CN107291988A - A kind of momenttum wheel installation interface equivalent excitation power acquisition methods - Google Patents

A kind of momenttum wheel installation interface equivalent excitation power acquisition methods Download PDF

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CN107291988A
CN107291988A CN201710376825.7A CN201710376825A CN107291988A CN 107291988 A CN107291988 A CN 107291988A CN 201710376825 A CN201710376825 A CN 201710376825A CN 107291988 A CN107291988 A CN 107291988A
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CN107291988B (en
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邹元杰
于登云
王泽宇
葛东明
朱卫红
张志娟
庞世伟
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Beijing Institute of Spacecraft System Engineering
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Abstract

A kind of momenttum wheel and the acquisition methods of spacecraft installation interface equivalent excitation power; the test data utilized need not in momenttum wheel main structure install sensor; the protection structure of momenttum wheel will not be broken; testing program is simple and easy to apply; using the mass matrix of momenttum wheel, Element of Stiffness Matrix as amendment object, the amount of calculation of amendment is reduced, analysis efficiency is improved; disturbance response analysis is carried out using momenttum wheel installation interface equivalent excitation power provided by the present invention, analysis precision is high.The momenttum wheel equivalent excitation power that method is obtained puts on the installation interface of momenttum wheel and spacecraft, coupling that can be between accurate response spacecraft and momenttum wheel, improves the precision of momenttum wheel perturbation analysis indication.

Description

A kind of momenttum wheel installation interface equivalent excitation power acquisition methods
Technical field
The present invention relates to a kind of momenttum wheel installation interface equivalent excitation power acquisition methods, the method that the present invention is provided is used to move The disturbance response analysis of amount wheel.
Background technology
The in-orbit microvibration of satellite will produce influence to tasks such as space science experiment, laser communication, optical remote sensings, wherein Momenttum wheel and control-moment gyro are important disturbing sources, thus momenttum wheel and control-moment gyro disturbance response analysis very It is important.Scholar both domestic and external has carried out measurement work to disturbing source very early, and research finds that the caused disturbance of part rotation can quilt Self structure is amplified.In order to remove this influence, the disturbance for having scholar to be proposed according to Masterson R A, Miller D W is humorous Ripple empirical model, relevant parameter is picked out using test data, and the perturbed force (square) fitted under different rotating speeds carries out disturbance point Analysis.Because disturbance source structure is to the amplification of disturbance, identification result is often bigger than normal；In view of immobile interface condition not Genuine interface condition when can accurately represent the source of disturbing on the spacecraft, Laila Mireille Elias et al. are proposed Static acceleration analytic approach, Zhe Zhang Guglielmo S et al. proposed a kind of based on dynamic acceleration in 2013 Flywheel micro-vibration analysis method.This method is using disturbing source and the acceleration of flexible foundation structure, the perturbed force obtained to measurement (square) is modified, and revised data reflect the coupled characteristic of the source of disturbing and spacecraft structure, can be loaded directly into dynamics Perturbation analysis is carried out in analysis model.Correction factor used in this method can be obtained by finite element analysis or test measurement, But be different, it is necessary to recalculate to different spacecraft correction factors, which increase the action of analysis and experiment and Its limited precision.
The content of the invention
Present invention solves the technical problem that being：Overcome prior art not enough, propose that a kind of momenttum wheel installation interface is equivalent and swash Power acquisition methods are encouraged, transmission characteristic of this method according to perturbed force in momenttum wheel in structure gives model analysis calculating Difference and model quality, Element of Stiffness Matrix between the rigid interface restraining force that draws, the rigid interface restraining force of experiment actual measurement Relation, and the element in matrix is modified as object, so that actual measurement mode is related to Analysis Mode, makes the meter of model Calculate result consistent with actual test result；In addition, the installation interface equivalent excitation power that invention is obtained is a kind of accurate micro-vibration Source decouples loading method, this method considers amplification of the momenttum wheel self structure to disturbance, it is adaptable to which spacecraft is disturbed Dynamic response is analyzed.
