阅读说明：本技术 气-液路耦合推进系统运载火箭的Pogo系统建模方法 (Pogo system modeling method of gas-liquid path coupling propulsion system carrier rocket ) 是由 赵旺 刘锦凡 谭述君 毛玉明 孙丹 朱春艳 狄文斌 于 2019-09-27 设计创作，主要内容包括：本发明公开了一种气-液路耦合推进系统运载火箭的Pogo系统建模方法,建立燃气发生器、涡轮、燃气导管、气-液型推力室等气体部件的二阶动力学模型,与建立的液路部分二阶动力学方程组装成完整的推进系统,从而建立起基于状态空间法的包含气路特性的Pogo状态空间模型。与传统的传递矩阵法相比,可以在计算推进系统频率的同时给出阻尼比；与传统的迭代法相比,状态空间法计算效率更高,而且不受迭代初值的影响,不会漏根；状态方程法可以考虑多模态、多耦合点、芯级与助推耦合作用等更多因素,结果更为精确。(The invention discloses a Pogo system modeling method of a gas-liquid path coupling propulsion system carrier rocket, which comprises the steps of establishing a second-order kinetic model of gas components such as a gas generator, a turbine, a gas guide pipe, a gas-liquid type thrust chamber and the like, and assembling the second-order kinetic model and an established second-order kinetic equation of a liquid path part into a complete propulsion system, thereby establishing a Pogo state space model containing gas path characteristics based on a state space method. Compared with the traditional transfer matrix method, the damping ratio can be given while the frequency of the propulsion system is calculated; compared with the traditional iteration method, the state space method has higher calculation efficiency, is not influenced by the initial value of the iteration, and cannot leak roots; the state equation method can consider more factors such as multimode, multi-coupling points, core-stage and boosting coupling effects, and the result is more accurate.)

1. A Pogo system modeling method of a gas-liquid path coupling propulsion system carrier rocket is characterized by comprising the following steps:

step 1: establishing a second-order dynamic model of a liquid path part of the propulsion system;

step 2: establishing a second-order dynamic model of each unit and boundary conditions of the gas path part;

and step 3: assembling the units of the gas circuit and the second order kinetic equation of the boundary condition according to the selection sequence of the state variables;

and 4, step 4: in order to enable the liquid circuit and the gas circuit to be assembled, the liquid circuit and the gas circuit need to be made to have the same state variable;

and 5: adjusting a gas path second-order kinetic equation of the propulsion system into a form suitable for being connected with a liquid path part;

step 6: assembling the liquid path second order kinetic equation coefficient matrix and the gas path second order kinetic equation coefficient matrix according to the selection sequence of the state variables, and assembling the two matrixes in the R_GFilling 1 in a place corresponding to the pressure at the tail end of the matrix liquid path, and deriving a second-order kinetic equation of the propulsion system containing the gas path characteristics;

and 7: solving first derivative on two sides of the structural vibration equation, and adjusting the structural vibration equation into a form suitable for being assembled with a propulsion system;

and 8: and coupling a second-order kinetic equation of the propulsion system with a structural vibration equation to derive a Pogo system model containing gas path characteristics.

2. The Pogo system modeling method of a gas-liquid path coupled propulsion system launch vehicle of claim 1, wherein the step 1 of establishing a propulsion system liquid path part second order dynamics model specifically comprises:

wherein the mass displacement u_LIs a state variable of the liquid path system,

3. The Pogo system modeling method of the gas-liquid path coupling propulsion system carrier rocket of claim 1, wherein in the step 2, Taylor second-order expansion is performed on a transmission matrix model of a gas path system precombustion chamber, a turbine, a rectifier grid, a gas guide pipe, a thrust chamber unit and boundary conditions, Laplace inverse transformation is performed to convert the expansion into second-order kinetic equation description, and the second-order kinetic equation description of each unit of a gas path part and the boundary conditions is derived.

