CN112432634A - Harmonic vibration force suppression method based on multi-synchronous rotation coordinate transformation - Google Patents
Harmonic vibration force suppression method based on multi-synchronous rotation coordinate transformation Download PDFInfo
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Abstract
The invention discloses a magnetic suspension rotor harmonic vibration force suppression method based on a multi-synchronous coordinate transformation method, which comprises the following steps: firstly, a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic is established, and then a magnetic suspension rotor harmonic vibration force suppression method based on MSRFT is adopted. The MSRFT can accurately restrain harmonic vibration force, and the controller is used for restraining vibration in X and Y directions at the same time, so that hardware computing resources are reduced, and the restraining dynamic process is accelerated. Meanwhile, the absolute stability of the system in a larger frequency range can be ensured by introducing a phase compensation angle. The MSRFT controller has a simple structure, is very convenient in practical application, can inhibit harmonic vibration force in the magnetic suspension rotor, and is suitable for inhibiting the harmonic vibration force of a magnetic suspension rotor system with mass unbalance and sensor harmonic.
Description
Technical Field
The invention relates to the technical field of suppression of harmonic vibration force of a magnetic suspension rotor, in particular to a suppression method of harmonic vibration force of a magnetic suspension rotor based on a multi-synchronous rotation coordinate transformation method, which is used for suppressing the harmonic vibration force of a rotor system of a magnetic suspension control moment gyroscope in a full working rotation speed range and providing technical support for application of the magnetic suspension control moment gyroscope on an ultra-static satellite platform and an ultra-stable satellite platform.
Background
The control moment gyroscope has the advantages of large output moment, high response speed and the like, and becomes a key attitude control inertia actuating mechanism of a high-performance satellite. Compared with the traditional mechanical bearing, the active magnetic bearing can realize the non-contact support of the rotor, so that the active magnetic bearing has the advantages of no friction, no need of lubrication, high rotating speed and the like, and the active vibration is controllable. Compared with the traditional mechanical gyro, the magnetic suspension control moment gyro has the characteristic of micro-vibration and is widely used on a high-performance satellite.
However, some vibration is inevitable in the magnetic suspension control moment gyro due to machining errors of the rotor and measurement errors of the sensor. The two main vibration sources are rotor mass unbalance and sensor harmonic waves, the mass unbalance is represented by the misalignment of the geometric center and the mass center of the rotor, when the rotor rotates at high speed, larger centrifugal force can be generated, the amplitude of the centrifugal force is increased along with the increase of the rotating speed, and the frequency is the same as the rotating frequency; the harmonic wave of the sensor comes from the roundness error of the measurement surface, so that the output signal of the displacement sensor contains harmonic interference with the same frequency and frequency multiplication of the rotating speed, and the electromagnetic coil of the magnetic bearing of the active magnetic suspension control moment gyroscope generates harmonic current with the same frequency and frequency multiplication, thereby generating harmonic vibration force. These harmonic vibration forces are transmitted to the satellite platform through the base, so that the pointing accuracy and attitude stability of the satellite are seriously affected.
In order to solve the vibration, a traditional method is to use a mechanical vibration isolation device, and a vibration source is isolated by adding a physical device, on one hand, the vibration isolation device is expensive, large in size and inconvenient to maintain, and the vibration isolation performance is inevitably influenced along with the increase of the service time; on the other hand, the vibration isolation device does not eliminate the vibration, but converts the low-frequency high-amplitude vibration into the high-frequency low-amplitude vibration, and the energy of the vibration is not reduced. Therefore, the active vibration control is carried out on the magnetic suspension control moment gyroscope, and the vibration is eliminated through a vibration control algorithm, so that the method for effectively solving the vibration is provided.
The active vibration control algorithm of the magnetic suspension control moment gyroscope mainly comprises a wave trap, a resonance controller, a repetitive controller, an LMS (least mean square) algorithm and the like, wherein the repetitive controller is an effective method for processing harmonic interference, but the repetitive control has the defect of low response speed, the LMS algorithm has large calculation amount and is not easy to realize, and although the wave trap and the resonance controller have simple structures, a plurality of controllers are required to be connected in parallel when a plurality of frequencies are processed, so that the calculation burden of the system is increased.
