CN101662248B - Special three-direction self-synchronizing vibrating screen and determining method of structural parameters thereof - Google Patents

Abstract

The invention relates to a special three-direction self-synchronizing vibrating screen and a determining method of structural parameters thereof, belonging to the engineering technical field of vibrating utilization. The invention provides a special three-direction self-synchronizing vibrating screen which can realize synchronization of two eccentric runners of a dual motor drive vibrating screen, and a determining method of structural parameters thereof; the special three-direction self-synchronizing vibrating screen comprises a supporting bracket, one end of a spring is fixed on the supporting bracket, the other end of the spring is fixed on a screen body which is provided with a screen cloth internally; two vibrating motors for driving the two eccentric runners are respectively arranged on the screen body, rotating centers of two vibrating motors are symmetrical by taking a mass center of the screen body as the center, and the rotating planes of the two eccentric runners are parallel. The determining method of structural parameters of the special three-direction self-synchronizing vibrating screen comprises the following steps: 1): building a mathematical model of a system; 2) quasi-stable state electro-magnetic torque of the asynchronous motor is determined; 3) the capture condition of the frequency of dual rotors and the synchronous stability condition are determined.

Description

Definite method of space three-direction self-synchronizing vibrating screen and structural parameters thereof

Technical field:

The invention belongs to vibration and utilize field of engineering technology, definite method of particularly a kind of space three-direction self-synchronizing vibrating screen and structural parameters thereof.

Background technology:

Find the earliest mechanical system vibration synchronia or motor synchronizing phenomenon be Huygnens (1629-1695).He once did such test, when two wall clocks hang on the thin plate that can swing simultaneously, and when satisfying certain condition, can observe two wall clocks and swing synchronously, and when being suspended to them on the wall, they can lose synchronously.From 1894 to nineteen twenty-two, many scientists, like Rayleigh, Vincent, Moler, Appletont, van der Pol has found synchronia in nonlinear circuit, and claims that this phenomenon is " frequency is captured ".The sixties in 20th century, doctor Blehman of the former Soviet Union has proposed the synchronous theory of two vibration generator bobbing machines.1980, it is synchronous that Japanese scholar Inoue and Araki etc. have studied 3 frequencys multiplication of plane vibration machine of Dual-motors Driving.1981, Chinese scholar was heard the Chinese toon academician of nation and is proposed, and in some non linear system, can realize that not only 3 frequencys multiplication are synchronous, and can realize that the frequency multiplication of each harmonic is synchronous that promptly 2 frequencys multiplication, 3 frequencys multiplication and n frequency multiplication are synchronous.In fact, a plurality of rotary bodies or a plurality of more general rotating machinery structure interrelate through the system, coupled dynamics of confirming, can realize a certain specific being synchronized with the movement.Therefore, characteristic, the coupling of studying coupling in this type systematic are the important contents of complication system science to the influence of system dynamics behavior.The theory of the dynamics of this type of mechanical mechanical-electric coupling has important significance for theories and actual application value to such mechanical system design and coupled synchronization design of Controller.

Since the sixties in last century; Chinese scholars has just been carried out number of research projects to the electromechanical coupling characteristics of motor synchronizing vibrating machine system; Obtained many critical achievements in research, as: plane motion bobbing machine motor synchronizing theory, spatial movement bobbing machine motor synchronizing theory, vibration generator gauche form bobbing machine motor synchronizing theory etc.But vibration is synchronously theoretical to be related to fewerly to motor characteristic, and the theoretical method of setting up is phase place dynamic method (Phase Dynamic Approach), is parameter with two eccentric rotor phase places only promptly.At first, this method has been ignored the characteristic that frequency is captured, if the mean angular velocity of two motors of supposition is constants, the phase difference of two vibration generators is then thought a parameter that can diminish.The differential equation that the different differential equation of motion of two vibration generators is merged into a phase difference only is fit to the synchronous vibration system that analysis has two Asynchronous Motor Driving of identical coupling vibration generator.And in the production real process, even the same a collection of motor of same model, its parameter difference is also unavoidable, even can not realize synchronous operation.Secondly, the dynamic characteristic shortcoming of asynchronous motor is considered.In fact, the self synchronous generation of vibrational system is the influence owing to the motor coupling, and the capture frequency of system also depends on the dynamic parameter of two asynchronous motors.At present; Existing Dual-motors Driving vibrating screen all adopts the Plane Installation mode; And the installation site adopts folk prescription to symmetrical mounting means, and two eccentric rotors are difficult to realize synchronously, cause the Dual-motors Driving circle shake sieve mostly adopt rigidity synchronously or unit drive; Increase the complexity of system, reduced system's traveling comfort.

Summary of the invention:

To the difficult synchronously problem of existing Dual-motors Driving vibrating screen two eccentric rotors, the present invention provides a kind of space three-direction self-synchronizing vibrating screen that Dual-motors Driving vibrating screen two eccentric rotors are synchronous and definite method of structural parameters thereof of realizing.

To achieve these goals, the present invention adopts following technical scheme, and a kind of space three-direction self-synchronizing vibrating screen comprises bracing frame, on bracing frame, is fixed with an end of spring, and the other end of spring is fixed on inside to have on the sieve nest of screen cloth; Sieve nest is provided with two and is respectively applied for the vibrating motor that drives two eccentric rotors, and the centre of gyration of two vibrating motors is about the barycenter symmetry of sieve nest, and the plane of rotation of two eccentric rotors is parallel to each other.

The angle δ of the plane of rotation of described eccentric rotor and horizontal plane is 0～45 °.

Definite method of described space three-direction self-synchronizing vibrating screen structural parameters comprises the steps:

Step 1: the Mathematical Modeling of setting up system;

Step 2: confirm asynchronous motor quasi-stable state electromagnetic torque;

Step 3: confirm birotor frequency capture conditions and synchronous stability condition;

1), sets up system frequency and capture equation;

2), confirm the system frequency capture conditions;

3), confirm the stability condition of system synchronization.

The Mathematical Modeling of the system described in the step 1 is:

${\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{yi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3-(m_1{l}_{z1}-m_2{l}_{z2})\stackrel{\&CenterDot;\&CenterDot;}{y}+(m_1{l}_{y1}-m_2{l}_{y2})\stackrel{\&CenterDot;\&CenterDot;}{z}+$

${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{yi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1-\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{yi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3+(m_1{l}_{z1}-m_2{l}_{z2})\stackrel{\&CenterDot;\&CenterDot;}{x}+(m_1{l}_{x1}-m_2{l}_{x2})\stackrel{\&CenterDot;\&CenterDot;}{z}+{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_2+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">k</figure-callout>}}_2{\mathrm{\&psi;}}_2=$

${\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1-\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{yi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2-(m_1{l}_{y1}-m_2{l}_{y2})\stackrel{\&CenterDot;\&CenterDot;}{x}-(m_1{l}_{x1}-m_2{l}_{x2})\stackrel{\&CenterDot;\&CenterDot;}{y}+{\mathrm{<figure-callout\; id="3"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">f</figure-callout>}}_3{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_3+{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">k</figure-callout>}}_3{\mathrm{\&psi;}}_3=$

${\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="J\; p"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; zi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{zi}}^{}+l_{\mathrm{yi}}^2),$ ${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2={\mathrm{<figure-callout\; id="2"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; xi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{xi}}^{}+l_{\mathrm{zi}}^2),$ ${\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3={\mathrm{<figure-callout\; id="3"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; xi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{xi}}^{}+l_{\mathrm{yi}}^2)$

$k_1=k_{\mathrm{<figure-callout\; id="2"\; label="y\; l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">y</figure-callout>}}{l}_z^{}{}_2^{}+k_{\mathrm{<figure-callout\; id="2"\; label="z\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">z</figure-callout>}}{l}_y^{}{}_2^{},$ $k_2=k_x{\mathrm{<figure-callout\; id="2"\; label="l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_z^{}+k_z{l}_x^2,$ $k_3=k_x{\mathrm{<figure-callout\; id="2"\; label="l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_y^{}+k_y{l}_x^2$

$f_1=f_{\mathrm{<figure-callout\; id="2"\; label="y\; l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">y</figure-callout>}}{l}_z^{}{}_2^{}+f_{\mathrm{<figure-callout\; id="2"\; label="z\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">z</figure-callout>}}{l}_y^{}{}_2^{},$ $f_2=f_x{\mathrm{<figure-callout\; id="2"\; label="l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_z^{}+f_z{l}_x^2,$ $f_3=f_x{\mathrm{<figure-callout\; id="2"\; label="l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_y^{}+f_y{l}_x^2.$

The Mathematical Modeling of the asynchronous motor quasi-stable state electromagnetic torque described in the step 2 is:

T _e＝T _e0-k _e0ε

${\mathrm{<figure-callout\; id="0"\; label="k\; e"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">k</figure-callout>}}_e=\frac{n_p^2{L}_m^2{\mathrm{<figure-callout\; id="0"\; label="U\; S"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">U</figure-callout>}}_S^0}{{L^2}_s{R}_r{\mathrm{\&omega;}}_s}\frac{1-{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-n_p{\mathrm{\&omega;}}_{m0})}^2}{{[1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{m0})}^2]}^2}\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}{{\mathrm{\&omega;}}_s},$ ${\mathrm{<figure-callout\; id="0"\; label="T\; e"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">T</figure-callout>}}_e=\frac{3n_p{L^2}_m{U}_{s0}^2}{2{L^2}_s{R}_r}\frac{{\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">r</figure-callout>}0}}{1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{r0})}^2}.$

System frequency capture conditions described in the step 3 is:

$m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_c\&GreaterEqual;(T_{e01}-T_{e02})-(f_1-f_2){\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}-\mathrm{\&eta;}{W}_{s2})$

$W_c=-r_m[\frac{\cos {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\sin }^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{\cos {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;}-r_{y1}\sin \mathrm{\&delta;})}^2-$

$\frac{\cos {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}({\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2+r_{x2}^2{\sin }^2\mathrm{\&delta;})-\frac{\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}({\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3+r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;})]$

μ _x＝1-(ω _nx/ω _m0) ²，μ _y＝1-(ω _ny/ω _m0) ²，μ _z＝1-(ω _nz/ω _m0) ²；

μ ₁＝1-(ω ₁/ω _m0) ²，μ ₂＝1-(ω ₂/ω _m0) ²，μ ₃＝1-(ω ₃/ω _m0) ²；

${\mathrm{<figure-callout\; id="1"\; label="W\; s"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_s=r_m[\frac{\sin {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}\sin {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\sin }^2\mathrm{\&delta;}\sin {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{\sin {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y1}\sin \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;})}^2+$

$\frac{\sin {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x2}^2{\sin }^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2)+\frac{\sin {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3)]$

W _s2＝ηW _s1。

The stability condition of the system synchronization described in the step 3 is:

ρ ₁＞0，ρ ₂＞0， ${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{\&rho;}}_{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">c</figure-callout>}}^2>0;$

${\mathrm{\&rho;}}_1=1-\frac{{\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2},$ ${\mathrm{\&rho;}}_2=\mathrm{\&eta;}(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2}),$ ${\mathrm{\&rho;}}_c=\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}^2{\cos }^2\left(2{\mathrm{\&alpha;}}_0\right)+{{\mathrm{<figure-callout\; id="2"\; label="W\; s"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_s}^2{\sin }^2\left(2{\mathrm{\&alpha;}}_0\right)};$

