A vibration analysis method for multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiment

Abstract

The present application belongs to the technical field of buffet detection of aircraft model support system in hypersonic flow field environment, and discloses a vibration analysis method of multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiment. Firstly, according to rigid body mechanics and elastic body theory, the engineering model is simplified into a two-degree-of-freedom rotational vibration system coupled by four flexible springs and a rigid plane, mainly by four flexible springs to transfer actuation energy and amplify vibration effect, one end of the spring is fixed to the excitation source, the other end is connected to the rigid plane, so as to produce directional actuation angle. The excitation source end can stably output sinusoidal periodic driving force by default, and the vibration control equation of the micro unit is listed according to d'Alembert principle and generalized Hooke's law; finally, the two-degree-of-freedom vibration equation is obtained according to the coupling coordination condition, and the real-time accurate analytical calculation of the whole vibration model is realized without relying on any force position measuring point.

Description

Technical Field

[0001] This invention belongs to the field of vibration detection technology for aircraft model support systems in hypersonic flow field environments, and relates to a vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments. Background Technology

[0002] Supersonic fighter jets, rockets, and other high-speed aircraft possess significant application value. As a crucial tool in aircraft research, dynamic derivative experiments in high-speed wind tunnel environments can simulate the motion of aircraft under high-frequency excitations during different airflow loads, thereby obtaining corresponding dynamic stability derivative data to aid aerodynamic design and prevent destructive vibrations leading to instability under high-speed conditions. Addressing the need for multi-degree-of-freedom high-frequency motion simulation experimental devices for dynamic derivative experiments, a tail-supported multi-degree-of-freedom high-frequency motion simulation system using piezoelectric ceramics as the excitation source and flexible springs as the vibration energy amplification device has been put into use. This system can meet the experimental requirements of multi-degree-of-freedom high-frequency motion simulation while accommodating the limitations of the confined experimental space of high-speed wind tunnels. To ensure the stability and reliability of the system's output vibration mode and to prevent large-scale uncontrollable vibrations under complex aerodynamic loads and high-speed, high-pressure flow fields, analyzing the vibration characteristics of the system's load model is of great significance for vibration control and system monitoring. While modal analysis and mode shape analysis, as mainstream analytical methods for related models, offer advantages such as fast response speed and high analytical accuracy, they also suffer from high sensitivity due to strong dependence on the arrangement of measurement points, making it easy for deviations in the measurement point positions to distort the analysis results. Furthermore, these methods struggle to effectively decouple the vibration characteristics of complex vibration systems from local to global perspectives, resulting in poor model integrity and significant technical challenges in reconstructing the system vibration equations, making the solution difficult. To address these issues and meet the analytical requirements of wind tunnel experimental models, it is urgent to develop and design a vibration analytical method for multi-degree-of-freedom high-frequency motion simulation devices designed for high-speed wind tunnel experiments.

[0003] The paper "Research on Vibration Characteristics of Dual-Rotor Systems for Aero-Engines" by Ma Pingping et al. investigated the vibration characteristics and typical vibration faults of dual-rotor systems for aero-engines. They established analytical dynamic models and rigid-flexible coupling simulation models for the dual-rotor system to study the coupled vibration and transmission laws of the system. However, this research suffers from a strong dependence on the arrangement of measurement points in the dynamic modeling of dual-rotor systems with misalignment faults, which can easily lead to distorted analysis results.

[0004] The patent "A Method and Device for Analyzing Nonlinear Vibration Characteristics of a Three-Dimensional Serpentine Structure" by Cao Shancheng et al. of Northwestern Polytechnical University (patent number CN202411907444.3) discloses a method and device for analyzing the nonlinear vibration characteristics of a three-dimensional serpentine structure. This device, based on engineering beam theory and the Lagrange equation, can comprehensively reflect the vibration mechanism of the structure under actual working conditions. However, the solution process involves multiple complex derivations. It requires obtaining the first-order buckling mode through finite element simulation, calculating energy parameters using von Kármán nonlinear beam theory, deriving the dimensionless control equation based on the Lagrange equation, and finally numerically solving the dynamic response equation. Each step requires high-performance computing equipment and specialized algorithms, making the solution difficult and hindering the achievement of the goal of decoupling the system's vibration and reconstructing the system's vibration equation.