The technical scheme that the present invention is solved is：A kind of momenttum wheel and the acquisition side of spacecraft installation interface equivalent excitation power Method, step is as follows：
(1) propose that the dynamics equations of momenttum wheel are as follows：
X (t) is the displacement of t momenttum wheel in formula,For the speed of t momenttum wheel,For t momentum The acceleration of wheel；{ f } is the excitation suffered by momenttum wheel；
In formula, m is the quality of momenttum wheel, and Irr is that momenttum wheel is used to around the rotation of radial direction.
C in formulaazAxially damped for momenttum wheel, c damps for the translation of momentum wheel radial, cd2For momenttum wheel damping of oscillations.
In formula, k in formulaazFor momenttum wheel axial rigidity, k is momentum wheel radial translation rigidity, kd2Swing firm for momenttum wheel Degree.ωrThe radial direction translation mode angular frequency of mould measurement determination is represented, according to engineering experience, preliminary design ωr=1e4 is obtained To k；Wherein represent the axial translation mode angular frequency of mould measurement determination, preliminary design ωaz=2e4 obtains kaz；Wherein ωswingRepresent mould measurement determination waves mode angular frequency, preliminary design ωswing=1.5e3 is obtained kd2。
(2) the i.e. C when not considering the damping of momenttum wheelf=0, momentum wheel frame fix when time domain kinetics equation be：
In formula, subscript f represents the rotor free degree, behalf gimbal freedom, MffFor rotor quality matrix, KffIt is firm for rotor Spend matrix,
F1(t) line number and MffIt is identical, F1(t) it is perturbed force, F suffered by momenttum wheel rotor2(t) line number and MssIt is identical, F2(t) For restraining force of the rigid interface to momenttum wheel.
The corresponding frequency domain kinetics equation of formula (2) is：
In formula, F1For the perturbed force suffered by the rotor in momenttum wheel, F2The rigid interface restraining force fixed for momenttum wheel, F1、F2It is ω function, i.e. F1=F1(ω)、F2=F2(ω).ω is that momenttum wheel vibrates circular frequency, xfFor xf(t) corresponding frequency Thresholding.
(3) transitive relation for the rigid interface restraining force that the perturbed force suffered by the rotor in momenttum wheel is fixed with momenttum wheel It is expressed as follows：
Ksf[-ω2Mff+Kff]-1{F1}={ F2}···············(31)
In formula, Ksf=-Kff, equation (4) is transformed to
In formula, E is unit matrix, its dimension and MffIt is identical；
Formula (5) conversion is obtained
(4) according to formula (6), the math equation for setting up the real structure of momenttum wheel is as follows：
In formula,For the mass matrix for real momenttum wheel of waiting to look for the truth,For the stiffness matrix for real momenttum wheel of waiting to look for the truth,For the true momenttum wheel rigid interface restraining force of measurement gained.
(5) formula (3) calculates gained { F2With measuring gainedIn the presence of certain error, if：
Wherein
Formula (7) subtracts formula (6) and using formula (8) and formula (9), obtains：
If KffFor parameter p1、p2、…、piFunction, i.e. Kff=Kff(p1,p2,…,pi), MffFor parameter pi+1、…、pn's Function, i.e. Mff=Mff(pi+1,pi+2,…,pn), to the matrix in formula (10)Carry out Taylor expansion：
Wherein Δ p1、Δp2、…、Δpi、Δpi+1、…、ΔpnFor Kff、MffMiddle p1、p2、…、pi、pi+1、…、pnWith structure The deviation of actual parameter.
Formula (11) is substituted into equation (10) to obtain
In formula,[S] represents sensitivity matrix：
(6) according to formula (12), equation is set up respectively at different frequencies omegas, it is as follows：
Solution formula (13), obtains momenttum wheel mass matrix MffWith momenttum wheel stiffness matrix KffThe correction value of parameterRevised mass matrix：
Stiffness matrix：
The undamped-free vibration equation for finally giving true momenttum wheel is：
Wherein
(7) when momenttum wheel, which is fixed on, carries out perturbed force measurement on force plate/platform, the free degree x (t) of momenttum wheel is divided into Two groups, it is respectively：The internal degree of freedom not being connected with rigid interface, i.e. rotor free degree xf(t) side and in rigid interface Boundary's free degree, i.e. gimbal freedom xs(t), i.e.,
WhereinFor the Mass matrix of momenttum wheel, C is momenttum wheel damping matrix, by engineering experience test to Go out,For momenttum wheel Stiffness Matrix.
