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CN107291988B - Method for acquiring equivalent excitation force of momentum wheel mounting interface - Google Patents

Method for acquiring equivalent excitation force of momentum wheel mounting interface Download PDF

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CN107291988B
CN107291988B CN201710376825.7A CN201710376825A CN107291988B CN 107291988 B CN107291988 B CN 107291988B CN 201710376825 A CN201710376825 A CN 201710376825A CN 107291988 B CN107291988 B CN 107291988B
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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 method for obtaining equivalent excitation force of a momentum wheel and a spacecraft installation interface utilizes test data without installing a sensor on a momentum wheel main structure and breaking a protection structure of the momentum wheel, is simple and easy to implement in a test scheme, takes mass matrix and rigidity matrix elements of the momentum wheel as correction objects, reduces the calculated amount of correction, improves the analysis efficiency, and is high in analysis precision when the equivalent excitation force of the momentum wheel installation interface provided by the invention is utilized to perform disturbance response analysis. The equivalent exciting force of the momentum wheel obtained by the method is applied to the installation interface of the momentum wheel and the spacecraft, so that the coupling effect between the spacecraft and the momentum wheel can be accurately reflected, and the precision of the disturbance analysis prediction of the momentum wheel is improved.

Description

一种动量轮安装界面等效激励力获取方法A method for obtaining equivalent excitation force of momentum wheel installation interface

技术领域technical field

本发明涉及一种动量轮安装界面等效激励力获取方法，本发明提供的方法用于动量轮的扰动响应分析。The invention relates to a method for obtaining the equivalent excitation force of a momentum wheel installation interface, and the method provided by the invention is used for the disturbance response analysis of the momentum wheel.

背景技术Background technique

卫星在轨微小振动将对空间科学试验、激光通信、光学遥感等任务产生影响，其中动量轮与控制力矩陀螺是重要的扰动源，因此动量轮与控制力矩陀螺的扰动响应分析十分重要。国内外的学者很早就对扰动源开展了测量工作，研究发现部件转动引起的扰动会被自身结构放大。为了去除这种影响，有学者根据Masterson R A,Miller D W提出的扰动谐波经验模型，利用试验数据辨识出相关参数，拟合出不同转速下的扰动力(矩)进行扰动分析。由于扰动源结构对扰动的放大作用，辨识结果往往是偏大的；考虑到固定界面条件并不能精确地代表扰源安装在航天器上时的真实界面条件，Laila Mireille Elias等人提出了静态加速性分析法,Zhe Zhang Guglielmo S等人在2013年提出了一种基于动态加速性的飞轮微振动分析方法。该方法利用扰动源与柔性基础结构的加速性，对测量得到的扰动力(矩)进行修正，修正后的数据反映了扰源与航天器结构的耦合特性，可直接加载于动力学分析模型上进行扰动分析。该方法所用的修正系数可通过有限元分析或者试验测量而得，但对不同的航天器修正系数是不同的，需要重新计算，这增加了分析与试验的工作内容且其精度有限。The small vibration of satellites in orbit will have an impact on tasks such as space science experiments, laser communication, and optical remote sensing. Among them, the momentum wheel and the control torque gyroscope are important disturbance sources. Therefore, the disturbance response analysis of the momentum wheel and the control torque gyroscope is very important. Scholars at home and abroad have carried out measurement work on the disturbance source for a long time, and the study found that the disturbance caused by the rotation of the component will be amplified by its own structure. In order to remove this influence, some scholars use the experimental data to identify the relevant parameters according to the disturbance harmonic empirical model proposed by Masterson R A and Miller D W, and fit the disturbance force (moment) at different speeds for disturbance analysis. Due to the amplifying effect of the disturbance source structure on the disturbance, the identification results are often too large; considering that the fixed interface conditions cannot accurately represent the real interface conditions when the disturbance source is installed on the spacecraft, Laila Mireille Elias et al. proposed a static acceleration In 2013, Zhe Zhang Guglielmo S et al proposed a flywheel microvibration analysis method based on dynamic acceleration. This method uses the acceleration of the disturbance source and the flexible infrastructure to correct the measured disturbance force (moment). The corrected data reflects the coupling characteristics of the disturbance source and the spacecraft structure, and can be directly loaded on the dynamic analysis model. Perform a perturbation analysis. The correction factor used in this method can be obtained by finite element analysis or test measurement, but the correction factor is different for different spacecraft and needs to be recalculated, which increases the work content of analysis and test and its accuracy is limited.

发明内容SUMMARY OF THE INVENTION

本发明解决的技术问题为：克服现有技术不足，提出一种动量轮安装界面等效激励力获取方法，这种方法根据扰动力在动量轮在结构内的传递特性，给出了模型分析计算得出的刚性界面约束力、试验实测的刚性界面约束力之间的差与模型质量、刚度矩阵元素的关系，并将矩阵内的元素作为对象进行修正，以使实测模态和分析模态相关，使模型的计算结果和实际测试结果一致；此外，发明获取的安装界面等效激励力是一种精确的微振动源解耦加载方法，这种方法考虑了动量轮自身结构对扰动的放大作用，适用于航天器的扰动响应分析。The technical problem solved by the invention is: to overcome the deficiencies of the prior art, a method for obtaining the equivalent excitation force at the installation interface of the momentum wheel is proposed. This method provides a model analysis calculation method according to the transmission characteristics of the disturbance force in the momentum wheel in the structure. The relationship between the obtained rigid interface restraint force and the difference between the rigid interface restraint force measured by the test and the model mass and stiffness matrix elements, and the elements in the matrix are used as objects to be corrected, so that the measured mode and the analysis mode are related. , so that the calculated results of the model are consistent with the actual test results; in addition, the equivalent excitation force of the installation interface obtained by the invention is an accurate decoupling loading method of the micro-vibration source, which takes into account the amplifying effect of the momentum wheel itself on the disturbance. , which is suitable for disturbance response analysis of spacecraft.

本发明解决的技术方案为：一种动量轮与航天器安装界面等效激励力的获取方法，步骤如下：The technical scheme solved by the present invention is: a method for obtaining the equivalent excitation force of the momentum wheel and the installation interface of the spacecraft, the steps are as follows:

(1)提出动量轮的结构动力学方程如下：(1) The structural dynamics equation of the momentum wheel is proposed as follows:

Figure GDA0002502571480000021
Figure GDA0002502571480000021

式中x(t)为t时刻动量轮的位移,

Figure GDA0002502571480000022
为t时刻动量轮的速度，
Figure GDA0002502571480000023
为t时刻动量轮的加速度；{f}为动量轮所受的激励；where x(t) is the displacement of the momentum wheel at time t,
Figure GDA0002502571480000022
is the velocity of the momentum wheel at time t,
Figure GDA0002502571480000023
is the acceleration of the momentum wheel at time t; {f} is the excitation of the momentum wheel;

Figure GDA0002502571480000024
Figure GDA0002502571480000024

式中，m为动量轮的质量，Irr为动量轮绕径向的转动惯。In the formula, m is the mass of the momentum wheel, and Irr is the rotational inertia of the momentum wheel around the radial direction.

Figure GDA0002502571480000031
Figure GDA0002502571480000031

式中caz为动量轮轴向阻尼，c为动量轮径向平动阻尼，cd2为动量轮摆动阻尼。where c az is the axial damping of the momentum wheel, c is the radial translation damping of the momentum wheel, and cd 2 is the swing damping of the momentum wheel.

