CN110569574B - Method for improving out-of-plane vibration stability of rotor by using sine-shaped magnetic poles of permanent magnet motor - Google Patents

Method for improving out-of-plane vibration stability of rotor by using sine-shaped magnetic poles of permanent magnet motor Download PDF

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CN110569574B
CN110569574B CN201910780474.5A CN201910780474A CN110569574B CN 110569574 B CN110569574 B CN 110569574B CN 201910780474 A CN201910780474 A CN 201910780474A CN 110569574 B CN110569574 B CN 110569574B
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magnetic pole
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王世宇
王哲人
柳金龙
王姚志豪
李海洋
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Tianjin University
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Abstract

The invention discloses a method for improving the stability of vibration outside a rotor surface of a sinusoidal magnetic pole of a permanent magnet motor, which comprises the following steps: respectively establishing a dynamic model of a sinusoidal magnetic pole and a uniform magnetic pole under a follow-up coordinate system; judging the combination relation between the vibration wave number and the number of the permanent magnets by the operational property of the trigonometric function, and classifying and calculating a characteristic equation of out-of-plane vibration; and calculating a characteristic value according to the characteristic equation to obtain an unstable domain. The invention provides a method for improving the stability of a common uniform magnetic pole, so that the designed permanent magnet motor can better meet engineering requirements.

Description

Method for improving out-of-plane vibration stability of rotor by using sine-shaped magnetic poles of permanent magnet motor
Technical Field
The invention relates to the field of vibration inhibition, in particular to a technology for improving stability of out-of-plane vibration of a permanent magnet motor rotor.
Background
In various engineering fields, for example: fans and water pumps in industry and agriculture, high-precision servo systems, spindle motors for hard disk drives, and the like. In actual operation, vibration and noise are often generated, and the stability of operation is affected. The vibration and noise of the motor mainly originate from electromagnetic vibration, and the stability of the motor is improved mainly by optimizing the opening width, the pole arc coefficient and the like at present, but the modes have cases and special cases, so that a commonly applicable technology for improving the stability is particularly required.
Document (Y.B.Yang, X.H.Wang, C.Q.Zhu.Reducing Cogging Torque by Adopting Isodiametric Permanent magnet.ieee Conference on Industrial Electronics & applications.ieee 2009.) reduces cogging torque by introducing equal diameter poles and is validated by finite elements. However, the method proposed by the authors is only for examples and does not reveal general laws.
Document (N.R.Tavana, A.Shoulaie.Pole-shape optimization of permanent-magnet linear synchronous motor for reduction of thrust energy Conversion & Management,2011,52 (1): 349-354.) designs circular arc poles to reduce the effects of pulsations, and then verifies by finite element methods to verify the correctness of the study. However, the arc-shaped magnetic pole designed by the document has a complex structure and is not easy to process.
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
The invention aims to overcome the defects in the prior art, and provides a method for improving the stability of rotor out-of-plane vibration of a sinusoidal magnetic pole of a permanent magnet motor, which solves the defect that the rotor out-of-plane vibration generates an unstable phenomenon, so that a designed rotor better meets engineering requirements, and is described in detail below:
the invention aims at realizing the following technical scheme:
a method for improving the stability of vibration outside a rotor surface by using a sinusoidal magnetic pole of a permanent magnet motor comprises the following steps:
(1) Respectively establishing a dynamic model of a sinusoidal magnetic pole and a uniform magnetic pole under a follow-up coordinate system;
(2) Judging the combination relation between the vibration wave number and the number of the permanent magnets by the operational property of the trigonometric function, and classifying and calculating a characteristic equation of out-of-plane vibration;
(3) And calculating a characteristic value according to the characteristic equation to obtain an unstable domain.
Further, the dynamics model specifically comprises:
Figure GDA0004227759630000021
wherein omega is the rotation speed, k t For centrifugal stiffness operator, k rp And k rs Respectively representing dynamic support stiffness and static support stiffness operators, k p Representing a magnetic stiffness operator; wherein the sine-type magnetic pole and the uniform-type magnetic pole dynamics equation are different from each other on a magnetic stiffness operator.
