In this paper, the subdomain analysis model of the eddy current brake (ECB) is established. By comparing with the finite element method, the accuracies of the subdomain model and the finite element model are verified. Furthermore, the resistance characteristics of radial, axial, and Halbach arrays under impact load are calculated and compared. The axial array has a large braking force coefficient but low critical velocity. The radial array has a low braking force coefficient but high critical velocity. The Halbach array has the advantages of the first two arrays. Not only the braking force coefficient is large, but also the critical speed is high. The parameter analysis of the Halbach array is further carried out. The inner tube thickness and air gap length are the sensitive factors of resistance characteristics. The demagnetization effect is significantly enhanced by the increase of the inner tube thickness. In order to ensure that the ECB does not overheat, the electromagnetic-thermal coupling model is established based on the heat transfer theory. The temperature rise of the inner tube is obvious while that of the permanent magnet is small. The temperature rise of the inner tube is more than 20 K each time, and that of the permanent magnet is less than 1 K each time.
Compared with the traditional viscous and viscoelastic dampers, the ECB operates without direct contact between the primary and secondary, does not depend on friction, and has no working fluid. The structure of ECB is simple and reliable. The ECB has high performance, good durability, easy maintenance, long working life, and other advantages [
The ECB can be divided into rotary type and linear type according to the motion form of the mover, and the rotary type can be classified as radial and axial. Shin et al. [
At present, the research of ECB is mostly used to suppress structural vibration or transfer torque as a coupler. A lot of research has been done on radial, axial, and linear plate ECB, but the research on tubular linear ECB is rare. It is a promising research direction to apply ECB to the braking of impact load. Under the impact load, the working speed of ECB will be relatively large, and the reliability requirement is higher. Compared with rotary ECB, linear ECB does not need an intermediate mechanical conversion device, and its structure is simpler and more reliable, especially suitable for high-speed operation. In addition, braking the impact load puts forward higher requirements for the energy consumption density of ECB. Compared with the linear plate ECB, there is no transverse end effect in the tubular linear ECB, so the permanent magnet utilization rate is high and the magnetic leakage is small, which can effectively improve the energy consumption density. However, due to its closed structure, the heat dissipation is poor. The change of temperature will change the constitutive relationship of the material and affect the performance of ECB. Therefore, the temperature change of ECB is a problem worthy of attention. Especially for the ECB with high energy consumption, it is necessary to conduct thermal analysis on the ECB.
In this paper, the tubular linear ECB is applied to the braking of impact load. The braking force characteristics of the axial array, radial array, and Halbach array under impact load are investigated to select the permanent magnet array suitable for impact load braking. The influence of design parameters on the braking force characteristics is further studied, which provides a valuable reference for the design of ECB. In order to prevent the ECB from overheating, the electromagnetic thermal field coupling model is established to study the transient heat transfer process.
The structure of the tubular linear ECB for impact load braking is shown in
For tubular linear ECB, radial array, axial array, or Halbach array are mainly used.
When the relative motion between primary and secondary occurs, the secondary cuts the magnetic force lines formed by the primary, thereby generating dynamic electromotive force. Under the action of the dynamic electromotive force, and charges move directionally, forming an eddy current. The interaction between the eddy current and the magnetic field generates the Lorentz force, which hinders the relative motion of the two. From the perspective of energy conversion, the mechanical energy is first converted into electrical energy, then converted into thermal energy through resistance for dissipation.
The quasi-static analytical model of ECB can be established by subdomain technology. Taking the Halbach array as an example,
The analytical model is divided into six subdomains.
Subdomain I: Shaft. Assuming that the conductivity is zero, the permeability is the same as air.
Subdomain II: Permanent magnet. Assuming that the conductivity is zero, the permeability is the same as air.
Subdomain III: Air gap.
Subdomain IV: Inner tube. High conductivity, permeability is the same as air.
Subdomain V: Outer tube. Ferromagnetic material, assuming that the constitutive relationship is linear.
Subdomain VI: Outside air.
Applying Maxwell’s equations and magnetic vector potential, the governing equation can be expressed as:
where
Since the conductivity of subdomains I, III and VI is zero, the governing equation can be simplified as the Laplace equation.
The constitutive relation of the permanent magnet can be described by residual magnetism
where
The distribution of remanence can be expressed by the Fourier series.
where
Substitute
According to Ohm’s Law and Faraday’s Law
The governing equation of Subdomain IV and V are derived as:
where
According to the assumption, all electromagnetic quantities have a period of
Substitute
Solving the above ordinary differential equation to obtain the general solution of each subdomain which is
where
The normal component of magnetic flux density on the interface is continuous, guaranteed by
Analytical expressions of the magnetic flux density and eddy current density of Subdomain IV and V can get as:
Finally, the Lorentz force can be calculated
The accuracy of the subdomain model will be verified by comparing it with the finite element model. The calculations are performed using the parameters given in
Parameter | Description | Value |
---|---|---|
Outer diameter of the shaft | 20 mm | |
Outer diameter of the permanent magnet | 61 mm | |
Outer diameter of the air gap | 63 mm | |
Outer diameter of the inner tube | 64 mm | |
Outer diameter of the outer tube | 68 mm | |
Axial length of the axially magnetized permanent magnet | 20 mm | |
Axial length of the radially magnetized permanent magnet | 20 mm | |
Conductivity of the inner tube | 36 MS/m | |
Conductivity of the outer tube | 11.2 MS/m | |
Permeability of the outer tube | 600 |
The results of the subdomain method and the finite element method are shown in
The main advantage of the subdomain model is the lower computational cost and high accuracy. However, it has some important disadvantages that limit its applicability. First, the complexity of the method depends on the number of subdomains, and boundary conditions of the subdomains must be simple. This makes it difficult to apply machines with complex geometric structures. The second is that this method is only applicable to quasi-static electromagnetic fields. The third is that the method assumes that the material’s constitutive relationship is linear. Although it is possible to calculate the nonlinear constitutive relationship through iteration, it will undoubtedly increase the calculation time greatly [
According to the calculation results of the subdomain model and the finite element model, the accuracy of the subdomain model and the finite element model are verified. The ECB studied in this paper has a very rapid speed change under the action of an intensive impact load, so it must be studied as a transient field. In addition, there is magnetic saturation. Therefore, the following sections will use the finite element method as the main research method.
