CN112613115B - Flexible spacecraft dynamics modeling method with friction boundary - Google Patents
Flexible spacecraft dynamics modeling method with friction boundary Download PDFInfo
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Abstract
The invention discloses a dynamic modeling method for a flexible spacecraft with a friction boundary, which comprises the following steps: constructing a flexible spacecraft energy functional with a friction boundary; simulating general boundary conditions of a flexible structure in the flexible spacecraft and constructing a displacement tolerance function of the flexible structure in the flexible spacecraft; solving a global analytic mode function of the flexible spacecraft according to the flexible spacecraft energy functional with the friction boundary and the displacement tolerance function; and obtaining a low-dimensional analysis rigid-flexible coupling dynamic model of the flexible spacecraft according to the modal function of the flexible spacecraft and analyzing the dynamic characteristics. The method solves the problem that the defect that the influence of boundary gap friction on the inherent characteristics of the spacecraft is not considered in the traditional method; the method has strong applicability, can be applied to general boundary conditions, and can be used for conveniently processing the spacecraft with the flexible body with a complex shape; a low-dimensional analytical dynamic model is obtained, and the design of a control law is facilitated.
Description
Technical Field
The invention relates to the technical field of flexible spacecraft dynamics modeling, in particular to a flexible spacecraft dynamics modeling method with a friction boundary, electronic equipment and a storage medium.
Background
With the development of aerospace technology, spacecraft tend to be large-sized, flexible and light. The large flexible spacecraft is limited by carrying envelope and unfolding modes, and the large flexible spacecraft needs to adopt an unfolding structural design mode. On one hand, the structure has inevitable gaps among the hinges after the unfolding and locking of the expandable structure due to factors such as processing, assembly, abrasion and the like; on the other hand, the spacecraft is subjected to alternating thermal loads when in orbit to enter and exit the earth shadow region, and enough clearance is reserved at the structural hinge joint during the design of the spacecraft structure in order to release the thermal stress generated inside the structure. The two factors cause that the gaps between the hinges are unavoidable after the spacecraft is unfolded and locked, and the friction effect caused by the gaps of the hinges can greatly influence the dynamic characteristics of the spacecraft, so that the attitude control precision of the spacecraft and the pointing precision of the effective load are influenced.
Currently, in the research on the dynamic modeling of the flexible spacecraft with the hinge gap, the research on the collision caused by the gap is more, and the research on the friction effect caused by the gap is less; in the existing spacecraft dynamics modeling considering clearance friction, friction is generally taken as a force constraint condition and applied to a dynamics equation as an external force, and the influence of the friction on the inherent characteristics of spacecraft dynamics is rarely researched; in the current spacecraft dynamics modeling research containing the clearance friction, most of the research is directed at specific boundary conditions and flexible structures with regular shapes, but no research is carried out on spacecrafts with general boundary conditions and flexible structures with complex irregular shapes; in the existing spacecraft dynamics modeling with clearance friction, a numerical calculation method is generally utilized, an analytic or semi-analytic method is rarely adopted, and the analytic or semi-analytic method can fully reveal the dynamics essence and rule of a system and is convenient for designing a control law; most of the current researches aim at the friction effect generated at the joint of the hinge with the gap, the spacecraft dynamics modeling technology with the friction boundary is rarely researched, and the friction effect at the boundary greatly increases the difficulty of the problem. In view of the above-mentioned aspects, current kinetic modeling techniques do not solve these problems well.
Disclosure of Invention
The invention aims to provide a dynamic modeling method, electronic equipment and a storage medium of a flexible spacecraft with a friction boundary, so as to solve the problem that the influence of boundary clearance friction on the inherent characteristics of the spacecraft is not considered in the traditional method.
In order to solve the above problems, the present invention is realized by the following technical scheme:
a method of modeling the dynamics of a flexible spacecraft having a friction boundary, comprising: s1, constructing a flexible spacecraft energy functional with a friction boundary. S2, simulating general boundary conditions of the flexible structure in the flexible spacecraft, and constructing a displacement tolerance function of the flexible structure in the flexible spacecraft. And S3, solving a global analytic mode function of the flexible spacecraft with the friction boundary according to the flexible spacecraft energy functional with the friction boundary and the displacement tolerance function. And S4, obtaining a low-dimensional analysis rigid-flexible coupling dynamic model of the flexible spacecraft according to the modal function of the flexible spacecraft and analyzing the dynamic characteristics.
