}^{2} + y_{1}^{2}} - n_{1}) H (\sqrt{x_{1}^{2} + y_{1}^{2}} - n_{1})}{λ_{2}^{2}} (cos ψ_{A} - μ sin ψ_{A}) \end{array}

(26)

\begin{array}{l} {\ddot{y}}_{1} + \frac{η_{1}}{α_{2} λ_{1}^{2}} (y_{1} - y_{2}) - \frac{η_{2}}{α_{2} λ_{1}^{2}} θ_{x} \\ = \frac{1}{4 (P - 1)} [{(\frac{1 - P y_{1} - D {\dot{y}}_{1}}{1 - y_{1}})}^{2} - {(\frac{1 + P y_{1} + D {\dot{y}}_{1}}{1 + y_{1}})}^{2}] \\ + \frac{α y_{1}}{4 (P - 1)} [{(\frac{1 - P x_{1} - D {\dot{x}}_{1}}{1 - x_{1}})}^{2} + {(\frac{1 + P x_{1} + D {\dot{x}}_{1}}{1 + x_{1}})}^{2}] \\ - \frac{(\sqrt{x_{1}^{2} + y_{1}^{2}} - n_{1}) H (\sqrt{x_{1}^{2} + y_{1}^{2}} - n_{1})}{λ_{2}^{2}} (cos ψ_{A} + μ sin ψ_{A}) \end{array}

(27)

\begin{array}{l} {\ddot{x}}_{2} + \frac{2 ζ}{λ_{1}} {\dot{x}}_{2} + \frac{η_{3}}{λ_{1}^{2}} x_{2} - \frac{η_{1}}{λ_{1}^{2}} x_{1} - \frac{η_{4}}{λ_{1}^{2}} x_{3} - \frac{η_{5}}{λ_{1}^{2}} θ_{y} \\ = U S^{2} \cos (S τ) \end{array}

(28)

\begin{array}{l} {\ddot{y}}_{2} + \frac{2 ζ}{λ_{1}} {\dot{y}}_{2} + \frac{η_{3}}{λ_{1}^{2}} y_{2} - \frac{η_{1}}{λ_{1}^{2}} y_{1} - \frac{η_{4}}{λ_{1}^{2}} y_{3} + \frac{η_{5}}{λ_{1}^{2}} θ_{x} \\ = U S^{2} sin (S τ) \end{array}

(29)

{\ddot{θ}}_{x} + 2 S {\dot{θ}}_{y} + 4 n_{2}^{2} \frac{η_{6}}{λ_{1}^{2}} θ_{x} - \frac{4 n_{2}^{2} η_{2}}{λ_{1}^{2}} y_{1} + 4 n_{2}^{2} \frac{η_{5}}{λ_{1}^{2}} y_{2} + 4 n_{2}^{2} \frac{η_{7}}{λ_{1}^{2}} y_{3} = 0

(30)

{\ddot{θ}}_{y} - 2 S {\dot{θ}}_{x} + 4 n_{2}^{2} \frac{η_{6}}{λ_{1}^{2}} θ_{y} + \frac{4 n_{2}^{2} η_{2}}{λ_{1}^{2}} x_{1} - 4 n_{2}^{2} \frac{η_{5}}{λ_{1}^{2}} x_{2} - 4 n_{2}^{2} \frac{η_{7}}{λ_{1}^{2}} x_{3} = 0

(31)

\begin{array}{l} {\ddot{x}}_{3} + \frac{η_{8}}{α_{2} λ_{1}^{2}} (x_{3} - x_{2}) - \frac{η_{9}}{α_{2} λ_{1}^{2}} θ_{y} \\ = \frac{α_{1}}{4 (1 - α_{1}) (P - 1)} [{(\frac{1 - P x_{3} - D {\dot{x}}_{3}}{1 - x_{3}})}^{2} - {(\frac{1 + P x_{3} + D {\dot{x}}_{3}}{1 + x_{3}})}^{2}] \\ + \frac{α α_{1} x_{3}}{4 (1 - α_{1}) (P - 1)} [{(\frac{1 - P y_{3} - D {\dot{y}}_{3}}{1 - y_{3}})}^{2} + {(\frac{1 + P y_{3} + D {\dot{y}}_{3}}{1 + y_{3}})}^{2}] \\ - \frac{α_{1} (\sqrt{x_{3}^{2} + y_{3}^{2}} - n_{1}) H (\sqrt{x_{3}^{2} + y_{3}^{2}} - n_{1})}{(1 - α_{1}) λ_{2}^{2}} (\cos ψ_{D} - μ \sin ψ_{D}) \end{array}

