Abstract
Active magnetic bearings (AMBs) have been extensively used in the vibration reduction of rotating machinery because of active control technology. In fact, the mechanical characteristics of the magnetic bearing are also important for vibration isolation as the main vibration transmission path. In this paper, a finite element (FE) model of a 30 kW magnetic suspended prototype pump suspended by 5-DOF AMBs was established to simulate the vibration transmission of the system. Two external excitations, unbalanced force of the rotor and hydrodynamic force acted on the impeller were considered. Based on the model, several main factors that affect the vibration transmission of magnetic suspended pump were investigated, such as mass distribution of the system, the equivalent stiffness of the AMBs and the rubber vibration isolators were investigated. Based on the above work, a new magnetic suspended pump was developed. The results of simulation and experiments show that the new design can effectively reduce the vibration transmission of the system.
Introduction
Rotating machinery is an important type of power equipment in modern industry, including blower, pump, motor, turbine, aero-engine, etc. Vibration and noise are generated during the operation of rotating machinery, which not only seriously affect the service life of the machinery, but also pollute the environment, causing harm to the physical and mental health of operators [1]. The main vibration sources of rotating machinery include the centrifugal force caused by the asymmetric structure, material defects and assembly errors and the external excitation caused by electromagnetic force and running load, etc. The vibration is transferred to the bottom of the machine through bearings and then to the base through flexible supports. With the increase of speed and power, vibration suppression of rotating machinery becomes more and more important and difficult. Compared with traditional mechanical bearing, magnetic bearing has no mechanical wear, no contact and can work under ultra-high speed, which is an ideal supporting element for rotating machinery, particularly suitable for active vibration control of rotor [2].
Generally speaking, the AMBs can reduce vibration from three aspects: (1) Realizing self-balance of rotor. The AMBs can reduce or eliminate the unbalance of rotor [3]; (2) Add feedforward controller in AMBs by studying the external excitation force of rotating machinery and establishing the comprehensive excitation model [4]; (3) Adjusting control parameters in feedback controller to change the equivalent stiffness and damping of the magnetic bearing, so as to improve the stability of the system. All of the control strategies mentioned above can be concluded in upgrading vibration transmission characteristics of rotating machinery and reducing the vibration transmission [5].
In recent years, many researches have focused on AMBs in order to suppress the vibration on rotating machinery. Shiqiang Zheng, et al. [6], proposed a feedforward control strategy combined with a novel adaptive notch filter to solve the rotor unbalance problem in magnetic suspended centrifugal compressors. The scheme applied a new type of adaptive notch filter to identify unbalance signal of displacement, and used it as a compensation signal to eliminate the unbalanced current components. At the same time, the controller contains a feed-forward control signal with an inverse power amplifier model to compensate the negative displacement stiffness caused by the effects of low-pass characteristic of power amplifier. Qi Chen, et al. [7], put forward another solution to the same problem. They proposed a double-loop compensation design approach based on the AMBs. The first circuit is to use the notch filter to measure the unbalanced characteristic displacement signal of the rotor system. The second circuit is a simple feedback loop at low speed, which is cut at high speed and replaced with an adaptive circuit to adjust the control current to suppress the influence of low pass characteristics of power amplifier at high speed. Zhang Kai, et al. [8], designed a comprehensive controller for the AMBs on a turbo molecular pump, using different methods to suppress different kinds of vibration. Santosh Shelke [9], raised a controller design methodology of four pole pair active magnetic bearings. Then they carried out investigations of the radial forces in two axes model as well as the relative performance response through the PID controller system.
Magnetic suspended pump.
