CN105446141A - Disk-shaped magnetic suspension rotor system dynamics modeling method and coupling dynamics equation set - Google Patents
Disk-shaped magnetic suspension rotor system dynamics modeling method and coupling dynamics equation set Download PDFInfo
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- CN105446141A CN105446141A CN201510974726.XA CN201510974726A CN105446141A CN 105446141 A CN105446141 A CN 105446141A CN 201510974726 A CN201510974726 A CN 201510974726A CN 105446141 A CN105446141 A CN 105446141A
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
Abstract
The invention relates to a disk-shaped magnetic suspension rotor system dynamics modeling method and a coupling dynamics equation set. The disk-shaped magnetic suspension rotor system dynamics modeling method includes the following steps that: step S1, the geometric model of a disk-shaped magnetic suspension rotor system is established according to the stressed conditions of the disk-shaped magnetic suspension rotor system; and step S2, the dynamics model of the disk-shaped magnetic suspension rotor system is established according to the geometric model. According to the disk-shaped magnetic suspension rotor system dynamics modeling method and coupling dynamics equation set provided by the invention, in the dynamics model, influence of factors such as the types of a magnetic suspension rotor and the directions of basic movement is considered, and therefore, the accuracy of the disk-shaped magnetic suspension rotor system can be improved.
Description
Technical field
The present invention relates to a kind of dynamic modeling method and the coupled dynamical equation group of plate-like magnetic suspension rotor system.
Background technology
Along with the development of magnetic levitation technology, be applied in increasing, such as vehicle-mounted flying wheel battery, submarine vibration and noise reducing, wind power generation etc.Be applied in the basis of the magnetic bearing of these occasions itself also in motion, and the magnetic suspension rotor bearing system dynamics model usually set up at present, all suppose that the bearing seat at two ends does not move, and to be in fact normally non-rigidly connected between magnetic bearing bearing (basis) with the earth, motion between them intercouples, influences each other, thus in structure with kinetically constitute bearing-rotor-basic coupled system.Because the installation quality of machine and long-term vibration will cause loosening between bearing seat and stator foundation, when the larger out-of-balance force produced when machine at high speeds rotates has exceeded the gravity of bearing seat, bearing seat will be lifted by the cycle, produce great vibrations, and quiet the touching of rotor can be caused to rub, so the dynamic behavior of research bearing-rotor-basic system is significant.At present about in the kinetic model research of bearing-rotor-basic system, mainly for the conventional mechanical such as sliding bearing and rolling bearing bearing or the modeling method of axle class rigid rotor system for magnetic bearing supporting, along with magnetic suspension shaft class rotor transverse bearing between the reduction of length of support and the increase of rotor radial size, when the axially mounting distance of magnetic suspension rotor is reduced to a certain degree, namely so-called plate-like magnetic suspension rotor is become, therefore, the modeling method for the disc-like rotor system of magnetic bearing supporting is also rare.
Summary of the invention
The object of this invention is to provide a kind of dynamic modeling method and control method of plate-like magnetic suspension rotor system, for solving the larger technical matters of model error that existing magnetic suspension rotor system easily causes for axle class rotor in Dynamic Modeling process.
In order to solve the problems of the technologies described above, the invention provides a kind of modeling method of plate-like magnetic suspension rotor system kinetic model, comprising the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model.
Further, described plate-like magnetic suspension rotor system comprises: plate-like magnetic suspension rotor, on supporting base, and same circumferentially equally distributed three magnetic bearing M
1, M
2, M
3and three eddy current displacement sensor S
1, S
2, S
3; According to the stressing conditions of plate-like magnetic suspension rotor system in described step S1, the method setting up the geometric model of plate-like magnetic suspension rotor system comprises:
Step S11, basic assumption, namely suppose that plate-like magnetic suspension rotor itself is rigid body, during the vibration of plate-like magnetic suspension rotor, angular displacement is very little, and the bearing of magnetic bearing itself is rigid body, and the bearing of magnetic bearing only exists the translation of vertical direction and horizontal direction;
Step S12, sets up three-dimensional coordinate system; Namely, true origin overlaps with plate-like magnetic suspension rotor barycenter, six-freedom degree is there is in plate-like magnetic suspension rotor in space, translation along z-axis and the rotation around x, y-axis are by three Active Magnetic Bearing Control, along 2 degree of freedom of x, y-axis translation by the centripetal effect force constraint of electromagnetic field, the degree of freedom of rotating around z-axis does not retrain;
Use z
s, θ
x, θ
ydescribe plate-like magnetic suspension rotor respectively along the translation of z-axis and the rotation around x, y-axis, work as θ
x, θ
yenough hour, then cos θ
x≌ 1, sin θ
x≌ θ
x, cos θ
y≌ 1, sin θ
y≌ θ
y.
