CN114296353A - Modal frequency calculation method for satellite with double axes SADA - Google Patents
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Abstract
A modal frequency calculation method for a satellite with a double-axis SADA (satellite-assisted Radar) relates to the technical field of spacecraft attitude control, and solves the problem that a calculation method of unconstrained modal frequency is needed. The method comprises the following steps: establishing a rigid-flexible coupling satellite kinetic equation with a flexible sailboard, linearizing, and obtaining a satellite structure kinetic characteristic equation by Laplace transform processing; establishing a dynamic equation of the SADA driven flexible sailboard, linearizing the dynamic equation, and obtaining a dynamic characteristic equation of the SADA driven flexible sailboard by Laplace transform processing; solving a dynamic characteristic equation of the flexible sailboard driven by the SADA to obtain the non-constrained modal frequency and damping ratio of the SADA and the flexible sailboard system, and accordingly solving a satellite structure dynamic characteristic equation to obtain the non-constrained modal frequency and damping ratio of the satellite system. The method is provided for the design problem of the flexible suppression control law of the rigid-flexible coupling satellite with the double-shaft SADA driving flexible sailboard, and the calculated satellite unconstrained modal frequency has a reference meaning.
Description
Technical Field
The invention relates to the technical field of spacecraft attitude control, in particular to a modal frequency calculation method for a satellite with a double-axis SADA.
Background
In order to realize high-precision remote sensing detection, the modern satellite needs to have enough pointing precision and stability and also needs to ensure the optical axis pointing precision and stability of the effective load. However, the platform is often provided with large flexible solar panels, antennas and other flexible accessories, and has the characteristics of more freedom degrees of the whole satellite, high flexibility, dense low-frequency modes, high modal coupling degree, small structural damping and the like. A Solar Array Drive System (SADS) generally consists of a Solar Array Drive Assembly (SADA) and a Solar Array Drive System (SADA). The internal disturbance factors of the SADA can generate an additional disturbance torque, once the flexible body is acted by the periodic excitation force, the vibration duration of the flexible body can be prolonged, and the flexible body can generate coupling vibration with other components, so that the satellite attitude control stability is poor.
The easiest way to suppress flexural oscillations is to compress the bandwidth of the closed loop control loop so that the system unconstrained modal frequency lies outside the control bandwidth. The rigid-flexible coupled satellite with the double-shaft SADA driven flexible sailboard has the advantages that the sailboard vibration fundamental frequency is low, the mass is relatively large, the rake angle is large, and the rotational inertia of the satellite body can be changed periodically.
Therefore, aiming at the problem of design of a flexible suppression control law of a rigid-flexible coupling satellite with a double-axis SADA driven flexible sailboard, a calculation method of the unconstrained modal frequency needs to be designed to provide input for the design of a satellite dynamics spectrum planning and control system.
Disclosure of Invention
In order to solve the above problems, the present invention provides a modal frequency calculation method for a satellite with a dual axis SADA.
The technical scheme adopted by the invention for solving the technical problem is as follows:
a modal frequency calculation method of a satellite with a double-axis SADA (satellite-associated data acquisition) comprises the following steps:
s1, establishing a rigid-flexible coupling satellite kinetic equation with a flexible sailboard by adopting a mixed coordinate method, and linearizing the rigid-flexible coupling satellite kinetic equation; establishing an SADA driving flexible sailboard dynamic equation, and linearizing the SADA driving flexible sailboard dynamic equation;
s2, performing Laplace transform processing on the linearized rigid-flexible coupling satellite kinetic equation to obtain a satellite structure kinetic characteristic equation, and performing Laplace transform processing on the linearized SADA driven flexible sailboard kinetic equation to obtain a dynamic characteristic equation of the SADA driven flexible sailboard;
s3, solving a dynamic characteristic equation of the flexible sailboard driven by the SADA to obtain the SADA and a kth-order unconstrained modal frequency omega of the flexible sailboard system'kAnd damping ratio xi'kFrom Ω'kAnd xi'kSolving a satellite structure dynamics characteristic equation to obtain a k' th order unconstrained modal frequency omega of the satellite systemk'Zeta damping ratiok'。
The invention has the beneficial effects that:
the invention discloses a modal frequency calculation method of a satellite with a double-shaft SADA (synthetic aperture radar), which is a novel method provided for the design problem of a flexible suppression control law of a rigid-flexible coupling satellite with a double-shaft SADA driven flexible sailboard, is a calculation method of unconstrained modal frequency, and provides input for the design of a satellite dynamics spectrum planning and control system. According to the invention, the SADA and the sailboard are taken as a whole, the modal frequency and the damping ratio of the system are closer to reality, and thus the calculated satellite unconstrained modal frequency has more reference significance.
