Rail-controlled direct-gas composite control elastic motion modeling method for anti-aircraft missile

Abstract

The present application belongs to the field of elastic motion modeling of air defense missiles, and particularly relates to an elastic motion modeling method for air defense missiles with trajectory control and direct-gas combined control, aiming at solving the problem that the traditional modeling method is insufficient in covering trajectory control force, thus being difficult to realize accurate modeling of the elastic motion of air defense missiles with trajectory control and direct-gas combined control. The present application comprises: calculating the kinetic energy T of the missile system caused by the bending deformation of the elastic missile body, the elastic potential energy U of the missile system, and the damping energy D of the missile system caused by the bending deformation of the elastic missile body; calculating the generalized force Q acting on the missile body corresponding to the virtual work through the virtual work principle i ; based on the kinetic energy T of the missile system, the elastic potential energy U of the missile system, the damping energy D of the missile system, and the generalized force Q i , constructing the dynamic equation of the elastic missile body through the Lagrange equation. The present application corrects and improves the problem of insufficient coverage of trajectory control force in the traditional modeling method, and has high accuracy and precision in the elastic motion modeling of air defense missiles with trajectory control and direct-gas combined control.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of elastic motion modeling of air defense missiles, and particularly relates to an elastic motion modeling method for a trajectory control type direct-gas composite control air defense missile. BACKGROUND

[0002] With the rapid development of missile technology, the demand for modeling of missile systems is increasing. Under the conditions of given target motion law, missile flight speed variation and control method, modeling of missile motion can obtain the variation in the flight process of the missile, and analyze the influence of various parameters on the motion of the missile, which is of great significance to the design and test of the missile.

[0003] In the elastic motion modeling of air defense missiles, various forces acting on the missile body in the flight process need to be considered. For a direct-gas composite control air defense missile using a trajectory control engine, the trajectory control engine generates a lateral force when working. The traditional air rudder control missile elastic motion modeling does not consider this factor, and it is difficult to realize high-precision and high-accuracy elastic motion modeling of the trajectory control type direct-gas composite control air defense missile. SUMMARY

[0004] In order to solve the above problems in the prior art, i.e., the traditional modeling method has insufficient coverage of trajectory control force, thereby making it difficult to realize accurate modeling of the elastic motion of the trajectory control type direct-gas composite control air defense missile, the application provides an elastic motion modeling method for a trajectory control type direct-gas composite control air defense missile, which comprises the following steps:

[0005] calculating the kinetic energy T of the missile system caused by the bending deformation of the elastic missile body, the elastic potential energy U of the missile system, and the damping energy D of the missile system caused by the bending deformation of the elastic missile body;

[0006] calculating the generalized force Q acting on the missile body corresponding to the virtual work by the virtual work principle i ; the generalized force Q i includes the aerodynamic force of the missile body the aerodynamic force of the rudder surface the inertia torque generated by the deflection of the rudder surface and the lateral force generated by the trajectory control engine

[0007] Based on the kinetic energy T of the missile system, the elastic potential energy U of the missile system, the damping energy D of the missile system, and the generalized force Q i , the dynamic equation of the elastic missile body is constructed by the Lagrange equation.

[0008] In some preferred embodiments, the kinetic energy T of the missile system caused by the bending deformation of the elastic missile body is calculated by the following method:

[0009] Based on the momentum equation, the initial expression of kinetic energy of the missile system is constructed between the kinetic energy and the elastic deformation displacement of the missile micro-element and the mass of the micro-element.

[0010] The natural bending mode is taken as the degree of freedom of the continuous elastic mechanics system, and the generalized coordinate is taken as q i , and the elastic deformation displacement of the missile micro-element at any point is expanded.

[0011] Based on the expanded elastic deformation displacement of the missile micro-element and the initial expression of kinetic energy, the kinetic energy T of the missile system caused by the elastic bending deformation of the missile elastic body is obtained.

