CN110110619B - Satellite micro-vibration source quantitative identification method based on sparse blind source separation - Google Patents
Satellite micro-vibration source quantitative identification method based on sparse blind source separation Download PDFInfo
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- G01—MEASURING; TESTING
- G01H—MEASUREMENT OF MECHANICAL VIBRATIONS OR ULTRASONIC, SONIC OR INFRASONIC WAVES
- G01H17/00—Measuring mechanical vibrations or ultrasonic, sonic or infrasonic waves, not provided for in the preceding groups
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Abstract
The invention discloses a satellite micro-vibration source quantitative identification method for sparse blind source separation, which comprises the steps of firstly, arranging acceleration sensors at different positions of a sensitive load position and a model surface of a satellite cabin structural model, collecting vibration signals of each vibration source during normal work, and ensuring that the number of observation signals is greater than that of the sources; then, using L1Norm construction reference sparse blind deconvolution algorithm reference L1And constructing a reference signal according to the time-frequency domain characteristic prior information of the vibration source by using a norm objective function, and searching an optimal solution of the separation signal by using a gradient descent method iterative optimization objective function to realize the extraction of the single vibration source signal. Finally, calculating the single-source response signal of each vibration source at the sensitive load by using a frequency domain single-source response signal solving method; and calculating the contribution of each vibration source at the sensitive load by using a vector projection-based contribution characterization method. The contribution evaluation index is an index for quantitatively identifying the satellite micro-vibration source and can provide a basis for micro-vibration inhibition.
Description
Technical Field
The invention relates to a quantitative identification method for a vibration source of mechanical equipment, in particular to a quantitative identification method for a micro-vibration source of a satellite based on sparse blind source separation.
Background
The satellite is used as key space equipment and plays an important role in national military national defense construction and national economic development. High resolution satellites are receiving wide attention from all countries in the world as the future development direction of satellites. However, the satellite micro-vibration severely restricts the improvement of the satellite resolution and other performances, so that the quantitative identification of the satellite micro-vibration source is carried out, the contribution of the main vibration source to the sensitive load is evaluated, a foundation and a basis can be provided for the satellite micro-vibration suppression work, and the method has obvious engineering application value
Because the satellite system has a complex structure, the vibration source signal is difficult to directly measure in the in-orbit operation process, and even the signal measured near the vibration source is a mixed signal with the adjacent vibration source, not a pure source signal. The micro-vibration sources distributed at different positions of the satellite are coupled to the sensitive load area in different paths and transmission, so even if the vibration magnitude of each vibration source is the same, the vibration sources can contribute to the sensitive load area differently. In addition, the actual satellite has a complex structure, the main micro-vibration sources are rotating parts such as a control moment gyroscope, a flywheel and the like, so that harmonic components of vibration source signals are more, the frequency band overlapping among the vibration source signals is strong, the identification of the micro-vibration source signals is more difficult, the sum of the energy of each response signal at an observation point when each vibration source operates in a sub-part mode is also caused to be unequal to the energy of a mixed response signal at the observation point when each vibration source operates simultaneously, and the real contribution of the vibration source at the observation point is difficult to accurately reflect by the contribution represented by the energy.
Disclosure of Invention
The invention aims to provide a satellite micro-vibration source quantitative identification method based on sparse blind source separation, which overcomes the defects of the prior art, has high efficiency, low cost and high accuracy, can provide a basis and a basis for satellite micro-vibration inhibition, and can meet the application requirements of actual engineering.
In order to achieve the purpose, the invention adopts the following technical scheme:
a satellite micro-vibration source quantitative identification method based on sparse blind source separation comprises the following steps:
(1) acquisition of vibration signals
Arranging acceleration sensors at different positions of a sensitive load part (a camera simulation part) of a satellite cabin structure model and the surface of the model, and acquiring vibration signals, namely observation signals, of each vibration source during normal work:
x(t)＝[x1(t),x2(t),...,xM(t),]T
where M is the number of observed signals, xm(t) is a vibration response signal collected at the mth observation point at the moment t, and M is 1-M;
ensuring that the number of observation signals is greater than the number N of vibration sources;
(2) extraction of single vibration source signal based on reference sparse blind deconvolution algorithm
By means of L1Norm construction reference sparse blind deconvolution algorithm reference L1A norm objective function, constructing a reference signal according to the prior information of the time-frequency domain characteristics of the vibration source, and acquiringIterative optimization of a target function by using a gradient descent method is used for searching an optimal solution of the separation signal, so that the extraction of the single-vibration source signal is realized;
(3) calculating the contribution of each vibration source to the sensitive load by using a frequency domain sparse contribution estimation method
Calculating single-source response signals of each vibration source at the sensitive load by using a frequency domain single-source response signal solving method; and calculating the contribution of each vibration source at the sensitive load by using a vector projection-based contribution characterization method.
