CN112327603A - Method for predicting thermal bending vibration in magnetic suspension bearing rotor system - Google Patents
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Abstract
The invention discloses a method for predicting thermal bending vibration in a magnetic suspension bearing rotor system, which comprises the steps of firstly determining the magnetic suspension bearing rotor system, and obtaining the unbalance response of a rotor by combining the supporting characteristics of a magnetic suspension bearing; then combining the characteristics of the electric control system, obtaining the iron loss distribution on the rotor journal by utilizing the unbalanced response, and obtaining the temperature distribution and the thermal deformation of the rotor by utilizing the iron loss distribution; then, a thermal-structure coupling method is used, and thermal bending-unbalanced coupling vibration of the rotor is obtained by combining thermal deformation of the rotor; and finally, continuously updating the conditions by utilizing the thermal bending-unbalance coupling vibration of the rotor, and repeating the iteration until the vibration is converged or diverged. The process is repeated in all working rotating speed ranges, the thermal bending vibration and the change process thereof in the magnetic suspension bearing rotor system can be predicted, and the influence of the thermal bending vibration on the system can be judged accordingly. The result obtained by the invention can provide reference for the rotor structure design, the dynamic design and the electric control system design of the magnetic suspension system.
Description
Technical Field
The invention relates to the field of magnetic suspension bearings, in particular to a method for predicting thermal bending vibration in a rotor system of a magnetic suspension bearing.
Background
In a magnetic suspension bearing-rotor system, due to the working principle and characteristics of a magnetic suspension bearing, the iron loss concentration of a rotor journal can occur, so that the temperature distribution and the thermal bending vibration are caused. The thermal bending vibration can affect the iron loss concentration and the temperature distribution degree, so that a closed loop feedback is formed, the change and fluctuation of the vibration amplitude and the phase are caused, and the equipment performance is affected.
However, the current research on the magnetic levitation rotor mostly focuses on modal analysis, strength analysis and vibration suppression, the conventional electromagnetic analysis generally considers that the iron loss is uniformly distributed, and the structural design and the dynamic design of the magnetic levitation rotor in practice do not take the thermal bending vibration generated by the concentrated iron loss into consideration. Thermal bending vibrations cannot be excluded or avoided at design time. In addition, most of the current researches on the thermal bending vibration of the rotor are researches for analyzing the change process of the thermal bending vibration starting from fault diagnosis, vibration characteristics and influence factors after the thermal bending occurs, and starting from the influence of the thermal bending vibration on temperature distribution.
Therefore, in the magnetic suspension rotor system, the characteristics of the magnetic suspension system are combined, and the prediction of the thermal bending vibration which may occur to the rotor has very important significance. Meanwhile, the characteristics and the change process of the thermal bending vibration of the magnetic suspension rotor can be researched to provide guidance for realizing vibration suppression and eliminating the thermal bending fault.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for predicting thermal bending vibration in a magnetic suspension bearing rotor system aiming at the defects involved in the background technology. Firstly, the unbalance response of the rotor is calculated by combining the bearing characteristics of the magnetic suspension bearing. And then, the unbalance response is utilized, and the working principle and the characteristics of the magnetic suspension bearing are combined to obtain the iron loss distribution of the rotor journal. And then, obtaining the temperature distribution and the thermal deformation of the rotor by using the iron loss as a heat source, establishing a rotor vibration equation under the action of temperature, and solving to obtain the thermal bending-unbalance coupling response. And finally, obtaining new iron loss distribution, temperature distribution, thermal deformation and vibration response by utilizing the thermal bending-unbalance coupling response, continuously updating the calculation conditions by using an iterative calculation method until the vibration is converged or diverged, and further obtaining the change process of the thermal bending-unbalance coupling vibration.