(32)

\begin{array}{l} {\ddot{y}}_{3} + \frac{η_{8}}{α_{2} λ_{1}^{2}} (y_{3} - y_{2}) + \frac{η_{9}}{α_{2} λ_{1}^{2}} θ_{x} \\ = \frac{α_{1}}{4 (1 - α_{1}) (P - 1)} [{(\frac{1 - P y_{3} - D {\dot{y}}_{3}}{1 - y_{3}})}^{2} - {(\frac{1 + P y_{3} + D {\dot{y}}_{3}}{1 + y_{3}})}^{2}] \\ + \frac{α α_{1} y_{3}}{4 (1 - α_{1}) (P - 1)} [{(\frac{1 - P x_{3} - D {\dot{x}}_{3}}{1 - x_{3}})}^{2} + {(\frac{1 + P x_{3} + D {\dot{x}}_{3}}{1 + x_{3}})}^{2}] \\ - \frac{α_{1} (\sqrt{x_{3}^{2} + y_{3}^{2}} - n_{1}) H (\sqrt{x_{3}^{2} + y_{3}^{2}} - n_{1})}{(1 - α_{1}) λ_{2}^{2}} (cos ψ_{D} + μ sin ψ_{D}) \end{array}

(33)

The coefficients ω_n, ω₁, ω₂, and η_k (k=1, 2 … 9) are given in Appendix. These equations include nonlinearity due to the magnetic bearing forces, nonlinearity arising from the geometric coupling of magnetic actuators and nonlinearity resulting from contact forces.

Figure 2.

The geometry of AMB and rotor.

Figure 3.

Rotor - backup bearing contact model (point A).

Experimental procedure

The experimental setup is shown in Figure 4. A 3-phase motor (ECHTOP type MS-632-4) and an inverter (LS type SV-IE5) are selected to drive the rotor with speeds of up to 3000 rpm. A lightweight, flexible coupling is used between the shaft and motor. So, the rotor can move freely in radial direction. The shaft has one disk at its midpoint. The rotor is supported vertically in two AMBs and two backup bearings. The stiffness of the backup bearing supports is adjustable.¹⁵

Figure 4.

Experiment setup of the flexible rotor–AMB system with backup bearings.

Each AMB has four actuators along with four inductive proximity sensors (type IPS-303-AV14-P) in the x and y directions. The control system records rotor displacement data from the experiments and adjusts the position of the rotor by varying the control currents. In order to achieve this, the STM32F103RET6 has been used. Eight switching amplifier boards (bridges between the control currents and the applied currents in the coils of the actuators) are designed and constructed for overcoming the coil inductance and resistance. The physical specifications of the flexible rotor–AMB system with backup bearings are available in Table 1.

Table 1.

Physical specifications of the flexible rotor–AMB system with backup bearings.

Parameter	Value
Area of one magnetic pole, A_p	10⁻³ m²
External damping coefficient, C	0.001 N.s./m
Derivative feedback gain, $\bar{D}$ .	0.4 A.s/m
Young’s modulus of the shaft, E	2×10¹¹ N/m²
Radial clearance of AMBs, g₀	2×10⁻³ m
Radial clearance of backup bearings, g₁	10⁻³ m
Bias magnetic coil current, I₀	3.82 A
Backup bearings stiffness, k_a	9.81×10⁵ N/m
Length of the shaft, L	0.6 m
Mass of each shaft journal, m₁, m₃	0.066 kg
Mass of disk, m₂	0.741 kg
Number of coil turns, N	100
Proportional feedback gain, $\bar{P}$ .	2100 A/m
Radius of shaft, r	3 mm
Radius of disk, r₁	55 mm
Position of disk, α₁L	0.3 m
Friction coefficient of the backup bearings, µ	0.2
Permeability of free space, µ₀	4π×10⁻⁷ H/m

