Airborne towed antenna modeling method based on Kane equation

Abstract

The invention discloses an airborne towed antenna modeling method based on a Kane equation. The method comprises the following steps of: (1) inputting motion parameters and initial motion conditions of the towed antenna into a dynamic simulation model of the towed antenna; (2) obtaining a kinetic equation by a Kane method; (3) carrying out coordinate conversion on the towed antenna; (4) calculating node speed and acceleration according to the coordinates obtained in the step (3); step (5), calculating a node partial velocity according to the node velocity obtained in the step (4); (6) calculating external force and inertia force borne by the nodes according to the node deviation velocity obtained in the step (5); (7) substituting each parameter into the kinetic equation in the step (2); (8) performing numerical solution according to the motion parameters and the initial conditions obtained in the step (1); and (9) outputting each motion parameter of the towed antenna. According to the method, the complexity of a kinetic equation is reduced, and the calculation speed of simulation is increased.

Description

Technical Field

[0001] The present invention belongs to the technical field of towed antennas, and particularly relates to a modeling method for airborne towed antennas based on Kane's equations. Background Art

[0002] The dynamic modeling and analysis of airborne towed antennas involve multiple aspects such as antenna structure, aerodynamics, and control systems.

[0003] Zheng Xiaohong et al. proposed a dynamic model and steady-state configuration calculation of a towed antenna in the literature "Dynamic Modeling and Simulation of Airborne Towed Antennas", and studied the influence of factors such as the speed, inclination angle, and cone bag stability of the carrier aircraft on its verticality. However, since this method is limited to the steady state, it can only model and analyze the motion of the towed antenna in the steady state.

[0004] Han Zhiren et al. proposed using Kane's method to perform transient dynamic modeling of towed antennas in the literature "Transient Dynamics and Safety Analysis of Airborne Dual Towed Antennas". However, Kane's method is restricted by the selection of generalized coordinates, and the calculated equations contain many parameters and are computationally complex.

[0005] Therefore, there is still room for optimization in the selection of generalized coordinates. Summary of the Invention

[0006] In order to overcome the deficiencies of the above-mentioned prior art, the purpose of the present invention is to provide a modeling method for airborne towed antennas based on Kane's equations, which combines Kane's method with appropriate generalized coordinates to establish a transient dynamic model of the airborne towed antenna in the sum-difference angle system, having the characteristics of simple model establishment, being able to simulate transient dynamics, and eliminating constraint forces; reducing the complexity of the dynamic equations and accelerating the simulation operation speed.

[0007] In order to achieve the above purpose, the technical solution adopted by the present invention is:

[0008] A modeling method for airborne towed antennas based on Kane's equations, comprising the following steps;

[0009] Step (1): Input the motion parameters and initial motion conditions of the towed antenna into the towed antenna dynamic simulation model;

[0010] Step (2): Obtain the dynamic equations of the airborne towed antenna by Kane's method;

[0011] Step (3): Perform coordinate transformation on the towed antenna;

[0012] Step (4): Calculate the node velocities and accelerations according to the coordinates obtained in step (3);

[0013] Step (5): Calculate the partial velocities of the nodes according to the node velocities obtained in step (4);

[0014] Step (6): Calculate the external force and inertial force acting on the node according to the partial velocity of the node obtained in step (5).

[0015] Step (7): Substitute each parameter into the dynamic equation in step (2).

[0016] Step (8): Perform numerical solution according to the motion parameters and initial conditions obtained in step (1).

[0017] Step (9): Output each motion parameter of the towed antenna.

[0018] The specific content of step (1) is as follows:

[0019] The motion parameters of the towed antenna system include the length of the towed antenna, the diameter of the antenna, the density of the antenna, the mass of the cone bag, the acceleration of gravity, the hovering height of the aircraft, the hovering radius, and the flight speed of the aircraft.

[0020] The initial motion condition of the towed antenna is the spatial coordinate of the towed antenna at the initial moment.

[0021] The specific content of step (2) is as follows:

[0022] The Kane method is a set of dynamic equations, and the number of equations in this set of equations is consistent with the degrees of freedom of the system. Each equation is the sum of the generalized active force and the generalized inertial force corresponding to each degree of freedom.