[0005] Based on the problems existing in the above technologies, it is necessary to propose a vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments. The method should be able to acquire vibration data using the provided experimental device, reconstruct the system vibration equation, and thus meet the analytical requirements of the wind tunnel experimental model, achieving rapid, high-precision, and low-cost vibration analysis. Summary of the Invention

[0006] The purpose of this invention is to provide a vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments. This method ensures the high-frequency response capability of the model while maintaining the reverse decoupling of the vibration system and the reconstruction of the vibration equation within a reasonable computational range, thus rapidly and accurately meeting the analytical requirements of wind tunnel experimental models.

[0007] The technical solution of the present invention:

[0008] A vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments is proposed. First, based on rigid body mechanics and elasticity theory, the engineering model is simplified into a two-degree-of-freedom rotational vibration system coupled to a rigid plane and four flexible springs. The system mainly uses the four flexible springs to transmit kinetic energy and amplify the vibration effect. One end of each flexible spring is fixed to the excitation source, and the other end is connected to the rigid plane, thus generating a directional actuation angle. It is assumed that the excitation source can stably output a sinusoidal driving force. The vibration control equations of the micro-element are derived based on d'Alembert's principle and the generalized Hooke's law. Finally, based on the coupling coordination conditions, the two-degree-of-freedom vibration equations are obtained. Thus, real-time and accurate analytical calculations of the entire two-degree-of-freedom rotational vibration system coupled to the rigid plane are achieved through direct calculation without relying on any force or position measurement points.

[0009] The specific steps are as follows:

[0010] Step 1: Simplify the experimental model;

[0011] The experimental model used in actual wind tunnel conditions is simplified and modeled to establish a two-degree-of-freedom rotational vibration system coupled with multiple flexible springs and a rigid plane. The fixed point on the bow shaft, which is fixed to the vibrating head, is set as the origin of the global inertial coordinate system. The excitation section and the vibrating head are hinged to the bow shaft via pins. When the excitation source actuates, causing the flexible springs to bend and deform, the axis around which the vibrating head and bow shaft rotate relative to each other is the x-axis, and the axis around which the excitation section and bow shaft rotate relative to each other is the y-axis. By default, the excitation section is only allowed to rotate about the y-axis passing through the origin of the global inertial coordinate system, and the vibrating head is only allowed to rotate about the x-axis passing through the origin of the global inertial coordinate system. Four independent excitation sources are fixedly installed on the excitation section, each capable of outputting an amplitude. A sinusoidal driving force with controllable frequency is used, and each excitation source is fixed to the rigid plane of the vibrating head through a flexible spring. It is assumed that the rigid plane has no elastic deformation of its own and only bears the transmission of force and motion. The two-degree-of-freedom rotational vibration system coupled with multiple flexible springs and rigid planes takes the rotation angle of the vibrating head around the x-axis and the rotation angle of the excitation segment around the y-axis as generalized coordinates. The four flexible springs only undergo axial tension or compression deformation. Their deformation and the two generalized coordinates satisfy a linear motion coordination relationship based on the assumption of small rotation angle. The sinusoidal driving force of the excitation source is transmitted to the rigid plane through the flexible springs, thereby realizing the closed-loop coupling of excitation input-flexible deformation-rigid body rotation. The modeling process also follows the assumptions of ideal rigid body, backlash-free and frictionless rotating pair, and linear viscous damping.

[0012] The second step is to define the generalized coordinates and model them step by step.

