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 the large-scale sloshing force of liquid fuel in a spacecraft storage tank. In this method, the liquid fuel in the spacecraft storage tank is equivalent to a centroid point, the mass of the centroid point is equal to the mass of all the liquid fuel, and the centroid point can only move within the center-of-mass constraint plane. According to the relative motion relationship between the centroid point and the centroid surface, different methods are used to calculate the force between the centroid point and the centroid surface, and the sloshing force of the liquid fuel in the tank can be obtained. This method can calculate the force and moment of liquid fuel sloshing in multiple storage tanks in the spacecraft. At the same time, this method is not limited by the motion conditions of the spacecraft and the size of the liquid sloshing amplitude, and can solve the existing simple pendulum model and other methods. The calculation of liquid slosh and large nonlinear liquid slosh under microgravity conditions has large errors, which cannot meet the requirements of engineering use.

Description

An Approximate Method for Obtaining the Large Sloshing Force of Liquid Fuel in Spacecraft Tank

technical field

The invention belongs to the field of dynamics and control, and relates to an approximate method for calculating the large-scale sloshing force of liquid fuel in a spacecraft storage tank.

Background technique

With the development of aerospace technology, large-scale spacecraft represented by GEO satellites carry more and more liquid chemical propellants in order to complete their tasks. There are potential impacts on the safety and reliability of the stages of launch, orbit change, and on-orbit operation. For high-precision spacecraft, it may also affect the on-orbit performance of its payload. Therefore, the quantitative calculation of the force and moment of liquid fuel sloshing in the spacecraft tank has become an important link in the design and analysis of spacecraft.

At present, commercial software based on computational fluid dynamics (CFD) methods, such as FLUENT, FLOW-3D, etc., can also perform numerical simulations on liquid sloshing in spacecraft tanks, but CFD methods have disadvantages such as large amount of calculation and low efficiency. It cannot meet the requirements of large-scale simulation analysis required for the development of aerospace engineering models. For small linear sloshing problems, aerospace engineering generally uses the equivalent mechanical model of a single pendulum for solution analysis, but the calculation accuracy of the equivalent mechanical model of a single pendulum for large nonlinear liquid sloshing is very low, which cannot meet the needs of aerospace engineering models.

Contents of the invention

The technical problem solved by the invention is to overcome the deficiencies of the prior art, provide an approximate method for calculating the large-scale sloshing force of the liquid fuel in the spacecraft storage tank, and solve the simulation analysis problem of the large-scale sloshing of the liquid fuel in the spacecraft storage tank. Quantitatively calculate the force and moment of the liquid fuel sloshing of the spacecraft under various working conditions, analyze the influence of the liquid fuel slosh on the attitude motion of the spacecraft, and provide a basis for the design of the control subsystem.

The technical solution of the present invention is: an approximate method for obtaining the large-scale sloshing force of the liquid fuel in the spacecraft storage tank, the steps are as follows:

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

2) According to the geometric shape of the storage tank and the filling capacity of the liquid fuel in the storage tank, the motion trajectory of the center of mass of the liquid fuel when the storage tank rotates around its geometric center point is calculated by using the finite element method, and the movement constraint surface of the center of mass is obtained;

3) Define three movement modes of the center of mass relative to the movement constraint surface of the center of mass: ①Free movement—the center of mass does not touch the movement constraint surface of the center of mass, and is controlled by inertial force to move freely inside it; ②Contact movement—the center of mass is in contact with Center-of-mass-constrained surface contact, movement on the center-of-mass surface is dominated by contact force and inertial force; ③Collision motion——transitional motion mode when the center of mass point changes from free motion to contact motion;

4) When the centroid point moves on the centroid surface, according to Newton's second law and the coordinate transformation relationship of the relative motion of the rigid body, the motion equation of the centroid point is derived;

5) Based on the three motion modes established in step 3), the calculation model of the force between the centroid point and the constrained surface of the centroid is established, wherein the calculation formula of the tangential force between the centroid point and the constrained surface is F _τ = μ| V _τ |; μ ≥ 0, where μ is the coefficient of friction, V _τ is the relative motion velocity between the centroid point and the constraining surface of the centroid; the calculation of the normal force during collision motion adopts the linear spring-damper based on Hertz theory commonly used in engineering Model calculation; when contacting motion, the acceleration of the center of mass point along the normal direction of the surface should be 0, and the normal force calculation formula is derived according to this constraint;

6) First determine the motion state of the centroid point, then calculate the normal force and tangential force by different methods, and finally perform vector synthesis calculation on the tangential force and normal force to obtain the force between the centroid point and the centroid surface, That is, the liquid shakes vigorously.

