Dynamic Behavior of Magnetic Bearing System Under Narrow-Band Excitation

The purpose of this paper is to investigate the vibration characteristics and motion of random magnetic bearing systems with narrow band noise. Firstly, the multiple scale method is used to obtain the averaged equations. Secondly, it is found that there exist different motions with the various excitation forces and permeability values by the bifurcation diagram and maximum Lyapunov exponent based on the resulting average equations. Finally, the influence of narrow-band excitation on magnetic bearings is verified, and the conclusion shows that the thickness of the limit cycle increases gradually as the bandwidth of narrow-band excitation changes in a small range.


Introduction
The use of magnetic bearings has been an interesting topic of research in the last few years. It has many advantages such as being able to rotate at a very high speed in a low friction environment, to provide high efficiency and a low maintenance demand. Therefore these magnetic bearings have been used in industrial application such as pumps, turbines and on energy-storing flywheels [1]. The research on magnetic bearings has always been a problem of great concern in the engineering community. The magnetic 1 3 bearing system is a broad interdisciplinary coverage system from the nonlinear structure design to nonlinear dynamic behaviors.
On the one hand, scholars aimed at the internal structure of magnetic bearings. These researches focused on extending its service life by updating the internal structure of magnetic bearings. Reference [2] proposed an iron loss prediction model for a radial magnetic bearing of turbomolecular pump. The high temperature superconducting (HTS) bearings which suitable for helium cold compressor were studied in reference [3]. Hutterer [4] described the development of a self-sensing unbalance rejection in combination with a self-sensing speed control of the motor controller. Sun [5] presented a new type of axial passive magnetic bearing (APMB) with low power consumption. The effectiveness of the proposed adaptive compensator for mass imbalance was carried out [6].
On the other hand, it is about the dynamic behavior of the magnetic bearing system. Most components of magnetic bearing systems are non-linear, so its dynamic behavior is extremely rich and interesting. The main research content of this category included many aspects [7][8][9][10], such as the characteristics of the system's stability and bifurcation [11,12], and motion state [13,14]. Among them, there were a lot of references on the vibration characteristics of magnetic bearings. Amer [15,16] investigated the response of the active magnetic bearings under the main resonance, and he drew the conclusions that there are different effects of system parameters. The multiple scale method was used to analyze the response of the magnetic bearing system in the primary and internal resonance models in reference [17].
According to the above researches, we can get that the research of magnetic bearing systems vibration is very meaningful. However, the external excitation force was considered as a constant value in most studies, and the random parameter factors are studied in this paper [18,19]. In this paper, we consider the magnetic bearing system with narrow-band excitationthe which is more common on the system than the Gaussian noise. Reference [20,21] describes the response of two systems under the influence of random narrow-band noise parameters.
The structure of this paper is as follows. After the introduction, we first establish a magnetic bearing model, which does not consider horizontal and vertical coupling under the excitation of narrow-band noise, and use a multiple scale method to derive the average equation of magnetic bearing system dynamics. Next, Sect. 3 studies the motion state of the corresponding magnetic bearing system. We discusses the analysis of the dynamic behavior of the system with the bandwidth of narrow-band excitation using a combination of numerical calculation and numerical simulation in Sect. 4.

System Model
In this paper, we neglect the coupling of the magnetic bearing system in the x and y directions, then the system model is simplified as the following equation The multiple scale methods are now widely used in the study of deterministic system vibration problems. In recent years, this method has gradually been extended to the field of random vibration of nonlinear systems [22]. In order to study the main resonance of the magnetic bearing model, we further process the model. Multiply the small parameter before the external excitation, non-linear term and resistance, then the model is converted into the following form Considering the main resonance of the system under the influence of random narrow-band excitation, the random narrow band process can be described by many models. In this paper, we consider the form of narrow band random excitation as follows: where f is the amplitude determined by the random excitation, Ω is the center frequency, W(t) is the standard Wierner process, and > 0 is the bandwidth of the random excitation. We only consider the case where is sufficiently small.
When using a multiple scale method to study system resonance, two different time scales can be used, and the approximate solution can be written as follows where T 0 = t, T 1 = t are fast and slow scales, respectively. And we can record the form as . Then ordinary differential operators can be written into partial differential operators, as shown below We take Eqs. (3) and (5) into Eq. (2), and compare the order of the coefficient on both sides of the equation to get the following equation and We consider the 1:1 main parameter resonance response of Eq. (2), namely Ω ≈ , and introduce the tuning parameter , which has the following expression Let the generalized solution of Eq. (6) be of the form is the slow amplitude of the response. Substituting Eqs. (8) and (9) into Eq. (7), we have By eliminating the duration term in the Eq. (10) and making the right side equal to zero, the following equation is obtained The polar coordinate form of a in the Eq. (10) can be expressed as follows Then we take Eq. (12) into Eq. (11), and separate the real and imaginary parts to get the average equation of the system as follows and We make = T 1 + W(T 1 ) − (T 1 ) . Let write a in Cartesian form as follows The system's Cartesian equation can be obtained:

The Motion of System with = 0
We take = 0 . Then Eqs. (13) and (14) can be written as and In order to obtain a steady-state response, we make � = � = 0 , then the above formula becomes Eliminating the trigonometric function in the equation, we get the frequency response curve equation of the system as follows: To take = 0.9, 1 = −0.9262, 2 = −0.0113, 3 = = 1 , we choice f as 0.05, 0.1, 0.3 respectively. According to the expression of the amplitude-frequency response curve Eq. (20), we can see that both the force f and the magnetic permeability will affect it. Then, we use numerical simulation to study the force f and the magnetic permeability .
In numerical simulations, we first consider the effect of the force f on the system. At this time, we let the magnetic permeability = 0.9 , and the values of the magnetic force f are 0.05, 0.1 and 0.3, respectively. The results of the numerical simulation are shown in Fig. 1. There are three graphs in Fig. 1, of which Fig. 1a is drawn according to the Eq. (20) of the amplitude-frequency response curve, Fig. 1b, c is phase diagram and time history diagram of the system corresponding to Fig. 1a made by Cartesian Eq. (15). We find that with the force f increase, the response amplitude of the frequency response curve is increasing.
Next, we fix the value of force f to study the effect of the magnetic permeability on the system. We take f = 0.3 , and the magnetic permeability various from 0.1, 0.3 and 0.9. The values of the remaining parameters are the same as those in Fig. 1. According to the frequency response curve and the Cartesian equation, the nonlinear dynamic behavior of the system under different magnetic permeability can be plotted. The results are shown in Fig. 2.
It is shown that the influence of the magnetic permeability on the system is different from the influence of the force f on the system. Compared with Fig. 1, it can be seen from Fig. 2a that the amplitude-frequency response curve is

3
decreasing as the magnetic permeability increases, and the shape of the amplitude-frequency response curve changes. The curve changes from a smooth curve to a steep curve with the magnetic permeability decreases. The fact can also be seen from the phase track of the system in Fig. 2b and time history in Fig. 2c. In Fig. 2b, the phase diagram of the system changes with the decrease of magnetic permeability, from a closed circular curve to a closed curved line, and then to many curved lines, it shows that the motion state of the system will change with the change of magnetic permeability. In order to verify this conclusion, we draw a bifurcation diagram and a graph of the maximum Lyapunov exponent of the system about the parameter magnetic permeability as shown in Fig. 3. Figure 3a shows that the chaos exists in the parameter ∈ [0, 0.2] of the system. We can judge the motion state of the system based on the maximum Lyapunov exponent. As we all know, when the maximum Lyapunov exponent of the system is less than zero, the system performs the cycle motion or quasi-periodic motion, the system is progressively stable; when the maximum Lyapunov exponential of the system is greater than zero, the system generates chaotic motion. From the Fig. 3b, the maximum Lyapunov exponential of the system is between

The Response of System with ≠ 0
Let us consider the effect of noise on the system, that is ≠ 0 . At this time, the solution of the equation can be assumed as follows Among them, the value of * is determined by Eq. (13), and x, y are perturbation terms. Bring Eq. (21) into Eq. (13), and ignore the non-linear terms about x, y to get the linear equation at ( * , * ) is as follows where Equation (22) can be written as the following Itô formula where A = (c ij ) 2×2 , X = (x, y) T , B = (0, ) T . Let Ex and Ex 2 be first-and secondorder steady-state moments, which can be calculated by the moment method. For steady-state moments, dEx∕dT 1 = dEy∕dT 1 = 0 , then Ex = Ey = 0 . we can find Next, we use numerical simulation to analyze the magnetic bearing model. First, because the derivative of the unit Wiener process is Gaussian white noise, and its spectrum is constant, it is easier to implement in numerical simulation. Usually, a pseudo-random signal is taken in numerical simulation as follows where k is an independent and normally distributed variable within (0, 2 ] , N is a large integer, and when N → ∞ , the random process (t) defined by the above (21) = * + x, = * + y. (22) x � = c 11 x + c 12 y, y � = c 21 x + c 22 y + W � , (25) (t) = formula will converge to a correlation function and spectral density ergodic Gaussian steady state process. At this point, the equation can be written as Let = 0.05 and = 0.1 be plotted separately, as shown in Fig. 4. It can be seen from the figure that as the value of increases, the amplitude of the limit cycle formed by the phase diagram of the system becomes larger. And the thickness of the limit cycle becomes thicker. This result is consistent with the result Eq. (24) in our theoretical calculations. With the increase of , both Ex 2 and Ey 2 increase. The results also show that narrow-band noise excitation does affect the dynamic behavior of magnetic bearing system.

Conclusion
In the present paper, we have investigated magnetic bearings under constant excitation and random excitation using the method of multiple scale. It has been shown that various excitation force f and magnetic permeability can lead to different influence when the bandwidth = 0 . The dynamic system under constant excitation exhibits the rich system motion state. Besides, the random excitation has been considered, and the method of perturbation and moment has been used to obtain the Ito formula and second-order steady-state moments, respectively. One can observe that the random noise W(t) can change the response of system from a limit cycle to a diffused limit cycle. And the diffused limit cycle will increase with the value of increasing.