Further, the method setting up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model in described step S2 comprises the steps:
Step S21, the basic dynamic equations that utilization Lagrange's equation sets up plate-like magnetic suspension rotor system is as follows:
In above formula (1), f
zfor the electromagnetic force in the z-direction that magnetic bearing produces, m
xfor the moment around x-axis that electromagnetic force produces, m
yfor the moment around y-axis that electromagnetic force produces, f
z dfor the external interference power in z direction, m
x dfor the disturbance torque around x-axis, m
y dfor the disturbance torque around y-axis;
Step S22, according to the stressing conditions of plate-like magnetic suspension rotor, magnetic bearing produce electromagnetism in the z-direction make a concerted effort, the moment around x direction and the relational expression between the moment around y direction and three electromagnetic forces that magnetic bearing produces as follows:
In above formula (2), f
1, f
2, f
3be respectively three magnetic bearing M
1, M
2, M
3the electromagnetic force produced, near equilibrium position, electromagnetic force being carried out linearization can obtain: f
k=k
ii
k+ k
xx
k, in formula: k
ifor the power-current coefficient of magnetic bearing, k
xfor the power-displacement coefficient of magnetic bearing, i
kfor controlling electric current, x
kfor electromagnet in magnetic bearing is to the displacement of plate-like magnetic suspension rotor, and the value of k is corresponding with magnetic bearing or eddy current displacement sensor, namely gets 1,2,3 respectively;
Step S23, by the geometric relationship of plate-like magnetic suspension rotor in space, can obtain electromagnet to the displacement of plate-like magnetic suspension rotor and the relational expression of disc-like rotor between locus:
I
1, i
2, i
3with x
1, x
2, x
3between relational matrix B calculated by the control system of plate-like magnetic suspension rotor, and to be designated as
Order
the kinetics equation of plate-like magnetic suspension rotor can be obtained, as follows:
M=diag (m, J in above formula (4)
x, J
y), m is the quality of plate-like magnetic suspension rotor, J
x, J
ybe respectively the moment of inertia of plate-like magnetic suspension rotor around x-axis, y-axis, q is the state variable of the plate-like magnetic suspension rotor of definition, q
bthe state variable of the supporting base of the plate-like magnetic suspension rotor system of definition, F
dit is the external interference moment battle array of plate-like magnetic suspension rotor system;
A is the relational matrix of the electromagnetic force of disc-like rotor Moment and magnetic bearing, namely
c is that electromagnet arrives the displacement of plate-like magnetic suspension rotor and disc-like rotor at spatial relation matrix, namely
Further, the method that the control system of described plate-like magnetic suspension rotor calculates relational matrix B comprises:
Step S231, described control system adopts PID controller, and its transport function is:
In above formula (5), K
p, K
i, K
dbe respectively the scale-up factor of PID controller, integral coefficient, differential coefficient, T
dfor the damping time constant of PID controller differentiation element, its corresponding differential equation is:
Step S232, sets up the differential equation of described control system intermediate power amplifier, namely
The transport function of power amplifier is reduced to first order inertial loop:
In formula: A
afor the gain T of power amplifier
afor the damping time constant of power amplifier;
Above formula (7) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (8), Uout is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system;
Step S233, sets up the differential equation of eddy current displacement sensor, and namely the transport function of displacement transducer is also reduced to first order inertial loop:
In above formula (9): A
sfor the gain of eddy current displacement sensor; T
sfor the damping time constant of eddy current displacement sensor; Above formula (9) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (10), q=(z
sθ
xθ
y) ' the be displacement vector at disc-like rotor barycenter place; ; L
sBbecause sensor and magnetic bearing non-concurrent point are installed and the coupled matrix of introducing;
Described relational matrix B is B=G
s(s) G
c(s) G
a(s) L
sB -1.
Further, described coupled matrix L
sBpreparation method as follows:
If eddy current displacement sensor and the radius of a circle residing for magnetic bearing are a, and draw the coordinate of the axial line of each eddy current displacement sensor and magnetic bearing in described three-dimensional coordinate system, namely
S
1：(-asin30°，-acos30°，0)
S
2：(-asin30°，acos30°，0)(10)
S
3：(a，0，0)
M
1：(-a，0，0)
M
2：(asin30°，acos30°，0)
M
3：(asin30°，-acos30°，0)(11)
If C
1, C
2, C
3for the point of 3 on plate-like magnetic suspension rotor, its projection on the x-y plane overlaps with the axial line of three eddy current displacement sensors respectively; δ
1, δ
2, δ
3be respectively plate-like magnetic suspension rotor that three eddy current displacement sensors measure along eddy current displacement sensor axial line to corresponding eddy current displacement sensor between distance, i.e. the measured value of eddy current displacement sensor, to obtain C on plate-like magnetic suspension rotor
1, C
2, C
3the coordinate of point in described three-dimensional coordinate system, namely
C
1：(-asin30°，-acos30°，δ
1)
C
2：(-asin30°，acos30°，δ
2)(12)
C
3：(a，0，δ
3)
The coordinate supposing any known point on certain moment plate-like magnetic suspension rotor is (x
0, y
0, z
0), plate-like magnetic suspension rotor law vector is { A ', B ', C ' }, then the equation of motion of plate-like magnetic suspension rotor is this moment:
A′(x-x
0)+B′(y-y
0)+C′(z-z
0)＝0(13)；
By C
1, C
2, C
3substitute into and can obtain into equation (13):
The homogeneous equation group about A ', B ', C ' be made up of formula (14) has the condition of untrivialo solution to be:
The equation of plate-like magnetic suspension rotor plane can be obtained by (15) formula:
Obtain the spatiality of plate-like magnetic suspension rotor, with obtain further 3 magnetic bearing place plate-like magnetic suspension rotors along magnetic bearing axial line to magnetic bearing between distance, and by this distance by the coordinate figure of the x-y plane residing for magnetic bearing, namely in formula (11), the value of x, y substitutes into formula (11) and tries to achieve corresponding z coordinate value, namely
Obtain:
And obtain:
and
Obtain plate-like magnetic suspension rotor along magnetic bearing axial line to magnetic bearing between distance Z
mk, to derive the control current i of arbitrary magnetic bearing
k.
And then obtain described relational matrix B.
Another aspect, present invention also offers the electromechanical Coupled Dynamics system of equations of a kind of magnetic bearing-plate-like magnetic suspension rotor-basic system, comprising:
The differential equation of the kinetics equation of plate-like magnetic suspension rotor, the differential equation corresponding to control system of described plate-like magnetic suspension rotor, described control system intermediate power amplifier, and the differential equation of eddy current displacement sensor.
Further, the method for building up of described electromechanical Coupled Dynamics system of equations comprises the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model; And
Step S3, obtains described electromechanical Coupled Dynamics system of equations.