The concrete method of step 5) is:

① During free movement, the normal force and tangential force are both 0;

② During collision motion, the calculation formula for the tangential force between the centroid point and the centroid constraint surface is: F _τ = μ|V _τ |; μ≥0, where μ is the friction coefficient, and V _τ is the centroid point relative to the centroid constraint surface The speed of motion; the calculation of normal force adopts the linear spring-damper model based on Hertz theory commonly used in engineering, and its calculation formula is:

Among them, k and c are the collision stiffness coefficient and collision damping coefficient respectively, δ is the embedding depth of the centroid point to the centroid plane, is the first derivative of the embedding depth.

③When contacting motion, the calculation formula of the tangential force between the centroid point and the constraining surface of the centroid is: F _τ = μ|V _τ |; μ≥0, where μ is the friction coefficient; the acceleration of the centroid point along the normal direction of the surface should be is 0, this constraint is expressed as:

Among them, N _i is the surface normal vector at the centroid point; Indicates its total derivative; i, j are positive integers, indicating the tank number, then the calculation formula for the normal force of a spacecraft with n tanks is:

Among them, n≥1; b _i =N _i (B _i ·n _i )+C _i , n _i and τ _i are the surface unit normal and tangent vectors of the point where m _i is located; Χ _i =ω×ρ _i ; Ω=ω×(ω×r _0,i ); m ₀ is the dry spacecraft, that is, the mass of the spacecraft without liquid fuel; I ₀ is the moment of inertia matrix of the dry spacecraft relative to its center of mass coordinate system; F _n,i and F _τ,i are the ith storage The normal and tangential components of the force of the liquid in the tank on the tank; F _B and M _B are the control force and control moment on the spacecraft, respectively; ω is the angular velocity of the dry spacecraft; r is the dry spacecraft relative to The vector radius of the origin of the inertial coordinate system; r _0,i is the vector radius of the point where the i-th centroid point m _i is located relative to the origin of the spacecraft centroid coordinate system, and there is: r _0,i = ρ _0,i + ρ _i , ρ _0,i is the radial vector of the geometric center of the centroid surface relative to the origin of the spacecraft’s centroid coordinate system; ρ _i is the radial vector of the point where _mi is located relative to the geometric center of the centroid surface.

Step 4) The specific method of establishing the equation of motion of the _centroid point mi is:

Among them, m ₀ is the mass of the dry spacecraft, that is, the spacecraft without liquid fuel; I ₀ is the moment of inertia matrix of the dry spacecraft relative to the barycentric coordinate system of the dry spacecraft; F _n,i and F _τ,i are respectively The force of the liquid in the i-th tank on the tank along the normal and tangential components; F _B and M _B are the control force and control torque on the spacecraft, respectively; ω is the angular velocity of the dry spacecraft; r is the dry The vector radius of the spacecraft relative to the origin of the inertial coordinate system; r _0,i is the vector radius of the point where the i-th centroid point m _i is located relative to the origin of the spacecraft’s centroid coordinate system, and: r _0,i = ρ _{0, i} +ρ _i , ρ _0,i is the radial vector of the geometric center of the centroid plane relative to the origin of the spacecraft's center of mass coordinate system; ρ _i is the radial vector of the point where _mi is located relative to the geometric center of the centroid plane.

The advantage of the present invention compared with prior art is:

The simulation method proposed in this patent is used in the dynamic analysis of liquid fuel sloshing in spacecraft storage tanks to solve the problem of large nonlinear liquid sloshing force and moment calculation in engineering. The existing simplified calculation models for force engineering of liquid sloshing are generally simple pendulum model and spring-mass model. These models are established based on the assumption of small linear sloshing of liquid in the tank, so they can only be applied to the solution of small linear sloshing. For large The solution accuracy of nonlinear sloshing and nonlinear sloshing problems under microgravity conditions is poor, which cannot meet the needs of spacecraft engineering. The method proposed in this patent can quantitatively calculate the liquid sloshing force and moment of a spacecraft with multiple storage tanks under different working conditions, and can guarantee calculation accuracy for both linear sloshing and nonlinear sloshing, and can be used for spacecraft control system design Provide input to verify the rationality of the control parameter settings.

Description of drawings

Figure 1 Modeling of spacecraft and storage tanks and description of their coordinate system;

Fig. 2 Logic diagram of the three movement modes of the centroid point.

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings.

The specific implementation methods of the approximate calculation method for the large-scale sloshing force of liquid fuel in the spacecraft tank are mainly as follows:

(1) Liquid fuel approximation method; the liquid fuel in the tank is equivalent to a mass point (called the centroid point), the mass of this mass point is equal to the total mass of the liquid fuel, and its motion range is limited to the center of mass motion constraint plane.

