CN106383971A - Improved analytical model of motor stator core vibration induced by magnetostriction - Google Patents

Abstract

The invention discloses an improved analytical model of motor stator core vibration induced by magnetostriction. According to the characteristics of a structure and magnetic field distribution of a radial flux motor, a stator core is divided into two subdomains: a yoke and a tooth; the vibration of the yoke is the vibration of a torus and the vibration of the tooth is the vibration of a rectangular block; based on a piezomagnetic equation and the Newton's law, vibration partial differential equations of the yoke and the tooth of the stator core along the radial and axial directions are established; and a vibration differential equation is solved by adopting a method of separation of variables, and undetermined coefficients are solved according to a boundary condition. The improved analytical model of the motor stator core vibration induced by the magnetostriction provided by the invention is accurate in calculation result and clear in physical conception, can be applied to calculating the vibration displacement, velocity and acceleration of the motor stator core induced by the magnetostriction, the stress and strain distribution characteristics, and is an effective method for motor vibration analysis.

Description

The electric machine stator iron vibration analysis model that improved magnetostriction causes

Technical field

The invention belongs to motor electromagnetic Vibration Analysis Technology field is and in particular to a kind of electricity that causes of improved magnetostriction Machine stator iron coring vibration analytical model.

Background technology

The vibration noise source of motor is mainly：Electromagnetic vibration noise source, mechanical vibration noise source and air force vibration are made an uproar Sound source.Electromagnetic vibration noise is divided into the vibration noise that electromagnetic force causes and the vibration noise that magnetostriction causes etc. again.Existing Achievement in research in, many scholars have ignored magnetostrictive effect, and studies emphatically the electric and magnetic oscillation causing due to electromagnetic force Noise.But in recent years, with the continuous development of experiment test technology, engineer applied problem become more meticulous research progressively deeply, And the constantly improve of FEM calculation, the progressively development that especially multiple physical field coupling calculates, magnetostrictive effect is for electricity The impact of dynamo-electric magnetic vibration is progressively taken seriously.

In motor, magnetostrictive effect is really a magnet machine coupling problem.Chinese scholars are in research mangneto at present During the motor electromagnetic vibration that flex effect causes, commonly used computational methods are FInite Element.During using finite element method analysis The foundation of magnetostriction model is broadly divided into two kinds, respectively the model based on percentage elongation and the model based on power.Based on elongation The model of rate is by the use of magnetostriction percentage elongation as major variable.It is by the use of magnetostrictive force as main based on the model of power Variable.Magnetostrictive force is the change in size same shape of generation being produced magnetic material under the influence of a magnetic field by magnetostriction Describing, this virtual power is referred to as magnetostrictive force to the equivalent power becoming.Although it is high that FInite Element has solving precision, its Take computer resource too many, the time that calculates is long, and result of calculation is susceptible to the impact of the factors such as mesh generation.At present only One open source literature data is vibrated using the motor electromagnetic that analytic method causes to magnetostriction and is studied, but this analytical Calculation mould Type only accounts for electric machine stator iron two dimensional surface internal vibration, does not account for the axial vibration of iron core.The method only considers simultaneously Stator core yoke portion torus stress tangentially, have ignored stress radially and axially, this simplified model can only be counted Calculate motor yoke toric Mean Oscillation situation it is impossible to calculate the internal vibration displacement of electric machine stator iron yoke portion, stress and The distribution character of strain.

Content of the invention

Goal of the invention

The deficiency existing for existing motor oscillating computational methods, the present invention proposes a kind of improved magnetostriction and causes Electric machine stator iron vibration analysis model.

Technical scheme

For solving above-mentioned technical problem, the present invention adopts the following technical scheme that：

A kind of electric machine stator iron vibration analysis model that improved magnetostriction causes it is characterised in that：According to radially The structure of flux electric machine and characteristics of magnetic field distribution, stator core are divided into yoke portion and two subdomains of teeth portion, yoke portion vibrates for circle The vibration of ring body, teeth portion vibrates the vibration for rectangular block；Set up vibration analysis model to comprise the following steps that：

1) cause electric machine stator iron vibration principle from magnetostriction, stator core is set up respectively based on pressure magnetic equation Yoke portion and the fundamental equation in teeth portion solution domain；

2) according to Newton's second law, set up stator core yoke portion respectively and teeth portion solves domain vibration equation；

3) adopt the separation of variable, introduce Bessel function, utilize the vibration of linear partial differential equation Superposition Principle micro- Divide equation, obtain the general solution of vibration equation；

4) determine the boundary condition of stator core yoke portion and teeth portion；

5) the Boundary Condition for Solving undetermined coefficient according to stator core yoke portion and teeth portion, obtains stator core yoke portion and teeth portion Solve the solution that domain meets boundary condition；

6) vibration displacement based on stator core yoke portion and teeth portion, is derived by the vibration speed of yoke portion and teeth portion further Degree, acceleration, strain and stress.