(8) the border free degree being connected according to the internal degree of freedom not being connected with rigid interface, with rigid interface carries out C Piecemeal：
Equation (15) is changed into the dynamics equations of momenttum wheel under rigid interface, as follows：
(9) under frequency domain, the dynamics equations of momenttum wheel under the rigid interface of step (8) are converted into：
In formula, xf、xs、F1、F2X is represented respectivelyf(t)、xs(t)、F1And F (t)2(t) corresponding domain complex amount.
(10) Dynamic Stiffness Matrix of momenttum wheel is setBy Z Formula (17) formula, i.e., can be converted into by piecemeal：
Wherein
The framework of momenttum wheel is fixed on force plate/platform, xs=0, formula (18) is substituted into, the constraint of interface is fixed Power F2, it is as follows
(11) spacecraft structure FEM model is set up, momenttum wheel is arranged on spacecraft, momenttum wheel and the boat is set up The coupled dynamical equation of its device, according to momenttum wheel rotor displacement xf, momentum wheel frame displacement xs, spacecraft node xk, will couple Kinetics equation carries out piecemeal：
In formula, subscript " k " represents the label of the node on spacecraft FEM model, xkFor the modal displacement of spacecraft；
Obtained by formula (20)：
IfThen
(12) in the momenttum wheel installation interface applying power-F of spacecraft2, momenttum wheel and spacecraft the coupled dynamical equation (20) it is changed into：
In formula,For momenttum wheel rotor displacement,For momentum wheel frame displacement,For spacecraft modal displacement.
Formula (19) is substituted into formula (24), obtained：
(13) reduced equation (22) and (26), equation (23) and (27), it is known that, now：
Understand when in spacecraft disturbance interface application force vectorCalculating obtains kinetics equation solution and equation (20) Solution is identical.So far, obtaining momenttum wheel installation interface equivalent excitation power is
(14) (27) are solved equation, the disturbance response caused during momenttum wheel work to spacecraft is obtained.
The advantage of the present invention compared with prior art is：
(1) momenttum wheel model modification method proposed by the invention provides method foundation for the Modifying model of momenttum wheel, The precision of momenttum wheel disturbance indication is necessarily improved after being corrected；
(2) test data that is utilized of the present invention need not in momenttum wheel main structure install sensor, will not break dynamic The protection structure of wheel is measured, testing program is simple and easy to apply；
(3) present invention reduces the calculating of amendment using the mass matrix of momenttum wheel, Element of Stiffness Matrix as amendment object Amount, improves analysis efficiency；
(4) disturbance response analysis, analysis essence are carried out using momenttum wheel installation interface equivalent excitation power provided by the present invention Degree is high.
(5) present invention is by deriving dynamics equations, it was found that disturbance is in the source of disturbing-measure interface-satellite mounting surface Interior transfer law, it is proposed that momenttum wheel installation interface equivalent excitation power acquisition methods.This method is according to perturbed force in momentum The transmission characteristic in structure is taken turns, rigid interface restraining force, the rigid boundary of experiment actual measurement that model analysis is calculated is given Difference and model quality, the relation of Element of Stiffness Matrix between the restraining force of face, and the element in matrix is repaiied as object Just, so that actual measurement mode is related to Analysis Mode, make the result of calculation of model consistent with actual test result；
(6) the installation interface equivalent excitation power that the present invention is obtained is a kind of accurate micro-vibration source decoupling loading method, this The method of kind considers amplification of the momenttum wheel self structure to disturbance, it is adaptable to the disturbance response analysis of spacecraft.
Brief description of the drawings
Fig. 1 is the perturbation response curve of certain point on spacecraft of the present invention；
Fig. 2 is that momenttum wheel of the present invention constitutes sketch；
Fig. 3 is the momenttum wheel system diagram under rigid interface of the present invention；
Fig. 4 is momenttum wheel of the present invention+celestial body structure coupled systems figure；
Fig. 5 is that momenttum wheel of the present invention decouples loading system figure.
