CN110457761B - Method for solving singularity problem of Pogo model - Google Patents

Method for solving singularity problem of Pogo model Download PDF

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CN110457761B
CN110457761B CN201910645955.5A CN201910645955A CN110457761B CN 110457761 B CN110457761 B CN 110457761B CN 201910645955 A CN201910645955 A CN 201910645955A CN 110457761 B CN110457761 B CN 110457761B
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谭述君
高强
刘锦凡
毛玉明
赵旺
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Dalian University of Technology
Shanghai Aerospace System Engineering Institute
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Abstract

The invention discloses a method for solving the problem of singularity of a Pogo model, which specifically comprises the following steps: step 1: generating system matrixes E and A of the Pogo state space model; step 2: solving the eigenvalue Lambda and the eigenvector phi; and step 3: arranging the eigenvalue Λ from small to large, and correspondingly arranging the eigenvector phi; and 4, step 4: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate a new eigenvalue matrix
Figure DDA0002133590410000011
And a feature vector matrix
Figure DDA0002133590410000012
And 5: solving the system matrix (E)T,AT) Characteristic value of (A)tAnd the eigenvector phit(ii) a Step 6: the characteristic value ΛtArranged from small to large, the eigenvector phitAre also arranged accordingly; and 7: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate an eigenvalue matrix
Figure DDA0002133590410000013
And a feature vector matrix
Figure DDA0002133590410000014
And 8: using feature vectors
Figure DDA0002133590410000015
And transforming the original state x into a state eta space. The nonsingular Pogo model derived by the method can be directly applied to time domain simulation and active suppression design, the applicability is wide, and repeated modeling work is avoided.

Description

Method for solving singularity problem of Pogo model
Technical Field
The invention belongs to the technical field of liquid carrier rocket Pogo modeling, and particularly relates to a method for solving the problem of singularity of a Pogo model.
Background
The Pogo vibration is unstable closed-loop self-excited vibration generated by the interaction of the longitudinal vibration of a structural system and the liquid path pulsation of a propulsion system in the launching process of a large liquid carrier rocket, and is also called as seesaw vibration or longitudinal coupling vibration. The Pogo vibration can not only deteriorate the low-frequency vibration environment of the rocket, but also cause the failure of flight because instruments and equipment on the rocket cannot work reliably; but also can cause disorder of the physiological system of the astronaut, such as blurred vision and the like. With the improvement of the demand of people on rocket carrying capacity, the size of the rocket is larger, the frequency of the rocket is lower, the problem of coupling with the frequency of a propulsion system is more serious, the United states space administration (NASA) has taken the suppression of Pogo vibration as an important index for liquid rocket design, the Pogo vibration becomes a problem which needs to be emphasized and solved by designers, and a Pogo model of a large-scale liquid rocket is the key for Pogo mechanism analysis and suppression design research.
The current representative Pogo model is a finite element modeling method proposed by the American trainee Rubin, a unified modeling frame is provided by the method, a state space method is adopted for description, and the method is suitable for modeling of complex three-dimensional pipelines. However, the Pogo state space model obtained by the method is singular and is mainly used for frequency domain analysis, the algebraic equations need to be manually reduced during time domain simulation to solve the problem of model singularity, and the method is not suitable for the requirements of rapid analysis and design of Pogo vibration suppression. Aiming at the singularity problem of the Pogo model, the Tanzjun and the Wangqing improve the kinetic equation of the propulsion system component, and derive a nonsingular Pogo state space model. However, this method relies on the description of the propulsion system components and is only applicable to purely liquid-based propulsion systems. For a liquid oxygen kerosene afterburning cycle engine system, a propulsion system adopting gas path-liquid path cross coupling transmission is influenced by gas path complexity, and a proper description mode is difficult to find to establish a Pogo nonsingular model. Considering that most of Pogo models established by scholars are singular, how to put forward a method for solving the problem of model singularity on the basis of the work is very important for subsequent time domain simulation and control system design.
Disclosure of Invention
Aiming at the problems, the invention provides a method for solving the problem of singularity of the Pogo model, the nonsingular Pogo model derived by the method can be directly applied to time domain simulation, the applicability is wide, and repeated modeling work is avoided.