[0013] The deflection angle, generalized velocity, and generalized acceleration of a two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane are defined; Deflection angle: For the vibrating head to rotate The deflection angle of the axis; For the excitation section The deflection angle of the axis; generalized velocity: For the vibrating head to rotate The angular velocity of the shaft's deflection. For the excitation section Angular velocity of axis deflection; generalized acceleration: For the excitation section The angular velocity acceleration of the shaft;

[0014] Assuming each component in a two-degree-of-freedom rotational vibration system coupled to a rigid plane is an ideal rigid body, and considering only the kinetic energy generated by rotation about an axis, and assuming that the kinetic energies of rotation about a fixed axis can be superimposed, calculate the total kinetic energy of the two-degree-of-freedom rotational vibration system coupled to a rigid plane. for:

[0015] (1)

[0016] In the formula: Let x be the moment of inertia of the excitation head about the x-axis; Let be the moment of inertia of the excitation segment about the y-axis;

[0017] Calculate the total potential energy of a two-degree-of-freedom rotational vibration system coupled with a flexible spring and a rigid plane. Considering only the elastic potential energy of four flexible springs, the elastic deformation of a single flexible spring is... :

[0018] (2)

[0019] In the formula: Let be the distance between the fixed point of the excitation section and the assumed origin of the global inertial coordinate system. This is the distance between the flexible spring fixing point of the vibrating head and the assumed origin of the global inertial coordinate system;

[0020] The total potential energy is the sum of the potential energies of the four flexible springs:

[0021]

[0022] (3)

[0023] In the formula: The axial stiffness of a single flexible spring; The vibration head is wound around Principal stiffness of the shaft and the winding of the excitation section Principal stiffness of the shaft; For the vibrating head to rotate Shaft and excitation section around The coupling stiffness of the shaft;

[0024] The two-degree-of-freedom rotational vibration system based on the coupling of multiple flexible springs and a rigid plane mainly operates in high-speed wind tunnel conditions. Therefore, wind resistance is considered as non-viscous damping. Under this condition, the energy dissipation generated during independent vibration of a single degree of freedom is defined as principal damping, and the energy dissipation generated during the transfer of energy between the two degrees of freedom is defined as coupling damping. Principal damping consists of two parts: structural viscous damping, which characterizes the energy dissipation caused by internal friction and hysteresis at the contact surface within the structure itself, and aerodynamic non-viscous damping, which describes the energy dissipation caused by aerodynamic phenomena. Coupling damping consists of two parts: viscous coupling damping, which causes cross-energy dissipation between the two degrees of freedom due to the viscous effect within the structure, and aerodynamic coupling damping, which causes energy dissipation due to mutual interference between the vibrations of the two degrees of freedom through the wind field. The dissipation function is composed of both principal damping and coupling damping, thus yielding the modified damping dissipation function. for:

[0025] (4)

[0026] In the formula: These are the structural viscous damping coefficient and the aerodynamic inviscous damping coefficient of the vibrating head about the x-axis, respectively. These are the structural viscous damping coefficient and the aerodynamic inviscous damping coefficient of the excitation section about the y-axis, respectively. This is the viscous coupling damping coefficient; This is the aerodynamic coupling damping coefficient;

[0027] Based on the principle of virtual work, the form of the incentive force is:

[0028] (5)

[0029] In the formula: For the first The excitation force at the end of each spring; The amplitude of the excitation source; The frequency of the excitation source; It is a time variable;

[0030] The resulting generalized incentive force is:

[0031] (6)

[0032] (7)

[0033] (8)

[0034] (9)

[0035] In the formula: It is the generalized excitation force amplitude of the vibrating head around the x-axis. Let be the generalized excitation force amplitude of the excitation segment about the y-axis. It is the generalized excitation force of the vibrating head around the x-axis. It is the generalized excitation force of the excitation segment about the y-axis;

[0036] According to the Lagrange equation:

[0037] (10)

[0038] In the formula: k can be either x or y;

[0039] Solving the above equations (1) to (9) simultaneously, simplifying them, and substituting them into the equations and into (10), we get:

[0040] (11)

[0041] (12)

[0042] In the formula: Let x be the moment of inertia of the vibrating head about the x-axis; Let be the moment of inertia of the excitation segment about the y-axis; The total damping of the vibrating head around the x-axis and the total damping of the excitation section around the y-axis; For coupling damping;