[0012] In some preferred embodiments, the expanded elastic deformation displacement of the missile micro-element is:

[0013]

[0014] Wherein, δ y (x, t) is the elastic deformation displacement δ y of the micro-element on the missile body, φ i (x) is the mode function satisfying the orthogonality condition, and n is the number of terms of the series expansion.

[0015] In some preferred embodiments, the kinetic energy T of the missile system caused by the elastic bending deformation of the missile elastic body is:

[0016]

[0017] Wherein, is the first derivative of the generalized coordinate q i , and M i is the generalized mass of the i-th mode.

[0018] In some preferred embodiments, the elastic potential energy U of the missile system is calculated by:

[0019] Based on the elastic potential energy equation, the initial expression of elastic potential energy of the missile system is constructed between the elastic potential energy and the elastic deformation displacement of the missile micro-element.

[0020] Based on the expanded elastic deformation displacement of the missile micro-element and the initial expression of elastic potential energy, the expanded elastic potential energy expression is obtained.

[0021] Combined with the one-dimensional beam elastic bending deformation differential equation and the boundary conditions of the initial expression of elastic potential energy, the elastic potential energy U of the missile system is obtained.

[0022] In some preferred embodiments, the expanded elastic potential energy expression is:

[0023]

[0024] where E is the elastic modulus, I is the area moment of inertia of the cross section about the center of the elastic body, and φ i (x) is the second derivative of φ i (x).

[0025] In some preferred embodiments, the boundary condition of the initial expression of the elastic potential energy is:

[0026]

[0027] In some preferred embodiments, the method for modeling the elastic motion of the trajectory-controlled direct-gas composite control anti-aircraft missile system is characterized in that the elastic potential energy U of the missile system is:

[0028]

[0029] In some preferred embodiments, the calculation method of the missile system damping energy D caused by the elastic bending deformation of the missile body is:

[0030] An initial expression of the damping energy of the missile system and the elastic deformation displacement of the microelement on the body is constructed based on the damping energy equation.

[0031] The damping coefficient of the body and the initial expression of the damping energy are combined to obtain the missile system damping energy D caused by the elastic bending deformation of the missile body.

[0032] In some preferred embodiments, the initial expression of the damping energy of the missile system and the elastic deformation displacement of the microelement on the body is:

[0033]

[0034] where c = 2ξ i ω i M i is the damping coefficient of the body, ξ i is the damping ratio of the i-th mode of the body.

[0035] In some preferred embodiments, the missile system damping energy D caused by the elastic bending deformation of the missile body is:

[0036]

[0037] In some preferred embodiments, the generalized force Q i acting on the missile body corresponding to the virtual work is:

[0038]

[0039] where f y(x, t) is the distributed force acting on the missile in the y-axis direction of the aerodynamic body-fixed coordinate system, δ y (x, t) is the distributed force acting on the missile in the y-axis direction of the aerodynamic body-fixed coordinate system, δ i (x) is a mode function satisfying the orthogonality condition.

[0040] In some preferred embodiments, the aerodynamic force of the missile body is:

[0041]

[0042] where V is the velocity in the elastic reference coordinate system of the missile body, θ is the pitch angle, Y α (x) is the aerodynamic force derivative varying with x.

[0043] In some preferred embodiments, the aerodynamic force of the rudder is:

[0044]

[0045] where δ z (t) is the rudder deflection angle required by the control signal, δ e (t) is the additional rudder deflection angle caused by the elastic deformation of the missile body, Y δ (x) is the rudder aerodynamic force derivative distributed along the x-axis of the elastic reference coordinate system of the missile body.

[0046] In some preferred embodiments, the inertia moment generated by the deflection of the rudder is:

[0047]

[0048] where J δ is the moment of inertia of a rudder relative to the rudder axis, δ is the rudder deflection angle, is the second derivative of δ with respect to time.

[0049] In some preferred embodiments, the lateral force generated by the trajectory control engine is:

[0050]

[0051] where P is the lateral force of the trajectory control engine, x p is the coordinate of the action point of the trajectory control engine on the x-axis of the elastic reference coordinate system of the missile body.