The invention adopts the following technical scheme for solving the technical problems:
a method for predicting thermal bending vibration in a magnetic suspension bearing rotor system comprises the following steps;
step 1), determining a magnetic suspension bearing rotor system, wherein the magnetic suspension bearing rotor system comprises a stator structure and parameters thereof, a rotor structure and parameters thereof, a control rule, an electric control system and parameters thereof;
step 2), solving the steady state unbalance response of the rotor by combining the support characteristics of the magnetic suspension bearing under the control rule;
step 3), solving the iron loss of the silicon steel sheet part at the position of the rotor journal by utilizing the vibration of the steady-state unbalance response;
step 4), obtaining the temperature distribution of the rotor by using iron loss;
step 5), solving the thermal deformation of the rotor by utilizing the temperature distribution of the rotor;
step 6), solving the unbalance and thermal bending coupling response by utilizing the temperature distribution and the thermal deformation of the rotor;
and 7) repeating the steps 3) to 6) until the vibration is converged or diverged to predict the thermal bending vibration of the rotor, and judging the influence of the thermal bending vibration on the system according to the amplitude of the thermal bending vibration.
As a further optimization scheme of the method for predicting the thermal bending vibration in the magnetic suspension bearing rotor system, the supporting characteristics of the magnetic suspension bearing in the step 2) comprise equivalent rigidity and equivalent damping;
the control rule of the magnetic suspension bearing is as follows:
the frequency domain expression is as follows:
wherein P (omega) and Q (omega) are respectively a real part and an imaginary part of G (j omega), G(s) is a transfer function of a control rule of the magnetic suspension bearing, I(s) is current output, X(s) is displacement input, s is an independent variable, bmIs a constant number, anThe constant G (j omega) is a frequency domain expression of a transfer function of a magnetic suspension bearing control rule, m is a constant, n is a constant, and n is more than or equal to m;
according to the theory of magnetic suspension bearings, the equivalent stiffness and the equivalent damping are as follows:
in the formula, ki、kxRespectively representing the current stiffness and the displacement stiffness of the magnetic suspension bearing, respectively representing the equivalent stiffness and the equivalent damping for k and c, and representing omega angular frequency;
the rotor steady state unbalance response is obtained by solving a rotor motion differential equation, wherein the rotor motion differential equation is as follows:
wherein [ M ]]、[C]、[G]、[K]Respectively a mass matrix, a damping matrix, a gyroscopic effect matrix and a rigidity matrix of the rotor, omega is the rotating speed of the rotor, i is an imaginary number unit, t is time,
is the generalized force vector applied to the rotor, where mj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(j ═ 1,2,3 … N) is the j-th shaft eccentricity, φj(j ═ 1,2,3 … N) is the j-th axial segment initial phase;
the solution to the equation is:
As a further optimization scheme of the method for predicting the thermal bending vibration in the magnetic bearing rotor system, the detailed steps of the step 3) are as follows:
the current in the stator coil of the magnetic suspension bearing comprises a bias current IbAnd control the current IcThe bias current is determined according to the bearing capacity of the magnetic suspension bearing, the control current is determined by an electric control system and vibration displacement, and the expression of the control current is as follows:
Ic(s)＝KsKAU(s)G(s)
in the formula, Ks、KARespectively gain of a displacement sensor and a power amplifier, U(s) is an expression of rotor vibration displacement, G(s) is a control rule transfer function expression, the control current expression is subjected to Laplace inverse transformation to obtain a time domain expression thereof, and then the total current I in the stator coils＝Ib+L-1[Ic(s)]In the formula, IbFor bias current, Ic(s) is a control current expression, L-1[]Representing the inverse laplacian transform.
Magnetic induction at rotor position
In the formula, delta0Is the air gap length, μ0For vacuum permeability, phi is the total magnetic flux of the magnetic circuit, ApIs the area of the magnetic pole, N is the number of turns of the coil, IsIs the total current in the coil, z is the rotor amplitude;
iron loss of silicon steel sheet parts includes hysteresis loss, eddy current loss and abnormal loss; hysteresis loss P generated by magnetization process of iron core material in alternating magnetic flux with frequency fh═ ^ f · SdV, where S is the area of the hysteresis loop, S ═ HdB, H is the magnetic field strength, B is the magnetic induction strength, and f is the frequency of the magnetic field variation;
loss of eddy current
Where f is the frequency of the magnetic field change, h1Is the thickness of the silicon steel sheet, BmThe magnetic induction intensity is V, the volume of the silicon steel sheet is V, and the rho is the resistivity of the silicon steel material;
abnormal loss Pe＝ke(fB)1.5。
In the formula, keThe abnormal loss coefficient, f, the frequency of the magnetic field change, and B, the magnetic induction.
