CN106709161A - Approximation method for obtaining large-amplitude sloshing acting force of liquid fuel in storage tanks of spacecraft - Google Patents

Abstract

The invention discloses an approximation method for obtaining large-amplitude sloshing acting force of liquid fuel in storage tanks of spacecraft. The method comprises the following steps of: carrying out equivalence on liquid fuel in a storage tank of a spacecraft to form a centroid point, wherein the quality of the centroid point is equal to the quality of all the liquid fuel and the centroid point only can move in a centroid point constraint surface; and calculating acting force between the centroid point and a centroid surface by adoption of different methods according to a relative movement relationship between the centroid point and the centroid surface, so as to obtain the sloshing acting force of the liquid fuel in the storage tank of the spacecraft. According to the method, the sloshing acting force and acting moment of liquid fuel in a plurality of storage tanks of spacecraft can be calculated; and the method is not limited by the movement condition of the spacecraft and the liquid sloshing amplitude size, so as to solve the problem that the existing methods such as a simple pendulum model and the like are big in error when being used for calculating the liquid sloshing and large-amplitude nonlinear liquid sloshing under low-gravity conditions and then the engineering requirements cannot be satisfied.

Description

Approximation method for obtaining large-amplitude shaking acting force of liquid fuel in spacecraft storage tank

Technical Field

The invention belongs to the field of dynamics and control, and relates to an approximation method for calculating a large-amplitude shaking acting force of liquid fuel in a spacecraft storage tank.

Background

With the development of aerospace technology, large-scale spacecrafts represented by GEO satellites are required to complete tasks, more and more liquid chemical propellants are carried, the shaking of liquid fuel in the storage tank can generate larger interference force, potential influence is provided on the safety and reliability of the spacecrafts in the stages of launching, orbital transfer, in-orbit operation and the like, and the load in-orbit working performance of the high-precision spacecrafts can be influenced. Therefore, the quantitative calculation of the shaking acting force and moment of the liquid fuel in the storage tank of the spacecraft becomes an important link for the design and analysis of the spacecraft.

At present, commercial software based on a Computational Fluid Dynamics (CFD) method, such as FLUENT, FLOW-3D and the like, can also carry out numerical simulation on liquid shaking in a storage tank of a spacecraft, but the CFD method has the defects of large calculation amount, low efficiency and the like, and cannot meet the requirement of large-scale simulation analysis required by aerospace engineering model development. For the problem of small-amplitude linear shaking, the aerospace engineering generally adopts a simple pendulum equivalent mechanical model for solving and analyzing, but the calculation precision of the simple pendulum equivalent mechanical model for large-amplitude nonlinear liquid shaking is very low, and the requirement of the aerospace engineering model cannot be met.

Disclosure of Invention

The technical problem solved by the invention is as follows: the method overcomes the defects of the prior art, provides an approximation method for calculating the large-amplitude shaking acting force of the liquid fuel in the spacecraft storage tank, and solves the simulation analysis problem of large-amplitude shaking of the liquid fuel in the spacecraft storage tank. And quantitatively calculating the action force and moment of the liquid fuel shaking of the spacecraft under various working conditions, analyzing the influence of the liquid fuel shaking on the attitude motion of the spacecraft, and providing a basis for the design of a control subsystem.

The technical scheme of the invention is as follows: an approximation method for obtaining the large-amplitude shaking acting force of liquid fuel in a spacecraft storage tank comprises the following steps:

1) liquid fuel in a spacecraft storage tank is equivalent to a mass center point, the mass of the mass center point is equal to the mass of all the liquid fuel, and the mass center point can only move in a mass center movement constraint plane;

2) calculating the motion track of the mass center of the liquid fuel when the storage tank rotates around the geometric central point of the storage tank by adopting a finite element method according to the geometric shape of the storage tank and the filling amount of the liquid fuel in the storage tank to obtain a mass center motion constraint surface;

3) three motion modes are defined by the centroid point relative to the centroid motion constraint plane: free motion, namely a mass center point is not contacted with a mass center motion constraint surface and is controlled by inertia force to move freely in the mass center point; secondly, the movement is related, namely the mass center point is in contact with the mass center constraint surface and moves under the common control of contact force and inertia force on the mass center surface; collision motion-a transition motion mode when the centroid point is transformed from free motion to linked motion;

4) when the centroid point moves on the centroid plane, a motion equation of the centroid point is deduced according to Newton's second law and the coordinate conversion relation of the relative motion of the rigid body;

