CN100354869C - Method for deciding radial rotation stability of magnetic suspension rotor system - Google Patents

Abstract

The present invention relates to a method for judging the stability of a magnetic suspension rotor system. Through building a dynamics differential equation model of the radial rotational motion of the magnetic suspension rotor system, the method converts a real coefficient two-variable equation to a complex coefficient univariable form by plurality conversion to obtain a univariable equivalent system, then draws the dual frequency Bode chart of the open loop transfer function of the equivalent complex coefficient system, judges rotor precession and nutation stability by the generalized classic frequency field stability criteria, and provides require conditions for designing correction links for enhancing system stability. The present invention converts the multivariable system to the equivalent univariable system so as to analyze stability by using the classical univariable control theory, and has better visual properties and robustness than usual multivariable methods, such as a state space analysis method, etc., and thus, the present invention is more suitable for the application of a real system.

Description

A kind ofly judge the radially method of rotational stability of magnetic suspension rotor system

Technical field

The present invention relates to a kind of magnetic suspension rotor system determination of stability method, can be used for the stability analysis that magnetic suspension rotor system radially rotates mode, and judge whether the precession of magnetic suspension rotor and nutating stablize.

Background technology

With respect to traditional mechanical ball bearing, outstanding advantages such as that magnetic bearing has is contactless, rigidity and damping active controllable, thereby there is not a friction and wear, also need not to lubricate, allow rotor high-speed rotation, little, the supporting precision height of vibration, be particularly suitable for super-clean environment equipment, high rotating speed equipment and require low vibration, high precision, long-life space equipment.At present, based on the magnetic suspension rotor system of magnetic bearing at hydro-extractor, high precision numerical control lathe, turbine, accumulated energy flywheel, and obtain increasingly extensive application in the spacecraft attitude control executing mechanism such as magnetically levitated flywheel and magnetic suspension control torque gyroscope, and the trend that replaces mechanical bearing is arranged.

In the process of design magnetic suspension rotor system, the stability of decision-making system is committed step wherein.Only know about magnetic suspension rotor system kinetic stability under the different rotating speeds in the whole range of speeds, could design effective controller, the stability of raising and safeguards system.But because the axially-movable of magnetic suspension rotor system and radially the equal decoupling zero of translation be single dof mobility, and stability and rotating speed are irrelevant, and radially there is the gyro coupling in the two-freedom rotation at the rotor time rotational, stability is closely related with rotating speed, thereby judges that the stability of radially rotating is the difficult point and the key of magnetic suspension rotor system stability problem.Gyro coupling makes radially rotates mode and is divided into precession and nutating, can further be summed up as the stability of rotor precession and nutating when judging different rotating speeds so judge rotational stability radially.

The existence of gyro coupling make magnetic suspension rotor system radially rotate subsystem become in the real number field can not decoupling zero two variable systems, adopt the state space analysis method that is suitable for multi-variable system to carry out stability analysis and judgement at present usually.But state-space method too relies on system model and mathematical analysis, does not possess the intuitive and the robustness of classical control theory, is not easy to practical application.For the more weak magnetic suspension rotor system of gyro coupling, can ignore the gyro coupling and approximately use classical single argument control theory and analyze, but this approximate processing method is not suitable for the magnetic suspension rotor system with strong gyroscopic coupling effect.

Summary of the invention

Technology of the present invention is dealt with problems: overcome the deficiencies in the prior art, provide a kind of have good robustness, directly perceived and be fit to real system a kind ofly judge the radially method of rotational stability of magnetic suspension rotor system.