[0052] In some preferred embodiments, the elastic body dynamics equation of the missile is:

[0053]

[0054] where Dαi is the generalized aerodynamic force coefficient of the i-th mode of vibration, D ωi is the generalized aerodynamic damping force coefficient of the i-th mode of vibration, D δi is the generalized aerodynamic force coefficient of the i-th mode of vibration, D is the generalized aerodynamic force coefficient of the i-th mode of vibration, D Pi is the generalized aerodynamic force coefficient of the i-th mode of vibration, D qij is the generalized aerodynamic force coefficient of the i-th mode of vibration, D

[0055] Advantages of the present application:

[0056] The method for modeling elastic motion of the trajectory control and direct air composite control anti-air missile according to the characteristics of the trajectory control and direct air composite control anti-air missile, completely describes each force generated in flight of the trajectory control and direct air composite control anti-air missile, and comprehensively analyzes and calculates the generalized force affecting the elastic vibration, including the direct lateral force generated by the trajectory control engine, and on this basis, establishes the elastic vibration dynamics equation of the missile body, completely describes the elastic vibration of the missile body of the trajectory control and direct air composite control anti-air missile, and perfects the elastic vibration dynamics equation of the traditional air control anti-air missile, and is accurate and precise. BRIEF DESCRIPTION OF DRAWINGS

[0057] Other features, objects and advantages of the present application will become more apparent from the following detailed description of non-limiting embodiments, made with reference to the following drawings:

[0058] Figure 1 is the aerodynamic fixed coordinate system O of the method for modeling elastic motion of the trajectory control and direct air composite control anti-air missile of the present application a x a y a z a and the elastic reference coordinate system O of the missile body e x e y e z e definition diagram;

[0059] Figure 2 is the missile body elastic deformation schematic diagram of the method for modeling elastic motion of the trajectory control and direct air composite control anti-air missile of the present application. DETAILED DESCRIPTION

[0060] The present application will be further described in detail below with reference to the drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the related application, and are not a limitation on the application. In addition, it should be noted that, in order to facilitate description, only the parts related to the application are shown in the drawings.

[0061] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict. The present application will be described in detail below with reference to the drawings and in combination with the embodiments.

[0062] The modeling method comprises:

[0063] The missile system kinetic energy T, the missile system elastic potential energy U and the missile system damping energy D caused by the bending deformation of the missile elastic body are calculated.

[0064] The generalized force Q acting on the missile body corresponding to the virtual work is calculated by the virtual work principle i ; the generalized force Q i includes the body aerodynamic force the aerodynamic force of the control surface the inertia torque generated by the deflection of the control surface and the lateral force generated by the track control engine

[0065] Based on the missile system kinetic energy T, the missile system elastic potential energy U and the missile system damping energy D, and the generalized force Q i , the missile elastic body dynamics equation is constructed by Lagrange equation.

[0066] In order to more clearly illustrate the track control type direct gas composite control anti-air missile elastic motion modeling method of the present application, the following drawings will expand the detailed description of each step in the embodiment of the present application.

[0067] As Figure 1 shown, the track control type direct gas composite control anti-air missile elastic motion modeling method of the present application is the aerodynamic solid coupling coordinate system O a x a y a z a and the body elastic reference coordinate system O e x e y e z e definition diagram, considering the similarity of y and z directions, taking y direction as an example to explain the elastic motion modeling method, the z direction can be modeled by referring to the method.

[0068] The track control type direct gas composite control anti-air missile elastic motion modeling method of the first embodiment of the present application is described in detail as follows:

[0069] Step S100, as Figure 2As shown in the figure, it is a missile elastic deformation schematic diagram of the rail-controlled direct air defense missile elastic motion modeling method of the application, on the basis of which the missile system kinetic energy T caused by the missile elastic body bending deformation, the missile system elastic potential energy U and the missile system damping energy D caused by the missile elastic body bending deformation are calculated.

[0070] Step S110, calculating the missile system kinetic energy T caused by the missile elastic body bending deformation.