[0023] The dynamic equation of the airborne towed antenna system obtained by the Kane method is:

[0024]

[0025] i is the node number of the micro-element segment after discretization of the towed antenna, m is the number of generalized coordinates of the towed antenna, that is, the degrees of freedom, and n is the number of discretized towed antenna segments. is the generalized active force acting on the i-th segment of the towed antenna in the m-th generalized coordinate. is the generalized inertial force acting on the i-th segment of the towed antenna in the m-th generalized coordinate.

[0026] The specific content of step (3) is as follows:

[0027] Discretize the towed antenna. The node of each micro-element segment is located at the end of the segment. The coordinates of each node in the spherical coordinate system are:

[0028]

[0029] Transform the coordinates of the towed antenna. The following transformation method is available:

[0030]

[0031] i is the node number of the discretized microelement segment of the towed antenna, θi is the elevation angle of the i-th microelement segment in the spherical coordinate system, is the azimuth angle of the i-th infinitesimal segment in the spherical coordinate system, α i and β i is the transformed generalized coordinate, α i is the sum angular coefficient, β i is the difference angle coefficient.

[0032] The step (4) is specifically as follows:

[0033] According to the transformed coordinates, write the node coordinates of the towed antenna, and take the first and second derivatives of the coordinates to obtain the node velocity and acceleration:

[0034]

[0035]

[0036]

[0037] v i is the velocity of the previous node, v i+1 is the current node speed, where v0 is the aircraft speed, l is the unit drag antenna length, and a i is the acceleration of the previous node, a i+1 is the current node acceleration, where a0 is the aircraft acceleration.

[0038] The step (5) is specifically as follows:

[0039] Take the partial derivative of each node's velocity with respect to the transformed generalized rate to obtain the partial velocity of the towed antenna system.

[0040]

[0041]

[0042] is the generalized coordinate α of the towed antenna j The partial velocity, is the generalized coordinate β of the towed antenna j The lateral velocity is , and l is the length of the unit towed antenna.

[0043] The step (6) is specifically as follows:

[0044] The external forces acting on the towed antenna include air resistance, wind force and its own gravity;

[0045] The air resistance is calculated by the following formula:

[0046] f ci =-0.5ρ airC D ld|v i |v i

[0047] The wind force received is calculated by the following formula:

[0048] f win = 0.5ρ air C D ld|V|V

[0049] ρ air is the air density, C D is the air resistance coefficient, l is the length of the unit antenna, d is the diameter of the antenna, v i is the node velocity, and V is the wind speed;

[0050] The gravity of each node is calculated by the following formula:

[0051] f gi = (0, 0, -mg)

[0052] The gravity of the cone bag is calculated by the following formula:

[0053] f M = (0, 0, -Mg)

[0054] m is the mass of the unit antenna, M is the mass of the cone bag, and g is the acceleration due to gravity;

[0055] According to D'Alembert's principle, the inertial force received by the towed antenna is calculated by the following formula:

[0056] f i = -ma i .

[0057] The specific content of step (7) is as follows:

[0058] and are the generalized active force and the generalized inertial force of the m-th generalized coordinate respectively, and the expressions are:

[0059]

[0060]

[0061] The generalized coordinate q = (α1, α2, α3, …, α n , β1, β2, β3, …, β n ), i is the node number, f i is the active force acting on node i, q m is the m-th generalized coordinate, a i and v i are the acceleration and velocity of node i in the Cartesian coordinate system respectively, mi is the mass of the i-th node, is the partial velocity of the velocity of the i-th node with respect to the generalized rate;

[0062] By using Kane's method to operate on the external force, inertial force and partial velocities of each node acting on the towed antenna, the dynamic equation of the towed antenna can be obtained. Since it is a system of nonlinear equations, numerical methods are used for solving.

[0063] The specific content of step (8) is as follows:

[0064] It can be known from step (2) and step (7) that the dynamic equation is a set of nonlinear differential equations. Substituting the parameters at each current moment, the and of each node can be obtained. Update the motion state of the node at the next moment, and the expression is:

[0065]

[0066]

[0067]

[0068]

[0069] i is the node number, t represents the current moment. t + 1 represents the next moment, and dt represents the time step.

[0070] The specific content of step (9) is as follows:

[0071] When the calculation process reaches the preset simulation time, end the iterative calculation and output the parameters of the motion state of the towed antenna, including the position, velocity, acceleration, etc. of the towed antenna.

