Modern control techniques can improve the performance and robustness of a rotor active magnetic bearing (AMB) system. Since those control methods usually rely on system models, it is important to obtain a precise rotor AMB analytical model. However, the interference fits and shrink effects of rotor AMB cause inaccuracy to the final system model. In this paper, an experiment based model updating method is proposed to improve the accuracy of the finite element (FE) model used in a rotor AMB system. Modelling error is minimized by applying a numerical optimization Nelder-Mead simplex algorithm to properly adjust FE model parameters. Both the error resonance frequencies and modal assurance criterion (MAC) values are minimized simultaneously to account for the rotor natural frequencies as well as for the mode shapes. Verification of the updated rotor model is performed by comparing the experimental and analytical frequency response. The close agreements demonstrate the effectiveness of the proposed model updating methodology.
As potential alternatives to conventional mechanical bearings, active magnetic bearings have been increasingly used in compressors, pumps, and many other high-speed rotating machineries [
The AMB system is open loop unstable and feedback control is needed for levitation. This function is commonly achieved by single input and single output (SISO) controller design such as proportional-integral-derivative (PID) controller [
For a rotor AMB system, its dynamic properties play a key role. The rotor model is often obtained by the finite element method (FEM) or the transfer matrix method. For a beam type structure, the transfer matrix and the FEM formulations are theoretically equivalent, but the FEM formulation is usually more numerically stable [
Dynamic modelling is an important consideration for structure design and almost always results in errors to some degree when compared to experimental result [
Although model updating approaches have been widely applied to other engineering field, few literatures can be found in applying model updating to rotor AMB system. Li et al. [
The MAC is widely used to evaluate the comparison of mode shapes. In this paper, after building a rotor finite element model, we update the rotor model using Nelder-Mead nonlinear unconstrained optimization method and provide an objective function, combining first four MAC values and bending frequency errors together. The advantages of this combination are that the mode shapes calculated by the updated model are highly accurate as well as the normal modal frequencies. By comparing the updated model’s dynamic characteristics with experimental data, the close agreements demonstrate that the proposed method is effective for AMB rotor model updating.
The remainder of the paper is organized as follows. Section
The experimental test rig for this study is five degrees of freedom (DOFs) rotor AMB system designed and built as a research platform at Nanjing University of Aeronautics and Astronautics, pictured in Figure
Specific details of the rotor.
Label | Name |
---|---|
A/L | Front/rear thrust AMB lamination stacks |
B/K | Magnetic isolation ring |
C/J | Front/rear radical AMB lamination stacks |
D/I | Front/rear sensor ring |
E | Rotor shaft |
F/H | Motor rotor fix stacks |
G | Motor rotor |
An overview of the rotor AMB test rig.
Rotor of the AMB system.
The nominal rotor finite element model has been created using beam finite elements. Previous experiences have confirmed that relatively simple beam element models are adequate for analyzing most rotors. Beam finite elements used currently have two or three nodes per element, with two nodes being most common since they are easy to model typical industrial rotor geometries. With four DOFs at each node it enables the simultaneous modelling of beam deflection in both horizontal and vertical planes [
The rotor finite element model is developed according to the geometrical and mass information, pictured in Figure
Theoretic mode shapes of free-free rotor.
The material of the rotor is a nickel-base superalloy with good corrosion resistance. The FE code compiled in MATLAB for this rotor does not contain the unknown interactions of shrink fit interfaces, inhomogeneous materials, small geometrical details, and so on, which cause potential errors to the modelling. For the rotor of this AMB system, its theoretical first four free-free bending modal frequencies and mode shapes calculated by the mathematical finite element model without model updating are listed and drawn in Table
Experimental and theoretical (no updating) modal frequencies.
Bending |
Calculated freq. (Hz) | Measured freq. (Hz) | Error |
---|---|---|---|
1 | 454.9 | 477.5 | 22.6 |
2 | 1137.6 | 1167.5 | 29.9 |
3 | 1802.7 | 1857.5 | 54.8 |
4 | 2733.2 | 2805 | 71.8 |
Modal experiment is implemented to update and verify the rotor mathematical model. The rotor is suspended vertically at the end with a string. One acceleration sensor is fixed on the rotor (node 40) and an instrumented impulse hammer is employed to impact the rotor at 17 different axial locations (nodes 1, 4, 7, 11, 15, 17, 20, 23, 29, 31, 35, 38, 40, 43, 46, 52, and 54). The noise and vibration analyser and the modal analysis software are employed to perform the modal test, pictured in Figure
Modal test devices.
