CN110046418B - Vibration characteristic analysis method for periodic stator of permanent magnet motor - Google Patents
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Abstract
The invention discloses a vibration characteristic analysis method of a periodic stator of a permanent magnet motor, which comprises the following steps: establishing a dynamic model of the periodic stator by means of a magnetic field follow-up coordinate system; solving the characteristic value of the circular stator body; solving a first order perturbation characteristic value of the periodic stator according to a perturbation method; analyzing the influence of the wave number and the number of the permanent magnets on the characteristic value according to the operation characteristics of the trigonometric function, and revealing the mapping relation between the key parameters, the modal characteristics and the dynamic stability; and solving the characteristic value of the periodic stator to obtain the natural frequency splitting and stabilizing rule. The invention establishes a dynamic model under a magnetic field follow-up coordinate system, and then provides a vibration analysis technology suitable for a periodic stator by adopting a perturbation method, so that the obtained result better meets the actual requirements of engineering.
Description
Technical Field
The invention relates to the field of permanent magnet motors, in particular to a vibration characteristic analysis method of a periodic stator of a permanent magnet motor.
Background
The power of the permanent magnet motor is from a few milliwatts to thousands of kilowatts, and the permanent magnet motor is widely applied to various engineering fields, such as: air conditioner and fan in household appliances, computer and printer in office supplies, robots and automation equipment in industrial production, warplanes and ships in military industry, auxiliary appliances and artificial hearts in medical treatment, etc. (Wang Xiuhe, etc. permanent magnet motor. Beijing: china electric Press, 2010: 8-9). Under the action of time-varying magnetic pull force, the stator generally vibrates significantly, thereby causing noise and dynamic stability problems. Because of the limitations of the existing stability analysis technology, a high-efficiency and accurate analysis and prediction technology aiming at actual working conditions is particularly needed. The document (Y.Lefevre, B.Davat, M.Lajoie-maznc. Determination of synchronous motor vibrations due to electromagnetic force harmonics. IEEE Transactions on Magnetics,1989,25 (4): 2974-2976) uses a finite element method to analyze electromagnetic noise, in particular by calculating magnetic pull forces acting on stator teeth, to predict vibration characteristics. Document (M.N.Anwar, I.Husain.Radial force calculation and acoustic noise prediction in switched reluctance machines. IEEE Transactions on Industry Applications,2000,36 (6): 1589-1597) analyzes transient electromagnetic field distributions, magnetic pull frequencies of stator teeth, and electromagnetic noise. In addition, the prior art also generally adopts a numerical method to predict the dynamic stability, and the method has lower calculation efficiency and cannot reveal the universality rule.
Disclosure of Invention
Aiming at the problem of the magnetic vibration of the periodic stator of the permanent magnet motor, the invention establishes a dynamic model under a magnetic field follow-up coordinate system, and then provides a vibration analysis technology suitable for the periodic stator by adopting a perturbation method, so that the obtained result better meets the actual requirements of engineering.
A method of analyzing vibration characteristics of a periodic stator of a permanent magnet motor, the method comprising the steps of:
establishing a dynamic model of the periodic stator by means of a magnetic field follow-up coordinate system; solving the characteristic value of the circular stator body;
solving a first order perturbation characteristic value of the periodic stator according to a perturbation method;
analyzing the influence of the wave number and the number of the permanent magnets on the characteristic value according to the operation characteristics of the trigonometric function, and revealing the mapping relation between the key parameters, the modal characteristics and the dynamic stability;
and solving the characteristic value of the periodic stator to obtain the natural frequency splitting and stabilizing rule.
The dynamic model of the periodic stator is specifically as follows:
wherein M is a quality operator, G is a gyroscopic operator, D is a centripetal stiffness operator, K, K (1) And K (2) Respectively represent the bending stiffness operator of the annular stator and the stiffness operator generated by magnetic pulling force.
Further, the characteristic values of the circular stator body are specifically as follows:
wherein n is the vibration wave number, k u And k v Tangential and radial stiffness, Ω respectively v Is a dimensionless rotating speed.