4. The method of claim 3, wherein the Pogo system modeling of the gas-liquid path coupled propulsion system launch vehicle,

(1) the dimensional second order kinetic equation for the prechamber unit is described as:

wherein

(2) the dimensional second order kinetic equation for a turbine unit is described as:

in the formula:

(3) the dimensional second order kinetic equation of the gas conduit unit is described as:

in the formula:

(4) the dimensional second order kinetic equation of the rectifier grid unit is described as follows:

in the formula:

(5) the dimensionalized second order kinetic equation of the thrust cell unit is described as:

the force of the thrust chamber on the structure is described as follows:

wherein Q_p′＝[-A_thC_f]，C_fIs the thrust coefficient, Ath is the thrust chamber throat area;

in the formula:

(6) boundary conditions need to be added at the tail end of the gas circuit, and the boundary conditions with dimension basic variables are as follows:

5. the Pogo system modeling method of a gas-liquid path coupled propulsion system launch vehicle of claim 1, wherein the propulsion system gas path portion equation including boundary conditions after the step 3 assembly is completed is:

wherein M'_G，R′_G，K′_GRespectively a mass array, a damping array and a rigidity array of the gas circuit system q'_GState variables of the gas circuit:

6. the Pogo system modeling method of a launch vehicle of a gas-liquid path coupled propulsion system according to claim 1, characterized in that step 4 is implemented by solving first derivatives on both sides of the propulsion system equation of the liquid path in such a way that the state variables of the liquid path become the same as the state variables of the gas path, and adjusting the second order kinetic equation of the liquid path of the propulsion system to a form suitable for connection with the gas path portion;

make the equation of the liquid path part become

Wherein M is_LIs a mass array; r_LIs a damping array; k_LIs a stiffness matrix; p_LIs the pressure intensity and mass flow q of each node of the liquid path unit_LAnd mass displacement u_LIn a relationship ofNamely, it is

7. The method of modeling the Pogo system of a gas-liquid coupled propulsion system launch vehicle of claim 1, wherein step 5 is to M'_G，R′_G，K′_GTwo columns 1 and 2 of the matrix are interchanged to

q′_G＝[P_G0,q_G0,P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^T

Coefficient matrix M 'derived for state variables'_G，R′_G，K′_GWritten as state variables

q_G＝[q_G0,P_G0,P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^T

Coefficient matrix M derived for basic variables_G，R_G，K_G。

8. The Pogo system modeling method of a gas-liquid path coupled propulsion system launch vehicle of claim 1, wherein the specific implementation manner of step 6 is as follows:

the liquid path and the gas path are assembled as follows:

wherein

Variable of state

q_L＝[q_L1,q_L2,q_L3…q_Ln,q_G0]^TIs the mass flow state variable of the liquid path;

q_G0，P_G0，q″_G＝[P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^Tis the state variable of the gas circuit;

the outlet quantity of the liquid path is the inlet quantity of the gas path, and the state variables of the liquid path and the gas path both contain q_G0I.e. the state variables of the liquid path and the gas path overlap.

9. The Pogo system modeling method of a gas-liquid path coupled propulsion system launch vehicle of claim 1, wherein the specific implementation manner of step 7 is as follows:

equation of vibration of structure

Two ends are derived to obtain

M_s，R_s，K_sA structural system mass array, a damping array and a rigidity array; q₀，Q₁，Q₂For coupling system matrices, Q_pTo be composed ofWhen being state variables Q_p' spreading matrix, q_sModal displacement of the structural system; v is a coupling system coefficient matrix; wherein the relationship between the q mass flow and the u mass displacement is

10. The Pogo system modeling method of a gas-liquid path coupled propulsion system launch vehicle of claim 1, wherein the specific implementation manner of step 8 is: taking state variablesCoupling a second order kinetic equation of the propulsion system with a structural vibration equation to obtain a Pogo system model can be written as:

wherein

Technical Field

The invention belongs to the field of dynamic modeling, and particularly relates to a Pogo system modeling method containing air path characteristics based on a state space method.

Background

The Pogo vibration modeling method comprises a single transmission method, a critical damping method, a matrix method and the like, wherein a state equation method proposed by Rubin is more advanced. However, the methods are all researches on a liquid path system, and the heavy carrier rockets and the like in China adopt novel high-pressure low-temperature liquid oxygen/kerosene afterburning cycle engines with gas path characteristics, the influence of the gas path entropy wave characteristics on the Pogo vibration modeling and stability is not negligible, and the literature data in the aspect is very lacking. According to the Liu jin Van, a transfer matrix method is used for carrying out stability analysis on a Pogo system with gas path characteristics, although the Pogo stability of a rocket can be well predicted, the method is only suitable for Pogo stability analysis of a single-path carrying rocket, and can only be equivalently treated as a single path for a rocket with multiple paths, so that the analysis effect is poor.