Figure GDA0002502571480000032
Figure GDA0002502571480000032

式中，式中kaz为动量轮轴向刚度，k为动量轮径向平动刚度，kd2为动量轮摆动刚度。

Figure GDA0002502571480000033
ωr代表模态测试确定的径向平动模态角频率，根据工程经验，初设ωr＝1e4得到k；
Figure GDA0002502571480000034
其中ωaz代表模态测试确定的轴向平动模态角频率，初设ωaz＝2e4得到kaz；
Figure GDA0002502571480000035
其中ωswing代表模态测试确定的摇摆模态角频率，初设ωswing＝1.5e3得到kd2。where k az is the axial stiffness of the momentum wheel, k is the radial translation stiffness of the momentum wheel, and kd 2 is the swing stiffness of the momentum wheel.
Figure GDA0002502571480000033
ω r represents the radial translation modal angular frequency determined by the modal test. According to engineering experience, k is obtained by initially setting ω r = 1e 4 ;
Figure GDA0002502571480000034
where ω az represents the axial translational modal angular frequency determined by the modal test, and ka az is obtained by initially setting ω az = 2e 4 ;
Figure GDA0002502571480000035
Where ω swing represents the angular frequency of the swing mode determined by the modal test, and kd 2 is obtained by initially setting ω swing =1.5e 3 .

(2)当不考虑动量轮的阻尼时即Cf＝0，动量轮框架固定时的时域动力学方程为：(2) When the damping of the momentum wheel is not considered, that is, C f = 0, the time-domain dynamic equation when the frame of the momentum wheel is fixed is:

Figure GDA0002502571480000036
Figure GDA0002502571480000036

式中，下标f代表转子自由度，s代表框架自由度，Mff为转子质量矩阵，Kff为转子刚度矩阵，

Figure GDA0002502571480000041
In the formula, the subscript f represents the degree of freedom of the rotor, s represents the degree of freedom of the frame, M ff is the rotor mass matrix, K ff is the rotor stiffness matrix,
Figure GDA0002502571480000041

Figure GDA0002502571480000042
Figure GDA0002502571480000043
F1(t)行数与Mff相同，F1(t)为动量轮转子所受扰动力，F2(t)行数与Mss相同，F2(t)为刚性界面对动量轮的约束力。
Figure GDA0002502571480000042
Figure GDA0002502571480000043
The number of rows of F 1 (t) is the same as that of M ff , F 1 (t) is the disturbance force on the rotor of the momentum wheel, the number of rows of F 2 (t) is the same as that of M ss , and F 2 (t) is the force of the rigid interface on the momentum wheel. binding force.

式(2)对应的频域动力学方程为：The frequency domain dynamic equation corresponding to formula (2) is:

Figure GDA0002502571480000044
Figure GDA0002502571480000044

式中，F1为动量轮内的转子所受到的扰动力，F2为动量轮固定的刚性界面约束力，F1的频域表达形式为F1(ω)、F2的频域表达形式为F2(ω)，ω为动量轮振动圆频率，xf为xf(t)对应的频域值。In the formula, F 1 is the disturbance force received by the rotor in the momentum wheel, F 2 is the rigid interface constraint force fixed by the momentum wheel, and the frequency domain expression form of F 1 is the frequency domain expression form of F 1 (ω) and F 2 is F 2 (ω), ω is the vibration circular frequency of the momentum wheel, and x f is the frequency domain value corresponding to x f (t).

(3)动量轮内的转子所受到的扰动力与动量轮固定的刚性界面约束力的传递关系表示如下：(3) The transmission relationship between the disturbance force received by the rotor in the momentum wheel and the rigid interface constraint force fixed by the momentum wheel is expressed as follows:

Ksf[-ω2Mff+Kff]-1{F1}＝{F2} ·················(31)K sf [-ω 2 M ff +K ff ] -1 {F 1 }={F 2 }

式中，Ksf＝-Kff，方程(4)变换为In the formula, K sf =-K ff , and equation (4) is transformed into

Figure GDA0002502571480000045
Figure GDA0002502571480000045

式中，E为单位矩阵，其维度与Mff相同；In the formula, E is the identity matrix, and its dimension is the same as Mff ;

对式(5)变换得到Transform equation (5) to get

Figure GDA0002502571480000046
Figure GDA0002502571480000046

(4)根据式(6)，建立动量轮的真实结构的数学方程如下：(4) According to formula (6), the mathematical equation to establish the real structure of the momentum wheel is as follows:

Figure GDA0002502571480000051
Figure GDA0002502571480000051

式中，

Figure GDA0002502571480000052
为待求真实动量轮的质量矩阵，
Figure GDA0002502571480000053
为待求真实动量轮的刚度矩阵，
Figure GDA0002502571480000054
为测量所得真实动量轮刚性界面约束力。In the formula,
Figure GDA0002502571480000052
is the mass matrix of the real momentum wheel to be found,
Figure GDA0002502571480000053
is the stiffness matrix of the real momentum wheel to be obtained,
Figure GDA0002502571480000054
is the rigid interface constraint force of the real momentum wheel obtained from the measurement.

(5)式(3)计算所得{F2}与测量所得

Figure GDA0002502571480000055
存在一定的误差，设：(5) Calculated {F 2 } from formula (3) and measured
Figure GDA0002502571480000055
There is a certain error, let:

Figure GDA0002502571480000056
Figure GDA0002502571480000056

Figure GDA0002502571480000057
Figure GDA0002502571480000057

其中

Figure GDA0002502571480000058
in
Figure GDA0002502571480000058

式(7)减去式(6)并利用式(8)与式(9)，得到：Subtracting Equation (6) from Equation (7) and using Equation (8) and Equation (9), we get:

Figure GDA0002502571480000059
Figure GDA0002502571480000059

设Kff表达为参数p1、p2、…、pi的函数，即Kff＝Kff(p1,p2,…,pi)，Mff表达为参数pi+1、…、pn的函数，即Mff＝Mff(pi+1,pi+2,…,pn)，对式(10)中的矩阵

Figure GDA00025025714800000510
进行泰勒展开：Let Kff be expressed as a function of parameters p 1 , p 2 ,..., p i , that is, K ff =K ff (p 1 ,p 2 ,...,pi ), and M ff be expressed as parameters p i +1 ,..., The function of pn , namely M ff =M ff (pi +1 ,pi +2 ,...,p n ), for the matrix in equation (10)
Figure GDA00025025714800000510
Perform Taylor expansion:

Figure GDA00025025714800000511
Figure GDA00025025714800000511

其中Δp1、Δp2、…、Δpi、Δpi+1、…、Δpn为Kff、Mff中p1、p2、…、pi、pi+1、…、pn与结构真实参数的偏差。where Δp 1 , Δp 2 , ..., Δpi , Δpi +1 , ..., Δpn are p 1 , p 2 , ..., p i , p i+1 , ..., pn and structure in K ff , M ff The deviation of the true parameters.

将式(11)代入方程(10)得到Substitute equation (11) into equation (10) to get

Figure GDA00025025714800000512
Figure GDA00025025714800000512

式中，

Figure GDA00025025714800000513
[S]代表灵敏度矩阵：In the formula,
Figure GDA00025025714800000513
[S] represents the sensitivity matrix:

Figure GDA0002502571480000061
Figure GDA0002502571480000061

(6)根据公式(12)，在不同的频率ω处分别建立方程，如下：(6) According to formula (12), establish equations respectively at different frequencies ω, as follows:

Figure GDA0002502571480000062
Figure GDA0002502571480000062

求解式(13)，得到动量轮质量矩阵Mff与动量轮刚度矩阵Kff参数的修正值

Figure GDA0002502571480000063
修正后的质量矩阵：Solve Equation (13) to get the modified values of the momentum wheel mass matrix M ff and the momentum wheel stiffness matrix K ff
Figure GDA0002502571480000063
Corrected mass matrix:

Figure GDA0002502571480000064
Figure GDA0002502571480000064

刚度矩阵：Stiffness Matrix:

Figure GDA0002502571480000065
Figure GDA0002502571480000065

最终得到真实动量轮的无阻尼自由振动方程为：Finally, the undamped free vibration equation of the real momentum wheel is obtained as:

Figure GDA0002502571480000066
Figure GDA0002502571480000066

其中

Figure GDA0002502571480000067
in
Figure GDA0002502571480000067

(7)当动量轮固定在测力平台上进行扰动力测量时，将动量轮的自由度x(t)分成两组，分别为：不与刚性界面相连的内部自由度，即转子自由度xf(t)和在刚性界面上的边界自由度，即框架自由度xs(t)，即

Figure GDA0002502571480000068
(7) When the momentum wheel is fixed on the force measuring platform for disturbance force measurement, the degrees of freedom x(t) of the momentum wheel are divided into two groups, namely: the internal degrees of freedom not connected to the rigid interface, namely the rotor degrees of freedom x f (t) and the boundary degrees of freedom on the rigid interface, i.e. the frame degrees of freedom x s (t), i.e.
Figure GDA0002502571480000068

Figure GDA0002502571480000069
Figure GDA0002502571480000069

其中

Figure GDA0002502571480000071
为动量轮的质量阵，C为动量轮阻尼矩阵，由工程经验验给出，
Figure GDA0002502571480000072
为动量轮刚度阵。in
Figure GDA0002502571480000071
is the mass matrix of the momentum wheel, C is the damping matrix of the momentum wheel, which is given by engineering experience,
Figure GDA0002502571480000072
is the momentum wheel stiffness matrix.