Above-mentioned steps 1) in, the electric machine stator iron vibration principle that causes from magnetostriction, built based on pressure magnetic equation Vertical electric machine stator iron yoke portion and teeth portion fundamental equation are as follows：

Electric machine stator iron yoke portion fundamental equation is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{yo\mathrm{ke}\_r}=s_{11}^H{\mathrm{\σ}}_{yo\mathrm{ke}\_r}+s_{12}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{21}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_\mathrm{\θ}}=s_{12}^H{\mathrm{\σ}}_{yo\mathrm{ke}\_r}+s_{22}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{22}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_z}=s_{33}^H{\mathrm{\σ}}_{yoke\_z}+d_{23}{H}_{yoke\_\mathrm{\θ}}\end{array}\right.;$

Electric machine stator iron teeth portion fundamental equation is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{teeth\_x}=\frac{{\mathrm{\σ}}_{teeth\_x}}{E}+d_{11}{H}_{teeth\_x}\\ {\mathrm{\ϵ}}_{teeth\_z}=\frac{{\mathrm{\σ}}_{teeth\_z}}{E}+d_{13}{H}_{teeth\_x}\end{array}\right..$

Above-mentioned steps 2) in, stator core yoke portion is set up according to Newton's second law and the vibration equation of teeth portion is as follows：

Stator core yoke portion torus oscillatory differential equation is：

$\{\begin{array}{c}\frac{{\∂}^2{u}_{yoke\_r}}{\∂t^2}=c_1^2\[\frac{{\∂}^2{u}_{yo\mathrm{ke}\_r}}{\∂r^2}+\frac{1}{r}\frac{\∂u_{yo\mathrm{ke}\_r}}{\∂r}-\frac{u_{yo\mathrm{ke}\_r}}{r^2}\]\\ \frac{{\∂}^2{u}_{yoke\_z}}{\∂t^2}=c_2^2\frac{{\∂}^2{u}_{yoke\_z}}{\∂z^2}\end{array};$

The oscillatory differential equation of stator core teeth portion is：

$\left\{\begin{array}{c}\frac{{\∂}^2{u}_{teeth\_x}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}\\ \frac{{\∂}^2{u}_{teeth\_z}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}\end{array}\right..$

Above-mentioned steps 3) in, using the separation of variable, introduce Bessel function, utilize linear partial differential equation principle of stacking Oscillatory differential equation is solved, obtains stator core yoke portion and the general solution of teeth portion is as follows：

The general solution in stator core yoke portion：

$\left\{\begin{array}{c}{u}_{yo\mathrm{ke}\_r}=\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ u_{yoke\_z}=({\mathrm{Acosk}}_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.;$

The general solution of stator core teeth portion：

$\{\begin{array}{c}{u}_{teeth\_x}=({\mathrm{Ccosk}}_{3}x+{\mathrm{Dsink}}_{3}x)e^{j\mathrm{\ω}t}\\ u_{teeth\_z}=({\mathrm{Fcosk}}_{3}z+{\mathrm{Gsink}}_{3}z)e^{j\mathrm{\ω}t}\end{array}.$

Above-mentioned steps 4) in, the boundary condition in stator core yoke portion is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{yoke\_r}{|}_{r=R_1}=0\\ {\mathrm{\σ}}_{yoke\_r}{|}_{r=R_{i1}}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=l_z}=0\end{array}\right.;$

The boundary condition of stator core teeth portion is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{teeth\_x}{|}_{x=0}=0\\ u_{teeth\_x}{|}_{x=h_t}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=l_z}=0\end{array}\right..$

When calculating stator core yoke portion and teeth portion vibration, the impact that teeth portion and winding vibrate to yoke portion is set to additional matter Amount, the impact that yoke portion vibrates to teeth portion is the initial displacement that yoke portion vibration displacement is set to teeth portion vibration.

Advantage and effect

Compared to the prior art, the present invention has advantages below：

(1) relation between each physical quantity, clear physics conception can clearly be reflected.

(2) amount of calculation is little, and calculating speed is fast, and result of calculation is accurately and reliably.

(3) can be used for the calculating analysis of the vibration displacement of electric machine stator iron, speed, acceleration, stress and strain, for meter Electric machine stator iron vibration a kind of effective method of offer that magnetostriction causes is provided.

Brief description

Fig. 1 is the structure chart of radial flux magneto, and in figure 1 is casing, and 2 is stator core, and 3 is rotor core.

Fig. 2 is analytical model Establishing process figure.

Fig. 3 is stator core yoke portion's torus and mass-element schematic diagram.Wherein, Fig. 3 a is yoke portion torus schematic diagram, figure 3b is yoke portion ring cross-section schematic diagram, and Fig. 3 c is mass-element schematic diagram.

Fig. 4 is stator core single tooth schematic diagram.

Fig. 5 is motor 3D electromagnetic finite element computation model in embodiment.

Fig. 6 is electric machine stator iron yoke portion B point position tangentially and radially magnetic flux density figure in embodiment.Wherein, Fig. 6 a is Tangential magnetic is close, and Fig. 6 b is that radial direction magnetic is close.

Fig. 7 is electric machine stator iron teeth portion A point position radially and tangentially magnetic flux density figure in embodiment.Wherein, Fig. 7 a is Radial direction magnetic is close, and Fig. 7 b is that tangential magnetic is close.

Fig. 8 is the radial vibration displacement diagram of electric machine stator iron yoke portion B point position in embodiment.