[0071] Step S111, constructing the kinetic energy initial expression between the missile system kinetic energy and the elastic deformation displacement of the body microelement and the microelement mass based on the momentum equation, as shown in formula (1):

[0072]

[0073] Wherein, δ y is the elastic deformation displacement of the microelement on the body in the y direction of the body elastic reference coordinate system, is the first order derivative of δ y with respect to time, m is the mass of the body microelement, x is the axial coordinate of the body elastic reference coordinate system, y is the upward direction coordinate perpendicular to the forward direction of the missile body, z is the right direction coordinate perpendicular to the forward direction of the missile body, and L is the length of the missile.

[0074] Step S112, taking the natural bending mode as the degree of freedom of the continuous elastic mechanics system, and taking the generalized coordinate as q i , expanding the elastic deformation displacement of the body microelement at any point, and the expanded elastic deformation displacement of the body microelement is in the form of series, as shown in formula (2):

[0075]

[0076] Wherein, φ i (x) is the i-th order mode function satisfying the orthogonality condition, and n is the number of series expansion terms.

[0077] The mode function φ i (x) satisfies the orthogonality condition as shown in formula (3):

[0078]

[0079] In the above formula, φ j (x) is the j-th order mode function satisfying the orthogonality condition, M i is the generalized mass of the i-th order mode, as shown in formula (4):

[0080]

[0081] Combined with formula (2) and formula (3), formula (5) is obtained:

[0082]

[0083] Step S113, based on the elastic deformation displacement of the extended missile micro-element and the initial expression of kinetic energy, the missile system kinetic energy T caused by the missile system elastic bending deformation is obtained, that is, formula (5) is substituted into formula (1), and formula (6) is obtained:

[0084]

[0085] wherein, is the generalized coordinate q i The first order derivative with respect to time.

[0086] Step S120, the missile system elastic potential energy U is calculated.

[0087] Step S121, based on the elastic potential energy equation, the elastic potential energy initial expression between the missile system elastic potential energy and the elastic deformation displacement of the missile micro-element is constructed, as shown in formula (7):

[0088]

[0089] wherein, E is the elastic modulus, and I is the sectional area moment of inertia about the center along the length of the missile body.

[0090] Step S122, based on the elastic deformation displacement of the extended missile micro-element and the elastic potential energy initial expression, that is, formula (2) is substituted into formula (7), and the extended elastic potential energy expression is obtained, as shown in formula (8):

[0091]

[0092] wherein, φ i (x) is the second order derivative of φ i (x), and the mode function satisfying the orthogonality condition.

[0093] Similarly, the mode function φ i (x) also satisfies the orthogonality condition shown in formula (3), and satisfies the one-dimensional beam elastic bending deformation differential equation.

[0094] Step S123, combined with the one-dimensional beam elastic bending deformation differential equation and the boundary condition of the elastic potential energy initial expression, the missile system elastic potential energy U is obtained.

[0095] The one-dimensional beam elastic bending deformation differential equation is shown in formula (9):

[0096]

[0097] wherein, ω i is the natural frequency of the i-th mode of the missile body.

[0098] The boundary condition of the initial expression of the elastic potential energy is shown in equation (10):

[0099]

[0100] Equation (7) can be simplified by twice stepwise integration of equation (8), combining equation (3) and equation (9), and considering the boundary condition equation (10), and finally the expression of the elastic potential energy U of the missile system is obtained as shown in equation (11):

[0101]

[0102] In step S130, the missile system damping energy D caused by the bending deformation of the elastic missile body is calculated.

[0103] In step S131, the initial expression of the damping energy between the missile system damping energy and the elastic deformation displacement of the infinitesimal on the missile body is constructed based on the damping energy equation, as shown in equation (12):

[0104]

[0105] where c=2ξ i ω i M i is the damping coefficient of the missile body, ξ i is the damping ratio of the i-th mode of the missile body.