Description of the pump
Meanwhile, some researchers paid attention to the structure design of the AMB itself. Alexei Filatov, et al. [10], studied the difference in vibration reduction among different composite structures of AMBs based on a 300 kW turbo compressor under the designed speed of 40,000 rpm. Three kind of actuator arrangements were covered: Heteropolar electrically-biased radial actuator and conventional electrically-biased axial actuator; Heteropolar electrically-biased radial actuator and electrically-biased axial actuator with low target OD; ‘Side-By-Side’ homopolar permanent-magnet-biased combination axial/radial actuator. According to the numerical calculations, for the prototype compressor, the third bearing has good performance, but its development is subject to price and technology. Yushi Wen [11] worked on the structural parameter design of thrust magnetic bearing and radial magnetic bearing. He established the optimization model of radial magnetic bearing and the finite element analysis system for the dynamic characteristics of magnetic bearing. On this basis, the selection of the default parameters of the optimization system was discussed. Jianqiang Zheng [12] established a 3D-FEM model of MBs to carry out the structural design of the magnetic bearing, in the simulation model, the nonlinear of the material and the unbalance of the rotor was considered.
Magnetic suspended pump model.
Model instroction
In summary, most of the researches concentrated on control methods or the structure of the AMBs to reduce vibration, however, the mechanical characteristics of the bearing and the whole structure are also important for vibration isolation since they are the main vibration transmission path.
Hydrodynamic force.
This article paid attention to the mechanical structure characteristics and established a finite element model based on a 30 kW prototype pump suspended by 5-DOF AMBs to investigate the key factors that affect system vibration. Based on this research, a dynamics optimized design was proposed. Then the corresponding mode shapes as well as the response of the seat vibration were calculated based on the FE model. Besides, corresponding experiments were carried out to verify the effectiveness of the optimal design. The result shows that the modal frequency can be avoided well and the seat acceleration vibration level can be reduced by approximately 2 dB.
Model establishment
The original magnetic suspended prototype pump shown in Fig. 1 was mounted vertically through four flexible rubber vibration isolators at the corners of the mounting flange. Its original mechanical bearings were replaced a 5-DOF AMBs system. More details about the pump are listed in Table 1.
Analytical and numerical solution of modal frequencies
Analytical and numerical solution of modal frequencies
The finite element model is shown in Fig. 2. The rotor is modeled by beam elements. The pump case is simplified as lumped mass. The AMBs are simplified as spring-dampers with equivalent stiffness and the damping related to rotational speed, and the flexible rubber vibration isolators are simplify as rubber vibration isolator three direction spring-dampers. All the components and the whole model have passed the grid independence validation. The whole model contains 27014 solid elements, 67 beam elements, 4 integrated mass elements and 14631 constraint elements. More details of the model can be found in Table 2.
Modes that may cause resonance.
Vibration response on seats in transient state.
Vibration response on seats in steady state.
Two external excitations were considered, including an unbalanced force of the rotor and a hydrodynamic force acted on the impeller. The unbalanced force can be obtained by ISO 1940 under the dynamic balancing precision of G6.3. The hydrodynamic force can be simplified as a time-domain force obtained from a CFD model. The frequency of unbalance force is 50 Hz and the main frequency components of the hydrodynamic force are 350 Hz and 700 Hz, the base frequency of which equals the rotating speed multiplying the number of blades. The fast Fourier transformation results of the hydrodynamic force is shown in Fig. 3. It gives a frequency-domain representation of the hydrodynamic force and may help to simplify the external force model.
FFT transformation of the vibration response.
Modal analysis
Based on the established model, it is easy to obtain the modal information of the pump (in Table 4). The first modal frequency is 4.976 Hz, that is close to the experimental result 5 Hz (obtained by model test). If regarding the pump as a rigid component, the analytical solution of the top six modal frequency can be obtained. The relative error is less than 5% (shown in Table 3).
Modes of the pump
Modes of the pump
Vibration displacement level of the seats before and after simplification
According to the modal analysis result of the pump, the modes which might be excited by external forces are the rigid modes of the rotor, the first and the second bend modes of the rotor (shown in Fig. 4).
Comparison between the original model and the simplified model
Comparison of vibration displacement level between the original model and the simplified model
The seat acceleration vibration level of the pump is shown in Fig. 5 and Fig. 6. It is obvious that the response consists of two parts. One is the free vibration that will decay over time because of the damping of the rubber vibration isolator, the other is the forced vibration response under unbalance force and hydrodynamic force. What we care about is the latter part.