5) establishing a calculation model of the acting force between the centroid point and the centroid constraint surface based on the three motion modes established in the step 3), wherein the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface is F_τ＝μ|V_τL, |; mu is more than or equal to 0, wherein mu is friction coefficient, V_τThe relative motion speed between the centroid point and the centroid constraint surface; normal force calculation during collision movement adopts common engineeringBased on the Hertz theory, calculating a linear spring-damping model; when in connection with movement, the acceleration of the centroid point along the normal direction of the curved surface is 0, and a normal force calculation formula is derived according to the constraint condition;

6) the motion state of the centroid point is determined, the normal force and the tangential force are calculated by different methods, and finally the vector synthesis calculation is carried out on the tangential force and the normal force to obtain the acting force between the centroid point and the centroid surface, namely the liquid shaking acting force.

The specific method of the step 5) comprises the following steps:

when the device moves freely, the normal force and the tangential force are both 0;

② when in collision movement, the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface is F_τ＝μ|V_τL, |; mu is more than or equal to 0, wherein mu is friction coefficient, V_τThe movement speed of the centroid point relative to the centroid constraint surface; the normal force calculation adopts a linear spring-damping model which is commonly used in engineering and is based on the Hertz theory, and the calculation formula is as follows:

wherein k and c are respectively a collision stiffness coefficient and a collision damping coefficient, which are the embedding depths of the barycentric points to the barycentric surface,the first derivative of the embedding depth.

③ when in contact with movement, the tangential acting force between the centroid point and the centroid constraint surface has the calculation formula of F_τ＝μ|V_τL, |; mu is more than or equal to 0, wherein mu is a friction coefficient; the acceleration of the centroid point along the normal direction of the curved surface should be 0, and this constraint is expressed as:

wherein N is_iIs a normal vector of the curved surface at the centroid point;represents the full derivative thereof; i. j is a positive integer and represents the number of the storage tank, and then the calculation formula of the normal force of the spacecraft with the n storage tanks is as follows:

wherein n is more than or equal to 1;b_i＝N_i(B_i·n_i)+C_i， n_iand τ_iAre respectively m_iThe surface unit normal and tangential vectors of the point;Χ_i＝ω×ρ_i；Ω＝ω×(ω×r_0,i)；m₀is the mass of a dry spacecraft, i.e. a spacecraft without liquid fuel; i is₀A rotational inertia matrix of the dry spacecraft relative to a centroid coordinate system of the dry spacecraft; f_n,iAnd F_τ,iThe components of the acting force of the liquid in the ith storage tank on the storage tank along the normal direction and the tangential direction respectively; f_BAnd M_BRespectively is control force and control moment to the spacecraft; omega is the angular velocity of the dry spacecraft; r is the vector of the dry spacecraft relative to the origin of the inertial coordinate system; r is_0,iIs the ith centroid point m_iPhase of the positionThe radius of the origin of the coordinate system of the mass center of the spacecraft is as follows: r is_0,i＝ρ_0,i+ρ_i，ρ_0,iThe radius of the geometric center of the mass center plane relative to the origin of the mass center coordinate system of the spacecraft is shown; rho_iIs m_iThe radial of the position point relative to the geometric center of the centroid plane.

Step 4) establishing a centroid point m_iThe specific method of the motion equation is as follows:

wherein m is₀Is the mass of a dry spacecraft, i.e. a spacecraft without liquid fuel; i is₀A rotational inertia matrix of the dry spacecraft relative to a dry spacecraft centroid coordinate system; f_n,iAnd F_τ,iThe components of the acting force of the liquid in the ith storage tank on the storage tank along the normal direction and the tangential direction respectively; f_BAnd M_BRespectively is control force and control moment to the spacecraft; omega is the angular velocity of the dry spacecraft; r is the vector of the dry spacecraft relative to the origin of the inertial coordinate system; r is_0,iIs the ith centroid point m_iThe vector of the position point relative to the origin of the spacecraft centroid coordinate system is as follows: r is_0,i＝ρ_0,i+ρ_i，ρ_0,iThe radius of the geometric center of the mass center plane relative to the origin of the mass center coordinate system of the spacecraft is shown; rho_iIs m_iThe radial of the position point relative to the geometric center of the centroid plane.