[0072] Advantages of the present invention:

[0073] 1. When modeling the airborne towed antenna, under the condition of determining the initial conditions, the motion state of the towed antenna at any moment can be simulated. The selection of the generalized coordinates is not limited to the coordinates in the geometric sense, but can be flexibly selected according to the constraint conditions of the system and the nature of the problem. By reasonably selecting the generalized coordinates, the constraint conditions of the system can be implicitly satisfied, thus avoiding explicit treatment of the constraint force, which makes the form of the dynamic equation more concise. Therefore, the generalized coordinates without geometric meaning - the sum angle system and the difference angle system are selected, reducing the complexity of the dynamic equation and accelerating the operation speed of the simulation.

[0074] 2. The traditional modeling method has the problem that the aircraft's motion trajectory is limited to circular motion. However, in the present invention, the motion trajectory of the aircraft can be arbitrarily selected, and the motion of the aircraft in space is transmitted to each node of the towed antenna according to the established constraint conditions. According to the spatial motion of the towed antenna system, by solving the dynamic equation of the towed antenna obtained by Kane's method, the motion state of the towed antenna can be obtained. At the same time, the method of the present invention is not only limited to the modeling of the towed antenna, but can also be extended to other rope dynamics problems, such as the in-air refueling pipe and other fields, and has strong versatility and popularization value. Description of the Drawings

[0075] Figure 1 It is a flowchart of a method for modeling an airborne towed antenna based on Kane's equation provided by an embodiment of the present invention.

[0076] Figure 2 It is an output result diagram of simulating the airborne towed antenna model by the method provided by an embodiment of the present invention. Detailed Embodiment

[0077] The present invention will be further described in detail below with reference to the drawings.

[0078] Please refer to Figure 1 , Figure 1 It is a flowchart of a method for modeling an airborne towed antenna based on Kane's equation provided by an embodiment of the present invention, including:

[0079] (1) Input the motion parameters and initial conditions of the towed antenna.

[0080] Input the motion parameters and initial conditions of the towed antenna into the dynamic simulation model of the towed antenna. The motion parameters of the towed antenna system include the length of the towed antenna, the diameter of the antenna, the density of the antenna, the mass of the cone bag, the acceleration due to gravity, the hovering height of the aircraft, the hovering radius, and the flight speed of the aircraft.

[0081] The initial motion condition of the towed antenna is the azimuth angle and elevation angle of each node on the towed antenna at time 0.

[0082] The dynamic simulation model of the towed antenna is the entire program, and the motion parameters and initial conditions of the towed antenna are for the numerical solution in step (8).

[0083] (2) Obtain the dynamic equation by Kane's method.

[0084] Kane's method is a set of dynamic equations. The number of equations in this system of equations is the same as the degree of freedom of the system, and each equation is the sum of the generalized active force and the generalized inertia force corresponding to each degree of freedom.

[0085] The dynamic equation of the airborne towed antenna system obtained by using Kane's method is:

[0086]

[0087] i is the node number of the micro - element segment after discretization of the towed antenna, m is the number of generalized coordinates of the towed antenna, and n is the number of discretized segments of the towed antenna. is the generalized active force acting on the i - th segment of the towed antenna in the m - th generalized coordinate. is the generalized inertial force acting on the i - th segment of the towed antenna in the m - th generalized coordinate.

[0088] In this step, the dynamic model of the towed antenna is systematically established by Kane's method, making the process of establishing the dynamic equation more organized and preparing for the parameter substitution in step (7).

[0089] (3) Coordinate transformation.

[0090] In a multi - body system, the multi - body system can be represented by multiple generalized coordinates. The number of generalized coordinates is not less than the degrees of freedom of the system. There are complex coupling relationships between each generalized coordinate and other generalized coordinates, and there are mutual influences and interactions among multiple coordinates, such that a change in one coordinate will affect the changes in other coordinates. This coupling phenomenon can be linear or non - linear. By appropriate coordinate transformation, the coupling relationships between various generalized coordinates can be weakened to achieve the purpose of simplifying the dynamic equation.

[0091] The coordinates of the towed antenna in the spherical coordinate system are:

[0092]

[0093] The coordinates of the towed antenna are transformed as follows:

[0094]

[0095] θ i is the elevation angle of the towed antenna node in the spherical coordinate system. is the azimuth angle of the towed antenna node in the spherical coordinate system, α i and β i are the transformed generalized coordinates, α i is the sum - angle coefficient, β i is the difference - angle coefficient.