[0043] Integrating equations (11) and (12), we obtain the matrix form of the two-degree-of-freedom vibration equation:

[0044]

[0045] (13)

[0046] In the formula: Let M be the inertia matrix; Let C be the damping matrix; Let K be the stiffness matrix;

[0047] The third step is to analyze and solve the two-degree-of-freedom vibration equation;

[0048] The vibration frequencies and mode shapes of a two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane are called modes, and the vector describing the mode shape is called the mode vector. The modal superposition method is a calculation method that decomposes the complex response of a two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane into a linear superposition of modal responses. By integrating the responses of each mode, the overall vibration characteristics of the system are obtained. The modal superposition method is used to analyze and solve the two-degree-of-freedom vibration equation obtained in the second step, transforming it into two independent single-degree-of-freedom equations. Then, through modal superposition, the generalized coordinates of the two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane are restored, thereby obtaining the rotational direction of the two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane. Axis and winding Deflection angle of the axis angular velocity of deflection angular acceleration of deflection First, we make the assumption of a proportional damping matrix, assuming that the damping matrix, inertia matrix, and stiffness matrix satisfy a linear proportional relationship; the proportional damping matrix is:

[0049] (14)

[0050] In the formula: ;

[0051] The two-degree-of-freedom vibration equation in the transformed equation (13) is:

[0052] (15)

[0053] (16)

[0054] (17)

[0055] In the formula: Physical coordinates; It is a generalized activation vector; Angular velocity; Angular acceleration;

[0056] By establishing an undamped homogeneous equation, the first and second natural frequencies of a two-degree-of-freedom rotational vibration system coupled with a flexible spring and a rigid plane are solved. Assuming the solution to the two-degree-of-freedom vibration equation is simple harmonic motion, substituting into equation (17) yields:

[0057] (18)

[0058] in, For modal vectors, The vibration frequency;

[0059] The characteristic equation that satisfies equation (18) Under the given conditions, the natural frequencies are solved to find the solution. The corresponding eigenvalues, and set the two eigenvalues ​​as follows: and Removing the external excitation term from equation (17), the external excitation term on the right-hand side of equation (18) is: Expand to get information about The quadratic equation;

[0060] (19)

[0061] Solving this quadratic equation yields the undamped natural frequency. sorted in descending order , These are the first-order and second-order natural frequencies, respectively.

[0062] The first-order natural frequency Substitute into equation (18) to solve for the first-order mode vector. :

[0063] (20)

[0064] (twenty one)

[0065] In the formula: First-order mode vector exist Components in the axial direction, First-order mode vector exist Components along the axial direction;

[0066] Solving the system of equations (21) yields the first-order mode vector. Similarly, the second-order natural frequency Substituting into equation (18) yields the second-order mode vector. The first-order mode vector and the second-order mode vector are combined to obtain the second-order mode matrix:

[0067] (twenty two)

[0068] In the formula: Second-order mode vector exist Components in the axial direction, Second-order mode vector exist Components along the axial direction;

[0069] Define modal coordinates Using modal orthogonality to eliminate coupling terms, substitute equation (22) into equation (17), and multiply the left side of both sides of the equation by... :

[0070]

[0071] (twenty three)

[0072] According to orthogonality, the coefficient matrices of all terms on the left side of the equation are diagonal matrices, and the right side of the equation represents the modal excitation vectors, which represent the components of the external excitation on each order of the modal vectors:

[0073] (twenty four)

[0074] Equation (24) can be decomposed into two independent single-degree-of-freedom forced vibration equations:

[0075] (25)

[0076] In the formula: , Modal mass describes the inertial characteristics of a two-degree-of-freedom rotational vibration system in its first and second modes. , Modal damping describes the damping characteristics of a two-degree-of-freedom rotational vibration system in the first and second modes. , Modal stiffness describes the stiffness characteristics of a two-degree-of-freedom rotational vibration system in its first and second modes. It is the sum of modal excitation forces; It is a first-order modal excitation force; It is a second-order modal excitation force;