Further, the root judges the combination relation between the vibration wave number and the number of the permanent magnets by means of the operation property of the trigonometric function, and calculates the characteristic equation of the out-of-plane vibration in a classified manner, wherein the characteristic equation is respectively as follows:
when 2N/N m When =int, the characteristic equation is
Figure GDA0004227759630000022
When 2N/N m When not equal to int, the characteristic equation is
Figure GDA0004227759630000023
Wherein N is the vibration wave number, N m The number of the magnetic poles is the integer, the w is the out-of-plane vibration displacement, the M is the mass matrix, the G is the gyro matrix and the K is c And K u The stiffness matrices are independent of and dependent on the combination.
Further, calculating a characteristic value according to a characteristic equation to obtain an unstable domain; and judging the effect of the sinusoidal magnetic pole on improving the stability according to the unstable region diagram.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
1. the invention firstly establishes a sine magnetic pole and a uniform magnetic pole dynamics equation respectively by means of a follow-up coordinate system, and then obtains a characteristic equation under the combination of different wave numbers and the number of magnetic poles according to a trigonometric function relation. Calculating a characteristic value according to a characteristic equation to obtain an unstable domain;
2. the invention adopts an analytic method to give out the characteristic value of the out-of-plane vibration of the rotor, and judges the dynamic stability of the system according to the characteristic value;
3. compared with the prior art, the invention has the characteristics of high efficiency, accuracy and universality, can reveal the relation between parameters, modal characteristics and dynamic stability according to the technology, and provides a technology for improving the stability of out-of-plane vibration of the rotor by changing the shape of the magnetic pole, thereby guiding the dynamic design of rotationally symmetrical machinery and improving the running stability and reliability.
Drawings
FIG. 1 is a schematic diagram of a sinusoidal pole rotor for a permanent magnet motor according to the present invention;
FIG. 2a shows the distribution of the unstable region under the remanence and the different rotational speeds of the sinusoidal magnetic poles when the vibration wave number is 2;
FIG. 2b shows the distribution of the unstable region under the remanence and the different rotational speeds of the sinusoidal magnetic poles when the vibration wave number is 3;
FIG. 3a shows the distribution of the unstable region under the remanence and different rotational speeds of the uniform magnetic pole when the vibration wave number is 2;
FIG. 3b shows the distribution of the unstable region under the remanence and different rotational speeds of the uniform magnetic pole when the vibration wave number is 3;
FIG. 4a shows the distribution of unstable regions at different rotational speeds and magnetizing thicknesses for sinusoidal poles with a vibration wave number of 2;
FIG. 4b shows the distribution of unstable regions at different rotational speeds and magnetizing thicknesses for sinusoidal poles with a vibration wave number of 3;
FIG. 5a shows the distribution of unstable regions at different speeds and magnetizing thicknesses for a uniform pole with a vibration wave number of 2;
FIG. 5b shows the distribution of unstable regions at different speeds and magnetizing thicknesses for a uniform pole with a vibration wave number of 3;
FIG. 6a is a graph showing the distribution of unstable regions at different rotational speeds and angles of sinusoidal magnetic poles at vibration wave number of 2;
FIG. 6b shows the distribution of unstable regions at different rotational speeds and angles of the sinusoidal magnetic poles at a vibration wave number of 3;
FIG. 7a is a graph showing the distribution of unstable regions at different rotational speeds and angles of the uniform magnetic poles at a vibration wave number of 2;
FIG. 7b shows the distribution of unstable regions at different rotational speeds and angles of the uniform magnetic poles at a vibration wave number of 3.
Detailed Description
The invention is described in further detail below with reference to the drawings and the specific examples. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
The embodiment of the invention provides a technology for improving the out-of-plane vibration stability of a rotor by using a sinusoidal magnetic pole of a permanent magnet motor.
The embodiment of the invention can be suitable for designing the magnetic pole of the permanent magnet motor.
The technical scheme of the embodiment of the invention is as follows: a technique for improving the stability of out-of-plane vibration of rotor by sine magnetic pole of permanent-magnet motor features that the stability of out-of-plane vibration of sine magnetic pole and uniform magnetic pole is compared.