Fig. 9 is the radial vibration displacement diagram of electric machine stator iron teeth portion A point position in embodiment.

Figure 10 is the radial vibration displacement of stator core yoke portion, strain and stress figure in embodiment.Wherein, Figure 10 a is radially Displacement, Figure 10 b is radial strain, and Figure 10 c is radial stress.

Figure 11 is the portion's tangential strain of stator core yoke and stress diagram in embodiment.Wherein, Figure 11 a is tangential strain, Figure 11 b For tangential stress.

Figure 12 is stator core yoke portion axial vibratory displacement, strain and stress figure in embodiment.Wherein, Figure 12 a is axially Displacement, Figure 12 b axial strain, Figure 12 c is axial stress.

Figure 13 is stator core teeth portion radial vibration displacement, strain and stress figure in embodiment.Wherein, Figure 13 a is radially Displacement, Figure 13 b radial strain, Figure 13 c is radial stress.

Figure 14 is stator core teeth portion axial vibratory displacement, strain and stress figure in embodiment.Wherein, Figure 14 a is axially Displacement, Figure 14 b axial strain, Figure 14 c is axial stress.

Specific embodiment：

The present invention proposes the electric machine stator iron vibration analysis model that a kind of improved magnetostriction causes, according to radially The structure of flux electric machine and characteristics of magnetic field distribution, stator core are divided into yoke portion and two subdomains of teeth portion, yoke portion vibrates for circle The vibration of ring body, teeth portion vibrates the vibration for rectangular block.Set up vibration analysis model to comprise the following steps that：

Step 1, causes electric machine stator iron vibration principle from magnetostriction, sets up stator respectively based on pressure magnetic equation Rear of core and the fundamental equation in teeth portion solution domain.

Step 2, according to Newton's second law, sets up stator core yoke portion respectively and teeth portion solves domain vibration equation.

Step 3, using the separation of variable, is introduced Bessel function, is shaken using linear partial differential equation Superposition Principle The dynamic differential equation, obtain the general solution of vibration equation.

Step 4, determines the boundary condition of stator core yoke portion and teeth portion.

Step 5, according to the Boundary Condition for Solving undetermined coefficient of stator core yoke portion and teeth portion, obtain stator core yoke portion and Teeth portion solves the solution that domain meets boundary condition.

Step 6, based on the vibration displacement in stator core yoke portion and teeth portion, is derived by the vibration of yoke portion and teeth portion further Speed, acceleration, strain and stress.

Below in conjunction with the accompanying drawings the present invention is described in detail：

Fig. 1 is structure chart, the structure according to radial flux motors and the characteristics of magnetic field distribution of radial flux motors, by stator Iron core is divided into yoke portion and two subdomains of teeth portion, and yoke portion vibrates for toric vibration, and teeth portion vibrates the vibration for rectangular block.For It is easy to analyze, make following basic assumption：

1) yoke portion magnetic field is main circumferentially, mainly considers circumferential component, ignores other components.Yoke portion magnetic field is radially even Distribution.

2) teeth portion magnetic field primarily radially, mainly considers radial component, ignores other components.Usual for middle and small motor For parallel teeth, the distribution of teeth portion field homogeneity.

Magnetostrictive effect is inherent character when object is magnetized, and Magnetostrictive Properties can use the tensor shape of pressure magnetic equation Formula is expressed as：

$\{\begin{array}{c}{\mathrm{\ϵ}}_i=s_{ij}^H{\mathrm{\σ}}_j+d_{ni}{H}_n\\ B_n=d_{mj}{\mathrm{\σ}}_j+{\mathrm{\μ}}_{mn}^{\mathrm{\σ}}{H}_n\end{array}\left|\begin{array}{c}i,j=1,\mathrm{......6}\\ m,n=1,\mathrm{......},3\end{array}\right.---\left(1\right);$

Wherein, ε_iFor strain tensor component；s^HFor the elastic constant under normal magnetic field；σ_jFor stress tensor component；D is mangneto Coefficient of dilatation；H_nFor magnetic intensity vector component；B_mFor magnetic flux density vector component；μ^σFor the pcrmeability under normal pressure power.

Fig. 2 is to set up analytical model flow chart, sets up the analytical model of stator core yoke portion and teeth portion separately below.

1st, the foundation of stator core yoke portion analytical model, comprises the following steps that：

For easy analysis, using cylindrical coordinate (O-r θ z).Along the circumferential direction component is H to yoke portion magnetic field intensity_{yoke_θ}.