Comparison of modal frequency between the original model and the simplified model
Comparison of modal frequency between the original model and the simplified model
In Fig. 7, during the time 0 ∼ 0.1 s, the external force in the x direction stimulated the horizontal vibration of the pump with a modal frequency of 20 Hz. The force in y direction stimulated the horizontal translation motion of the pump with frequencies of 5 Hz, 7 Hz and 11 Hz. The corresponding motions are the pump oscillation modes in the x and z directions and the translation mode in the y direction. The exciting force in the z direction stimulated the horizontal motion mode of water pump in the z direction with frequency of 11 Hz. These excitation vibration were caused by the sudden application of the excitation force, which decayed over time. The damping time was about 0.8 s, that means the calculation of the first 0.8 s is not important. By 0.8 s, these free vibration responses have almost disappeared. At this time, the main vibrations are caused by hydrodynamic force and unbalanced force.
Vibration acceleration level of the seat after simplification
The influence of mass increase on modal frequencies
The model presented above is still too complex to be used for optimized design. According to the dynamic analysis, the model can be simplified from two aspects. Firstly, the decay time may be cut down by modifying the external force model. Secondly, the FE model can be further simplified according to the multi-body dynamics to reduce the complex structure, so as to reduce computational expense.
The influence of mass increase on seats response
The influence of mass increase on seats response
It can be seen from Fig. 3 that the hydrodynamic force is quite complicated in time domain. However, its frequency-domain expression is relatively concise. According to the FFT result (in Fig. 3), the hydrodynamic force can be approximately considered as the superposition of several sine forces on the basis of constant forces in three directions. The steady state response of the system is a small vibration at the equilibrium position. In fact, the constant force merely affects the equilibrium position of the system. In order to simplify the external force model, it is deemed that the influence of the equilibrium position is negligible. After simplification, sine forces are directly applied to the system. The responses are shown in Table 5.
Vibration transmission path 1.
Vibration transmission path 2.
Influence of front bearing stiffness.
In order to reduce computational expense, we further simplified the FE model on the premise of high calculation precision, which laid a foundation for studying the vibration influence factors of magnetic suspended pump.
In simplification, rear stator components, front stator components, motor stator, flange and the pump case were regarded as rigid bodies. The rotor-bearing system and motor shell were regarded as flexible bodies, with motor shell simulated by shell elements. In this way, the freedom of the system is greatly reduced. As shown in Table 6, the number of elements in the simplified model is 2.77% of that in the original model, which improves the computation efficiency.
Influence of rear bearing stiffness.
Influence of bearing stiffness.
In order to verify the rationality and accuracy of the simplified model, the error between the two models is calculated, and the cause of relative error is analyzed. The results of modal analysis, and transient dynamic analysis are shown in Table 7 and Table 8.
Table 8 compares the first 20 vibration modes. Because the 16th, 17th and 21st modes disappeared, the number was extended to 23. The maximum difference in the first 20 vibration modes is less than 4%, and the maximum error occurs in the mode that the rotor oscillating around x or z axis. It is because the simplified model partially increases the stiffness of the system. The comparison of transient dynamic analysis results are shown in Table 7. The response error is less than 2.5%, indicating that the simplification has no obvious effects on the response of the system. At the same time, it also indicates that the deformation of front/rear stator components and flange is small.
Based on the new model, the seat acceleration vibration response can be calculated (shown in Table 9).
Vibration path
In order to carry on the vibration isolation design to the water pump, several structural factors are analyzed from the perspective of vibration transmission path.
As shown in Fig. 8, both hydrodynamic and unbalanced forces are transferred to the seat through the rotor. Hydrodynamic force also has another transmission path, which is transferred to the base through the pump case, as demonstrated in Fig. 9. However, the second transmission path involves complicated fluid-solid coupling problem, which is beyond the scope of this paper. This article only takes the first path into account.