As a further optimization scheme of the method for predicting the thermal bending vibration in the magnetic bearing rotor system, the detailed steps of the step 4) are as follows:
according to the Fourier theorem and the law of energy conservation, the law that the temperature of each position inside an object changes along with time, namely a heat conduction differential equation, is as follows:
wherein rho is the material density, c is the specific heat capacity, q isvIs the heat of formation of an internal heat source, lambdax、λy、λzThe thermal conductivity coefficients in the x, y and z directions are respectively, T is temperature, and T is time;
the thermal boundary conditions comprise the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor, the convective heat transfer coefficient of the end surface of the silicon steel sheet part and the surface of the mandrel, wherein the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor is a preset constant, and the convective heat transfer coefficients of the end surface of the silicon steel sheet part and the surface of the mandrel are preset
In the formula, n is the rotating speed, and d is the diameter of the cylindrical surface;
as a further optimization scheme of the method for predicting the thermal bending vibration in the magnetic bearing rotor system, the step 5) comprises the following detailed steps:
the thermal deformation at the position where the rotor protrudes out of the bearing by a length L is expressed as:
in the formula, yaFor thermal deformation of the rotor at a location where the length of the bearing extends to L, where α is the coefficient of thermal expansion of the material and L is the coefficient of thermal expansion of the materialbIs journal length, R is journal position radius, Δ T is journal temperature difference, and L is extension from journal position;
as a further optimization of the method for predicting thermal bending vibration in a magnetic bearing rotor system of the present invention, the detailed steps of step 6) are as follows:
the unbalance amount and thermal bending coupling response are obtained by solving a rotor vibration equation under the action of temperature stress, the uneven expansion of the rotor is restrained to generate thermal stress, and a basic equation of a thermal stress unit is as follows:
in the formula, σTFor the thermal stress unit equation, E (T) is the elastic modulus varying with temperature, alpha is the linear expansion coefficient, T is the temperature of the rotor during operation, T0Is the initial ambient temperature, mu is the poisson's ratio;
according to the darenbell principle, the equation of the rotor vibration under the action of temperature stress is as follows:
wherein [ M ]]、[G]、[C]、[K0]Respectively a mass matrix, a Coriolis force matrix, a damping matrix and a rigidity matrix of the rotor system;
{ U } are acceleration vector, velocity vector and displacement vector of the rotor, respectively; [ K ]1]A stiffness matrix caused by temperature stress;
{ P } is the external load vector, including the unbalanced load vector of the rotor itself and the load vector caused by thermal bending.
Wherein, ydj(j ═ 1,2,3 … N) is the amount of thermal bending deformation of the j-th shaft segment, mj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(j ═ 1,2,3 … N) is the j-th shaft eccentricity, φj(j ═ 1,2,3 … N) is the j-th axial segment initial phase angle, and i is in imaginary units.
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
according to the working principle and characteristics of the magnetic suspension bearing, the thermal bending generated by concentrating the iron loss of the rotor journal is considered, and the thermal bending and unbalance coupling response of the rotor is obtained by utilizing a thermal-structure coupling method in combination with the supporting characteristic of the magnetic suspension bearing and the control rule of an electric control system. Meanwhile, closed loop feedback formed after thermal bending is simulated by using an iterative calculation method, and qualitative and quantitative analysis can be performed on factors influencing the thermal bending of the rotor. The results of the analytical calculations may provide a reference for the magnetic bearing-rotor system to avoid thermal bending vibrations during design.