Technical solution of the present invention is: a kind ofly judge the radially method of rotational stability of magnetic suspension rotor system, its characteristics are to comprise the following steps:

(1) sets up closed loop rotor dynamics Differential Equation Model

Definition rotor coordinate system oxyz, the o point is positioned at rotor centroid, and x and y axle are along rotor radial and only follow the radially rotation of rotor and do not follow rotation, and the z axle is along rotor axial.According to gyro technology equation, the parameter substitution following formula of magnetic suspension rotor system is obtained the dynamic differential equation group of rotor radial rotational motion:

$\left\{\begin{array}{c}{J}_y\stackrel{\·\·}{\mathrm{\β}}-H\stackrel{\·}{\mathrm{\α}}-2k_h{l}_m^2\mathrm{\β}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\β}+p_{\mathrm{dy}}\\ J_x\stackrel{\·\·}{\mathrm{\α}}+H\stackrel{\·}{\mathrm{\β}}-2k_h{l}_m^2\mathrm{\α}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\α}+p_{\mathrm{dx}}\end{array}\right.$

Ka Erdan angle α, β represent the rotor radial relative stator around x in the following formula, the angular displacement that the y axle rotates, J _x=J _yAnd J _zBe respectively rotor radial and axial moment of inertia, H=J _zΩ=2 π J _zF _rBe rotor angular momentum, Ω (rad/s of unit) and F _r(Hz of unit) is rotor speed, p _DxAnd p _DyBe the disturbing moment of rotor radial, k _iAnd k _hBe the displacement rigidity and the current stiffness of magnetic bearing, k _sBe the sensitivity of magnetic bearing displacement transducer, l _mAnd l _sBe respectively magnetic bearing and displacement transducer distance, g to rotor center _cAnd g _wInput-output transformation operator for controller and power amplifier promptly has $L\left[g_c\left(\frac{d}{\mathrm{dt}}\right)\right]=g_c\left(s\right),$ $L\left[g_w\left(\frac{d}{\mathrm{dt}}\right)\right]=g_w\left(s\right).$ Here L represents Laplace transformation, and s is an operator, g _c(s) and g _w(s) be the transport function of controller and power amplifier.

(2) real coefficient two variable equations are converted into complex coefficient single argument form

Introduce the j of imaginary unit, make J _Rr=J _x=J _y, =α+j β, p _d=p _Dx+ jp _Dy, first formula of differential equation group be multiply by j be added to second formula again, real coefficient two variate models are transformed to following complex coefficient single argument form:

Doing Laplace transformation again obtains:

Utilizing following formula is a univariate unity feedback system of complex coefficient with the original system equivalence, and its equivalent controlled device and equivalent control channel transfer function are respectively:

$g_{\mathrm{oeff}}\left(s\right)=\frac{1}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-{2k}_h{l}_m^2}$

g _ceff(s)＝2l _ml _sk _ik _sg _w(s)g _c(s)

Equivalence open-loop transfer function and secular equation are respectively:

$g_{\mathrm{OL}}\left(s\right)=\frac{2l_m{l}_s{k}_i{k}_s{g}_w\left(s\right)g_c\left(s\right)}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2}$

1+g _OL(s)＝0

(3) the positive real part limit of open loop number is that the Q value is calculated

Be similar to classical frequency field stability criterion, use the present invention and judge that magnetic suspension rotor radially at first will know the positive real part limit number of open cycle system during rotational stability.Because g _Ceff(s) each link is the minimum phase link, thereby g _OL(s) positive real part limit only may be from g _Oeff(s).Order $J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2=0,$ Solve open loop pole:

Wherein ${\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="C20061001157800131.png,C20061001157800141.png"\; state="\{\{state\}\}">H</figure-callout>}}_0={2l}_m\sqrt{2J_{\mathrm{rr}}{k}_h},$ Corresponding rotor speed is ${\mathrm{<figure-callout\; id="0"\; label="F\; r"\; filenames="C20061001157800131.png,C20061001157800141.png"\; state="\{\{state\}\}">F</figure-callout>}}_r=l_m\sqrt{2J_{\mathrm{rr}}{k}_h}/\left({\mathrm{\&pi;J}}_z\right).$ Represent the positive real part limit of open loop number with Q, according to following formula, as long as know rotor speed F _r, F then _r＜F _R0The time Q=1, and F _r＞F _R0The time Q=0.