Numerical results and discussion

For numerical analysis of the non-dimensional equations (26–33), they are transformed into the first order differential equations. The direct numerical integration is performed by variable step Runge–Kutta method. In order to eliminate the transient part of the response, the first few hundred time-series data of the integration are discarded and the next few hundred time-series of steady state response are used for analysis. The bifurcation diagrams, dynamic orbits, FFT plots, Poincaré maps, and maximum Lyapunov exponent are obtained from the steady state response.

A bifurcation plot is a visual summary of the steady state solution versus the control parameter. The bifurcation diagram can be employed for exploring the routes to chaos. In this paper, the speed parameter S, is used as bifurcation control parameter. The corresponding variations of the dimensionless vibrational response x₁ versus S are plotted to form the bifurcation diagram. The solution trajectories in the dynamic orbit provide easy visualization of the periodic or non periodic motions. FFT plot shows the frequency content of the motion. The Poincare´ map includes the time-series at a constant interval of 2π/Ω. The Poincare´ map of a chaotic motion has a geometrically fractal structure.²

The bifurcation diagrams in Figure 5 describe the effect of the speed parameter S on the response of the magnetically supported flexible rotor system for different values of U=0.05, U=0.1, and U=0.15.

Figure 5.

Bifurcation diagrams of x₁(nT) versus speed parameter S for (a) U=0.05 (b) U=0.1 (c) U=0.15.

For unbalance parameter U=0.05 (Figure 5(a)), the periodic motion is found at S=[0.005 ∼ 0.15]. However, a jump phenomenon happens at S=0.15. The system motion loses its regularity at S=[0.155 ∼ 0.27], but periodic vibrations will be exhibited when S=[0.275 ∼ 0.835]. The periodic motion changes to nonperiodic attractor at S=0.840. This trend will continue up to S=1.205. Finally, the rotor is transferred to periodic regime for S≥1.210.

For unbalance parameter U=0.1 (Figure 5(b)), the periodic motion is found at S=[0.005 ∼ 0.175]. However, a jump phenomenon happens at S=0.145. The system motion loses its regularity at S=[0.18 ∼ 0.27], but periodic vibrations will be exhibited when S=[0.275 ∼ 0.740]. The sudden bifurcation from periodic into nonperiodic regime is observed at S=0.745. This trend will continue up to S=0.79. The irregular motion of the system transforms into a sub-harmonic motion with the period 8T at S=[0.795 ∼ 0.815]. The motion of system at S=[0.82 ∼ 0.90] undergoes another sub-harmonic motion with the period 4T. Nonperiodic motion of the system reappears at S=[0.905 ∼ 1.045]. From S=1.05 to S=1.185, the nonperiodic motion is replaced by a 5T-periodic motion. At S=[1.19 ∼ 1.315] the system gets into nonperiodic motion. Finally, the rotor is transferred to periodic regime for S≥1.320.

For unbalance parameter U=0.15 (Figure 5(c)), the periodic motion is found at S=[0.005 ∼ 0.195]. However, a jump phenomenon happens at S=0.165. As the speed parameter S is increased, the motion of system becomes nonperiodic at S=[0.2 ∼ 0.31]. The nonperiodic motion of the system changes to a sub-harmonic motion with the period 3T at S=[0.315 ∼ 0.35]. For speed parameter in the range S=[0.355 ∼ 0.405], nonperiodic vibration occurs in the rotor response. At S=[0.41 ∼ 0.74], the system repeats periodic motion. The response of system jumps to irregular attractor at S=0.745. This trend will continue up to S=0.775. The nonperiodic motion of system at S=[0.78 ∼ 0.875] is replaced by the periodic motion. Nonperiodic motion of system reappears at S=[0.88 ∼ 1.39]. Finally, the rotor is transferred to periodic regime for S≥1.395.