$W_s=\mathrm{\&eta;}{r}_m[\frac{\sin {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{\cos {\mathrm{\&delta;}}^2\sin {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}+\frac{\sin {\mathrm{\&delta;}}^2\sin {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{\sin {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}({{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z}^2\cos {\mathrm{\&delta;}}^2+{r_{y1}}^2\sin {\mathrm{\&delta;}}^2-r_{y1}{r}_{z1}\sin \mathrm{\&delta;}\cos \mathrm{\&delta;}$

$-r_{y1}{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;}\sin \mathrm{\&delta;})-\frac{\sin {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{z2}{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z+{r_{x2}}^2\sin {\mathrm{\&delta;}}^2)-\frac{\sin {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}({{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y}^2+{r_{x3}}^2\cos {\mathrm{\&delta;}}^2)]$

${\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c=r_m[\frac{\cos {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\sin }^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{\cos {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y1}\sin \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;})}^2+$

$\frac{\cos {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x2}^2{\sin }^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2)+\frac{\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3)]$

W _c2＝ηW _c1，W _ccos(2α ₀)＞0，W _c＞0，2α ₀∈(-π/2，π/2)。

Parameter declaration:

m ₁-eccentric block 1 quality, m ₂-eccentric block 2 quality, m-bobbing machine physique amount;

m _i-eccentric block i quality, i=1,2; The mass of vibration that M-is total, M=m+m ₁+ m ₂The r-eccentric arm;

J ₁, J ₂, J ₃-vibrate body respectively about x, y, the moment of inertia of z axle; J ₀₁, J ₀₂The moment of inertia of-axle 1 and axle 2;

k ₁, k ₂, k ₃-ψ ₁, ψ ₂And ψ ₃The spring rate of three directions; f _{ψ 1}, f _{ψ 2}, F _{ψ 3}-ψ ₁, ψ ₂And ψ ₃The damping coefficient of three directions;

T _E01, T _E02The stable state electromagnetic torque of- motor 1,2; k _x, k _y, k _z-x, y, the spring rate of three directions of z;

f _x, f _y, f _z-x, y, the resistance coefficient of three directions of z; f _D1, f _D2Coefficient of friction on-axle 1 and the axle 2;

δ ₁The angle on the Plane of rotation of-vibration generator 1 and oxy plane, δ ₂The angle on the Plane of rotation of-vibration generator 2 and oxy plane;

The phase place of

-eccentric rotor 1, the phase place of -eccentric rotor 2;

l _X1The center of-motor 1 is to the distance of z axle, l _X2The center of-motor 2 is to the distance of z axle, l _Y1The center of-motor 1 is to the distance of x axle, l _Y2The center of-motor 2 is to the distance of x axle, l _Z1The center of-motor 1 is to the distance of y axle, l _Z2The center of-motor 2 is to the distance of y axle, l _xThe radius of turn of-body on the x direction, l _yThe radius of turn of-body on the y direction, l _zThe radius of turn of-body on the z direction;

ψ ₁-body is around the corner of axle, ψ ₂-motor 1 is around the corner of axle, ψ ₃-motor 2 is around the corner of axle;

J _P1, J _P2, J _P3-vibration body is respectively about x ", y ", the z " moment of inertia of axle;

K _iThe stiffness matrix of-spring i, K _i=diag (k _x/ 8, k _y/ 8, k _z/ 8);

F _iThe damping matrix of-spring i, F _i=diag (f _x/ 8, f _y/ 8, f _z/ 8);

x _KiThe elongation of-spring, x _Ki0The initial elongation amount of-spring; Q _iThe generalized force of-vibrating screen, q _iThe generalized coordinates of-vibrating screen; m ₀₁The quality of-motor 1, m ₀₂The quality of-motor 2;

T _e-motor quasi-stable state electromagnetic torque; σ-asynchronous motor leakage inductance coefficient, $\mathrm{\&sigma;}=1-{\mathrm{<figure-callout\; id="2"\; label="L\; m"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">L</figure-callout>}}_m^{}/L_s{L}_s;$

T _E0-rotor electrical angle speed is ω _R0The time electromagnetic torque; k _E0The stiffness coefficient of-steady state point electrical angle speed;

ε-rotating speed is at ω _R0Near the minor fluctuations coefficient that becomes slowly that produces; τ _r-rotor time constant, τ _r=L _r/ R _r

R _r-rotor equivalent resistance; L _s, L _r, L _m-stator inductance, rotor equivalent inductance, the mutual inductance between stator and the rotor;

n _p-number of pole-pairs; ω _s-mains supply frequency; U _S0-terminal voltage; ω _R0The electrical angle speed of-steady state point asynchronous motor rotor;

ω _M0-system reaches the rotating speed of synchronous operation state; W _c-two vibration generator phase angles coupling cosine function coefficient;

f ₁, f ₂The moment of resistance coefficient of- axle 1,2; W _S1, W _S2The sinusoidal function coefficient at- vibration generator 1 and 2 phase angle;

η=m ₂/ m ₁The mass ratio of-eccentric block 2 and eccentric block 1; r _m=m ₁/ M-eccentric block and body mass ratio;

ω _NxNatural frequency on the x of-system direction, ω _NyNatural frequency on the y of-system direction, ω _NzNatural frequency on the z of-system direction, ω ₁, ω ₂, ω ₃The natural frequency of-system;

γ _x, γ _y, γ _z, r ₁, γ ₂, γ ₃-x, y, z, ψ ₁, ψ ₂And ψ ₃The angle of retard of direction; The angle on the Plane of rotation of δ-vibration generator and oxy plane; α ₀Phase difference during-synchronism stability;

W _C1, W _C2- vibration generator 1 and 2 phase place cosine of an angle function coefficient;

W _sThe sinusoidal function coefficient of the coupling at-two vibration generator phase angles;

r _Xi=l _x/ l _Ei, r _Yi=l _y/ l _Ei, r _Zi=l _z/ l _Ei, i=1,2,3-X, Y, the phase equivalent radius of turn of Z direction;

$l_{e1}=\sqrt{{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1/M},$ $l_{e2}=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2/M},$ $l_{e3}=\sqrt{{\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3/M}$ -X, Y, Z direction equivalence radius of turn;

T _E1The electromagnetic torque of-motor 1, T _E2The electromagnetic torque of-motor 2, T _L1The moment of resistance of-motor 1, T _L2The moment of resistance of-motor 2;

The phase place of-body, the phase difference of α-rotor; ω _nThe natural frequency of-system;

k _E01The stiffness coefficient of the steady state point electrical angle speed of-motor 1, k _E02The stiffness coefficient of the steady state point electrical angle speed of-motor 2.

Beneficial effect of the present invention:

(1) because the sieve nest of vibrating screen of the present invention is provided with two vibrating motors, be built in the barycenter of the line of centres of the eccentric rotor in two vibrating motors respectively, and the plane of rotation of two eccentric rotors is parallel to each other through sieve nest; Just make vibrating screen of the present invention can realize synchronous vibration;

(2) introduce W in the Structure of Vibrating Screen determination method for parameter of the present invention _S1, W _S2, W _C1, W _C2, W _sAnd W _c6 dimensionless groups characterize the Coupled Dynamics characteristic of two vibration generators, derive dynamic symmetry dimensionless factor

Thereby make the present invention can be to structure the asymmetric and inconsistent system of two motor parameters keep dynamic symmetric ability to analyze;

(3) introduce dimensionless coupled rotation inertia in the Structure of Vibrating Screen determination method for parameter of the present invention $H_{}^{}$ ${}_0^{\&prime;}=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{\&rho;}}_c^{\&prime;2},$ Make Structure of Vibrating Screen determination method for parameter of the present invention more directly perceived to the analysis of the vibrating screen stability of synchronization.

Description of drawings:

Fig. 1 is the structural representation of vibrating screen of the present invention;

Fig. 2 is the vertical view of Fig. 1;

Fig. 3 is the left view of Fig. 1;

Fig. 4 is the structural representation of vibrating motor of the present invention;

Fig. 5 is the program flow diagram of Structure of Vibrating Screen determination method for parameter of the present invention;

Fig. 6 (a) and (b), (c) are the mechanical model figure of vibrating screen of the present invention;

Fig. 7 is the sketch map of three different reference frames of three phase squirrel cage asynchronous machine;

Fig. 8 (a) and (b), (c), (d) are that the vibrational system of different parameters is at η-r _X1Synchronous scope is realized on the plane;

Fig. 9 (a) and (b), (c), (d) are each state variation curve charts of two vibrating motors of the present invention symmetry dynamic symmetry coefficient when installing;

Figure 10 is a motor synchronizing stability boundary sketch map, wherein,

(a) be to work as H ₀=0 (H ₀'=0), H ₁=0 (H ₁'=0), during H=0 (H '=0), at r _m-r _L31R in the plane _L31maxCurve chart;

(b) be to work as H ₀=0 o'clock,

And r _mConcern sketch map;

Figure 11 is the Computer Simulation figure of vibrating screen of the present invention, wherein,

(a) be the Computer Simulation figure of two motor speeds;

(b) be the Computer Simulation figure of two vibration generator phase differences;

(f) be ψ ₁Deflection calculation of displacement machine analogous diagram;

(h) be ψ ₂Deflection calculation of displacement machine analogous diagram;

(i) be ψ ₃Deflection calculation of displacement machine analogous diagram;

Figure 12 is another Computer Simulation figure of vibrating screen of the present invention, wherein,

(a) be the Computer Simulation figure of two motor speeds;

(b) be the Computer Simulation figure of two vibration generator phase differences;

Wherein, among Fig. 1-Fig. 4,1-vibrating motor, 2-screen cloth, 3-sieve nest, 4-spring, 5-bracing frame, 6-eccentric rotor.

Embodiment:

Like Fig. 1, Fig. 2, Fig. 3, shown in Figure 4, a kind of space three-direction self-synchronizing vibrating screen comprises bracing frame 5, on bracing frame 5, is fixed with an end of spring 4, and the other end of spring 4 is fixed on inside to have on the sieve nest 3 of screen cloth 2; Sieve nest 3 is provided with two barycenter symmetries that are respectively applied for the centre of gyration of vibrating motor 1, two vibrating motor 1 that drives two eccentric rotors 6 about sieve nest 3, and the plane of rotation of two eccentric rotors 6 is parallel to each other.

The angle δ of the plane of rotation of described eccentric rotor 6 and horizontal plane is 0～45 °; Can realize the elliptic motion of body in horizontal plane and the sinusoidal motion of vertical direction; When δ=0, body movement is the elliptic motion in the horizontal plane.

As shown in Figure 5, definite method of described space three-direction self-synchronizing vibrating screen structural parameters comprises the steps:

Step 1: the Mathematical Modeling of setting up system:

The mechanical model of vibrating screen is as shown in Figure 6, and shown in Fig. 6 (a), this model comprises that a rigidity body and two are respectively by the eccentric block of Induction Motor Drive.The rigidity body is supported by upper and lower two elastic foundations, and they comprise the spring of the vibration isolation that four symmetries are installed.Shown in Fig. 6 (b), o ₁, o ₂Be respectively the pivot of vibration generator 1 and 2.Line o ₁o ₁' and o ₂o ₂' all being parallel to the x axle, the Plane of rotation of vibration generator 1 is crossed line o ₁o ₁', and with the angle on oxy plane be δ ₁The Plane of rotation of vibration generator 2 is crossed line o ₂o ₂', and with the angle on oxy plane be δ ₂Diaxon is done revolution in the same way.Shown in Fig. 6 (c), the static barycenter G of frame is projected as the o point the z axle, and oxyz is a static coordinate system; This static coordinate is as the z axle with the frame central line; Dynamic coordinate system Gx " y " z " be that former coordinate system shifts and gets, and with former coordinate system keeping parallelism, dynamic coordinate system is fixed on the frame.