Drawings
FIG. 1(a) is a view showing a structure of a stator, and FIG. 1(b) is a view showing a structure of a rotor;
FIG. 2 is a cloud of rotor journal cross-sectional core loss distributions;
FIG. 3 is a cloud of rotor temperature profiles;
FIG. 4(a) is a schematic view of a circumferential temperature distribution of a rotor journal, and FIG. 4(b) is a schematic view of a circumferential temperature difference distribution of the rotor journal;
FIG. 5(a) is a schematic view showing the x-direction thermal deformation of the rotor, and FIG. 5(b) is a schematic view showing the y-direction thermal deformation of the rotor;
FIG. 6(a) is a schematic diagram of circumferential temperature variation of a rotor journal, and FIG. 6(b) is a schematic diagram of circumferential temperature difference variation of the rotor journal;
FIG. 7(a) is a schematic diagram showing the variation of the amplitude of the unbalance-thermal bending coupled vibration of the rotor, and FIG. 7(b) is a schematic diagram showing the variation of the phase of the unbalance-thermal bending coupled vibration of the rotor;
FIG. 8(a) is a graph showing the comparison between the magnitude of the unbalanced-thermal bending coupled vibration and the unbalanced response at the rotation frequency of 1-450 Hz, and FIG. 8(b) is a graph showing the comparison between the phase of the unbalanced-thermal bending coupled vibration and the unbalanced response at the rotation frequency of 1-450 Hz;
fig. 9 is a flow chart of the present invention.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the present invention may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. In the drawings, components are exaggerated for clarity.
As shown in FIG. 9, the present invention discloses a method for predicting thermal bending vibration in a rotor system of a magnetic suspension bearing, comprising the following steps:
the first step is as follows: according to the actual working requirement, determining a magnetic suspension bearing rotor system, which comprises a stator structure and parameters thereof, a rotor structure and parameters thereof, a control rule, an electric control system and parameters thereof.
The stator and rotor structures are shown in fig. 1(a) and 1(b), respectively. The control strategy selects a PID control strategy, the electric control system comprises a switch type power amplifier and an eddy current type displacement sensor, and the stator, the rotor, the controller and electric control related parameters are shown in a table 1.
TABLE 1 stator and rotor structural parameters and electric control system parameters
Secondly, solving the steady state unbalance response of the rotor by combining the support characteristics of the magnetic suspension bearing under the control strategy;
the transfer function G(s) of the magnetic suspension bearing electric control system under the PID control law can be expressed as follows:
in the formula, Kp、Ki、KdRespectively proportional, integral, differential coefficients, TfFor the cut-off period of an incomplete differential filter, KsAnd KARespectively, the gain coefficients of the displacement sensor and the power amplifier.
According to the theory of magnetic suspension bearings, the equivalent stiffness and equivalent damping of a magnetic suspension bearing can be expressed as:
wherein G (j omega) is a frequency domain expression of a transfer function of the control law, omega is an angular frequency, and kxAnd kiThe specific expressions are as follows:
in the formula, mu0For vacuum permeability, S is the area of a single magnetic pole, N is the number of turns of the coil, IbFor bias current, δ0Is the air gap size. In combination with the parameters, k, in Table 1xAnd kiRespectively is kx＝5.7955×106N/m，ki＝448.0745N/A。
And combining the formula 1 and the formula 2, the equivalent stiffness and the equivalent damping expression of the magnetic suspension bearing under the PID control strategy are shown as a formula 4.
In the formula, kxAnd kiRespectively the displacement stiffness and the current stiffness of the magnetic bearing, Kp、Ki、KdRespectively proportional, integral, differential coefficients, TfFor the cut-off period of an incomplete differential filter, KsAnd KARespectively, the gain coefficients of the displacement sensor and the power amplifier.
In combination with the equivalent stiffness and damping coefficient of the magnetic suspension bearing, the unbalance response of the magnetic suspension rotor can be obtained by solving a rotor motion differential equation.