A comparison of the bifurcation diagrams shown in Figure 5 reveals that by increasing the unbalance eccentricity (U=0.05, U=0.1, and U=0.15), the first irregular motion initiates at much higher speed parameters (S=0.155, S=0.18, and S=0.2). However, the higher values of unbalance eccentricity lead to smaller range of periodic motions. Also, the irregular motions in the flexible rotor–AMB with backup bearings can be eliminated by increasing the speed parameter S.

For the experimental verification of the simulated results at U=0.05, U=0.1, and U=0.15, a mass unbalance 5 gr is added at three radial positions to the center of the disk, respectively. The remaining physical parameters are kept constant in each experiment (as listed in Table 1) and the rotational speed is varied. After reaching the desired rotational speed, experimental data of the steady state vibration have been collected using the LabView data acquisition system.

Figure 6 exhibits a comparison between the theoretical and experimental results of the flexible rotor–AMB with backup bearings at U=0.05, S=1.0; U=0.1, S=0.2; U=0.1, S=1.5; U=0.15, and S=0.325. A full experimental explanation of Figure 6 is contained in Table 2.

Figure 6.

Comparison of theoretical and experimental results of the flexible rotor–AMB with backup bearings at U=0.05, S=1.0; U=0.1, S=0.2; U=0.1, S=1.5; U=0.15, S=0.325. (a) Theoretical orbit plot at U=0.05, S=1.0, (b) Experimental orbit plot at U=0.05, S=1.0, (c) Theoretical FFT plot at U=0.05, S=1.0, (d) Experimental FFT plot at U=0.05, S=1.0, (e) Theoretical orbit plot at U=0.1, S=0.2, (f) Experimental orbit plot at U=0.1, S=0.2, (g) Theoretical FFT plot at U=0.1, S=0.2, (h) Experimental FFT plot at U=0.1, S=0.2, (i) Theoretical orbit plot at U=0.1, S=1.5, (j) Experimental orbit plot at U=0.1, S=1.5, (k) Theoretical FFT plot at U=0.1, S=1.5, (l) Experimental FFT plot at U=0.1, S=1.5, (m) Theoretical orbit plot at U=0.15, S=0.325, (n) Experimental orbit plot at U=0.15, S=0.325, (o) Theoretical FFT plot at U=0.15, S=0.325, (p) Experimental FFT plot at U=0.15, S=0.325.

Table 2.

Summary of the experimental extraction.

Experiment	Figure concerned	Parameter U	Mass eccentricity u (mm)	Radial position to the center of the disk (mm)	Parameter S	Rotational speed Ω (rpm)
1	6 (b), 6(d)	0.05	0.1	14.8	1.0	1772
2	6 (f), 6(h)	0.1	0.2	29.6	0.2	355
3	6 (j), 6(l)	0.1	0.2	29.6	1.5	2658
4	6 (n), 6(p)	0.15	0.3	44.5	0.325	576

Comparison of the theoretical and experimental results shows a good degree of similarity. Complex looped pattern (Figure 6(a), (b), (e), and (f)), one circle (Figure 6(i) and (j)), and multiple ellipsoids (Figure 6(m) and (n)) in the orbit plots reveal the irregular (quasi-periodic or chaotic), periodic, and sub-harmonic motions, respectively.

The evidence of many frequency components (Figure 6(c), (d), (g), and (h)), a single basic frequency (Figure 6(k) and (l)) and distinct peaks (Figure 6(o) and (p)) in the FFT plots show the irregular, periodic, and sub-harmonic motions, respectively.

Figure 7 illustrates the Poincaré map of m₁ at U=0.05, S=1.0; U=0.1, S=0.2; U=0.1, S=1.5; U=0.15, and S=0.325. Regular pattern of points (Figure 7(a)), unevenly scattered points (Figure 7(b)), one distinct point (Figure 7(c)), and three distinct points (Figure 7(d)) in the Poincaré maps represent the quasi-periodic, chaotic, periodic, and 2T-periodic, respectively.