Preferably, the step S1 includes: the total kinetic energy T of the flexible spacecraft in the flexible spacecraft energy functional with the friction boundary is as follows:
T＝T 1 +T 2
in the formula,
is the kinetic energy of the flexible structure in the flexible spacecraft;
wherein m is g Is the mass of the central rigid body, x r 、y r And z r Is the position coordinate of the point o in the inertial coordinate system;
representing the angular velocity vector, J, of the flexible spacecraft in an inertial coordinate system 1 ＝diag(J x ,J y ,J z ) Representing the moment of inertia of the central rigid body; a is the width of the plate when the flexible structure is a flexible plate structure, and h is the plate thickness; f (x) represents a smooth curve function in the form of a parabola, describing the boundary curve of a continuous plate, i.e. f (x) = c 1 x 2 +c 2 Wherein c is 1 And c 2 Is a curve parameter;
represents another boundary curve of the panel,
x, y and z represent the coordinate directions of the length, width and height of the flexible plate structure respectively, u represents the displacement in the x direction in the body-following coordinate system, v represents the displacement in the y direction in the body-following coordinate system, and w represents the displacement in the z direction in the body-following coordinate system; ρ represents the density of the flexible sheet structure and t represents the movement time;
potential energy V generated by bending deformation of flexible structure of flexible spacecraft in flexible spacecraft energy functional with friction boundary p Comprises the following steps:
wherein D = Eh 3 /12(1-υ 2 ) Is the bending stiffness; e and upsilon are respectively Young modulus and Poisson ratio;
potential energy V generated by surface strain in flexible structure of flexible spacecraft in flexible spacecraft energy functional with friction boundary ε Comprises the following steps:
wherein F Tx And F Ty Representing mid-plane internal forces, F, in the x and y directions of the sheet, respectively Txy Indicating the plane offset force or longitudinal shearing force of the thin plate; epsilon x And ε y Representing positive strain, gamma, in the x and y directions of the sheet, respectively xy Indicating the shear strain of the middle plane of the sheet;
the flexible spacecraft energy functional with the friction boundary also comprises work V performed by the friction force in the longitudinal plane caused by the sliding of the hinge on the friction boundary of the flexible structure f Comprises the following steps:
wherein, P x Representing the friction force on the friction boundary of the thin plate;
Preferably, in the step S1, an integral domain discrete method is adopted to calculate a double integral in the flexible spacecraft energy functional with the friction boundary.
Preferably, the step S2 includes: and respectively and uniformly arranging transverse displacement springs and rotation restraint springs on four transverse boundaries of the flexible plate structure, and simulating any transverse elastic boundary condition by setting different spring stiffness values.
Normal direction and tangential direction constraint linear springs are respectively and uniformly arranged on four in-plane boundaries of the flexible plate structure, and the simulation of the elastic boundary conditions in any plane is realized by setting different spring stiffness values.
Preferably, the displacement tolerance function includes:
describing the lateral displacement w (x, y, t) of the flexible plate structure by using a two-dimensional improved Fourier series expansion:
in the formula, λ m ＝mπ/a，λ n N = n pi/b, m, n, and s represent the number of expansion series terms; i represents an imaginary unit; a and b represent the length and width of the rectangular thin plate, respectively; a. The mn Is a Fourier coefficient, c sm And d sn Coefficient of the auxiliary series, omega, respectively 1 Is made inherentA circular frequency; xi shape sa (x) And xi sb (y) auxiliary functions associated with x and y, respectively;
describing the longitudinal displacement u (x, y, t) of the flexible structure by adopting two-dimensional improved Fourier series expansion:
wherein,
B mn denotes the Fourier coefficient, ω 2 Is the natural circular frequency, λ am ＝mπ/a，λ bn ＝nπ/b；
In the flexible spacecraft, the central rigid body displacement q coupled with the vibration of the flexible structure 0 And a rotation angle theta 0 Respectively, as follows:
where w denotes the circle frequency, t denotes the movement time, X 0 、Y 0 、Z 0 、
And
are all unknown coefficients.
Preferably, the step S3 includes: the total potential energy of the flexible spacecraft with the friction boundary also comprises elastic potential energy V stored by the boundary spring k The method comprises two parts: transverse boundary spring stored elastic potential energy V kw Elastic potential energy V stored by spring in boundary between surfaces ku Respectively as follows:
the lagrangian function L of the flexible spacecraft can be obtained as:
L＝V p +V ε +V k +V f -T
according to the Rayleigh-Ritz method, the unknown Fourier expansion coefficient is subjected to an extreme value calculation to obtain the following result:
the dynamic characteristic equation of the flexible spacecraft with the friction boundary can be obtained as follows:
in the formula,
in order to be a matrix of the stiffness of the system,
for the quality matrix, ω is the natural circular frequency and X is the column vector containing all the unknown coefficients, of the form:
wherein M and N are the truncation numbers of M and N in the series expansion of w respectively,
and
the truncation numbers of m and n in the series expansion of u are respectively; a. The 00 ,A 01 ,…A MN ,c 10 ,c 12 ,…,c 4M ,d 10 ,d 11 ,…d 4M Respectively the parameters in the Fourier series expansion of the lateral displacement w,
parameters in a Fourier series expansion for the longitudinal displacement u;
for the j-th order system natural frequency w j (j＝1,…,N t ),N t For the intercepted modal number, the natural frequency w of the flexible spacecraft with the friction boundary can be obtained by solving a characteristic equation j Corresponding feature vectors; then, a j order modal function expression of the transverse displacement and the longitudinal displacement of the flexible structure can be respectively obtained through a displacement tolerance function formula;
the lateral displacement mode function expression:
the longitudinal displacement modal function expression:
the j-th order rigid-flexible coupling mode of the flexible spacecraft is as follows:
preferably, the step S4 includes: and forming a rigid-flexible coupling mode matrix phi by using the rigid-flexible coupling mode function, wherein the displacement of the flexible spacecraft with the friction boundary is as follows:
[x r ,y r ,z r ,θ x ,θ y ,θ z ,w,u] T ＝[x rr ,y rr ,z rr ,θ xr ,θ yr ,θ zr ,0,0] T +Φq(t)
where Φ is the rigid-flexible coupling mode matrix, take the first N t Order mode, phi = [ phi ] 1 ,Φ 2 ,L,Φ j ,LΦ Nt ](ii) a q (t) is a generalized coordinate, q (t) = [ q (t) = 1 (t),q 2 (t),L,q Nt (t)] T ；x r 、y r And z r Representing the position coordinate, theta, of the center o of the satellite coordinate system in the inertial coordinate system x 、θ y And theta z Representing the attitude angle of the central rigid body; x is the number of rr 、y rr 、z rr 、θ xr 、θ yr And theta zr Representing the integral large-scale rigid motion of the spacecraft, and being irrelevant to the time t;
the discrete rigid-flexible coupling dynamic model of the flexible spacecraft with the friction boundary is obtained by using the Hamilton principle as follows:
wherein, M 1 、C 1 And K 1 Respectively obtaining a spacecraft integral mass matrix, a damping matrix and a rigidity matrix after the dispersion of a rigid-flexible coupling mode function; q represents a generalized coordinate;
when the system is excited by an external load, a rigid-flexible coupling dynamic model of the flexible spacecraft with the friction boundary is solved through a fourth-order Runge-Kutta method, the dynamic response of the system can be obtained, and the boundary friction effect is further analyzed.
