This paper is to be submitting an identification method for the unbalance parameters of a rotor-bearing system. In the method, the unbalance parameters identification problem is formulated as the unbalance force reconstruction which belongs to solving deconvolution problem, in which the unbalance force is expressed in the time domain. The unbalance response is expressed by the convolution integral of Green’s function and the unbalance force. In order to avoid the unstable solution arising from the noisy responses and the deconvolution, a regularization method is adopted to stabilize the solution. Meanwhile, a searching of the sensitive measured point has also been carried out to confirm the robustness of the method. Numerical example and a test rig have been used to illustrate the proposed method.
In rotating machinery systems, undesirable vibration is caused by unbalance parameters which are inevitable due to asymmetric geometry, material inhomogeneity, manufacturing tolerances, and elastic deformations of the flexible rotor shaft during operations. However, reliable estimates of unbalance parameters are difficult to obtain with theoretical models due to many impact factors. On the contrary, identification methods based on experimental data from actual test conditions provide reliable unbalance characterization which avoids the complexity of exact system modeling and analysis. It is for this reason that designers of rotating machinery mostly rely on the experimentally identified unbalance parameters in their instability analysis.
Over the years, identification of the unbalance parameters is fundamental for the balance of a rotor-bearing system. The method of mechanical balancing is adopted to isolate the vibration arising from the inherent rotor unbalance, in which small amounts of unbalance mass are added and/or removed at specific locations to gain a satisfactory state of balance for a machine. Thus, identification technology of the unbalance parameters is improved with the development of the balancing technology [
Some researchers introduced optimization algorithm and inversed technology to identify the unbalance parameters. Han et al. [
In the paper, the unbalance parameters identification is formulated as the unbalance force identification problem. An identification method for the unbalance parameters of a rotor-bearing system based on the unbalance force reconstruction is proposed to avoid the ill condition from the measured response and the deconvolution. In the method, the unbalance force acting on rotor-bearing systems is to be determined based on the measured time domain responses coming from transient analysis. The ill condition problem of the unbalance force reconstruction will be treated by the Tikhonov regularization which can provide efficient and numerically stable solutions. In order to search for the sensitive measured point, the Finite Element Method (FEM) is adopted for the forward analysis to obtain transient responses with assumed unbalance parameters. Finally, numerical example and a test rig have been used to illustrate the identification method for the unbalance parameters.
The linearized motion equation of rotor-bearing systems is expressed as
In the paper, the rotor model of rotor-bearing systems is built by Euler beam elements and the consistent matrices’ approach described by Genta [
In fact, the unbalance force caused by the unbalance parameters can be structured by
In many practical situations, the radius
Considering the system that acted on the unbalance force is a liner and time-invariant system, the transient response of (
When this convolution integral in the time domain is discretized to
Equation (
Because the rotor-bearing system is linear, the total displacement can be gained by the linear superposition method. When a rotor-bearing system is acted on multi-source unbalance forces in two radial directions (
For presentation purposes, (
In order to reconstruct
In general, the identified unbalance force is inaccurate because of the noise data of measured displacement which can produce amplified errors of identified result in inverse problems. And Green’s matrix is sensitive to these errors.
Considering the noise data of measured displacement, (
When there is no additional information in the process of unbalanced force identification, a singular value decomposition (SVD) in the method of regularization is used to deal with the noise data of measured displacement [
Through (
If the filter function is selected by (
In actual practice, the ill posed inverse problem can be dealt with by regularization described above. It is often effective to select effective measurements that are sensitive to the parameter variation to avoid the ill posedness. The proposed solving strategy for identifying the unbalance parameters of a rotor-bearing system based on the unbalance force reconstruction is outlined in Figure
The identification method for the unbalance parameters of a rotor-bearing system.
In this method, firstly, transient response of numerical calculation based on the FEM can be obtained with assumed unbalance parameters and displacement responses numerically generated are treated as actual measurements after adding the measurement noise. Numerical simulation model of rotor-bearing systems is used to search for a strong correlation point between the displacement response and the unbalanced force by comparison of identified unbalance parameters with the assumed ones. Then, the response of a unit pulse generated numerically and the experimental response of the test rig at reference measured points are used to gain the unbalance force reconstruction, and the regularization method is adopted to deal with ill posedness. Finally, the unbalance parameters of the rotor-bearing system are identified from the unbalance force reconstruction based on the experimental data.
Before applying the proposed method to the actual test data, using the direct numerical simulation to search for strong correlation point between the displacement response and the unbalanced force, meanwhile the numerical simulation is used to examine the reliability and accuracy of the proposed method. A numerical simulation coming from the literature [
Parameters of the numerical model.