Further, described plate-like magnetic suspension rotor system comprises: plate-like magnetic suspension rotor, on supporting base, and same circumferentially equally distributed three magnetic bearing M
1, M
2, M
3and three eddy current displacement sensor S
1, S
2, S
3;
According to the stressing conditions of plate-like magnetic suspension rotor system in described step S1, the method setting up the geometric model of plate-like magnetic suspension rotor system comprises:
Step S11, basic assumption, namely suppose that plate-like magnetic suspension rotor itself is rigid body, during the vibration of plate-like magnetic suspension rotor, angular displacement is very little, and the bearing of magnetic bearing itself is rigid body, and the bearing of magnetic bearing only exists the translation of vertical direction and horizontal direction;
Step S12, sets up three-dimensional coordinate system; Namely, true origin overlaps with plate-like magnetic suspension rotor barycenter, six-freedom degree is there is in plate-like magnetic suspension rotor in space, translation along z-axis and the rotation around x, y-axis are by three Active Magnetic Bearing Control, along 2 degree of freedom of x, y-axis translation by the centripetal effect force constraint of electromagnetic field, the degree of freedom of rotating around z-axis does not retrain;
Use z
s, θ
x, θ
ydescribe plate-like magnetic suspension rotor respectively along the translation of z-axis and the rotation around x, y-axis, work as θ
x, θ
yenough hour, then cos θ
x≌ 1, sin θ
x≌ θ
x, cos θ
y≌ 1, sin θ
y≌ θ
y.
Further, the method setting up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model in described step S2 comprises the steps:
Step S21, the basic dynamic equations that utilization Lagrange's equation sets up plate-like magnetic suspension rotor system is as follows:
In above formula (1), f
zfor the electromagnetic force in the z-direction that magnetic bearing produces, m
xfor the moment around x-axis that electromagnetic force produces, m
yfor the moment around y-axis that electromagnetic force produces, f
z dfor the external interference power in z direction, m
x dfor the disturbance torque around x-axis, m
y dfor the disturbance torque around y-axis;
Step S22, according to the stressing conditions of plate-like magnetic suspension rotor, magnetic bearing produce electromagnetism in the z-direction make a concerted effort, the moment around x direction and the relational expression between the moment around y direction and three electromagnetic forces that magnetic bearing produces as follows:
In above formula (2), f
1, f
2, f
3be respectively three magnetic bearing M
1, M
2, M
3the electromagnetic force produced, near equilibrium position, electromagnetic force being carried out linearization can obtain: f
k=k
ii
k+ k
xx
k, in formula: k
ifor the power-current coefficient of magnetic bearing, k
xfor the power-displacement coefficient of magnetic bearing, i
kfor controlling electric current, x
kfor electromagnet in magnetic bearing is to the displacement of plate-like magnetic suspension rotor, and the value of k is corresponding with magnetic bearing or eddy current displacement sensor, namely gets 1,2,3 respectively;
Step S23, by the geometric relationship of plate-like magnetic suspension rotor in space, can obtain electromagnet to the displacement of plate-like magnetic suspension rotor and the relational expression of disc-like rotor between locus:
I
1, i
2, i
3with x
1, x
2, x
3between relational matrix B calculated by the control system of plate-like magnetic suspension rotor, and to be designated as
Order
the kinetics equation of plate-like magnetic suspension rotor can be obtained, as follows:
M=diag (m, J in above formula (4)
x, J
y), m is the quality of plate-like magnetic suspension rotor, J
x, J
ybe respectively the moment of inertia of plate-like magnetic suspension rotor around x-axis, y-axis, q is the state variable of the plate-like magnetic suspension rotor of definition, q
bthe state variable of the supporting base of the plate-like magnetic suspension rotor system of definition, F
dit is the external interference moment battle array of plate-like magnetic suspension rotor system;
A is the relational matrix of the electromagnetic force of disc-like rotor Moment and magnetic bearing, namely
c is that electromagnet arrives the displacement of plate-like magnetic suspension rotor and disc-like rotor at spatial relation matrix, namely
The method that the control system of described plate-like magnetic suspension rotor calculates relational matrix B comprises:
Step S231, described control system adopts PID controller, and its transport function is:
In above formula (5), K
p, K
i, K
dbe respectively the scale-up factor of PID controller, integral coefficient, differential coefficient, T
dfor the damping time constant of PID controller differentiation element, its corresponding differential equation is:
Step S232, sets up the differential equation of described control system intermediate power amplifier, namely
The transport function of power amplifier is reduced to first order inertial loop:
In formula: A
afor the gain T of power amplifier
afor the damping time constant of power amplifier;
Above formula (7) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (8), Uout is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system;
Step S233, sets up the differential equation of eddy current displacement sensor, and namely the transport function of displacement transducer is also reduced to first order inertial loop:
In above formula (9): A
sfor the gain of eddy current displacement sensor; T
sfor the damping time constant of eddy current displacement sensor; Above formula (9) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (10), q=(z
sθ
xθ
y) ' the be displacement vector at disc-like rotor barycenter place; L
sBbecause sensor and magnetic bearing non-concurrent point are installed and the coupled matrix of introducing;
Described relational matrix B is B=G
s(s) G
c(s) G
a(s) L
sB -1;
Described coupled matrix L
sBpreparation method as follows:
If eddy current displacement sensor and the radius of a circle residing for magnetic bearing are a, and draw the coordinate of the axial line of each eddy current displacement sensor and magnetic bearing in described three-dimensional coordinate system, namely
S
1：(-asin30°，-acos30°，0)
S
2：(-asin30°，acos30°，0)(10)
S
3：(a，0，0)
M
1：(-a，0，0)
M
2：(asin30°，acos30°，0)
M
3：(asin30°，-acos30°，0)(11)
If C
1, C
2, C
3for the point of 3 on plate-like magnetic suspension rotor, its projection on the x-y plane overlaps with the axial line of three eddy current displacement sensors respectively; δ
1, δ
2, δ
3be respectively plate-like magnetic suspension rotor that three eddy current displacement sensors measure along eddy current displacement sensor axial line to corresponding eddy current displacement sensor between distance, i.e. the measured value of eddy current displacement sensor, to obtain C on plate-like magnetic suspension rotor
1, C
2, C
3the coordinate of point in described three-dimensional coordinate system, namely
C
1：(-asin30°，-acos30°，δ
1)
C
2：(-asin30°，acos30°，δ
2)(12)
C
3：(a，0，δ
3)
The coordinate supposing any known point on certain moment plate-like magnetic suspension rotor is (x
0, y
0, z
0), plate-like magnetic suspension rotor law vector is { A ', B ', C ' }, then the equation of motion of plate-like magnetic suspension rotor is this moment:
A′(x-x
0)+B′(y-y
0)+C′(z-z
0)＝0(13)；
By C
1, C
2, C
3substitute into and can obtain into equation (13):
The homogeneous equation group about A ', B ', C ' be made up of formula (14) has the condition of untrivialo solution to be:
The equation of plate-like magnetic suspension rotor plane can be obtained by (15) formula:
Obtain the spatiality of plate-like magnetic suspension rotor, with obtain further 3 magnetic bearing place plate-like magnetic suspension rotors along magnetic bearing axial line to magnetic bearing between distance, and by this distance by the coordinate figure of the x-y plane residing for magnetic bearing, namely in formula (11), the value of x, y substitutes into formula (11) and tries to achieve corresponding z coordinate value, namely
Obtain:
And obtain:
and
Obtain plate-like magnetic suspension rotor along magnetic bearing axial line to magnetic bearing between distance Z
mk, to derive the control current i of arbitrary magnetic bearing
k.