In order to achieve the purpose, the technical scheme of the application is as follows: a method for solving the problem of singularity of a Pogo model specifically comprises the following steps:
step 1: generating system matrixes E and A of the Pogo state space model;
step 2: carrying out eigenvalue analysis on a system matrix (E, A) of the Pogo state space model, and solving an eigenvalue Lambda and an eigenvector phi;
and step 3: arranging the eigenvalue Lambda from small to large according to a model, and correspondingly arranging the eigenvector phi and the eigenvalue; setting the dimension of the matrix E as n and the rank of the matrix E as m;
and 4, step 4: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate a new eigenvalue matrix
Figure BDA0002133590390000021
And a feature vector matrix
Figure BDA0002133590390000022
And 5: solving the system matrix (E)T,AT) Characteristic value of (A)tAnd the eigenvector phit；
Step 6: the characteristic value ΛtArranging the characteristic vectors phi from small to large according to the modetArranged corresponding to the eigenvalues;
and 7: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate an eigenvalue matrix
Figure BDA0002133590390000023
And a feature vector matrix
Figure BDA0002133590390000024
And 8: using feature vectors
Figure BDA0002133590390000025
And transforming the original state x into a state eta space.
Further, a Pogo state space model is established in the step 1:
Figure BDA0002133590390000031
in which the state variable x is composed of a propulsion system variable p and a structural system variable q, i.e. x ═ pT,qT]TE and A are system matrices, f is an external disturbance or control force;
Figure BDA00021335903900000311
is the first derivative of the state variable x;
the solving of the eigenvalue Λ and the eigenvector Φ in the step 2 specifically comprises:
EΦΛ＝AΦ (2)
in step 3, the eigenvalues Λ are arranged from small to large, and the eigenvectors Φ are correspondingly arranged, specifically:
Figure BDA0002133590390000032
Figure BDA0002133590390000033
further, a new eigenvalue matrix is generated in step 4
Figure BDA0002133590390000034
And a feature vector matrix
Figure BDA0002133590390000035
The method specifically comprises the following steps:
Figure BDA0002133590390000036
Figure BDA0002133590390000037
the formula (2) is then the following formula,
Figure BDA0002133590390000038
further, the system matrix in step 5 (E)T,AT) Characteristic value of (A)tAnd the eigenvector phitThe method specifically comprises the following steps:
ETΦtΛt＝ATΦt (8)。
further, the characteristic value Λ is processed in step 6tArranging the characteristic vectors phi from small to large according to the modetAnd correspondingly arranging, specifically:
Figure BDA0002133590390000039
Figure BDA00021335903900000310
according to the matrix eigenvalue theory, Λt＝Λ。
Further, in step 7, a matrix of eigenvalues is generated
Figure BDA0002133590390000041
And a feature vector matrix
Figure BDA0002133590390000042
The method specifically comprises the following steps:
Figure BDA0002133590390000043
Figure BDA0002133590390000044
then equation (8) becomes:
Figure BDA0002133590390000045
further, the step 8 is implemented by the following steps:
Figure BDA0002133590390000046
can obtain the product
Figure BDA0002133590390000047
Can obtain the product
Figure BDA0002133590390000048
Left-multiplying both sides of equation (16) simultaneously
Figure BDA0002133590390000049
Obtaining:
Figure BDA00021335903900000410
due to the fact that
Figure BDA00021335903900000411
So that the Pogo model described by state η is non-singular, and so both sides of equation (17) are simultaneously left-multiplied
Figure BDA00021335903900000412
Obtaining:
Figure BDA00021335903900000413
the nonsingular Pogo model described by the state eta solves the problem of singularity of the original Pogo model,
Figure BDA00021335903900000414
is the first derivative of state η;
further, η initial value η (t)0) Determination, when solving eta (t) based on differential equation (18), it is necessary to determine an initial value eta (t)0) Because of the original state initial value x (t)0) As is known, it is obtained from the inverse of the transformation (14),
Figure BDA0002133590390000051
as a further step, the system matrices E and a are asymmetric real matrices, so the eigenvalue Λ and eigenvector matrix Φ may be complex numbers, if complex, then necessarily appear in pairs, corresponding to the second order vibration equations of the structural or propulsion system; the eigenvalues and eigenvectors of the matrix are thus described as
Figure BDA0002133590390000052
Φ＝[Φr+iΦi,Φr-iΦi] (21)
Wherein, ΛrAnd ΛiAre the real and imaginary part, phi, of the eigenvalue Λ, respectivelyrAnd phiiRespectively a real part and an imaginary part of the eigenvector matrix phi, wherein i is an imaginary unit;
finishing to obtain:
Figure BDA0002133590390000053
thus, the eigenvalues and eigenvectors can be described by real numbers, i.e.
Figure BDA0002133590390000054
As a further step, the above method is adopted to perform order reduction processing on the Pogo model: if only the feature values of interest are retained in equations (5), (6) and (11), (12), then the derived Pogo model (18) is a reduced order model.