[0096] The coordinate transformation in this step can achieve the purpose of weakening the coupling relationships of the generalized coordinates and, combined with Kane's method, can simplify the dynamic equation of the towed antenna.

[0097] (4) Calculate the node velocity and acceleration.

[0098] The motion states of the towed antenna system include velocity and acceleration. The Kane method requires projecting the velocity in Euclidean geometry into the transformed generalized coordinate space, that is, obtaining the partial velocity. The purpose of this step is to find the velocity and acceleration of the towed antenna system to prepare for calculating the partial velocity and generalized inertial force later.

[0099] According to the transformed coordinates, write out the node coordinates of the towed antenna, and take the first and second derivatives of the coordinates respectively to obtain the velocity and acceleration of the node as follows:

[0100]

[0101]

[0102] v i is the velocity of the previous node, v i+1 is the velocity of the current node, where v0 is the velocity of the aircraft, l is the length of the unit towed antenna, a i is the acceleration of the previous node, a i+1 is the acceleration of the current node, where a0 is the acceleration of the aircraft.

[0103] (5) Calculate the partial velocity of the node.

[0104] In step (3), the coordinates were transformed. Usually, the velocity of the towed antenna system in three-dimensional space is obtained, while the Kane method requires the velocity in the transformed coordinates, that is, the partial velocity. This step calculates the partial velocity of the towed antenna to prepare for step (7).

[0105] Take the partial derivatives of the velocities of each node with respect to the transformed generalized rates to obtain the partial velocity of the towed antenna system.

[0106]

[0107]

[0108] is the partial velocity of the towed antenna with respect to the generalized coordinate α j of, is the partial velocity of the towed antenna with respect to the generalized coordinate β j of, and l is the length of the unit towed antenna.

[0109] (6) Calculate the external forces and inertial forces acting on the node.

[0110] For the towed antenna system, the external forces acting on it can change its motion state, and the inertial forces acting on it can maintain its motion state. The external forces and inertial forces are very important for studying the motion state of the towed antenna. This step obtains the external forces and inertial forces acting on the towed antenna. The external forces include aerodynamic forces and gravity.

[0111] The air resistance on the towed antenna is calculated by the following formula:

[0112] f ci = -0.5ρ air C D ld|v i |v i

[0113] The wind force on the towed antenna is calculated by the following formula:

[0114] f win = 0.5ρ air C D ld|V|V

[0115] ρ air is the air density, C D is the air resistance coefficient, l is the length of the unit antenna, d is the diameter of the antenna, v i is the node velocity, and V is the wind speed.

[0116] The gravity of each node is calculated by the following formula:

[0117] f gi = (0, 0, -mg)

[0118] The gravity of the cone bag is calculated by the following formula:

[0119] f M = (0, 0, -Mg)

[0120] m is the mass of the unit antenna, M is the mass of the cone bag, and g is the acceleration due to gravity.

[0121] According to D'Alembert's principle, the inertial force on the towed antenna is calculated by the following formula:

[0122] f i = -ma i

[0123] (7) Substitute each parameter into the dynamic equation.

[0124] The dynamic parameters of the towed antenna system are obtained from the previous steps. Substitute each parameter into the dynamic equation in step (2), and the generalized inertial force and generalized active force in the dynamic equation can be calculated.

[0125] and are the generalized active force and generalized inertial force of the m-th generalized coordinate respectively, and the expressions are:

[0126]

[0127]

[0128] The generalized coordinates q = (α1, α2, α3, …, α n , β1, β2, β3, …, β n ), where i is the node number, and f i is the active force acting on node i, q m is the m-th generalized coordinate, a i and v i are the acceleration and velocity of node i in the Cartesian coordinate system respectively, m i is the mass of the i-th node, is the partial velocity of the velocity of the i-th node with respect to the generalized rate.

[0129] In this step, the dynamic equation of the towed antenna can be obtained. Since it is a system of nonlinear equations and difficult to obtain an analytical solution, a numerical method is used for solution.

[0130] (8) Numerical solution.