Compared with the prior art, the invention has the advantages that:

the simulation method provided by the patent is used for the analysis of the liquid fuel shaking dynamics of the storage tank of the spacecraft, and the calculation problem of large nonlinear liquid shaking force and moment in engineering is solved. The existing simplified calculation models for liquid shaking acting force engineering are generally simple pendulum models and spring-mass models, and the models are established on the basis of the assumption of small-amplitude linear shaking of liquid in a storage tank, so that the simplified calculation models can only be suitable for solving the small-amplitude linear shaking, have poor solving precision for the problems of large-amplitude nonlinear shaking and nonlinear shaking under the microgravity condition, and cannot meet the use requirements of spacecraft engineering. The method provided by the patent can quantitatively calculate the liquid shaking acting force and moment of the spacecraft with the plurality of storage tanks under different working conditions, can ensure the calculation precision for linear shaking and nonlinear shaking, can provide input for the design of a spacecraft control system, and verifies the reasonability of the setting of control parameters.

Drawings

FIG. 1 spacecraft and tank modeling and coordinate system description thereof;

FIG. 2 is a logical relationship diagram of three motion modes of a centroid point.

Detailed Description

The present invention is described in further detail below with reference to the attached drawing figures.

The specific implementation mode of the approximate calculation method for the large-amplitude shaking acting force of the liquid fuel in the spacecraft storage tank mainly comprises the following steps:

(1) liquid fuel approximation method; the liquid fuel in the tank is equivalent to a mass point (referred to as the centroid point) having a mass equal to the total mass of the liquid fuel and a range of motion defined within the centroid motion constraint.

(2) Centroid motion constraint surface, as shown in FIG. 1, OXYZ as inertial coordinate system, ② O_bX_bY_bZ_bIs a coordinate system of the center of mass of the spacecraft, an origin O_bAt the center of mass of the spacecraft ③ o_tx_ty_tz_tIs a coordinate system of the tank, origin O_tIs positioned at the geometric center of the storage box. Calculating the motion track of the mass center of the liquid fuel when the storage tank rotates around the geometric central point of the storage tank by adopting a finite element method according to the geometric shape of the storage tank and the filling amount of the liquid fuel in the storage tank to obtain a mass center motion constraint surface; for the common axial symmetric storage box on the spacecraft, the mass center transportThe dynamic constraint surface is generally an ellipsoid surface and can be expressed by the following formula:

in the above formula, S represents a centroid motion constraint surface, (x, y, z) is coordinate values of any point on the centroid motion constraint surface, and a and b represent a major semi-axis and a minor semi-axis of the ellipsoid respectively.

(3) Liquid fuel centroid movement; the motion range of the centroid point is limited in the centroid constraint plane, but the motion direction of the centroid point is not limited, so that the centroid point has three motion modes relative to the centroid motion constraint plane: free motion, namely a mass center point is not contacted with a mass center motion constraint surface and is controlled by inertia force to move freely in the mass center point; secondly, the movement is related, namely the mass center point is in contact with the mass center constraint surface and moves under the common control of contact force and inertia force on the mass center surface; collision motion-a transition motion pattern when the centroid point transitions from free motion to connected motion. The logical relationship between the judgment and the mutual conversion of the three motion modes is shown in figure 2. In fig. 2, d represents the shortest distance from the centroid point to the centroid motion constraint plane; r represents the acting force vector of the centroid point to the centroid motion constraint surface; n represents the surface unit normal vector at the centroid point.

(4) A centroid point equation of motion; when the centroid point moves on the centroid plane, the centroid point m can be deduced according to Newton's second law and the coordinate conversion relation of the relative motion of the rigid body_iThe equation of motion of (a) is:

wherein m is₀The mass of a dry spacecraft (without liquid fuel); i is₀A rotational inertia matrix of the dry spacecraft relative to a centroid coordinate system of the dry spacecraft; f_n,iAnd F_τ,iThe components of the acting force of the liquid in the ith storage tank on the storage tank along the normal direction and the tangential direction respectively;F_Band M_BRespectively is control force and control moment to the spacecraft; omega is the angular velocity of the dry spacecraft; r is the vector of the dry spacecraft relative to the origin of the inertial coordinate system; r is_0,iIs the ith centroid point m_iThe vector of the position point relative to the origin of the spacecraft centroid coordinate system is as follows: r is_0,i＝ρ_0,i+ρ_i，ρ_0,iThe radius of the geometric center of the mass center plane relative to the origin of the mass center coordinate system of the spacecraft is shown; rho_iIs m_iThe radial of the position point relative to the geometric center of the centroid plane.