The first order perturbation characteristic value is specifically:
wherein B is M And C M The method comprises the following steps of:
B M ＝A un (r n0 -iΩ v n)
wherein A is un Is the amplitude, mu 0 Is vacuum magnetic permeability; h is a 0 Is the radial thickness of the magnetic pole; phi is magnetic flux; d, d 0 An air gap between the stator and the rotor; n (N) m The number of the magnetic poles is; r is the radius of a neutral circle; a section moment of inertia of the annular stator; n (N) m The number of the sector permanent magnets is that gamma is the central angle, and the-represents conjugation;
in the method, in the process of the invention,
according to the operational characteristics of the trigonometric function, the influence of the wave number and the number of the permanent magnets on the characteristic value is analyzed, and the mapping relation of the key parameters, the modal characteristics and the dynamic stability is disclosed as follows:
when 2N/N m When the characteristic values are not equal to the integer, the sine and cosine characteristic values of the forward traveling wave and the backward traveling wave are respectively equal, are pure imaginary numbers, have no natural frequency splitting and are in a stable state;
when 2N/N m When the wave length is equal to the integer, the sine and cosine eigenvalues of the forward wave and the backward wave are all unequal, and the system generates natural frequency splitting. The magnetic flux, the rotational speed, the number of permanent magnets, the central angle, the length of the air gap, the radial and tangential support stiffness all affect the steady state of the stator.
The characteristic value of the periodic stator is solved, and the natural frequency splitting and stabilizing rule is obtained specifically as follows:
according to the vibration propagation direction and the vibration mode, respectively obtaining a characteristic value of the cosine of the backward traveling wave and a characteristic value of the sine of the backward traveling wave; forward wave
Cosine eigenvalue, sine eigenvalue.
The technical scheme provided by the invention has the beneficial effects that:
1. firstly, establishing a dynamic model by means of a magnetic field follow-up coordinate system, then solving the characteristic value of an annular stator body, solving the characteristic value of a stator in a corresponding period according to a perturbation method, and predicting the natural frequency splitting and dynamic stability rule according to the characteristic value;
2. according to the invention, natural frequency splitting is inhibited and system stability is improved by changing parameters such as magnetic flux, rotating speed, central angle of a permanent magnet, air gap length, radial and tangential support rigidity and the like of a permanent magnet motor;
3. the invention adopts an analysis method to give out the characteristic value of the periodic stator of the permanent magnet motor, and predicts the natural frequency splitting and dynamic stability rules according to the characteristic value;
4. compared with the prior art, the technology provided by the invention has the characteristics of high efficiency, accuracy, universality and the like. According to the technology, the mapping relation between key parameters, modal characteristics and dynamic stability can be revealed, the vibration condition can be estimated in the design stage, the dynamic design of the permanent magnet motor is guided, and finally the running stability and reliability are improved.
Drawings
FIG. 1 is a schematic diagram of a periodic stator of a permanent magnet motor provided by the invention;
FIG. 2 is a schematic diagram of a law of variation of real and imaginary parts of eigenvalues with magnetic flux obtained according to the technique provided by the present invention;
FIG. 3 is a schematic diagram of a change rule of real and imaginary parts of a characteristic value obtained according to the technology provided by the invention along with a central angle of a sector permanent magnet;
FIG. 4 is a schematic diagram of a variation rule of real and imaginary parts of a characteristic value with a rotation speed, which is obtained according to the technology provided by the invention;
FIG. 5 is a schematic diagram of a variation rule of real and imaginary parts of a characteristic value with an air gap length obtained according to the technique provided by the invention;
FIG. 6 is a schematic diagram of a variation rule of real and imaginary parts of a characteristic value obtained according to the technology provided by the invention with radial support rigidity of a stator;
fig. 7 is a schematic diagram of a variation rule of real and imaginary parts of a characteristic value obtained according to the technology provided by the invention along with the tangential support rigidity of a stator.
In the figure, BTW C Representing the cosine of the backward wave, BTW S Representing the sine of the backward traveling wave, FTW C Representing the cosine of the forward wave, FTW S The forward wave is sinusoidal.
To show the mapping between key parameters and dynamic stability, the embodiments in FIGS. 2-7 are all 2N/N m An integer.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be described in further detail below.
Example 1
The embodiment of the invention provides a vibration analysis method with strong applicability and specially aiming at a periodic stator of a permanent magnet motor. Firstly, establishing a dynamic model of a periodic stator by means of a magnetic field follow-up coordinate system, and then calculating characteristic values by adopting a perturbation method so as to predict modal characteristics and dynamic stability. The method can also be used for predicting the vibration of typical periodic structures such as stator and rotor of other types of rotating motors and annular components in micro devices.