Drawings
FIG. 1 is a schematic diagram of a satellite with dual axis SADA.
FIG. 2 is a flowchart of a modal frequency calculation method for a SADA satellite with two axes according to the present invention.
Fig. 3 is a front 5 th order mode shape diagram of a solar panel of the modal frequency calculation method with a biaxial SADA satellite according to the present invention.
FIG. 4 is an unconstrained modal frequency diagram of a satellite with a single windsurfing board calculated by the modal frequency calculation method of the satellite with the biaxial SADA.
FIG. 5 is an unconstrained modal frequency diagram of a satellite with a double-sided windsurfing board calculated by the modal frequency calculation method of the satellite with the double-axis SADA.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The modal frequency calculation method of the SADA satellite with the double shafts can calculate the unconstrained modal frequency of a plurality of windsurfing boards.
The schematic diagram of a satellite with double-sided sailboards is shown in figure 1, and a satellite body B is fixedly connected with an O of the satellitebXbYbZbRectangular coordinate system is satellite body coordinate system, origin of coordinates ObAt the center of mass of the satellite, + XbAxis is in the same direction as the satellite flight direction, + ZbThe axis is perpendicular to the butt joint surface of the satellite butt joint ring and the transition section of the carrier and points to the optical camera, + YbThe axes are determined according to the right hand rule. And when the earth orientation has no attitude deviation, the earth orientation is superposed with the satellite orbit coordinate system. The SADA-A shaft is used for driving a sailboard to wind a satellite body coordinate system YbThe shaft rotates at an orbital angular velocity to track the sun rays, and the rotation range is-180 degrees to +180 degrees; SADA-B axis is used for driving sailboard to wind satellite body coordinate system XbThe shaft swings to a fixed angle, so that the solar rays vertically irradiate the sailboard, and the rotation range is 0-40 degrees. Two sailboards, i.e. flexible sailboards A1And a flexible sailboard A2。
The calculation flow chart of the modal frequency calculation method of the SADA satellite with two axes is shown in FIG. 2, and the method comprises the following steps:
s1, establishing a rigid-flexible coupling satellite kinetic equation with a flexible sailboard by adopting a mixed coordinate method, and linearizing the rigid-flexible coupling satellite kinetic equation; establishing an SADA driving flexible sailboard dynamic equation, and linearizing the SADA driving flexible sailboard dynamic equation;
s2, performing Laplace transform processing on the linearized rigid-flexible coupling satellite kinetic equation to obtain a satellite structure kinetic characteristic equation, and performing Laplace transform processing on the linearized SADA driven flexible sailboard kinetic equation to obtain a dynamic characteristic equation of the SADA driven flexible sailboard;
s3, solving the dynamic characteristic equation of the SADA driving flexible sailboard to obtain the SADAAnd a k-th order unconstrained modal frequency omega of flexible sailboard system'kAnd SADA and flexible sail panel system kth order unconstrained modal damping ratio ξ'kFrom Ω'kAnd xi'kSolving a satellite structure dynamic characteristic equation to obtain a k' th order unconstrained modal frequency zeta of the satellite systemk'And the k-th order unconstrained modal damping ratio Zeta of the satellite systemk'。
The present invention is described in detail below, wherein the rigid-flexible coupled satellite dynamics equations and the SADA driven flexible windsurfing dynamics equations do not distinguish between the precedence relationship, and S1 is described in detail below according to steps one through four.