The kinetic energy T of vibrational system can represent as follows:

$T=\frac{1}{2}m({\stackrel{\&CenterDot;}{x}}^2+{\stackrel{\&CenterDot;}{y}}^2+{\stackrel{\&CenterDot;}{z}}^2)+\frac{1}{2}({\mathrm{<figure-callout\; id="1"\; label="J\; P"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_P{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_1^2+{\mathrm{<figure-callout\; id="2"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_p{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_2^2+{\mathrm{<figure-callout\; id="3"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_p{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_3^2)+\frac{1}{2}{m}_1{\stackrel{\&CenterDot;}{x}}_1^T{\stackrel{\&CenterDot;}{x}}_1+\frac{1}{2}{m}_2{\stackrel{\&CenterDot;}{x}}_2^T{\stackrel{\&CenterDot;}{x}}_2---\left(1\right)$

The potential energy V of system is:

$V=\frac{1}{2}\underset{i=1}{\overset{8}{\mathrm{\&Sigma;}}}{(x_{\mathrm{ki}}-x_{\mathrm{ki}0})}^T{K}_i(x_{\mathrm{ki}}-x_{\mathrm{ki}0})---\left(2\right)$

The dissipation function D of system is:

$D=\frac{1}{2}\underset{i=1}{\overset{8}{\mathrm{\&Sigma;}}}{\stackrel{\&CenterDot;}{x}}_{\mathrm{ki}}^T{F}_i{\stackrel{\&CenterDot;}{x}}_{\mathrm{ki}}---\left(3\right)$

Application of Lagrange's Equations is set up the equation of motion:

$\frac{d}{\mathrm{dt}}\frac{\&PartialD;(T-V)}{\&PartialD;{\stackrel{\&CenterDot;}{q}}_i}-\frac{\&PartialD;(T-V)}{\&PartialD;q_i}+\frac{\&PartialD;D}{\&PartialD;{\stackrel{\&CenterDot;}{q}}_i}=Q_i---\left(4\right)$

Then the system motion differential equation can be simplified as follows:

${\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{yi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3-(m_1{l}_{z1}-m_2{l}_{z2})\stackrel{\&CenterDot;\&CenterDot;}{y}+(m_1{l}_{y1}-m_2{l}_{y2})\stackrel{\&CenterDot;\&CenterDot;}{z}+---\left(5\right)$

${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{yi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1-\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{yi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3+(m_1{l}_{z1}-m_2{l}_{z2})\stackrel{\&CenterDot;\&CenterDot;}{x}+(m_1{l}_{x1}-m_2{l}_{x2})\stackrel{\&CenterDot;\&CenterDot;}{z}+{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_2+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">k</figure-callout>}}_2{\mathrm{\&psi;}}_2=$

${\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_3+\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{xi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_1-\left(\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i{l}_{\mathrm{yi}}{l}_{\mathrm{zi}}\right){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&psi;}}}_2-(m_1{l}_{y1}-m_2{l}_{y2})\stackrel{\&CenterDot;\&CenterDot;}{x}-(m_1{l}_{x1}-m_2{l}_{x2})\stackrel{\&CenterDot;\&CenterDot;}{y}+{\mathrm{<figure-callout\; id="3"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">f</figure-callout>}}_3{\stackrel{\&CenterDot;}{\mathrm{\&psi;}}}_3+{\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">k</figure-callout>}}_3{\mathrm{\&psi;}}_3=$

${\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="J\; p"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; zi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{zi}}^{}+l_{\mathrm{yi}}^2),$ ${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2={\mathrm{<figure-callout\; id="2"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; xi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{xi}}^{}+l_{\mathrm{zi}}^2),$ ${\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3={\mathrm{<figure-callout\; id="3"\; label="J\; p"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_p+\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{m}_i({\mathrm{<figure-callout\; id="2"\; label="l\; xi"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{xi}}^{}+l_{\mathrm{yi}}^2);$

${\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=k_{\mathrm{<figure-callout\; id="2"\; label="y\; l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">y</figure-callout>}}{l}_z^{}{}_2^{}+k_{\mathrm{<figure-callout\; id="2"\; label="z\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">z</figure-callout>}}{l}_y^{}{}_2^{},$ ${\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=k_{\mathrm{<figure-callout\; id="2"\; label="x\; l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">x</figure-callout>}}{l}_z^{}{}_2^{}+k_z{l}_x^2,$ ${\mathrm{<figure-callout\; id="3"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">k</figure-callout>}}_3=k_{\mathrm{<figure-callout\; id="2"\; label="x\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">x</figure-callout>}}{l}_y^{}{}_2^{}+k_y{l}_x^2;$

${\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_1=f_y{l}_{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">z</figure-callout>}}^2+f_{\mathrm{<figure-callout\; id="2"\; label="z\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">z</figure-callout>}}{l}_y^{}{}_2^{},$ ${\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2=f_{\mathrm{<figure-callout\; id="2"\; label="x\; l\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">x</figure-callout>}}{l}_z^{}{}_2^{}+f_z{l}_x^2,$ ${\mathrm{<figure-callout\; id="3"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">f</figure-callout>}}_3=f_{\mathrm{<figure-callout\; id="2"\; label="x\; l\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">x</figure-callout>}}{l}_y^{}{}_2^{}+f_y{l}_x^2.$

Step 2: confirm asynchronous motor quasi-stable state electromagnetic torque:

The sketch map of three different reference frames of three phase squirrel cage asynchronous machine is as shown in Figure 7: the stator reference system (ar, br), the rotor synchronous coordinate system (d, q) and any reference system (α, β).With asynchronous motor at rotor synchronous coordinate system (d; The q axle of the state equation q) is taken on the direction of stator voltage

, derives the Mathematical Modeling of motor quasi-stable state electromagnetic torque:

T _e＝T _e0-k _e0ε (6)

${\mathrm{<figure-callout\; id="0"\; label="k\; e"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">k</figure-callout>}}_e=\frac{n_p^2{L}_m^2{\mathrm{<figure-callout\; id="0"\; label="U\; S"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">U</figure-callout>}}_S^0}{{L^2}_s{R}_r{\mathrm{\&omega;}}_s}\frac{1-{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-n_p{\mathrm{\&omega;}}_{m0})}^2}{{[1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{m0})}^2]}^2}\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}{{\mathrm{\&omega;}}_s},$ ${\mathrm{<figure-callout\; id="0"\; label="T\; e"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">T</figure-callout>}}_e=\frac{3n_p{L^2}_m{U}_{s0}^2}{2{L^2}_s{R}_r}\frac{{\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">r</figure-callout>}0}}{1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{r0})}^2}.$

Step 3: confirm birotor frequency capture conditions and synchronous stability condition:

1, set up system frequency and capture equation:

This moment, eccentric rotor 1 was ahead of

is α if the average phase of two eccentric rotors is established for

; Eccentric rotor 2 lags behind is α, and then the phase place of two eccentric rotors and speed are expressed as respectively:

If establishing the instantaneous mean speed of two eccentric rotors is ω _M0, coefficient of variation is ε ₁, two eccentric rotors are ε to the transient fluctuation coefficient of average phase ₂

$\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}={\mathrm{\&epsiv;}}_2{\mathrm{\&omega;}}_{m0}---\left(8\right)$

Then the instantaneous angular velocity of two eccentric rotors and angular acceleration are expressed as:

If when t → ∞, system is at T=2 π/ω _M0The average of rotating speed and phase fluctuation coefficient is 0 in cycle, that is:

${\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1=0$

${\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2=0---\left(10\right)$

Because the asynchronous motor working speed is a little less than synchronous speed, so systematic steady state has δ＜＜1, ε＜＜1 when moving.Therefore, can ignore

And supposition m ₁=m ₀, m ₂=η m ₀(0＜η≤1), introduce following dimensionless group:

${\mathrm{\&omega;}}_{\mathrm{nx}}=\sqrt{\frac{k_x}{M}},$ ${\mathrm{\&xi;}}_x=\frac{f_x}{2\sqrt{m{k}_x}},$ ${\mathrm{\&omega;}}_{\mathrm{ny}}=\sqrt{\frac{k_y}{M}},$ ${\mathrm{\&xi;}}_y=\frac{f_y}{2\sqrt{m{k}_y}},$ ${\mathrm{\&omega;}}_{\mathrm{nz}}=\sqrt{\frac{k_z}{M}},$ ${\mathrm{\&xi;}}_z=\frac{f_z}{2\sqrt{m{k}_z}},$

${\mathrm{\&omega;}}_1=\sqrt{\frac{k_1}{{\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">J</figure-callout>}}_1}},$ ${\mathrm{\&xi;}}_1=\frac{{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{2\sqrt{J_1{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}},$ ${\mathrm{\&omega;}}_2=\sqrt{\frac{k_2}{{\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">J</figure-callout>}}_2}},$ ${\mathrm{\&xi;}}_2=\frac{{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2}{2\sqrt{J_2{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}},$ ${\mathrm{\&omega;}}_3=\sqrt{\frac{k_3}{{\mathrm{<figure-callout\; id="3"\; label="J"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">J</figure-callout>}}_3}},$ ${\mathrm{\&xi;}}_3=\frac{{\mathrm{<figure-callout\; id="3"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">f</figure-callout>}}_3}{2\sqrt{J_3{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}},$ $r_m=\frac{m_0}{M},$

$l_{e1}=\sqrt{\frac{J_1}{M}},$ $l_{e2}=\sqrt{\frac{J_2}{M}},$ $l_{e3}=\sqrt{\frac{J_3}{M}},$ $r_{\mathrm{lxji}}=\frac{l_{\mathrm{xi}}}{l_{\mathrm{ej}}},$ $r_{\mathrm{lyji}}=\frac{l_{\mathrm{yi}}}{l_{\mathrm{ej}}},$ $r_{\mathrm{lzji}}=\frac{l_{\mathrm{zi}}}{l_{\mathrm{ej}}},$

i＝1，2，j＝1，2，3

Then the differential equation of system can be write as following form:

To off-resonance vibrational system, i.e. ω _M0＞(4～5) ω _n, and the damping ratio of vibrational system less (ξ＜0.07) is ignored damping to magnitude determinations, can get vibration generator 1 and in the response of x direction be:

$x_{01}=\frac{r{r}_m\cos {\mathrm{\&delta;}}_1}{1-{({\mathrm{\&omega;}}_{\mathrm{nx}}/{\mathrm{\&omega;}}_{m0}(1+{\mathrm{\&epsiv;}}_1+{\mathrm{\&epsiv;}}_2))}^2}----\left(12\right)$

Vibrational system is ω in rotor frequency _M0(1+ ε ₁+ ε ₂) the available rotor velocity of response when driving is ω _M0The time Taylor launch expression, ignore high-order term, can get:

$x_{01}=\frac{r{r}_m\cos \mathrm{\&delta;}}{1-{({\mathrm{\&omega;}}_{\mathrm{nx}}/{\mathrm{\&omega;}}_{m0})}^2}[1-\frac{2{({\mathrm{\&omega;}}_{\mathrm{nx}}/{\mathrm{\&omega;}}_{m0})}^2}{1-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="nx"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">nx</figure-callout>}}^2/{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2}({\mathrm{\&epsiv;}}_1+{\mathrm{\&epsiv;}}_2)\left]\right)---\left(13\right)$

Generally, asynchronous motor operate as normal slippage is usually less than 2% to 8%, therefore

|ε ₁+ε ₂|＜0.1 (14)

The fluctuation of vibrational system mechanical angle speed is ignored to the influence of response, and system responses is following:

In the formula, μ _x=1-(ω _Nx/ ω _M0) ², μ _y=1-(ω _Ny/ ω _M0) ², μ _z=1-(ω _Nz/ ω _M0) ², μ ₁=1-(ω ₁/ ω _M0) ², μ ₂=1-(ω ₂/ ω _M0) ², μ ₃=1-(ω ₃/ ω _M0) ²π-γ _x, π-γ _y, π-γ _z, π-γ ₁, π-γ ₂And π-γ ₃Represent x respectively, y, z, ψ ₁, ψ ₂And ψ ₃The phase angle of direction.