Due to the adoption of the technical scheme, the invention can obtain the following technical effects:
1. the method can process a common Pogo singular model to obtain a nonsingular Pogo model, has wide applicability and avoids repeated modeling work.
2. The nonsingular Pogo model derived by the method can be directly applied to time domain simulation.
3. The nonsingular Pogo model derived by the method can be used for Pogo active suppression design.
4. The method can be used for order reduction of the Pogo model.
5. The method can adopt complex number operation and can also be completed based on real number operation.
Drawings
FIG. 1 is a view of a rocket propulsion system of a certain type;
FIG. 2 is an external force disturbing diagram;
FIG. 3 is a graph of the structural damping ratio of the Pogo system;
FIG. 4 is a structural response diagram of the Pogo system;
FIG. 5 is a graph of the structural damping ratio of the Pogo system;
fig. 6 is a structural response diagram of the Pogo system.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it should be understood that the described examples are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
Example 1
The embodiment provides a method for solving the problem of singularity of a Pogo model, which specifically comprises the following steps:
(1) and generating system matrixes E and A of the Pogo state space model. A Rubin method or other methods are adopted to establish a Pogo state space model,
Figure BDA0002133590390000061
in which the state variable x is composed of a propulsion system variable p and a structural system variable q, i.e. x ═ pT,qT]TE and A are system matrices and f is an external disturbance or control force.
Since the matrix E is singular, the equation is generally only used for frequency domain analysis, and is difficult to be directly used in time domain simulation and control system design.
(2) Eigenvalue analysis is performed on the system matrix (E, A) of the Pogo state space equation, the eigenvalue Λ and the eigenvector Φ are solved according to the following eigenvalue equation,
EΦΛ＝AΦ \*MERGEFORMAT (1.2)
(3) and arranging the eigenvalue Lambda from small to large according to a model, and correspondingly arranging the eigenvector Phi. Since the matrix E is singular, there are infinite generalized eigenvalues of the system matrix (E, a). If the dimension of matrix E is n and the rank of matrix E is m, the system matrix (E, A) has m infinite generalized eigenvalues, the described eigenvalues are arranged from small to large, and the eigenvectors are correspondingly adjusted as follows
Figure BDA0002133590390000071
Figure BDA0002133590390000072
(4) Reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate a new eigenvalue matrix
Figure BDA0002133590390000073
And a feature vector matrix
Figure BDA0002133590390000074
Since the last m eigenvalues are infinite and the corresponding eigenvectors have no meaning, only the first n-m eigenvalues and their corresponding eigenvectors, i.e., the first n-m eigenvalues and their corresponding eigenvectors, are retained
Figure BDA0002133590390000075
Figure BDA0002133590390000076
Then the process of the first step is carried out,
Figure BDA0002133590390000077
(5) solving the system matrix (E) usingT,AT) Characteristic value of (A)tAnd the eigenvector phit，
ETΦtΛt＝ATΦt \*MERGEFORMAT (1.8)
(6) The characteristic value ΛtArranged from small to large, the eigenvector phitCorresponding arrangements are also made. Also, due to the matrix ETIs singular, resulting in a system matrix (E)T,AT) There are infinite generalized eigenvalues. Because, the matrix ETIs m, then the system matrix (E)T,AT) There are m infinite generalized eigenvalues, the described eigenvalues are arranged from small to large, and the eigenvectors are correspondingly adjusted as follows
Figure BDA0002133590390000081
Figure BDA0002133590390000082
According to the matrix eigenvalue theory, Λt＝Λ。
(7) Reserving the first n-m characteristic values and corresponding characteristic vectors to generate
Figure BDA0002133590390000083
And
Figure BDA0002133590390000084
similarly, since the last m eigenvalues are infinite, the corresponding eigenvector has no meaning, and thus only the first n-m eigenvalues and their corresponding eigenvectors, i.e., the first n-m eigenvalues and their corresponding eigenvectors, are retained
Figure BDA0002133590390000085
Figure BDA0002133590390000086
Then the formula becomes
Figure BDA0002133590390000087
(8) Using feature vectors
Figure BDA0002133590390000088
Transforming the original state x to a state eta space,
Figure BDA0002133590390000089
substituting formula into the formula to obtain
Figure BDA00021335903900000810
Substituting formula into the formula to obtain
Figure BDA00021335903900000811
Simultaneous left multiplication of two sides of formula
Figure BDA00021335903900000812
To obtain
Figure BDA00021335903900000813
Because of the fact that
Figure BDA00021335903900000814
So that the Pogo model described by state η is non-singular. Thus, two sides of the formula are simultaneously left-handed
Figure BDA00021335903900000815
So as to obtain the compound with the characteristics of,
Figure BDA0002133590390000091
the non-singular Pogo model described by the state eta solves the problem of singularity of the original Pogo model. After η is obtained by the solution, the solution of the original state x can be obtained by using the transformation expression. Time domain simulation, control system design and the like can be developed based on the transformed nonsingular model.