Wherein [ M ]]、[C]、[G]、[K]Respectively a mass matrix, a damping matrix, a gyroscopic effect matrix and a rigidity matrix of the rotor, omega is the rotating speed of the rotor, i is an imaginary number unit, t is time,
is the generalized force vector, m, to which the rotor is subjectedj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(j ═ 1,2,3 … N) is the j-th shaft eccentricity, φj(j ═ 1,2,3 … N) is the j-th axial segment initial phase. At a rotational frequency of 70Hz, the amplitude of the rotor at the location of the cantilever-end magnetic bearing was 32.45 μm, direction 3.174 °.
The third step: solving iron loss of rotor journal position silicon steel sheet part by using vibration of steady state unbalance response
When the magnetic suspension bearing works, the current in the stator coil of the magnetic suspension bearing comprises direct current bias current 1.5A and alternating control current, the control current is determined by an electric control system and vibration displacement, and the expression of the control current is as follows by combining a formula (1):
and performing Laplace inverse transformation on the control current to obtain a time domain expression of the control current, wherein the current in the stator coil is as follows:
Is＝Ib+L-1[Ic(s)] (7)
the magnetic induction at the rotor is then:
wherein the content of the first and second substances,
for the total magnetic flux of the magnetic circuit, ApIs the area of the magnetic pole, delta0Is the air gap length, μ0For vacuum permeability, ApIs the area of the magnetic pole, N is the number of turns of the coil, IsIs the total current in the coil and z is the rotor amplitude.
The iron loss at the rotor journal, including hysteresis loss, eddy current loss and abnormal loss, is respectively:
Ph＝∫∫∫f·SdV (9)
Pe＝ke(fB)1.5 (11)
wherein S is the area of the hysteresis loop, S ═ r ═ HdB, H is the magnetic field strength, B is the magnetic induction, f is the frequency, H is the magnetic induction1Is the thickness of the silicon steel sheet, V is the volume of the silicon steel sheet, rho is the resistivity of the silicon steel material, keIs an abnormal loss factor. A cloud of the distribution of the core loss in the cross-section of the rotor journal location is thus obtained as shown in figure 2.
Fourthly, obtaining the temperature distribution of the rotor by using iron loss;
assuming that no iron loss distribution difference exists in the axial direction of the rotor journal assembly, namely the iron loss of any cross section of the assembly is consistent, the rule that the temperature of each position inside changes along with time, namely a heat conduction differential equation, is as follows:
wherein rho is the material density, c is the specific heat capacity, q isvIs the heat of formation of an internal heat source, lambdax、λy、λzThermal conductivity in x, y and z directions respectivelyAnd (4) counting.
The thermal boundary conditions comprise the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor and the convective heat transfer coefficient of the end surface of the silicon steel sheet part and the surface of the mandrel, wherein the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor is usually 3-10W/m2And x deg.c, the latter can be calculated by formula 13.
Wherein n is the rotation speed and d is the diameter of the cylindrical surface. According to the formula, the convection heat transfer coefficients of the end surface of the silicon steel sheet part and the surface of the mandrel are respectively 99.01W/m2X.degree.C. and 82.3W/m2X deg.C. The rotor temperature distribution is shown in fig. 3, fig. 4(a) is a schematic diagram of the circumferential temperature distribution of the rotor journal, and fig. 4(b) is a schematic diagram of the circumferential temperature difference distribution of the rotor journal.
And fifthly, solving the thermal deformation of the rotor by utilizing the temperature distribution of the rotor.
Assuming that the second stepped shaft plane at the leftmost end of the rotor is fixed, the thermal bending of the rotor can be obtained by using the temperature distribution of the rotor obtained in the fourth step as a load, fig. 5(a) is a schematic diagram of the thermal deformation of the rotor in the x direction, and fig. 5(b) is a schematic diagram of the thermal deformation of the rotor in the y direction.
Sixthly, solving the coupling response of the unbalance and the thermal bending by utilizing the temperature distribution and the thermal deformation of the rotor;
the unbalance amount and the thermal bending coupling response can be obtained by solving a rotor vibration equation under the action of temperature stress.