(1) fundamental equation

Based on pressure magnetic equation, the Basic equation group in stator core yoke portion is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{yo\mathrm{ke}\_r}=s_{11}^H{\mathrm{\σ}}_{yo\mathrm{ke}\_r}+s_{12}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{21}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_\mathrm{\θ}}=s_{12}^H{\mathrm{\σ}}_{yo\mathrm{ke}\_r}+s_{22}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{22}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_z}=s_{33}^H{\mathrm{\σ}}_{yoke\_z}+d_{23}{H}_{yoke\_\mathrm{\θ}}\end{array}\right.---\left(2\right);$

According to the physical property of material, elastic constant s₁₁=s₂₂=s₃₃It is assumed that magnetostriction changes for equal-volume, press magnetic Coefficient d₂₁=d₂₂, d₂₃=-d₂₂/2.Replace elastic constant with Young's moduluss E and Poisson's ratio αThen haveSubstitution formula (2)：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{yoke\_r}=\frac{E}{1-{\mathrm{\α}}^2}({\mathrm{\ϵ}}_{yoke\_r}+{\mathrm{\α\ϵ}}_{yoke\_\mathrm{\θ}})-\frac{d_{22}E}{1-{\mathrm{\α}}^2}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\σ}}_{yoke\_\mathrm{\θ}}=\frac{E}{1-{\mathrm{\α}}^2}({\mathrm{\ϵ}}_{yoke\_\mathrm{\θ}}+{\mathrm{\α\ϵ}}_{yoke\_r})-\frac{d_{22}E}{1-{\mathrm{\α}}^2}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\σ}}_{yoke\_z}={\mathrm{E\ϵ}}_{yoke\_z}-d_{23}{\mathrm{EH}}_{yoke\_\mathrm{\θ}}\end{array}\right.---\left(3\right);$

According to annulus Theory of Vibration, the stator core yoke portion each component of toric strain is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{yo\mathrm{ke}\_r}=\frac{\∂u_{yo\mathrm{ke}\_r}}{\∂r}\\ {\mathrm{\ϵ}}_{yo\mathrm{ke}\_\mathrm{\θ}}=\frac{u_{yo\mathrm{ke}\_r}}{r}\\ {\mathrm{\ϵ}}_{yo\mathrm{ke}\_z}=\frac{\∂u_{yo\mathrm{ke}\_z}}{\∂z}\end{array}\right.---\left(4\right);$

Wherein, u_{yoke_r}For yoke portion along r direction vibration displacement, u_{yoke_z}For yoke portion vibration displacement in the z-direction.

(2) vibration equation

Stator core yoke portion annulus body unit fritter is as shown in figure 3, stator teeth and winding are to stator yoke when calculating Impact considered with an additional mass, here introduce stator core quality additional coefficient Δ_w, it is

${\mathrm{\Δ}}_w=1+\frac{M_t+M_w}{M_c}---\left(5\right);$

Wherein, M_c, M_t, M_wIt is respectively the quality of stator yoke, teeth portion and winding.

If the density of stator core is ρ, the quality of unit fritter is ρ Δ_wRd θ drdz, according to Newton's second law, abbreviation Obtaining stator core yoke portion torus oscillatory differential equation is：

$\{\begin{array}{c}\frac{{\∂}^2{u}_{yoke\_r}}{\∂t^2}=c_1^2\[\frac{{\∂}^2{u}_{yo\mathrm{ke}\_r}}{\∂r^2}+\frac{1}{r}\frac{\∂u_{yo\mathrm{ke}\_r}}{\∂r}-\frac{u_{yo\mathrm{ke}\_r}}{r^2}\]\\ \frac{{\∂}^2{u}_{yoke\_z}}{\∂t^2}=c_2^2\frac{{\∂}^2{u}_{yoke\_z}}{\∂z^2}\end{array}---\left(6\right);$

Wherein,

$c_1=\sqrt{\frac{E}{{\mathrm{\ρ\Δ}}_w(1-{\mathrm{\α}}^2)}},c_2=\sqrt{\frac{E}{{\mathrm{\ρ\Δ}}_w}}---\left(7\right).$

(3) general solution of vibration equation

Stator yoke iron core is in alternating magnetic field H_{yoke_θ}=H_je^jωtThe lower generation forced vibration of effect, using the separation of variable, Introduce Bezier (Bessel) function, using linear partial differential equation principle of stacking, equation is solved.Vibration equation can be obtained General solution be：

$\left\{\begin{array}{c}{u}_{yo\mathrm{ke}\_r}=\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ u_{yoke\_z}=({\mathrm{Acosk}}_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(8\right);$

Wherein, J₁(k₁R) it is first kind first-order bessel function, Y₁(k₁R) be Equations of The Second Kind first-order bessel function, M, N, A, B are the constant being determined by boundary condition,

$k_1=\frac{\mathrm{\ω}}{c_1},k_2=\frac{\mathrm{\ω}}{c_2}---\left(9\right).$

(4) boundary condition

Stator core yoke portion is radially that machinery is free, is zero in border upper stress.Stator core yoke portion is machine vertically Tool free boundary condition, is zero in border upper stress.If R₁For stator core outer radius, R_i1For in stator core yoke portion annulus body Radius, its boundary condition is represented by：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{yoke\_r}{|}_{r=R_1}=0\\ {\mathrm{\σ}}_{yoke\_r}{|}_{r=R_{i1}}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=l_z}=0\end{array}\right.---\left(10\right);$

(5) undetermined coefficient and the solution meeting boundary condition are solved

According to the toric boundary condition in stator core yoke portion, can obtain：

$\left\{\begin{array}{c}M=\frac{P_4-P_2}{P_1{P}_4-P_2{P}_3}I\\ N=\frac{P_1-P_3}{P_1{P}_4-P_2{P}_3}I\\ A=\frac{d_{23}{H}_j({\mathrm{cosk}}_2{l}_z-1)}{k_2{\mathrm{sink}}_2{l}_z}\\ B=\frac{d_{23}{H}_j}{k_2}\end{array}\right.---\left(11\right);$