Initial value of eta (t)0) And (4) determining. When solving η (t) based on differential equation, it is necessary to determine the initial value η (t)0). Because of the original state initial value x (t)0) As is known, and therefore can be obtained from the inverse of the transform,
Figure BDA0002133590390000092
the real number of the operation process is as follows: since E and a are asymmetric real matrices, the eigenvalue Λ and eigenvector matrix Φ may be complex, and if complex must occur in pairs, corresponding to the second order vibration equations of the structural or propulsion system. Thus the eigenvalues and eigenvectors of the matrix can be described as
Figure BDA0002133590390000093
Φ＝[Φr+iΦi,Φr-iΦi] \*MERGEFORMAT (1.21)
Wherein, ΛrAnd ΛiAre the real and imaginary part, phi, of the eigenvalue Λ, respectivelyrAnd phiiRespectively the real and imaginary parts of the eigenvector matrix phi.
Is finished to obtain
Figure BDA0002133590390000094
Thus, the eigenvalues and eigenvectors in the preceding steps can be described in real numbers, i.e.
Figure BDA0002133590390000095
The above method can be used for order reduction of Pogo models: if only the feature values of interest are retained in equations (1.5), (1.6) and (1.11), (1.12), then the derived Pogo model (1.18) is a reduced order model.
Example 2
The Pogo system of a rocket of a certain type is analyzed, and the propulsion system is an air-liquid path coupling propulsion system as shown in figure 1. The system matrix E of the original state x in the formula obtained by modeling by a traditional method is 60 x 60 dimensions, wherein the rank of the E matrix is 56, and the E matrix is not full rank and is singular. By adopting the method, the simulation is carried out by transforming the state to the eta space. Two working conditions were simulated.
(1) Working condition 1
The nominal value of the system parameter is adopted in the working condition 1, and the eigenvalue analysis result of the system matrix (E, a) of the original state space x is shown in fig. 3, so that the structural modal damping ratios are all larger than zero, and the system is stable. According to the method, the simulation can be carried out by transforming to the state eta space, the simulation result is transformed back to the original state space x, the external force disturbance is shown in figure 2, the simulation result is shown in figure 4, it can be seen that under random disturbance, the structural response of the Pogo system is not diverged and is consistent with the frequency domain analysis result, and the correctness of the nonsingular Pogo model obtained by using the method is proved.
(2) Working condition 2
Under the working condition 2, on the basis of the nominal value of the system parameter, a larger deviation is superposed on the structural modal quality parameter in the vicinity of 30 seconds, and at the moment, the characteristic value analysis result of the system matrix (E, A) of the original state space x is shown in fig. 5, so that the 1 st-order modal damping ratio of the structure is smaller than zero in the vicinity of 30 seconds, and at the moment, the system is unstable. According to the method, the model can be transformed to a state eta space for simulation, a simulation result is further transformed back to an original state space x in an inverse mode, external force disturbance which is the same as that under the working condition 1 is taken as shown in a figure 2, and a simulation result is shown in a figure 6, so that the structural response of the Pogo system is shown to be vibration divergence (amplitude divergence is two orders of magnitude) in the vicinity of 30 seconds under the same random interference, and the accuracy of the nonsingular Pogo model obtained by the method is proved to be consistent with the frequency domain analysis result.
The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes performed by the present invention as described in the specification and the accompanying drawings, or applied directly or indirectly to other related technical fields, are also included in the scope of the present invention.