And then obtain described relational matrix B.
Further, described electromechanical Coupled Dynamics system of equations is obtained in described step S3, namely
The above-mentioned formula of simultaneous (4), (6), (8) and (10), set up described electromechanical Coupled Dynamics system of equations, namely
In formula:
B＝G
s(s)G
c(s)G
a(s)L
SB -1；
The invention has the beneficial effects as follows, the invention provides plate-like magnetic suspension rotor system dynamic modeling method and the coupled dynamical equation group, in kinetic model, consider the impact of the factor such as the type of magnetic suspension rotor, the direction of foundation motion, improve the accuracy of plate-like magnetic suspension rotor system model.
Accompanying drawing explanation
Below in conjunction with drawings and Examples, the present invention is further described.
Fig. 1 is plate-like magnetic suspension rotor system structural drawing of the present invention;
Fig. 2 (a) is the coordinate diagram of whole plate-like magnetic suspension rotor system force analysis;
Fig. 2 (b) is the distribution plan of magnetic bearing and sensor relative position;
Fig. 2 (c) is the geometric model figure of plate-like magnetic suspension rotor system;
Fig. 3 is plate-like magnetic suspension rotor control system block diagram.
In figure: plate-like magnetic suspension rotor 1, supporting base 2, magnetic bearing 3, electromagnet 301, eddy current displacement sensor 4.
Embodiment
In conjunction with the accompanying drawings, the present invention is further detailed explanation.These accompanying drawings are the schematic diagram of simplification, only basic structure of the present invention are described in a schematic way, and therefore it only shows the formation relevant with the present invention.
Embodiment 1
The present embodiment 1 provides a kind of modeling method of plate-like magnetic suspension rotor system kinetic model, comprises the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model.
Concrete, as shown in Fig. 1 and Fig. 2 (a), Fig. 2 (b) He Fig. 2 (c), described plate-like magnetic suspension rotor system comprises: plate-like magnetic suspension rotor, on supporting base, and same circumferentially equally distributed three magnetic bearing M
1, M
2, M
3and three eddy current displacement sensor S
1, S
2, S
3.
Further, according to the stressing conditions of plate-like magnetic suspension rotor system in described step S1, the method setting up the geometric model of plate-like magnetic suspension rotor system comprises:
Step S11, basic assumption, namely suppose that plate-like magnetic suspension rotor itself is rigid body, during the vibration of plate-like magnetic suspension rotor, angular displacement is very little, and the bearing of magnetic bearing itself is rigid body, and the bearing of magnetic bearing only exists the translation of vertical direction and horizontal direction;
Step S12, sets up three-dimensional coordinate system; Namely, true origin overlaps with plate-like magnetic suspension rotor barycenter, six-freedom degree is there is in plate-like magnetic suspension rotor in space, translation along z-axis and the rotation around x, y-axis are by three Active Magnetic Bearing Control, along 2 degree of freedom of x, y-axis translation by the centripetal effect force constraint of electromagnetic field, the degree of freedom of rotating around z-axis does not retrain;
Use z
s, θ
x, θ
ydescribe plate-like magnetic suspension rotor respectively along the translation of z-axis and the rotation around x, y-axis, work as θ
x, θ
yenough hour, then cos θ
x≌ 1, sin θ
x≌ θ
x, cos θ
y≌ 1, sin θ
y≌ θ
y.