Step one, establishing a rigid-flexible coupling satellite dynamic model with a flexible sailboard
Adopting a hybrid coordinate method to establish the attitude motion of the satellite with the flexible sailboard: the central rigid body is represented by euler angles, which generally describe the attitude of the rigid body, and the flexible appendages are represented by discrete modal coordinates. The following are defined:
whole star moment of inertia matrix Is，
Angular velocity vector omega of whole-satellite relative inertial systemsThe rotation angular velocity vector omega of the ith (i is a positive integer) flexible sailboard relative to the central rigid bodyiAAnd modal frequency omega of ith SADA and flexible sailboard system'iModal damping ratio xi of ith SADA to flexible windsurfing board systemi', ith SADA and flexible sailboard system modal coordinate etaiI th flexible sailboard moment of inertia matrix IiaThe coupling coefficient B of the self vibration and the translation of the ith flexible sailboarditranThe coupling coefficient F of the vibration and rotation of the ith flexible sailboardiAThe coupling coefficient F between the vibration of the ith flexible sailboard and the rotation of the stariSThe coupling coefficient R of the i-th flexible sailboard rotation and the star rotationiasThe ith flexible sailboard rotation and the whole star rotation are coupled with an inertial dyadic RisaThe moment T of interaction of the ith flexible windsurfing board with the star at the point of attachmentipMoment of coupling of sailboard flexible vibration to star
X denotes the diagonally symmetric matrix form of the vector,
represents omegasIn the form of an oblique-symmetrical matrix, the moment coupling the rotational movement of the sailboard to the star
Flywheel control moment TwAngular momentum of flywheel hwThe external disturbance torque T of the whole satellited；
The dynamic equation of the rigid-flexible coupling satellite with i flexible sailboards is
Step two: rigid-flexible coupling satellite kinetic equation linearization
The on-orbit mode of the satellite is the inherent characteristic and is not related to the external environment action, and the kinetic equation (1) can be simplified into
For small attitude angles, the dynamical model can be linearized near the desired trajectory. The small attitude angle means that the satellite attitude is in a basic stable state, and the deviation of the actual attitude angle and the expected attitude is small and within +/-5 degrees. According to a kinematic equation, converting the angular velocity omega of the star bodysConversion into Euler angles
Is expressed as
Wherein, ω is0Representing the orbital angular velocity of the satellite.
And substituting the formula (3) into the formula (2) to obtain the linearized dynamic model of the rigid-flexible coupling satellite with the flexible sailboard.
The linearized rigid-flexible coupling satellite dynamic model with the flexible sailboard is
Where q represents a coordinate variable and L, M and N represent a coefficient matrix for the coordinate variable q. When the satellite has two flexible sailboards, let the coordinate variable q be [ theta eta [ ]1 η2]T。
The sailboard takes the first n-order mode (n is a positive integer), the coefficient matrixes are all 2n + 3-order square matrixes, and the expressions are respectively
Wherein E isnRepresenting an n-order identity matrix.