In the modeling method for liquid rocket Pogo vibration, the equation of state method is a relatively perfect method at present. The method comprises the steps of dividing a pipeline into a plurality of basic units by using a finite element modeling thought for reference, uniformly describing the basic units and elements such as a pump, a pressure accumulator, a thrust chamber and the like by using a second-order differential equation, and constructing a second-order differential equation of Pogo vibration of the whole system by coupling with a structural vibration equation, so that Pogo stability analysis is converted into a characteristic value problem of a generalized matrix. The method is easy to popularize to the modeling of a complex three-dimensional pipeline, has strong universality, can conveniently analyze the bundled liquid rocket with a plurality of boosters and a plurality of engines, and is successfully applied to the Atlas-II/Centaurand rocket in the United states and the research on the Pogo problem of CZ-2F rocket in China. The state space model established by the method can be directly used for frequency domain analysis, and compared with the traditional transfer matrix method, the method can calculate the frequency of the propulsion system and give a damping ratio; compared with the traditional iteration method, the state space method has higher calculation efficiency, is not influenced by the initial value of the iteration, and cannot leak roots; the equation of state method can consider more factors such as multimode, multi-coupling point, core stage and boosting coupling effect, and the result is more accurate. But the equation of state method has no relevant report for Pogo modeling of a gas-liquid path propulsion system.

Disclosure of Invention

The application provides a Pogo system modeling method of a gas-liquid path coupling propulsion system carrier rocket, which is characterized in that a second-order dynamic model of gas components such as a gas generator, a turbine, a gas guide pipe and a gas-liquid type thrust chamber is established, and the second-order dynamic model and an established second-order dynamic equation of a liquid path part are assembled into a complete propulsion system, so that a Pogo state space model containing gas path characteristics based on a state space method is established.

In order to achieve the purpose, the technical scheme of the invention is as follows: a Pogo system modeling method of a gas-liquid path coupling propulsion system carrier rocket comprises the following steps:

step 1: establishing a second-order dynamic model of a liquid path part of the propulsion system;

step 2: establishing a second-order dynamic model of each unit and boundary conditions of the gas path part;

and step 3: assembling the units of the gas circuit and the second order kinetic equation of the boundary condition according to the selection sequence of the state variables;

and 4, step 4: in order to enable the liquid circuit and the gas circuit to be assembled, the same state variables are needed for the liquid circuit and the gas circuit;

and 5: adjusting a gas path second-order kinetic equation of the propulsion system into a form suitable for being connected with a liquid path part;

step 6: assembling the coefficient matrix of the second order kinetic equation of the liquid path and the coefficient matrix of the second order kinetic equation of the gas path according to the selection sequence of the state variables, and assembling the coefficient matrix of the second order kinetic equation of the gas path and the coefficient matrix of the second order kinetic equation of the gas path in the R sequence_GAnd filling 1 in a place corresponding to the pressure at the tail end of the matrix liquid path, and deriving a second-order kinetic equation of the propulsion system containing the gas path characteristics.

And 7: solving first derivative on two sides of the structural vibration equation, and adjusting the structural vibration equation into a form suitable for being assembled with a propulsion system;

and 8: and coupling a second-order kinetic equation of the propulsion system with a structural vibration equation to derive a Pogo system model containing gas path characteristics.

Further, the step 1 of establishing a second-order dynamic model of the liquid path part of the propulsion system specifically comprises the following steps:

wherein the mass displacement u_LIs a state variable of the liquid path system,

for the flow velocity of the fluid relative to the wall of the tube, P_LIs the pressure at each node of the liquid path unit, and u_L＝[u₁u₂… u_i…]^T，u₁u₂… u_i… is the mass displacement at each node; p_G0The pressure intensity of the tail end node of the liquid path is also the pressure intensity of the initial end of the gas path; q. q.s_sModal displacement of the structural system;

is u_LA first derivative;is u_LA second derivative;

is q_sA first derivative;

is q_sA second derivative; m_LIs a mass array; r_LIs a damping array; k_LIs a stiffness matrix; u shape_0L，U_1L，U_2LCoefficient matrixes for different sections of the propulsion system; rho is mass density; a is the sectional area of the pipeline.

Further, step 2, carrying out Taylor second-order expansion on the pre-combustion chamber, the turbine, the rectifier grid, the gas guide pipe, the thrust chamber unit of the gas path system and the transfer matrix model of the boundary condition, carrying out Laplace inverse transformation to convert the expansion into a second-order kinetic equation description, and deriving the second-order kinetic equation description of each unit and the boundary condition of the gas path part.