Through the analysis of the vibration transmission path, vibration influence factors can be investigated from three aspects: rotor-bearing system, the pump body and rubber vibration isolator. As the rotor size is hard to change, this article just studied the influence of bearing stiffness in rotor-bearing system. About the pump body, this article discussed the influence of mass distribution on vibration. As for rubber vibration isolators, the type and mounting were explored.
Research on factors influencing the vibration
Influence of bearing stiffness
The influence of the front bearing stiffness on the modes of pump only reflects on the third and fifth modes, that is, the horizontal translation motion modes of water pump in two radial directions (Fig. 10a). The relevant modal frequencies upgrades with the increase of the stiffness, and the growth rate decline gradually. When pump moves in the x or z direction, there is a slight relative motion between the rotor and the rest. With the increase of bearing stiffness, the relative motion is restrained and the overall stiffness increases. If the bearing stiffness attains a certain magnitude, the rotor is nearly fixed and the contribution of the further increase to the overall stiffness is negligible.
In terms of rotor modes, modal frequency of the oscillation mode swinging around the rear bearing rises with the increase of the front bearing stiffness (Fig. 10b). While the front bearing stiffness exceeds 107 N/m, the “oscillation modes” transform into the overall movement of the pump. The vibration of the rotor will drive the entire pump body swinging. In addition, the rotor first bending modal frequency will ascend with the increase of bearing stiffness after 107 N/m.
In Fig. 10c, at 4 × 106 N/m and 5 × 107 N/m, the radial seats response has obvious peak. At 4 × 106 N/m, the modal frequency of rotor oscillating around the rear bearing are 49.936 Hz and 50.628 Hz, and the unbalanced force will cause resonance. At 5 × 107 N/m, the first bending modal frequency of the rotor are 345.993 Hz and 355.604 Hz, and the hydrodynamic force will cause resonance. Axial response also changed with the stiffness of radial bearing. Because the response contains the component of radial oscillating mode.
The influence of rear bearing stiffness on vibration is similar to that of front bearing (Fig. 11). In Fig. 11c, the modal frequency of the rotor oscillation mode around the front bearing is always around 50 Hz, so the response is relatively large.
Influence of front stator components mass.
Influence of back stator components mass.
Provided that two radial bearings have the same stiffness, we can obtain the influence of the bearing stiffness on the pump mode and on the response. Comparing Fig. 12 with Fig. 11 and Fig. 10, the influence can be approximately regarded as the sum of that of front and rear bearing stiffness. As stiffness changes, the modal frequency of the pump translation mode and the rotor oscillation mode changes.
Influence of rubber vibration isolators type and mounting.
In addition, the conclusion about axial bearing can also be obtained. The increase of the axial bearing stiffness only affects the modal frequency of the rotor axial vibration, and has little contribution to response.
In terms of modal frequency, the mass of the front and back stator components have great effects on the modes of oscillation and translation of pump, that is, the top six modes of the system. The front stator components mass will change the three translation motion modal frequencies. The back stator components mass will change the two oscillation motion modal frequencies and the translation motion on the y axis. In fact, the horizontal “translational vibration” of the whole pump are not absolute translational motion, but oscillations the rotating center of which is away from the pump itself. From this point of view, influence of front and rear components mass on modal frequency has certain similarity and symmetry.
Optimized design parameters and original design parameters
Optimized design parameters and original design parameters
Comparison of modal frequencies between optimized design and original design
As for response, the influence of front and rear components mass is different. With the increase of the front components mass, the radial response decrease and the axial response increase. In contrast, with the increase of the back components mass, the radial response of the seats is unchangeable and the axial response decrease.
In addition, the simulation shows that the thickness of the motor shell has little influence on modes and response. The position of the junction box, although inducing asymmetry of the mass distribution of the motor shell, does not have great effects on vibration, due to its small weight (3.75 kg).
Therefore, it can be concluded that for vertically mounted pumps, the mass at the top and bottom has a great influence on vibration, while the influence of the middle part is relatively small (in Table 10 and Table 11). From this point of view, the vertically mounted pump can be approximately regarded as an inverted pendulum.