Further, the step (2) specifically comprises:
(2.1) Using L1Norm construction blind deconvolution algorithm reference L1The norm objective function J (y, r) is:
J(y,r)＝||F(y)||1+λE{(y-r)2}
wherein y is the split signal; r is a reference signal; f (-) is a discrete Fourier transform of a variable; λ is a scale factor; e {. is the mathematical expectation of the variable;
(2.2) constructing a reference signal r according to the prior information of the equipment vibration source;
(2.3) setting the separation filter length L, rewriting x (t) obtained in the step (1) into a time lag form
The blind deconvolution model is converted into a standard ICA model:
wherein the content of the first and second substances,
a separation filter of dimension 1 × NL;
at this time, L with respect to y1Norm numberThe standard function is converted into
The objective function of (2):
the above equation can be expressed by fourier transform theory as:
wherein Re is
A matrix formed by the real parts after Fourier transformation; im is
A matrix formed by the imaginary parts after Fourier transformation;
(2.4) iterative optimization of reference L by gradient descent method1Norm objective function by dividing vector
Iteratively updating and searching the optimal solution of the separation signal y corresponding to the minimum objective function,
the iterative update of (2) is called:
where k represents the kth iterative update and g is the objective function J (y, r) pair
A is the iteration step of the algorithm;
each iteration obtains a new
After that, it needs to be normalized:
iterate until convergence to obtain a separation vector
Then the signals are separated into
And the extraction of the single-vibration source signal is realized.
The extraction of different source signals is realized by constructing different reference signals.
Further, the step (3) is specifically:
(3.1) let y (t) be the extracted separation signal, which can be obtained by the blind deconvolution theory, and a filter h (tau) exists and satisfies the following relational expression:
x(t)＝h(τ)*y(t)
wherein, is convolution operation;
based on the convolution theory, the convolution operation form is expressed as a product form in a frequency domain, and a matrix relation between a frequency domain value X (omega) of an observation signal at each frequency point omega, a filter frequency domain value H (omega) and a separation signal frequency domain value Y (omega) can be obtained:
X(ω)＝H(ω)Y(ω)
a mixing matrix H (omega) can be obtained at each frequency point in a frequency domain, signals are sparse signals in the frequency domain, the value of an observation signal at the frequency point omega is either zero or close to zero, or the signal is a response generated by a certain vibration source signal, so most values in the matrix H (omega) at each frequency point omega are 0, the proportion of energy of components at the frequency omega in each source signal in the total energy of the source signal is calculated, and the source signal with the largest proportion is the generation source of the observation signal at the frequency point omega.
Hypothesis frequency point omega0Where the observed signal is from YnThen the amplitude of the other observation signals at the point is 0 and at the frequency point omega0The following can be obtained:
solving the above formula to obtain the frequency point omega0Mixing matrix H (ω)0)；
The same method is adopted to solve the mixing matrix H (omega) at each frequency point, and the single-source response of each vibration source at the position of the camera simulation part can be solved in the frequency domain after the mixing matrix is obtained:
wherein the content of the first and second substances,
a response signal at the mth observation for the nth source; hmn(ω) is the mth row and nth column element in the mixing matrix H (ω);
(3.2) Single Source response Signal
Observing signal X at sensitive loadmProjection Z ofmnAs a contribution of the nth vibration source to the mth observation, ZmnAt XmThe proportion of the total carbon is the contribution amount evaluation index cmn：
The contribution amount evaluation index reflects the contribution and influence of each vibration source on the vibration of the sensitive load, and is an index for quantitatively identifying the micro-vibration source of the satellite.