[0106] In step S132, the missile system damping energy D caused by the bending deformation of the elastic missile body is obtained by combining the damping coefficient of the missile body and the initial expression of the damping energy, as shown in equation (13):

[0107]

[0108] In step S200, the generalized force Q i acting on the missile body corresponding to the virtual work is calculated by the virtual work principle; the generalized force Q i includes the aerodynamic force of the missile body, the aerodynamic force of the control surface, the inertia torque generated by the deflection of the control surface, and the lateral force generated by the orbit control engine.

[0109] Let f y (x,t) be the distributed force acting on the missile in the y-axis direction of the aerodynamic fixed coordinate system, and the elastic deformation displacement δ y is the virtual displacement corresponding to the distributed force acting in the y-axis direction, and the virtual work is obtained from equation (2) and the virtual work principle as shown in equation (14):

[0110] ​

[0111] Since the i-th generalized coordinate q i The corresponding generalized force is Thus the virtual work corresponds to the generalized force Q i As shown in equation (15):

[0112]

[0113] Step S210, calculate the aerodynamic force Q iα .

[0114] The aerodynamic force of the missile body is determined by the angle of attack in the aerodynamic fixed coordinate system. For an elastic missile body, the angle of attack is composed of three parts:

[0115] 1. The rigid angle of attack a(t);

[0116] 2. The additional acceleration of the rigid body rotating around the center of mass in the elastic reference coordinate system x of the missile body, as shown in equation (16):

[0117]

[0118] Where V is the speed in the elastic reference coordinate system of the missile body, θ is the pitch angle, x cg is the center of mass of the missile, then the local angle of attack at this point is as shown in equation (17):

[0119]

[0120] 3. The additional angle of attack caused by the elastic motion of the body, the distribution along the length of the missile body is as shown in equation (18):

[0121]

[0122] Substitute equation (2) into equation (18) to obtain equation (19):

[0123]

[0124] Combine the above formulas to obtain the total angle of attack, as shown in equation (20):

[0125]

[0126] The aerodynamic load density determined by the total angle of attack is as shown in equation (21):

[0127]

[0128] Where Y α (x) is the derivative of the aerodynamic force with respect to x.

[0129] Substitute equation (21) into equation (15) to obtain the aerodynamic force of the missile body as shown in equation (22)

[0130]

[0131] Step S220, calculating the rudder aerodynamic force

[0132] For the elastic missile, the rudder deflection angle in the aerodynamic fixed coordinate system is shown in equation (23):

[0133] δ(t) = δ z (t) + δ e (t) (23)

[0134] Wherein, δ z (t) is the rudder deflection angle required by the control signal, δ e (t) is the additional rudder deflection angle caused by the elastic deformation of the missile.

[0135] The additional rudder deflection angle δ e (t) is shown in equation (24):

[0136]

[0137] Wherein, x δ is the coordinate of the rudder axis on the x-axis of the missile elastic reference coordinate system.

[0138] The load density of the rudder aerodynamic force is shown in equation (25):

[0139]

[0140] Wherein, Y δ (x) is the rudder aerodynamic force derivative distributed along the x-axis of the missile elastic reference coordinate system.

[0141] Substitute equation (25) into equation (15) to obtain the rudder aerodynamic force shown in equation (26):

[0142]

[0143] Step S230, calculating the inertia moment generated by the rudder deflection

[0144] The load density of the inertia moment generated by one rudder deflection is shown in equation (27):

[0145]

[0146] Wherein, J δ is the moment of inertia of one rudder relative to the rudder axis, δ is the rudder deflection angle, is the second derivative of δ with respect to time.

[0147] Thus, the inertial moment generated by the rudder deflection As shown in equation (28):

[0148]

[0149] Where, φ i ′(x) is the first derivative of φ i (x).

[0150] Step S240, calculate the lateral force generated by the orbit control engine

[0151] The load density of the lateral force generated by the orbit control engine is shown in equation (29):

[0152]

[0153] Where, P is the lateral force of the orbit control engine, x p is the coordinate of the orbit control engine action point on the x-axis of the elastic reference coordinate system of the missile body.