Further, (1) the dimensionalized second order kinetic equation of the prechamber unit is described as:

wherein

Respectively the steady state gas pressure, flow, temperature and mixing ratio at the outlet of the pre-combustion chamber,

steady state pressure and flow, p, respectively, of the prechamber inlet_G2、q_G2、T_G2、 K_G2Respectively representing the pressure, flow, temperature and mixing ratio of the gas at the outlet of the precombustion chamber; k_ggThe rated mixing ratio of the gas generator; tau is_ΓCombustion time lag of the gas generator; psi is the slope of the combustion product temperature versus the mixing ratio of the propellant components, and

P_G0，q_G0for the pressure at the end of the liquid path and the pulsating quantity, τ, of the liquid oxidizing agent flow entering the generator through the nozzle_ggTime from gas formation to gas generator outlet; k is a radical of_ggIs a gas adiabatic index; t is the combustion product temperature;

are the partial derivative symbols.

(2) The dimensional second order kinetic equation for a turbine unit is described as:

in the formula:respectively the steady state gas pressure, flow, temperature and mixing ratio at the turbine inlet,

respectively steady state gas pressure, flow, temperature and mixing ratio, p, at the turbine outlet_G2、q_G2、T_G2、K_G2Turbine inlet gas pressure, flow, temperature and mixing ratio, respectively; p is a radical of_G3、q_G3、T_G3、K_G3Respectively turbine outlet gas pressure, flow, temperature and mixing ratio;andrespectively the static temperature of the inlet and the outlet of the turbine; ε is the slope of the gas flow through the turbine versus pressure.

(3) The dimensional second order kinetic equation of the gas conduit unit is described as:

in the formula:respectively the steady state gas pressure, flow, temperature and mixing ratio at the inlet of the gas conduit,

respectively, steady state gas pressure, flow, temperature and mixing ratio, p, of the gas conduit outlet_G3、q_G3、T_G3、K_G3Respectively being gas conduitsInlet gas pressure, flow, temperature and mixing ratio; p is a radical of_G4、q_G4、T_G4、K_G4Respectively the pressure, flow, temperature and mixing ratio of the gas at the outlet of the gas conduit; k is a radical of_gdThe heat insulation index of the gas is the gas insulation index of the gas guide pipe; tau is_gdThe residence time of the fuel gas in the fuel gas guide pipe;

(4) the dimensional second order kinetic equation of the rectifier grid unit is described as follows:

in the formula:

respectively the steady state gas pressure, flow, temperature and mixing ratio at the inlet of the flow straightener,

respectively the steady state gas pressure, flow, temperature and mixing ratio, p, of the outlet of the rectifier grid_G4、q_G4、T_G4、K_G4Respectively the pressure, flow, temperature and mixing ratio of the gas at the inlet of the rectifier grid, p_G5、q_G5、T_G5、K_G5Respectively the pressure, flow, temperature and mixing ratio of the gas at the outlet of the rectifier grid; epsilon is the slope of the gas flow rate-pressure ratio curve passing through the rectifier grid.

(5) The dimensionalized second order kinetic equation of the thrust cell unit is described as:

the gas-liquid type thrust chamber is different from other gas circuit units, and has the acting force on the structure besides the kinetic equation, and the acting force on the structure is described as follows:

wherein Q_p′＝[-A_thC_f]，C_fIs the thrust coefficient, Ath is the thrust chamber throat area;

in the formula:

respectively the steady state gas pressure, flow, temperature and mixing ratio at the inlet of the thrust chamber,

respectively the steady state gas pressure, flow, p of the outlet of the thrust chamber_cFor pulsating pressure in the combustion chamber, q_mcIs the total pulsating flow of the combustion chamber; p is a radical of_G5、q_G5、T_G5、K_G5Respectively the pressure, flow, temperature and mixing ratio of the gas at the inlet of the thrust chamber; k_ggThe rated mixing ratio of the fuel gas entering the combustion chamber from the fuel gas conduit through the nozzle; tau is_cThe residence time of the fuel gas in the combustion chamber; k is a radical of_cIs the adiabatic index of the gas in the combustion chamber; psi_cIs the slope of the curve of the gas temperature in the combustion chamber as a function of the mixing ratio of the components, and

q_mfgis a steady oxygen-enriched gas flow entering the combustion chamber from a gas conduit through a nozzle; k_mcThe rated mixing ratio of the combustion chamber; q. q.s_mfcIs the steady state liquid fuel flow into the combustion chamber; a. the_q＝q_mfg+(1+K_gg)q_mfc；τ_TcThe time for the liquid fuel to convert to gas in the combustion chamber.