The periodic stator consists of an annular stator body, a permanent magnet, a radial support and a tangential support; the periodic stator is under the action of rotary magnetic tension; the vibration analysis technique is basically characterized in that: the dynamic stability analysis and prediction of the periodic stator are realized by adopting a magnetic field follow-up coordinate system, and the method comprises the following specific steps:
(1) By means of a magnetic field follow-up coordinate system, a dynamic model of the periodic stator is established according to the Hamilton principle:
wherein M is a quality operator, G is a gyroscopic operator, D is a centripetal stiffness operator, K, K (1) And K (2) The rigidity operators respectively represent bending rigidity operators of the annular stator and rigidity operators generated by magnetic tension, and specifically comprise:
fig. 1 is a periodic stator of a permanent magnet motor. Magnetic field follow-up coordinate system
Rotated at an angular velocity omega. O represents the geometric centroid of the statorPosition. The radial thickness, the axial height, the density, the Young's modulus, the section moment of inertia and the neutral circle radius of the stator body are h, b, rho, E, I and R respectively; />
Is the tangential vibration quantity; epsilon is a dimensionless small parameter; />
Is the position angle of the servo system; omega shape v Is a dimensionless rotating speed; mu (mu) 0 Is vacuum magnetic permeability; h is a 0 Is the radial thickness of the magnetic pole; phi is magnetic flux; d, d 0 An air gap between the stator and the rotor; n (N) m The number of the magnetic poles is; />
Is a dirac function; />
Is a dirac function.
The outside of the rotor is provided with N m And the central angles of the sector permanent magnets are gamma. While assuming that one end of the first permanent magnet is located on the polar axis. The outside of the annular stator is provided with a uniform support (not shown) with tangential and radial stiffness k respectively u And k v 。I(I＝bh 3 And/12) is the cross-sectional moment of inertia of the annular stator.
(2) Solving the eigenvalues of the annular stator body, assuming for this purpose intermediate variables:
in the method, in the process of the invention,
in imaginary units，r n0 Is characteristic value, A un The amplitude value is represented by "-" and the conjugate is represented by "n" and the vibration wave number is represented by "n". Defining an inner product:
wherein x is a complex variable; y is a complex variable.
Then substituting the hypothesis into the dynamics model of the step (1) and combining with
As an inner product, the characteristic value can be obtained:
wherein n is the vibration wave number.
(3) Solving a first order perturbation characteristic value of the periodic stator according to a perturbation method; for this purpose it is assumed that:
r n ＝r n0 +εr n1 (14)
wherein r is n1 Is a first order perturbation characteristic value. Substituting the hypothesis into the power model of the step (1) and then combining with
The first order perturbation characteristic value of the periodic stator can be obtained by simplifying the inner product:
wherein B is M And C M The method comprises the following steps of:
B M ＝A un (r n0 -iΩ v n) (17)
wherein Φ is the magnetic flux.
(4) According to the operational characteristics of the trigonometric function, there are:
the influence of the wave number and the number of the permanent magnets on the characteristic value can be analyzed, so that the mapping relation between the key parameters, the modal characteristics and the dynamic stability is revealed.
(5) Solving the characteristic value of the periodic stator so as to obtain the natural frequency splitting and stabilizing rule, which comprises the following steps:
assume the intermediate variables:
two cases are available depending on the direction of vibration propagation and the vibration mode: backward wave (Im (r) n0 )＝Ω v n+a n ) And forward traveling wave (Im (r) n0 )＝Ω v n-a n ). Cosine amplitude of known response is
While the amplitude of the sinusoidal response is
The characteristic value of the backward wave cosine is as follows:
the characteristic values of the backward traveling wave sine are as follows:
the characteristic value of the forward wave cosine is as follows:
the characteristic value of the forward wave sine is as follows:
when 2N/N m When the characteristic values of the forward traveling wave and the backward traveling wave are equal to each other and are pure imaginary numbers, and the system has no natural frequency splitting and is in a stable state;
when 2N/N m When the wave length is equal to the integer, the sine and cosine eigenvalues of the forward wave and the backward wave are all unequal, and the system generates natural frequency splitting. The magnetic flux, the rotational speed, the number of permanent magnets, the central angle, the length of the air gap, the radial and tangential support stiffness all affect the steady state of the stator.