The method setting up the kinetic model of plate-like magnetic suspension rotor system in described step S2 according to described geometric model comprises the steps:
3 magnetic bearings, in conjunction with the mechanical construction drawing of plate-like magnetic suspension rotor system and the supporting principle of magnetic bearing, are equivalent to stiffness and damping system, foundation motion z by step S21
brepresent.In conjunction with actual conditions, compared with the foundation motion of vertical direction, the impact of horizontal foundation motion on magnetic suspension rotor system dynamics is less, therefore only the motion of supporting base in z direction is considered, its geometric model is as shown in Fig. 2 (c), and the basic dynamic equations that utilization Lagrange's equation sets up plate-like magnetic suspension rotor system is as follows:
In above formula (1), f
zfor the electromagnetic force in the z-direction that magnetic bearing produces, m
xfor the moment around x-axis that electromagnetic force produces, m
yfor the moment around y-axis that electromagnetic force produces, f
z dfor the external interference power in z direction, m
x dfor the disturbance torque around x-axis, m
y dfor the disturbance torque around y-axis;
Step S22, according to the stressing conditions of plate-like magnetic suspension rotor, magnetic bearing produce electromagnetism in the z-direction make a concerted effort, the moment around x direction and the relational expression between the moment around y direction and three electromagnetic forces that magnetic bearing produces as follows:
In above formula (2), f
1, f
2, f
3be respectively three magnetic bearing M
1, M
2, M
3the electromagnetic force produced, near equilibrium position, electromagnetic force being carried out linearization can obtain: f
k=k
ii
k+ k
xx
k, in formula: k
ifor the power-current coefficient of magnetic bearing, k
xfor the power-displacement coefficient of magnetic bearing, i
kfor controlling electric current, x
kfor electromagnet in magnetic bearing is to the displacement of plate-like magnetic suspension rotor, and the value of k is corresponding with magnetic bearing or eddy current displacement sensor, namely gets 1,2,3 respectively;
Step S23, by the geometric relationship of plate-like magnetic suspension rotor in space, can obtain electromagnet to the displacement of plate-like magnetic suspension rotor and the relational expression of disc-like rotor between locus:
I
1, i
2, i
3with x
1, x
2, x
3between relational matrix B calculated by the control system of plate-like magnetic suspension rotor, and to be designated as
Order
the kinetics equation of plate-like magnetic suspension rotor can be obtained, as follows:
M=diag (m, J in above formula (4)
x, J
y), m is the quality of plate-like magnetic suspension rotor, J
x, J
ybe respectively the moment of inertia of plate-like magnetic suspension rotor around x-axis, y-axis, q is the state variable of the plate-like magnetic suspension rotor of definition, q
bthe state variable of the supporting base of the plate-like magnetic suspension rotor system of definition, F
dit is the external interference moment battle array of plate-like magnetic suspension rotor system;
A is the relational matrix of the electromagnetic force of disc-like rotor Moment and magnetic bearing, namely
c is that electromagnet arrives the displacement of plate-like magnetic suspension rotor and disc-like rotor at spatial relation matrix, namely
The method that the control system of described plate-like magnetic suspension rotor calculates relational matrix B comprises:
As shown in Figure 3, wherein, U
othe reference voltage corresponding to plate-like magnetic suspension rotor ideal position, U
sthe voltage corresponding to plate-like magnetic suspension rotor physical location recorded, U
ethe difference of the reference voltage corresponding to plate-like magnetic suspension rotor ideal position and voltage corresponding to the plate-like magnetic suspension rotor physical location that records, U
outit is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system.
Step S231, described control system adopts PID controller, and its transport function is:
In above formula (5), K
p, K
i, K
dbe respectively the scale-up factor of PID controller, integral coefficient, differential coefficient, T
dfor the damping time constant of PID controller differentiation element, its corresponding differential equation is:
Step S232, sets up the differential equation of described control system intermediate power amplifier, namely
The transport function of power amplifier is reduced to first order inertial loop:
In formula: A
afor the gain T of power amplifier
afor the damping time constant of power amplifier;
Above formula (7) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (8), Uout is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system;
Step S233, sets up the differential equation of eddy current displacement sensor, and namely the transport function of displacement transducer is also reduced to first order inertial loop:
In above formula (9): A
sfor the gain of eddy current displacement sensor; T
sfor the damping time constant of eddy current displacement sensor; Above formula (9) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (10), q=(z
sθ
xθ
y) ' the be displacement vector at disc-like rotor barycenter place; ; L
sBbecause sensor and magnetic bearing non-concurrent point are installed and the coupled matrix of introducing;
Described relational matrix B is B=G
s(s) G
c(s) G
a(s) L
sB -1.
From Fig. 2 (b), 3 current vortex sensor S
1, S
2, S
3with the displacement that detected displacement is not magnetic bearing place plate-like magnetic suspension rotor, the displacement of each magnetic bearing place plate-like magnetic suspension rotor just must can be translated into through corresponding calculating.
Therefore, described coupled matrix L
sBpreparation method as follows:
If eddy current displacement sensor and the radius of a circle residing for magnetic bearing are a, and draw the coordinate of the axial line of each eddy current displacement sensor and magnetic bearing in described three-dimensional coordinate system, namely
S
1：(-asin30°，-acos30°，0)
S
2：(-asin30°，acos30°，0)(10)
S
3：(a，0，0)
M
1：(-a，0，0)
M
2：(asin30°，acos30°，0)
M
3：(asin30°，-acos30°，0)(11)
If C
1, C
2, C
3for the point of 3 on plate-like magnetic suspension rotor, its projection on the x-y plane overlaps with the axial line of three eddy current displacement sensors respectively; δ
1, δ
2, δ
3be respectively plate-like magnetic suspension rotor that three eddy current displacement sensors measure along eddy current displacement sensor axial line to corresponding eddy current displacement sensor between distance, i.e. the measured value of eddy current displacement sensor, to obtain C on plate-like magnetic suspension rotor
1, C
2, C
3the coordinate of point in described three-dimensional coordinate system, namely
C
1：(-asin30°，-acos30°，δ
1)
C
2：(-asin30°，acos30°，δ
2)(12)
C
3：(a，0，δ
3)
The coordinate supposing any known point on certain moment plate-like magnetic suspension rotor is (x
0, y
0, z
0), plate-like magnetic suspension rotor law vector is { A ', B ', C ' }, then the equation of motion of plate-like magnetic suspension rotor is this moment:
A′(x-x
0)+B′(y-y
0)+C′(z-z
0)＝0(13)；
By C
1, C
2, C
3substitute into and can obtain into equation (13):
The homogeneous equation group about A ', B ', C ' be made up of formula (14) has the condition of untrivialo solution to be:
The equation of plate-like magnetic suspension rotor plane can be obtained by (15) formula:
Obtain the spatiality of plate-like magnetic suspension rotor, with obtain further 3 magnetic bearing place plate-like magnetic suspension rotors along magnetic bearing axial line to magnetic bearing between distance, and by this distance by the coordinate figure of the x-y plane residing for magnetic bearing, namely in formula (11), the value of x, y substitutes into formula (11) and tries to achieve corresponding z coordinate value, namely
Obtain:
And obtain:
and
Obtain plate-like magnetic suspension rotor along magnetic bearing axial line to magnetic bearing between distance Z
mk, to derive the control current i of arbitrary magnetic bearing
k.
And then obtain described relational matrix B.
Embodiment 2
On embodiment 1 basis, the present embodiment 2 additionally provides the electromechanical Coupled Dynamics system of equations of a kind of magnetic bearing-plate-like magnetic suspension rotor-basic system, and it comprises:
The differential equation of the kinetics equation of plate-like magnetic suspension rotor, the differential equation corresponding to control system of described plate-like magnetic suspension rotor, described control system intermediate power amplifier, and the differential equation of eddy current displacement sensor.