(8)按照不与刚性界面相连的内部自由度、与刚性界面连接的边界自由度将C进行分块：(8) Divide C into blocks according to the internal degrees of freedom not connected to the rigid interface and the boundary degrees of freedom connected to the rigid interface:

Figure GDA0002502571480000073
Figure GDA0002502571480000073

方程(15)变为刚性界面下动量轮的结构动力学方程，如下：Equation (15) becomes the structural dynamics equation of the momentum wheel under rigid interface as follows:

Figure GDA0002502571480000074
Figure GDA0002502571480000074

(9)在频域下，将步骤(8)的刚性界面下动量轮的结构动力学方程转化为：(9) In the frequency domain, transform the structural dynamics equation of the momentum wheel under the rigid interface in step (8) into:

Figure GDA0002502571480000075
Figure GDA0002502571480000075

式中，xf、xs、F1、F2分别表示xf(t)、xs(t)、F1(t)和F2(t)对应的频域复数量。In the formula, x f , x s , F 1 , and F 2 represent the frequency-domain complex quantities corresponding to x f (t), x s (t), F 1 (t), and F 2 (t), respectively.

(10)设动量轮的动刚度矩阵

Figure GDA0002502571480000076
将Z分块，即将公式(17)式可转化为：(10) Set the dynamic stiffness matrix of the momentum wheel
Figure GDA0002502571480000076
Divide Z into blocks, that is, formula (17) can be transformed into:

Figure GDA0002502571480000077
Figure GDA0002502571480000077

其中

Figure GDA0002502571480000078
in
Figure GDA0002502571480000078

Figure GDA0002502571480000079
Figure GDA0002502571480000079

动量轮的框架固定于测力平台上，xs＝0，代入公式(18)，得到固定界面处的约束力F2，如下The frame of the momentum wheel is fixed on the force measuring platform, x s = 0, and substituting into formula (18), the binding force F 2 at the fixed interface is obtained, as follows

Figure GDA00025025714800000710
Figure GDA00025025714800000710

(11)建立航天器结构有限元模型，将动量轮安装在航天器上，建立动量轮与该航天器的耦合动力学方程，按照动量轮转子位移xf、动量轮框架位移xs、航天器节点xk，将耦合动力学方程进行分块：(11) Establish a finite element model of the spacecraft structure, install the momentum wheel on the spacecraft, and establish the coupled dynamics equation between the momentum wheel and the spacecraft. According to the momentum wheel rotor displacement x f , the momentum wheel frame displacement x s , the spacecraft Node x k , which blocks the coupled dynamics equations:

Figure GDA0002502571480000081
Figure GDA0002502571480000081

式中，下标“k”代表航天器有限元模型上的节点的标号，xk为航天器的节点位移；In the formula, the subscript "k" represents the label of the node on the finite element model of the spacecraft, and x k is the node displacement of the spacecraft;

由式(20)得到：It can be obtained by formula (20):

Figure GDA0002502571480000082
Figure GDA0002502571480000082

Figure GDA0002502571480000083
Figure GDA0002502571480000083

Figure GDA0002502571480000084
Figure GDA0002502571480000084

设

Figure GDA0002502571480000085
则
Figure GDA0002502571480000086
Assume
Figure GDA0002502571480000085
but
Figure GDA0002502571480000086

(12)在航天器的动量轮安装界面施加力-F2，动量轮和航天器耦合动力学方程(20)变为：(12) When a force -F 2 is applied on the installation interface of the momentum wheel of the spacecraft, the coupling dynamic equation (20) of the momentum wheel and the spacecraft becomes:

Figure GDA0002502571480000087
Figure GDA0002502571480000087

式中，

Figure GDA0002502571480000088
为动量轮转子位移，
Figure GDA0002502571480000089
为动量轮框架位移，
Figure GDA00025025714800000810
为航天器节点位移。In the formula,
Figure GDA0002502571480000088
is the rotor displacement of the momentum wheel,
Figure GDA0002502571480000089
is the displacement of the momentum wheel frame,
Figure GDA00025025714800000810
is the node displacement of the spacecraft.

将式(19)代入式(24)，得到：Substituting equation (19) into equation (24), we get:

Figure GDA00025025714800000811
Figure GDA00025025714800000811

Figure GDA00025025714800000812
Figure GDA00025025714800000812

Figure GDA00025025714800000813
Figure GDA00025025714800000813

(13)对比方程(22)与(26)、方程(23)与(27)，可知，此时：(13) Comparing equations (22) and (26), and equations (23) and (27), it can be known that at this time:

Figure GDA00025025714800000814
Figure GDA00025025714800000814

可知当在航天器扰动界面施加力向量

Figure GDA00025025714800000815
计算得到动力学方程解与方程(20)解相同。至此，得到动量轮安装界面等效激励力为
Figure GDA00025025714800000816
It can be seen that when a force vector is applied to the disturbance interface of the spacecraft
Figure GDA00025025714800000815
The calculated solution of the kinetic equation is the same as that of equation (20). So far, the equivalent excitation force of the momentum wheel installation interface is obtained as
Figure GDA00025025714800000816

(14)解方程(27)，得到动量轮工作时给航天器造成的扰动响应。(14) Equation (27) is solved to obtain the disturbance response to the spacecraft when the momentum wheel works.

本发明与现有技术相比的优点在于：The advantages of the present invention compared with the prior art are:

(1)本发明所提出的动量轮模型修正方法为动量轮的模型修正提供了方法依据，经修正后必然提高动量轮扰动预示的精度；(1) The momentum wheel model correction method proposed by the present invention provides a method basis for the model correction of the momentum wheel, and the accuracy of the disturbance prediction of the momentum wheel must be improved after the correction;

(2)本发明所利用的试验数据不需要在动量轮主结构上安装传感器，不会打破动量轮的保护结构，试验方案简单易行；(2) The test data used in the present invention does not need to install sensors on the main structure of the momentum wheel, and the protective structure of the momentum wheel will not be broken, and the test plan is simple and easy to implement;

(3)本发明以动量轮的质量矩阵、刚度矩阵元素作为修正对象，降低了修正的计算量，提高了分析效率；(3) The present invention uses the mass matrix and stiffness matrix elements of the momentum wheel as the correction object, which reduces the calculation amount of the correction and improves the analysis efficiency;

(4)利用本发明所提供的动量轮安装界面等效激励力进行扰动响应分析，分析精度高。(4) The perturbation response analysis is carried out by using the equivalent excitation force of the momentum wheel installation interface provided by the present invention, and the analysis precision is high.