Experimental free-free rotor mode shapes.
The modal assurance criterion [
Comparison for MAC value.
Bending |
MAC |
MAC |
---|---|---|
1 | 0.998 | 0.999 |
2 | 0.989 | 0.986 |
3 | 0.985 | 0.984 |
4 | 0.979 | 0.987 |
Many sources of uncertainties exist in the rotor modelling including unknown structure features and simplifications. To update the rotor finite element model so it can fit the experimental measurements, we need to select model design variables carefully. For the rotor studied here, its physical properties such as mass, geometry, and polar and transverse moments of inertia properties can be determined accurately but the rotor stiffness of some region is difficult to obtain since the front/rear radical and thrust AMB lamination stacks, sensors reference stacks and motor, and so on are shrunk fit with the rotor, which contribute to the bending stiffness. Therefore, in the rotor finite elements model, the modulus of elasticity for these uncertain areas (pictured in Figure
Model updating variables (stiffness values) corresponding to the physical rotor.
Updating variable | Description/position |
---|---|
|
Thrust AMB lamination stacks |
|
Radical AMB lamination stacks |
|
Sensor ring |
|
Motor fix stacks |
|
Motor rotor |
FE rotor model indicating locations of the 6 moduli of elasticity design variables.
Initialization of the updating process begins with the definition of an error function. Since the finite element model is different from actual system, the model updating method tries to minimize the difference. Here first four modal frequency errors and corresponding MAC are selected as updating target which are written as follows:
Weight factors for error targets.
Weight factor | Value |
---|---|
|
8 |
|
3 |
|
2 |
|
1 |
|
1 |
The smaller the error value, the more accurate the AMB rotor model. The essence of the minimization of
Optimization of the error function is an important step in the model updating. The Nelder-Mead simplex method is an optimization algorithm solving the multidimensional nonlinear problem, which is employed in this paper. It was first proposed by Nelder and Mead [
For an optimization problem involving
Firstly, replace the worst point
Second, if the reflected point
Third, if the reflected point
Fourth, if the reflected point
After the steps above, the method forms a new simplex by replacing the worst
The basic idea of the simplex method is to use the simplex peak to calculate the peak function values and compare them with each other. Then it finds a favourable search direction, step length, and a better point to replace the inferior peak point, which produces a new simplex to replace the original simplex. Thus, the simplex will continuously approach the minimum value of the objective function until it finds the minimum point.
After about 1500 iterations, an adequate solution is obtained, which are plotted in Figure
Comparison for modal frequencies.
Bending |
Updated |
Measured freq. (Hz) | Error |
---|---|---|---|
1 | 477.5 | 477.5 | 0 |
2 | 1167.5 | 1167.5 | 0 |
3 | 1855.4 | 1857.5 | 1.1 |
4 | 2827.7 | 2805 | 22.7 |
The modulus of elasticity values before and after updating.
Verification of the updated rotor AMB model is achieved by comparing the rotor analytical and experimental frequency response functions (FRFs). For the model FRFs calculation, a proportional damping can be assumed as
Figure
Experimental and nonupdated model frequency response comparison.
Experimental and updated model frequency response comparison.
The rotor dynamic properties play a key role in the rotor AMB system. In order to obtain an accurate rotor mathematical model, model updating techniques are often employed to update the unconstrained model. Generally, models are corrected by applying an appropriate error function in terms of resonance frequencies; however, mode shape accuracy has been shown to be insufficient. In this paper, a model updating approach considering both resonance and mode shape is proposed and the mode shape error target adopted in this paper provides high sensitivity when the MAC value is not close to one. The interference fits and shrink effects on the rotor shaft are considered during the variable selection and Nelder-Mead simplex method is employed to optimize the FE rotor model. In order to make sure the updated model agrees with the actual rotor, the FRFs comparison between updated model response and experimental data is performed. The results show that the updated theoretical model is fairly accurate compared to experimental data, which indicates the updating method proposed in this paper is feasible for the rotor AMB system.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research has been supported by the Natural Science Foundation of China (51275240, 51205186).