The magnetic suspended dual-rotor system (MSDS) can effectively increase the thrust weight ratio of aeroengines. However, the MSDS dynamic characteristics have rarely been investigated. In this research, a MSDS with the outer rotor supported by two active magnetic bearings (AMBs) is designed, and the PID control is employed. The Riccati transfer matrix method using complex variables is adopted to establish the MSDS dynamic model. Subsequently, the influences of AMBs’ control parameters on the MSDS dynamic characteristics are explored. According to the analysis, two rigid mode shapes remain unchanged with the variation of the relationship between their corresponding damped critical speeds (DCSs). Moreover, the rigid DCSs disappear with large derivative coefficient. Eventually, the validity of the dynamic model and the appearance of rigid DCSs are verified.
Considering many advantages of active magnetic bearings (AMBs) over conventional bearings such as contactless operation, low power consumption, high reliability, adjustable dynamic characteristics, and extended life [
It is already known that the prediction of critical speed of rotating systems is indispensable to the rotor design. With negligible damping in the bearing-rotor system, undamped resonant frequencies are calculated as the conventional critical speeds. Generally, damping is the basic parameter of rotor dynamic analysis, and it possesses a great effect on the dynamic characteristics of various types of bearing-rotor systems. In the general case, with the consideration of damping, it is the damped natural frequencies that are to be determined, namely, the damped critical speeds (DCSs), which establish the actual critical speeds of bearing-rotor systems, including the stiffening effect of the damping in the bearings. These results give a more realistic base than the conventional critical speed calculation for assessing any potential trouble from passing through or operating close to a critical speed [
As one of the key components of MSDS, in the last few decades, AMBs have been under heavy development and found applications in rotating machinery where conventional bearings might not be feasible. Researches on the bearing characteristics of AMBs have drawn an increasing attention. In addition to the bearing capacity models of AMBs taken flux leakage [
On the other hand, the dual-rotor structure, namely, the core component of the MSDS, is widely used in aeroengines. Researches on the dynamic characteristics of dual-rotor systems are of great significance for lowering vibration and noise and enhancing their overall performance. Considerable theoretical work has been carried out on dual-rotor systems supported by mechanical bearings, covering various aspects like critical speeds, unbalance response [
In this research, a structure of MSDS is designed, whose inner and outer rotors are interlinked by two rolling bearings as interbearings and the outer rotor is supported by two AMBs with PID control. Then, the Riccati transfer matrix method using complex variables and the fully coupled unit model are employed to develop the dynamic equation of the system, which is solved by Muller’s quadratic interpolation technique. Subsequently, the influences of the AMBs’ control parameters on the DCSs and stability are carried out. Finally, the dynamic model and some of the conclusions are verified.
The structure and theoretical model of the MSDS are depicted in Figure
(a) Structure and (b) theoretical model of the MSDS.
The bearing characteristics are the foundation of the rotor dynamics analysis for either AMBs or traditional mechanical bearings. In mechanical bearings, the stiffness and the damping are the commonly used parametric representations for bearing characteristics, and they are also used for AMBs, namely, the equivalent stiffness and the equivalent damping [
For an eight-pole AMB with differential drive mode, the basic control loop and control block diagram in the
(a) Basic control loop and (b) control block diagram of the AMB.
Equation (
The mathematical model of the MSDS is shown in Figure The position where the magnetic force is applied is considered as the place where the displacement is tested by the sensor The dynamic properties of bearings are represented by 8 linearized coefficients, four ( Flexibilities and cross-coupling force effects of the foundation are ignored
In the MSDS, there are sixteen state variables, two deflections, two slopes, two bending moments, and two shear force components for inner and outer rotors, respectively. The Riccati transfer matrix method divides the sixteen state variables
(a) Fully coupled unit model and (b) its sign convention of the MSDS.