(5)本发明通过推导结构动力学方程，发现了扰动在扰源-测量界面-卫星安装面内的传递规律，提出了动量轮安装界面等效激励力获取方法。这种方法根据扰动力在动量轮在结构内的传递特性，给出了模型分析计算得出的刚性界面约束力、试验实测的刚性界面约束力之间的差与模型质量、刚度矩阵元素的关系，并将矩阵内的元素作为对象进行修正，以使实测模态和分析模态相关，使模型的计算结果和实际测试结果一致；(5) The present invention discovers the transmission law of disturbance in the disturbance source-measurement interface-satellite installation surface by deriving the structural dynamic equation, and proposes a method for obtaining the equivalent excitation force of the momentum wheel installation interface. According to the transmission characteristics of the disturbance force in the momentum wheel in the structure, this method gives the relationship between the rigid interface restraint force calculated by the model analysis and the difference between the rigid interface restraint force measured by the test and the model mass and stiffness matrix elements , and modify the elements in the matrix as objects, so that the measured mode and the analysis mode are related, so that the calculation results of the model are consistent with the actual test results;

(6)本发明获取的安装界面等效激励力是一种精确的微振动源解耦加载方法，这种方法考虑了动量轮自身结构对扰动的放大作用，适用于航天器的扰动响应分析。(6) The equivalent excitation force of the installation interface obtained by the present invention is an accurate decoupling loading method of the micro-vibration source. This method considers the amplifying effect of the momentum wheel's own structure on the disturbance, and is suitable for the analysis of the disturbance response of the spacecraft.

附图说明Description of drawings

图1为本发明航天器上某点的扰动响应曲线；Fig. 1 is the disturbance response curve of a certain point on the spacecraft of the present invention;

图2为本发明动量轮构成简图；Figure 2 is a schematic diagram of the composition of the momentum wheel of the present invention;

图3为本发明刚性界面下的动量轮系统图；Fig. 3 is the momentum wheel system diagram under the rigid interface of the present invention;

图4为本发明动量轮+星体结构耦合系统图；Fig. 4 is the coupling system diagram of the momentum wheel + star structure of the present invention;

图5为本发明动量轮解耦加载系统图。FIG. 5 is a diagram of the decoupling loading system of the momentum wheel of the present invention.

具体实施方式Detailed ways

下面结合附图和具体实施例对本发明进行详细说明。The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

一种动量轮与航天器安装界面等效激励力的获取方法，利用的试验数据不需要在动量轮主结构上安装传感器，不会打破动量轮的保护结构，试验方案简单易行，以动量轮的质量矩阵、刚度矩阵元素作为修正对象，降低了修正的计算量，提高了分析效率，利用本发明所提供的动量轮安装界面等效激励力进行扰动响应分析，分析精度高。将方法获得的动量轮等效激励力施加于动量轮与航天器的安装界面，能够准确反应航天器与动量轮间的耦合作用，提高动量轮扰动分析预示的精度。A method for obtaining the equivalent excitation force at the installation interface of a momentum wheel and a spacecraft. The test data used does not need to install sensors on the main structure of the momentum wheel, and will not break the protective structure of the momentum wheel. The test scheme is simple and easy to implement. The mass matrix and stiffness matrix elements of the present invention are used as the correction objects, which reduces the calculation amount of correction and improves the analysis efficiency. The equivalent excitation force of the momentum wheel obtained by the method is applied to the installation interface of the momentum wheel and the spacecraft, which can accurately reflect the coupling effect between the spacecraft and the momentum wheel, and improve the prediction accuracy of the momentum wheel disturbance analysis.

动量轮的建模与加载方法对于整个微振动分析模型而言是最为重要的输入，十几年来一直是国内外微振动领域的研究热点之一。从原理上讲，动量轮本身也是柔性体，扰源作用的大小受到星体结构动态特性的影响，其实际作用机理是扰源与星体结构的耦合作用，需要采用耦合分析。利用本发明所提供的等效激励力获取方法进行分析，计算响应与真实响应一致。The modeling and loading method of the momentum wheel is the most important input for the whole micro-vibration analysis model, and it has been one of the research hotspots in the field of micro-vibration at home and abroad for more than ten years. In principle, the momentum wheel itself is also a flexible body, and the magnitude of the disturbance source is affected by the dynamic characteristics of the star structure. The actual mechanism of action is the coupling effect of the disturbance source and the star structure, which requires coupling analysis. The equivalent excitation force acquisition method provided by the present invention is used for analysis, and the calculated response is consistent with the real response.

本发明的步骤如下：The steps of the present invention are as follows:

为简化动量轮的动力学模型，认为其质量集中于转子，模型包含10个自由度。In order to simplify the dynamic model of the momentum wheel, it is considered that its mass is concentrated in the rotor, and the model contains 10 degrees of freedom.

(1)提出动量轮的结构动力学方程如下：(1) The structural dynamics equation of the momentum wheel is proposed as follows:

Figure GDA0002502571480000101
Figure GDA0002502571480000101

式中x(t)为t时刻动量轮的位移,

Figure GDA0002502571480000102
为t时刻动量轮的速度，
Figure GDA0002502571480000103
为t时刻动量轮的加速度；{f}为动量轮所受的激励；where x(t) is the displacement of the momentum wheel at time t,
Figure GDA0002502571480000102
is the velocity of the momentum wheel at time t,
Figure GDA0002502571480000103
is the acceleration of the momentum wheel at time t; {f} is the excitation of the momentum wheel;

Figure GDA0002502571480000111
Figure GDA0002502571480000111

式中，m为动量轮的质量，Irr为动量轮绕径向的转动惯。In the formula, m is the mass of the momentum wheel, and Irr is the rotational inertia of the momentum wheel around the radial direction.

Figure GDA0002502571480000112
Figure GDA0002502571480000112

式中caz为动量轮轴向阻尼，c为动量轮径向平动阻尼，cd2为动量轮摆动阻尼。where c az is the axial damping of the momentum wheel, c is the radial translation damping of the momentum wheel, and cd 2 is the swing damping of the momentum wheel.

Figure GDA0002502571480000113
Figure GDA0002502571480000113

式中，式中kaz为动量轮轴向刚度，k为动量轮径向平动刚度，kd2为动量轮摆动刚度；

Figure GDA0002502571480000114
ωr代表模态测试确定的径向平动模态角频率，根据工程经验，初设ωr＝1e4得到k；
Figure GDA0002502571480000115
其中ωaz代表模态测试确定的轴向平动模态角频率，初设ωaz＝2e4得到kaz；
Figure GDA0002502571480000121
其中ωswing代表模态测试确定的摇摆模态角频率，初设ωswing＝1.5e3得到kd2；where k az is the axial stiffness of the momentum wheel, k is the radial translation stiffness of the momentum wheel, and kd 2 is the swing stiffness of the momentum wheel;
Figure GDA0002502571480000114
ω r represents the radial translation modal angular frequency determined by the modal test. According to engineering experience, k is obtained by initially setting ω r = 1e 4 ;
Figure GDA0002502571480000115
where ω az represents the axial translational modal angular frequency determined by the modal test, and ka az is obtained by initially setting ω az = 2e 4 ;
Figure GDA0002502571480000121
where ω swing represents the angular frequency of the swing mode determined by the modal test, and kd 2 is obtained by initially setting ω swing = 1.5e 3 ;

(2)当不考虑动量轮的阻尼时即Cf＝0，动量轮框架固定时的时域动力学方程为：(2) When the damping of the momentum wheel is not considered, that is, C f = 0, the time-domain dynamic equation when the frame of the momentum wheel is fixed is:

Figure GDA0002502571480000122
Figure GDA0002502571480000122

式中，下标f代表转子自由度，s代表框架自由度，Mff为转子质量矩阵，Kff为转子刚度矩阵，

Figure GDA0002502571480000123
In the formula, the subscript f represents the degree of freedom of the rotor, s represents the degree of freedom of the frame, M ff is the rotor mass matrix, K ff is the rotor stiffness matrix,
Figure GDA0002502571480000123

Figure GDA0002502571480000124
Figure GDA0002502571480000125
F1(t)行数与Mff相同，F1(t)为动量轮转子所受扰动力，F2(t)行数与Mss相同，F2(t)为刚性界面对动量轮的约束力。
Figure GDA0002502571480000124
Figure GDA0002502571480000125
The number of rows of F 1 (t) is the same as that of M ff , F 1 (t) is the disturbance force on the rotor of the momentum wheel, the number of rows of F 2 (t) is the same as that of M ss , and F 2 (t) is the force of the rigid interface on the momentum wheel. binding force.