In another aspect, the present invention also provides an electronic device comprising a processor and a memory, the memory having stored thereon a computer program which, when executed by the processor, implements the method as described above.
In yet another aspect, the present invention also provides a readable storage medium, in which a computer program is stored, which, when executed by a processor, implements a method as described above.
The invention has at least one of the following advantages:
the method overcomes the defect that the influence of boundary gap friction on the inherent characteristics of the spacecraft is not considered in the traditional method, and solves the global analytic mode function of the spacecraft with the friction boundary; the method has strong applicability, can be applied to general boundary conditions, and can be used for conveniently processing the spacecraft with a flexible body in a complex shape; the low-dimensional analytic dynamic model is obtained, the problem of order reduction of the traditional flexible spacecraft model is solved, and the design of a control law is facilitated. The method has more engineering practicability particularly for the dynamic modeling of the complex spacecraft with the friction boundary. When the flexible spacecraft system is excited by an external load, the dynamic equation is solved through a four-order Runge-Kutta method numerical value, the dynamic response of the system can be obtained, and then the influence rules of boundary friction effects (such as gap size, friction normal contact force, friction coefficient, assembly pretightening force and the like), cross section geometrical characteristics, longitudinal and transverse coupling effects and the like on the attitude motion and flexible vibration of the spacecraft are analyzed, so that technical support is provided for the overall structure design and control law design of the spacecraft.
Drawings
FIG. 1 is a schematic diagram of a simulated structure of a flexible spacecraft with a friction boundary according to an embodiment of the present invention;
FIG. 2 is a flow chart of a method for modeling the dynamics of a flexible spacecraft including a friction boundary according to an embodiment of the present invention;
fig. 3 is a schematic view illustrating a spacecraft flexible structure integral domain discretization provided in an embodiment of the present invention;
FIG. 4 is a schematic diagram of a spacecraft flexible structure model under any lateral boundary condition provided by an embodiment of the invention;
fig. 5 is a schematic diagram of a spacecraft flexible structure model under any longitudinal boundary condition according to an embodiment of the present invention.
Detailed Description
The flexible spacecraft dynamics modeling method with friction boundary, the electronic device and the storage medium provided by the invention are further described in detail in the following with reference to the attached drawings and the detailed description. The advantages and features of the present invention will become more apparent from the following description. It is to be noted that the drawings are in a very simplified form and are all used in a non-precise scale for the purpose of facilitating and distinctly aiding in the description of the embodiments of the present invention. To make the objects, features and advantages of the present invention comprehensible, reference is made to the accompanying drawings. It should be understood that the structures, ratios, sizes, and the like shown in the drawings and described in the specification are only used for matching with the disclosure of the specification, so as to be understood and read by those skilled in the art, and are not used to limit the implementation conditions of the present invention, so that the present invention has no technical significance, and any structural modification, ratio relationship change or size adjustment should still fall within the scope of the present invention without affecting the efficacy and the achievable purpose of the present invention.
As shown in fig. 1, a schematic diagram of a rigid-flexible coupled spacecraft with a friction boundary is given. The flexible spacecraft with the friction boundary comprises a central rigid body and a flexible body (flexible structure), wherein the flexible body is illustrated by taking a thin plate as an example, and transverse boundary conditions and longitudinal boundary conditions around the flexible body are simulated by uniformly arranged springs respectively. In order to avoid the thermal expansion deformation of the flexible body structure of the spacecraft, a gap delta is reserved in the x direction of the sliding hinge on the right end boundary of the spacecraft during design so as to release the thermal stress of the structure. The clearance delta is large enough that the hinge can only slip or stick without collision. F t Indicating the pre-tightening force, m, to which the hinge is subjected at the friction boundary t Representing the concentrated mass attached to the hinge and mu the coefficient of friction.