Disclosure of Invention
The purpose of the invention is as follows: the method for suppressing the harmonic vibration force of the magnetic suspension rotor based on the multi-synchronous rotation coordinate transformation method overcomes the defects of the prior art, the bearing force constructed by current and displacement according to an electromagnetic force model is used as the input of a control algorithm, so that the vibration force can be completely suppressed, the vibration force in two directions can be simultaneously suppressed by one controller by utilizing the orthogonal characteristic of X, Y direction signals, the calculation resources are reduced, the dynamic response speed is improved, and the absolute stability of the full working frequency band is realized by introducing a phase compensation angle to compensate the phase in different frequency bands.
The technical scheme adopted by the invention is as follows: a magnetic suspension rotor harmonic vibration force suppression method based on a multi-synchronous coordinate transformation method comprises the following steps:
step (1) establishing a full-active magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic
The invention is applied to an active magnetic bearing system in a magnetic suspension control moment gyro. Let N be the geometric center of the stator, NXY be the inertial coordinate system, C and O be the center of mass and the geometric center of the rotor, respectively, and O ε η be the rotational coordinate system. The method mainly aims at the suppression of the vibration force of two degrees of freedom of radial translation, so that only the modeling of two degrees of freedom of translation is considered. From newton's second law, the following kinetic equation is obtained:
where m is the rotor mass, X (t), Y (t) represent the translational displacements of the rotor center of mass in the X and Y directions, respectively, fx(t),fy(t) resultant forces of bearing forces in X-direction and Y-direction, respectively, fax(t),fbx(t),fay(t),fby(t) is the bearing force of the four pairs of radial magnetic bearings. When the rotor moves in a small displacement, the nonlinear bearing force can be approximately linearized and can be expressed as follows:
wherein Ki,KhCurrent stiffness and displacement stiffness, iax(t),ibx(t),iay(t),iby(t) four pairs of radial magnetic bearing coil currents, xa(t),xb(t),ya(t),yb(t) displacement under a bearing coordinate system; subscripts a, b denote the A, B ends of the rotor system;
because the mass unbalance makes geometric center and center of mass misalignment, displacement sensor measures as rotor geometric center displacement, has following relation:
X(t)＝x(t)+Θx(t)
Y(t)＝y(t)+Θy(t)
wherein X (t), Y (t) respectively represent the displacement of the geometric center, and are obtained by the geometric relationship:
Θx(t),Θy(t) represents the amount of unbalance and has the following formFormula (II):
Θx(t)＝ecos(Ωt+χ)
Θy(t)＝esin(Ωt+χ)
e is the magnitude of the unbalance, χ is the initial phase, and Ω is the rotor rotational speed. It can be seen that the unbalanced mass can generate co-frequency interference signals, so that the active magnetic bearing generates co-frequency vibration force.
Because the displacement sensor has sensor harmonic waves, the output signals of the four pairs of displacement sensors are not the real displacement signals of the rotor, and the displacement sensors provide the following signals:
xas(t),xbs(t),yas(t),ybs(t) is the displacement sensor output signal; daxs(t),dbxs(t),days(t),dbys(t) is the sensor harmonic interference signal, which can be expressed as follows:
wherein s isai,sbiIs the harmonic amplitude of the sensor, χiThe initial phase is I, the harmonic order is I, and the sensor harmonic can generate an interference signal with the same frequency and frequency multiplication with the rotating speed, so that the magnetic bearing system generates harmonic vibration force.