The motor rotor structure consists of an equivalent outer ring, a web plate equivalent support and uniformly dispersed permanent magnets; the structure is subject to self-rotation; the elastic vibration analysis technique is basically characterized in that: the dynamic stability analysis and prediction of the annular periodic structure are realized by adopting a follow-up coordinate system, and the method comprises the following specific steps:
(A1) Figure 1 shows the rotation of the rotor of a permanent magnet motor around a spatial axis and a coordinate system,
Figure GDA0004227759630000031
is a follow-up coordinate system. The radius, width and thickness of the outer ring neutral circle are R, b and h respectively, and the inner diameter and the outer diameter of the web are R respectively a And R is b Young's modulus is E. Evenly distributing N on the outer ring m The number of poles, i (i=1, 2.) represents the i-th pole, the position of which is described by an H (, step function) function, then +.>
Figure GDA0004227759630000032
And->
Figure GDA0004227759630000033
The lower edge and the upper edge of the ith magnetic pole are respectively arranged, the lower edge of the first magnetic pole is positioned on the polar axis, and the included angle of the magnetic poles is gamma, so the magnetic pole has +.>
Figure GDA0004227759630000034
And->
Figure GDA0004227759630000035
By means of a follow-up coordinate system, a kinetic model of the annular periodic structure is established according to the Hamilton principle:
Figure GDA0004227759630000041
wherein t is time,
Figure GDA0004227759630000042
is the position angle, w is the out-of-plane vibration displacement, omega is the rotation speed, k t For centrifugal stiffness operator, k rp And k rs Respectively representing dynamic support stiffness and static support stiffness operators, k p Representing the magnetic stiffness operator. Wherein the sine-type magnetic pole and the uniform-type magnetic pole dynamics equation are different from each other on a magnetic stiffness operator.
For a sinusoidal magnetic pole,
Figure GDA0004227759630000043
for a uniform type of pole,
Figure GDA0004227759630000044
in the formula, h m0 And d 0 Respectively the maximum magnetizing thickness of the sinusoidal magnetic pole and the maximum distance between the stator and the rotor, B r And h m The residual magnetism and magnetizing thickness mu of the permanent magnet are respectively 0 For vacuum permeability, δ is the length of the air gap between the stator and the rotor.
Discrete by first-order Galerkin
Figure GDA0004227759630000045
Wherein W (t) represents an out-of-plane vibration complex function, "-" represents a conjugate, n represents a vibration wave number, i represents an imaginary unit, converting a partial differential equation into a normal differential equation,
S 1 +S 2 +S 3 +S 4 ＝0 (5)
for sinusoidal poles, the magnetic field is, in the formula,
Figure GDA0004227759630000046
Figure GDA0004227759630000047
Figure GDA0004227759630000048
Figure GDA0004227759630000049
Figure GDA00042277596300000410
for a uniform magnetic pole, the magnetic pole, in the formula,
Figure GDA00042277596300000411
Figure GDA00042277596300000412
Figure GDA0004227759630000051
Figure GDA0004227759630000052
Figure GDA0004227759630000053
combining (4) with
Figure GDA00042277596300000511
Performing an inner product operation to define an inner product operation,
Figure GDA0004227759630000054
(A2) Further, the root judges the combination relation between the vibration wave number and the number of the permanent magnets by means of the operation property of the trigonometric function, and then calculates the characteristic value of the out-of-plane vibration in a classified manner, wherein the trigonometric function has the following properties:
Figure GDA0004227759630000055
where int represents an integer. When 2N/N m When =int, the characteristic equation is
Figure GDA0004227759630000056
When 2N/N m When not equal to int, the characteristic equation is
Figure GDA0004227759630000057
Wherein w is an axial vibration displacement matrix, M is a unit mass matrix, G is a secondary diagonal gyro matrix, K c And K u The stiffness matrices are independent of and dependent on the combination. For the purpose of the matrix of gyroscopes,
G 12 ＝-G 21 ＝2nΩ (20)
in the stiffness matrix K c In the process, the liquid crystal display device comprises a liquid crystal display device,
Figure GDA0004227759630000058
for sinusoidal poles, the magnetic field is, in the formula,
Figure GDA0004227759630000059
for a uniform magnetic pole, the magnetic pole, in the formula,
Figure GDA00042277596300000510
in the stiffness matrix K u For sinusoidal poles, in the formula,
Figure GDA0004227759630000061
for a uniform magnetic pole, the magnetic pole, in the formula,
Figure GDA0004227759630000062
(A3) Solving the characteristic value of the out-of-plane vibration of the rotor of the permanent magnet motor, wherein the characteristic solutions of the formulas (18) and (19) are provided as
Figure GDA0004227759630000063
In which W is Re And W is Im Vibration amplitude is the real part and the imaginary part, lambda is the eigenvalue, and beta is the phase. For further analytical analysis, the eigenvalues are written in the form of real and imaginary parts,
λ＝λ Re +iλ Im (27)
wherein lambda is Re And lambda (lambda) Im And substituting the formula (27) into the formula (26) for the real part and the imaginary part of the eigenvalue, so that the real part and the imaginary part of the external vibration eigenvalue in different combinations can be obtained.