式(2)对应的频域动力学方程为：The frequency domain dynamic equation corresponding to formula (2) is:

Figure GDA0002502571480000126
Figure GDA0002502571480000126

式中，F1为动量轮内的转子所受到的扰动力，F2为动量轮固定的刚性界面约束力，F1的频域表达形式为F1(ω)、F2的频域表达形式为F2(ω)。ω为动量轮振动圆频率，xf为xf(t)对应的频域值。In the formula, F 1 is the disturbance force received by the rotor in the momentum wheel, F 2 is the rigid interface constraint force fixed by the momentum wheel, and the frequency domain expression form of F 1 is the frequency domain expression form of F 1 (ω) and F 2 is F 2 (ω). ω is the vibration circular frequency of the momentum wheel, and x f is the frequency domain value corresponding to x f (t).

(3)动量轮内的转子所受到的扰动力与动量轮固定的刚性界面约束力的传递关系表示如下：(3) The transmission relationship between the disturbance force received by the rotor in the momentum wheel and the rigid interface constraint force fixed by the momentum wheel is expressed as follows:

Ksf[-ω2Mff+Kff]-1{F1}＝{F2} ·················(4)K sf [-ω 2 M ff +K ff ] -1 {F 1 }={F 2 }

式中，Ksf＝-Kff，方程(4)变换为In the formula, K sf =-K ff , and equation (4) is transformed into

Figure GDA0002502571480000131
Figure GDA0002502571480000131

式中，E为单位矩阵，其维度与Mff相同；In the formula, E is the identity matrix, and its dimension is the same as Mff ;

对式(5)变换得到Transform equation (5) to get

Figure GDA0002502571480000132
Figure GDA0002502571480000132

(4)根据式(6)，建立动量轮的真实结构的数学方程如下：(4) According to formula (6), the mathematical equation to establish the real structure of the momentum wheel is as follows:

Figure GDA0002502571480000133
Figure GDA0002502571480000133

式中，

Figure GDA0002502571480000134
为待求真实动量轮的质量矩阵，
Figure GDA0002502571480000135
为待求真实动量轮的刚度矩阵，
Figure GDA0002502571480000136
为测量所得真实动量轮刚性界面约束力。In the formula,
Figure GDA0002502571480000134
is the mass matrix of the real momentum wheel to be found,
Figure GDA0002502571480000135
is the stiffness matrix of the real momentum wheel to be obtained,
Figure GDA0002502571480000136
is the rigid interface constraint force of the real momentum wheel obtained from the measurement.

(5)(3)计算所得{F2}与测量所得

Figure GDA0002502571480000137
存在一定的误差，设：(5)(3) Calculated {F 2 } and measured
Figure GDA0002502571480000137
There is a certain error, let:

Figure GDA0002502571480000138
Figure GDA0002502571480000138

Figure GDA0002502571480000139
Figure GDA0002502571480000139

其中

Figure GDA00025025714800001310
in
Figure GDA00025025714800001310

式(7)减去式(6)并利用式(8)与式(9)，得到：Subtracting Equation (6) from Equation (7) and using Equation (8) and Equation (9), we get:

Figure GDA00025025714800001311
Figure GDA00025025714800001311

设Kff表达为参数p1、p2、…、pi的函数，即Kff＝Kff(p1,p2,…,pi)，Mff表达为参数pi+1、…、pn的函数，即Mff＝Mff(pi+1,pi+2,…,pn)，可以对式(10)中的矩阵

Figure GDA00025025714800001312
进行泰勒展开：Let Kff be expressed as a function of parameters p 1 , p 2 ,..., p i , that is, K ff =K ff (p 1 ,p 2 ,...,pi ), and M ff be expressed as parameters p i +1 ,..., The function of p n , that is, M ff =M ff (pi +1 ,pi +2 ,...,p n ), can be used for the matrix in equation (10)
Figure GDA00025025714800001312
Perform Taylor expansion:

Figure GDA00025025714800001313
Figure GDA00025025714800001313

其中Δp1、Δp2、…、Δpi、Δpi+1、…、Δpn为Kff、Mff中p1、p2、…、pi、pi+1、…、pn与结构真实参数的偏差。where Δp 1 , Δp 2 , ..., Δpi , Δpi +1 , ..., Δpn are p 1 , p 2 , ..., p i , p i+1 , ..., pn and structure in K ff , M ff The deviation of the true parameters.

将式(11)代入方程(10)得到Substitute equation (11) into equation (10) to get

Figure GDA0002502571480000141
Figure GDA0002502571480000141

式中，

Figure GDA0002502571480000142
[S]代表灵敏度矩阵：In the formula,
Figure GDA0002502571480000142
[S] represents the sensitivity matrix:

Figure GDA0002502571480000143
Figure GDA0002502571480000143

(6)根据公式(12)，在不同的频率ω处分别建立方程，如下：(6) According to formula (12), establish equations respectively at different frequencies ω, as follows:

Figure GDA0002502571480000144
Figure GDA0002502571480000144

求解式(13)，得到动量轮质量矩阵Mff与动量轮刚度矩阵Kff参数的修正值

Figure GDA0002502571480000145
修正后的质量矩阵：Solve Equation (13) to get the modified values of the momentum wheel mass matrix M ff and the momentum wheel stiffness matrix K ff
Figure GDA0002502571480000145
Corrected mass matrix:

Figure GDA0002502571480000146
Figure GDA0002502571480000146

刚度矩阵：Stiffness Matrix:

Figure GDA0002502571480000147
Figure GDA0002502571480000147

最终得到真实动量轮的无阻尼自由振动方程为：Finally, the undamped free vibration equation of the real momentum wheel is obtained as:

Figure GDA0002502571480000148
Figure GDA0002502571480000148

其中

Figure GDA0002502571480000149
in
Figure GDA0002502571480000149

至此完成了对动量轮结构动力学模型的修正，修正后必然提高动量轮扰动预示的精度；本发明所利用的试验数据不需要在动量轮主结构上安装传感器，不会打破动量轮的保护结构；这个修正过程以动量轮的质量矩阵、刚度矩阵元素作为修正对象，降低了修正的计算量，提高了分析效率。So far, the correction of the dynamic model of the momentum wheel structure has been completed, and the accuracy of the momentum wheel disturbance prediction will inevitably be improved after the correction; the test data used in the present invention does not require the installation of sensors on the main structure of the momentum wheel, and the protection structure of the momentum wheel will not be broken. ; This correction process takes the mass matrix and stiffness matrix elements of the momentum wheel as the correction object, which reduces the calculation amount of correction and improves the analysis efficiency.

(7)优选利用瑞士Kistler公司生产的9255B型测力平台，进行动量轮扰动力(矩)测量。动量轮构成简图见说明书附图2，动量轮包括转子和框架，测量时框架固定于测力平台上，工作框架固定于航天器上。动量轮扰动测量状态见说明书附图3。此时，动量轮固定在测力平台上进行扰动力测量时。将动量轮自由度x(t)分成两组，分别为：不与刚性界面相连的内部自由度(即转子自由度)xf(t)和在刚性界面上的边界自由度(即框架自由度)xs(t)，即

Figure GDA0002502571480000151
(7) It is preferable to use the 9255B type force measuring platform produced by Kistler Company of Switzerland to measure the disturbance force (moment) of the momentum wheel. The schematic diagram of the momentum wheel is shown in Figure 2 of the description. The momentum wheel includes a rotor and a frame. During measurement, the frame is fixed on the force measuring platform, and the working frame is fixed on the spacecraft. The measurement state of the momentum wheel disturbance is shown in Figure 3 of the specification. At this time, the momentum wheel is fixed on the force measuring platform to measure the disturbance force. The momentum wheel degrees of freedom x(t) are divided into two groups: the internal degrees of freedom not connected to the rigid interface (i.e. the rotor degrees of freedom) x f (t) and the boundary degrees of freedom on the rigid interface (i.e. the frame degrees of freedom). )x s (t), i.e.
Figure GDA0002502571480000151

Figure GDA0002502571480000152
Figure GDA0002502571480000152

其中

Figure GDA0002502571480000153
为动量轮的质量阵，C为动量轮阻尼矩阵，由工程经验给出。国内某型号动量轮的阻尼参数矩阵结果如下：in
Figure GDA0002502571480000153
is the mass matrix of the momentum wheel, and C is the damping matrix of the momentum wheel, which is given by engineering experience. The damping parameter matrix results of a certain type of momentum wheel in China are as follows:

Figure GDA0002502571480000154
Figure GDA0002502571480000154

Figure GDA0002502571480000155
为动量轮刚度阵。
Figure GDA0002502571480000155
is the momentum wheel stiffness matrix.