Step (2) designs a magnetic suspension rotor harmonic vibration force suppression method based on a multi-synchronous rotation coordinate transformation method
The controller takes the same-frequency vibration force and frequency multiplication current as input, is connected into an original closed loop system in a parallel mode, and the output of the controller is fed back to the power amplifier input end of an original control system, and the design of the module mainly comprises the following two aspects:
the method for transforming the multi-synchronous rotation coordinate comprises the following steps: according to different forms of the same-frequency and frequency-doubled vibration force generated by an actual magnetic suspension rotor system, the same-frequency vibration comprises a current stiffness force and a displacement stiffness force, and the frequency-doubled vibration only comprises the current stiffness force; according to a model of system electromagnetic force, current and displacement are used for constructing same-frequency vibration force to be used as input of a multiple synchronous rotation coordinate transformation same-frequency force suppression part, and frequency multiplication vibration can be suppressed by suppressing frequency multiplication current, so that the current is used as input of the frequency multiplication force suppression part;
theoretical analysis and proof show that the stable condition of the system with multi-synchronous rotation coordinate transformation is obtained; and designing a corresponding phase compensation angle according to the closed-loop characteristic of the actual magnetic suspension rotor system, and realizing the absolute stability of the system in the working rotating speed range through the phase compensation angle.
Further, the vibration force suppression algorithm in the step (2) is as follows:
the synchronous rotating coordinate transformation for realizing harmonic suppression mainly comprises three parts, namely, a disturbing signal is transformed from a static coordinate system to a rotating coordinate system through synchronous rotating coordinate transformation, at the moment, a same frequency/frequency multiplication disturbing signal is represented as a direct current quantity under the rotating coordinate system, then the direct current quantity is identified through low-pass filtering, and finally, the same frequency/frequency multiplication disturbing signal under the static coordinate system is obtained through synchronous rotating coordinate inverse transformation. The output is fed back to a closed loop structure formed by the output end of the power amplifier, so that the same frequency/frequency multiplication vibration force can be effectively inhibited.
Because the phases of all frequency bands of the system are inconsistent, a phase compensation angle needs to be introduced for phase compensation in order to ensure absolute stability of the system within the full working frequency band range. The transformation equation of the synchronous rotation coordinate with the phase compensation angle is as follows:
wherein u is1(t),u2(t) is the input signal, udc1(t),udc2(t) is a synchronous rotation coordinate transformation output signal; omega is the rotational speed of the rotor and,
is a phase compensation angle for ensuring closed loop stability, i is a positive integer, and is synchronously rotated when the harmonic frequency of the input signal is i times of omegaConverting the coordinate into a direct current signal, outputting the direct current signal, extracting harmonic components in the original signal through low-pass filtering, and setting a low-pass filter to have the following transfer function:
where k is the low pass filter gain factor, ωcIs the cut-off frequency;
suppose the signal after low-pass filtering is
Let udc1(t),udc2(t),
Respectively u as a laplace transformdc1(s),udc2(s),
The following equation holds true:
the above formula is rewritten as follows:
inverse laplace transform of the above equation can yield the following differential equation:
let k be k ωcThe above equation is written in matrix form as follows:
the low-pass filtered signal needs to be reversely transformed to the original stationary coordinate system by the synchronous rotating coordinate, and the transformation relation is expressed as follows:
x1(t),x2(t) is the output signal after the synchronous rotation coordinate inverse transformation;
from the above equations, the following state space expressions can be solved:
the transfer function matrix is derived from the state space expression:
the corresponding input and output relationship is as follows:
u1(t),u2(t) are the orthogonal sinusoidal signals, assuming the form:
Amand γ represents the amplitude and initial phase of the signal, respectively;
from the above formula, the following relationship exists:
iΩu1(s)＝su2(s)
an equivalent single input transfer function can be derived from the above equation:
when s is j ω, it can be found that when ω is i Ω, there are:
when ω iscAnd k, the synchronous rotation coordinate transformation can generate large gain at harmonic frequency, and the harmonic component can be effectively inhibited after the synchronous rotation coordinate transformation and the original system form a closed loop. Thereby eliminating the harmonic vibration forces of the system.