Compared with the prior art, the invention has the following beneficial technical effects:
the invention adoptsThe extraction of vibration source signals is realized by a reference sparse blind deconvolution algorithm, the contribution of each vibration source to a sensitive load is calculated by a frequency domain sparse contribution estimation method, the quantitative identification of the satellite micro-vibration sources is realized, and a basis is provided for micro-vibration suppression. Its advantage is that in the extraction of vibration source signal, the L for measuring the sparsity of signal is chosen1The norm is used as a target function, the accuracy is higher when the source signal containing harmonic waves and overlapped frequency bands is solved, and the precision is further improved by introducing a reference signal; in the aspect of single-source response solving, a frequency domain single-source response signal extraction method is used for solving, and effective calculation can be realized when only a signal frequency spectrum is known; in the aspect of contribution calculation, the contribution characterization method based on vector projection can reflect the contribution of each vibration source at the sensitive load more truly. In conclusion, the method quantitatively identifies the contribution of each vibration source to the sensitive load point by using the vibration signals acquired at the sensitive load position and the surface of the satellite cabin structure, has the characteristics of simplicity, high efficiency, easiness in implementation, high accuracy and the like, can provide a foundation and basis for satellite micro-vibration inhibition work, and has important engineering practical value.
Drawings
FIG. 1 is a flow chart of the present invention for the quantitative identification of a satellite micro-vibration source based on sparse blind source separation;
FIG. 2 is a vibration experimental apparatus for equipment operation; wherein, 1, an acceleration sensor; 2. a data acquisition system; 3. a computer; 4. a speed regulator; 5. a satellite cabin structure model; 6. a first vibration exciter; 7. a second vibration exciter; 8. a first power amplifier; 9. a second power amplifier; 10. a signal generator.
FIG. 3 is a time domain plot of the x-direction vibration response signal collected by the apparatus of FIG. 2; wherein, (a) is a time domain waveform diagram of a response signal at the top of the camera simulation piece; (b) a time domain waveform diagram of the root response signal of the camera simulation piece; (c) a time domain oscillogram of a response signal collected by a certain sensor on the box body; the abscissa in the figure represents time in units of s; the ordinate represents the vibration amplitude in mg.
FIG. 4 is a graph of the frequency spectrum of the vibration response signal of FIG. 3; the abscissa in the figure represents frequency in Hz; the ordinate represents the amplitude in mg.
FIG. 5 is a time domain waveform of a source signal separated using a reference sparse blind deconvolution algorithm; wherein, (a) and (b) correspond to the single vibration source signals of the two vibration exciters respectively; (c) a single vibration source signal corresponding to the motor; the abscissa in the figure represents time in units of s; the ordinate represents the vibration amplitude in mg.
FIG. 6 is a spectrum diagram of the source signal of FIG. 5; the abscissa in the figure represents frequency in Hz; the ordinate represents the amplitude in mg.
FIG. 7 is a frequency spectrum diagram of response signals of each vibration source at a sensitive load point, which is obtained by a frequency domain single-source response signal solving method; wherein, (a) and (b) correspond to the single-source response signal frequency spectrums of the two vibration exciters respectively; (c) a single-source response signal frequency spectrum of the corresponding motor; the abscissa in the figure represents frequency in Hz; the ordinate represents the amplitude in mg.
Detailed Description
The invention is described in detail below with reference to the following figures and detailed description:
referring to fig. 1, a flow chart of quantitative identification of a micro-vibration source of a satellite is shown, wherein each vibration source of a structural model of a satellite cabin section works normally, vibration signals at a sensitive load and different positions on the surface of the model are collected, and the number of observation signals is ensured to be larger than that of source signals; by means of L1Norm construction reference sparse blind deconvolution algorithm reference L1Constructing a reference signal according to the prior information of the vibration source by using a norm objective function, and iteratively optimizing a reference L by using a gradient descent method1A norm target function, namely searching the optimal solution of the corresponding separation signal y when the target function is minimum, and realizing the extraction of the single-vibration source signal; respectively calculating the response of each vibration source at the sensitive load point by a frequency domain single-source response signal solving method; and calculating the contribution of each micro-vibration source by using a contribution characterization method based on vector projection. The contribution amount evaluation index reflects the contribution and influence of each vibration source on the vibration of the sensitive load, and can provide a basis and basis for vibration suppression.