(8)按照不与刚性界面相连的内部自由度、与刚性界面连接的边界自由度将、C、进行分块：(8) According to the internal degrees of freedom not connected to the rigid interface and the boundary degrees of freedom connected to the rigid interface, divide C, into blocks:

Figure GDA0002502571480000161
Figure GDA0002502571480000161

方程(15)变为Equation (15) becomes

Figure GDA0002502571480000162
Figure GDA0002502571480000162

(9)在频域下，将步骤(8)的刚性界面下动量轮的结构动力学方程转化为：(9) In the frequency domain, transform the structural dynamics equation of the momentum wheel under the rigid interface in step (8) into:

Figure GDA0002502571480000163
Figure GDA0002502571480000163

式中，xf、xs、F1、F2分别表示xf(t)、xs(t)、F1(t)和F2(t)对应的频域复数量。In the formula, x f , x s , F 1 , and F 2 represent the frequency-domain complex quantities corresponding to x f (t), x s (t), F 1 (t), and F 2 (t), respectively.

(10)设动量轮的动刚度矩阵

Figure GDA0002502571480000164
将Z分块，即将公式(17)式可转化为：(10) Set the dynamic stiffness matrix of the momentum wheel
Figure GDA0002502571480000164
Divide Z into blocks, that is, formula (17) can be transformed into:

Figure GDA0002502571480000165
Figure GDA0002502571480000165

其中

Figure GDA0002502571480000166
in
Figure GDA0002502571480000166

Figure GDA0002502571480000167
Figure GDA0002502571480000167

当动量轮固定于测力平台上，xs＝0，代入公式(18)，得到固定界面处的约束力F2，如下：When the momentum wheel is fixed on the force-measuring platform, x s =0, and substituting into formula (18), the binding force F 2 at the fixed interface is obtained, as follows:

Figure GDA0002502571480000168
Figure GDA0002502571480000168

(11)建立航天器的有限元模型，将动量轮安装在航天器上(见说明书附图4)，建立动量轮与该航天器的耦合动力学方程，按照动量轮转子位移xf、动量轮框架位移xs与航天器节点xk，将耦合动力学方程进行分块：(11) Establish the finite element model of the spacecraft, install the momentum wheel on the spacecraft (see Figure 4 in the description), establish the coupled dynamics equation between the momentum wheel and the spacecraft, according to the rotor displacement x f of the momentum wheel, the momentum wheel The frame displacement x s and the spacecraft node x k block the coupled dynamics equations:

Figure GDA0002502571480000169
Figure GDA0002502571480000169

式中，下标“k”代表航天器有限元模型上的节点的标号。xk为航天器的节点位移。In the formula, the subscript "k" represents the label of the node on the finite element model of the spacecraft. x k is the nodal displacement of the spacecraft.

由式(20)得到：It can be obtained by formula (20):

Figure GDA0002502571480000171
Figure GDA0002502571480000171

Figure GDA0002502571480000172
Figure GDA0002502571480000172

Figure GDA0002502571480000173
Figure GDA0002502571480000173

设

Figure GDA0002502571480000174
则
Figure GDA0002502571480000175
Assume
Figure GDA0002502571480000174
but
Figure GDA0002502571480000175

(12)在航天器的动量轮安装界面施加力-F2(如图3和图5所示)，动量轮和航天器耦合动力学方程(20)变为：(12) When force -F 2 is applied on the installation interface of the momentum wheel of the spacecraft (as shown in Figures 3 and 5), the coupling dynamics equation (20) of the momentum wheel and the spacecraft becomes:

Figure GDA0002502571480000176
Figure GDA0002502571480000176

式中，

Figure GDA0002502571480000177
为动量轮转子位移，
Figure GDA0002502571480000178
为动量轮框架位移，
Figure GDA0002502571480000179
为航天器节点位移。In the formula,
Figure GDA0002502571480000177
is the rotor displacement of the momentum wheel,
Figure GDA0002502571480000178
is the displacement of the momentum wheel frame,
Figure GDA0002502571480000179
is the node displacement of the spacecraft.

将式(19)代入式(24)，得到：Substituting equation (19) into equation (24), we get:

Figure GDA00025025714800001710
Figure GDA00025025714800001710

Figure GDA00025025714800001711
Figure GDA00025025714800001711

Figure GDA00025025714800001712
Figure GDA00025025714800001712

(13)对比方程(22)与(26)、方程(23)与(27)，可知，此时：(13) Comparing equations (22) and (26), and equations (23) and (27), it can be known that at this time:

Figure GDA00025025714800001713
Figure GDA00025025714800001713

可知当在航天器扰动界面施加力向量

Figure GDA00025025714800001714
计算得到动力学方程解与方程(20)解相同。至此，得到动量轮安装界面等效激励力为
Figure GDA00025025714800001715
It can be seen that when a force vector is applied to the disturbance interface of the spacecraft
Figure GDA00025025714800001714
The calculated solution of the kinetic equation is the same as that of equation (20). So far, the equivalent excitation force of the momentum wheel installation interface is obtained as
Figure GDA00025025714800001715

(14)解方程(27)，得到动量轮工作给航天器造成的扰动响应，如图1所示。(14) Solve equation (27) to obtain the disturbance response caused by the momentum wheel work to the spacecraft, as shown in Figure 1.

图1为航天器上某点的扰动响应曲线，其中横坐标表示ω/2π，纵坐标为航天器上某节点的加速度幅值。Figure 1 shows the disturbance response curve of a point on the spacecraft, where the abscissa represents ω/2π, and the ordinate represents the acceleration amplitude of a node on the spacecraft.

以国内某型号动量轮为对象，利用测力平台进行扰动力测量。利用本发明所提供的动量轮安装界面等效激励力获取方法获取等效激励力，然后进行扰动响应分析，分析得到的航天器某位置仿真结果与试验测量结果进行对比如下表：。Taking a domestic model of momentum wheel as the object, the disturbance force measurement is carried out by using the force measuring platform. The equivalent excitation force is obtained by using the method for obtaining the equivalent excitation force of the momentum wheel installation interface provided by the present invention, and then the disturbance response analysis is performed.

表1响应分析结果Table 1 Response analysis results

Figure GDA0002502571480000181
Figure GDA0002502571480000181

如表1所示，利用本文所提方法获得的激励力进行动量轮扰动响应分析，得到的航天器某点加速度响应误差最大不超过5％，说明发明所提供方法是合理有效的，精度较高。As shown in Table 1, the momentum wheel disturbance response analysis is carried out using the excitation force obtained by the method proposed in this paper, and the obtained acceleration response error at a certain point of the spacecraft does not exceed 5%, indicating that the method provided by the invention is reasonable and effective, and the accuracy is high .