Embodiment
The present invention is described in detail with specific embodiment below in conjunction with the accompanying drawings.
The acquisition methods of a kind of momenttum wheel and spacecraft installation interface equivalent excitation power, the test data utilized need not be Install sensor in momenttum wheel main structure, will not break the protection structure of momenttum wheel, and testing program is simple and easy to apply, with momenttum wheel Mass matrix, Element of Stiffness Matrix reduce the amount of calculation of amendment, improve analysis efficiency, utilize this hair as amendment object Bright provided momenttum wheel installation interface equivalent excitation power carries out disturbance response analysis, and analysis precision is high.By moving that method is obtained Amount wheel equivalent excitation power puts on the installation interface of momenttum wheel and spacecraft, coupling that can be between accurate response spacecraft and momenttum wheel Cooperation is used, and improves the precision of momenttum wheel perturbation analysis indication.
The modeling of momenttum wheel and loading method are mostly important input, more than ten for whole micro-vibration analysis model One of study hotspot in always domestic and international micro-vibration field over year.It is, in principle, that momenttum wheel is also flexible body in itself, source is disturbed The size of effect is influenceed by celestial body Structure dynamic characteristics, and its actual mechanism of action is that the source of disturbing and the coupling of celestial body structure are made With, it is necessary to using coupling analysis.Analyzed using equivalent excitation power acquisition methods provided by the present invention, calculate response and true Real response is consistent.
The step of the present invention is as follows：
To simplify the kinetic model of momenttum wheel, it is believed that its mass concentration includes 10 frees degree in rotor, model.
(1) propose that the dynamics equations of momenttum wheel are as follows：
X (t) is the displacement of t momenttum wheel in formula,For the speed of t momenttum wheel,For t momentum The acceleration of wheel；{ f } is the excitation suffered by momenttum wheel；
In formula, m is the quality of momenttum wheel, and Irr is that momenttum wheel is used to around the rotation of radial direction.
C in formulaazAxially damped for momenttum wheel, c damps for the translation of momentum wheel radial, cd2For momenttum wheel damping of oscillations.
In formula, k in formulaazFor momenttum wheel axial rigidity, k is momentum wheel radial translation rigidity, kd2Swing firm for momenttum wheel Degree.ωrThe radial direction translation mode angular frequency of mould measurement determination is represented, according to engineering experience, preliminary design ωr=1e4 is obtained To k；Wherein represent the axial translation mode angular frequency of mould measurement determination, preliminary design ωaz=2e4 obtains kaz；Wherein ωswingRepresent mould measurement determination waves mode angular frequency, preliminary design ωswing=1.5e3 is obtained kd2。
(2) the i.e. C when not considering the damping of momenttum wheelf=0, momentum wheel frame fix when time domain kinetics equation be：
In formula, subscript f represents the rotor free degree, behalf gimbal freedom, MffFor rotor quality matrix, KffIt is firm for rotor Spend matrix,
F1(t) line number and MffIt is identical, F1(t) it is perturbed force, F suffered by momenttum wheel rotor2(t) line number and MssIt is identical, F2(t) For restraining force of the rigid interface to momenttum wheel.
The corresponding frequency domain kinetics equation of formula (2) is：
In formula, F1For the perturbed force suffered by the rotor in momenttum wheel, F2The rigid interface restraining force fixed for momenttum wheel, F1、F2It is ω function, i.e. F1=F1(ω)、F2=F2(ω).ω is that momenttum wheel vibrates circular frequency, xfFor xf(t) corresponding frequency Thresholding.
(3) transitive relation for the rigid interface restraining force that the perturbed force suffered by the rotor in momenttum wheel is fixed with momenttum wheel It is expressed as follows：
Ksf[-ω2Mff+Kff]-1{F1}={ F2}····················(4)
In formula, Ksf=-Kff, equation (4) is transformed to
In formula, E is unit matrix, its dimension and MffIt is identical；
Formula (5) conversion is obtained
(4) according to formula (6), the math equation for setting up the real structure of momenttum wheel is as follows：
In formula,For the mass matrix for real momenttum wheel of waiting to look for the truth,For the stiffness matrix for real momenttum wheel of waiting to look for the truth,For the true momenttum wheel rigid interface restraining force of measurement gained.