(6) Boundary conditions need to be added at the tail end of the gas circuit, and the boundary conditions with dimension basic variables are as follows:

further, the equation of the gas path part of the propulsion system including the boundary conditions after the assembly in step 3 is as follows:

wherein M'_G，R′_G，K′_GRespectively a mass array, a damping array and a rigidity array of the gas circuit system q'_GState variables of the gas circuit:

further, step 4, solving first derivatives on both sides of the liquid path propulsion system equation by changing the state variable of the liquid path to be the same as the state variable of the gas path, and adjusting the second order kinetic equation of the liquid path of the propulsion system to be suitable for being connected with the gas path part;

make the equation of the liquid path part become

Wherein M is_LIs a mass array; r_LIs a damping array; k_LIs a stiffness matrix; p_LIs the pressure intensity and mass flow q of each node of the liquid path unit_LAnd mass displacement u_LIn a relationship of

Namely, it is

The flow rate of the fluid relative to the pipe wall is shown, and A is the sectional area of the pipeline.

Further, step 5 is to M'_G，R′_G，K′_GExchanging two columns of 1 and 2 of the matrix, and replacing the original two columns with the original two columns

q′_G＝[P_G0,q_G0,P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^T

Coefficient matrix M 'derived for state variables'_G，R′_G，K′_GWritten as state variables

q_G＝[q_G0,P_G0,P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^T

Coefficient matrix M derived for basic variables_G，R_G，K_G。

Further, the specific implementation manner of step 6 is:

the liquid path and the gas path are assembled as follows:

wherein

Variable of state

q_L＝[q_L1,q_L2,q_L3…q_Ln,q_G0]^TIs the mass flow state variable of the liquid path;

q_G0，P_G0，q″_G＝[P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^Tis the state variable of the gas circuit;

the outlet quantity of the liquid path is the inlet quantity of the gas path, and the state variables of the liquid path and the gas path both contain q_G0I.e. the state variables of the liquid path and the gas path overlap.

Further, the specific implementation manner of step 7 is:

equation of vibration of structure

Two ends are derived to obtain

M_s，R_s，K_sA structural system mass array, a damping array and a rigidity array; q₀，Q₁，Q₂For coupling system matrices, Q_pTo be composed of

When being state variables Q_p' spreading matrix, q_sStructural system modal shift; v is a coupling system coefficient matrix; wherein the relationship between the q mass flow and the u mass displacement is

Namely, it is

Further, the specific implementation manner of step 8 is: taking state variables

Coupling a second order kinetic equation of the propulsion system with a structural vibration equation to obtain a Pogo system model can be written as:

wherein

Through the technical scheme, the invention can obtain the following effects: compared with the traditional transfer matrix method, the damping ratio can be given while the frequency of the propulsion system is calculated; compared with the traditional iterative method, the state space method has higher calculation efficiency, is not influenced by the initial iteration value, and does not leak roots; the equation of state method can consider multimodality, multiple coupling points, core level and boosting coupling and other factors, and the result is more accurate.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for the present invention will be briefly described below.

FIG. 1 is a layout diagram of units of a liquid path system in which only an oxygen path is considered by a rocket of a certain type in the embodiment;

FIG. 2 is a layout diagram of the units of the air path system in the embodiment.

The sequence numbers in the figures illustrate: 1 a first section of corrugated pipe, 2 a second section of straight pipe, 3 a third section of corrugated pipe, 4 a fourth section of straight pipe, 5 a three-way-accumulator super unit, 6 a swinging hose corrugated pipe, 7 a pre-pressurizing pump, 8 a pump pipe, 9 a pump pipe, 10 a main pump, 11 a pump rear pipe, 12 a pump rear pipe, b-valve super unit, 13 a pre-combustion chamber, 14 a turbine, 15 a gas guide pipe, 16 a rectifying grid and 17 a thrust chamber.

Detailed Description

In this embodiment, the above steps are explained by taking a certain type of gas-liquid path coupled propulsion carrier rocket as an example:

step 1: establishing a second-order dynamic model of a liquid path part of a propulsion system: the layout of each unit of the liquid path system of a certain rocket only considering an oxygen path is shown in figure 1: the system comprises a first section of corrugated pipe, a second section of straight pipe, a third section of corrugated pipe, a fourth section of straight pipe, a three-way-accumulator super unit, a swinging hose corrugated pipe, a pre-pressing pump, an inter-pump pipe, a main pump, a post-pump pipe and a post-pump pipe b-valve super unit which are arranged in sequence;

the liquid path part model is

Wherein the second straight tube section is divided into 3 units, so M_L，R_L，K_LIs a 15 × 15 matrix, U_0L，U_1L，U_2LIs a 15 × 4 matrix, u_L＝[u_L1u_L2… u_L15]^TDimension number is 15 × 1. q. q.s_s＝[q_s1q_s2q_s3q_s4]^TDimension 4 × 1.