The method for realizing the quantitative identification of the satellite micro-vibration source based on the sparse blind source separation is implemented according to the following specific steps:
(1) acquisition of vibration signals
Referring to fig. 2, the experimental object is a satellite cabin structural model. The cavity material is an aluminum honeycomb sandwich plate, the bottom of the model is supported and placed on an optical vibration isolation table through four rubber air springs so as to eliminate the influence of ground vibration, and the top of the model is a camera simulation piece which is a sensitive load part of the device. The vibration source comprises 4 vibration motors and two vibration exciters. The vibration sources operate under a set working condition, and acceleration sensors are used for collecting vibration response signals at sensitive loads (cameras) and on the surface of the box body, so that the number of observation signals is larger than that of the sources.
(2) Extraction of single vibration source signal based on reference sparse blind deconvolution algorithm
First, using L1Norm construction blind deconvolution algorithm reference L1The norm objective function J (y, r) is:
J(y,r)＝||F(y)||1+λE{(y-r)2}
wherein y is the split signal; r is a reference signal; f (-) is a discrete Fourier transform of a variable; λ is a scale factor; e {. is the mathematical expectation of the variable;
secondly, constructing a reference signal according to prior information of a vibration source of the equipment, taking a square wave signal with a fundamental frequency of a source signal frequency to be extracted as the reference signal for the vibration source signal which is a periodic signal with a single frequency, extracting time-frequency characteristics of the vibration source from an observation signal collected from a sensitive load for the source signal with a complex frequency, and constructing the reference signal by using the frequency and the phase of a main component to obtain a reference signal r;
then, a separation filter length L is set, and x (t) obtained in the step (1) is rewritten into a time lag form
The blind deconvolution model is converted into a standard ICA model:
wherein the content of the first and second substances,
a separation filter of dimension 1 × NL;
at this time, L with respect to y1Norm objective function is converted to
The objective function of (2):
the above equation can be expressed by fourier transform theory as:
wherein Re is
A matrix formed by the real parts after Fourier transformation; im is
A matrix formed by the imaginary parts after Fourier transformation;
finally, the reference L is iteratively optimized by using a gradient descent method1Norm objective function by dividing vector
Iteratively updating and searching the optimal solution of the separation signal y corresponding to the minimum objective function,
is updated by iterationCalled:
where k represents the kth iterative update and g is the objective function J (y, r) pair
A is the iteration step of the algorithm;
each iteration obtains a new
After that, it needs to be normalized:
iterate until convergence to obtain a separation vector
Then the signals are separated into
And the extraction of the single-vibration source signal is realized.
The extraction of different source signals is realized by constructing different reference signals.
(3) And calculating the contribution of each vibration source to the sensitive load by using a frequency domain sparse contribution estimation method.
y (t) is the extracted separation signal, which can be obtained by the blind deconvolution theory, and a filter h (tau) exists which satisfies the following relation:
x(t)＝h(τ)*y(t)
wherein, is convolution operation;
based on the convolution theory, the convolution operation form is expressed as a product form in a frequency domain, and a matrix relation between a frequency domain value X (omega) of an observation signal at each frequency point omega, a filter frequency domain value H (omega) and a separation signal frequency domain value Y (omega) can be obtained:
X(ω)＝H(ω)Y(ω)
a mixing matrix H (omega) can be obtained at each frequency point in a frequency domain, signals are sparse signals in the frequency domain, the value of an observation signal at the frequency point omega is either zero or close to zero, or the signal is a response generated by a certain vibration source signal, so most values in the matrix H (omega) at each frequency point omega are 0, the proportion of energy of components at the frequency omega in each source signal in the total energy of the source signal is calculated, and the source signal with the largest proportion is the generation source of the observation signal at the frequency point omega.
Hypothesis frequency point omega0Where the observed signal is from YnThen the amplitude of the other observation signals at the point is 0 and at the frequency point omega0The following can be obtained:
solving the above formula to obtain the frequency point omega0Mixing matrix H (ω)0)；
The same method is adopted to solve the mixing matrix H (omega) at each frequency point, and the single-source response of each vibration source at the position of the camera simulation part can be solved in the frequency domain after the mixing matrix is obtained:
wherein the content of the first and second substances,
a response signal at the mth observation for the nth source; hmn(ω) is the mth row and nth column element in the mixing matrix H (ω);
single source response signal
Observing signal X at sensitive loadmProjection Z ofmnAs a contribution of the nth vibration source to the mth observation, ZmnAt XmThe proportion of the total weight is the contribution amount evaluationIndex cmn：
The contribution amount evaluation index reflects the contribution and influence of each vibration source on the vibration of the sensitive load, is an index for quantitatively identifying the micro-vibration source of the satellite, and can provide a basis for micro-vibration inhibition.