The basic principle of the invention is as follows: the magnetic suspension control moment gyroscope is supported by a magnetic suspension bearing, and for a magnetic suspension rotor, the main vibration sources are mass unbalance and sensor harmonic waves. Due to mass unbalance and the existence of sensor harmonic waves, harmonic vibration force is contained in the magnetic suspension rotor system. The harmonic vibration force is transmitted to the spacecraft through the base, and the performance of the spacecraft platform is seriously influenced. The invention provides a harmonic vibration force suppression algorithm based on a multi-synchronous rotation coordinate transformation method by establishing a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic aiming at the magnetic suspension rotor harmonic vibration force of a magnetic suspension control moment gyroscope. The orthogonal characteristic of the output signals of the displacement sensors in the X direction and the Y direction is utilized, the controller is used for simultaneously inhibiting the vibration force in the two directions, a plurality of algorithms with different frequencies are connected in parallel, the inhibition of common-frequency and frequency-doubling vibration is realized, a phase compensation angle is introduced, and the stability of the system in a full rotating speed range is ensured by changing the phase angle in different frequency bands.
Compared with the prior art, the invention has the advantages that:
1) most of the traditional magnetic suspension rotor vibration suppression methods are zero current control, most of vibration force is suppressed by suppressing harmonic current, but the suppression of harmonic current can only suppress current stiffness force, and the participated displacement stiffness force cannot be suppressed. The invention uses current and displacement to construct vibration force as input signal, and can realize the suppression of all vibration force.
2) The invention utilizes the orthogonal characteristic of the output signal of the displacement sensor, and uses one controller to simultaneously realize the vibration suppression in two directions, thereby reducing the system computing resource and improving the dynamic response speed. And different phase angle sizes are selected in different frequency sections through the phase compensation angle, so that the stability of the system in a full rotating speed range can be ensured.
Drawings
FIG. 1 is a flow chart of the present invention;
fig. 2 is a schematic structural diagram of a magnetic suspension rotor system, wherein 1 is an active magnetic bearing, 2 is a rotor, 3 is a geometric axis of the rotor, and 4 is an inertial axis of the rotor;
FIG. 3 is a block diagram of a basic control system of a magnetic levitation rotor;
FIG. 4 is a schematic diagram of a principle of synchronous selective coordinate transformation, wherein 5 is a sensor probe, and 2 is a rotor;
FIG. 5 is a schematic block diagram of a synchronous selective coordinate transformation algorithm;
FIG. 6 is a block diagram of a multi-synchronous rotational coordinate transformation method and a master controller compound control system;
FIG. 7 is a simplified block diagram of an equivalent single-input synchronous rotating coordinate transformation method and a master controller compound control system.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments, and all other embodiments obtained by a person skilled in the art based on the embodiments of the present invention belong to the protection scope of the present invention without creative efforts.
According to an embodiment of the present invention, as shown in fig. 1, an implementation process of a magnetic suspension rotor harmonic vibration force suppression method based on a multi-synchronous rotation coordinate transformation method is as follows: firstly, establishing a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic waves; and then designing a controller based on a multi-synchronous rotation coordinate transformation method to suppress harmonic vibration force.
Step (1) establishing a magnetic suspension rotor dynamic model containing mass unbalance and sensor harmonic
The invention is applied to an active magnetic bearing system in a magnetic suspension control moment gyro. The structural schematic diagram is shown in fig. 2, wherein N is the geometric center of the stator, NXY is the inertial coordinate system, C and O are the mass center and the geometric center of the rotor, respectively, and O epsilon η is the rotational coordinate system. The method mainly aims at the suppression of the vibration force of two degrees of freedom of radial translation, so that only the modeling of two degrees of freedom of translation is considered. From newton's second law, the following kinetic equation is obtained:
where m is the rotor mass, X (t), Y (t) represent the translational displacements of the rotor center of mass in the X and Y directions, respectively, fx(t),fy(t) resultant forces of bearing forces in X-direction and Y-direction, respectively, fax(t),fbx(t),fay(t),fby(t) is the bearing force of four pairs of radial magnetic bearings (A, B two pairs at each end, only one pair being visible at each end of A, B, only one pair being shown in the figure). When the rotor moves in a small displacement, the nonlinear bearing force can be approximately linearized and can be expressed as follows:
wherein Ki,KhCurrent stiffness and displacement stiffness, iax(t),ibx(t),iay(t),iby(t) four pairs of radial magnetic bearing coil currents, xa(t),xb(t),ya(t),ybAnd (t) is the displacement of the bearing coordinate system.