(A4) And predicting the vibration instability rule according to the solved characteristic value of the out-of-plane vibration of the rotor of the permanent magnet motor. The influence of the sinusoidal magnetic pole on the improvement of the stability can be judged by comparing the unstable regions of the sinusoidal magnetic pole and the uniform magnetic pole.
Aiming at the characteristics of the vibration equation, the embodiment of the invention provides a technology for improving the out-of-plane vibration stability of a rotor by using a sinusoidal magnetic pole of a permanent magnet motor, which can obtain a characteristic value in an analytic form and predict the dynamic stability according to the characteristic value, thereby providing a technology for improving the stability, and the specific process is as follows:
(B1) Respectively establishing a dynamic model of a sinusoidal magnetic pole and a uniform magnetic pole under a follow-up coordinate system;
(B2) Judging the combination relation between the vibration wave number and the number of the permanent magnets by the operational property of the trigonometric function, and classifying and calculating a characteristic equation of out-of-plane vibration;
(B3) And calculating a characteristic value according to the characteristic equation to obtain an unstable domain.
The specific steps of the technology for improving the vibration stability of the rotor face by considering the sine-shaped magnetic poles of the permanent magnet motor are as follows:
(C1) Respectively establishing a dynamic model of a sinusoidal magnetic pole and a uniform magnetic pole under a follow-up coordinate system;
(C2) Judging the combination relation between the vibration wave number and the number of the permanent magnets by the operational property of the trigonometric function, and classifying and calculating a characteristic equation of out-of-plane vibration;
assuming that the characteristic equation of the dynamic equation in the step (C1) is respectively a formula (18) and a formula (19) under the combination of different wave numbers and the number of magnetic poles, and further revealing the vibration instability rule according to the virtual part and the real part of the characteristic value by solving the characteristic value, thereby providing a technology for improving the stability.
(C3) Taking the ring-shaped periodic structure parameters in table 1 as an example, the characteristic values are calculated by a numerical method.
TABLE 1 basic parameters of cyclic structures
Figure GDA0004227759630000071
(C4) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of different rotating speeds and unstable regions under residual magnetism of the sinusoidal magnetic pole is shown in the figure 2 a.
(C5) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of different rotating speeds and unstable regions under residual magnetism of the sinusoidal magnetic pole is shown in the figure 2 b.
(C6) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of different rotating speeds and unstable regions under residual magnetism of the uniform magnetic pole is shown in the figure 3 a.
(C7) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of different rotating speeds and unstable regions under residual magnetism of the uniform magnetic pole is shown in the figure 3 b.
(C8) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of unstable regions under different rotating speeds and magnetizing thicknesses of the sinusoidal magnetic poles is shown in the figure 4 a.
(C9) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of unstable regions under different rotating speeds and magnetizing thicknesses of the sinusoidal magnetic poles is shown in the figure 4 b.
(C10) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of unstable regions under different rotating speeds and magnetizing thicknesses of the uniform magnetic poles is shown in the figure 5 a.
(C11) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of unstable regions under different rotating speeds and magnetizing thicknesses of the uniform magnetic poles is shown in the figure 5 b.
(C12) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of unstable regions under different rotating speeds and included angles of the sinusoidal magnetic poles is shown in the figure 6 a.
(C13) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of unstable regions under different rotating speeds and included angles of the sinusoidal magnetic poles is shown in the figure 6 b.
(C14) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 2, the distribution of unstable regions under different rotating speeds and included angles of the magnetic poles of the uniform magnetic poles is shown in the figure 7 a.
(C15) And (3) according to the characteristic value of out-of-plane vibration obtained in the step (C2), when the vibration wave number is 3, the distribution of unstable regions under different rotating speeds and included angles of the magnetic poles of the uniform magnetic poles is shown in the figure 7 a.