[0154] Substitute equation (29) into equation (15) to obtain the lateral force generated by the orbit control engine shown in equation (30)

[0155]

[0156] In summary, the aerodynamic force of the missile body The rudder aerodynamic force The inertial moment generated by the rudder deflection And the lateral force generated by the orbit control engine The final virtual work corresponds to the generalized force Q acting on the missile body i As shown in equation (31):

[0157]

[0158] Step S300, based on the kinetic energy T of the missile system, the elastic potential energy U of the missile system, and the damping energy D of the missile system, and the generalized force Q i , the Lagrange equation is used to construct the dynamic equation of the elastic missile body.

[0159] Substitute the expressions of T, U, D obtained in step S100 and the expression of Qi obtained in step S200 into the Lagrange equation, as shown in equation (32):

[0160]

[0161] Some parameters are defined as shown in equations (33)-(39):

[0162]

[0163]

[0164]

[0165]

[0166]

[0167]

[0168] D Pi = φ i (x P ) / M i (39)

[0169] Finally, the missile elastic body dynamics equation is shown as equation (40):

[0170]

[0171] wherein, D αi is the i-th mode generalized aerodynamic force coefficient, D ωi is the i-th mode generalized aerodynamic damping force coefficient, D δi is the i-th mode generalized air rudder control force coefficient, D δi is the i-th mode generalized air rudder control force influence coefficient, D Pi is the i-th mode generalized trajectory control force coefficient, D qij is the coupling generalized force coefficient of the j-th mode additional aerodynamic force on the i-th mode, D qij is the coupling generalized force coefficient of the j-th mode additional aerodynamic damping force on the i-th mode.

[0172] Although each step is described in the above-mentioned order in the above-mentioned embodiment, it can be understood by those skilled in the art that, in order to achieve the effect of the embodiment, the different steps do not have to be executed in such an order, they can be executed simultaneously (in parallel) or in a reversed order, and these simple changes are within the protection scope of the present application.

[0173] The trajectory control type straight-air composite control anti-air missile elastic motion modeling system of the second embodiment of the present application comprises:

[0174] A system kinetic energy, potential energy and damping energy calculation module is configured to calculate the missile system kinetic energy T caused by the bending deformation of the missile elastic body, the missile system elastic potential energy U and the missile system damping energy D caused by the bending deformation of the missile elastic body.

[0175] The generalized force calculation module is configured to calculate a generalized force Q corresponding to the virtual work on the missile body by a virtual work principle i ; the generalized force Q i includes a missile body aerodynamic force a control surface aerodynamic force an inertial torque generated by control surface deflection and a lateral force generated by the orbit control engine

[0176] The modeling module is configured to construct a missile elastic body dynamics equation by a Lagrange equation based on a missile system kinetic energy T, a missile system elastic potential energy U, a missile system damping energy D, and the generalized force Q i

[0177] Those skilled in the art can clearly understand that, for the convenience and brevity of description, the specific working process and related description of the system described above can refer to the corresponding process in the foregoing method embodiments, which will not be described here.

[0178] It should be noted that the trajectory control type straight air composite control anti-air missile elastic motion modeling system provided in the foregoing embodiments is only used as an example for the division of the foregoing functional modules, and in actual applications, the foregoing functions can be completed by different functional modules according to needs, that is, the modules or steps in the embodiments of the present application are further decomposed or combined, for example, the modules in the foregoing embodiments can be combined into one module, or can be further split into multiple sub-modules to complete all or part of the functions described above. The names of the modules and steps involved in the embodiments of the present application are only used to distinguish the modules and steps, and are not considered as improper limitations on the present application.

[0179] The terms "first", "second", and the like are used to distinguish similar objects, and are not used to describe or indicate a particular order or sequence.

[0180] The term "comprising" or any other similar term is intended to cover non-exclusive inclusion, so that a process, method, article or device / apparatus including a series of elements includes not only those elements, but also other elements not explicitly listed, or inherent elements of the process, method, article or device / apparatus.