Step three: establishing a dynamic model of SADA (software-aided design) driven flexible sailboard
And establishing a dynamic model of the two-phase hybrid stepping motor under the drive of sine and cosine currents. The definition is as follows: number of teeth Z of motor rotorrCommanded rotational angle of motor θcMechanical angle of rotation theta of motormCurrent amplitude I of phase A and phase B0Phase of current I of phase AA＝I0cos(Zrθc) Phase B current IB＝I0sin(Zrθc) D, electromagnetic moment coefficient KmPositioning moment coefficient D, total electromagnetic moment T of motore＝-K(θm-θc)-Dsin(4Zrθm)，K＝KmI0ZrCoulomb friction torque coefficient TcViscous damping coefficient sigma, motor friction torque
Load moment TlMoment of inertia J of the motor about the axis of rotationmA matrix J of rotational inertia of the motor,
SADA reduction ratio j, i flexible windsurfing board modal frequency omegaiModal damping ratio xi of the ith flexible windsurfing boardiThe ith flexible windsurfing board-to-star conversion matrix Tisa，TisaMechanical rotation angle theta of motor along with SADAmMay vary. SADA load moment
θmIs thetamVector extending into three dimensions, thetacIs thetacVector extending into three dimensions, the direction of rotation of the SADA being along the Y axis of the star, then θm＝[0 θm 0]；θc＝[0 θc 0]。
The dynamic equation of the SADA driven flexible sailboard is
Where η represents the windsurfing board modal coordinates.
Step four: SADA driven flexible sailboard dynamic equation linearization
Let coordinate variable τ be [ θ ═ dm ηi]TAfter the formula (5) is linearized
The sailboard takes the first n-order mode, and the value of n is equal to the second stepCoefficient matrix of the value of middle n and coordinate variable tau
Are all n +3 order square matrixes, and control variable coefficients
Is an n × 3 matrix, and the expressions are respectively
Wherein E represents an identity matrix.
Step five: satellite unconstrained modal frequency resolution
And (5) obtaining a satellite structure dynamics characteristic equation after the Laplace transform is adopted in the equation (4):
Ls2+Ms+N＝0 (7)
wherein s represents a complex variable.
Similarly, after the equation (6) is processed by laplace transform, a dynamic characteristic equation of the flexible sailboard driven by the SADA is obtained:
according to SADA and flexible windsurfing parameters, i.e. according to
Solving equation (8) can obtain n +3 pairs of complex characteristic values lambda'k,
Thereby having the formula of
Obtaining the K-th order unconstrained modal frequency omega of the SADA and flexible sailboard system'kAnd SADA and flexible sail panel system kth order unconstrained modal damping ratio ξ'kK is a positive integer, k represents the order of the unrestrained modal frequencies of the SADA and flexible sailboard system, and the total n +3 order of the unrestrained modal frequencies in formula (9), wherein lambda'kRepresents the characteristic value of the k-th order of equation (8).
Prepared from omega'k、ξ′kSubstituting into the satellite structure dynamics characteristic equation (7) to obtain 2n +3 pairs of complex characteristic values lambdak',
And then to
Obtaining the k' th order unconstrained modal frequency omega of the satellite systemk'And the k' th order unconstrained modal damping ratio zeta of the satellite systemk'K 'is a positive integer, k' represents the order of the unconstrained modal frequency of the satellite system, and the total 2n +3 orders of unconstrained modal frequency in formula (10), wherein λk'The characteristic value of the k' th order of the formula (9) is expressed.
The invention has the following effects:
the natural frequency of satellite dynamics is objective, the satellite cannot vibrate under the condition of no disturbance, and once a periodic disturbance source is coupled with the natural frequency of a platform, resonance is caused, and the satellite attitude stability is seriously influenced. The research on the frequency characteristic of the whole satellite system dynamics system with a plurality of flexible accessories is a premise for developing the design of a flexible vibration suppression control law and has important significance for the development of satellite attitude control. The invention discloses a modal frequency calculation method of a satellite with a double-shaft SADA (synthetic aperture radar), which is a novel method provided for the design problem of a flexible suppression control law of a rigid-flexible coupling satellite with a double-shaft SADA driven flexible sailboard, is a calculation method of unconstrained modal frequency, and provides input for the design of a satellite dynamics spectrum planning and control system. According to the invention, the SADA and the sailboard are taken as a whole, the modal frequency and the damping ratio of the system are closer to reality, and thus the calculated satellite unconstrained modal frequency has more reference significance.