As can be seen from fig. 2, since the mass imbalance causes the geometric center to be misaligned with the mass center, the displacement sensor measures the displacement of the geometric center of the rotor, which has the following relationship:
X(t)＝x(t)+Θx(t)
Y(t)＝y(t)+Θy(t)
wherein X (t), Y (t) respectively represent the displacement of the geometric center, and are obtained by the geometric relationship:
Θx(t),Θy(t) represents the amount of unbalance, and has the following form:
Θx(t)＝ecos(Ωt+χ)
Θy(t)＝esin(Ωt+χ)
e is the magnitude of the unbalance, χ is the initial phase, and Ω is the rotor rotational speed. It can be seen that the unbalanced mass can generate co-frequency interference signals, so that the active magnetic bearing generates co-frequency vibration force.
Because the displacement sensor has sensor harmonic waves, the output signals of the four pairs of displacement sensors are not the real displacement signals of the rotor, and the displacement sensors provide the following signals:
xas(t),xbs(t),yas(t),ybs(t) is the displacement sensor output signal; daxs(t),dbxs(t),days(t),dbys(t) is the sensor harmonic interference signal, which can be expressed as follows:
wherein s isai,sbiIs the harmonic amplitude of the sensor, χiIs the initial phase, i is the harmonic order, the sensor harmonic will generate the interference of the same frequency and frequency multiplication with the rotation speedThe signal makes the magnetic bearing system generate harmonic vibration force. The control principle of a magnetic levitation rotor system with mass unbalance and sensor harmonics is shown in FIG. 3, where Gc(s),Gw(s) denotes the controller transfer function and the power amplifier transfer function, respectively, KsRepresenting the sensor gain factor.
Step (2): magnetic suspension rotor harmonic vibration force suppression method based on multi-synchronous rotation coordinate transformation method
The controller takes the same-frequency vibration force and frequency multiplication current as input, is connected into an original closed loop system in a parallel mode, and the output of the controller is fed back to the power amplifier input end of an original control system, and the design of the module mainly comprises the following two aspects:
a Multi-Synchronous rotation coordinate Transformation method (Multi-Synchronous rotation Frame Transformation, MSRFT): according to different forms of the same-frequency and frequency-doubled vibration force generated by an actual magnetic suspension rotor system, the same-frequency vibration comprises a current stiffness force and a displacement stiffness force, and the frequency-doubled vibration only comprises the current stiffness force; according to a model of system electromagnetic force, current and displacement are used for constructing same-frequency vibration force to be used as input of a multiple synchronous rotation coordinate transformation same-frequency force suppression part, and frequency multiplication vibration can be suppressed by suppressing frequency multiplication current, so that the current is used as input of the frequency multiplication force suppression part;
theoretical analysis and proof show that the stable condition of the system with multi-synchronous rotation coordinate transformation is obtained; and designing a corresponding phase compensation angle according to the closed-loop characteristic of the actual magnetic suspension rotor system, and realizing the absolute stability of the system in the working rotating speed range through the phase compensation angle.