The relationship between braking force and speed of different arrays are shown in
In order to compare the actual braking effect of different arrays of ECB, the transient braking process was calculated. The intensive impact load
where
The braking force generated by the ECB during actual braking is shown in
Comparing the performance of the axial, radial, and Halbach arrays of the ECB, the Halbach array has advantages over both the axial array and the radial array. The braking force coefficient is large and the critical speed is high. In the same volume, the braking force provided by the Halbach array is much greater than that of the axial array and the radial array, the actual performance is the best.
The performance of the ECB is affected by many parameters, such as the air gap distance, the thickness of the inner tube, the thickness of the outer tube, and the size of the permanent magnet. The analysis in the previous section shows that the performance of the Halbach array is better than axial or radial arrays. This section takes the Halbach array ECB as the research object, then analyzes in detail the effects of the outer tube thickness, inner tube thickness, and air gap distance on the performance of the ECB.
The distribution of magnetic induction lines of the static magnetic field of the Halbach array ECB is shown in
The resultant resistance curves of the ECB under different outer tube thicknesses are shown in
Air gap distance is a critical parameter. The air gap magnetic flux density is closely related to the air gap distance. As shown in
As mentioned in the first section, the kinetic energy of the moving part is finally converted into thermal energy dissipation. Temperature changes will change the conductivity of the inner tube, the conductivity and permeability of the outer tube, and the performance of the permanent magnet. When the temperature exceeds the Curie temperature of the permanent magnet, it will cause irreversible loss of the permanent magnet and seriously affect the performance of the ECB. In this section, the electromagnetic-thermal coupling model is established based on the heat transfer characteristics of the ECB, then the thermal field of the ECB is calculated.
The heat source of the ECB is mainly concentrated in the inner tube. The heat is dissipated by the forced convection of the air gap between the primary and secondary, heat conduction inside the inner and outer tubes, and natural convection on the outer surface of the outer tube. The control equation of convection heat dissipation is shown in
where
The primary and secondary form a concentric annular space (air gap). This annular space can be assumed to perform forced convection heat transfer when the ECB is in operation. The Reynolds number determines the state of forced convection (laminar or turbulent flow) in the air gap. The Reynolds number can be calculated by the following formula:
where
The outer surface of the outer tube is in natural convective heat transfer in a large space. The
where
where
At standard atmospheric pressure, the
The eddy current calculated by the electromagnetic field is used as the heat source of the thermal field, boundary conditions of the thermal field are set according to the obtained heat transfer coefficient, and the ambient temperature is set as 298.15 K, then the temperature change of the ECB is solved.
Curves of the maximum temperature of the ECB and the maximum temperature of the permanent magnet are shown in
Assume that the ECB runs 8 times per minute, so the working interval is 7.5 s. The thermal field result of the previous work is used as the initial condition for the calculation of the next thermal field. The calculated maximum temperature change curve of the ECB with the continuous operation is shown in
First time | Second time | Third time | Fourth time | Fifth time | |
---|---|---|---|---|---|
Inner tube (K) | 22 | 21.89 | 21.76 | 20.84 | 20.74 |
Permanent magnet (K) | 0.35 | 0.41 | 0.56 | 0.61 | 0.69 |
This paper analyzes the resistance characteristics of three types of ECB. The parameter analysis of the Halbach array ECB is further conducted. Finally, the electromagnetic-thermal field coupling model is established, then the temperature change of the ECB is calculated. The main conclusions are as follows:
Braking force calculated by the subdomain model and the finite element model are in good agreement, assuming that the constitutive relationship of ferromagnetic material is linear. However, the actual braking force is lower than the braking force calculated by the linear assumption due to magnetic saturation. Compared with radial array and axial array, Halbach array permanent magnet ECB has advantages of large braking force coefficient and high critical speed. Increasing the thickness of the inner or/and outer tubes will increase the braking force coefficient and decrease the critical speed. Under the game between these two indicators, there is an optimal value for the thickness of the inner and outer tubes. The smaller the air gap distance, the higher the air gap magnetic flux density. The highest temperature appears in the inner tube, and the temperature of the permanent magnet does not change much. Assuming that the working interval is 7.5 s, the temperature of the inner tube rises about 21 K each time it works.
In the future, experimental studies can be carried out to verify the accuracy of the model in this paper. During continuous operation, the temperature rise of ECB is obvious. It is a meaningful and interesting research work to add a cooling system to the original structure.