Wherein,

$\left\{\begin{array}{c}{P}_1=k_1{J}_0\left(k_1{R}_1\right)-\frac{1-\mathrm{\α}}{R_1}{J}_1\left(k_1{R}_1\right)\\ P_2=k_1{Y}_0\left(k_1{R}_1\right)-\frac{1-\mathrm{\α}}{R_1}{Y}_1\left(k_1{R}_1\right)\\ P_3=k_1{J}_0\left(k_1{R}_{i1}\right)-\frac{1-\mathrm{\α}}{R_{i1}}{J}_1\left(k_1{R}_{i1}\right)\\ P_4=k_1{Y}_0\left(k_1{R}_{i1}\right)-\frac{1-\mathrm{\α}}{R_{i1}}{Y}_1\left(k_1{R}_{i1}\right)\\ I=(1+\mathrm{\α})d_{22}{H}_j\end{array}\right.---\left(12\right);$

Constant M, N, A, B are substituted into formula (8), that is, is met the solution of boundary condition.

(6) yoke portion vibration characteristics

Based on stator core yoke portion vibration displacement, the vibration velocity obtaining stator core yoke portion is：

$\left\{\begin{array}{c}{v}_{yo\mathrm{ke}\_r}=\frac{{\mathrm{du}}_{yo\mathrm{ke}\_r}}{dt}=\mathrm{\ω}\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ v_{yoke\_z}\frac{{\mathrm{du}}_{yoke\_z}}{dt}=\mathrm{\ω}({\mathrm{Acosk}}_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(13\right);$

The acceleration of vibration in stator core yoke portion is

$\left\{\begin{array}{c}{a}_{yo\mathrm{ke}\_r}=\frac{d^2{u}_{yo\mathrm{ke}\_r}}{{\mathrm{dt}}^2}={\mathrm{\ω}}^2\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ a_{yoke\_z}\frac{d^2{u}_{yoke\_z}}{{\mathrm{dt}}^2}={\mathrm{\ω}}^2({\mathrm{Acosk}}_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(14\right);$

The flexible strain in stator core yoke portion is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{yoke\_r}=\frac{\∂u_{yoke\_r}}{\∂r}=\left(M\right(k_1{J}_0\left(k_{1}r\right))+N(k_1{Y}_0\left(k_{1}r\right)-\frac{1}{r}{Y}_1\left(K_{1}r\right)\left)\right)e^{j\mathrm{\ω}t}\\ {\mathrm{\ϵ}}_{yoke\_\mathrm{\θ}}=\frac{u_{yoke\_r}}{r}=\frac{\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}}{r}\\ {\mathrm{\ϵ}}_{yoke\_z}=\frac{\∂u_{yoke\_z}}{\∂z}=({\mathrm{Bcosk}}_{2}z-{\mathrm{Asink}}_{2}z)k_2{e}^{j\mathrm{\ω}t}\end{array}\right.---\left(15\right);$

The dilation matrices in stator core yoke portion are：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{yoke\_r}=\frac{E}{1-{\mathrm{\α}}^2}\left(\begin{array}{c}M\left(k_1{J}_0\right(k_{1}r)-(1-\mathrm{\α}\left)\frac{J_1\left(k_{1}r\right)}{r}\right)+N\left(k_1{Y}_0\right(k_{1}r)-(1-\mathrm{\α}\left)\frac{Y_1\left(k_{1}r\right)}{r}\right)\\ +(1-\mathrm{\α})d_{22}{H}_{yoke\_\mathrm{\θ}}\end{array}\right)\\ {\mathrm{\σ}}_{yoke\_\mathrm{\θ}}=\frac{E}{1-{\mathrm{\α}}^2}\left(\begin{array}{c}M\left({\mathrm{\αk}}_1{J}_0\right(k_{1}r)+(1-\mathrm{\α}\left)\frac{J_1\left(k_{1}r\right)}{r}\right)+N\left({\mathrm{\αk}}_1{Y}_0\right(k_{1}r)+(1-\mathrm{\α}\left)\frac{Y_1\left(k_{1}r\right)}{r}\right)\\ -(1+\mathrm{\α})d_{22}{H}_{yoke\_\mathrm{\θ}}\end{array}\right)\\ {\mathrm{\σ}}_{yoke\_z}=E({\mathrm{Bcosk}}_{2}z-{\mathrm{Asink}}_{2}z)k_2{e}^{j\mathrm{\ω}t}-d_{23}{\mathrm{EH}}_{yoke\_\mathrm{\θ}}\end{array}\right.---\left(16\right).$

2nd, the foundation of stator core teeth portion analytical model, comprises the following steps that：

(1) fundamental equation

For easy analysis, using rectangular coordinate system.Stator core single tooth schematic diagram is as shown in figure 4, the height h of tooth_t In the x-direction, the width b of tooth_tIn the y-direction, the axial length l of tooth_zIn the z-direction.Stator core teeth portion magnetic field radial component H_{teeth_x}.The fundamental equation of stator core teeth portion is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{teeth\_x}=\frac{{\mathrm{\σ}}_{teeth\_x}}{E}+d_{11}{H}_{teeth\_x}\\ {\mathrm{\ϵ}}_{teeth\_z}=\frac{{\mathrm{\σ}}_{teeth\_z}}{E}+d_{13}{H}_{teeth\_x}\end{array}\right.---\left(17\right);$