[0077] For each equation in the equations for forced vibration with a single degree of freedom, normalize it to the standard form for single-degree-of-freedom vibration:

[0078] (26)

[0079] In the formula: for The natural frequencies of the first mode; Modal damping ratio, describing Energy dissipation capability of first mode; The modal excitation amplitude reflects the external excitation at... Excitation intensity on the first mode;

[0080] The general solution to forced vibration consists of the homogeneous solution of the transient response and the particular solution of the steady-state response; where the homogeneous solution... for:

[0081] (27)

[0082] In the formula: The damped natural frequency; The constants are determined by the initial conditions; if we assume the initial conditions are 0, i.e., when the system is initially at rest, the homogeneous solution simplifies to:

[0083] (28)

[0084] A particular solution representing the steady-state response and continuous sinusoidal oscillation. for:

[0085] (29)

[0086] In the formula: This represents the steady-state amplitude of the modal. In response to the phase difference of the delayed excitation, its value is 90° in the case of resonance;

[0087] Modal superposition is performed to obtain the general solution of the coordinates of the two modes. Substitute into the coordinate transformation formula: Thus, the final general solution for the physical coordinates is obtained:

[0088] (30).

[0089] The beneficial effects of this invention are as follows: The vibration analysis method for multi-degree-of-freedom high-frequency motion simulation devices for high-speed wind tunnel experiments mainly focuses on modeling and analyzing two-degree-of-freedom rotational vibration systems coupled with multiple flexible springs and rigid planes required for dynamic derivative experiments. Addressing the problems of traditional analytical methods, such as strong dependence on the arrangement of measuring points, poor model integrity, and high solution difficulty, this method can achieve rapid and accurate analysis of multi-degree-of-freedom high-frequency motion simulation devices while ensuring stable and reliable system output vibration modes and safe and controllable amplitude, thus meeting the requirements of dynamic derivative experiments in high-speed wind tunnel environments. Attached Figure Description

[0090] Figure 1 This is a schematic diagram of the vibration system experimental setup consisting of the excitation section, the bow shaft, and the vibrating head model.

[0091] Figure 2 These are a cross-sectional view of the vibration system experimental setup and a schematic diagram of the established right-handed coordinate system;

[0092] Figure 3 This is an analytical flowchart of a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments;

[0093] Figure 4 This is a graph showing the two-degree-of-freedom motion angle response obtained from MATLAB software analysis; where (a) is the rotation angle of the vibrating head. (t) Time-domain response, (b) is the rotation angle of the excitation segment. (t) Time-domain response.

[0094] In the diagram: 1-bow shaft, 2-excitation section, 3-vibration head, 4-flexible spring. Detailed Implementation

[0095] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0096] Example

[0097] A two-degree-of-freedom rotational vibration system, consisting of a bow shaft 1 fixedly connected to an excitation section 2 and a vibrating head 3 respectively, coupled to a rigid plane, is selected for analytical modeling. The excitation section 2 and the vibrating head 3 are coupled and fixedly connected by a flexible spring 4. To simplify the theoretical calculation process, the fixed point on the bow shaft 1 that is fixed to the vibrating head 3 is set as the origin of the global inertial coordinate system. Furthermore, the excitation section is only allowed to rotate about the y-axis passing through the origin, and the vibrating head is only allowed to rotate about the x-axis passing through the origin. This example illustrates the wide application of this method in the analytical calculation of vibration in wind tunnel experiments.

[0098] A vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments, such as... Figure 3 As shown, the specific steps are as follows:

[0099] The first step is to assume relevant system parameters and set the initial vibration state of the system;

[0100] Make assumptions about the relevant parameters, including the moment of inertia. ; Proportional Damping ,but The initial condition is assumed to be a static state.

[0101] The second step is to construct the two-degree-of-freedom vibration equations and solve for the modal parameters.