(2) Center of mass motion constraint surface; as shown in Figure 1, OXYZ is the inertial coordinate system; ② O _b X _b Y _b Z _b is the spacecraft center of mass coordinate system, and the origin O _b is located at the center of mass of the spacecraft; ③ o _t x _t y _t z _t is the tank coordinate system, and the origin O _t is located at the geometric center of the tank. According to the geometric shape of the tank and the filling amount of the liquid fuel in the tank, the finite element method is used to calculate the movement track of the center of mass of the liquid fuel when the tank rotates around its geometric center point, and the motion constraint surface of the center of mass is obtained; for the axes commonly used on spacecraft For symmetrical tanks, the center-of-mass motion constraint surface is generally an ellipsoid, which can be expressed by the following formula:

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

(3) Movement of the centroid point of liquid fuel; the movement range of the centroid point is limited within the center-of-mass constraint surface, but its movement direction is not limited, so there are three motion modes for the center-of-mass point relative to the movement constraint surface of the center-of-mass point: ①Free movement—the center-of-mass point does not move Contact with the center-of-mass motion constraint surface, and move freely within it dominated by inertial force; ②Contact motion——the center of mass point is in contact with the center-of-mass constraint surface, and moves under the control of both contact force and inertial force on the center-of-mass surface; ③Collision motion— —Transitional motion mode when the center of mass transitions from free motion to linked motion. See Figure 2 for the judgment of the three motion modes and the logical relationship between them. In Figure 2, d represents the shortest distance from the centroid point to the centroid motion constraining surface; R represents the force vector of the centroid point on the centroid motion constraining surface; n represents the surface unit normal vector at the centroid point.

(4) Equation of motion of the center of mass; when the center of mass moves on the surface of the center of mass, according to Newton's second law and the coordinate transformation relationship of the relative motion of the rigid body, the equation of motion of the center of mass _mi can be derived as:

Among them, m ₀ is the mass of the dry spacecraft (without liquid fuel); I ₀ is the moment of inertia matrix of the dry spacecraft relative to its center of mass coordinate system; F _n,i and F _τ,i are the ith tank The normal and tangential components of the force of the inner liquid on the tank; F _B and M _B are the control force and control moment on the spacecraft, respectively; ω is the angular velocity of the dry spacecraft; r is the relative inertia of the dry spacecraft The vector radius of the origin of the coordinate system; r _0,i is the vector radius of the point where the i-th centroid point m _i is located relative to the origin of the spacecraft centroid coordinate system, and: r _0,i = ρ _0,i + ρ _i , ρ _0,i is the radial vector of the geometric center of the centroid plane relative to the origin of the spacecraft barycentric coordinate system; ρ _i is the radial vector of the point where _mi is located relative to the geometric center of the centroid plane.

(5) The calculation model of the force between the center of mass point and the center of mass constraint surface; based on the three relative motion modes established in step (3), the calculation model of the force between the center of mass point and the center of mass constraint surface is established:

① During free movement, the normal force and tangential force are both 0;

②The formula for calculating the tangential force between the center of mass point and the constraining surface of the center of mass during collision motion is:

F _τ = μ|V _τ |; μ≥0 (3)

Among them, μ is the friction coefficient, which is related to the properties of the liquid and the tank; the normal force calculation adopts the linear spring-damping model based on the Hertz theory commonly used in engineering, and its calculation formula is:

Among them, k and c are the collision stiffness coefficient and collision damping coefficient respectively, δ is the embedding depth of the centroid point to the centroid plane, is the first derivative of the embedding depth.

③ In connection motion, the formula for calculating the tangential force between the center of mass point and the constraining surface of the center of mass is the same as that in collision motion, which is formula (3);

The calculation method of the normal force during the contact motion is as follows. The acceleration of the centroid point along the normal direction of the surface should be 0. This constraint can be expressed as:

Among them, N _i is the surface normal vector at the centroid point, represents its total derivative. Based on the relevant theories of analytic geometry, combined with the spaceflight motion equation and the centroid point motion equation (2), the calculation formula for the normal force of the spacecraft with n storage tanks (n≥1) can be obtained as follows:

in, b _i =N _i (B _i ·n _i )+C _i , n _i and τ _i are the surface unit normal and tangent vectors of the point where m _i is located; Χ _i =ω×ρ _i ; Ω=ω×(ω×r _0,i );

(6) The tangential force is calculated by the formula (3). The calculation of the normal force needs to determine the motion state of the center of mass point first. If the center of mass point and the center of mass surface are in a collision motion state, the normal force is calculated by the formula (4). If the center of mass If the point and the centroid plane are in a state of joint motion, the normal force is calculated by formula (6), and finally the vector synthesis calculation is performed on the tangential force and the normal force to obtain the force between the centroid point and the centroid plane, that is, the liquid sloshing effect 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.