With instantaneous mean speed ω _M0Coefficient of variation ε ₁Transient fluctuation coefficient ε with average phase ₂For variable is set up system's average differential equation.

Ask a total differential to get to the time first formula of formula (16):

Formula (17) is asked total differential one time to the time again, omit ε ₁, ε ₂The second order high-order term:

In like manner, obtain

With

(i=1,2,3), and ask monocycle mean value, omit ε ₁, ε ₂High-order term:

$(J_{01}+m_1{r}^2){\mathrm{\&omega;}}_{m0}({\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_1+{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_2)+{\mathrm{<figure-callout\; id="1"\; label="f\; d"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_d{\mathrm{\&omega;}}_{m0}(1+{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1+{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2)=T_{e1}-{\mathrm{<figure-callout\; id="1"\; label="T\; L"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">T</figure-callout>}}_L$

$(J_{02}+m_2{r}^2){\mathrm{\&omega;}}_{m0}({\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_1-{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_2)+{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2{\mathrm{\&omega;}}_{m0}(1+{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1-{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2)=T_{e2}-T_{L2}---\left(19\right)$

Because ε ₁＜＜1, ε ₂＜＜1, and with respect to ω _M0Be slow variable element, so can be in a swing circle with ε ₁, ε ₂,

Get its intermediate value separately with α

With

Obtaining the average differential equation is:

${\stackrel{\&OverBar;}{T}}_{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">L</figure-callout>}1}={\mathrm{\&chi;}}_{11}^{\&prime;}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_1+{\mathrm{\&chi;}}_{12}^{\&prime;}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_2+{\mathrm{\&chi;}}_{11}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1+{\mathrm{\&chi;}}_{12}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2+{\mathrm{\&chi;}}_a+{\mathrm{\&chi;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}1}$

${\stackrel{\&OverBar;}{T}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">L</figure-callout>}2}={\mathrm{\&chi;}}_{21}^{\&prime;}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_1+{\mathrm{\&chi;}}_{21}^{\&prime;}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}}_2+{\mathrm{\&chi;}}_{21}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1+{\mathrm{\&chi;}}_{22}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2-{\mathrm{\&chi;}}_a+{\mathrm{\&chi;}}_{f2}---\left(20\right)$

Wherein: ω _M0

${\mathrm{\&chi;}}_a=\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_c\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c)$

${\mathrm{\&chi;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}1}=\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2({\stackrel{\&CenterDot;}{W}}_{s1}+W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s))$

${\mathrm{\&chi;}}_{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}2}=\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2({\mathrm{\&eta;}W}_{s2}+W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s))$

${\mathrm{\&chi;}}_{11}^{\&prime;}=-\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}({\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c+W_s\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{12}^{\&prime;}=-\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}(W_{c1}-W_s\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)+W_c\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{21}^{\&prime;}=-\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}(\mathrm{\&eta;}{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c+W_s\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{22}^{\&prime;}=-\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}({-\mathrm{\&eta;W}}_{c2}-W_s\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{11}=m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}+W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)+W_c\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s))$

${\mathrm{\&chi;}}_{12}=m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}-W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{21}=m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(\mathrm{\&eta;}{W}_{s2}-W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

${\mathrm{\&chi;}}_{22}=m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(-\mathrm{\&eta;}{W}_{s2}-W_s\cos (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_s)-W_c\sin (2\mathrm{\&alpha;}+{\mathrm{\&theta;}}_c))$

$a_s={\mathrm{\&eta;r}}_m[\frac{{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}+\frac{{\sin \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2{\sin \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{\sin {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}(r_{z11}{r}_{z12}{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2+)$

$r_{y11}{r}_{y12}{\sin \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2-r_{y11}{r}_{z12}{\sin \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2-r_{y12}{r}_{z11}{\cos \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2)-\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{z21}{r}_{z22}+$

$r_{x21}{r}_{x22}{\sin \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2)-\frac{{\sin \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{y31}{r}_{y32}+r_{x32}{r}_{x31}{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2)]$

$b_s={\mathrm{\&eta;r}}_m[\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{z21}{r}_{x22}{\sin \mathrm{\&delta;}}_2-r_{z22}{r}_{x21}{\sin \mathrm{\&delta;}}_1)+\frac{\sin {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x32}{r}_{y31}{\cos \mathrm{\&delta;}}_2-r_{x31}{r}_{y32}{\cos \mathrm{\&delta;}}_1)]$

$a_c=-{\mathrm{\&eta;r}}_m[\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}+\frac{{\sin \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2{\cos \mathrm{\&delta;}}_z}{{\mathrm{\&mu;}}_z}-\frac{{\cos \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}(r_{z11}{r}_{z12}{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2+)$

$r_{y11}{r}_{y12}{\sin \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2-r_{z11}{r}_{y12}{\cos \mathrm{\&delta;}}_1{\sin \mathrm{\&delta;}}_2-r_{y11}{r}_{z12}{\sin \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2)-\frac{{\cos \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{z21}{r}_{z22}+)$

$r_{x21}{r}_{x22}{\sin \mathrm{\&delta;}}_1\sin {\mathrm{\&delta;}}_2)-\frac{{\cos \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{y31}{y}_{y32}+r_{x31}{r}_{x32}{\cos \mathrm{\&delta;}}_1{\cos \mathrm{\&delta;}}_2)]$

$b_c={\mathrm{\&eta;r}}_m[\frac{{\cos \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x21}{r}_{22}{\sin \mathrm{\&delta;}}_1-r_{x22}{r}_{z21}{\sin \mathrm{\&delta;}}_2)+\frac{{\cos \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{y31}{r}_{x32}{\cos \mathrm{\&delta;}}_2-r_{x31}{r}_{y32}{\cos \mathrm{\&delta;}}_1)]$

$W_s=\sqrt{a_{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="b\; s"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">b</figure-callout>}}_s^{}},$ ${\mathrm{\&theta;}}_s=\left\{\begin{array}{cc}\arctan (-b_s/a_s),& a_s\&GreaterEqual;0\\ \mathrm{\&pi;}-\arctan (-b_s/a_s),& a_s<0\end{array}\right.$

$W_c=\sqrt{a_{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">c</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="b\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">b</figure-callout>}}_c^{}},$ ${\mathrm{\&theta;}}_c=\left\{\begin{array}{cc}\arctan (b_c/a_c),& a_c\&GreaterEqual;0\\ \mathrm{\&pi;}+\arctan (b_c/a_c),& a_c<0\end{array}\right.$

$W_{s1}=r_m[\frac{{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_1\sin {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{\mathrm{si}{n}^2{\mathrm{\&delta;}}_1{\sin \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{{\sin \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y11}{\sin \mathrm{\&delta;}}_1-r_{z11}{\cos \mathrm{\&delta;}}_1)}^2+$

$\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x21}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}}_1+r_{z21}^2)+\frac{{\sin \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x31}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_1+r_{y31}^2)]$

$W_{s2}=\mathrm{\&eta;}{r}_m[\frac{{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_2\sin {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{\mathrm{si}{n}^2{\mathrm{\&delta;}}_2{\sin \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{{\sin \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y12}{\sin \mathrm{\&delta;}}_2-r_{z12}{\cos \mathrm{\&delta;}}_2)}^2+$

$\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x22}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}}_2+r_{z22}^2)+\frac{{\sin \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x32}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_2+r_{y32}^2)]$

${\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c=r_m[\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_1\cos {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{\mathrm{si}{n}^2{\mathrm{\&delta;}}_1{\cos \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{{\cos \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y11}{\sin \mathrm{\&delta;}}_1-r_{z11}{\cos \mathrm{\&delta;}}_1)}^2+$

$\frac{{\cos \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x21}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}}_1+r_{z21}^2)+\frac{{\cos \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x31}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_1+r_{y31}^2)]$

${\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c=\mathrm{\&eta;}{r}_m[\frac{\cos {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_2\cos {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}}_2\cos {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{\cos {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{(r_{y12}\sin {\mathrm{\&delta;}}_2-r_{z12}\cos {\mathrm{\&delta;}}_2)}^2+$

$\frac{\cos {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x22}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}}_2+r_{z22}^2)+\frac{\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x32}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}}_2+r_{y32}^2)]$

The Plane of rotation of vibration generator 1 and the angle of horizontal plane are δ ₁, the Plane of rotation of vibration generator 2 and the angle of horizontal plane are δ ₂δ under the parallel situation ₁=δ ₂, promptly be δ, but theory analysis starts with from the system configuration generality, so set 2 values, be convenient to explanation.

In the time of theory analysis, the structural parameters of setting are asymmetric installations, so setting is: $r_{\mathrm{Lxji}}=\frac{l_{\mathrm{Xi}}}{l_{\mathrm{Ej}}},$ $r_{\mathrm{Lyji}}=\frac{l_{\mathrm{Yi}}}{l_{\mathrm{Ej}}},$ $r_{\mathrm{Lzji}}=\frac{l_{\mathrm{Zi}}}{l_{\mathrm{Ej}}},$ I=1,2, j=1,2,3; And under the symmetric case: l _X1=l _X2=l _x, ly, lz are also similar.So r _Xi=l _x/ l _Ei, r _Yi=l _y/ l _Ei, r _Zi=l _z/ l _Ei, i=1,2,3-X, Y, the phase equivalent radius of turn of Z direction; The symmetrical, parallel words of installing: b then _s=0, b _c=0, θ _c=0, θ _s=0.