[0102] Based on the system parameters set in the first step and the default initial static state, substitute them into the two-degree-of-freedom vibration equation (13):

[0103]

[0104] Solve for the modal parameters, let Substituting the parameters, we get:

[0105] First-order natural frequency:

[0106]

[0107] Second-order natural frequency:

[0108]

[0109] 1st-order mode vector:

[0110]

[0111] 2nd order mode vector:

[0112]

[0113] The modal equation parameters can be obtained from the natural frequencies and modal vectors obtained above, and substituted into equation (23):

[0114]

[0115] The solution yields:

[0116] Modal quality:

[0117]

[0118]

[0119] Modal stiffness:

[0120]

[0121]

[0122] Modal damping:

[0123]

[0124] First-order damping ratio:

[0125]

[0126] Second-order damping ratio:

[0127]

[0128] Modal excitation:

[0129]

[0130]

[0131] The third step is to solve for the steady-state response based on the required modal parameters.

[0132] The steady-state response results are as follows:

[0133] First-order mode steady-state amplitude and phase difference:

[0134]

[0135] Second-order modal steady-state amplitude and phase difference:

[0136]

[0137] Physical coordinate steady-state response:

[0138]

[0139] Thus, the calculation of an embodiment of a vibration analysis method for a multi-degree-of-freedom high-frequency motion simulation device for high-speed wind tunnel experiments is completed.

Claims

1. A multi-degree-of-freedom high-frequency motion simulation device vibration analysis method for high-speed wind tunnel experiments, characterized by, Comprising the following steps: First step, simplify the experimental model; The actual wind tunnel operating conditions are simplified to model the experimental model, and a two-degree-of-freedom rotational vibration system coupled with multiple flexible springs and rigid planes is established. The fixed point on the bow shaft connected with the vibration head is set as the origin of the global inertial coordinate system, and the excitation section and the vibration head are respectively connected with the bow shaft through the pin to form the hinge constraint. When the excitation source is actuated to drive the flexible spring to bend and deform, the axis around which the vibration head and the bow shaft realize relative rotation is the x-axis, and the axis around which the excitation section and the bow shaft realize relative rotation is the y-axis. It is assumed that the excitation section is only allowed to rotate around the y-axis passing through the origin of the global inertial coordinate system, and the vibration head is only allowed to rotate around the x-axis passing through the origin of the global inertial coordinate system. Four independent excitation sources are fixedly installed on the excitation section, each of which can output a sine driving force with controllable amplitude and frequency, and each excitation source is correspondingly fixedly connected with the rigid plane of the vibration head through a flexible spring. It is assumed that the rigid plane has no elastic deformation itself and only bears the transmission of force and motion. The two-degree-of-freedom rotational vibration system coupled with multiple flexible springs and rigid planes takes the rotation angle of the vibration head around the x-axis and the rotation angle of the excitation section around the y-axis as the generalized coordinates, and the four flexible springs only undergo axial tensile or compressive deformation. The deformation amount and the two generalized coordinates satisfy the linear motion coordination relationship based on the small angle assumption, the sine driving force of the excitation source is transmitted to the rigid plane through the flexible spring, and then the closed-loop coupling of excitation input-flexible deformation-rigid body rotation is realized. The modeling process also follows the assumptions of ideal rigid body, no-gap no-friction rotary pair and linear viscous damping; Second step, define the generalized coordinates and model step by step to obtain the two-degree-of-freedom vibration equation; Third step, analysis and solution of the two-degree-of-freedom vibration equation.

2. The vibration analysis method for a multi-degree-of-freedom high-frequency motion simulator for a high-speed tunnel experiment according to claim 1, characterized in that, The generalized coordinates in the second step are defined as follows: The deflection angle, generalized velocity and generalized acceleration of the two-degree-of-freedom rotational vibration system coupled by the multi-flexible spring and the rigid plane are defined; the deflection angle: is the deflection angle of the vibration head around axis; is the deflection angle of the vibration exciting section around axis; the generalized velocity: is the deflection angular velocity of the vibration head around axis, is the deflection angular velocity of the vibration exciting section around axis; the generalized acceleration: is the deflection angular acceleration of the vibration exciting section around axis.