Further, the vibration force suppression algorithm in the step (2) is as follows:
the synchronous rotating coordinate transformation for realizing harmonic suppression mainly comprises three parts, namely, a disturbing signal is transformed from a static coordinate system to a rotating coordinate system through synchronous rotating coordinate transformation, at the moment, a same frequency/frequency multiplication disturbing signal is represented as a direct current quantity under the rotating coordinate system, then the direct current quantity is identified through low-pass filtering, and finally, the same frequency/frequency multiplication disturbing signal under the static coordinate system is obtained through synchronous rotating coordinate inverse transformation. The schematic diagram is shown in FIG. 4M is the geometric center of the rotor, C is the center of mass, epsilon is the vibration quantity, the rotor rotates around the center of mass due to the self-centering principle when rotating at high speed, and CXsYsBeing a stationary coordinate system, CXrYrAs a rotating coordinate system, xs,ysIs a vibration quantity coordinate, x, of a stationary coordinate systemr,yrThe vibration amount coordinate under the rotating coordinate is a constant flow amount, and the harmonic component can be processed by low-pass filtering in the rotating coordinate system and synchronous rotating coordinate inverse transformation. The output is fed back to a closed loop structure formed by the output end of the power amplifier, so that the same frequency/frequency multiplication vibration force can be effectively inhibited.
1. Synchronous rotating coordinate transformation algorithm analysis
Because the phases of all frequency bands of the system are inconsistent, a phase compensation angle needs to be introduced for phase compensation in order to ensure absolute stability of the system within the full working frequency band range. As shown in fig. 5, the synchronous rotating coordinate transformation equation with the phase compensation angle is as follows:
wherein u is1(t),u2(t) is the input signal, udc1(t),udc2(t) is a synchronous rotation coordinate transformation output signal; omega is the rotational speed of the rotor and,
the phase compensation angle is used for ensuring the stability of a closed loop, when the harmonic frequency of an input signal is k times of omega, the input signal is output as a direct current signal after synchronous rotating coordinate transformation, the harmonic component in the original signal can be extracted through low-pass filtering, and a low-pass filter is provided with the following transfer function:
where k is the low pass filter gain factor, ωcIs the cut-off frequency;
suppose the signal after low-pass filtering is
Order to
Respectively u as a laplace transformdc1(s),udc2(s),
The following equation holds true:
the above formula is rewritten as follows:
inverse laplace transform of the above equation can yield the following differential equation:
let k be k ωcThe above equation is written in matrix form as follows:
the low-pass filtered signal needs to be reversely transformed to the original stationary coordinate system by the synchronous rotating coordinate, and the transformation relation is expressed as follows:
x1(t),x2(t) is the inverse of the synchronized rotation coordinateA transformed output signal;
from the above equations, the following state space expressions can be solved:
the transfer function matrix is derived from the state space expression:
the corresponding input and output relationship is as follows:
u1(t),u2(t) are the orthogonal sinusoidal signals, assuming the form:
Amand γ represents the amplitude and initial phase of the signal, respectively;
from the above formula, the following relationship exists:
iΩu1(s)＝su2(s)
an equivalent single input transfer function can be derived from the above equation:
when s is j ω, it can be found that when ω is i Ω, there are:
when ω iscK, can be regarded as synchronous rotation coordinatesThe transformation generates a large gain at the harmonic frequency, and the effective suppression of harmonic components can be realized after the transformation and the original system form a closed loop.
As shown in fig. 6, the co-frequency vibration force and the frequency doubling current are used as the input of the multi-synchronous rotation coordinate transformation, the output is fed back to the input end of the power amplifier and added with the control signal of the main controller, and the harmonic vibration force of the system is eliminated.
2. Stability analysis
Because the X direction and the Y direction have symmetrical structures and the same parameters, the stability analysis is carried out by taking the X direction as an example, the stability analysis idea of the invention is that the insertion of each controller is based on a first stabilization system, namely the insertion of the same-frequency vibration force algorithm is based on the stability of the original system, and the insertion of the frequency doubling suppression algorithm is based on the system stability of the adding same-frequency suppression algorithm. The invention only carries out expansion analysis on the stability condition of the same-frequency suppression algorithm insertion system, and the frequency multiplication algorithm is similar when being inserted, and the simplified block diagram is shown in figure 7.
The closed-loop characteristic polynomial of the system obtained from fig. 7 is:
1+2KsKiGc(s)Gw(s)P(s)-2KhP(s)+2KiGw(s)GSRFx(s)＝0
the above equation can be converted into:
wherein the content of the first and second substances,
is a system function.