With the addition and subtracting each other respectively of two formulas of formula (20), and consider ${\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}\mathrm{\&epsiv;}=\stackrel{\&CenterDot;}{\mathrm{\&alpha;}},$ After the arrangement:

$A\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=\mathrm{B\&epsiv;}+u---\left(21\right)$

In the formula, $\mathrm{\&epsiv;}={\left\{\begin{array}{cc}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1& {\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2\end{array}\right\}}^T,$ U={u ₁u ₂} ^T,

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=\frac{T_{e01}}{m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}+\frac{T_{e02}}{m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}}-\frac{{\mathrm{<figure-callout\; id="1"\; label="f\; d"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_d+f_{d2}}{m_0{r}^2}-\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}{2}(W_{s1}+\mathrm{\&eta;}{W}_{s2})-{\mathrm{\&omega;}}_{m0}{W}_s\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s),$

${\mathrm{<figure-callout\; id="1"\; label="u"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">u</figure-callout>}}_1=\frac{T_{e01}}{m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}}-\frac{T_{e02}}{m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{m0}}-\frac{f_{d1}-f_{d2}}{m_0{r}^2}-\frac{{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}{2}(W_{s1}-\mathrm{\&eta;}{W}_{s2})-{\mathrm{\&omega;}}_{m0}{W}_c\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c),$

${\mathrm{\&rho;}}_1=1-\frac{{\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2},$ ${\mathrm{\&kappa;}}_1=\frac{k_{e01}}{m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2}+\frac{f_1}{m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}+W_{s1},$

${\mathrm{\&rho;}}_2=\mathrm{\&eta;}(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2}),$ ${\mathrm{\&kappa;}}_2=\frac{k_{e02}}{m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2}+\frac{f_2}{m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}}+\mathrm{\&eta;}{W}_{s2},$

$A=\left[\begin{array}{cc}{\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2+W_c\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c)& {\mathrm{\&rho;}}_1-{\mathrm{\&rho;}}_2+W_s\sin (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s)\\ {\mathrm{\&rho;}}_1-{\mathrm{\&rho;}}_2-W_s\sin (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s)& {\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2-W_c\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c)\end{array}\right],$

$B=-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}\left[\begin{array}{cc}{\mathrm{\&kappa;}}_1+{\mathrm{\&kappa;}}_2-2W_s\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s)& {\mathrm{\&kappa;}}_1-{\mathrm{\&kappa;}}_2-2W_c\sin (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c)\\ {\mathrm{\&kappa;}}_1-{\mathrm{\&kappa;}}_2+2W_c\sin (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c)& {\mathrm{\&kappa;}}_1+{\mathrm{\&kappa;}}_2+2W_s\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s)\end{array}\right].$

and two motor mean angular velocities are disturbance parameters; If formula (21) null solution exists and is stable; Then this system can realize that frequency captures, and this method converts the motor synchronizing problem of two-shipper vibrational system into the existence and the stability problem of the average differential equation null solution of perturbation parameter.Formula (21) is the frequency of system and captures equation.

2, confirm the system frequency capture conditions:

Formula (10) substitution formula (21) rearranged:

$(T_{e01}+T_{e02})-({\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_1+f_2){\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2[W_{s1}+\mathrm{\&eta;}{W}_{s2}+2W_s\cos (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_s)]=0---\left(22\right)$

$m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_c\sin (2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}+{\mathrm{\&theta;}}_c)=(T_{e01}-T_{e02})-(f_1-f_2){\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}-\mathrm{\&eta;}{W}_{s2})---\left(23\right)$

T _E01+ T _E02Be Motor Drive moment sum, second is that last term is the loading moment summation that acts on two motors because motor reel is fricative resistance torque.Formula (23) is called the quasi-stationary torque equilibrium equation of vibrational system.

The difference that frequency is captured moment, two motor residual electricity magnetic torques is following:

$T_S=m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_c---\left(24\right)$

T _Difference＝T _Residual1-T _Residual2 (25)

In the formula, $T_{\mathrm{Residual}1}=T_{e01}-{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_1{\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_{s1}$ With $T_{\mathrm{Residual}2}=T_{e02}-{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">f</figure-callout>}}_2{\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{\mathrm{\&eta;\; W}}_{s2}$ Be respectively two motor residual electricity magnetic torques.

Because | sin (2 α+θ _c) |≤1, therefore, the frequency capture conditions of system is:

T _S≥|T _Difference| (26)

Vibrational system realizes that the condition that frequency is captured is: system captures the difference moment that moment is greater than or equal to these two motors.

Then the system frequency capture conditions is:

$m_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{W}_c\&GreaterEqual;(T_{e01}-T_{e02})-(f_1-f_2){\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}-\mathrm{\&eta;}{W}_{s2})$

$W_c={-r}_m[\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}{\cos \mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;}{\cos \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{{\cos \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; y"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_y\sin \mathrm{\&delta;})}^2-$

$\frac{{\cos \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}({\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2+r_{x2}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;})-\frac{\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}({\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3+r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;})]$

μ _x＝1-(ω _nx/ω _m0) ²，μ _y＝1-(ω _ny/ω _m0) ²，μ _z＝1-(ω _nz/ω _m0) ²；

μ ₁＝1-(ω ₁/ω _m0) ²，μ ₂＝1-(ω ₂/ω _m0) ²，μ ₃＝1-(ω ₃/ω _m0) ²；

$W_{s1}=r_m[\frac{{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}{\sin \mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;}{\sin \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{{\sin \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="r\; y"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_y\sin \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;})}^2+$

$\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x2}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2)+\frac{{\sin \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3)$

W _s2＝ηW _s1。

3, confirm the stability condition of system synchronization:

Exist by formula (21) $\stackrel{\&OverBar;}{\mathrm{\&alpha;}}={\mathrm{\&alpha;}}_0$ Place's lienarized equation (22) and (23), and consider $\stackrel{\&OverBar;}{\mathrm{\&alpha;}}={\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^{*}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2,$ $(\mathrm{\&Delta;\; \&alpha;}=\stackrel{\&OverBar;}{\mathrm{\&alpha;}}-{\mathrm{\&alpha;}}_0),$ Get $z={\left\{\begin{array}{ccc}{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1& {\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2& \stackrel{\&OverBar;}{\mathrm{\&alpha;}}\end{array}-{\mathrm{\&alpha;}}_0\right\}}^T,$ Then

$\stackrel{\&CenterDot;}{z}=\mathrm{Cz}---\left(27\right)$

Wherein, C=A ^'-1B ',

$A^{\&prime;}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& 0\\ a_{21}& a_{22}& 0\\ 0& 0& 1\end{array}\right],$ $B^{\&prime;}=-{\mathrm{\&omega;}}_{m0}\left[\begin{array}{ccc}{b}_{11}& b_{12}& -{2W}_s\sin (2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)\\ b_{21}& b_{12}& {2W}_c\cos ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)\\ 0& -1& 0\end{array}\right].$

Suppose z=vexp (λ t), substitution formula (27), determinant equation is found the solution det (C-λ I)=0, and the characteristic value that gets characteristic equation is following:

λ ³+c ₁λ ²+c ₂λ+c ₃＝0 (28)

Wherein, c ₁=4 ω _M0H ₁/ H ₀, ${\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">c</figure-callout>}}_2=2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">H</figure-callout>}}_2/{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0,$ ${\mathrm{<figure-callout\; id="3"\; label="c"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">c</figure-callout>}}_3={2\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^3{\mathrm{<figure-callout\; id="3"\; label="H"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">H</figure-callout>}}_3/{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0,$ And

${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)+W_s^2{\sin }^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)$

H ₁＝ρ ₁κ ₂+ρ ₂κ ₁+W _sW _ccos(4α ₀+θ _s+θ _c) (29)

${\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">H</figure-callout>}}_2={\mathrm{\&kappa;}}_1{\mathrm{\&kappa;}}_2+({\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2)\cos ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)+({\mathrm{\&rho;}}_1-{\mathrm{\&rho;}}_2)W_s\sin ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)-W_s^2-$

$W_s^2{\sin }^2(2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2(2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)$

H ₃＝(κ ₁+κ ₂)W _ccos(2α ₀+θ _c)+(κ ₁-κ ₂)W _ssin(2α ₀+θ _s)-2W _sW _ccos(4α ₀+θ _s+θ _c)

Then the stability condition of system synchronization is:

ρ ₁＞0，ρ ₂＞0， ${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{\&rho;}}_{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">c</figure-callout>}}^2>0;$

${\mathrm{\&rho;}}_1=1-\frac{{\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2},$ ${\mathrm{\&rho;}}_2=\mathrm{\&eta;}(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c}{2}),$ ${\mathrm{\&rho;}}_c=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2\left({2\mathrm{\&alpha;}}_0\right)+W_s^2{\sin }^2\left({2\mathrm{\&alpha;}}_0\right)};$

$W_s=\mathrm{\&eta;}{r}_m[\frac{{\sin \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\cos \mathrm{\&delta;}}^2\sin {\mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}+\frac{\sin {\mathrm{\&delta;}}^2\sin {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{{\sin \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}({{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z}^2\cos {\mathrm{\&delta;}}^2+{{\mathrm{<figure-callout\; id="1"\; label="r\; y"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_y}^2\sin {\mathrm{\&delta;}}^2-r_{y1}{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\sin \mathrm{\&delta;}\cos \mathrm{\&delta;}$

$-r_{y1}{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;}\sin \mathrm{\&delta;})-\frac{{\sin \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{z2}{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z+{r_{x2}}^2\sin {\mathrm{\&delta;}}^2)-\frac{\sin {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}({{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y}^2+{r_{x3}}^2\cos {\mathrm{\&delta;}}^2)]$

${\mathrm{<figure-callout\; id="1"\; label="W\; c"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">W</figure-callout>}}_c=r_m[\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_y}{{\mathrm{\&mu;}}_y}+\frac{{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;}\cos {\mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}+\frac{\cos {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="r\; y"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_y\sin \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;})}^2+$

$\frac{\cos {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}(r_{x2}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2)+\frac{\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}(r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;}+{\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3)]$

W _c2＝ηW _c1，W _ccos(2α ₀)＞0，W _c＞0，2α ₀(-π/2，π/2)。

In engineering, the maximum damping coefficient of vibrational system is 0.14M ω _nOr 0.14J ω _nFor example, ξ≤0.07.Therefore, with c ₁, c ₂And c ₃W in the expression formula _cCompare W _sVery little, can ignore.H then ₀, H ₁, H ₂And H ₃Can be reduced to:

${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0^{\&prime;}=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)$

H ₁′＝ρ ₁κ ₂+ρ ₁κ ₂

${\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">H</figure-callout>}}_2^{\&prime;}=2{\mathrm{\&kappa;}}_1{\mathrm{\&kappa;}}_2+({\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2)W_c\cos ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)$

H ₃′＝(κ ₁+κ ₂)W _c?cos(2α ₀+θ _c) (30)

Utilize the Routh-Hurwitz criterion, have:

Work as c ₁＞0, c ₃＞0, c ₁c ₂＞c ₃(31)

Trivial solution z _iThe=0th, stable, then:

H ₀＞0, H ₁'＞0, H ₃'＞0 and 4H ₁' H ₂'-H ₀' H ₃'＞0 (32)

H ₁'＜0, H ₁'＜0, H ₃'＜0 and 4H ₁' H ₂'-H ₀' H ₃'＞0 (33)

Because κ ₁＞0 and κ ₂＞0, H ₀'＞0 and H ₁'＞0,

ρ ₁＞0, ρ ₂＞0 and $4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\mathrm{Cos}}^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)>0---\left(34\right)$

Because H ₃'＞0,

W _ccos(2α ₀+θ _c)＞0 (35)

H ₀', H ₁', H ₂' and H ₃' expression formula substitution 4H ₁' H ₂'-H ₀' H ₃'＞0, rearrange:

$[4{\mathrm{\&rho;}}_1^2{\mathrm{\&kappa;}}_2+4{\mathrm{\&rho;}}_2^2{\mathrm{\&kappa;}}_2+({\mathrm{\&kappa;}}_1+{\mathrm{\&kappa;}}_2){\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)]W_c\cos (2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)>---\left(36\right)$

$-4({\mathrm{\&rho;}}_1{\mathrm{\&kappa;}}_2+{\mathrm{\&rho;}}_2{\mathrm{\&kappa;}}_1)({2\mathrm{\&kappa;}}_1{\mathrm{\&kappa;}}_2+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}+{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\cos }^2(2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c))$

Work as W _cCos (2 α ₀+ θ _c)＞0, the left side of formula (36) is greater than 0; Work as ρ ₁＞0, ρ ₂＞0, the right of inequality (36) is less than 0.Therefore, inequality (34) and formula (35) satisfy inequality (36).