The uneven expansion of the rotor is restrained to generate thermal stress, and the basic equation of a thermal stress unit is as follows:
wherein σTFor the thermal stress unit equation, E (T) is the elastic modulus varying with temperature, alpha is the linear expansion coefficient, T is the temperature of the rotor during operation, T0At the initial ambient temperature, μ is the poisson's ratio.
According to the darenbell principle, the equation of the rotor vibration under the action of temperature stress is as follows:
wherein [ M]、[G]、[C]、[K0]Respectively a mass matrix, a Coriolis force matrix, a damping matrix and a rigidity matrix of the rotor system;
{ U } are acceleration vector, velocity vector and displacement vector of the rotor, respectively; [ K ]1]A stiffness matrix caused by temperature stress;
{ P } is the external load vector, including the unbalanced load vector of the rotor itself and the load vector caused by thermal bending.
Wherein, ydj(j ═ 1,2,3 … N) th axial segment thermal bending deflection, mj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(j ═ 1,2,3 … N) is the j-th shaft eccentricity, φj(j ═ 1,2,3 … N) is the j-th axial segment initial phase angle, and i is in imaginary units. When the temperature difference of the rotor journal obtained in the fifth step was about 0.18 deg.c, the maximum thermal deformation was about 0.25 μm, and the bending direction was about 17.335 deg., the unbalance-thermal bending coupled vibration of the rotor was 3.63 deg. in the 33.28 μm direction. At this time, the amplitude of the unbalanced response is small, the rotating speed is low, the generated temperature difference is small, the bending deformation is small, and therefore the vibration change is small.
And seventhly, repeating the third step to the sixth step by utilizing the unbalance amount and the thermal bending coupling response until the vibration converges or diverges. And repeating the process in all working rotating speed ranges, predicting the thermal bending vibration change of the rotor, and judging the influence of the thermal bending vibration on the system.
And (3) utilizing the unbalance amount and the thermal bending coupling response as the iron loss calculation condition, obtaining new temperature distribution, new thermal deformation and thermal bending-unbalance coupling vibration, continuously updating the calculation condition, repeating the third step to the sixth step until the vibration is converged or diverged, thus obtaining the thermal bending vibration of the magnetic suspension rotor and the change process thereof, and judging the influence of the thermal bending vibration on the system. Fig. 6(a) shows a schematic diagram of the circumferential temperature variation of the rotor journal at a rotation frequency of 70Hz, fig. 6(b) shows a schematic diagram of the circumferential temperature difference variation of the rotor journal at a rotation frequency of 70Hz, fig. 7(a) shows a schematic diagram of the amplitude variation of the unbalance-thermal bending coupled vibration of the rotor, and fig. 7(b) shows a schematic diagram of the phase variation of the unbalance-thermal bending coupled vibration of the rotor. As can be seen from the figure, the temperature difference of the rotor journal fluctuates around 0.18 ℃, the thermal bending vibration is finally stabilized at 33.3 μm, and the phase is about 3.6 degrees. Namely, the rotor does not generate obvious thermal bending vibration phenomenon due to iron loss concentration under the rotation frequency of 70 Hz.
In the working speed range, the thermal bending vibration of the magnetic suspension rotor can be predicted by the method at each rotating speed. In order to illustrate the characteristics of the thermal bending vibration of the magnetic suspension rotor, the thermal bending vibration of the rotor is calculated and the variation characteristics of the thermal bending vibration are analyzed within the rotating frequency range of 1-450 Hz by using the method. FIG. 8(a) is a graph showing the comparison between the magnitude of the unbalanced-thermal bending coupled vibration and the unbalanced response at the rotation frequency of 1 to 450Hz, and FIG. 8(b) is a graph showing the comparison between the phase of the unbalanced-thermal bending coupled vibration and the unbalanced response at the rotation frequency of 1 to 450 Hz. As can be seen from the figure, the thermal bending vibration divergence of the test rotor system does not occur in the range of 1-450 Hz, and the vibration change caused by thermal bending is very small in most cases, so that the design of the rotor system is reasonable, and the obvious thermal bending vibration phenomenon caused by concentrated iron loss can not occur.