(5) (3) calculate gained { F2With measuring gainedIn the presence of certain error, if：
Wherein
Formula (7) subtracts formula (6) and using formula (8) and formula (9), obtains：
If KffFor parameter p1、p2、…、piFunction, i.e. Kff=Kff(p1,p2,…,pi), MffFor parameter pi+1、…、pn's Function, i.e. Mff=Mff(pi+1,pi+2,…,pn), can be to the matrix in formula (10)Carry out Taylor expansion：
Wherein Δ p1、Δp2、…、Δpi、Δpi+1、…、ΔpnFor Kff、MffMiddle p1、p2、…、pi、pi+1、…、pnWith structure The deviation of actual parameter.
Formula (11) is substituted into equation (10) to obtain
In formula,[S] represents sensitivity matrix：
(6) according to formula (12), equation is set up respectively at different frequencies omegas, it is as follows：
Solution formula (13), obtains momenttum wheel mass matrix MffWith momenttum wheel stiffness matrix KffThe correction value of parameterRevised mass matrix：
Stiffness matrix：
The undamped-free vibration equation for finally giving true momenttum wheel is：
Wherein
So far the amendment to momenttum wheel model via dynamical response is completed, momenttum wheel disturbance indication is necessarily improved after amendment Precision；The test data that is utilized of the present invention need not in momenttum wheel main structure install sensor, momenttum wheel will not be broken Protection structure；This makeover process reduces amendment using the mass matrix of momenttum wheel, Element of Stiffness Matrix as amendment object Amount of calculation, improves analysis efficiency.
(7) the 9255B type force plate/platforms produced preferably by Kistler companies of Switzerland, carry out momenttum wheel perturbed force (square) Measurement.Momenttum wheel constitutes sketch and sees Figure of description 2, and momenttum wheel includes rotor and framework, and framework is fixed on dynamometry and put down during measurement On platform, working frame is fixed on spacecraft.Momenttum wheel disturbance measuring state is shown in Figure of description 3.Now, momenttum wheel is fixed on When carrying out perturbed force measurement on force plate/platform.Momenttum wheel free degree x (t) is divided into two groups, is respectively：It is not connected with rigid interface Internal degree of freedom (i.e. the rotor free degree) xf(t) the border free degree (i.e. gimbal freedom) x and in rigid interfaces(t), I.e.
WhereinFor the Mass matrix of momenttum wheel, C is momenttum wheel damping matrix, is provided by engineering experience. The damping parameter matrix result of certain domestic model momenttum wheel is as follows：
For momenttum wheel Stiffness Matrix.
(8) according to the internal degree of freedom not being connected with rigid interface, the border free degree being connected with rigid interface general, C, enter Row piecemeal：
Equation (15) is changed into
(9) under frequency domain, the dynamics equations of momenttum wheel under the rigid interface of step (8) are converted into：
In formula, xf、xs、F1、F2X is represented respectivelyf(t)、xs(t)、F1And F (t)2(t) corresponding domain complex amount.
(10) Dynamic Stiffness Matrix of momenttum wheel is setWill Formula (17) formula, i.e., can be converted into by Z piecemeals：
Wherein
When momenttum wheel is fixed on force plate/platform, xs=0, formula (18) is substituted into, the restraining force F of interface is fixed2, It is as follows：
(11) FEM model of spacecraft is set up, momenttum wheel is arranged on spacecraft (see Figure of description 4), is set up The coupled dynamical equation of momenttum wheel and the spacecraft, according to momenttum wheel rotor displacement xf, momentum wheel frame displacement xsWith spacecraft Node xk, the coupled dynamical equation is subjected to piecemeal：
In formula, subscript " k " represents the label of the node on spacecraft FEM model.xkFor the modal displacement of spacecraft..