Step 2: and establishing a second-order dynamic model of each unit and boundary conditions of the gas path part. The layout of each unit of the gas circuit system is shown in fig. 2, and a second-order kinetic equation of each unit of the gas circuit system is written as follows:

(1) the second order kinetic equation of the precombustion chamber is

Wherein

(2) The second order kinetic equation of the turbine is

Wherein

(3) The second order kinetic equation of the gas conduit is

Wherein

(4) The second order kinetic equation of the rectifier grid is

Wherein

(5) The second order kinetic equation of the thrust chamber is

Wherein

The force of the thrust chamber on the structure is described as follows:

wherein Q_p′＝[-A_thC_f]。

(6) The second order kinetic equation of the boundary condition is

And step 3: and assembling the units of the gas circuit and the second order kinetic equation of the boundary condition according to the selection sequence of the state variables.

The assembled gas path part model is as follows:

wherein q'_GState variables of the gas circuit:

q′_G＝[P_G0q_G0p_G2q_G2T_G2K_G2p_G3q_G3T_G3K_G3

p_G4q_G4T_G4K_G4p_G5q_G5T_G5K_G5p_cq_mc]^T

and 4, step 4: in order to enable the assembly of the liquid circuit and the gas circuit parts, the same state variables are required for the liquid circuit and the gas circuit.

Solving the first derivative on both sides of the equation of the liquid path part to make the state variable of the equation of the liquid path part consistent with that of the gas path part, i.e.

Whereinq_s＝[q_s1q_s2q_s3q_s4]^T。

And 5: and adjusting a gas path second-order kinetic equation of the propulsion system into a form suitable for being connected with the liquid path part. To M'_G，R′_G，K′_GThe two columns 1 and 2 of the matrix are interchanged.

The gas circuit part model at this moment is:

M_Gq_G+R_Gq_G+K_Gq_G＝0

wherein

q_G＝[q_G0P_G0p_G2q_G2T_G2K_G2p_G3q_G3T_G3K_G3

p_G4q_G4T_G4K_G4p_G5q_G5T_G5K_G5p_cq_mc]^T

Step 6: assembling the coefficient matrix of the second order kinetic equation of the liquid path and the coefficient matrix of the second order kinetic equation of the gas path according to the selection sequence of the state variables, and assembling the coefficient matrix of the second order kinetic equation of the gas path and the coefficient matrix of the second order kinetic equation of the gas path in the R sequence_GAnd filling 1 in a place corresponding to the pressure at the tail end of the matrix liquid path, and deriving a second-order kinetic equation of the propulsion system containing the gas path characteristics.

The liquid path and the gas path are assembled as follows:

wherein

Variable of stateThe dimension is 34 × 1. M_L，R_L，K_LIs a 15 × 15 matrix, M_G，R_G，K_GIs a 19 × 20 matrix, M_p，R_p，K_pIs a 34 × 34 matrix, U₀， U₁，U₂Is a 34 × 4 matrix, q_L＝[q_L1,q_L2,q_L3…,q_G0]^TIs a state variable of mass flow of the fluid path, q_G0＝q_L15；q_G0，P_G0，q″_G＝[P_G2,q_G2,T_G2,K_G2…P_G5,q_G5,T_G5,K_G5]^TIs the state variable of the gas circuit;

and 7: and solving first derivative on two sides of the structural vibration equation, and adjusting the structural vibration equation into a form suitable for being assembled with the propulsion system.

Equation of vibration of structure

Two ends are derived to obtain

In this example, the first four modes of the structure are selected to couple with the propulsion system, so M_s，R_s， K_sV is a 4X 4 matrix, Q₀，Q₁，Q₂，Q_pA 4 x 34 matrix.

And 8: and coupling a second-order kinetic equation of the propulsion system with a structural vibration equation to derive a Pogo system model containing gas path characteristics.

Coupling a second order kinetic equation of the propulsion system with a structural vibration equation to obtain a Pogo system model can be written as:

wherein

A_psThe dimension is 80 x 80.

So far, the Pogo system modeling including the gas path characteristics based on the state equation method is completed by taking a certain type of gas-liquid path coupling propulsion carrier rocket as an example.