According to the method, a reference sparse blind deconvolution algorithm is utilized to extract each single vibration source signal from vibration signals acquired from the sensitive load position of the satellite cabin structure model and the surface of the box body, the contribution of each micro-vibration source to the sensitive load point is calculated by a frequency domain sparse contribution estimation method, and a foundation and a basis are conveniently and effectively provided for vibration suppression of the satellite. Therefore, the method for realizing the quantitative identification of the satellite micro-vibration source based on the sparse blind source separation and the frequency domain sparse contribution estimation method is an effective technical approach.
A specific application example process is given below, while verifying the effectiveness of the invention in engineering applications:
the experimental device is shown in fig. 2, and the experimental object is a satellite cabin structure model 5. The cavity material is an aluminum honeycomb sandwich plate, the bottom of the model is supported and placed on an optical vibration isolation table through four rubber air springs so as to eliminate the influence of ground vibration, and the top of the model is a camera simulation piece which is a sensitive load part of the device. The vibration source comprises two vibration exciters and 4 vibration motors, output signals of the two vibration exciters are respectively controlled by a signal generator 10, a power amplifier 8 and a power amplifier 9, and the rotating speed of the motors is controlled by a speed regulator 4. In this example, two vibration exciters and a vibration motor are started as a vibration source, the output signal of the first vibration exciter 6 is a harmonic signal containing frequency components of 30Hz, 60Hz and 90Hz, the output signal of the second vibration exciter 7 is a harmonic signal containing frequency components of 40Hz, 80Hz and 120Hz, and the a-plane motor is operated at a rotation speed of 2000 r/min. The vibration response signals of the top of the camera simulation piece (which is a sensitive load part of the device), the root of the camera simulation piece, the A surface of the box body and the C surface of the box body are measured by the acceleration sensor 1, the vibration acceleration signals are collected by the data collection system 2, the collected vibration acceleration signal data are stored by the computer 3, the sampling frequency is 5000Hz, the sampling time is 6s, and in order to better display the characteristics of time domain signals, a time domain oscillogram only displays 0.5s of data. The acceleration sensor can acquire vibration response signals in three directions, wherein the direction perpendicular to the A surface is the x direction, the direction perpendicular to the B surface is the y direction, and the direction perpendicular to the ground is the z direction. Firstly, carrying out quantitative identification on the micro-vibration source in the x direction. The time domain waveform diagram of the collected x-direction vibration response signal and the spectrogram thereof are shown in fig. 3 and 4. The time domain oscillogram and the frequency spectrum of the single vibration source signal obtained by the separation by the reference sparse blind deconvolution algorithm are shown in fig. 5 and fig. 6. As can be seen from the figure, the vibration signals of the vibration sources are successfully separated. Then, a spectrogram of the response signal of each vibration source at the sensitive load point is obtained by a frequency domain unit response signal solving method, which is shown in fig. 7. And finally, calculating the contribution of each micro-vibration source at the sensitive load by using a contribution characterization method based on vector projection. The contribution amounts of the micro-vibration sources in the y direction and the z direction at the sensitive load are calculated by the same method, the real vibration response signals of the vibration sources at the sensitive load can be obtained by the independent operation of the vibration sources, so that the real values of the contribution amounts are obtained, the estimated values and the real values of the contribution amounts of the micro-vibration sources in different directions at the sensitive load are shown in the table 1, and the effectiveness of the method is verified by comparison. The contribution amount evaluation index reflects the contribution and influence of each vibration source on the vibration of the sensitive load, is an index for quantitatively identifying the micro-vibration source of the satellite, and can provide a basis for micro-vibration inhibition.