(4) the double frequency Bode that draws equivalent complex coefficient single-variable system schemes

Bode figure in the classical control theory only comprises the amplitude versus frequency characte L (ω) and the phase-frequency characteristic φ (ω) of real coefficient transport function.For complex coefficient transport function g _OL(s), because negative frequency characteristic g _OL(j ω) and positive frequency characteristic g _OL(j ω) is asymmetric, thereby the Bode figure of complex coefficient transport function is except g _OLOutside the L (ω) and φ (ω) of (j ω), also to comprise g _OLThe amplitude versus frequency characte L of (j ω) (ω) and phase-frequency characteristic φ (ω).Divide another name g _OL(j ω) and g _OL(j ω) Bode figure is negative figure of Bode frequently and the positive figure of Bode frequently, and is referred to as g _OL(j ω) double frequency Bode figure.

Make s=j ω and s=-j ω respectively, draw g in ω＞0 zone _OL(j ω) and g _OLThe Bode of (j ω) figure comprises that amplitude versus frequency characte L (ω) and L (ω), and phase-frequency characteristic φ (ω) and φ (ω), obtains complex coefficient single-variable system g _OL(s) double frequency Bode figure.

(5) rotational motion determination of stability radially

Double frequency Bode figure stability criterion with the complex coefficient single-variable system is: use L _N+And L _P+Represent L (ω)＞0 and the ω interval of L (ω)＞0, N respectively _N+And N _N-φ is (ω) at L in expression _N+Just passing through and bearing the number of times that passes through (2k+1) π line, N _P+And N _P-Expression φ (ω) is at L _P+Just passing through and bearing the number of times that passes through (2k+1) π line, k=0,1,2..., N is N=N for always passing through number of times _N++ N _P+-N _N--N _P-, Z represents the positive real part limit of closed-loop system number, and Z=Q-N 〉=0 is then always arranged, and the stable necessary and sufficient condition of closed-loop system is Z=0.

(6) precession and nutating determination of stability

If F _r＞F _R0And system's instability, can also further distinguish is precession instability or nutating instability.Double frequency Bode figure stability criterion with the complex coefficient single-variable system is: with g _OL(s) double frequency Bode figure is divided into ω＜ω _LhLow-frequency range and ω＞ω _LhHigh band, ω _LhBe F _rThe imaginary part absolute value of secular equation root during=0Hz.The number of times that always passes through of definition low-frequency range is N with positive real part closed-loop pole number _lAnd Z _l, the analog value of high band is N _hAnd Z _hUse double frequency Bode figure stability criterion respectively in low-frequency range and high band, if low-frequency range Z _l=-N _l=1, then precession instability; If high band Z _h=-N _h=1 nutating instability.

The present invention's advantage compared with prior art is: by being a complex coefficient single-variable system with the original system equivalence, thereby can continue to use classical single argument control theory and carry out stability analysis, simultaneously also inherited the intuitive that classical control theory has and the advantage of robustness, be more suitable for application in real system.The present invention can also promote the symmetric rotor system that is used for other support patterns simultaneously.

Description of drawings

Fig. 1 is a process flow diagram of the present invention;

Fig. 2 is magnetic levitation closed loop rotor-support-foundation system of the present invention and coordinate system definition;

Fig. 3 is the present invention's equivalence complex coefficient single-variable system block diagram;

Fig. 4 is the open loop pole (F of equivalent system under the different rotating speeds of the present invention _r=0～400Hz);

Fig. 5 rotates mode double frequency Bode figure (F for magnetic suspension rotor system of the present invention _r=0Hz);

Fig. 6 rotates mode double frequency Bode figure (F for magnetic suspension rotor system of the present invention _r=80Hz);

Fig. 7 rotates mode double frequency Bode figure (F for magnetic suspension rotor system of the present invention _r=160Hz);

Fig. 8 rotates mode double frequency Bode figure (F for magnetic suspension rotor system of the present invention _r=240Hz);

Fig. 9 rotates mode double frequency Bode figure (F for magnetic suspension rotor system of the present invention _r=400Hz).

Embodiment

Introduce content of the present invention in conjunction with the magnetic levitation high-speed rotor system that a kind of magnetic suspension control torque gyroscope is used.