Obtained by formula (20)：
IfThen
(12) in the momenttum wheel installation interface applying power-F of spacecraft2(as shown in Figure 3 and Figure 5), momenttum wheel and spacecraft The coupled dynamical equation (20) is changed into：
In formula,For momenttum wheel rotor displacement,For momentum wheel frame displacement,For spacecraft modal displacement.
Formula (19) is substituted into formula (24), obtained：
(13) reduced equation (22) and (26), equation (23) and (27), it is known that, now：
Understand when in spacecraft disturbance interface application force vectorCalculating obtains kinetics equation solution and equation (20) Solution is identical.So far, obtaining momenttum wheel installation interface equivalent excitation power is
(14) (27) are solved equation, momenttum wheel is obtained and works the disturbance response caused to spacecraft, as shown in Figure 1.
Fig. 1 is the perturbation response curve of certain point on spacecraft, and wherein abscissa represents the π of ω/2, and ordinate is on spacecraft The acceleration amplitude of certain node.
Using certain domestic model momenttum wheel as object, perturbed force measurement is carried out using force plate/platform.Using provided by the present invention Momenttum wheel installation interface equivalent excitation power acquisition methods obtain equivalent excitation power, then carry out disturbance response analysis, analyze To spacecraft position simulation result and test measurement result contrasted such as following table：.
The response analysis result of table 1
As shown in table 1, the exciting force obtained using this paper institutes extracting method carries out momenttum wheel disturbance response analysis, obtained boat Its device point acceleration responsive error maximum is no more than 5%, illustrates that invention institute offer method is rationally effective, precision is higher.

Claims (2)

1. a kind of momenttum wheel and the acquisition methods of spacecraft installation interface equivalent excitation power, it is characterised in that step is as follows：
(1) propose that the dynamics equations of momenttum wheel are as follows：
<mrow> <msub> <mi>M</mi> <mi>f</mi> </msub> <mo>{</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>+</mo> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>{</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>+</mo> <msub> <mi>K</mi> <mi>f</mi> </msub> <mo>{</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mo>{</mo> <mi>f</mi> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
X (t) is the displacement of t momenttum wheel in formula,For the speed of t momenttum wheel,For t momenttum wheel Acceleration；{ f } is the excitation suffered by momenttum wheel；
<mrow> <msub> <mi>M</mi> <mi>f</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi>I</mi> <mi>r</mi> <mi>r</mi> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi>I</mi> <mi>r</mi> <mi>r</mi> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
In formula, m is the quality of momenttum wheel, and Irr is that momenttum wheel is used to around the rotation of radial direction；
<mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>c</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>c</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>c</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>c</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>c</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>c</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>cd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
C in formulaazAxially damped for momenttum wheel, c damps for the translation of momentum wheel radial, cd2For momenttum wheel damping of oscillations；
<mrow> <msub> <mi>K</mi> <mi>f</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow> 1
In formula, k in formulaazFor momenttum wheel axial rigidity, k is momentum wheel radial translation rigidity, kd2Rigidity is swung for momenttum wheel；ωrThe radial direction translation mode angular frequency of mould measurement determination is represented, according to engineering experience, preliminary design ωr=1e4 is obtained k；Wherein represent the axial translation mode angular frequency of mould measurement determination, preliminary design ωaz=2e4 obtains kaz；Wherein ωswingRepresent mould measurement determination waves mode angular frequency, preliminary design ωswing=1.5e3 is obtained kd2；
(2) the i.e. C when not considering the damping of momenttum wheelf=0, momentum wheel frame fix when time domain kinetics equation be：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>{</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>{</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
In formula, subscript f represents the rotor free degree, behalf gimbal freedom, MffFor rotor quality matrix, KffFor rotor rigidity square Battle array,
<mrow> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mi>m</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi>I</mi> <mi>r</mi> <mi>r</mi> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi>I</mi> <mi>r</mi> <mi>r</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
<mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mi>k</mi> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>z</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msup> <mi>kd</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo>{</mo> <mi>f</mi> <mo>}</mo> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
F1(t) line number and MffIt is identical, F1(t) it is perturbed force, F suffered by momenttum wheel rotor2(t) line number and MssIt is identical, F2(t) it is firm Restraining force of the property interface to momenttum wheel；
The corresponding frequency domain kinetics equation of formula (2) is：
<mrow> <mo>(</mo> <mo>-</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mi>f</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mo>...</mo> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow>