Example 2
According to the basic characteristics of the periodic stator of the permanent magnet motor, the invention provides a perturbation analysis method. Firstly, establishing a mathematical model in a formula (1), and considering the concrete steps of a vibration analysis prediction technology without extension deformation as follows:
(S1) establishing a dynamic model of the periodic stator by means of a magnetic field follow-up coordinate system.
And (S2) solving the characteristic value of the circular stator body.
Assuming that the response of the kinetic model in step (S1) is equations (10) and (11), the inner product is defined as equation (12).
Substituting the hypothesis into the kinetic model of step 1 (1) and then correlating with
And (3) taking an inner product to obtain the characteristic value of the annular stator body:
(S3) solving the first order perturbation characteristic value of the periodic stator according to a perturbation method, assuming intermediate variables in formulas (14) and (15) for the first order perturbation characteristic value, substituting the assumption of the intermediate variables into the dynamics model of the step (1), and then combining
The inner product is used for simplifying to obtain the first order perturbation characteristic value of the periodic stator, as shown in formulas (16) - (18).
The characteristic value of the periodic stator can be obtained according to the above equation in combination with (S2).
(S4) according to the operation characteristics of the trigonometric function, the method comprises the following steps:
according to the method, the influence rule of basic parameters such as vibration wave number, permanent magnet number and the like on the characteristic value can be predicted, and the mapping relation between the parameters, the modal characteristics and the stability is further disclosed.
(S5) taking the periodic stator in Table 1 as an example, calculating the characteristic value by combining the numerical method.
TABLE 1 basic parameters of periodic structures
And (S6) predicting the natural frequency splitting and dynamic stability rules according to the characteristic values obtained in the step (S5), wherein specific results are shown in the accompanying figures 2-7.
Fig. 2 shows the law of variation of real and imaginary parts of eigenvalues with magnetic flux obtained by the method provided by the invention. With the increase of magnetic flux, the splitting degree of the natural frequencies of the forward traveling wave and the backward traveling wave becomes larger, the cosine mode of the backward traveling wave and the sine mode of the forward traveling wave are in a stable state, and the sine mode of the backward traveling wave and the cosine mode of the forward traveling wave are in an unstable state. Therefore, in order to reduce natural frequency splitting and improve stability, a smaller magnetic flux should be selected according to circumstances.
Fig. 3 shows the law of variation of real and imaginary parts of characteristic values obtained by the method according to the invention along with the angle of the permanent magnet. With the increase of the angle of the permanent magnet, the back traveling wave cosine mode and the front traveling wave sine mode are in a stable state, and the back traveling wave sine mode and the front traveling wave cosine mode are in an unstable state. The natural frequency of the forward and backward traveling waves is not split when pi/4. Around pi/4, the natural frequency splits to a lesser extent. In order to suppress or eliminate natural frequency splitting, an appropriate permanent magnet angle should be selected according to circumstances.
Fig. 4 shows the law of variation of real and imaginary parts of characteristic values with rotation speed obtained by the method according to the present invention. With the increase of the rotating speed, the back traveling wave cosine mode and the front traveling wave sine mode are in a stable state, and the back traveling wave sine mode and the front traveling wave cosine mode are in an unstable state. Although the natural frequencies of the forward traveling wave and the backward traveling wave are split, the degree of the natural frequency splitting is kept unchanged along with the increase of the rotating speed, which shows that the rotating speed only affects the natural frequency of the stator of the permanent magnet motor and has no effect on the frequency splitting. According to the situation, proper rotating speed is selected to adjust the natural frequency of the stator, so that resonance is avoided.
Fig. 5 shows the variation law of real and imaginary parts of the eigenvalues with the air gap obtained by the method provided by the invention. With the increase of the air gap, the back traveling wave cosine mode and the front traveling wave sine mode are in a stable state, and the back traveling wave sine mode and the front traveling wave cosine mode are unstable and tend to be in a stable state. The smaller the air gap, the greater the degree of frequency splitting. In selecting the air gap parameter, the air gap should be greater than 0.5mm in order to reduce frequency splitting and improve system stability.
Fig. 6 shows the variation law of real and imaginary parts of characteristic values obtained by the method according to the invention along with the radial support rigidity. With the increase of the radial support rigidity, the back traveling wave cosine mode and the front traveling wave sine mode are in a stable state, and the back traveling wave sine mode and the front traveling wave cosine mode are in an unstable state. The degree of natural frequency splitting becomes smaller as the support stiffness increases. In order to suppress natural frequency splitting, the radial support rigidity of the stator should be appropriately improved.