The method for building up of described electromechanical Coupled Dynamics system of equations comprises the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model; And
Step S3, obtains described electromechanical Coupled Dynamics system of equations.
Described step S1, step S2 are see the relevant discussion of embodiment 1.
Further, described electromechanical Coupled Dynamics system of equations is obtained in described step S3, namely
The above-mentioned formula of simultaneous (4), (6), (8) and (10), set up described electromechanical Coupled Dynamics system of equations, namely
In formula:
B＝G
s(s)G
c(s)G
a(s)L
SB -1；
With above-mentioned according to desirable embodiment of the present invention for enlightenment, by above-mentioned description, relevant staff in the scope not departing from this invention technological thought, can carry out various change and amendment completely.The technical scope of this invention is not limited to the content on instructions, must determine its technical scope according to right.
Claims (10)
1. a modeling method for plate-like magnetic suspension rotor system kinetic model, is characterized in that, comprises the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model.
2. the modeling method of plate-like magnetic suspension rotor system kinetic model according to claim 1, is characterized in that,
Described plate-like magnetic suspension rotor system comprises: plate-like magnetic suspension rotor, on supporting base, and same circumferentially equally distributed three magnetic bearing M
1, M
2, M
3and three eddy current displacement sensor S
1, S
2, S
3;
According to the stressing conditions of plate-like magnetic suspension rotor system in described step S1, the method setting up the geometric model of plate-like magnetic suspension rotor system comprises:
Step S11, basic assumption, namely suppose that plate-like magnetic suspension rotor itself is rigid body, during the vibration of plate-like magnetic suspension rotor, angular displacement is very little, and the bearing of magnetic bearing itself is rigid body, and the bearing of magnetic bearing only exists the translation of vertical direction and horizontal direction;
Step S12, sets up three-dimensional coordinate system; Namely, true origin overlaps with plate-like magnetic suspension rotor barycenter, six-freedom degree is there is in plate-like magnetic suspension rotor in space, translation along z-axis and the rotation around x, y-axis are by three Active Magnetic Bearing Control, along 2 degree of freedom of x, y-axis translation by the centripetal effect force constraint of electromagnetic field, the degree of freedom of rotating around z-axis does not retrain;
Use z
s, θ
x, θ
ydescribe plate-like magnetic suspension rotor respectively along the translation of z-axis and the rotation around x, y-axis, work as θ
x, θ
yenough hour, then
3. the modeling method of plate-like magnetic suspension rotor system kinetic model according to claim 2, is characterized in that,
The method setting up the kinetic model of plate-like magnetic suspension rotor system in described step S2 according to described geometric model comprises the steps:
Step S21, the basic dynamic equations that utilization Lagrange's equation sets up plate-like magnetic suspension rotor system is as follows:
In above formula (1), f
zfor the electromagnetic force in the z-direction that magnetic bearing produces, m
xfor the moment around x-axis that electromagnetic force produces, m
yfor the moment around y-axis that electromagnetic force produces, f
z dfor the external interference power in z direction, m
x dfor the disturbance torque around x-axis, m
y dfor the disturbance torque around y-axis;
Step S22, according to the stressing conditions of plate-like magnetic suspension rotor, magnetic bearing produce electromagnetism in the z-direction make a concerted effort, the moment around x direction and the relational expression between the moment around y direction and three electromagnetic forces that magnetic bearing produces as follows:
In above formula (2), f
1, f
2, f
3be respectively three magnetic bearing M
1, M
2, M
3the electromagnetic force produced, near equilibrium position, electromagnetic force being carried out linearization can obtain: f
k=k
ii
k+ k
xx
k, in formula: k
ifor the power-current coefficient of magnetic bearing, k
xfor the power-displacement coefficient of magnetic bearing, i
kfor controlling electric current, x
kfor electromagnet in magnetic bearing is to the displacement of plate-like magnetic suspension rotor, and the value of k is corresponding with magnetic bearing or eddy current displacement sensor, namely gets 1,2,3 respectively;
Step S23, by the geometric relationship of plate-like magnetic suspension rotor in space, can obtain electromagnet to the displacement of plate-like magnetic suspension rotor and the relational expression of disc-like rotor between locus:
I
1, i
2, i
3with x
1, x
2, x
3between relational matrix B calculated by the control system of plate-like magnetic suspension rotor, and to be designated as
Order
the kinetics equation of plate-like magnetic suspension rotor can be obtained, as follows:
M=diag (m, J in above formula (4)
x, J
y), m is the quality of plate-like magnetic suspension rotor, J
x, J
ybe respectively the moment of inertia of plate-like magnetic suspension rotor around x-axis, y-axis, q is the state variable of the plate-like magnetic suspension rotor of definition, q
bthe state variable of the supporting base of the plate-like magnetic suspension rotor system of definition, F
dit is the external interference moment battle array of plate-like magnetic suspension rotor system;
A is the relational matrix of the electromagnetic force of disc-like rotor Moment and magnetic bearing, namely
C is that electromagnet arrives the displacement of plate-like magnetic suspension rotor and disc-like rotor at spatial relation matrix, namely
4. the modeling method of plate-like magnetic suspension rotor system kinetic model according to claim 3, is characterized in that,
The method that the control system of described plate-like magnetic suspension rotor calculates relational matrix B comprises:
Step S231, described control system adopts PID controller, and its transport function is:
In above formula (5), K
p, K
i, K
dbe respectively the scale-up factor of PID controller, integral coefficient, differential coefficient, T
dfor the damping time constant of PID controller differentiation element, its corresponding differential equation is:
Step S232, sets up the differential equation of described control system intermediate power amplifier, namely
The transport function of power amplifier is reduced to first order inertial loop:
In formula: A
afor the gain T of power amplifier
afor the damping time constant of power amplifier;
Above formula (7) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (8), Uout is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system;
Step S233, sets up the differential equation of eddy current displacement sensor, and namely the transport function of displacement transducer is also reduced to first order inertial loop:
In above formula (9): A
sfor the gain of eddy current displacement sensor; T
sfor the damping time constant of eddy current displacement sensor; Above formula (9) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (10), q=(Z
sθ
xθ
y) ' the be displacement vector at disc-like rotor barycenter place; ; L
sBbecause sensor and magnetic bearing non-concurrent point are installed and the coupled matrix of introducing;
Described relational matrix B is B=G
s(s) G
c(s) G
a(s) L
sB -1.