Claims (6)

1. A method for solving the problem of singularity of a Pogo model is used for analyzing a gas-liquid path coupling propulsion system, and is characterized by comprising the following steps:
step 1: generating system matrixes E and A of the Pogo state space model;
step 2: carrying out eigenvalue analysis on a system matrix (E, A) of the Pogo state space model, and solving an eigenvalue Lambda and an eigenvector phi;
and step 3: arranging the eigenvalue Lambda from small to large according to a model, and correspondingly arranging the eigenvector phi and the eigenvalue; setting the dimension of the matrix E as n and the rank of the matrix E as m;
and 4, step 4: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate a new eigenvalue matrix
Figure FDA0002815107920000011
And a feature vector matrix
Figure FDA0002815107920000012
And 5: solving the system matrix (E)T,AT) Characteristic value of (A)tAnd the eigenvector phit；
Step 6: the characteristic value ΛtPress dieArranging the eigenvectors phi from small to largetArranged corresponding to the eigenvalues;
and 7: reserving the first n-m eigenvalues and corresponding eigenvectors thereof to generate an eigenvalue matrix
Figure FDA0002815107920000013
And a feature vector matrix
Figure FDA0002815107920000014
And 8: using feature vectors
Figure FDA0002815107920000015
Transforming the original state x to a state eta space;
the Pogo state space model established in the step 1 is as follows:
Figure FDA0002815107920000016
in which the state variable x is composed of a propulsion system variable p and a structural system variable q, i.e. x ═ pT,qT]TE and A are system matrices, f is an external disturbance or control force;
Figure FDA0002815107920000017
is the first derivative of the state variable x;
the solving of the eigenvalue Λ and the eigenvector Φ in the step 2 specifically comprises:
EΦΛ＝AΦ (2)
in step 3, the eigenvalues Λ are arranged from small to large, and the eigenvectors Φ are correspondingly arranged, specifically:
Figure FDA0002815107920000021
Figure FDA0002815107920000022
generating a new eigenvalue matrix in step 4
Figure FDA0002815107920000023
And a feature vector matrix
Figure FDA0002815107920000024
The method specifically comprises the following steps:
Figure FDA0002815107920000025
Figure FDA0002815107920000026
the formula (2) is then the following formula,
Figure FDA0002815107920000027
the step 8 is realized by the following steps:
Figure FDA0002815107920000028
can obtain the product
Figure FDA0002815107920000029
Can obtain the product
Figure FDA00028151079200000210
Left-multiplying both sides of equation (16) simultaneously
Figure FDA00028151079200000211
Obtaining:
Figure FDA00028151079200000212
due to the fact that
Figure FDA00028151079200000213
So that the Pogo model described by the state η is nonsingular, and thus, the two sides of the formula are simultaneously left-multiplied
Figure FDA00028151079200000214
Obtaining:
Figure FDA00028151079200000215
the non-singular Pogo model described by the state eta solves the problem of singularity of the original Pogo model.
2. The method for solving the singularity problem of the Pogo model in claim 1, wherein the system matrix (E) in step 5T,AT) Characteristic value of (A)tAnd the eigenvector phitThe method specifically comprises the following steps:
ETΦtΛt＝ATΦt (8)。
3. the method for solving the singularity problem of the Pogo model in claim 1, wherein the eigenvalue Λ in step 6tArranging the characteristic vectors phi from small to large according to the modetAnd correspondingly arranging, specifically:
Figure FDA0002815107920000031
Figure FDA0002815107920000032
according to the matrix eigenvalue theory, Λt＝Λ。
4. The method for solving the singularity problem of the Pogo model according to claim 3, wherein the eigenvalue matrix is generated in step 7
Figure FDA0002815107920000033
And a feature vector matrix
Figure FDA0002815107920000034
The method specifically comprises the following steps:
Figure FDA0002815107920000035
Figure FDA0002815107920000036
then equation (8) becomes:
Figure FDA0002815107920000037
5. the method for solving the singularity problem of the Pogo model in claim 1, wherein η (t) is an initial value η (t)0) Determination, when solving eta (t) based on differential equation (18), it is necessary to determine an initial value eta (t)0) Because of the original state initial value x (t)0) As is known, it is obtained from the inverse of the transformation (14),
Figure FDA0002815107920000038
6. the method for solving the singularity problem of the Pogo model according to claim 1, wherein the system matrices E and a are asymmetric real matrices, so the eigenvalue Λ and the eigenvector matrix Φ may be complex numbers, if complex, then necessarily occur in pairs, corresponding to the second order vibration equation of the structural system or the propulsion system; the eigenvalues and eigenvectors of the matrix are thus described as
Figure FDA0002815107920000041
Φ＝[Φr+iΦi,Φr-iΦi] (21)
Wherein, ΛrAnd ΛiAre the real and imaginary part, phi, of the eigenvalue Λ, respectivelyrAnd phiiRespectively a real part and an imaginary part of the eigenvector matrix phi, wherein i is an imaginary unit;
finishing to obtain:
Figure FDA0002815107920000042
thus, the eigenvalues and eigenvectors are described by real numbers, i.e.
Figure FDA0002815107920000043
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