The length of the side of the central rigid body is 2r 0 The length of the flexible body (flexible structure) is a in the x direction and b in the y direction. f (x) represents a smooth curve function in the form of a parabola, describing the boundary curve equation of a continuous sheet, i.e., f (x) = c 1 x 2 +c 2 Wherein c is 1 And c 2 Are curve parameters.
Another boundary curve equation of the thin plate is shown,
are parameters. Defining O-XYZ as inertial coordinate system, O-XYZ as random coordinate system fixed on central rigid body, where x, y and z represent length, width and thickness directions of flexible body, u represents displacement in x direction, v represents displacement in y direction, and w represents z directionAnd (4) displacement in the direction of the arrow.
As shown in fig. 2, a flowchart of a method for modeling dynamics of a flexible spacecraft with a friction boundary according to the present embodiment is provided, including:
s1, constructing a flexible spacecraft energy functional with a friction boundary.
With continued reference to fig. 1, a position vector r of any point P on the flexible spacecraft in the random coordinate system is:
r＝(r 0 +x+u)i+vj+wk (1)
where i, j, and k are unit vectors along the x, y, and z axes of the satellite coordinate system, 2r 0 The length of the central rigid body.
The flexible spacecraft energy functional with a friction boundary comprises: the total kinetic energy T of the flexible spacecraft; potential energy V generated by bending deformation of flexible structure of flexible spacecraft p (ii) a Potential energy V generated by surface strain in flexible structure of flexible spacecraft ε (ii) a At the friction boundary of the flexible structure, the work V exerted by the friction force in the longitudinal plane caused by the sliding of the hinge f (ii) a And the elastic potential energy V stored by the subsequent boundary spring k 。
Wherein the total kinetic energy T of the flexible spacecraft is:
T＝T 1 +T 2 (2)
in the formula, T 1 Representing kinetic energy of flexible body, T 2 Representing the kinetic energy of the central rigid body.
In the formula, m g Is the mass of the central rigid body, x r 、y r And z r Is the position coordinate of the point o in the inertial coordinate system;
representing the angular velocity vector, J, of the spacecraft in an inertial coordinate system 1 ＝diag(J x ,J y ,J z ) Representing the moment of inertia of the central rigid body; when the flexible structure is a flexible plate structure, the width of the plate is shown, and the thickness of the plate is h; f (x) represents a smooth curve function in the form of a parabola to describe the boundary curve of the continuous sheet, i.e., f (x) = c 1 x 2 +c 2 Wherein c is 1 And c 2 Is a curve parameter;
another boundary curve of the thin plate is shown,
is a parameter; u represents the displacement in the x direction in the body coordinate system, v represents the displacement in the y direction in the body coordinate system, and w represents the displacement in the z direction in the body coordinate system; ρ represents the density of the flexible sheet and t represents the movement time.
Wherein, the potential energy V generated by the bending deformation of the flexible structure (thin plate) of the flexible spacecraft p Comprises the following steps:
wherein D = Eh 3 /12(1-υ 2 ) Is the bending stiffness; e and υ are young's modulus and poisson ratio, respectively.
Potential energy V generated by surface strain in flexible structure (thin plate) of flexible spacecraft ε Comprises the following steps:
in the formula, F Tx And F Ty Representing mid-plane internal forces, F, in the x and y directions of the sheet, respectively Txy Indicating the plane offset force or longitudinal shearing force of the thin plate; epsilon x And ε y Representing positive strain, gamma, in the x and y directions of the sheet, respectively xy Indicating the planar shear strain in the sheet.
Work V exerted by friction forces in the longitudinal plane caused by hinge slip at the flexible structure (sheet) friction boundary f Comprises the following steps:
in the formula, P x Indicating the frictional force experienced at the frictional boundary of the flexible structure (sheet).
Work done by friction V f In the expression (c), the friction force received at the friction boundary of the thin plate
Wherein F t Indicating the pre-tightening force, m, to which the hinge is subjected at the friction boundary t Representing the concentrated mass of the connection to the hinge, mu the coefficient of friction,
the in-plane velocity at the friction boundary is indicated,
representing the lateral acceleration at the friction boundary. From P x The expression (F) shows the pre-tightening force t Mass m of concentration t The hinge gap value delta and the friction coefficient mu all have an influence on the inherent properties of the flexible structure.
In the process of deducing the energy functional, f (x) = c 1 x 2 +c 2 A boundary curve equation in the form of a parabola. Due to the complexity of the energy functional integrand, the analytical curve integration is difficult to realize for the curve boundary, and the calculation efficiency is extremely low, so that the double integration in the flexible spacecraft energy functional with the friction boundary is calculated by adopting an integration domain discrete method, so that the calculation efficiency is greatly improved. Below with V p For example, let
Then the
Equally dividing the flexible body integral domain into G parts along the x direction under a random coordinate system, then
In the formula x i The abscissa representing the left end point of the ith division of the integral domain along the x-direction, as shown in fig. 3; the integral domain is also equally divided into G in the y-direction 1 Portion is then
y ij Denotes x = x i The ordinate of the lower end point equally divided in the y direction j. Take the integral point as
Wherein
As shown in fig. 3. Then the
The work done by the friction force on the friction boundary of the flexible body is brought into the total energy functional of the spacecraft, the influence of the boundary friction effect on the spacecraft dynamics can be fully reflected, and the problem of the dynamics inherent characteristics of the spacecraft with the friction boundary can be solved by an approximate analysis method. In addition, the double integral in the energy functional is calculated by an integral domain discrete method, so that the calculation amount can be greatly reduced, the calculation efficiency is improved, and the method can be popularized to a more complex curve shape boundary equation.