Claims (2)

1.一种动量轮与航天器安装界面等效激励力的获取方法，其特征在于步骤如下：1. a method for obtaining the equivalent excitation force of a momentum wheel and a spacecraft installation interface, is characterized in that the steps are as follows: (1)提出动量轮的结构动力学方程如下：(1) The structural dynamics equation of the momentum wheel is proposed as follows:
Figure FDA0002502571470000011
Figure FDA0002502571470000011
 式中x(t)为t时刻动量轮的位移,
Figure FDA0002502571470000012
为t时刻动量轮的速度，
Figure FDA0002502571470000013
为t时刻动量轮的加速度；{f}为动量轮所受的激励；where x(t) is the displacement of the momentum wheel at time t,
Figure FDA0002502571470000012
is the velocity of the momentum wheel at time t,
Figure FDA0002502571470000013
is the acceleration of the momentum wheel at time t; {f} is the excitation of the momentum wheel;
Figure FDA0002502571470000014
Figure FDA0002502571470000014
 式中，m为动量轮的质量，Irr为动量轮绕径向的转动惯；In the formula, m is the mass of the momentum wheel, and Irr is the rotational inertia of the momentum wheel around the radial direction;
Figure FDA0002502571470000015
Figure FDA0002502571470000015
 式中caz为动量轮轴向阻尼，c为动量轮径向平动阻尼，cd2为动量轮摆动阻尼；where c az is the axial damping of the momentum wheel, c is the radial translation damping of the momentum wheel, and cd 2 is the swing damping of the momentum wheel;
Figure FDA0002502571470000021
Figure FDA0002502571470000021
 式中，式中kaz为动量轮轴向刚度，k为动量轮径向平动刚度，kd2为动量轮摆动刚度；
Figure FDA0002502571470000022
ωr代表模态测试确定的径向平动模态角频率，根据工程经验，初设ωr＝1e4得到k；
Figure FDA0002502571470000023
其中ωaz代表模态测试确定的轴向平动模态角频率，初设ωaz＝2e4得到kaz；
Figure FDA0002502571470000024
其中ωswing代表模态测试确定的摇摆模态角频率，初设ωswing＝1.5e3得到kd2；where k az is the axial stiffness of the momentum wheel, k is the radial translation stiffness of the momentum wheel, and kd 2 is the swing stiffness of the momentum wheel;
Figure FDA0002502571470000022
ω r represents the radial translation modal angular frequency determined by the modal test. According to engineering experience, k is obtained by initially setting ω r = 1e 4 ;
Figure FDA0002502571470000023
where ω az represents the axial translational modal angular frequency determined by the modal test, and ka az is obtained by initially setting ω az = 2e 4 ;
Figure FDA0002502571470000024
where ω swing represents the angular frequency of the swing mode determined by the modal test, and kd 2 is obtained by initially setting ω swing = 1.5e 3 ; (2)当不考虑动量轮的阻尼时即Cf＝0，动量轮框架固定时的时域动力学方程为：(2) When the damping of the momentum wheel is not considered, that is, C f = 0, the time-domain dynamic equation when the frame of the momentum wheel is fixed is:
Figure FDA0002502571470000025
Figure FDA0002502571470000025
 式中，下标f代表转子自由度，s代表框架自由度，Mff为转子质量矩阵，Kff为转子刚度矩阵，
Figure FDA0002502571470000026
In the formula, the subscript f represents the degree of freedom of the rotor, s represents the degree of freedom of the frame, M ff is the rotor mass matrix, K ff is the rotor stiffness matrix,
Figure FDA0002502571470000026
Figure FDA0002502571470000027
Figure FDA0002502571470000028
F1(t)行数与Mff相同，F1(t)为动量轮转子所受扰动力，F2(t)行数与Mss相同，F2(t)为刚性界面对动量轮的约束力；
Figure FDA0002502571470000027
Figure FDA0002502571470000028
The number of rows of F 1 (t) is the same as that of M ff , F 1 (t) is the disturbance force on the rotor of the momentum wheel, the number of rows of F 2 (t) is the same as that of M ss , and F 2 (t) is the force of the rigid interface on the momentum wheel. binding force; 式(2)对应的频域动力学方程为：The frequency domain dynamic equation corresponding to formula (2) is:
Figure FDA0002502571470000031
Figure FDA0002502571470000031
 式中，F1为动量轮内的转子所受到的扰动力，F2为动量轮固定的刚性界面约束力，F1的频域表达形式为F1(ω)、F2的频域表达形式为F2(ω)，ω为动量轮振动圆频率，xf为xf(t)对应的频域值；In the formula, F 1 is the disturbance force received by the rotor in the momentum wheel, F 2 is the rigid interface constraint force fixed by the momentum wheel, and the frequency domain expression form of F 1 is the frequency domain expression form of F 1 (ω) and F 2 is F 2 (ω), ω is the vibration circular frequency of the momentum wheel, and x f is the frequency domain value corresponding to x f (t); (3)动量轮内的转子所受到的扰动力与动量轮固定的刚性界面约束力的传递关系表示如下：(3) The transmission relationship between the disturbance force received by the rotor in the momentum wheel and the rigid interface constraint force fixed by the momentum wheel is expressed as follows: Ksf[-ω2Mff+Kff]-1{F1}＝{F2}······························(4)K sf [-ω 2 M ff +K ff ] -1 {F 1 }={F 2 } (4) 式中，Ksf＝-Kff，方程(4)变换为In the formula, K sf =-K ff , and equation (4) is transformed into
Figure FDA0002502571470000032
Figure FDA0002502571470000032
 式中，E为单位矩阵，其维度与Mff相同；In the formula, E is the identity matrix, and its dimension is the same as Mff ; 对式(5)变换得到Transform equation (5) to get
Figure FDA0002502571470000033
Figure FDA0002502571470000033
 (4)根据式(6)，建立动量轮的真实结构的数学方程如下：(4) According to formula (6), the mathematical equation to establish the real structure of the momentum wheel is as follows:
Figure FDA0002502571470000034
Figure FDA0002502571470000034
 式中，
Figure FDA0002502571470000035
为待求真实动量轮的质量矩阵，
Figure FDA0002502571470000036
为待求真实动量轮的刚度矩阵，
Figure FDA0002502571470000037
为测量所得真实动量轮刚性界面约束力；In the formula,
Figure FDA0002502571470000035
is the mass matrix of the real momentum wheel to be found,
Figure FDA0002502571470000036
is the stiffness matrix of the real momentum wheel to be obtained,
Figure FDA0002502571470000037
is the rigid interface constraint force of the real momentum wheel obtained from the measurement; (5)式(3)计算所得{F2}与测量所得
Figure FDA0002502571470000038
存在一定的误差，设：(5) Calculated {F 2 } from formula (3) and measured
Figure FDA0002502571470000038
There is a certain error, let:
Figure FDA0002502571470000039
Figure FDA0002502571470000039
Figure FDA00025025714700000310
Figure FDA00025025714700000310
 其中
Figure FDA00025025714700000311
in
Figure FDA00025025714700000311
 式(7)减去式(6)并利用式(8)与式(9)，得到：Subtracting Equation (6) from Equation (7) and using Equation (8) and Equation (9), we get:
Figure FDA0002502571470000041
Figure FDA0002502571470000041
 设Kff表达为参数p1、p2、…、pi的函数，即Kff＝Kff(p1,p2,…,pi)，Mff表达为参数pi+1、…、pn的函数，即Mff＝Mff(pi+1,pi+2,…,pn)，对式(10)中的矩阵
Figure FDA0002502571470000042
进行泰勒展开：Let Kff be expressed as a function of parameters p 1 , p 2 ,..., p i , that is, K ff =K ff (p 1 ,p 2 ,...,pi ), and M ff be expressed as parameters p i +1 ,..., The function of pn , namely M ff =M ff (pi +1 ,pi +2 ,...,p n ), for the matrix in equation (10)
Figure FDA0002502571470000042
Perform Taylor expansion:
Figure FDA0002502571470000043
Figure FDA0002502571470000043
 其中Δp1、Δp2、…、Δpi、Δpi+1、…、Δpn为Kff、Mff中p1、p2、…、pi、pi+1、…、pn与结构真实参数的偏差；where Δp 1 , Δp 2 , ..., Δpi , Δpi +1 , ..., Δpn are p 1 , p 2 , ..., p i , p i+1 , ..., pn and structure in K ff , M ff the deviation of the true parameters; 将式(11)代入方程(10)得到Substitute equation (11) into equation (10) to get
Figure FDA0002502571470000044
Figure FDA0002502571470000044
 式中，