In the prior art, a method for calculating the in-orbit modal frequency of a satellite with different rotation angles of a solar wing in a document [ Zhangjiang, Limenyu, Lvwang, Du Jiuji ]]Spacecraft engineering, 2017,26(04):35-40.]In the method, a dynamic characteristic equation of the satellite structure with the single-side sailboard is established by using a hybrid coordinate method, the modal frequency of the flexible sailboard at different rotation angles is calculated, but a gyro moment term is ignored in a model, and the dynamic characteristic of the SADA is not considered. Document [ ShiGuiguo, Zhuqinghua, Zhang Zilong, a control strategy study for actively suppressing flexible vibration of solar sailboard [ J]Shanghai Spaceflight, 2016,33(03):61-66.]A coupling dynamic model of the flexible sailboard driven by the stepping motor is established, the modal frequency and the damping ratio of the system are given, and the influence of the star attitude motion on the sailboard modal is not considered. According to the method for calculating the frequency of the unconstrained modal of the satellite, provided by the invention, the nutation motion characteristic of the SADA-B axis at a large angle is considered, and a coupling moment item I of attitude dynamics is added into a linear models(specifically, coefficient matrixes P and R in the second step) consider the influence of the attitude motion of the star body on the sailboard mode, and the calculated satellite unconstrained mode frequency has more reference significance. The method can calculate the change rule of the system unconstrained modal frequency along with the SADA corner when the satellite has a single-side sailboard and a double-side sailboard.
The application of the invention is specifically illustrated below.
The satellite has a biaxial SADA, the quality parameters are shown in tables 1 and 2, the constrained mode parameters and the mode shape of 5 th order in front of the windsurfing board are shown in tables 3 and 3, and the mode shape is obtained by HEXAGON software in FIG. 3.
TABLE 1 satellite quality characterization parameters
TABLE 2 SADA-A Axis parameters
| Parameter(s) | Numerical value |
| Number of |
50 |
| |
100 |
| Two-phase current amplitude I0/A | 0.6 |
| Moment of inertia J of motorm/kg.cm2 | 0.2 |
| Electromagnetic moment coefficient Km/Nm.A-1 | 0.3 |
| Viscous damping coefficient sigma/kg.m2.s-1 | 0.05 |
TABLE 3 solar sailboard flexibility parameters (first 5 th order)
Let the SADA-a axis be 0 °, consider the modal frequencies of a satellite with a single-sided sailboard and a double-sided sailboard, respectively, and the B axis at 0 ° and 40 ° states, respectively, as shown in comparison tables 4 and 5.
TABLE 4 modal frequency contrast (SADA-A, B for axes angles of 0 degree)
TABLE 5 modal frequency contrast (SADA-A axis angle of 0 degree, B axis angle of 40 degree)
When the angle of the A axis is changed from-180 degrees to 180 degrees, the change curve of the whole star unconstrained modal frequency is shown in fig. 4 and 5.
From the calculation results, it can be seen that: 1) as the load inertia increases, the modal frequency of the motor decreases; when the SADA-B axis is at 40 degrees, the modal frequency increases significantly when the motor drives a flexible load as compared to a rigid load. 2) When the B axis is at 0 degree, the 4 th order modal frequency of the coupled sailboard and the motor changes; when the B axis is at 40 degrees, modal frequencies of orders 2, 3 and 4 after the sailboard is coupled with the motor are changed, which is determined by the coupling coefficient of the sailboard; 3) when the satellite is provided with the unilateral sailboard, the frequency of each order of the whole satellite non-constrained mode is improved compared with the frequency of the constrained mode of the sailboard, and the fundamental frequency is improved by about 1 time; 4) when the satellite is provided with double-side sailboards, the frequencies of the 1 st order and the 2 nd order of the whole satellite in the non-constrained modes are basically unchanged, and the frequencies are in a decreasing trend from the 3 rd order; 5) when the angle of the B axis is kept unchanged, along with the increase of the angle of the A axis, the variation of the rotational inertia of the X axis and the Z axis of the whole satellite is large, and the 1 st, 2 nd and 4 th order unconstrained modal frequencies show two periods of fluctuation when a single sailboard is provided; the 3 rd and 5 th order unconstrained modal frequencies related to torsion are basically unchanged because the rotational inertia of the Y axis is unchanged; 6) when the axis B is a small angle, the periodic variation amplitude of each order of the whole satellite non-constrained modal frequency along with the angle of the axis A is larger; when the B-axis is at a large angle, the amplitude of the periodic variation is small.