When k is 0, s is- ωc+ -j omega, when k → 0 there is a closed loop pole at-omegacMoving in the ± j Ω region, in order to ensure a sufficient stability margin of the system, k is used as an independent variable and s is used as a dependent variable, and the differential is obtained when k is 0 and s is j Ω, thereby obtaining the following formula:
to ensure that the closed-loop eigenfunctions all follow the s-left half-plane after the algorithm is added to the system, the following conditions need to be satisfied:
because of omegacOmega, so
I.e. the above formula is about
The following stability conditions can be obtained:
the phase compensation angle is properly adjusted according to the phases of the system function at different frequencies, so that the system can be stable in the full working rotating speed range.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, but various changes may be apparent to those skilled in the art, and it is intended that all inventive concepts utilizing the inventive concepts set forth herein be protected without departing from the spirit and scope of the present invention as defined and limited by the appended claims.
Claims (2)
1. A magnetic suspension rotor harmonic vibration force suppression method based on a multi-synchronous coordinate transformation method is characterized by comprising the following steps: the method comprises the following steps:
step (1): establishing a magnetic suspension rotor dynamics model containing mass unbalance and sensor harmonic
For an active magnetic bearing system in a magnetic suspension control moment gyroscope, setting N as a geometric center of a stator, NXY as an inertial coordinate system, C and O as a mass center and a geometric center of a rotor respectively, and O epsilon eta as a rotating coordinate system, and carrying out modeling of two translational degrees of freedom aiming at the suppression of the vibration force of the two translational degrees of freedom in radial direction; from newton's second law, the following kinetic equation is obtained:
where m is the rotor mass, X (t), Y (t) represent the translational displacements of the rotor center of mass in the X and Y directions, respectively, fx(t),fy(t) resultant forces of bearing forces in X-direction and Y-direction, respectively, fax(t),fbx(t),fay(t),fby(t) is the bearing force of the four pairs of radial magnetic bearings, and the nonlinear bearing force is approximately linearized when the rotor performs a displacement motion less than a predetermined threshold as follows:
wherein Ki,KhCurrent stiffness and displacement stiffness, iax(t),ibx(t),iay(t),iby(t) four pairs of radial magnetic bearing coil currents, xa(t),xb(t),ya(t),yb(t) displacement under a bearing coordinate system; subscripts a, b denote the A, B ends of the rotor system;
because the mass unbalance makes geometric center and center of mass misalignment, displacement sensor measures as rotor geometric center displacement, has following relation:
X(t)＝x(t)+Θx(t)
Y(t)＝y(t)+Θy(t)
wherein X (t), Y (t) respectively represent the displacements of the geometric center in the X direction and the Y direction, and can be obtained by the geometric relationship:
Θx(t),Θy(t) represents the amount of unbalance, and has the following form;
Θx(t)＝ecos(Ωt+χ)
Θy(t)＝esin(Ωt+χ)
e is the amplitude of unbalance, chi is the initial phase, omega is the rotation speed of the rotor, and the mass unbalance can generate a same-frequency interference signal, so that the active magnetic bearing can generate a same-frequency vibration force;
because the displacement sensor has sensor harmonic waves, the output signals of the four pairs of displacement sensors are not the real displacement signals of the rotor, and the displacement sensors provide the following signals:
xas(t),xbs(t),yas(t),ybs(t) is the displacement sensor output signal; daxs(t),dbxs(t),days(t),dbys(t) is the sensor harmonic interference signal, expressed as follows:
wherein s isai,sbiIs the harmonic amplitude of the sensor, χiThe sensor harmonic can generate an interference signal with the same frequency and frequency multiplication with the rotating speed, so that a magnetic bearing system generates harmonic vibration force;
step (2): harmonic vibration force suppression method based on multi-synchronous rotation coordinate transformation
Designing a controller, wherein the controller takes the same-frequency vibration force and frequency multiplication current as input, is connected into an original closed-loop system in a parallel mode, and outputs and feeds back the output to the power amplifier input end of the original control system, and the controller comprises the following two aspects:
the method for transforming the multi-synchronous rotation coordinate comprises the following steps: according to different forms of the same-frequency and frequency-doubled vibration force generated by an actual magnetic suspension rotor system, the same-frequency vibration comprises a current stiffness force and a displacement stiffness force, and the frequency-doubled vibration only comprises the current stiffness force; according to a model of system electromagnetic force, current and displacement are used for constructing same-frequency vibration force to be used as input of a multiple synchronous rotation coordinate transformation same-frequency force suppression part, and frequency multiplication vibration can be suppressed by suppressing frequency multiplication current, so that the current is used as input of the frequency multiplication force suppression part;
secondly, calculating to obtain the stable condition of the system with multi-synchronous rotation coordinate transformation; and designing a corresponding phase compensation angle according to the closed-loop characteristic of the actual magnetic suspension rotor system, and realizing the absolute stability of the system in the working rotating speed range through the phase compensation angle.