Comparison of acceleration response between optimized design and original design
Actually, the influence of mass distribution is more useful in structure design part. Good mass distribution design can reduce difficulties for the subsequent controller parameter design. Fig. 13 and Fig. 14 illustrate that mass distribution do affect some modal frequencies, but quantitatively limited. Considering the mass of the stator components may hardly change over 30 percent, by adjusting the mass distribution, we can reduce response by up to 0.5 dB.
The type of rubber vibration isolators used by the pump was BE-120, and its x, y and z direction was the same as Fig. 1 and Fig. 2. The x and z direction can be exchanged during the actual mounting. With the growth of the type number, the stiffness of the isolators increases. The horizontal axis of the coordinate in Fig. 15 is the number of the isolators type.
Test rig.
Figure 15a depicts the variation of modal frequencies with the change of the isolators type which mounted in the original direction. If the two radial directions are exchanged, there is no significant difference. The type of the isolators only affects the modal frequency of the top six vibration modes of the pump.
It can be seen that under the original mounting direction, radial vibration in the orthogonal direction will have certain difference, and this difference will decline with the increase of isolators stiffness. If the x and z mounting directions are exchanged, there is little influence on response in the radial orthogonal directions. With the increase of the stiffness, the radial vibration is almost unchanged, but the axial vibration will be greatly reduced. If only axial response is focused, it is a good choice to change the mounting direction of the isolators.
If the mounting method of water pump is changed to side-fixed support, its first six modes (rigid modes) will change greatly. However, it cannot solve the resonance problem in this prototype pump.
The mechanical design of the front and rear components of the magnetic suspended pump has completed, which is difficult to change. Therefore, the mass distribution was not changed in the optimized design. Active magnetic bearing stiffness can be controlled by the controller parameters (such as proportion, integral and differential parameters in PID controller), and its range covering 106–108 N/m. Through the research on the effects of various factors on vibration, this article got a set of well-designed parameters to reduce the vibration of the prototype pump. The optimized and original parameters are shown in Table 12.
Experimental response (The horizontal axis is in Hz and the vertical axis is in m/s2).
By comparing the modal frequencies of the optimized design and the original design of the dangerous vibration modes (No. 8,9,13,14,18,19 mode in Table 4), it is found that the optimized design can avoid the rotor oscillation modal frequency from 50 Hz well but does not improve the resonance at 700 Hz very much because the rotor second bending mode cannot be adjusted by bearing stiffness (in Table 13). To further avoid the rotor second bending mode from 700 Hz, the structure of the rotor should be changed. After the redesign, the seats response of the magnetic suspended pump can decrease by about 2 dB (in Table 14).
To verify the proposed design, experiments have been carried out on the pump test rig at Tsinghua University, pictured in Fig. 16. The measuring system adopted LC8024 vibration monitoring fault diagnosis system and its matching magnetic adsorption piezoelectric accelerometers.
Figure 17 depicts that compared with original design, the optimized design performed better in vibration isolation in the low frequency band, especially below 500 Hz. As for 500–1000 Hz band, the optimized design inherits the vibration isolation effect of the original design. From the frequency-response curve of the original design, the maximum response peak is at 50 Hz, followed by 350 Hz and the peak at 700 Hz is not obvious. Actually, the optimized design significantly reduce the response peak at 50 Hz and shows good performance in vibration isolation.
Additionally, the y-direction response of measure point 4 is relatively small both in original design and in optimized design. Although this response is larger in optimized design, it does not influence the overall effect of vibration reduction.
Conclusion
A simplified model with high accuracy of an magnetic suspended prototype pump with 5-DOF AMBs system was established in this paper. Based on the model, the main factors that affect the vibration isolation of the system were investigated to discuss the design method of magnetic suspended water pump. An optimized design was proposed and the dangerous modes were avoided. Compared with the original design, the optimized design can reduce vibration response of 2 dB through simulation and experiments results.
This paper provides an accurate and effective optimized design for magnetic suspended rotating machinery, has great practical application value.
Footnotes
Acknowledgements
This work was supported by the National Key R&D Program of China [2018YFB2000100] and Tsinghua University Initiative Scientific Research Program.