Thus obtaining teeth portion stress is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{teeth\_x}={\mathrm{E\ϵ}}_{teeth\_x}-{\mathrm{Ed}}_{11}{H}_{teeth\_x}\\ {\mathrm{\σ}}_{teeth\_z}={\mathrm{E\ϵ}}_{teeth\_z}-{\mathrm{Ed}}_{13}{H}_{teeth\_x}\end{array}\right.---\left(18\right).$

(2) vibration equation

Stator core teeth portion with the strain in z direction is in the x-direction：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{teeth\_x}=\frac{\∂u_{teeth\_x}}{\∂x}\\ {\mathrm{\ϵ}}_{teeth\_z}=\frac{\∂u_{teeth\_z}}{\∂z}\end{array}\right.---\left(19\right);$

Wherein, u_{teeth_x}For teeth portion vibration displacement in the x-direction, u_{teeth_z}For teeth portion vibration displacement in the z-direction.

The quality of stator core teeth portion fritter is ρ dxdydz, according to Newton's second law, obtains stator core tooth through abbreviation The oscillatory differential equation in portion is：

$\left\{\begin{array}{c}\frac{{\∂}^2{u}_{teeth\_x}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}\\ \frac{{\∂}^2{u}_{teeth\_z}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}\end{array}\right.---\left(20\right);$

Wherein,

$c_3=\sqrt{\frac{E}{\mathrm{\ρ}}}---\left(21\right).$

(3) general solution of vibration equation

Stator core teeth portion is in alternating magnetic field H_{teeth_x}=H_te^jωtIn the presence of produce forced vibration, at this moment obtain formula (20) general solution is：

$\left\{\begin{array}{c}{u}_{yo\mathrm{ke}\_r}=\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ u_{yoke\_z}=({\mathrm{Acosk}}_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(22\right);$

Wherein,

$k_3=\frac{\mathrm{\ω}}{c_3}---\left(23\right).$

(4) boundary condition

One end that stator core teeth portion is connected with yoke portion is fixed constraint boundary condition, and one end that teeth portion is connected with air gap is Mechanical free boundary condition, teeth portion is mechanical free boundary condition vertically, and the boundary condition of stator core teeth portion can represent For：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{teeth\_x}{|}_{x=0}=0\\ u_{teeth\_x}{|}_{x=h_t}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=l_z}=0\end{array}\right.---\left(24\right).$

(5) undetermined coefficient and the solution meeting boundary condition are solved

Substitute into boundary condition, determine that undetermined coefficient C, D, F, G are：

$\left\{\begin{array}{c}C=-\frac{d_{11}{H}_t}{k_3}\·\frac{{\mathrm{sink}}_3{h}_t}{{\mathrm{cosk}}_3{h}_t}\\ D=\frac{d_{11}{H}_t}{k_3}\\ F=\frac{d_{13}{H}_t({\mathrm{cosk}}_3{l}_z-1)}{k_3{\mathrm{sink}}_3{l}_z}\\ G=\frac{d_{13}{H}_t}{k_3}\end{array}\right.---\left(25\right);$

Coefficient C, D, F, G that above formula is determined substitute into formula (22), that is, be met the solution of boundary condition.

Because stator core teeth portion quality is less than stator yoke, the vibration in stator core yoke portion is very big on teeth portion impact, yoke The impact to teeth portion vibration for portion's vibration is the initial displacement that yoke portion vibration displacement is set to teeth portion vibration.By yoke portion radially and The vibration of axial direction is superimposed along the vibration in the high direction of tooth and axial direction with teeth portion respectively, thus obtaining the vibration displacement of stator core teeth portion For：

$\left\{\begin{array}{c}{u}_{total\_r}=u_{teeth\_x}+u_{yoke\_r}\\ =({\mathrm{Ccosk}}_{3}x+{\mathrm{Dsink}}_{3}x)e^{j\mathrm{\ω}t}+\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ u_{total\_z}=u_{teeth\_z}+u_{yoke\_z}\\ =({\mathrm{Fcosk}}_{3}z+{\mathrm{Gsink}}_{3}z)e^{j\mathrm{\ω}t}+({\mathrm{Acosk}}_{2}z+{\mathrm{Bsink}}_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(26\right);$

The vibration velocity of stator core teeth portion is：

$\left\{\begin{array}{c}{v}_{total\_r}=\frac{{\mathrm{du}}_{total\_r}}{dt}=\mathrm{\ω}({\mathrm{Ccosk}}_{3}x+{\mathrm{Dsink}}_{3}x)e^{j\mathrm{\ω}t}+\mathrm{\ω}\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ v_{total\_z}=\frac{{\mathrm{du}}_{total\_z}}{dt}=\mathrm{\ω}({\mathrm{Fcosk}}_{3}z+{\mathrm{Gsink}}_{3}z)e^{j\mathrm{\ω}t}+\mathrm{\ω}({\mathrm{Acosk}}_{2}z+{\mathrm{Bsink}}_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(27\right);$