In the fully coupled unit model, it is assumed that the inner and outer rotors are divided into many elements
Considering the MSDS represented by
The partitioned matrices
Introducing a Riccati transformation at station
According to equations (
Since the left-hand side boundary conditions are
With the right-hand side boundary conditions
However, there are many opposite-sign-infinite-type singularities in the residual quantity curve of Δ1, and this makes the opposite-sign intervals and root-existence intervals not one-to-one correspondences and results in wrong roots and leaking roots. In order to avoid wrong roots and leaking roots, the technique described in [
For a given operating condition, the solution to the elliptical motion is assumed as
As mentioned above,
Obviously,
To demonstrate the application of the above analysis, simulation is carried out to investigate the dynamic characteristics of MSDS. The basic dimensions (unit, mm) of the dual-rotor are depicted in Figure
Model parameters of the AMB system.
Parameter | Description | Value |
---|---|---|
|
Vacuum permeability | 4π × 10−7 H/m |
|
Maximum bearing capacity | 240 N |
|
Nominal air-gap length | 0.5 mm |
|
Coil turn | 160 |
|
Bias current | 3.0 A |
|
Maximum control current | 3.0 A |
|
Sectional area of stator pole | 2.47 × 10−4 m2 |
|
Half angle between two poles | π/8 rad |
|
Gain of the power amplifier | 0.6 A/V |
|
Gain of the displacement sensor | 8 × 103 V/m |
In a dual-rotor system, co-rotation and counterrotation are two possible operation modes. The former one is with both rotors co-rotating and the latter one is with the rotors counterrotating with respect to each other. The main difference between the two operation modes is caused by gyroscopic moment. Due to the fact that gyroscopic moment has a minor effect on the amplitudes of DCS characteristics and it has little effect on the trends of DCS characteristics with the variations of AMBs’ control parameters, the typical case of co-rotation is considered in this research. The speed ratio
(a) Campbell diagram and (b) logarithmic decrements of the dual-rotor system.
Damped critical speeds (r/min).
Order | Inner rotor excitation | Outer rotor excitation |
---|---|---|
1 | 8132 | 8132 |
2 | 9763 | 9356 |
3 | 31315 | 31203 |
Figure
The first three mode shapes of the MSDS are shown in Figure
Mode shapes of the dual-rotor system (—, inner rotor; ---, outer rotor). Mode shapes excited by the (a) inner rotor and (b) outer rotor.
As stated above, the equivalent stiffness and equivalent damping are dependent on the controller with the AMB’s size and structure determined, and they generally have substantial effects on the DCSs, vibration modes, and system stability. Therefore, it is necessary to investigate the influence of PID control parameters on the MSDS dynamic characteristics.
The effect of proportional coefficient on the MSDS dynamic characteristics with
The effect of proportional coefficient on the MSDS dynamic characteristics. (a) Campbell diagram. (b) Logarithmic decrements.
The effect of derivative coefficient on the MSDS dynamic characteristics. (a) Campbell diagram. (b) Logarithmic decrements.
The effect of integral coefficient on the MSDS dynamic characteristics. (a) Campbell diagram. (b) Logarithmic decrements.
The effect of derivative coefficient with
The effect of integral coefficient with
From the above analysis, the proportional and derivative coefficients possess significant influence on the dynamic characteristics of the MSDS rigid modes, while the integral coefficient has a minor effect on the rigid modes. However, these PID control parameters have little influence on the dynamic characteristics of the first-order bending mode. In order to further explore the influence law of these parameters on the dynamic characteristics in detail, more cases are performed from the perspective of DCS characteristics.
The effect of derivative coefficient with
The effect of derivative coefficient on DCS characteristics (
In Figure
Mode shapes of MSDS with
Mode shapes of MSDS with
When the other parameters remain the same, the effects of derivative coefficient with
The effect of derivative coefficient on DCS characteristics (
The effect of derivative coefficient on DCS characteristics (
Another interesting observation in Figures
In this part, two cases corresponding to
The effect of proportional coefficient on DCS characteristics (
The effect of proportional coefficient on DCS characteristics (
In Figure
Similarly, two cases corresponding to
The effect of integral coefficient on DCS characteristics (
The effect of integral coefficient on DCS characteristics (
In Figure
In order to verify the above analysis, the validity of the model and the characteristic analysis are carried out. The model verification is performed by the DCSs of Lund’s rotor [
As stated above, a dual-rotor system gets degenerated into two straight rotors by setting all parameters related to interbearings equal to zero. The model is validated by the investigation of DCSs of Lund’s rotor [
DCSs of Lund’s rotor.