Figure FDA0002502571470000045
[S]代表灵敏度矩阵：In the formula,
Figure FDA0002502571470000045
[S] represents the sensitivity matrix:
Figure FDA0002502571470000046
Figure FDA0002502571470000046
 (6)根据公式(12)，在不同的频率ω处分别建立方程，如下：(6) According to formula (12), establish equations respectively at different frequencies ω, as follows:
Figure FDA0002502571470000047
Figure FDA0002502571470000047
 求解式(13)，得到动量轮质量矩阵Mff与动量轮刚度矩阵Kff参数的修正值
Figure FDA0002502571470000051
修正后的质量矩阵：Solve Equation (13) to get the modified values of the momentum wheel mass matrix M ff and the momentum wheel stiffness matrix K ff
Figure FDA0002502571470000051
Corrected mass matrix:
Figure FDA0002502571470000052
Figure FDA0002502571470000052
 刚度矩阵：Stiffness Matrix:
Figure FDA0002502571470000053
Figure FDA0002502571470000053
 最终得到真实动量轮的无阻尼自由振动方程；Finally, the undamped free vibration equation of the real momentum wheel is obtained; (7)当动量轮固定在测力平台上进行扰动力测量时，将动量轮的自由度x(t)分成两组，分别为：不与刚性界面相连的内部自由度，即转子自由度xf(t)和在刚性界面上的边界自由度，即框架自由度xs(t)，即
Figure FDA0002502571470000054
(7) When the momentum wheel is fixed on the force measuring platform for disturbance force measurement, the degrees of freedom x(t) of the momentum wheel are divided into two groups, namely: the internal degrees of freedom not connected to the rigid interface, namely the rotor degrees of freedom x f (t) and the boundary degrees of freedom on the rigid interface, i.e. the frame degrees of freedom x s (t), i.e.
Figure FDA0002502571470000054
Figure FDA0002502571470000055
Figure FDA0002502571470000055
 其中
Figure FDA0002502571470000056
为动量轮的质量阵，C为动量轮阻尼矩阵，
Figure FDA0002502571470000057
为动量轮刚度阵；in
Figure FDA0002502571470000056
is the mass matrix of the momentum wheel, C is the damping matrix of the momentum wheel,
Figure FDA0002502571470000057
is the momentum wheel stiffness matrix; (8)按照不与刚性界面相连的内部自由度、与刚性界面连接的边界自由度将C进行分块：(8) Divide C into blocks according to the internal degrees of freedom not connected to the rigid interface and the boundary degrees of freedom connected to the rigid interface:
Figure FDA0002502571470000058
Figure FDA0002502571470000058
 方程(15)变为刚性界面下动量轮的结构动力学方程，如下：Equation (15) becomes the structural dynamics equation of the momentum wheel under rigid interface as follows:
Figure FDA0002502571470000059
Figure FDA0002502571470000059
 (9)在频域下，将步骤(8)的刚性界面下动量轮的结构动力学方程转化为：(9) In the frequency domain, transform the structural dynamics equation of the momentum wheel under the rigid interface in step (8) into:
Figure FDA0002502571470000061
Figure FDA0002502571470000061
 式中，xf、xs、F1、F2分别表示xf(t)、xs(t)、F1(t)和F2(t)对应的频域复数量；where x f , x s , F 1 , and F 2 represent the frequency-domain complex quantities corresponding to x f (t), x s (t), F 1 (t) and F 2 (t), respectively; (10)设动量轮的动刚度矩阵
Figure FDA0002502571470000062
将Z分块，即将公式(17)式可转化为：(10) Set the dynamic stiffness matrix of the momentum wheel
Figure FDA0002502571470000062
Divide Z into blocks, that is, formula (17) can be transformed into:
Figure FDA0002502571470000063
Figure FDA0002502571470000063
 其中
Figure FDA0002502571470000064
in
Figure FDA0002502571470000064
Figure FDA0002502571470000065
Figure FDA0002502571470000065
 动量轮的框架固定于测力平台上，xs＝0，代入公式(18)，得到固定界面处的约束力F2，如下The frame of the momentum wheel is fixed on the force measuring platform, x s = 0, and substituting into formula (18), the binding force F 2 at the fixed interface is obtained, as follows
Figure FDA0002502571470000066
Figure FDA0002502571470000066
 (11)建立航天器结构有限元模型，将动量轮安装在航天器上，建立动量轮与该航天器的耦合动力学方程，按照动量轮转子位移xf、动量轮框架位移xs、航天器节点xk，将耦合动力学方程进行分块：(11) Establish a finite element model of the spacecraft structure, install the momentum wheel on the spacecraft, and establish the coupled dynamics equation between the momentum wheel and the spacecraft. According to the momentum wheel rotor displacement x f , the momentum wheel frame displacement x s , the spacecraft Node x k , which blocks the coupled dynamics equations:
Figure FDA0002502571470000067
Figure FDA0002502571470000067
 式中，下标“k”代表航天器有限元模型上的节点的标号，xk为航天器有限元模型上的节点位移；In the formula, the subscript "k" represents the label of the node on the finite element model of the spacecraft, and x k is the node displacement on the finite element model of the spacecraft; 由式(20)得到：It can be obtained by formula (20):
Figure FDA0002502571470000068
Figure FDA0002502571470000068
Figure FDA0002502571470000069
Figure FDA0002502571470000069
Figure FDA00025025714700000610
Figure FDA00025025714700000610
 设
Figure FDA0002502571470000071
则
Figure FDA0002502571470000072
Assume
Figure FDA0002502571470000071
but
Figure FDA0002502571470000072
 (12)在航天器的动量轮安装界面施加力-F2，动量轮和航天器耦合动力学方程(20)变为：(12) When a force -F 2 is applied on the installation interface of the momentum wheel of the spacecraft, the coupling dynamic equation (20) of the momentum wheel and the spacecraft becomes:
Figure FDA0002502571470000073
Figure FDA0002502571470000073
 式中，
Figure FDA0002502571470000074
为动量轮转子位移，
Figure FDA0002502571470000075
为动量轮框架位移，
Figure FDA0002502571470000076
为航天器节点位移；In the formula,
Figure FDA0002502571470000074
is the rotor displacement of the momentum wheel,
Figure FDA0002502571470000075
is the displacement of the momentum wheel frame,
Figure FDA0002502571470000076
is the node displacement of the spacecraft; 将式(19)代入式(24)，得到：Substituting equation (19) into equation (24), we get:
Figure FDA0002502571470000077
Figure FDA0002502571470000077
Figure FDA0002502571470000078
Figure FDA0002502571470000078
Figure FDA0002502571470000079
Figure FDA0002502571470000079
 (13)对比方程(22)与(26)、方程(23)与(27)，可知，此时：(13) Comparing equations (22) and (26), and equations (23) and (27), it can be known that at this time:
Figure FDA00025025714700000710
Figure FDA00025025714700000710
 可知当在航天器扰动界面施加力向量
Figure FDA00025025714700000711
计算得到动力学方程解与方程(20)解相同；至此，得到动量轮安装界面等效激励力为
Figure FDA00025025714700000712
It can be seen that when a force vector is applied to the disturbance interface of the spacecraft
Figure FDA00025025714700000711
The solution of the dynamic equation obtained by calculation is the same as that of equation (20); so far, the equivalent excitation force of the installation interface of the momentum wheel is obtained as
Figure FDA00025025714700000712
 (14)解方程(27)，得到动量轮工作时给航天器造成的扰动响应。(14) Equation (27) is solved to obtain the disturbance response to the spacecraft when the momentum wheel works. 2.根据权利要求1所述的一种动量轮与航天器安装界面等效激励力的获取方法，其特征在于：步骤(6)最终得到真实动量轮的无阻尼自由振动方程为：2. the acquisition method of the equivalent excitation force of a kind of momentum wheel according to claim 1 and spacecraft installation interface, it is characterized in that: the undamped free vibration equation that step (6) finally obtains true momentum wheel is:
Figure FDA00025025714700000713
Figure FDA00025025714700000713
 其中
Figure FDA00025025714700000714
in
Figure FDA00025025714700000714
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