S2, simulating general boundary conditions of the flexible structure in the flexible spacecraft, and constructing a displacement tolerance function of the flexible structure in the flexible spacecraft.
For a spacecraft in which the flexible body has a friction boundary at the free end, there is a coupling phenomenon of the lateral bending vibration of the flexible body (thin plate) and the in-plane vibration due to the effect of the in-plane friction force. In-plane friction can have an effect on lateral bending vibrations. In the embodiment, the transverse displacement springs and the rotation restraining springs are respectively and uniformly arranged on the four transverse boundaries of the flexible plate structure, and the simulation of any transverse elastic boundary condition is realized by setting different spring stiffness values, as shown in fig. 4, wherein k is tx0 、k txa 、k ty0 、k ty1 Respectively, x =0, x = a, y = f (x), y = f 1 (x) Stiffness value of edgewise transverse displacement spring, K tx0 、K txa 、K ty0 、K ty1 Respectively, x =0, x = a, y = f (x), y = f 1 (x) Stiffness values of edgewise rotation constraints. When k is tx0 、k txa 、k ty0 、k ty1 、K tx0 、K txa 、K ty0 、K ty1 When the values are all infinite, representing the boundary condition of the fixed support of the four transverse edges of the thin plate, and when the values are all 0, representing the free boundary condition of the four transverse edges of the thin plate; when k is tx0 、k txa 、k ty0 、k ty1 Take infinity, K tx0 、K txa 、K ty0 、K ty1 When 0 is taken, the boundary condition of the four sides of the transverse direction of the thin plate is shown.
Normal and tangential constraint linear springs are uniformly arranged on four in-plane boundaries of the thin plate structure respectively, and as shown in FIG. 5, simulation of an arbitrary in-plane elastic boundary condition is realized by setting different spring stiffness values.
Wherein k is px0 、k pxa 、k py0 、k py1 Respectively, x =0, x = a, y = f (x), y = f 1 (x) Stiffness value of edge normal linear displacement spring, K px0 、K pxa 、K py0 、K py1 Respectively, x =0, x = a, y = f (x), y = f 1 (x) Stiffness values of the edgewise tangential linear displacement springs. When k is px0 、k pxa 、k py0 、k py1 、K px0 、K pxa 、K py0 、K py1 When the four edges in the thin plate surface are all 0, the four-edge free boundary condition in the thin plate surface is represented; when k is px0 、k pxa 、k py0 、k py1 Take infinity, K px0 、K pxa 、K py0 、K py1 Take 0 or k px0 、k pxa 、k py0 、k py1 Take 0,K px0 、K pxa 、K py0 、K py1 When the infinite is taken, the boundary condition of the four sides in the thin plate surface is expressed.
For flexible spacecraft, the flexible body needs to satisfy the following boundary conditions: the left end is fixedly supported and the right end is free in the direction parallel to the x direction; the y-direction of the flexible body is free-free. Thus, k tx0 、K tx0 、k px0 And K px1 Infinity is taken, and the stiffness of the remaining border springs is taken as 0.
The displacement tolerance function includes: the thin plate adopts two-dimensional improved Fourier series expansion to describe the transverse displacement w of the plate, and adopts two-dimensional improved Fourier series expansion to describe the longitudinal displacement and the rigid displacement of the attitude motion of the u spacecraft.
Wherein the two-dimensional improved Fourier series expansion is adopted to describe the transverse displacement w (x, y, t) of the flexible structure as follows:
in the formula, λ m ＝mπ/a，λ n N = n pi/b, m, n, and s represent the number of expansion series terms; i represents an imaginary unit; a and b represent the length and width of the rectangular thin plate, respectively; a. The mn Is a Fourier coefficient, c sm And d sn Coefficient of the auxiliary series, omega, respectively 1 Is the natural circular frequency. Xi sa (x) And xi sb (y) auxiliary functions relating to x and y, respectively, and xi sa (x) Expressed as:
auxiliary function xi sb (y) is expressed as:
by introducing the auxiliary series, the problem that the derivative of the vibration displacement is discontinuous at the boundary is solved. The displacement function can simultaneously satisfy the displacement boundary condition and the force boundary condition, can be suitable for any elastic boundary condition, and can improve the convergence of the series.
Wherein the longitudinal displacement u (x, y, t) described by the two-dimensional improved Fourier series expansion is expressed as follows:
wherein,
B mn denotes the Fourier coefficient, ω 2 Is the natural circular frequency, λ am ＝mπ/a，λ bn And n pi/b, wherein m, n, i, t, a, b, x and y are defined as above.