Work as H ₀'＜0, H ₁'＜0, ρ is arranged ₁κ ₂+ ρ ₂κ ₁＜0; Work as H ₃'＜0, W is arranged _cCos (2 α ₀+ θ _c)＜0.In this case, the left side of inequality (36) is less than 0, and the right is greater than 0.Therefore, H ₀'＜0, H ₁'＜0 and H ₃'＜0 can not satisfy 4H ₁' H ₂'-H ₀' H ₃'＞0.Work as W _c＞0,2 α ₀∈ (pi/2, pi/2) satisfies inequality (35); W _c＜0,2 α ₀∈ (pi/2,3 pi/2s) also satisfies inequality (35).Therefore, inequality (35) is the phase difference stability condition of two vibration generators, and inequality (34) is a stability of synchronization condition.

Below in conjunction with two vibration generator Coupled Dynamics specificity analysises and Computer Simulation and analyze the present invention is further specified:

One. two vibration generator Coupled Dynamics specificity analysises

1. define dimensionless moment of inertia and coupling inertia

In the expression formula of the moment of inertia of the load torque of two motors and two vibration generators,

characterized the coupling of two vibration generators.

When vibrational system synchronously operates in stable state, promptly ${\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_1={\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_2=0$ The time, the average load torque that vibrational system acts on two motors can be write as:

${\stackrel{\&OverBar;}{T}}_{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">L</figure-callout>}1}=\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(W_{s1}+W_s\cos ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)+W_c\sin (2{\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c))---\left(37\right)$

${\stackrel{\&OverBar;}{T}}_{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">L</figure-callout>}2}=\frac{1}{2}{m}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="0"\; label="m"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">m</figure-callout>}0}^2(\mathrm{\&eta;}{W}_{s2}+W_s\cos ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)-W_c\sin ({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c))---\left(38\right)$

H ₀' (H ₀), ρ ₁With

Determined the self synchronous stability of two motors.ρ ₁And ρ ₂Be respectively the dimensionless moment of inertia of vibration generator 1 and 2 in the equal differential equation of disturbance parameter.The instantaneous moment of inertia of two vibration generators can be write as:

J ₁′＝m ₀r ²ρ ₁＝m ₀r ²(1-W _c1/2) (39)

J ₂′＝m ₂r ²ρ ₂＝m ₀r ²η(1-W _c2/2) (40)

According to H ₀' (H ₀) expression formula, the coupled rotation inertia of two vibration generators is defined as:

J _c'=m ₀r ²ρ _c' or j _c=m ₀r ²ρ _c(41)

Wherein, ρ _c'=W _cCos (2 α ₀+ θ _c) (simplification), ${\mathrm{\&rho;}}_c=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="W\; c"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">W</figure-callout>}}_c^{}{\mathrm{Cos}}^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_c)+W_s^2{\mathrm{Sin}}^2({2\mathrm{\&alpha;}}_0+{\mathrm{\&theta;}}_s)}$ The dimensionless coupled rotation inertia that is called two vibration generators.

${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0^{\&prime;}=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{\&rho;}}_c^{\&prime;2}$ Perhaps ${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="H2009101876962E00061.png,H2009101876962E00071.png"\; state="\{\{state\}\}">H</figure-callout>}}_0=4{\mathrm{\&rho;}}_1{\mathrm{\&rho;}}_2-{\mathrm{\&rho;}}_c^2---\left(42\right)$

2. system load torque and dynamic symmetrical analysis

W _S1, W _S2, W _C1, W _C2, W _sAnd W _cBe dimensionless group, characterize the Coupled Dynamics characteristic of two vibration generators.According to their expression formula, W _S1And W _S2Be defined as the sinusoidal function coefficient (CSEPA) at the phase angle of vibration generator 1 and 2 respectively.W _C1And W _C2Be respectively the phase place cosine of an angle function coefficient (CCEPA) of vibration generator 1 and 2.Ws is called the sinusoidal function coefficient (CCSEPA) of coupling at two vibration generator phase angles.W _cThe coupling cosine function coefficient (CCCEPA) that is called two vibration generator phase angles.

See formula (37) and (38), the load torque that vibrational system acts on two motors is made up of three parts.First, kinetic energy and CSEPA generation by vibration generator have characterized the effect of vibration generator to motor movement, are called the sine effect torque (TSEPA) at phase angle.Latter two part has been described a stimulus movement that vibration generator produces another electric motor load torque, i.e. the coupled load torque of two vibration generators.Second portion is the sinusoidal effect of the coupling that is called the phase angle torque (TCSEPA) with respect to CCSEPA; Third part is the torque (TCCEPA) with respect to the coupling cosine effect that is called the phase angle of CCCEPA.It should be noted that the numerical value jack per line of the TCSEPA of two motors, the numerical value contrary sign of the TCCEPA of two motors, promptly TCSEPA is not the load torque of vibrational system.TCCEPA is a load torque for the leading motor of phase place, and the increase of limit angles speed is a driving torque for the motor of phase lag, the minimizing of limit angles speed.In addition, the numerical value of TCCEPA equals Along with sin (2 α+θ _c) proportionally increase.If the parameter of vibrational system satisfies formula (25), 2 α ₀Must satisfy inequality (26), two motors produce synchronously.

Therefore, T _SBe that two motors are realized synchronous key factor.When the position that two excitations are installed is about frame mass centre fully during symmetry, promptly

l _x1＝l _x2，l _y1＝l _y2，l _z1＝l _z2，δ ₁＝δ ₂，η＝1 (43)

Then

${\mathrm{\&theta;}}_c=\left\{\begin{array}{cc}\mathrm{\&pi;},& a_c<0\\ 0,& a_c\&GreaterEqual;0\end{array}\right.,$ ${\mathrm{\&theta;}}_s=\left\{\begin{array}{cc}0,& a_s\&GreaterEqual;0\\ \mathrm{\&pi;},& a_s<0\end{array}\right.---\left(44\right)$

Work as a _x＜0, and the vibrational system parameter satisfies inequality (26), T _SDriving two vibration generators moves between π at phase difference.T _SBig more, phase difference approaches π more, especially when the quality of two vibration generators and the parameter of electric machine equal consistent the time.In this case, the operation of two motor stabilizings, and phase difference is π; This means that also two vibration generators still keep symmetry in running.This statement of facts has the vibrational system of two vibration generators that the symmetric ability of maintenance is arranged in running.If system parameters is asymmetric, T _SPhase difference is-θ when driving the operation of two vibration generators _cVibrational system can make two vibration generators keep the characteristic of certain phase difference to be referred to as the dynamic symmetry of system.Phase difference-θ _cBe system dynamics symmetry angle, T _SBe called system dynamics symmetry moment.Obviously, the motor synchronizing of two excitations comes from the common dynamic symmetry of system.

The system load torque comprises three part TSEPA, TCSEPA and TCCEPA.When system parameters symmetry fully, formula (43) is satisfied in the installation site of two vibration generators, and the parameter of electric machine is consistent, and the two phase place difference is π (a _c＜0) or 0 (a _c＞0).Work as a _c＜0, two vibration generators operation phase difference is stabilized in π, and TCCCEPA is 0.X, ψ is cancelled out each other in the vibration of y and z direction ₁, ψ ₂And ψ ₃The vibration mutual superposition of direction.TCSEPA and TSEPA sum are 2 times of load torque, and this also is because ψ ₁, ψ ₂And ψ ₃Vibration cause by a vibration generator.Work as a _c＞0, x, the vibration mutual superposition of y and z direction, ψ ₁, ψ ₂And ψ ₃The vibration of direction is cancelled out each other.TCSEPA and TSEPA sum are 2 times of load torque, and this also is that the vibration of y and z direction is caused by a vibration generator owing to x.When phase difference was not 0, dynamically symmetry moment produced, and ordered about phase difference and be tending towards 0.When the parameter of electric machine was inconsistent, the difference of two motor residual electricity magnetic torques was not 0, and phase difference can be stabilized in a value that satisfies inequality (26).

In this case, the sine value of TCCEPA and phase difference is proportional, so phase difference is exactly the load torque of two motors and the sign of electromagnetic torque asymmetry.When system parameters was asymmetric, dynamic symmetric moment can be ordered about phase difference and approached-θ _cTCSEPA value and sin (2 α ₀+ θ _s) proportional.Therefore, θ _sIt is the deflection of two electric motor load torques stack.Therefore, TCSEPA has characterized when system configuration the rule of the load torque of two motors when asymmetric and two motor parameters are inconsistent, and is consistent with the rule of linear system motion stack.

The torque of systemic effect on two motors can be divided into two parts, load torque and dynamic symmetric torque.Significantly, the latter is big more, and the former is more little, and system keeps dynamic symmetric ability just strong more.Dynamically the symmetry dimensionless factor can define as follows:

$\mathrm{\&zeta;}=\frac{{2W}_c}{W_{s1}+\mathrm{\&eta;}{W}_{s2}+W_s\cos ({\mathrm{\&theta;}}_s-{\mathrm{\&theta;}}_c)}---\left(45\right)$

When two motors reach synchronous rotation; The supply of electric power of a motor is cut off; Be that power supply is only supplied a motor,, can the motor energy of power supply be passed to the motor of stopping power supply if that frequency is captured moment is enough big; And can enough overcome passive electric motor load torque, then two motor continue synchronous operation.This special vibration synchronous transmission that is called as synchronously.The condition of vibration synchronous transmission is: frequency is captured the electromagnetic torque of moment more than or equal to 2 times power supply motor.