It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the prior art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.
The above-mentioned embodiments, objects, technical solutions and advantages of the present invention are further described in detail, it should be understood that the above-mentioned embodiments are only illustrative of the present invention and are not intended to limit the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (6)
1. A method for predicting thermal bending vibration in a magnetic suspension bearing rotor system is characterized by comprising the following steps;
step 1), determining a magnetic suspension bearing rotor system, wherein the magnetic suspension bearing rotor system comprises a stator structure and parameters thereof, a rotor structure and parameters thereof, a control rule, an electric control system and parameters thereof;
step 2), solving the steady state unbalance response of the rotor by combining the support characteristics of the magnetic suspension bearing under the control rule;
step 3), solving the iron loss of the silicon steel sheet part at the position of the rotor journal by utilizing the vibration of the steady-state unbalance response;
step 4), obtaining the temperature distribution of the rotor by using iron loss;
step 5), solving the thermal deformation of the rotor by utilizing the temperature distribution of the rotor;
step 6), solving the unbalance and thermal bending coupling response by utilizing the temperature distribution and the thermal deformation of the rotor;
and 7) repeating the steps 3) to 6) until the vibration is converged or diverged to predict the thermal bending vibration of the rotor, and judging the influence of the thermal bending vibration on the system according to the amplitude of the thermal bending vibration.
2. The method for predicting thermal bending vibration in a rotor system of a magnetic suspension bearing as claimed in claim 1, wherein the support characteristics of the magnetic suspension bearing in step 2) include equivalent stiffness and equivalent damping;
the control rule of the magnetic suspension bearing is as follows:
the frequency domain expression is as follows:
wherein P (omega) and Q (omega) are respectively a real part and an imaginary part of G (j omega), G(s) is a transfer function of a control rule of the magnetic suspension bearing, I(s) is current output, X(s) is displacement input, s is an independent variable, bmIs a constant number, anThe constant G (j omega) is a frequency domain expression of a transfer function of a magnetic suspension bearing control rule, m is a constant, n is a constant, and n is more than or equal to m;
according to the theory of magnetic suspension bearings, the equivalent stiffness and the equivalent damping are as follows:
in the formula, ki、kxRespectively representing the current stiffness and the displacement stiffness of the magnetic suspension bearing, respectively representing the equivalent stiffness and the equivalent damping for k and c, and representing omega angular frequency;
the rotor steady state unbalance response is obtained by solving a rotor motion differential equation, wherein the rotor motion differential equation is as follows:
wherein [ M ]]、[C]、[G]、[K]Respectively a mass matrix, a damping matrix, a gyroscopic effect matrix and a rigidity matrix of the rotor, omega is the rotating speed of the rotor, i is an imaginary number unit, t is time,
is the generalized force vector, m, to which the rotor is subjectedj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(i-1, 2,3 … N) is the j-th shaft eccentricity, φj(i ═ 1,2,3 … N) is the j-th axial segment initial phase;
the solution to the equation is:
3. Method for predicting thermal bending vibrations in a magnetic bearing rotor system according to claim 2, characterized in that the detailed steps of step 3) are as follows:
the current in the stator coil of the magnetic suspension bearing comprises a bias current IbAnd control the current IcThe bias current is determined according to the bearing capacity of the magnetic suspension bearing, the control current is determined by an electric control system and vibration displacement, and the expression of the control current is as follows:
Ic(s)＝KsKAU(s)G(s)
in the formula, Ks、KARespectively gain of a displacement sensor and a power amplifier, U(s) is an expression of rotor vibration displacement, G(s) is a control rule transfer function expression, the control current expression is subjected to inverse Laplace transformation to obtain a time domain expression thereof, and then the total current I in the stator coils＝Ib+L-1[Ic(s)]In the formula, IbFor bias current, Ic(s) is a control current expression, L-1[]Representing the inverse laplacian transform.