(5) An acting force calculation model between the centroid point and the centroid constraint surface; establishing a calculation model of the acting force between the centroid point and the centroid constraint surface based on the three relative motion modes established in the step (3):

when the device moves freely, the normal force and the tangential force are both 0;

secondly, the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface during collision motion is as follows:

F_τ＝μ|V_τ|；μ≥0 (3)

where μ is the coefficient of friction, related to the properties of the liquid and the reservoir; the normal force calculation adopts a linear spring-damping model which is commonly used in engineering and is based on the Hertz theory, and the calculation formula is as follows:

wherein k and c are respectively a collision stiffness coefficient and a collision damping coefficient, which are the embedding depths of the barycentric points to the barycentric surface,the first derivative of the embedding depth.

During the contact movement, the tangential force calculation formula between the centroid point and the centroid constraint surface is the same as that during the collision movement, namely formula (3);

the normal force calculation method in connection with the motion is as follows, the acceleration of the centroid point along the normal direction of the curved surface should be 0, and the constraint condition can be expressed as follows:

wherein N is_iIs the normal vector of the curved surface at the centroid point,representing the full derivative thereof. By the correlation theory of analytic geometry, and by combining an aerospace motion equation and a centroid point motion equation (2), a normal force calculation formula of the spacecraft with n storage boxes (n is more than or equal to 1) can be obtained through derivation:

wherein,b_i＝N_i(B_i·n_i)+C_i， n_iand τ_iAre respectively m_iThe surface unit normal and tangential vectors of the point;Χ_i＝ω×ρ_i；Ω＝ω×(ω×r_0,i)；

(6) the tangential force is obtained by calculating according to a formula (3), the motion state of a centroid point is required to be determined by calculating the normal force, if the centroid point and the centroid surface are in a collision motion state, the normal force is calculated according to a formula (4), if the centroid point and the centroid surface are in a contact motion state, the normal force is calculated according to a formula (6), and finally the tangential force and the normal force are subjected to vector synthesis calculation to obtain the acting force between the centroid point and the centroid surface, namely the liquid shaking acting force.

Claims (3)

1. An approximation method for obtaining the large-amplitude shaking acting force of liquid fuel in a spacecraft storage tank is characterized by comprising the following steps:

1) liquid fuel in a spacecraft storage tank is equivalent to a mass center point, the mass of the mass center point is equal to the mass of all the liquid fuel, and the mass center point can only move in a mass center movement constraint plane;

2) calculating the motion track of the mass center of the liquid fuel when the storage tank rotates around the geometric central point of the storage tank by adopting a finite element method according to the geometric shape of the storage tank and the filling amount of the liquid fuel in the storage tank to obtain a mass center motion constraint surface;

3) three motion modes are defined by the centroid point relative to the centroid motion constraint plane: free motion, namely a mass center point is not contacted with a mass center motion constraint surface and is controlled by inertia force to move freely in the mass center point; secondly, the movement is related, namely the mass center point is in contact with the mass center constraint surface and moves under the common control of contact force and inertia force on the mass center surface; collision motion-a transition motion mode when the centroid point is transformed from free motion to linked motion;

4) when the centroid point moves on the centroid plane, a motion equation of the centroid point is deduced according to Newton's second law and the coordinate conversion relation of the relative motion of the rigid body;

5) establishing a calculation model of the acting force between the centroid point and the centroid constraint surface based on the three motion modes established in the step 3), wherein the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface is F_τ＝μ|V_τL, |; mu is more than or equal to 0, wherein mu is friction coefficient, V_τThe relative motion speed between the centroid point and the centroid constraint surface; calculating the normal force during collision motion by adopting a linear spring-damping model commonly used in engineering and based on the Hertz theory; when in connection with movement, the acceleration of the centroid point along the normal direction of the curved surface is 0, and a normal force calculation formula is derived according to the constraint condition;

6) the motion state of the centroid point is determined, the normal force and the tangential force are calculated by different methods, and finally the vector synthesis calculation is carried out on the tangential force and the normal force to obtain the acting force between the centroid point and the centroid surface, namely the liquid shaking acting force.