Wherein, for the flexible spacecraft with friction boundary, the central rigid body displacement q mutually coupled with the vibration of the flexible structure 0 And a rotation angle theta 0 Respectively, as follows:
wherein w represents the circular frequency, t represents the motion time, X 0 、Y 0 、Z 0 、
And
are all unknown coefficients.
Therefore, the method for simulating the boundary conditions in the transverse direction and the plane of the spacecraft flexible body structure by adopting the uniform springs can realize the simulation of any elastic boundary conditions by setting different spring stiffness values, and has great advantages for processing the flexible spacecraft which has complex boundary conditions and needs to solve the approximate analytic mode.
And S3, solving a global analytic mode function of the flexible spacecraft with the friction boundary according to the flexible spacecraft energy functional with the friction boundary and the displacement tolerance function.
And according to a Rayleigh-Ritz method, a dynamic characteristic equation of the flexible spacecraft is deduced by using a function extreme value principle.
Specifically, after the boundary condition of the flexible body is simulated by adopting the uniform spring, the total potential energy of the flexible spacecraft with the friction boundary also comprises elastic potential energy V stored by the boundary spring k . Elastic potential energy V k The method comprises two parts: transverse boundary spring stored elastic potential energy V kw Elastic potential energy V stored by spring in boundary of surface ku Respectively is as follows:
V k ＝V kw +V ku (17)
substituting the expressions of transverse displacement, longitudinal displacement and rigid body displacement in the step S2 into an energy functional expression of the spacecraft to obtain a Lagrangian function L of the flexible spacecraft of
L＝V p +V ε +V k +V f -T (18)
According to the Rayleigh-Ritz method, the unknown Fourier expansion coefficient is subjected to extreme value calculation to obtain the following result:
the characteristic equation of the spacecraft with the friction boundary can be obtained as follows:
wherein,
in order to be a matrix of the stiffness of the system,
for the quality matrix, ω is the natural circular frequency and X is the column vector containing all the unknown coefficients, of the form:
where M and N are the truncated numbers of M and N, respectively, in a series expansion of w,
and
the truncations of m and n in the series expansion of u, respectively. A. The 00 ,A 01 ,…A MN ,c 10 ,c 12 ,…,c 4M ,d 10 ,d 11 ,…d 4M Respectively the parameters in the Fourier series expansion of the lateral displacement w,
parameters in the Fourier series expansion for the longitudinal displacement u.
From this, it can be seen that the natural frequency w of the j-th order system j (j＝1,…,N t ),N t For the intercepted modal number, the natural frequency w of the flexible spacecraft with the friction boundary can be obtained by solving the characteristic equation (formula 20) j The corresponding feature vector is shown in equation (21). Then, the j-th order modal function expressions of the transverse displacement and the longitudinal displacement of the flexible structure can be obtained through displacement tolerance function expressions (10) and (13) respectively:
transverse displacement modal function expression
The following were used:
longitudinal displacement modal function expression
The following:
the j-th order rigid-flexible coupling mode phi of the flexible spacecraft j Comprises the following steps:
the double integral in the energy functional is calculated by using an integral domain discrete method, so that the calculated amount can be greatly reduced, the calculation efficiency is improved, and the method can be popularized to a more complex curve shape boundary equation. A Rayleigh-Ritz method is utilized to obtain a characteristic equation of the spacecraft with the friction boundary, so that an analytic mode function of the spacecraft with the friction boundary is obtained.
And S4, obtaining a low-dimensional analysis rigid-flexible coupling dynamic model (discrete dynamic model) of the flexible spacecraft according to the mode function of the flexible spacecraft and analyzing the dynamic characteristics.
By using the analytic rigid-flexible coupling mode function obtained in step S3, the spacecraft displacement can be expressed as
[x r ,y r ,z r ,θ x ,θ y ,θ z ,w,u] T ＝[x rr ,y rr ,z rr ,θ xr ,θ yr ,θ zr ,0,0] T +Φq(t) (24)
Where Φ is the rigid-flexible coupling mode matrix, take the first N t Order mode, phi = [ phi ] 1 ,Φ 2 ,L,Φ j ,LΦ Nt ](ii) a q (t) is a generalized coordinate, q (t) = [ q (t) = 1 (t),q 2 (t),L,q Nt (t)] T ；x r 、y r And z r Representing the position coordinate, theta, of the center o of the satellite coordinate system in the inertial coordinate system x 、θ y And theta z Representing the attitude angle of the central rigid body; x is the number of rr 、y rr 、z rr 、θ xr 、θ yr And theta zr The large-scale rigid motion of the whole spacecraft is represented and is irrelevant to time t; w represents the lateral displacement of the flexible body, and u represents the longitudinal displacement of the flexible body.
Substituting the formula into an expression of an energy functional of the spacecraft, and obtaining a discrete dynamic model of the flexible spacecraft by using a Hamilton principle:
in the formula, M 1 、C 1 And K 1 And respectively representing generalized coordinates by using a spacecraft integral mass matrix, a damping matrix and a rigidity matrix q after modal dispersion.