[0131] The previous steps obtain a system of dynamic equations. The number of unknowns in this system of equations is the same as the number of equations, indicating that it can be solved. However, each independent variable in the equation is multiplied by a nonlinear coefficient, making it very difficult to obtain an analytical expression. Numerical solution is carried out by programming.

[0132] From steps (2) and (7), it can be seen that the dynamic equation is a system of nonlinear differential equations. Substituting the parameters at each current moment can be regarded as a system of linear equations. Thus, the and of each node are calculated to update the motion state of the node at the next moment. The expression is:

[0133]

[0134]

[0135]

[0136]

[0137] where i is the node number, t represents the current moment, t + 1 represents the next moment, and dt represents the time step.

[0138] (9) Output the motion parameters of the towed antenna.

[0139] After the calculation process reaches the preset simulation time, the iterative calculation is terminated and the parameters of the motion state of the towed antenna are output, including the position, velocity, acceleration, etc. of the towed antenna.

[0140] The advantages of the present invention can be further illustrated by the following simulation experiments:

[0141] 1. Simulation conditions of the towed antenna:

[0142] When the drag antenna is in normal operation, it is suspended from an aircraft, and the aircraft's motion trajectory is as follows:

[0143]

[0144] Assume that the wind speed varies with altitude, and the calculation method of the wind speed is:

[0145] V = h 0.2375

[0146] The remaining parameters are shown in Table 1:

[0147] Table 1 Parameters of the airborne drag antenna

[0148]

[0149] 2. Initial conditions for simulation:

[0150] The initial conditions for the dynamic simulation of the drag antenna are shown in Table 2:

[0151] Table 2 Initial conditions for simulation

[0152]

[0153]

[0154] The method of the present invention is used for the simulation of this airborne drag antenna.

[0155] 3. Simulation results:

[0156] A method for modeling an airborne drag antenna based on Kane's equation of the present invention is adopted. Please refer to Figure 2 , Figure 2 which is the simulation result diagram of the airborne drag antenna by the method of the present invention. Figure 2 It is the three-dimensional spatial position diagram of the end of the drag antenna output by the simulation. From Figure 2 it can be seen that the position change trend of the end of the drag antenna in the x direction first increases and then decreases, and finally fluctuates up and down around a certain value. The position change trend in the y direction is all increasing, and finally fluctuates up and down around a certain value, while there are relatively large fluctuations in the position change in the z direction.

[0157] By attaching Figure 2 , the conclusion that can be drawn is that this modeling method has a certain degree of credibility, the spatial position convergence of the end of the drag antenna is good, and the motion in each direction is relatively stable, indicating that the dynamic equation obtained by this method is solvable and the solution does not show a divergent phenomenon, which conforms to the physical laws under the action of external forces on the system.

[0158] The parts not described in detail in this implementation scheme belong to the commonly known and frequently used means in this industry, and will not be described one by one here. The above examples are only illustrative of the present invention and do not constitute a limitation on the protection scope of the present invention. Any design identical or similar to the present invention falls within the protection scope of the present invention.

Claims

1. An airborne towed antenna modeling method based on Kane's equation, characterized in that, It includes the following steps; Step (1): Input the motion parameters and initial motion conditions of the towed antenna into the dynamic simulation model of the towed antenna; Step (2): Obtain the dynamic equation of the airborne towed antenna by Kane's method; Step (3): Perform coordinate transformation on the towed antenna; Step (4): Calculate the node velocity and acceleration according to the coordinates obtained in Step (3); Step (5): Calculate the partial velocity of the node according to the node velocity obtained in Step (4); Step (6): Calculate the external force and inertial force acting on the node according to the partial velocity of the node obtained in Step (5); Step (7): Substitute each parameter into the dynamic equation in Step (2); Step (8): Perform numerical solution according to the motion parameters and initial conditions obtained in Step (1); Step (9): Output the motion parameters of each part of the towed antenna.

2. The airborne towed antenna modeling method based on Kane's equation according to claim 1, wherein The specific content of Step (1) is as follows: The motion parameters of the towed antenna system include the length of the towed antenna, the diameter of the antenna, the density of the antenna, the mass of the cone bag, the acceleration due to gravity, the hovering height of the aircraft, the hovering radius, and the flight speed of the aircraft; The initial motion condition of the towed antenna is the spatial coordinates of the towed antenna at the initial moment.

3. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that The specific content of Step (2) is as follows: The dynamic equation of the airborne towed antenna system obtained by Kane's method is: i is the node number of the infinitesimal segment after discretization of the towed antenna, m is the number of generalized coordinates of the towed antenna, i.e., the degrees of freedom, and n is the number of discretized segments of the towed antenna. is the generalized active force acting on the i-th segment of the towed antenna in the m-th generalized coordinate. is the generalized inertial force acting on the i-th segment of the towed antenna in the m-th generalized coordinate.

4. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, wherein, The specific content of Step (3) is as follows: Discretize the towed antenna, and the node of each microelement segment is located at the end of the segment. The coordinates of each node in the spherical coordinate system are: Transform the coordinates of the towed antenna, and there is the following transformation method: i is the node number of the infinitesimal segment after discretization of the towed antenna, θ i is the elevation angle of the i-th infinitesimal segment in the spherical coordinate system, is the azimuth angle of the i-th infinitesimal segment in the spherical coordinate system, α i and β i are the transformed generalized coordinates, α i is the sum angle coefficient, β i is the difference angle coefficient.

5. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that, The specific content of Step (4) is as follows: According to the transformed coordinates, write out the node coordinates of the towed antenna, and take the first derivative and the second derivative of the coordinates respectively to obtain the velocity and acceleration of the node as: v i is the speed of the previous node, v i+1 is the speed of the current node, where v0 is the speed of the aircraft, l is the length of the unit towed antenna, a i is the acceleration of the previous node, a i+1 is the acceleration of the current node, where a0 is the acceleration of the aircraft.

6. The airborne towed antenna modeling method based on Kane's equation according to claim 1, wherein The specific content of Step (5) is as follows: Take the partial derivative of the velocity of each node with respect to the transformed generalized rate to obtain the partial velocity of the towed antenna system; is the partial velocity of the towed antenna with respect to the generalized coordinate α j , is the partial velocity of the towed antenna with respect to the generalized coordinate β j , and l is the length of the unit towed antenna.

7. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that, The specific content of Step (6) is as follows: The external forces acting on the towed antenna include air resistance, wind force, and its own gravity; The air resistance is calculated by the following formula: f ci = -0.5ρ air C D ld|v i |v i The wind force is calculated by the following formula: f win = 0.5ρ air C D ld|V|V ρ air is the air density, C D is the air drag coefficient, l is the length of the unit antenna, d is the antenna diameter, v i is the node velocity, and V is the wind speed; The gravity of each node is calculated by the following formula: f gi = (0, 0, -mg) The gravity of the cone bag is calculated by the following formula: f M = (0, 0, -Mg) m is the mass of the unit antenna, M is the mass of the cone bag, and g is the acceleration due to gravity; According to D'Alembert's principle, the inertial force acting on the towed antenna is calculated by the following formula: f i = -ma i .

8. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that, The specific content of Step (7) is as follows: and are respectively the generalized active force and the generalized inertial force of the m-th generalized coordinate, and the expressions are as follows: The generalized coordinates \(q = (\alpha_1,\alpha_2,\alpha_3,\cdots,\alpha n ,\beta_1,\beta_2,\beta_3,\cdots,\beta n ), where \(i\) is the node number, \(f i \) is the active force acting on node \(i\), \(q m \) is the \(m\)-th generalized coordinate, \(a i \) and \(v i \) are the acceleration and velocity of node \(i\) in the Cartesian coordinate system respectively, \(m i \) is the mass of the \(i\)-th node, \) is the partial velocity of the velocity of the \(i\)-th node with respect to the generalized rate; Perform operations on the external force, inertial force, and partial velocity of each node of the towed antenna using Kane's method to obtain the dynamic equation of the towed antenna, and use numerical methods for solution.

9. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that, The specific content of Step (8) is as follows: As can be seen from steps (2) and (7), the kinetic equations are a set of nonlinear differential equations. Substituting the parameters at each current moment, the and node motion states at the next moment can be updated, and the expression is: i is the node number, t represents the current moment. t + 1 represents the next moment, and dt represents the time step.

10. A method for modeling an airborne towed antenna based on Kane's equation according to claim 1, characterized in that, The specific content of Step (9) is as follows: When the calculation process reaches the preset simulation time, end the iterative calculation and output the parameters of the motion state of the towed antenna, including the position, velocity, acceleration, etc. of the towed antenna.

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