3. The vibration analysis method of a multi-degree-of-freedom high-frequency motion simulator for a high-speed tunnel experiment according to claim 2, characterized in that, The step-by-step modeling in the second step is as follows: In the two-degree-of-freedom rotational vibration system with a default multi-flexible spring coupled with a rigid plane, each part is an ideal rigid body, only the kinetic energy generated by rotation around the axis is considered, and the rotational kinetic energy of the fixed axis can be superimposed, and the total kinetic energy of the two-degree-of-freedom rotational vibration system with a multi-flexible spring coupled with a rigid plane is calculated Is: （1） In the formulae: is the moment of inertia of the exciter head about the x-axis; is the moment of inertia of the exciter section about the y-axis; Total potential energy of a two-degree-of-freedom rotational vibration system with multiple flexible springs coupled to a rigid plane The elastic potential energy of only four flexible springs is considered, where the elastic deformation of a single flexible spring is : （2） wherein: is the distance from the fixed point of the excitation section to the origin of the assumed global inertial coordinate system, is the distance from the fixed point of the flexible spring of the vibration head to the origin of the assumed global inertial coordinate system; Then the total potential energy is the sum of the potential energy of the four flexible springs: （3） In the formula: is the axial stiffness of the single flexible spring; is the main stiffness of the vibration head around the axis and the main stiffness of the excitation section around the axis; is the coupling stiffness of the vibration head around the axis and the excitation section around the axis; The two-degree-of-freedom rotational vibration system based on the coupling of multi-flexible springs and rigid plane mainly serves in the working condition of high-speed wind tunnel, so the wind resistance is considered as non-viscous damping. In this condition, the energy dissipation generated by single degree of freedom independent vibration is defined as the main damping, and the energy dissipation generated in the process of energy transmission between two degrees of freedom is defined as the coupling damping. The main damping is composed of the structural viscous damping representing the energy dissipation caused by material internal friction and contact surface hysteresis effect, and the aerodynamic non-viscous damping describing the energy dissipation caused by aerodynamic phenomenon. The coupling damping is composed of the viscous coupling damping representing the cross energy dissipation caused by the internal viscous effect between two degrees of freedom, and the aerodynamic coupling damping representing the energy dissipation caused by the mutual interference of the vibration of two degrees of freedom through the wind field. The dissipation function is composed of the main damping and the coupling damping, so the modified damping dissipation function is obtained as follows: ​ （4） In the formula: respectively are the structural viscous damping coefficient and the aerodynamic non-viscous damping coefficient of the vibration head around the x axis; respectively are the structural viscous damping coefficient and the aerodynamic non-viscous damping coefficient of the excitation section around the y axis; is the viscous coupling damping coefficient; is the aerodynamic coupling damping coefficient; Through the virtual work principle, the excitation force is: （5） wherein: is the excitation force at the th spring end position; is the amplitude of the excitation source; is the actuation frequency of the excitation source; is the time variable; The generated generalized excitation force is: （6） （7） （8） （9） wherein: is the generalized excitation force amplitude of the vibration head around the x axis, is the generalized excitation force amplitude of the excitation section around the y axis, is the generalized excitation force of the vibration head around the x axis, is the generalized excitation force of the excitation section around the y axis; According to the Lagrange equation: （10） In the formula: k is x or y; By combining the above equations (1)-(9), simplifying and substituting equation (10), equation (10) is obtained: （11） （12） In the formula: is the moment of inertia of the vibration head about the x axis; is the moment of inertia of the excitation section about the y axis; is the total damping of the vibration head about the x axis and the excitation section about the y axis; is the coupling damping; Integrating equations (11) and (12), the two-degree-of-freedom vibration equation in matrix form is obtained: （13） wherein: is the mass matrix M; is the damping matrix C; is the stiffness matrix K.