The above equation (25) is a rigid-flexible coupling kinetic equation in a low-dimensional analytic form. When the flexible spacecraft system is excited by an external load, the dynamic equation (formula 23) is solved through a fourth-order Runge-Kutta method numerical value, the dynamic response of the flexible spacecraft system can be obtained, and then the influence rules of boundary friction effects (such as gap size, friction normal contact force, friction coefficient, assembling pretightening force and the like) and cross section geometric characteristics, longitudinal and transverse coupling effects and the like on spacecraft attitude motion and flexible vibration are analyzed, so that technical support is provided for the overall structure design and control law design of the spacecraft.
Therefore, the rigid-flexible coupling kinetic equation in the low-dimensional analytic form is established by utilizing the Hamilton principle, the problem of order reduction of the traditional flexible spacecraft model is solved, and the design of the control law is facilitated. The method has more engineering practicability particularly for dynamic modeling of the complex spacecraft with the friction boundary.
In another aspect, the present embodiment also provides an electronic device, which includes a processor and a memory, where the memory stores a computer program, and the computer program, when executed by the processor, implements the method as described above.
In yet another aspect, the present embodiment further provides a readable storage medium, in which a computer program is stored, and the computer program, when executed by a processor, implements the method as described above.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrases "comprising a," "8230," "8230," or "comprising" does not exclude the presence of additional like elements in a process, method, article, or apparatus that comprises the element.
It should be noted that the apparatuses or methods disclosed in the embodiments herein may be implemented in other ways. The apparatus embodiments described above are merely illustrative, and for example, the flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of apparatus, methods and computer program products according to various embodiments herein. In this regard, each block in the flowchart or block diagrams may represent a module, a program, or a portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.
While the present invention has been described in detail with reference to the preferred embodiments, it should be understood that the above description should not be taken as limiting the invention. Various modifications and alterations to this invention will become apparent to those skilled in the art upon reading the foregoing description. Accordingly, the scope of the invention should be determined from the following claims.
Claims (7)
1. A method for modeling the dynamics of a flexible spacecraft with a friction boundary, comprising:
s1, constructing a flexible spacecraft energy functional with a friction boundary;
s2, simulating general boundary conditions of a flexible structure in the flexible spacecraft, and constructing a displacement tolerance function of the flexible structure in the flexible spacecraft;
s3, solving a global analytic mode function of the flexible spacecraft with the friction boundary according to the flexible spacecraft energy functional with the friction boundary and the displacement tolerance function;
s4, obtaining a low-dimensional analysis rigid-flexible coupling dynamic model of the flexible spacecraft according to the modal function of the flexible spacecraft and analyzing the dynamic characteristics;
the step S1 includes:
the total kinetic energy T of the flexible spacecraft in the flexible spacecraft energy functional with the friction boundary is as follows:
T＝T 1 +T 2
in the formula,
is the kinetic energy of the flexible structure in the flexible spacecraft;
wherein m is g Is the mass of the central rigid body, x r 、y r And z r Is the position coordinate of the point o in the inertial coordinate system;
representing the angular velocity vector, J, of the flexible spacecraft in an inertial coordinate system 1 ＝diag(J x ,J y ,J z ) Representing the moment of inertia of the central rigid body; a is the length of the plate when the flexible structure is a flexible plate structure, and h is the plate thickness; f (x) represents a smooth curve function in the form of a parabola to describe the boundary curve of a continuous plate, i.e. f (x) = c 1 x 2 +c 2 Wherein c is 1 And c 2 Is a curve parameter;
represents another boundary curve of the panel,
x, y and z represent the coordinate directions of the length, width and height of the flexible plate structure respectively, u represents the displacement in the x direction in the satellite coordinate system, namely longitudinal displacement, v represents the displacement in the y direction in the satellite coordinate system, and w represents the displacement in the z direction in the satellite coordinate system, namely transverse displacement; ρ represents the density of the flexible sheet structure and t represents the movement time;
the flexible junction of the flexible spacecraft in the flexible spacecraft energy functional with the friction boundaryPotential energy V generated by bending deformation of structure p Comprises the following steps:
wherein D = Eh 3 /12(1-υ 2 ) Is the bending stiffness; e and upsilon are respectively Young modulus and Poisson ratio;
potential energy V generated by surface strain in flexible structure of flexible spacecraft in flexible spacecraft energy functional with friction boundary ε Comprises the following steps:
wherein F Tx And F Ty Representing mid-plane internal forces, F, in the x and y directions of the sheet, respectively Txy Indicating the plane offset force or longitudinal shearing force of the middle surface of the thin plate; epsilon x And ε y Denotes the positive strain, ε, in the x and y directions of the sheet, respectively xy Indicating the shear strain of the middle plane of the sheet;
the flexible spacecraft energy functional with the friction boundary also comprises work V performed by the friction force in the longitudinal plane caused by the sliding of the hinge on the friction boundary of the flexible structure f Comprises the following steps:
wherein, P x Representing the friction force on the friction boundary of the thin plate;
2. The method for modeling the dynamics of a flexible spacecraft including a friction boundary as recited in claim 1, wherein said step S2 comprises:
respectively and uniformly arranging transverse displacement springs and rotation restraint springs on four transverse boundaries of the flexible plate structure, and simulating any transverse elastic boundary condition by setting different spring stiffness values;
normal and tangential constraint linear springs are uniformly arranged on four in-plane boundaries of the flexible plate structure respectively, and the simulation of the elastic boundary conditions in any plane is realized by setting different spring stiffness values.