Parts | Parameters | Specifications | ||
---|---|---|---|---|
Shaft | Length (mm) | 425.0 | ||
Diameter (mm) | 10.0 | |||
Mass (kg) | 4 | |||
Diametral inertia (kg⋅m2) | 0.0046 | |||
Polar inertia (kg⋅m2) | 0.00786 | |||
|
||||
Disk | Diameter (mm) | 74.0 | ||
Thickness (mm) | 25.0 | |||
|
||||
The sliding bearing | Stiffness coefficients | Damping coefficients | ||
(MN⋅m−1) | (kN⋅s⋅m−1) | |||
|
||||
The first bearing |
|
24.10 |
|
71.30 |
|
−3.19 |
|
34.20 | |
|
72.20 |
|
48.30 | |
|
45.40 |
|
55.40 | |
|
||||
The second bearing |
|
25.80 |
|
72.90 |
|
−3.78 |
|
38.00 | |
|
81.20 |
|
52.00 | |
|
52.60 |
|
63.60 |
The model for the numerical simulation.
Statically determinate rotor model
The FEM model
The displacement responses in the time domain at nodes 2 and 20 are collected at 4000 rpm by using assumed unbalance parameters. 5% Gaussian noise is added in the displacement responses to simulate the actual measurement noise and to check the robustness of the proposed method for identifying the unbalance parameters. And the identification method was proposed to determine the multisource unbalance forces. The identified unbalance parameters will be compared with the corresponding actual unbalance parameters and the unbalance parameters identified by the impulse response in the literature [
The comparison results of the numerical simulation in Figures
Comparison of the unbalance parameters from numerical simulation.
Unbalance parameters position | Assumed unbalance, magnitude@phase |
Identified unbalance (from literature [ |
Identified unbalance, magnitude@phase |
---|---|---|---|
Disc 1 | 4.50@30 | 4.42@29.65 | 4.51@30.05 |
Disc 2 | 2.20@60 | 2.18@59.72 | 2.19@59.84 |
The identified unbalance forces on the first disk.
The horizontal unbalance force
The vertical unbalance force
The identified unbalance forces on the second disk.
The horizontal unbalance force
The vertical unbalance force
It is aimed at gaining the unbalance parameters of a rotor-bearing system from the measured displacement response. The test rig considered in this paper shown in Figure
Parameters of the test rig.
Parts | Parameters | Specifications | ||
---|---|---|---|---|
Shaft | Length (mm) | 490.0 | ||
Diameter (mm) | 10.0 | |||
Density (kg(m3)−1) | 7850.0 | |||
Young’s modulus (Gpa) | 205.8 | |||
Poisson ratio | 0.3 | |||
|
||||
Disk | Diameter (mm) | 80.0 | ||
Thickness (mm) | 15.0 | |||
|
||||
The sliding bearing | Stiffness coefficients | Damping coefficients | ||
(MN⋅m−1) | (kN⋅s⋅m−1) | |||
|
||||
The first bearing |
|
49.513 |
|
200.21 |
|
−29.591 |
|
21.507 | |
|
6.4475 |
|
702.16 | |
|
35.744 |
|
272.48 | |
|
||||
The second bearing |
|
200.021 |
|
874.03 |
|
−69.727 |
|
390.64 | |
|
51.759 |
|
361.34 | |
|
0.1 |
|
244.16 |
The test rig and schematic diagram.
Responses at the rotor spin speed of 2000 rpm.
The horizontal displacement response
The vertical displacement response
The displacement responses in the time history from the experimental data are used as inputs and the experimental unbalance parameters have been identified. Figure
The identified unbalance forces of the test rig.
The identified horizontal unbalance force
The identified vertical unbalance force
A comparison of the vibration amplitudes before and after balancing proved the usefulness of the proposed method when the actual unbalance parameters are unknown. The unbalance mass of 1.4734 g is added to the disk in Figure
Highest amplitude of measured points before and after balancing.
Highest amplitude- |
The effect of equilibrium |
Highest amplitude- |
The effect of equilibrium | ||
---|---|---|---|---|---|
Before balancing | After balancing | Before balancing | After balancing | ||
57.1744 | 23.5264 | 58.85 | 73.8255 | 25.2785 | 65.76 |
Experimental unbalance responses: before and after the balancing procedure.
The vertical displacement response
The horizontal displacement response
An identification method for the unbalance parameters of a rotor-bearing system is proposed. The main advantage of this method is that the states of a few measured points need to be measured only by one test which greatly reduced the number of physical experiments and cost and it is robust to the noise data of measured displacement, which is very critical to the practical situation. And the proposed method can be accurate and effective to obtain the dynamic load determined difficultly by some traditional methods.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work has been supported by a project supported by Scientific Research Fund of Hunan Provincial Education Department, China (Grant no. 15B057), Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant no. 20130161130001), National Natural Science Foundation of China (Grant nos. 11202076, 11202073), and the Changzhou City Science and Technology Support Program (CE20140027).