In formula, F1For the perturbed force suffered by the rotor in momenttum wheel, F2The rigid interface restraining force fixed for momenttum wheel, F1、F2 It is ω function, i.e. F1=F1(ω)、F2=F2(ω)；ω is that momenttum wheel vibrates circular frequency, xfFor xf(t) corresponding frequency domain Value；
(3) transitive relation for the rigid interface restraining force that the perturbed force suffered by the rotor in momenttum wheel is fixed with momenttum wheel is represented It is as follows：
Ksf[-ω2Mff+Kff]-1{F1}={ F2}…………………………(4)
In formula, Ksf=-Kff, equation (4) is transformed to
<mrow> <mo>{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>}</mo> <mo>=</mo> <mrow> <mo>(</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>-</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
In formula, E is unit matrix, its dimension and MffIt is identical；
Formula (5) conversion is obtained
<mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>}</mo> <mo>+</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
(4) according to formula (6), the math equation for setting up the real structure of momenttum wheel is as follows：
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>}</mo> <mo>+</mo> <mo>{</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mi>X</mi> </msubsup> <mo>}</mo> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>{</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mi>X</mi> </msubsup> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
In formula,For the mass matrix for real momenttum wheel of waiting to look for the truth,For the stiffness matrix for real momenttum wheel of waiting to look for the truth,To survey The true momenttum wheel rigid interface restraining force of amount gained；
(5) formula (3) calculates gained { F2With measuring gainedIn the presence of certain error, if：
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>...</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mo>{</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mi>X</mi> </msubsup> <mo>}</mo> <mo>=</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>+</mo> <mi>&amp;Delta;</mi> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein
Formula (7) subtracts formula (6) and using formula (8) and formula (9), obtains：
<mrow> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>}</mo> <mo>+</mo> <mo>{</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mi>x</mi> </msubsup> <mo>}</mo> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mi>E</mi> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mi>&amp;Delta;</mi> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
If KffFor parameter p1、p2、…、piFunction, i.e. Kff=Kff(p1,p2,…,pi), MffFor parameter pi+1、…、pnFunction, That is Mff=Mff(pi+1,pi+2,…,pn), to the matrix in formula (10)Carry out Taylor expansion：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>&amp;Delta;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </mfrac> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;Delta;p</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </mfrac> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;Delta;p</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>...</mn> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;Delta;p</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <msub> <mi>&amp;Delta;p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mfrac> <mrow> <mo>&amp;part;</mo> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </mfrac> <msub> <mi>&amp;Delta;p</mi> <mi>n</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>...</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Wherein Δ p1、Δp2、…、Δpi、Δpi+1、…、ΔpnFor Kff、MffMiddle p1、p2、…、pi、pi+1、…、pnIt is true with structure The deviation of parameter；
Formula (11) is substituted into equation (10) to obtain
<mrow> <mo>&amp;lsqb;</mo> <mi>S</mi> <mo>&amp;rsqb;</mo> <mo>{</mo> <mi>&amp;Delta;</mi> <mi>p</mi> <mo>}</mo> <mo>=</mo> <mo>&amp;lsqb;</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mi>E</mi> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mi>&amp;Delta;</mi> <mo>{</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>}</mo> <mo>...</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
In formula,[S] represents sensitivity matrix：
<mrow> <mo>&amp;lsqb;</mo> <mi>S</mi> <mo>&amp;rsqb;</mo> <mo>=</mo> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </mfrac> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <mo>...</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <mo>{</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>}</mo> <mo>+</mo> <mo>{</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mi>x</mi> </msubsup> <mo>}</mo> <mo>)</mo> </mrow> <mo>...</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
(6) according to formula (12), equation is set up respectively at different frequencies omegas, it is as follows：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>{</mo> <mrow> <mi>&amp;Delta;</mi> <mi>p</mi> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;omega;</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mi>E</mi> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;omega;</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mi>E</mi> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;omega;</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mi>E</mi> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
Solution formula (13), obtains momenttum wheel mass matrix MffWith momenttum wheel stiffness matrix KffThe correction value of parameterRevised mass matrix：
<mrow> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow>
Stiffness matrix：
<mrow> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> <mo>=</mo> <msub> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;p</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
Finally give the undamped-free vibration equation of true momenttum wheel；