The calculation result shows that: for the satellite with the unilateral sailboard, the frequency of the unconstrained mode is higher than that of the constrained mode, the mode is more obvious in the first orders, and the fundamental frequency is increased by about 1 time; the unconstrained mode corresponds to reduced constraint, making the flexure attachment more free to vibrate. Meanwhile, the influence of the coupling coefficient of the sailboard on the frequency has a certain corresponding relation: the larger the coupling coefficient is, the stronger the coupling effect of the order mode and the whole star is, and the order mode is easily influenced by the star body; for the satellite with the double-side sailboard, the frequency of the unconstrained mode is not improved compared with that of the constrained mode, and the new mode is considered to be added on the basis of the adjacent constrained mode.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
Claims (10)
1. A modal frequency calculation method of a satellite with a double-axis SADA is characterized by comprising the following steps:
s1, establishing a rigid-flexible coupling satellite kinetic equation with a flexible sailboard by adopting a mixed coordinate method, and linearizing the rigid-flexible coupling satellite kinetic equation; establishing an SADA driving flexible sailboard dynamic equation, and linearizing the SADA driving flexible sailboard dynamic equation;
s2, performing Laplace transform processing on the linearized rigid-flexible coupling satellite kinetic equation to obtain a satellite structure kinetic characteristic equation, and performing Laplace transform processing on the linearized SADA driven flexible sailboard kinetic equation to obtain a dynamic characteristic equation of the SADA driven flexible sailboard;
s3, solving a dynamic characteristic equation of the flexible sailboard driven by the SADA to obtain the SADA and a kth-order unconstrained modal frequency omega of the flexible sailboard system'kAnd damping ratio xi'kFrom Ω'kAnd xi'kSolving a satellite structure dynamics characteristic equation to obtain a k' th order unconstrained modal frequency omega of the satellite systemk′Zeta damping ratiok′。
2. The modal frequency calculation method of a satellite with two axes SADA of claim 1, wherein the dynamical equation of the rigid-flexible coupling satellite with i flexible sailboards is as follows:
wherein, IsA matrix of the moment of inertia of the whole star is represented,
ωsrepresenting the angular velocity vector of the whole satellite relative to the inertial system,
represents omegasIn the form of an obliquely symmetrical matrix of hwWhich represents the angular momentum of the flywheel,
representing the moment at which the windsurfing board flexural vibrations are coupled to the star,
representing the moment, T, by which the rotating motion of the sailboard is coupled to the stardRepresenting the external disturbance moment, T, of the whole satellitewIndicating flywheel control moment, omegaiARepresents the rotation angular velocity vector of the i-th flexible sailboard relative to the central rigid body, i is a positive integer and omega'iRepresenting the modal frequency, ξ, of the ith SADA and flexible windsurfing systemsi' represents the modal damping ratio, η, of the ith SADA to the flexible windsurfing systemiRepresenting the modal coordinates of the ith SADA and flexible windsurfing systems, FiSExpressing the coupling coefficient of the i-th flexible sailboard vibration and the star rotation, RiasExpressing the coupling coefficient of the i-th flexible sailboard rotation and the star rotation, RisaRepresenting the inertia vector of the coupling of the I-th flexible windsurfing board rotation and the whole star rotation, IiaRepresenting the i-th flexible windsurfing board moment of inertia matrix, FiAThe vibration and rotation coupling coefficient of the ith flexible sailboard is shown, and the interaction moment T of the ith flexible sailboard and the star body at the connecting point is shownip。
3. The method of claim 2, wherein the process of linearizing the equations of dynamics of the rigid-flexible coupled satellite comprises:
in the case where the in-orbit mode of the satellite is its inherent characteristic, equation (1) is simplified as:
will omegasConversion into Euler angles
The expression of (a) is:
wherein, ω is0Represents the orbital angular velocity of the satellite;
and substituting the formula (3) into the formula (2) to obtain a linear rigid-flexible coupling satellite kinetic equation with the flexible sailboard.