The process flow diagram of determination of stability as shown in Figure 1, at first set up the radially rotational motion dynamic differential equation model of magnetic suspension rotor system, adopt plural conversion that real coefficient two variable equations are converted into complex coefficient single argument form, obtain complex coefficient single argument equivalent system, draw the double frequency Bode figure of equivalent complex coefficient system open-loop transfer function then, utilize frequency field stability criterion to judge the stability of radially rotational motion, if it is stable, then rotor precession and nutating are all stablized, otherwise further judge it is precession instability or nutating instability.

Magnetic suspension rotor system as shown in Figure 2, this system is made of displacement transducer, controller, power amplifier, electromagnet and rotor, the magnetic bearing of the rotor A of only drawing among the figure, B two ends y direction, the x direction is similar with it.Each degree of freedom of magnetic bearing is all by the rotor displacement on this degree of freedom of displacement sensor, if rotor is not on given zero-bit, then error signal by magnetic bearing controller according to set control law computing after, by power amplifier output corresponding control current, drive the suitable magnetic attraction rotor of magnetic bearing electromagnet generation and be returned on the given position.U among the figure ₀The bias voltage of representing corresponding power amplifier input.

Equivalence complex coefficient single-variable system block diagram as shown in Figure 3, wherein $g_{\mathrm{oeff}}\left(s\right)=\frac{1}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-{2k}_h{l}_m^2}$ And g _Ceff(s)=2l _ml _sk _ik _sg _w(s) g _c(s) be respectively the transport function of equivalent controlled device and simulator, p _dBe respectively disturbance input and system's output with .

F _rThe open loop pole of the complex coefficient single-variable system of system's equivalence during=0～400Hz as shown in Figure 4, " * " number expression limit wherein.Because the stiffness term in the equivalent object is for negative, thus the imaginary part of open loop pole be on the occasion of, limit all is in the first half of complex plane.F _R0=115Hz, F _r＜F _R0The time open loop pole be two plural numbers about imaginary axis symmetry, F _r＞F _R0After be two pure imaginary numbers, imaginary part raises with rotating speed respectively and reduces and raise.So F _r＜F _R0The time Q=1, and F _r＞F _R0The time Q=0.

Draw g respectively in ω＞O zone _OL(j ω) and g _OLThe Bode of (j ω) figure, ω=2 π f wherein obtain the double frequency Bode figure of magnetic suspension rotor system, and as Fig. 5～shown in Figure 9, corresponding rotating speed is respectively F _r=0,80,160,240 and 400Hz.Transverse axis is frequency f (Hz of unit) among the figure, and last figure is an amplitude versus frequency characte, and the longitudinal axis is log gain (dB of unit), and figure below is a phase-frequency characteristic, and the longitudinal axis is phase angle (unit " ° ").Each arrow is represented to pass through for half time in the phase-frequency characteristic, upwards for just passing through, passes through for bearing downwards, and half of the difference of the two promptly always passed through the times N value.Use L _N+And L _P+Representing respectively to gain in negative frequency and the positive frequency amplitude versus frequency characte is higher than the frequency field of 0dB, and Fig. 5～Fig. 9 is described as follows:

A) F _rDuring=0Hz (Fig. 5), Q=1, N=1-1-0.5-0.5=1, Z=1-1=0, system stability.

B) F _rDuring=80Hz (Fig. 6), Q=1, N=1-1-0.5-0.5=1, Z=1-1=0, systems stabilisation.

C) F _rDuring=160Hz (Fig. 7), Q=0, N=1+1-0.5-1.5=0, Z=0-0=0, system stability.

D) F _rDuring=240Hz (Fig. 8), Q=0, N=1+1-0.5-2.5=-1, Z=0-(1)=1, system's instability.Adopt criterion further to examine or check low-frequency range characteristic and high band characteristic, the frequency boundary of the two is F _Lh=ω _Lh/ (2 π)=95Hz.Wherein on the low-frequency range characteristic, N _l=1+1-0.5-1.5=0, Z _l=0, and on the high-band frequency characteristic, N _h=0+0-0-1=-1, Z _h=1, thereby be the nutating instability.