[0181] So far, the technical solutions of the present application have been described in combination with the preferred embodiments shown in the drawings, but those skilled in the art can easily understand that the protection scope of the present application is obviously not limited to these specific embodiments. Those skilled in the art can make equivalent changes or replacements to related technical features without departing from the principles of the present application, and the technical solutions after the changes or replacements will fall within the protection scope of the present application.​

Claims

1. A rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method, characterized in that, The modeling method comprises: calculating missile system kinetic energy T caused by missile elastic body bending deformation, missile system elastic potential energy U and missile system damping energy D caused by missile elastic body bending deformation; The generalized force Q acting on the missile body corresponding to the virtual work is calculated by the virtual work principle i ; the generalized force Q i includes the missile body aerodynamic force the control surface aerodynamic force the inertia moment generated by the control surface deflection and the lateral force generated by the orbit control engine Based on the kinetic energy T, the elastic potential energy U and the damping energy D of the missile system, and the generalized force Q i , the elastic missile body dynamics equation is constructed by Lagrange equation The calculation method of the missile system elastic potential energy U is: constructing an elastic potential energy initial expression between missile system elastic potential energy and elastic deformation displacement of a missile body microelement based on an elastic potential energy equation; obtaining an extended elastic potential energy expression based on the extended elastic deformation displacement of the missile body microelement and the elastic potential energy initial expression, the extended elastic potential energy expression being: where E is the modulus of elasticity, I is the area moment of inertia of the cross section about the center of the elastic body, φ i the second derivative of (x) with respect to x, L is the elastic length i the second derivative of (x) with respect to x, L is the elastic length obtaining the missile system elastic potential energy U in combination with a one-dimensional beam elastic bending deformation differential equation and a boundary condition of the elastic potential energy initial expression, the boundary condition of the elastic potential energy initial expression being:

2. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 1, characterized in that, The calculation method of the missile system kinetic energy T caused by missile elastic body bending deformation is: constructing a kinetic energy initial expression between missile system kinetic energy and elastic deformation displacement of a missile body microelement and microelement mass based on a momentum equation; The inherent bending mode is taken as the degree of freedom of a continuous elastic mechanics system, and the generalized coordinate is taken as q i , to extend the elastic deformation displacement of a micro-element of the elastic body at any point. obtaining the missile system kinetic energy T caused by missile elastic body bending deformation based on the extended elastic deformation displacement of the missile body microelement and the kinetic energy initial expression.

3. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 2, characterized in that, The extended elastic deformation displacement of the missile body microelement is: where δ y (x, t) is the elastic deformation displacement δ y of the microelement of the elastic body extended on the basis, φ i (x) is the mode function satisfying the orthogonality condition, and n is the number of terms of the series expansion.

4. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 3, characterized in that, The missile system kinetic energy T caused by missile elastic body bending deformation is: wherein, is the first derivative of the generalized coordinate q i , M i is the generalized mass of the i-th mode.

5. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 1, characterized in that, The missile system elastic potential energy U is: where M i is the generalized mass of the ith mode, ω i is the natural frequency of the ith mode of the elastic body.

6. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 1, characterized in that, The calculation method of the missile system damping energy D caused by missile elastic body bending deformation is: constructing a damping energy initial expression between missile system damping energy and elastic deformation displacement of a missile body microelement based on a damping energy equation; obtaining the missile system damping energy D caused by missile elastic body bending deformation in combination with a missile body damping coefficient and the damping energy initial expression.

7. The rail-controlled direct-gas composite control anti-air defense missile elastic motion modeling method according to claim 6, characterized in that, The damping energy initial expression between the missile system damping energy and the elastic deformation displacement of the missile body microelement is: where c = 2ξ i ω i M i is the damping coefficient of the elastic body, ξ i is the damping ratio of the i-th mode of the elastic body, M i is the generalized mass of the i-th mode, is the first derivative of the elastic deformation displacement δ y of the microelement on the elastic body.

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