Fig. 7 shows the law of variation of real and imaginary parts of characteristic values obtained by the method according to the invention along with tangential support rigidity. With the increase of the tangential support rigidity, the back traveling wave cosine mode and the front traveling wave sine mode are in a stable state, and the back traveling wave sine mode and the front traveling wave cosine mode are in an unstable state. The degree of natural frequency splitting becomes smaller as the support stiffness increases. In order to suppress natural frequency splitting, the tangential support stiffness of the stator should be suitably increased.
In summary, the embodiment of the invention provides an analysis technology for predicting the periodic stator vibration law of a permanent magnet motor. The technology uses a magnetic field follow-up coordinate system and adopts a perturbation method to obtain a characteristic value, so that the modal and dynamic stability are predicted, the accuracy, the calculation efficiency and the universality are obviously improved, and the actual requirements of engineering are better met.
Those skilled in the art will appreciate that the drawings are schematic representations of only one preferred embodiment, and that the above-described embodiment numbers are merely for illustration purposes and do not represent advantages or disadvantages of the embodiments.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.
Claims (4)
1. A method for analyzing vibration characteristics of a periodic stator of a permanent magnet motor, the method comprising the steps of:
establishing a dynamic model of the periodic stator by means of a magnetic field follow-up coordinate system; solving the characteristic value of the circular stator body;
solving a first order perturbation characteristic value of the periodic stator according to a perturbation method;
analyzing the influence of the wave number and the number of the permanent magnets on the characteristic value of the circular stator body according to the operation characteristics of the trigonometric function, and revealing the mapping relation between the key parameters, the modal characteristics and the dynamic stability;
solving a first order perturbation characteristic value of the periodic stator to obtain a natural frequency splitting and stabilizing rule;
the first order perturbation characteristic value of the periodic stator is solved, and the natural frequency splitting and stabilizing rule is obtained specifically as follows:
according to the vibration propagation direction and the vibration mode, respectively obtaining a characteristic value of the cosine of the backward traveling wave and a characteristic value of the sine of the backward traveling wave; characteristic values of forward wave cosine and characteristic values of sine;
the first order perturbation characteristic value is specifically:
wherein B is M And C M The method comprises the following steps of:
B M ＝A un (r n0 -iΩ v n)
wherein A is un Is the amplitude, mu 0 Is vacuum magnetic permeability; h is a 0 Is the radial thickness of the magnetic pole; phi is magnetic flux; d, d 0 An air gap between the stator and the rotor; n (N) m The number of the magnetic poles is; r is the radius of a neutral circle; a section moment of inertia of the annular stator; gamma is the central angle and-represents conjugation; e is Young's modulus, I is section moment of inertia;
in the method, in the process of the invention,
2. the method for analyzing vibration characteristics of a periodic stator of a permanent magnet motor according to claim 1, wherein the dynamic model of the periodic stator is specifically:
wherein M is a quality operator, G is a gyroscopic operator, D is a centripetal stiffness operator, K, K (1) And K (2) Respectively represent the bending stiffness operator of the annular stator and the stiffness operator generated by magnetic pulling force.
3. The method for analyzing vibration characteristics of a periodic stator of a permanent magnet motor according to claim 1, wherein the characteristic values of the circular stator body are specifically as follows:
wherein n is the vibration wave number, k u And k v Tangential and radial stiffness, Ω respectively v Is a dimensionless rotating speed.
4. The method for analyzing vibration characteristics of a periodic stator of a permanent magnet motor according to claim 1, wherein the analyzing the influence of the wave number and the number of permanent magnets on the characteristic value of the circular stator body according to the operational characteristics of the trigonometric function reveals that the mapping relation between the key parameter and the modal characteristic and the dynamic stability is specifically as follows:
when 2N/N m When the characteristic values are not equal to the integer, the sine and cosine characteristic values of the forward traveling wave and the backward traveling wave are respectively equal, are pure imaginary numbers, have no natural frequency splitting and are in a stable state;
when 2N/N m When the wave length is equal to the integer, the sine and cosine eigenvalues of the forward wave and the backward wave are all unequal, and the system generates natural frequency splitting; the magnetic flux, the rotational speed, the number of permanent magnets, the central angle, the length of the air gap, the radial and tangential support stiffness all affect the steady state of the stator.
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