(7) when momenttum wheel, which is fixed on, carries out perturbed force measurement on force plate/platform, the free degree x (t) of momenttum wheel is divided into two groups, Respectively：The internal degree of freedom not being connected with rigid interface, i.e. rotor free degree xf(t) border and in rigid interface is free Degree, i.e. gimbal freedom xs(t), i.e.,
<mrow> <mi>M</mi> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>C</mi> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>K</mi> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
WhereinFor the Mass matrix of momenttum wheel, C is momenttum wheel damping matrix,For momentum Take turns Stiffness Matrix；
(8) the border free degree being connected according to the internal degree of freedom not being connected with rigid interface, with rigid interface is divided C Block：
<mrow> <mi>C</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow>
Equation (15) is changed into the dynamics equations of momenttum wheel under rigid interface, as follows：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
(9) under frequency domain, the dynamics equations of momenttum wheel under the rigid interface of step (8) are converted into：
<mrow> <mo>(</mo> <mo>-</mo> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>i</mi> <mi>&amp;omega;</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mi>f</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>s</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow>
In formula, xf、xs、F1、F2X is represented respectivelyf(t)、xs(t)、F1And F (t)2(t) corresponding domain complex amount；
(10) Dynamic Stiffness Matrix of momenttum wheel is setBy Z points Formula (17) formula, i.e., can be converted into by block：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mi>f</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>s</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
Wherein
The framework of momenttum wheel is fixed on force plate/platform, xs=0, formula (18) is substituted into, the restraining force F of interface is fixed2, It is as follows
<mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
(11) spacecraft structure FEM model is set up, momenttum wheel is arranged on spacecraft, momenttum wheel and the spacecraft is set up The coupled dynamical equation, according to momenttum wheel rotor displacement xf, momentum wheel frame displacement xs, spacecraft node xk, power will be coupled Learn equation and carry out piecemeal：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>x</mi> <mi>f</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>s</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mi>F</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
In formula, subscript " k " represents the label of the node on spacecraft FEM model, xkFor the section on spacecraft FEM model Point displacement；
Obtained by formula (20)：
<mrow> <msub> <mi>x</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>-</mo> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>x</mi> <mi>s</mi> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>x</mi> <mi>s</mi> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mo>(</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>k</mi> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>...</mo> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow>
IfThen
(12) in the momenttum wheel installation interface applying power-F of spacecraft2, momenttum wheel and spacecraft the coupled dynamical equation (20) become For：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>f</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>s</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>...</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
In formula,For momenttum wheel rotor displacement,For momentum wheel frame displacement,For spacecraft modal displacement；
Formula (19) is substituted into formula (24), obtained：
<mrow> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>f</mi> </msub> <mo>=</mo> <mo>-</mo> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>s</mi> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow> 5
<mrow> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <msubsup> <mi>Z</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow> <mi>k</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>s</mi> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>T</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>...</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
(13) reduced equation (22) and (26), equation (23) and (27), it is known that, now：
<mrow> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>s</mi> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;OverBar;</mo> </mover> <mi>f</mi> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>f</mi> </msub> </mrow>
Understand when in spacecraft disturbance interface application force vectorCalculating obtains kinetics equation solution and solves phase with equation (20) Together；So far, obtaining momenttum wheel installation interface equivalent excitation power is
(14) (27) are solved equation, the disturbance response caused during momenttum wheel work to spacecraft is obtained.
2. a kind of momenttum wheel according to claim 1 and the acquisition methods of spacecraft installation interface equivalent excitation power, it is special Levy and be：The undamped-free vibration equation that step (6) finally gives true momenttum wheel is：
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>M</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>{</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>f</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> <mi>X</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>K</mi> <mrow> <mi>s</mi> <mi>s</mi> </mrow> <mi>X</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>{</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mn>0</mn> </mrow>
Wherein
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