3. A method of modeling the dynamics of a flexible spacecraft including a friction boundary as recited in claim 2, wherein said displacement allowance function in said step S2 comprises:
describing the lateral displacement w (x, y, t) of the flexible plate structure by using a two-dimensional improved Fourier series expansion:
in the formula of lambda m ＝mπ/a，λ n N = n pi/b, m, n, and s represent the number of expansion series terms; i represents an imaginary unit; a and b represent the length and width of the rectangular thin plate, respectively; a. The mn Is a Fourier coefficient, c sm And d sn Coefficient of the auxiliary series, omega, respectively 1 Is the natural circular frequency; xi sa (x) And xi sb (y) auxiliary functions associated with x and y, respectively;
describing the longitudinal displacement u (x, y, t) of the flexible structure by using a two-dimensional improved Fourier series expansion:
wherein,
B mn denotes the Fourier coefficient, ω 2 Is the natural circular frequency, λ am ＝mπ/a，λ bn ＝nπ/b；
In the flexible spacecraft, the central rigid body displacement q coupled with the vibration of the flexible structure 0 And a rotation angle theta 0 Respectively, as follows:
q 0 ＝[X 0 Y 0 Z 0 ] T sinωt,
where ω denotes the circular frequency, t denotes the movement time, X 0 、Y 0 、Z 0 、
And
are all unknown coefficients.
4. A method for modeling the dynamics of a flexible spacecraft including a friction boundary as recited in claim 3, wherein said step S3 comprises:
the total potential energy of the flexible spacecraft with the friction boundary also comprises elastic potential energy V stored by the boundary spring k The method comprises two parts: transverse boundary spring stored elastic potential energy V kw Elastic potential energy V stored by spring in boundary of surface ku Respectively is as follows:
V k ＝V kw +V ku
the lagrangian function L of the flexible spacecraft can be found as:
L＝V p +V ε +V k +V f -T
according to the Rayleigh-Ritz method, the unknown Fourier expansion coefficient is subjected to extreme value calculation to obtain the following result:
the dynamic characteristic equation of the flexible spacecraft with the friction boundary can be obtained as follows:
in the formula,
is a matrix of the stiffness of the system,
for the quality matrix, ω is the natural circular frequency and X is the column vector containing all the unknown coefficients, of the form:
wherein M and N are the numbers of truncations of M and N in the series expansion of w,
and
the truncation numbers of m and n in the series expansion of u are respectively; a. The 00 ,A 01 ,…A MN ,c 10 ,c 12 ,…,c 4M ,d 10 ,d 11 ,…d 4M Respectively the parameters in the Fourier series expansion of the lateral displacement w,
parameters in a Fourier series expansion for the longitudinal displacement u;
for the j-th order system natural frequency w j (j＝1,…,N t ),N t For the intercepted modal number, the natural frequency w of the flexible spacecraft with the friction boundary can be obtained by solving a characteristic equation j A corresponding feature vector; then, a j order modal function expression of the transverse displacement and the longitudinal displacement of the flexible structure can be respectively obtained through a displacement tolerance function formula;
the transverse displacement modal function expression:
the longitudinal displacement mode function expression is as follows:
the j-th order rigid-flexible coupling mode of the flexible spacecraft is as follows:
5. the method of modeling the dynamics of a flexible spacecraft including a friction boundary of claim 4, wherein said step S4 includes:
and forming a rigid-flexible coupling mode matrix phi by using the rigid-flexible coupling mode function, wherein the displacement of the flexible spacecraft with the friction boundary is as follows:
[x r ,y r ,z r ,θ x ,θ y ,θ z ,w,u] T ＝[x rr ,y rr ,z rr ,θ xr ,θ yr ,θ zr ,0,0] T +Φq(t)
where Φ is the rigid-flexible coupling mode matrix, take the first N t The mode of the order is that,
q (t) is a generalized coordinate which,
x r 、y r and z r Representing the position coordinate, theta, of the center o of the satellite coordinate system in the inertial coordinate system x 、θ y And theta z Representing the attitude angle of the central rigid body; x is the number of rr 、y rr 、z rr 、θ xr 、θ yr And theta zr Representing the integral large-scale rigid motion of the spacecraft, and being irrelevant to the time t;
the discrete rigid-flexible coupling dynamic model of the flexible spacecraft with the friction boundary is obtained by using the Hamilton principle as follows:
wherein M is 1 、C 1 And K 1 Respectively obtaining a spacecraft integral mass matrix, a damping matrix and a rigidity matrix after the dispersion of the rigid-flexible coupling mode function; q represents a generalized coordinate;
when the system is excited by external load, the rigid-flexible coupling dynamic model of the flexible spacecraft with the friction boundary is solved through a fourth-order Runge-Kutta method numerical value, the dynamic response of the system can be obtained, and the boundary friction effect is further analyzed.
6. An electronic device comprising a processor and a memory, the memory having stored thereon a computer program which, when executed by the processor, implements the method of any of claims 1 to 5.
7. A readable storage medium, in which a computer program is stored which, when being executed by a processor, carries out the method of any one of claims 1 to 5.
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