2. The approximate method of deriving a substantial sloshing force of a liquid fuel in a spacecraft tank of claim 1, wherein: the specific method of the step 5) comprises the following steps:

when the device moves freely, the normal force and the tangential force are both 0;

secondly, during collision movement, the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface is as follows:

F_τ＝μ|V_τl, |; mu is more than or equal to 0, wherein mu is friction coefficient, V_τThe movement speed of the centroid point relative to the centroid constraint surface; the normal force calculation adopts a linear spring-damping model which is commonly used in engineering and is based on the Hertz theory, and the calculation formula is as follows:

$F_n=k\mathrm{\δ}+c\stackrel{\·}{\mathrm{\δ}}$

wherein k and c are respectively a collision stiffness coefficient and a collision damping coefficient, which are the embedding depths of the barycentric points to the barycentric surface,the first derivative of the embedding depth.

And thirdly, when in contact with the movement, the calculation formula of the tangential acting force between the centroid point and the centroid constraint surface is as follows:

F_τ＝μ|V_τl, |; mu is more than or equal to 0, wherein mu is a friction coefficient; the acceleration of the centroid point along the normal to the surface should be 0,

this constraint is expressed as:

${\stackrel{\·\·}{\mathrm{\ρ}}}_i\·N_i+{\stackrel{\·}{\mathrm{\ρ}}}_i\·{\stackrel{\·}{N}}_i=0$

wherein N is_iIs a normal vector of the curved surface at the centroid point;represents the full derivative thereof; i. j is a positive integer and represents the number of the storage tank, and then the calculation formula of the normal force of the spacecraft with the n storage tanks is as follows:

wherein n is more than or equal to 1;b_i＝N_i(B_i·n_i)+C_i， n_iand τ_iAre respectively m_iThe surface unit normal and tangential vectors of the point;Χ_i＝ω×ρ_i；Ω＝ω×(ω×r_0,i)；m₀is the mass of a dry spacecraft, i.e. a spacecraft without liquid fuel; i is₀A rotational inertia matrix of the dry spacecraft relative to a centroid coordinate system of the dry spacecraft; f_n,iAnd F_τ,iThe components of the acting force of the liquid in the ith storage tank on the storage tank along the normal direction and the tangential direction respectively; f_BAnd M_BRespectively is control force and control moment to the spacecraft; omega is the angular velocity of the dry spacecraft; r is the vector of the dry spacecraft relative to the origin of the inertial coordinate system; r is_0,iIs the ith centroid point m_iThe vector of the position point relative to the origin of the spacecraft centroid coordinate system is as follows: r is_0,i＝ρ_0,i+ρ_i，ρ_0,iThe radius of the geometric center of the mass center plane relative to the origin of the mass center coordinate system of the spacecraft is shown; rho_iIs m_iThe radial of the position point relative to the geometric center of the centroid plane.

3. The approximate method of deriving a substantial sloshing force of a liquid fuel in a spacecraft tank of claim 1, wherein: step 4) establishing a centroid point m_iThe specific method of the motion equation is as follows:

$\begin{array}{c}{\stackrel{\·\·}{\mathrm{\ρ}}}_i=r_{0,i}\×I_0^{-1}\[M_B-\mathrm{\ω}\×\left(I_0\·\mathrm{\ω}\right)+\underset{j=1}{\overset{n}{\Σ}}{r}_{0,j}\×\left(F_{n,j}+F_{\mathrm{\τ},j}\right)\]-\left(F_{n,i}+F_{\mathrm{\τ},i}\right)/m_i\\ -\stackrel{\·\·}{r}-2\mathrm{\ω}\×{\stackrel{\·}{\mathrm{\ρ}}}_i-\mathrm{\ω}\×(\mathrm{\ω}\×r_{0,i})+F_{b,i}/m_i\end{array}$

wherein m is₀Is the mass of a dry spacecraft, i.e. a spacecraft without liquid fuel; i is₀A rotational inertia matrix of the dry spacecraft relative to a dry spacecraft centroid coordinate system; f_n,iAnd F_τ,iThe components of the acting force of the liquid in the ith storage tank on the storage tank along the normal direction and the tangential direction respectively; f_BAnd M_BRespectively is control force and control moment to the spacecraft; omega is the angular velocity of the dry spacecraft; r is the vector of the dry spacecraft relative to the origin of the inertial coordinate system; r is_0,iIs the ith centroid point m_iThe vector of the position point relative to the origin of the spacecraft centroid coordinate system is as follows: r is_0,i＝ρ_0,i+ρ_i，ρ_0,iThe radius of the geometric center of the mass center plane relative to the origin of the mass center coordinate system of the spacecraft is shown; rho_iIs m_iThe radial of the position point relative to the geometric center of the centroid plane.