5. the modeling method of plate-like magnetic suspension rotor system kinetic model according to claim 4, is characterized in that,
Described coupled matrix L
sBpreparation method as follows:
If eddy current displacement sensor and the radius of a circle residing for magnetic bearing are a, and draw the coordinate of the axial line of each eddy current displacement sensor and magnetic bearing in described three-dimensional coordinate system, namely
S
1:(-asin30°,-acos30°,0)
S
2:(-asin30°,acos30°,0)(10)
S
3:(a,0,0)
M
1:(-a,0,0)
M
2:(asin30°,acos30°,0)
M
3:(asin30°,-acos30°,0)(11)
If C
1, C
2, C
3for the point of 3 on plate-like magnetic suspension rotor, its projection on the x-y plane overlaps with the axial line of three eddy current displacement sensors respectively; δ
1, δ
2, δ
3be respectively plate-like magnetic suspension rotor that three eddy current displacement sensors measure along eddy current displacement sensor axial line to corresponding eddy current displacement sensor between distance, i.e. the measured value of eddy current displacement sensor, to obtain C on plate-like magnetic suspension rotor
1, C
2, C
3the coordinate of point in described three-dimensional coordinate system, namely
C
1:(-asin30°,-acos30°,δ
1)
C
2:(-asin30°,acos30°,δ
2)(12)
C
3:(a,0,δ
3)
The coordinate supposing any known point on certain moment plate-like magnetic suspension rotor is (x
0, y
0, z
0), plate-like magnetic suspension rotor law vector be A', B', C'}, then the equation of motion of plate-like magnetic suspension rotor is this moment:
A'(x-x
0)+B'(y-y
0)+C'(z-z
0)＝0(13)；
By C
1, C
2, C
3substitute into and can obtain into equation (13):
The homogeneous equation group about A', B', C' be made up of formula (14) has the condition of untrivialo solution to be:
The equation of plate-like magnetic suspension rotor plane can be obtained by (15) formula:
Obtain the spatiality of plate-like magnetic suspension rotor, with obtain further 3 magnetic bearing place plate-like magnetic suspension rotors along magnetic bearing axial line to magnetic bearing between distance, and by this distance by the coordinate figure of the x-y plane residing for magnetic bearing, namely in formula (11), the value of x, y substitutes into formula (11) and tries to achieve corresponding z coordinate value, namely
Obtain:
And obtain:
and
Obtain plate-like magnetic suspension rotor along magnetic bearing axial line to magnetic bearing between distance Z
mk, to derive the control current i of arbitrary magnetic bearing
k.
And then obtain described relational matrix B.
6. an electromechanical Coupled Dynamics system of equations for magnetic bearing-plate-like magnetic suspension rotor-basic system, is characterized in that, comprising:
The differential equation of the kinetics equation of plate-like magnetic suspension rotor, the differential equation corresponding to control system of described plate-like magnetic suspension rotor, described control system intermediate power amplifier, and the differential equation of eddy current displacement sensor.
7. electromechanical Coupled Dynamics system of equations according to claim 6, is characterized in that,
The method for building up of described electromechanical Coupled Dynamics system of equations comprises the steps:
Step S1, according to the stressing conditions of plate-like magnetic suspension rotor system, sets up the geometric model of plate-like magnetic suspension rotor system;
Step S2, sets up the kinetic model of plate-like magnetic suspension rotor system according to described geometric model; And
Step S3, obtains described electromechanical Coupled Dynamics system of equations.
8. electromechanical Coupled Dynamics system of equations according to claim 7, is characterized in that,
Described plate-like magnetic suspension rotor system comprises: plate-like magnetic suspension rotor, on supporting base, and same circumferentially equally distributed three magnetic bearing M
1, M
2, M
3and three eddy current displacement sensor S
1, S
2, S
3;
According to the stressing conditions of plate-like magnetic suspension rotor system in described step S1, the method setting up the geometric model of plate-like magnetic suspension rotor system comprises:
Step S11, basic assumption, namely suppose that plate-like magnetic suspension rotor itself is rigid body, during the vibration of plate-like magnetic suspension rotor, angular displacement is very little, and the bearing of magnetic bearing itself is rigid body, and the bearing of magnetic bearing only exists the translation of vertical direction and horizontal direction;
Step S12, sets up three-dimensional coordinate system; Namely, true origin overlaps with plate-like magnetic suspension rotor barycenter, six-freedom degree is there is in plate-like magnetic suspension rotor in space, translation along z-axis and the rotation around x, y-axis are by three Active Magnetic Bearing Control, along 2 degree of freedom of x, y-axis translation by the centripetal effect force constraint of electromagnetic field, the degree of freedom of rotating around z-axis does not retrain;
Use z
s, θ
x, θ
ydescribe plate-like magnetic suspension rotor respectively along the translation of z-axis and the rotation around x, y-axis, work as θ
x, θ
yenough hour, then
9. electromechanical Coupled Dynamics system of equations according to claim 8, is characterized in that,
The method setting up the kinetic model of plate-like magnetic suspension rotor system in described step S2 according to described geometric model comprises the steps:
Step S21, the basic dynamic equations that utilization Lagrange's equation sets up plate-like magnetic suspension rotor system is as follows:
In above formula (1), f
zfor the electromagnetic force in the z-direction that magnetic bearing produces, m
xfor the moment around x-axis that electromagnetic force produces, m
yfor the moment around y-axis that electromagnetic force produces, f
z dfor the external interference power in z direction, m
x dfor the disturbance torque around x-axis, m
y dfor the disturbance torque around y-axis;
Step S22, according to the stressing conditions of plate-like magnetic suspension rotor, magnetic bearing produce electromagnetism in the z-direction make a concerted effort, the moment around x direction and the relational expression between the moment around y direction and three electromagnetic forces that magnetic bearing produces as follows:
In above formula (2), f
1, f
2, f
3be respectively three magnetic bearing M
1, M
2, M
3the electromagnetic force produced, near equilibrium position, electromagnetic force being carried out linearization can obtain: f
k=k
ii
k+ k
xx
k, in formula: k
ifor the power-current coefficient of magnetic bearing, k
xfor the power-displacement coefficient of magnetic bearing, i
kfor controlling electric current, x
kfor electromagnet in magnetic bearing is to the displacement of plate-like magnetic suspension rotor, and the value of k is corresponding with magnetic bearing or eddy current displacement sensor, namely gets 1,2,3 respectively;
Step S23, by the geometric relationship of plate-like magnetic suspension rotor in space, can obtain electromagnet to the displacement of plate-like magnetic suspension rotor and the relational expression of disc-like rotor between locus:
I
1, i
2, i
3with x
1, x
2, x
3between relational matrix B calculated by the control system of plate-like magnetic suspension rotor, and to be designated as
Order
the kinetics equation of plate-like magnetic suspension rotor can be obtained, as follows:
M=diag (m, J in above formula (4)
x, J
y), m is the quality of plate-like magnetic suspension rotor, J
x, J
ybe respectively the moment of inertia of plate-like magnetic suspension rotor around x-axis, y-axis, q is the state variable of the plate-like magnetic suspension rotor of definition, q
bthe state variable of the supporting base of the plate-like magnetic suspension rotor system of definition, F
dit is the external interference moment battle array of plate-like magnetic suspension rotor system;
A is the relational matrix of the electromagnetic force of disc-like rotor Moment and magnetic bearing, namely
C is that electromagnet arrives the displacement of plate-like magnetic suspension rotor and disc-like rotor at spatial relation matrix, namely
The method that the control system of described plate-like magnetic suspension rotor calculates relational matrix B comprises:
Step S231, described control system adopts PID controller, and its transport function is:
In above formula (5), K
p, K
i, K
dbe respectively the scale-up factor of PID controller, integral coefficient, differential coefficient, T
dfor the damping time constant of PID controller differentiation element, its corresponding differential equation is:
Step S232, sets up the differential equation of described control system intermediate power amplifier, namely
The transport function of power amplifier is reduced to first order inertial loop:
In formula: A
afor the gain T of power amplifier
afor the damping time constant of power amplifier;
Above formula (7) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (8), Uout is the control voltage obtained after the computing of plate-like magnetic suspension rotor control system;
Step S233, sets up the differential equation of eddy current displacement sensor, and namely the transport function of displacement transducer is also reduced to first order inertial loop:
In above formula (9): A
sfor the gain of eddy current displacement sensor; T
sfor the damping time constant of eddy current displacement sensor; Above formula (9) is carried out Laplace inverse transformation, and can obtain its differential equation is:
In above formula (10), q=(Z
sθ
xθ
y) ' the be displacement vector at disc-like rotor barycenter place; L
sBbecause sensor and magnetic bearing non-concurrent point are installed and the coupled matrix of introducing;
Described relational matrix B is B=G
s(s) G
c(s) G
a(s) L
sB -1;
Described coupled matrix L
sBpreparation method as follows:
If eddy current displacement sensor and the radius of a circle residing for magnetic bearing are a, and draw the coordinate of the axial line of each eddy current displacement sensor and magnetic bearing in described three-dimensional coordinate system, namely
If C
1, C
2, C
3for the point of 3 on plate-like magnetic suspension rotor, its projection on the x-y plane overlaps with the axial line of three eddy current displacement sensors respectively; δ
1, δ
2, δ
3be respectively plate-like magnetic suspension rotor that three eddy current displacement sensors measure along eddy current displacement sensor axial line to corresponding eddy current displacement sensor between distance, i.e. the measured value of eddy current displacement sensor, to obtain C on plate-like magnetic suspension rotor
1, C
2, C
3the coordinate of point in described three-dimensional coordinate system, namely
The coordinate supposing any known point on certain moment plate-like magnetic suspension rotor is (x
0, y
0, z
0), plate-like magnetic suspension rotor law vector is { A ', B ', C ' }, then the equation of motion of plate-like magnetic suspension rotor is this moment:
A′(x-x
0)+B′(y-y
0)+C′(z-z
0)＝0(13)；
By C
1, C
2, C
3substitute into and can obtain into equation (13):
The homogeneous equation group about A ', B ', C ' be made up of formula (14) has the condition of untrivialo solution to be:
The equation of plate-like magnetic suspension rotor plane can be obtained by (15) formula:
Obtain the spatiality of plate-like magnetic suspension rotor, with obtain further 3 magnetic bearing place plate-like magnetic suspension rotors along magnetic bearing axial line to magnetic bearing between distance, and by this distance by the coordinate figure of the x-y plane residing for magnetic bearing, namely in formula (11), the value of x, y substitutes into formula (11) and tries to achieve corresponding z coordinate value, namely
Obtain:
And obtain:
and
Obtain plate-like magnetic suspension rotor along magnetic bearing axial line to magnetic bearing between distance Z
mk, to derive the control current i of arbitrary magnetic bearing
k.
And then obtain described relational matrix B.
10. electromechanical Coupled Dynamics system of equations according to claim 9, is characterized in that,
Described electromechanical Coupled Dynamics system of equations is obtained, namely in described step S3
The above-mentioned formula of simultaneous (4), (6), (8) and (10), set up described electromechanical Coupled Dynamics system of equations, namely
In formula:
B＝G
s(s)G
c(s)G
a(s)L
SB -1；
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| CN109063356A (en) * | 2018-08-15 | 2018-12-21 | 东南大学 | A kind of high-speed electric main shaft rotor-bearing-enclosure system dynamic design approach |
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