TABLE 1 calculation results of the contribution of each micro-vibration source in different directions at the sensitive load
Claims (3)
1. A satellite micro-vibration source quantitative identification method based on sparse blind source separation is characterized by comprising the following steps:
(1) acquisition of vibration signals
Arranging acceleration sensors at different positions of the sensitive load position and the model surface of the satellite cabin structure model, and acquiring vibration signals, namely observation signals, of each vibration source during normal work:
x(t)＝[x1(t),x2(t),...,xM(t),]T
where M is the number of observed signals, xm(t) is a vibration response signal collected at the mth observation point at the moment t, and M is 1-M;
ensuring that the number of observation signals is greater than the number N of vibration sources;
(2) extraction of single vibration source signal based on reference sparse blind deconvolution algorithm
By means of L1Norm construction reference sparse blind deconvolution algorithm reference L1Constructing a reference signal according to time-frequency domain characteristic prior information of the vibration source by using a norm objective function, and searching an optimal solution of a separation signal by using a gradient descent method to iteratively optimize the objective function so as to extract a single vibration source signal;
the step (2) specifically comprises the following steps:
(2.1) Using L1Norm construction blind deconvolution algorithm reference L1The norm objective function J (y, r) is:
J(y,r)＝||F(y)||1+λE{(y-r)2}
wherein y is the split signal; r is a reference signal; f (-) is a discrete Fourier transform of a variable; λ is a scale factor; e {. is the mathematical expectation of the variable;
(2.2) constructing a reference signal r according to the prior information of the equipment vibration source;
(2.3) setting the separation filter length L, rewriting x (t) obtained in the step (1) into a time lag form
The blind deconvolution model is then converted into a standard ICA model:
wherein the content of the first and second substances,
a split vector of dimension 1 × NL;
at this time, L with respect to y1Norm objective function is converted to
The objective function of (2):
the above equation is expressed by fourier transform theory as:
wherein Re is
A matrix formed by the real parts after Fourier transformation; im is
A matrix formed by the imaginary parts after Fourier transformation;
(2.4) iterative optimization of reference L by gradient descent method1Norm objective function by dividing vector
Iteratively updating and searching the optimal solution of the separation signal y corresponding to the minimum objective function;
(3) calculating the contribution of each vibration source to the sensitive load by using a frequency domain sparse contribution estimation method
Calculating single-source response signals of each vibration source at the sensitive load by using a frequency domain single-source response signal solving method; and calculating the contribution of each vibration source at the sensitive load by using a vector projection-based contribution characterization method.
2. The method for quantitatively identifying the micro-vibration source of the satellite based on the sparse blind source separation as claimed in claim 1, wherein in the step (2.4), the method comprises
The iterative update process of (a) is:
where k represents the kth iterative update and g is the objective function J (y, r) pair
A is the iteration step of the algorithm;
each iteration obtains a new
Then, it is normalized:
iterate until convergence to obtain a separation vector
Then the signals are separated into
The extraction of the single vibration source signal is realized; by constructing different reference messagesThe number enables extraction of different source signals.
3. The method for quantitatively identifying the micro-vibration source of the satellite based on the sparse blind source separation according to claim 1, wherein the step (3) specifically comprises the following steps:
(3.1) let y (t) be the extracted separation signal, and according to the blind deconvolution theory, a filter h (tau) is present and satisfies the following relation:
x(t)＝h(τ)*y(t)
wherein, is convolution operation;
based on the convolution theory, expressing the convolution operation form as a product form in the frequency domain to obtain a matrix relation among the frequency domain value X (omega) of the observation signal at each frequency point omega, the filter frequency domain value H (omega) and the frequency domain value Y (omega) of the separation signal:
X(ω)＝H(ω)Y(ω)
obtaining a filter frequency domain value H (omega) at each frequency point in a frequency domain, calculating the proportion of energy of components at the frequency omega in each source signal to the total energy of the source signal, wherein the source signal with the maximum proportion is the generation source of an observation signal at the frequency point omega;
hypothesis frequency point omega0Where the observed signal is from YnThen the amplitude of the other observation signals at the point is 0 and at the frequency point omega0The following can be obtained:
solving the above formula to obtain the frequency point omega0Mixing matrix H (ω)0)；
And solving a mixed matrix H (omega) at each frequency point by adopting the same method, and solving the single-source response of each vibration source at the camera simulation part in a frequency domain after obtaining the mixed matrix:
wherein the content of the first and second substances,
a response signal at the mth observation for the nth source; hmn(ω) is the mth row and nth column element in the mixing matrix H (ω);
(3.2) Single Source response Signal
Observing signal X at sensitive loadmProjection Z ofmnAs a contribution of the nth vibration source to the mth observation, ZmnAt XmThe proportion of the total carbon is the contribution amount evaluation index cmn：
The contribution amount evaluation index reflects the contribution and influence of each vibration source on the vibration of the sensitive load, and is an index for quantitatively identifying the micro-vibration source of the satellite.
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