Magnetic induction at rotor position
In the formula, delta0Is the air gap length, μ0For vacuum permeability, phi is the total magnetic flux of the magnetic circuit, ApIs the area of the magnetic pole, N is the number of turns of the coil, IsIs the total current in the coil, z is the rotor amplitude;
iron loss of silicon steel sheet parts includes hysteresis loss, eddy current loss and abnormal loss; hysteresis loss P generated by magnetization process of iron core material in alternating magnetic flux with frequency fh═ ^ f · SdV, where S is the area of the hysteresis loop, S ═ HdB, H is the magnetic field strength, B is the magnetic induction strength, and f is the frequency of the magnetic field variation;
loss of eddy current
Where f is the frequency of the magnetic field change, h1Is the thickness of the silicon steel sheet, BmThe magnetic induction intensity is V, the volume of the silicon steel sheet is V, and the rho is the resistivity of the silicon steel material;
abnormal loss Pe＝ke(fB)1.5。
In the formula, keThe abnormal loss coefficient, f, the frequency of the magnetic field change, and B, the magnetic induction.
4. The method for predicting thermal bending vibration in a rotor system of an magnetic suspension bearing of claim 3, wherein the detailed steps of the step 4) are as follows:
according to the Fourier theorem and the law of energy conservation, the law that the temperature of each position inside an object changes along with time, namely a heat conduction differential equation, is as follows:
wherein rho is the material density, c is the specific heat capacity, q isvIs the heat of formation of an internal heat source, lambdax、λy、λzThe thermal conductivity coefficients in the x, y and z directions are respectively, T is temperature, and T is time;
the thermal boundary conditions comprise the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor, the convective heat transfer coefficient of the end surface of the silicon steel sheet part and the surface of the mandrel, wherein the convective heat transfer coefficient of the circumferential surface of the silicon steel sheet part at the air gap of the stator and the rotor is a preset constant, and the convective heat transfer coefficients of the end surface of the silicon steel sheet part and the surface of the mandrel are preset
In the formula, n is the rotating speed, and d is the diameter of the cylindrical surface;
5. the method for predicting thermal bending vibration in a rotor system of an magnetic suspension bearing of claim 4, wherein the detailed steps of the step 5) are as follows:
the thermal deformation at the position where the rotor protrudes out of the bearing by a length L is expressed as:
in the formula, yaFor thermal deformation of the rotor at a location where the length of the bearing extends to L, where α is the coefficient of thermal expansion of the material and L is the coefficient of thermal expansion of the materialbIs journal length, R is journal position radius, Δ T is journal temperature difference, and L is extension from journal position;
6. the method for predicting thermal bending vibration in a rotor system of an magnetic suspension bearing of claim 5, wherein the detailed steps of the step 6) are as follows:
the unbalance amount and thermal bending coupling response are obtained by solving a rotor vibration equation under the action of temperature stress, the uneven expansion of the rotor is restrained to generate thermal stress, and a basic equation of a thermal stress unit is as follows:
in the formula, σTFor the thermal stress unit equation, E (T) is the elastic modulus varying with temperature, alpha is the linear expansion coefficient, T is the temperature of the rotor during operation, T0Is the initial ambient temperature, mu is the poisson's ratio;
according to the darenbell principle, the equation of the rotor vibration under the action of temperature stress is as follows:
wherein [ M ]]、[G]、[C]、[K0]Respectively a mass matrix, a Coriolis force matrix, a damping matrix and a rigidity matrix of the rotor system;
{ U } are acceleration vector, velocity vector and displacement vector of the rotor, respectively; [ K ]1]A stiffness matrix caused by temperature stress;
{ P } is the external load vector, including the unbalanced load vector of the rotor itself and the load vector caused by thermal bending.
Wherein, ydj(j ═ 1,2,3 … N) is the amount of thermal bending deformation of the j-th shaft segment, mj(j ═ 1,2,3 … N) for the j-th shaft segment mass, ej(i-1, 2,3 … N) is the j-th shaft eccentricity, φj(j ═ 1,2,3 … N) is the j-th axial segment initial phase angle, and i is in imaginary units.
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