4. The method for calculating modal frequency of a satellite with two axes SADA of claim 3, wherein the linearized rigid-flexible coupled satellite dynamic model with flexible sailboards is
Where q represents a coordinate variable and L, M and N represent a coefficient matrix for the coordinate variable q.
5. The modal frequency calculation method of a satellite with two axes SADA of claim 4, wherein when the satellite has two flexible sailboards, let the coordinate variable q ═ 2θ η1 η2]T，
The sailboard takes the first n-order mode (n is a positive integer), the coefficient matrixes are all 2n + 3-order square matrixes, and the expressions are respectively
Wherein E isnRepresenting an n-order identity matrix.
6. The modal frequency calculation method for a biaxial SADA satellite as recited in claim 2, wherein the SADA driven flexible windsurfing board dynamics equation is as follows:
wherein J represents a motor moment of inertia matrix,
Jmrepresenting the moment of inertia, T, of the motor about the axis of rotationisaRepresenting the i-th flexible windsurfing board to star conversion matrix, j representing the SADA reduction ratio, thetamExpressing the mechanical rotation angle of the motor, eta expressing the modal coordinate of the sailboard, sigma expressing the viscous damping coefficient, and K being equal to KmI0Zr，KmRepresenting the electromagnetic moment coefficient, I0Represents the current amplitudes of the two phases of A phase and B phase, ZrIndicating number of motor rotor teeth, thetacRepresents the motor command angle, ΩiMode shape representing the ith flexible windsurfing boardFrequency xiiThe modal damping ratio of the ith flexible windsurfing board is shown.
7. The method of claim 6, wherein the linear SADA driven flexible windsurfing kinetic equation is a modal frequency calculation method for a biaxial SADA satellite
Let coordinate variable τ be [ θ ═ dm ηi]TIs obtained by linearizing the formula (5)
The windsurfing board takes the front n-order mode, n is a positive integer, wherein,
and E denotes an identity matrix.
8. The modal frequency calculation method for a biaxial SADA satellite as recited in claim 7, wherein the dynamic characteristic equation of the SADA driven flexible windsurfing board is as follows:
wherein s represents a complex variable;
solving equation (8) can obtain a complex characteristic value lambda'k,
Thereby having the formula of
Obtaining the K-th order unconstrained modal frequency omega of the SADA and flexible sailboard system'kAnd SADA and flexible sail panel system kth order unconstrained modal damping ratio ξ'kWherein k is a positive integer.
9. The method of claim 4, wherein the equation for the structural dynamics of the satellite is
Ls2+Ms+N＝0 (7)
Wherein s represents a complex variable.
10. The modal frequency calculation method of a biaxial SADA satellite according to claim 9, being based on Ω'kAnd xi'kSolving the formula (7) to obtain a complex eigenvalue lambdak′,
And then by
Obtaining the k' th order unconstrained modal frequency Zeta of the satellite systemk′And the k' th order unconstrained modal damping ratio zeta of the satellite systemk′And k' is a positive integer.
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| CN107036761A (en) * | 2016-11-11 | 2017-08-11 | 大连理工大学 | A kind of large angle maneuver lower band flexible appendage spacecraft rotary inertia in-orbit identification method |
| WO2019076015A1 (en) * | 2017-10-16 | 2019-04-25 | 东南大学 | Dual modal equation based dynamic response analysis method under random noise environment |
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