The acceleration of vibration of stator core teeth portion is：

$\left\{\begin{array}{c}{a}_{total\_r}=\frac{d^2{u}_{total\_r}}{{\mathrm{dt}}^2}={\mathrm{\ω}}^2({\mathrm{Ccosk}}_{3}x+{\mathrm{Dsink}}_{3}x)e^{j\mathrm{\ω}t}+{\mathrm{\ω}}^2\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ a_{total\_z}=\frac{d^2{u}_{total\_z}}{{\mathrm{dt}}^2}={\mathrm{\ω}}^2({\mathrm{Fcosk}}_{3}z+{\mathrm{Gsink}}_{3}z)e^{j\mathrm{\ω}t}+{\mathrm{\ω}}^2({\mathrm{Acosk}}_{2}z+{\mathrm{Bsink}}_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.---\left(28\right);$

The strain of stator core teeth portion is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{teeth\_x}=\frac{\∂u_{teeth\_x}}{\∂x}=({\mathrm{Dcosk}}_{3}x-{\mathrm{Csink}}_{3}x){\mathrm{ke}}^{j\mathrm{\ω}t}\\ {\mathrm{\ϵ}}_{teeth\_z}=\frac{\∂u_{teeth\_z}}{\∂z}=({\mathrm{Gcosk}}_{3}z-{\mathrm{Fsink}}_{3}z)k_3{e}^{j\mathrm{\ω}t}\end{array}\right.---\left(29\right);$

The stress of stator core teeth portion is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{teeth\_x}=E({\mathrm{Dcosk}}_{3}x-{\mathrm{Csink}}_{3}x)k_3{e}^{j\mathrm{\ω}t}-d_{11}{\mathrm{EH}}_{teeth\_x}\\ {\mathrm{\σ}}_{teeth\_z}=E({\mathrm{Gcosk}}_{3}z-{\mathrm{Fsink}}_{3}z)k_3{e}^{j\mathrm{\ω}t}-d_{13}{\mathrm{EH}}_{teeth\_x}\end{array}\right.---\left(30\right).$

Embodiment：

The present invention is tested using the electric machine stator iron vibration analysis computation model that FInite Element causes to magnetostriction Card.Carry out quantitative analysiss, this 2.1kW Rated motor frequency is taking a 2.1kW radial flux non-crystaline amorphous metal magneto as a example 267Hz, 8 pole 36 groove.During analysis, two methods adopt same performance parameter.Using 2D finite element, the electromagnetic field of magneto is entered Row analysis, a quarter periodic model setting up motor is as shown in Figure 5.It is calculated electric machine stator iron yoke portion B point position to cut To changing over curve with radial magnetic flux density as shown in fig. 6, electric machine stator iron teeth portion A point position radially and tangentially magnetic flux It is as shown in Figure 7 that density changes over curve.It can be seen that electric machine stator iron yoke portion magnetic field is predominantly tangential from Fig. 6 and 7, Teeth portion magnetic field is mainly radially.It is respectively adopted 3D FInite Element and analytical calculation model to electric machine stator iron teeth portion and yoke portion Vibration displacement is calculated.Obtain the radial vibration displacement of electric machine stator iron yoke portion B point position and teeth portion A point position respectively such as Shown in Fig. 8 and Fig. 9.From result of calculation as can be seen that the analytical model of yoke portion and teeth portion and FEM calculation value are coincide well.

Calculate the vibration characteristics of motor taking 2.1kW radial flux non-crystaline amorphous metal magneto as a example.Stator core yoke portion footpath As shown in Figure 10 with radical length change curve to vibration displacement, strain and stress.The portion's tangential strain of stator core yoke and stress As shown in figure 11 with radical length change curve.Stator core yoke portion axial vibratory displacement, strain and stress become with axial length Change curve as shown in figure 12.Stator core teeth portion radial vibration displacement, strain and stress are with radical length change curve such as Figure 13 Shown.Stator core teeth portion axial vibratory displacement, strain and stress are as shown in figure 14 with axial length change curve.

Claims (6)

1. a kind of improved magnetostriction causes electric machine stator iron vibration analysis model it is characterised in that：According to radial direction magnetic The structure of three-way motor and characteristics of magnetic field distribution, stator core are divided into yoke portion and two subdomains of teeth portion, yoke portion vibrates for annulus The vibration of body, teeth portion vibrates the vibration for rectangular block；Set up vibration analysis model to comprise the following steps that：

1) cause electric machine stator iron vibration principle from magnetostriction, stator core yoke portion is set up respectively based on pressure magnetic equation Solve the fundamental equation in domain with teeth portion；

2) according to Newton's second law, set up stator core yoke portion respectively and teeth portion solves domain vibration equation；

3) adopt the separation of variable, introduce Bessel function, vibrate differential side using linear partial differential equation Superposition Principle Journey, obtains the general solution of vibration equation；

4) determine the boundary condition of stator core yoke portion and teeth portion；

5) the Boundary Condition for Solving undetermined coefficient according to stator core yoke portion and teeth portion, obtains stator core yoke portion and teeth portion solves Domain meets the solution of boundary condition；

6) vibration displacement based on stator core yoke portion and teeth portion, be derived by further yoke portion and teeth portion vibration velocity, plus Speed, strain and stress.