It is necessary to verify the dynamic model of the MSDS as the modelling of the dual-rotor structure is simplified. The model is verified by FEM due to its high precision [
The FEM model of the MSDS.
In the case of interbearing stiffness
Results comparison between FEM and the Riccati transfer matrix method.
Order | Damped critical speeds (Hz) | Logarithmic decrements | ||||
---|---|---|---|---|---|---|
Riccati | FEM | Error (%) | Riccati | FEM | Error (%) | |
1 | 135.53 | 133.16 | 1.78 | 4.05 | 3.90 | 3.85 |
2 | 162.72 | 143.65 | 12.28 | 4.86 | 4.79 | 1.46 |
3 | 521.92 | 514.41 | 1.46 | 0.093 | 0.081 | 14.82 |
Comparison of unbalance responses between FEM and the proposed model of the MSDS. Unbalance response of (a) disk 1 and (b) disk 3.
It is known from Figure
With
Transient response of disk 1. (a)
Transient response of disk 3. (a)
In this research, a MSDS consisting of inner and outer rotors, two interbearings to connect the two rotors, and two AMBs to suspend the outer rotor is designed. With PID control adopted for the AMBs, the dynamic model of the MSDS is developed by the Riccati transfer matrix method using complex variables and the fully coupled unit model. On this basis, the dynamic characteristics of the MSDS are investigated.
According to the analysis, the dynamic characteristics of the MSDS rigid modes are more sensitive to the proportional and derivative coefficients and less sensitive to the integral coefficient. In contrast, the dynamic characteristics of first-order bending mode are insensitive to the PID control parameters. Moreover, two interesting phenomena are captured. The first one is that the mode shapes corresponding to the first two order DCSs still remain invariant even when the value of the first order DCS is greater than that of the second order. The second one is that the first two order DCSs disappear when the derivative coefficient is greater than a certain value, which indicates that the vibration response due to rigid modes can be eliminated or reduced by increasing the equivalent damping of AMBs. Finally, the dynamic model is verified by FEM and the second phenomenon is tested by the transient response simulation of the MSDS.
In Section
Cross-sectional area of shaft, m2
8 × 8 partitioned matrices
Damping coefficients of bearings to support inner rotor, N·s/m; 2nd and 3rd indexes are force direction and velocity direction, respectively
Damping coefficients of bearings to support outer rotor, N·s/m,
8 × 8 partitioned matrices
8 × 1 displacement vector with constituents:
Young’s modulus, Pa
8 × 1 force vector with constituents:
Shear modulus, Pa
Imaginary unit
Sectional moment of inertia of inner and outer rotors, respectively, m−4
Polar and transverse moment of inertia, respectively, kg·m2
Bearing stiffness coefficients for the inner rotor, N/m; 2nd and 3rd indexes are force direction and displacement direction, respectively
Bearing stiffness coefficients for the outer rotor, N/m
Interbearing stiffness coefficients, N/m
The inner and outer rotor lengths, respectively, m
The inner and outer rotor masses, respectively, kg
Bending moments, N·m, 1st index is moment direction
Shear force,
λ +
Time, s
8 × 8 partitioned matrices
Radial shaft displacements, m
16 × 1 state vector with state variables:
Cross-sectional shape factor for shear deformation of the shaft
The frequency equation with singularity
The nonsingularity frequency equation
Angular displacement for inner and outer rotors in the
Angular displacement for inner and outer rotors in the
Real part of complex frequency
Poisson’s ratio
Density of the shaft, kg/m3
Imaginary part of complex frequency
Angular speed of the shaft, rad/s Subscripts
A massless elastic beam
Interbearing
An element defined in Section
Inner rotor
Rotor station number
Outer rotor
Left-hand side of an element
Right-hand side of an element.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (51405351), major projects of Natural Science Foundation of Hubei Province (2014CFA013), China Scholarship Council (no. 201808420108), and Hubei Youth Science and Technology Morning Light Project (2018B22).
Figure 1: the schematic residual quantity curve of classic RTMM. Figure 2: the schematic residual quantity curve of the improved RTMM by the authors.