In summary, the invention provides a technology for improving the stability of out-of-plane vibration of a rotor by using sinusoidal magnetic poles of a permanent magnet motor. The technology uses a follow-up coordinate system, adopts an analysis method to obtain the characteristic value of the system, improves the accuracy, the calculation efficiency and the universality, and better meets the actual requirements of engineering.
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 invention is not limited to the embodiments described above. The above description of specific embodiments is intended to describe and illustrate the technical aspects of the present invention, and is intended to be illustrative only and not limiting. Numerous specific modifications can be made by those skilled in the art without departing from the spirit of the invention and scope of the claims, which are within the scope of the invention.

Claims (1)

1. A method for improving the stability of vibration outside a rotor surface by using a sinusoidal magnetic pole of a permanent magnet motor is characterized by comprising the following steps:
(1) Respectively establishing a dynamic model of a sinusoidal magnetic pole and a uniform magnetic pole under a follow-up coordinate system; the dynamics model is specifically as follows:
Figure FDA0004221327420000011
wherein t is time,
Figure FDA0004221327420000012
is the position angle, omega is the rotating speed, k t For centrifugal stiffness operator, k rp And k rs Respectively representing dynamic support stiffness and static support stiffness operators, k p Representing a magnetic stiffness operator; wherein the sine magnetic pole and the uniform magnetic pole dynamics equation are different from each other in the magnetic stiffness operator;
Figure FDA0004221327420000013
is a follow-up coordinate system; the radius, width and thickness of the outer ring neutral circle are R, b and h respectively, and the inner diameter and the outer diameter of the web are R respectively a And R is b Young's modulus E; evenly distributing N on the outer ring m The number of poles, i (i=1, 2.) represents the i-th pole, the position of which is described by an H (, step function) function, then +.>
Figure FDA0004221327420000014
And->
Figure FDA0004221327420000015
The lower edge and the upper edge of the ith magnetic pole are respectively arranged, the lower edge of the first magnetic pole is positioned on the polar axis, the included angle of the magnetic poles is gamma, and the included angle is +.>
Figure FDA0004221327420000016
And->
Figure FDA0004221327420000017
For a sinusoidal magnetic pole,
Figure FDA0004221327420000018
for a uniform type of pole,
Figure FDA0004221327420000019
in the formula, h m0 And d 0 Respectively the maximum magnetizing thickness of the sinusoidal magnetic pole and the maximum distance between the stator and the rotor, B r And h m The residual magnetism and magnetizing thickness mu of the permanent magnet are respectively 0 Vacuum magnetic permeability, delta is the length of an air gap between the stator and the rotor;
discrete by first-order Galerkin
Figure FDA00042213274200000110
Wherein W (t) represents an out-of-plane vibration complex function, "-" represents a conjugate, n represents a vibration wave number, i represents an imaginary unit, converting a partial differential equation into a normal differential equation,
S 1 +S 2 +S 3 +S 4 ＝0
for sinusoidal poles, the magnetic field is, in the formula,
Figure FDA00042213274200000111
Figure FDA0004221327420000021
Figure FDA0004221327420000022
Figure FDA0004221327420000023
Figure FDA0004221327420000024
for a uniform magnetic pole, the magnetic pole, in the formula,
Figure FDA0004221327420000025
Figure FDA0004221327420000026
Figure FDA0004221327420000027
Figure FDA0004221327420000028
Figure FDA0004221327420000029
will be described in
Figure FDA00042213274200000210
And->
Figure FDA00042213274200000214
Performing an inner product operation to define an inner product operation,
Figure FDA00042213274200000211
(2) Judging the combination relation between the vibration wave number and the number of the permanent magnets by the operational property of the trigonometric function, and classifying and calculating a characteristic equation of out-of-plane vibration; the characteristic equations are respectively:
when 2N/N m When =int, the characteristic equation is
Figure FDA00042213274200000212
When 2N/N m When not equal to int, the characteristic equation is
Figure FDA00042213274200000213
Wherein N is the vibration wave number, N m The number of the magnetic poles is the integer, the w is the out-of-plane vibration displacement, the M is the mass matrix, the G is the gyro matrix and the K is c And K u Stiffness matrices that are not affected by the combination and are affected by the combination, respectively;
(3) Calculating a characteristic value according to the characteristic equation to obtain an unstable region, and judging the effect of the sinusoidal magnetic pole on improving the stability according to the unstable region diagram.
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