2. the electric machine stator iron vibration analysis model that improved magnetostriction causes as claimed in claim 1, its feature exists In：Step 1) in, the electric machine stator iron vibration principle that causes from magnetostriction, based on pressure magnetic establishing equation motor stator Rear of core and teeth portion fundamental equation are as follows：

Electric machine stator iron yoke portion fundamental equation is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{yoke\_r}=s_{11}^H{\mathrm{\σ}}_{yoke\_r}+s_{12}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{21}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_\mathrm{\θ}}=s_{12}^H{\mathrm{\σ}}_{yoke\_r}+s_{22}^H{\mathrm{\σ}}_{yoke\_\mathrm{\θ}}+d_{22}{H}_{yoke\_\mathrm{\θ}}\\ {\mathrm{\ϵ}}_{yoke\_z}=s_{33}^H{\mathrm{\σ}}_{yoke\_z}+d_{23}{H}_{yoke\_\mathrm{\θ}}\end{array}\right.;$

Electric machine stator iron teeth portion fundamental equation is：

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{teeth\_x}=\frac{{\mathrm{\σ}}_{teeth\_x}}{E}+d_{11}{H}_{teeth\_x}\\ {\mathrm{\ϵ}}_{teeth\_z}=\frac{{\mathrm{\σ}}_{teeth\_z}}{E}+d_{13}{H}_{teeth\_x}\end{array}\right..$

3. the electric machine stator iron vibration analysis model that improved magnetostriction causes as claimed in claim 1, its feature exists In：Step 2) in, stator core yoke portion is set up according to Newton's second law and the vibration equation of teeth portion is as follows：

Stator core yoke portion torus oscillatory differential equation is：

$\left\{\begin{array}{c}\frac{{\∂}^2{u}_{yoke\_r}}{\∂t^2}=c_1^2\[\frac{{\∂}^2{u}_{yoke\_r}}{\∂r^2}+\frac{1}{r}\frac{\∂u_{yoke\_r}}{\∂r}-\frac{u_{yoke\_r}}{r^2}\]\\ \frac{{\∂}^2{u}_{yoke\_z}}{\∂t^2}=c_2^2\frac{{\∂}^2{u}_{yoke\_z}}{\∂z^2}\end{array}\right.;$

The oscillatory differential equation of stator core teeth portion is：

$\left\{\begin{array}{c}\frac{{\∂}^2{u}_{teeth\_x}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_x}}{\∂x^2}\\ \frac{{\∂}^2{u}_{teeth\_z}}{\∂t^2}=\frac{E}{\mathrm{\ρ}}\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}=c_3^2\frac{{\∂}^2{u}_{teeth\_z}}{\∂z^2}\end{array}\right..$

4. the electric machine stator iron vibration analysis model that improved magnetostriction causes as claimed in claim 1, its feature exists In：Step 3) in, using the separation of variable, introduce Bessel function, utilize linear partial differential equation principle of stacking micro- to vibrating Divide equation to be solved, obtain stator core yoke portion and the general solution of teeth portion is as follows：

The general solution in stator core yoke portion：

$\left\{\begin{array}{c}{u}_{yoke\_r}=\left({\mathrm{MJ}}_1\right(k_{1}r)+{\mathrm{NY}}_1(k_{1}r\left)\right)e^{j\mathrm{\ω}t}\\ u_{yoke\_z}=(A\cos k_{2}z+B\sin k_{2}z)e^{j\mathrm{\ω}t}\end{array}\right.;$

The general solution of stator core teeth portion：

$\left\{\begin{array}{c}{u}_{teeth\_x}=\left({\mathrm{Ccosk}}_{3}x+{\mathrm{Dsink}}_{3}x\right)e^{j\mathrm{\ω}t}\\ u_{teeth\_z}=\left({\mathrm{Fcosk}}_{3}z+{\mathrm{Gsink}}_{3}z\right)e^{j\mathrm{\ω}t}\end{array}\right..$

5. the electric machine stator iron vibration analysis model that improved magnetostriction causes as claimed in claim 1, its feature exists In：Step 4) in, the boundary condition in stator core yoke portion is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{yo\mathrm{ke}\_r}{|}_{r=R_1}=0\\ {\mathrm{\σ}}_{yoke\_r}{|}_{r=R_{i1}}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{yoke\_z}{|}_{z=l_z}=0\end{array}\right.;$

The boundary condition of stator core teeth portion is：

$\left\{\begin{array}{c}{\mathrm{\σ}}_{teeth\_x}{|}_{x=0}=0\\ u_{teeth\_x}{|}_{x=h_t}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=0}=0\\ {\mathrm{\σ}}_{teeth\_z}{|}_{z=l_z}=0\end{array}\right..$

6. the electric machine stator iron vibration analysis model that improved magnetostriction causes as claimed in claim 1, its feature exists In：When calculating stator core yoke portion and teeth portion vibration, the impact that teeth portion and winding vibrate to yoke portion is set to additional mass, yoke portion Impact to teeth portion vibration is the initial displacement that yoke portion vibration displacement is set to teeth portion vibration.