2. The method for suppressing the harmonic vibration force of the magnetic suspension rotor based on the multi-synchronous coordinate transformation method as claimed in claim 1, wherein: the magnetic suspension rotor harmonic vibration force suppression algorithm in the step (2) is as follows:
the synchronous rotating coordinate transformation for realizing harmonic suppression mainly comprises three parts, namely a disturbance signal is transformed from a static coordinate system to a rotating coordinate system through synchronous rotating coordinate transformation, at the moment, a same frequency/frequency multiplication interference signal is represented as a direct current quantity under the rotating coordinate system, then the direct current quantity is identified through low-pass filtering, finally, the same frequency/frequency multiplication interference signal under the static coordinate system is obtained through synchronous rotating coordinate inverse transformation, and the output is fed back to a closed loop structure formed by the output end of a power amplifier for suppressing the same frequency/frequency multiplication vibration force;
for the inconsistency of the phases of all frequency bands of the system, a phase compensation angle is introduced for phase compensation, so that the absolute stability of the system in the full working frequency band range is ensured, and a synchronous rotating coordinate transformation equation with the phase compensation angle is as follows:
wherein u is1(t),u2(t) is the input signal, udc1(t),udc2(t) is a synchronous rotation coordinate transformation output signal; omega is the rotational speed of the rotor and,
the phase compensation angle is used for ensuring the stability of a closed loop, i is a positive integer, when the harmonic frequency of an input signal is i times of omega, the input signal is output as a direct current signal after synchronous rotating coordinate transformation, harmonic components in an original signal can be extracted through low-pass filtering, and a low-pass filter is provided with the following transfer functions:
where k is the low pass filter gain factor, ωcIs the cut-off frequency;
suppose the signal after low-pass filtering is
And make udc1(t),udc2(t),
Respectively u as a laplace transformdc1(s),udc2(s),
The following equation holds true:
the above formula is rewritten as follows:
inverse laplace transform of the above equation can yield the following differential equation:
let k be k ωcThe above equation is written in matrix form as follows:
the low-pass filtered signal needs to be reversely transformed to the original stationary coordinate system by the synchronous rotating coordinate, and the transformation relation is expressed as follows:
x1(t),x2(t) is the output signal after the synchronous rotation coordinate inverse transformation;
the following state space expression is solved from the above equations:
the transfer function matrix is derived from the state space expression:
the corresponding input and output relationship is as follows:
u1(t),u2(t) are orthogonal sinusoidal signals, provided thatThe following forms:
Amand γ represents the amplitude and initial phase of the signal, respectively;
from the above formula, the following relationship exists:
iΩu1(s)＝su2(s)
an equivalent single input transfer function can be derived from the above equation:
when s is j ω, it can be found that when ω is i Ω, there are:
when ω iscAnd < kappa >, the synchronous rotation coordinate transformation is considered to generate a preset gain at the harmonic frequency, and the harmonic component can be effectively inhibited after the synchronous rotation coordinate transformation and the original system form a closed loop, so that the harmonic vibration force of the system is eliminated.
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