E) F _rDuring=400Hz (Fig. 9), Q=0, N=0+0-0.5-1.5=-2, Z=0-(2)=2, system's instability.Adopt criterion further to examine or check low-frequency range characteristic and high band characteristic, wherein on the low-frequency range characteristic, N _l=0+0-0.5-0.5=-1, Z _l=1, and on the high-band frequency characteristic, N _h=0+0-0-1=-1, Z _h=1, thereby precession and nutating are all unstable.

Stability analysis result based on double frequency Bode figure shows F _r=0,80, system stability during 160Hz, F _r=240 o'clock nutating instabilities, and F _rPrecession and nutating are all unstable during=400Hz.

Utilize the magnetic suspension rotor system of this magnetic suspension control torque gyroscope to carry out the experiment of examination precession and nutating stability.Magnetic suspension rotor system is from F during experiment _rBegin raising speed during=0Hz.System is stable when low speed, but raising speed is to F _rThe nutating unstability takes place during=175Hz.Adopt correction link compensation high frequency to lag behind (but not influencing low frequency characteristic) afterwards, it is stable that system is recovered again.Continue rising rotor speed, system is at F _rThe precession unstability appears during=250Hz.Experimental result shows that the magnetic suspension rotor system of not doing compensation is at F _rNutating instability behind the＞175Hz, and F _rPrecession and chapter are all unstable behind＞the 250Hz.Experiment show the present invention can correctly judge the stability that magnetic suspension rotor system radially rotates.

Claims (4)

1, a kind ofly judges the radially method of rotational stability of magnetic suspension rotor system, it is characterized in that comprising the following steps:

(1) set up the radially rotational motion dynamic differential equation model of magnetic suspension rotor system according to gyro technology equation:

$\left\{\begin{array}{c}{J}_y\stackrel{..}{\mathrm{\&beta;}}-H\stackrel{.}{\mathrm{\&alpha;}}-2k_h{l}_m^2\mathrm{\&beta;}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\&beta;}+p_{\mathrm{dy}}\\ J_x\stackrel{..}{\mathrm{\&alpha;}}+H\stackrel{.}{\mathrm{\&beta;}}-2k_h{l}_m^2\mathrm{\&alpha;}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\&alpha;}+p_{\mathrm{dx}}\end{array}\right.$

Ka Erdan angle α, β represent the rotor radial relative stator around x in the following formula, the angular displacement that the y axle rotates, J _x=J _yAnd J _zBe respectively rotor radial and axial moment of inertia, H=J _zΩ=2 π J _zF _rBe rotor angular momentum, Ω and F _rBe rotor speed, p _DxAnd p _DyBe the disturbing moment of rotor radial, k _iAnd k _hBe the displacement rigidity and the current stiffness of magnetic bearing, k _sBe the sensitivity of magnetic bearing displacement transducer, l _mAnd l _sBe respectively magnetic bearing and displacement transducer distance, g to rotor center _cAnd g _wInput-output transformation operator for controller and power amplifier promptly has $L\left[g_w\left(\frac{d}{\mathrm{dt}}\right)\right]=g_w\left(s\right),$ L represents Laplace transformation, and s is an operator, g _c(s) and g _w(s) be the transport function of controller and power amplifier;

(2) adopt plural conversion that the two variable equations of the real coefficient in the step (1) are converted into complex coefficient single argument form, obtain complex coefficient single argument equivalent system:

Introduce the j of imaginary unit, make J _Rr=J _x=J _y, =α+j β, p _d=p _Dx+ jp _Dy, first formula of differential equation group be multiply by j be added to second formula again, real coefficient two variate models are transformed to following complex coefficient single argument form:

Doing Laplace transformation again obtains:

Utilizing following formula is a univariate unity feedback system of complex coefficient with the original system equivalence, and its equivalent controlled device and equivalent control channel transfer function are respectively:

$g_{\mathrm{oeff}}\left(s\right)=\frac{1}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2}$

g _ceff(s)＝2l _ml _sk _ik _sg _w(s)g _c(s)

Equivalence open-loop transfer function and secular equation are respectively:

$g_{\mathrm{OL}}\left(s\right)=\frac{2l_m{l}_s{k}_i{k}_s{g}_w\left(s\right)g_c\left(s\right)}{J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2}$

1+g _OL(s)＝0；

(3) the positive real part limit of the open loop calculating that is the Q value:

Order $J_{\mathrm{rr}}{s}^2-\mathrm{jHs}-2k_h{l}_m^2=0,$ Solve open loop pole:

Angular momentum wherein $H_0=2l_m\sqrt{2J_{\mathrm{rr}}{k}_h},$ Corresponding rotor speed is $F_{r0}=l_m\sqrt{2J_{\mathrm{rr}}{k}_h}/\left(\mathrm{\&pi;}{J}_z\right).$ Represent the positive real part limit of open loop number, rotor speed F with Q _r＜F _R0The time Q=1, and F _r＞F _R0The time Q=0;

(4) the double frequency Bode figure of the equivalent complex coefficient of drafting system open-loop transfer function:

Make s=j ω and s=-j ω respectively, draw g in ω＞0 zone _OL(j ω) and g _OLThe Bode of (j ω) figure comprises that amplitude versus frequency characte L (ω) and L (ω), and phase-frequency characteristic φ (ω) and φ (ω), obtains complex coefficient single-variable system g _OL(s) double frequency Bode figure;

(5), judge the stability of radially rotational motion according to the double frequency Bode figure stability criterion of complex coefficient single-variable system;

(6) if radially rotational motion instability according to the double frequency Bode figure stability criterion of complex coefficient single-variable system, is further judged the stability of precession and nutating.

2, the judgement magnetic suspension rotor system according to claim 1 method of rotational stability radially is characterized in that: the method for radially rotational motion dynamic differential equation model that described step (1) is set up magnetic suspension rotor system is as follows:

Definition rotor coordinate system oxyz, the o point is positioned at rotor centroid, x and y axle are along rotor radial and only follow the radially rotation of rotor and do not follow rotation, the z axle is along rotor axial, according to gyro technology equation, the parameter substitution following formula of magnetic suspension rotor system is obtained the dynamic differential equation group of rotor radial rotational motion:

$\left\{\begin{array}{c}{J}_y\stackrel{..}{\mathrm{\&beta;}}-H\stackrel{.}{\mathrm{\&alpha;}}-2k_h{l}_m^2\mathrm{\&beta;}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\&beta;}+p_{\mathrm{dy}}\\ J_x\stackrel{..}{\mathrm{\&alpha;}}+H\stackrel{.}{\mathrm{\&beta;}}-2k_h{l}_m^2\mathrm{\&alpha;}=-2l_m{l}_s{k}_i{k}_s{g}_w{g}_c\mathrm{\&alpha;}+p_{\mathrm{dx}}\end{array}\right.\&CenterDot;$

3, the judgement magnetic suspension rotor system according to claim 1 method of rotational stability radially is characterized in that: the double frequency Bode figure stability criterion of complex coefficient single-variable system is in the described step (5): use L _N+And L _P+Represent L (ω)＞0 and the ω interval of L (ω)＞0, N respectively _N+And N _N-φ is (ω) at L in expression _N+Just passing through and bearing the number of times that passes through (2k+1) π line, N _P+And N _P-Expression φ (ω) is at L _P+Just passing through and bearing the number of times that passes through (2k+1) π line, k=0,1,2..., N is N=N for always passing through number of times _N++ N _P+-N _N--N _P-, Z represents the positive real part limit of closed-loop system number, and Z=Q-N 〉=0 is then always arranged, and the stable necessary and sufficient condition of closed-loop system is Z=0.

4, the judgement magnetic suspension rotor system according to claim 1 method of rotational stability radially is characterized in that: (6) precession of described step and nutating determination of stability method: with g _OL(s) double frequency Bode figure is divided into ω＜ω _LhLow-frequency range and ω＞ω _LhHigh band, ω _LhBe F _rThe imaginary part absolute value of secular equation root during=0Hz, the number of times that always passes through of definition low-frequency range is N with positive real part closed-loop pole number _lAnd Z _l, the analog value of high band is N _hAnd Z _hUse double frequency Bode figure stability criterion respectively in low-frequency range and high band, if low-frequency range Z _l=-N _l=1, then precession instability; If high band Z _h=-N _h=1 nutating instability.

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