3. influence the analysis of stability of synchronization parameter

The instantaneous relatively moment of inertia of two vibration generators reduces along with self-priming effect, and the CCEPA's of reduction and its each excitation is half the proportional.The condition of synchronism stability: the dimensionless moment of inertia of two motors is all greater than 0, and their 4 times product greater than their dimensionless coupling inertia square.

r _m, η, r _Xji, r _YjiAnd r _ZjiIt is the major parameter that has influence on the stability of synchronization.l _E01, l _E02And l _E03Be respectively frame x ", the radius of turn of equal value of y " and z "-direction.In addition, suppose that two motors are identical with the installation site of two vibration generators.If the distance of two vibration generators and frame mass centre be respectively l with

Then vibration generator 1 is at ψ ₁, ψ ₂And ψ ₃The dimensionless distance of direction (spherical coordinates) can roughly be expressed as:

$r_{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">l</figure-callout>}1}^2={\left(\frac{l}{l_{e1}}\right)}^2=\frac{1-(l_{e01}^2/{\mathrm{<figure-callout\; id="1"\; label="l\; e"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">l</figure-callout>}}_e^1)}{r_m[(1+{\mathrm{\&eta;}}_{01})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_1{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_1)+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_2+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_2){\mathrm{\&zeta;}}^2]}$

$r_{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}2}^2={\left(\frac{l}{l_{e2}}\right)}^2=\frac{1-(l_{e02}^2/{\mathrm{<figure-callout\; id="2"\; label="l\; e"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}}_e^2)}{r_m[(1+{\mathrm{\&eta;}}_{01})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_1{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_1)+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_2+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_2){\mathrm{\&zeta;}}^2]}$

$r_{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">l</figure-callout>}3}^2={\left(\frac{l}{l_{e3}}\right)}^2=\frac{1-(l_{e03}^2/{\mathrm{<figure-callout\; id="3"\; label="l\; e"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}_e^3)}{r_m[(1+{\mathrm{\&eta;}}_{01}){\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&phi;}+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02}){\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{\&zeta;}}^2]}---\left(46\right)$

Wherein, η ₀₁=m ₀₁/ m ₀, η ₀₂=m ₀₂/ m ₀,

When

φ ₁, φ ₂, θ ₁And θ ₂Value is given regularly, l _E1, l _E2And l _E3It is the monotonically increasing function of independent variable l. $\underset{l\&RightArrow;+\&infin;}{\lim l_{\mathrm{ei}}}=+\&infin;,$ i＝1，2，3。Therefore, can get:

$r_{\mathrm{<figure-callout\; id="1"\; label="l"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">l</figure-callout>}1\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}^2=\underset{l\&RightArrow;+\&infin;}{\lim }{\mathrm{<figure-callout\; id="1"\; label="r\; l"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_l=\frac{1}{r_m[(1+{\mathrm{\&eta;}}_{01})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_1{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_1)+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_2+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_2){\mathrm{\&zeta;}}^2]}$

$r_{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">l</figure-callout>}2\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}^2=\underset{l\&RightArrow;+\&infin;}{\lim }{\mathrm{<figure-callout\; id="2"\; label="r\; l"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_l=\frac{1}{r_{m.}[(1+{\mathrm{\&eta;}}_{01})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&phi;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_1)+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02})({\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_2+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_2){\mathrm{\&zeta;}}^2]}$

$r_{\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">l</figure-callout>}3\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}^2=\underset{l\&RightArrow;+\&infin;}{\lim }{\mathrm{<figure-callout\; id="3"\; label="r\; l"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_l=\frac{1-(l_{e03}^2/{\mathrm{<figure-callout\; id="3"\; label="l\; e"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">l</figure-callout>}}_e^3)}{r_m[(1+{\mathrm{\&eta;}}_{01}){\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&phi;}+(\mathrm{\&eta;}+{\mathrm{\&eta;}}_{02}){\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{\&zeta;}}^2]}---\left(47\right)$

Therefore, two vibration generators are at ψ ₁, ψ ₂And ψ ₃The dimensionless coordinate of direction is respectively:

${\mathrm{<figure-callout\; id="1"\; label="r\; xj"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{xj}}^{1\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_1{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_1$

${\mathrm{<figure-callout\; id="1"\; label="r\; yj"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{yj}}^{1\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_1{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_1$

${\mathrm{<figure-callout\; id="1"\; label="r\; zj"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{zj}}^{1\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_1---\left(48\right)$

${\mathrm{<figure-callout\; id="2"\; label="r\; xj"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{xj}}^{2\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{\&zeta;}}^2{\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&theta;}}_2$

${\mathrm{<figure-callout\; id="1"\; label="r\; yj"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{yj}}^{1\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{\&zeta;}}^2{\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&phi;}}_2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&theta;}}_2$

${\mathrm{<figure-callout\; id="2"\; label="r\; zj"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{zj}}^{2\mathrm{<figure-callout\; id="2"\; label="max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">max</figure-callout>}}={\mathrm{\&zeta;}}^2{\mathrm{<figure-callout\; id="2"\; label="r\; lj\; max"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_{\mathrm{lj}\max }^{}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&phi;}}_2,j=\mathrm{1,2,3}.$

If r _Xjimax ², r _Yjimax ²And r _Zjimax ²(i=1,2; J=1,2,3) satisfy inequality (34), system will be in the synchronism stability state always.

Two. the typical motion of giving an example track

If two vibration generators symmetry is installed, i.e. l _X1=l _X2=l _x, l _Y1=l _Y2=l _y, l _Z1=l _Z2=l _z, δ ₁=δ ₂=δ and η=1, then b _c=0, b _s=0; ECCEPA can be expressed as:

$W_c=-r_m[\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&delta;}\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}+\frac{{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&delta;}\cos \mathrm{\&gamma;}}_z}{{\mathrm{\&mu;}}_z}-\frac{{\cos \mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}{({\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z\cos \mathrm{\&delta;}-{\mathrm{<figure-callout\; id="1"\; label="r\; y"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_y\sin \mathrm{\&delta;})}^2----\left(49\right)$

$\frac{{\cos \mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}({\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2+r_{x2}^2{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\mathrm{\&delta;})-\frac{{\cos \mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3}({\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3+r_{x3}^2{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&delta;})]$

Wherein, r _Xi=l _Xi/ l _Ei, r _Yi=l _Yi/ l _Ei, r _Zi=l _Zi/ l _Ei, i=1,2,3.

Work as W _c＜0, θ _c=π, θ _s=0, system is at x, and y and z direction are done oscillating motion; Work as W _c＞0, θ _c=0, θ _s=π, system are at x, and y and z direction are done linear oscillator; That is: do elliptic motion on the xy plane, do straight-line oscillation in the z direction, promptly the vibrating screen of this moment is the vibrating screen of one three direction motion.The ratio of the amplitude of y direction and z direction amplitude can be through the adjustment of δ angle.

When δ=0, ECCEPA can be expressed as follows:

$W_c=-r_m(\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_x}+\frac{{\cos \mathrm{\&gamma;}}_x}{{\mathrm{\&mu;}}_y}-\frac{{\mathrm{<figure-callout\; id="1"\; label="r\; z"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^1\cos {\mathrm{\&gamma;}}_1}{{\mathrm{\&mu;}}_1}-\frac{{\mathrm{<figure-callout\; id="2"\; label="r\; z"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}_z^2\cos {\mathrm{\&gamma;}}_2}{{\mathrm{\&mu;}}_2}-\frac{({\mathrm{<figure-callout\; id="3"\; label="r\; y"\; filenames="H2009101876962E00011.png,H2009101876962E00081.png"\; state="\{\{state\}\}">r</figure-callout>}}_y^3+r_{x3}^2)\cos {\mathrm{\&gamma;}}_3}{{\mathrm{\&mu;}}_3})---\left(50\right)$

In this case, work as W _c＞0, frame is done elliptic motion on the x-y plane.

Three. Computer Simulation and analysis

1. initialization system parameter

The vibrational system parameter is following: total mass of vibration M=2400kg, and the eccentric arm r=0.2m of two eccentric blocks, the vibration body is about x " and, y ", " moment of inertia of axle is respectively z: J _P1=750kgm ², J _P2=980kgm ², J _P3=1800kgm ²X, y, z, ψ ₁, ψ ₂, ψ ₃The spring rate of direction is respectively: k _x=1247kN/m, k _y=1247kN/m, k _z=1247kN/m, k ₁=370kNm/rad, k ₂=500kNm/rad, k ₃=935kNm/rad.X, y, z, ψ ₁, ψ ₂, ψ ₃The resistance coefficient of direction is respectively: f _x=f _y=f _z=7.66kNs/m, f _{ψ 1}=2.4kNs/rad, f _{ψ 2}=3.1kNms/rad, f _{ψ 3}=5.5kNms/rad.Some of system

Calculating parameter is: μ _x=0.93, μ _y=0.93, μ _z=0.93, μ ₁=0.93, μ ₂=0.95, μ ₃=0.94.Moment of resistance coefficient f on the axle 1 ₁=0.01, the moment of resistance coefficient f on the axle 2 ₂=0.005; Coefficient of friction f on axle 1 and the axle 2 _D1=f _D2The critical damping of=0.001, six direction of vibration is 0.07.

Table 1 induction motor parameter

2. the vibrational system frequency is captured the analysis of ability

System realizes the factor of synchronous condition: the difference of two motor residual electricity magnetic torques and the installation site of two vibration generators.Suppose that two motors move ω under nominal load _e=max{ ω _E1, ω _E2, and T is being realized within the locking range in two vibration generator installation sites _S=T _DifferenceThe eccentric arm of two vibration generators can be adjusted through the expression formula that is obtained by formula (22) distortion in order to satisfy the needs of full load operation.

${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2=\frac{2(T_{e01}+T_{e02})-2({\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="H2009101876962E00022.png,H2009101876962E00041.png"\; state="\{\{state\}\}">f</figure-callout>}}_1+f_2){\mathrm{\&omega;}}_e}{m_1{\mathrm{\&omega;}}_e^2(W_{s1}+\mathrm{\&eta;}{W}_{s1}+2W_s\cos ({\mathrm{\&theta;}}_s-{\mathrm{\&theta;}}_c))}---\left(51\right)$

$|(T_{e01}-T_{e02})-(f_1-f_2){\mathrm{\&omega;}}_{m0}-\frac{1}{2}{m}_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_e^2(W_{s1}-{\mathrm{\&eta;W}}_{s2})|=m_0{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="H2009101876962E00011.png,H2009101876962E00072.png"\; state="\{\{state\}\}">r</figure-callout>}}^2{\mathrm{\&omega;}}_e^2{W}_c---\left(52\right)$

If m ₁, η, r _Lx, θ ₁, θ ₂, δ ₁, δ ₂, φ ₁And φ ₂Be given.The situation that two vibration generators symmetries is installed is promptly: δ ₁=δ ₂, θ ₂=π+θ ₁, ζ=1 and φ ₂=π-φ ₁In order to compare this space system and the synchronous at present ability of plane motion system, set r _X1As realizing synchronous range parameter.The vibrational system of different parameters is at η-r _X1The plane realizes that synchronous scope is as shown in Figure 8, and wherein, Fig. 8 (a) has represented to work as r _m=0.01, r _m=0.05 and r _m=0.1 at δ ₁=δ ₂=0, θ ₁=0, θ ₂=π, φ ₁=φ ₂Locking range during=pi/2.In this case, vibrational system is a kind of plane motion.