Figure 7.

Poincaré map of m₁ at (a) U=0.05, S=1.0 (b) U=0.1, S=0.2 (c) U=0.1, S=1.5 (d) U=0.15, S=0.325.

The largest Lyapunov exponent can be used as a beneficial technique to confirm the existence of chaos in the dynamical systems. The Lyapunov exponent associated with a trajectory is a quantity of the average rate of the exponential expansion and contraction of close trajectories. If the largest Lyapunov exponent is positive the dynamical response of CNT is chaotic.^2,16 In this paper, the largest Lyapunov exponent is calculated using the Wolf’s algorithm.¹⁷

According to Figure 8, the maximum Lyapunov exponents is positive which confirms the system behaves chaotic motion over the intervals U=0.15, S=[0.2 ∼ 0.31]; U=0.15, S=[0.355 ∼ 0.405], U=0.15, S=[0.745 ∼ 0.775], U=0.15, S=[0.88 ∼ 1.39], U=0.10, S=[0.18 ∼ 0.27], U=0.10, S=[0.745 ∼ 0.79], U=0.10, S=[0.905 ∼ 1.045], U=0.10, S=[1.19 ∼ 1.315], and U=0.05, S=[0.155 ∼ 0.27].

Figure 8.

Maximum Lyapunov exponents at (a) U=0.15, S=0.25 (b) U=0.15, S=0.375 (c) U=0.15, S=0.75 (d) U=0.15, S=1.2 (e) U=0.1, S=0.2 (f) U=0.1, S=0.75 (g) U=0.1, S=0.95 (h) U=0.1, S=1.25 (i) U=0.05, S=0.25.

Conclusions

This paper established an experimental and theoretical framework for the nonlinear dynamic analysis of flexible rotors supported by AMBs and backup bearings. The effects corresponding to the gyroscopic moments, flexibility of the shaft, and geometric coupling of the magnetic actuators were included in the theoretical model. A unique test rig was designed for the experimental validations. The bifurcation diagrams, dynamic orbits, FFT plots, Poincaré maps, and maximum Lyapunov exponent were used to identify the dynamic behavior of the system. The simulated results were in good agreement with the test. Following key findings have been achieved from this research:

Occurrence of the first irregular motion can be delayed by increasing the unbalance eccentricity.

A wider range of periodic motion can be achieved by decreasing the unbalance eccentricity.

The chaotic motions of flexible rotor–AMB with backup bearings were more frequently observed at lower rotational speeds. So, it is possible to suppress chaos in the system by increasing the rotational speed.

Thus, by changing the system parameters to suitable values, the system responses can avoid undesirable behavior.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Reza Ebrahimi

Appendix

\begin{array}{l} ϖ_{n}^{2} = \frac{μ_{0} N^{2} A_{p} I_{0}^{2} (P - 1)}{m_{1} g_{0}^{3}} ϖ_{1}^{2} = \frac{3 E I}{L^{3} m_{2}} ϖ_{2}^{2} = \frac{k_{a}}{m_{1}} \\ η_{1} = \frac{1}{α_{1}^{3}} η_{2} = \frac{1}{α_{1}^{2}} η_{3} = \frac{1 + 3 α_{1}^{2} - 3 α_{1}}{α_{1}^{3} {(1 - α_{1})}^{3}} \\ η_{4} = \frac{1}{{(1 - α_{1})}^{3}} η_{5} = \frac{(1 - 2 α_{1})}{α_{1}^{2} {(1 - α_{1})}^{2}} η_{6} = \frac{1}{α_{1} (1 - α_{1})} \\ η_{7} = \frac{1}{{(1 - α_{1})}^{2}} η_{8} = \frac{α_{1}}{{(1 - α_{1})}^{4}} η_{9} = α_{1} η_{4} \end{array}

Appendix

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