η-r _X1Variation relation shown in Fig. 8 (a), r _mTo realizing not having synchronously influence.With η-r _X1The plane is divided into I, II, and four zones of III and IV, at I, II zone two motors can be realized synchronously, and at III, IV zone two motors then can not be realized synchronously.In the I zone (first area synchronously), phase difference 2 α of synchronism stability ₀∈ (pi/2,3 pi/2s), in the II zone (second area synchronously), phase difference 2 α of synchronism stability ₀∈ (pi/2, pi/2); Work as T _Diffemece=0, III and IV zone become point (η=0.39, a r _X1=1.41) the quality ratio that, that is to say two vibration generators of two motors exists an optimum value to make vibrational system realize that synchronizing capacity strengthens.Fig. 8 (b) has represented to work as r _m=0.01, δ ₁=δ ₂=0, φ ₁=φ ₂During=pi/2, (θ ₁, θ ₂)=(0, π), the locking range of (π/4,5 π/4) and (π/3,4 π/3), comparison diagram 8 (b) and Fig. 8 (a), the first area is along with θ synchronously ₁(0≤θ ₁≤pi/2) increase and reducing, and synchronously second area along with θ ₁Increase and increase.Fig. 8 (c) has represented to work as δ ₁=δ ₂=0, θ ₁=π/4, θ ₂=5 π/4 o'clock, (φ ₁, φ ₂)=(pi/2, pi/2), (3 π/8,5 π/8), the locking range of (π/4,3 π/4) and (π/6,5 π/6).Shown in Fig. 8 (c), the first area is along with φ synchronously ₁(0≤φ ₁＜pi/2) increase and reducing, and synchronously second area along with φ ₁Increase and increase.In engineering, the elliptic motion of vibrating screen needs the phase difference of vibration generator near zero (second area is synchronous); But because the restriction of vibrating screen mesh structure, the radius of turn of equal value of z axle is the half the of its width.Therefore, the installation site of two vibration generators just makes r _X1Be difficult to satisfy second area synchronously, that is: when two vibration generators are installed on x axle or the y axle.The structure of vibrating screen of the present invention can reduce the requirement for the installation site, such as in the x direction, if this direction distance reaches requirement, can allow the distance on y and z direction to change to some extent.

Fig. 8 (d) has represented to work as θ ₁=π/4, θ ₂=5 π/4, φ ₁=π/4, φ ₂=3 π/4 o'clock, δ ₁=δ ₂=0, δ ₁=δ ₂=π/8, δ ₁=δ ₂=π/4 and δ ₁=δ ₂The locking range of=3 π/8.Work as δ ₁=δ ₂=0 o'clock, the second retaining zone scope was maximum, and works as δ ₁=δ ₂It is minimum that=π/4 o'clock, this zone reach.Comparison diagram 8 (d), Fig. 8 (b) and Fig. 8 (c), visible, δ ₁, δ ₂To the influence of locking range than θ ₁, θ ₂, φ ₁, φ ₂This Several Parameters is much smaller to its influence.See Fig. 8 (c) and Fig. 8 (d), work as φ ₁≠ δ ₁The time, area I II and IV permeate individual regional.

3. the dynamic symmetry specificity analysis of vibrational system

Fig. 9 has shown the curve chart of the dynamic symmetry coefficient ξ under different system parameter when two motors symmetry is installed.As shown in Figure 9, every curve all has a zero point, the r that each zero point is corresponding _X31Value is all used r _X310Expression, near zero point, vibrational system can not realize synchronously.In order to guarantee the circular motion of rigid frame, two vibration generators are installed must satisfy r _X31＞r _X310The ξ value is big more, and the stability of synchronization is big more.r _X31Depart from r _X310Far away more, the ξ value is big more.Two vibration generators depart from x axle, r more _X310Be worth more little.This shows: this space mounting structure of two vibration generators than folk prescription to mounting structure can reduce rigid frame realize circular motion apart from aspect requirement.Fig. 9 (c) has represented the influence of δ to ξ, can know that by figure this influence is very little; Fig. 9 (d) has represented the influence of two vibration generator mass ratio η to ξ, and when η=1, ζ reaches maximum, and along with η reduces, ζ also reduces.This shows: system structural strong more, the dynamic symmetry coefficient of system is also strong more.

4. vibrational system stability of synchronization surface analysis

In order to confirm stability boundary, suppose that the η value is given, ask r according to inequality (32) and (33) _m, r _L31Value is to satisfy (H ₀'=0), H ₁=0 (H ₁'=0), H=4H ₁H ₂-H ₀H ₃=0 (4H ₁' H ₂'-H ₀' H ₃'=0).Suppose and ignore r _L1, r _L2And r _L3Fluctuation, μ ₁, μ ₂And μ ₃Value is 0.95.δ is worked as in Figure 10 (a) expression ₁=δ ₂=π/6, θ ₁=π/4, θ ₂=5 π/4, φ ₁=3 π/8 and φ ₂=5 π/8 o'clock, H ₀=0 (H ₀'=0), H ₁=0 (H ₁'=0), H=0 (H '=0) is at r _m-r _L31R in the plane _{L31 max}Curve chart.H is worked as in Figure 10 (b) expression ₀=0 o'clock,

And r _mRelation.

5. system emulation result

When eccentric block quality is: m ₁=45kg (r _m=0.017), m ₂=45kg (η=1); Eccentric arm is: r=0.2m (J ₁=J ₂=1.8kgm ²).The parameter of vibrational system is: l _X1=l _X2=1m (r _X31≈ 1.3), l _Y1=l _Y2=1m (θ ₁=π/4, θ ₂=5 π/4), l _Z1=l _Z2=0.52m (φ ₁=5 π/8, φ ₂=7 π/8) and δ ₁=δ ₂=π/4 o'clock, system reaches (θ synchronously in second area _c=0).Can know from Figure 11: during stable state, at x, the motion of y and z direction rigid frame response is made a circulation, and at ψ ₁, ψ ₂And ψ ₃Direction should be 0 mutually.This simulation result has proved the correctness of above-mentioned theory demonstration.Shown in figure 11, when two motors start simultaneously, the angular acceleration of motor 2 is littler than motor 1, because the electromagnetic torque of vibration generator 2 is littler than vibration generator 1.Cause the system resonance velocity band when speed of gyration surpasses, two vibration generators cause system at x, y, z, ψ ₁, ψ ₂, ψ ₃The resonance response of six direction.The resonance response of system has produced dynamic symmetric moment, plays a part the adjustment electric motor load torque.When $0<2\stackrel{\&OverBar;}{\mathrm{\&alpha;}}<\mathrm{\&pi;}({\mathrm{\&theta;}}_c=0)$ The time, dynamically symmetric moment is a load torque for motor 1, and is driving moments for motor 2.And this moment approaches pi/2 along with phase difference and increases gradually, and therefore, motor 1 rotating speed successively decreases, and motor 2 rotating speeds increase progressively, and this just causes motor 2 bigger than the rotating speed of motor 1, and phase difference also can become negative.In this case, dynamically symmetric moment is exactly a load torque for motor 2, and is driving moment for motor 1.Therefore, phase difference is cyclic variation near 0, and the difference of the two rotating speed of motor cycle of also doing changes, shown in Figure 11 (a) and Figure 11 (b).Along with two rotating speed of motor increase, the high frequency response that has evoked system, dynamically symmetric moment is also increasing thereupon simultaneously.The amplitude of phase difference also reduces rapidly, and final phase difference is stabilized near 2.1 °, and the synchronous speed of two motors reaches 985.6r/min constantly at t=3.5s.When t=5s, motor 2 power supplies, dynamically symmetric moment begins adjustment, and phase difference is from 2.1 ° to 2.8 °, and synchronous speed reduces to 981.2r/min, but two motors still keep synchronously, and vibrational system is in the synchronism stability state.

When eccentric block quality is: m ₁=45kg (r _m=0.017), m ₂=30kg (η=0.667), eccentric arm is: r=0.2m (J ₁=1.8kgm ², J ₂=1.2kgm ²).The parameter of vibrational system is: l _X1=l _X2=1m, (r _X31≈ 1.3), l _Y1=l _Y2=1m (θ ₁=π/4, θ ₂=5 π/4), l _Z1=l _Z2=0.52m (φ ₁=5 π/8, φ ₂=7 π/8) and δ ₁=δ ₂=π/4 o'clock, simulation result is shown in figure 12, and it is synchronous that system also can reach.

Can be found out that by simulation result under identical structural parameters condition, two quality ratio η are big more, the stability of synchronization is good more.Along with the increase of η value, the two phase place difference is tending towards 0 gradually, x, y, z, ψ ₁, ψ ₂, ψ ₃The amplitude of direction reduces gradually, and when η=1, amplitude all reaches minimum value, especially ψ ₁, ψ ₂, ψ ₃The amplitude of three directions is tending towards 0 basically.The η value reduces; Be that mass ratio is more greatly different; System's difficulty more reaches stable, and this is because vibrational system increases adjusting moment through control phase difference sine value, but when the phase difference sine value increases to 1; Still can not find the balance point of system's equalising torque, the synchronous regime of system will lose stable.But, because the control phase that system does not stop in order to reach synchronism stability is poor, just caused the cycle of phase difference to change, other physical quantity is cyclic variation thereupon also.The simulation result of this computer has also confirmed the correctness of theoretical analysis result.

Claims (2)

1. a space three-direction self-synchronizing vibrating screen is characterized in that comprising bracing frame, on bracing frame, is fixed with an end of spring, and the other end of spring is fixed on inside to have on the sieve nest of screen cloth; Sieve nest is provided with two and is respectively applied for the asynchronous machine that drives two eccentric rotors; The centre of gyration of two asynchronous machines is about the barycenter symmetry of sieve nest; And the plane of rotation of two eccentric rotors is parallel to each other, and the structural parameters of this space three-direction self-synchronizing vibrating screen are confirmed according to the following steps

Step 1: the Mathematical Modeling of setting up system;

Step 2: confirm asynchronous motor quasi-stable state electromagnetic torque;

Step 3: confirm birotor frequency capture conditions and synchronous stability condition;

1), sets up system frequency and capture equation;

2), confirm the system frequency capture conditions;

3), confirm the stability condition of system synchronization.

2. space according to claim 1 three-direction self-synchronizing vibrating screen is characterized in that the Mathematical Modeling of confirming the asynchronous motor quasi-stable state electromagnetic torque described in the structural parameters step 2 is:

T _e＝T _e0-k _e0ε

$k_{e0}=\frac{n_p^2{L}_m^2{U}_{S0}^2}{{L^2}_s{R}_r{\mathrm{\&omega;}}_s}\frac{1-{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-n_p{\mathrm{\&omega;}}_{m0})}^2}{{[1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{m0})}^2]}^2}\frac{{\mathrm{\&omega;}}_{m0}}{{\mathrm{\&omega;}}_s},$ $T_{e0}=\frac{{3n}_p{L^2}_m{U}_{s0}^2}{2{L^2}_s{R}_r}\frac{{\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{r0}}{1+{\mathrm{\&sigma;}}^2{\mathrm{\&tau;}}_r^2{({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_{r0})}^2}$

In the formula, T _e-motor quasi-stable state electromagnetic torque; T _E0-rotor electrical angle speed is ω _R0The time electromagnetic torque; k _E0The stiffness coefficient of-steady state point electrical angle speed; ε-rotating speed is at ω _R0Near the minor fluctuations coefficient that becomes slowly that produces; σ-asynchronous motor leakage inductance coefficient,

τ _r-rotor time constant, τ _r=L _r/ R _rR _r-rotor equivalent resistance; L _s, L _r, L _m-stator inductance, rotor equivalent inductance, the mutual inductance between stator and the rotor; n _p-number of pole-pairs; ω _s-mains supply frequency; U _S0-terminal voltage; ω _R0The electrical angle speed of-steady state point asynchronous motor rotor; ω _M0-system reaches the rotating speed of synchronous operation state.

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