4. The vibration analysis method of a multi-degree-of-freedom high-frequency motion simulator for a high-speed tunnel experiment according to claim 3, characterized in that, The two-degree-of-freedom vibration equation in equation (13) is converted as follows: The vibration frequencies and mode shapes of a two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane are called modes, and the vector describing the mode shape is called the mode vector. The modal superposition method is a calculation method that decomposes the complex response of a two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane into a linear superposition of modal responses. By integrating the responses of each mode, the overall vibration characteristics of the system are obtained. The modal superposition method is used to analyze and solve the two-degree-of-freedom vibration equation obtained in the second step, transforming it into two independent single-degree-of-freedom equations. Then, through modal superposition, the generalized coordinates of the two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane are restored, thereby obtaining the rotational direction of the two-degree-of-freedom rotational vibration system coupled to a flexible spring and a rigid plane. Axis and winding Deflection angle of the axis angular velocity of deflection angular acceleration of deflection First, we make the assumption of a proportional damping matrix, assuming that the damping matrix, inertia matrix, and stiffness matrix satisfy a linear proportional relationship; the proportional damping matrix is: （14） In the formulae: ; By establishing a non-damped homogeneous equation, the first and second order natural frequencies of the two-degree-of-freedom rotational vibration system coupled with multiple flexible springs and rigid planes are solved, and it is assumed that the solution of the two-degree-of-freedom vibration equation is a simple harmonic motion, which is substituted into equation (17) to obtain: （15） （16） （17） wherein: is the physical coordinate; is the generalized excitation vector; is the angular velocity; is the angular acceleration; According to the orthogonality, the coefficient matrix of each term on the left side of the equation is a diagonal matrix, and the right side of the equation is a modal excitation vector, which represents the components of the external excitation on each modal vector: （18） wherein is the modal vector, is the vibration frequency; satisfying the characteristic equation of formula (18) Under the condition that the natural frequency is solved The corresponding eigenvalue is solved, and the two eigenvalues are respectively and ; The external excitation term of formula (17) is removed, so the external excitation term of the right end of formula (18) is Expansion to get the quadratic equation about ; （19） Solving this quadratic equation gives the undamped natural frequency , in descending order , are the 1st and 2nd natural frequencies, respectively. The first order natural frequency Substitute equation (18) into equation (17) to solve the first order modal vector : （20） （21） In the formula: is a first-order modal vector In component in the axial direction, is a first-order modal vector In component in the axial direction; Solving the equation set (21) obtains the first order modal vector ; similarly, substituting the second order natural frequency into equation (18) obtains the second order modal vector , and combining the first order modal vector and the second order modal vector obtains the second order modal matrix: （22） wherein: is a 2nd order modal vector in the component in the axial direction, is a 2nd order modal vector in the component in the axial direction; Defining modal coordinates Substituting equation (22) into equation (17) and multiplying both sides of the equation by : （23） Equation (24) is decomposed into two independent single-degree-of-freedom forced vibration equations: （24） For each equation in the single-degree-of-freedom forced vibration equation, the standard form of single-degree-of-freedom vibration is standardized: （25） wherein: , is the modal mass, describing the inertia characteristics of the two-degree-of-freedom rotational vibration system at the 1st and 2nd modal orders; , is the modal damping, describing the damping characteristics of the two-degree-of-freedom rotational vibration system at the 1st and 2nd modal orders; , is the modal stiffness, describing the stiffness characteristics of the two-degree-of-freedom rotational vibration system at the 1st and 2nd modal orders; is the total of the modal excitation forces; is the 1st modal excitation force; is the 2nd modal excitation force; ​ (26) wherein: is the natural frequency of the n-th mode; the modal damping ratio, describing the energy dissipation capability of the n-th mode; is the modal excitation amplitude, reflecting the excitation strength of the external excitation on the n-th mode; is the modal excitation amplitude, reflecting the excitation strength of the external excitation on the n-th mode; The general solution of forced vibration consists of the homogeneous solution of the transient response and the particular solution of the steady-state response; where the homogeneous solution... for: （27） wherein: is the damping natural frequency; is a constant to be determined, both of which are determined by initial conditions; if the initial conditions are assumed to be 0, i.e., the system is initially at rest, the homogeneous solution simplifies to: （28） Representative steady-state response, particular solution to a sustained sinusoidal vibration Is: （29） wherein: is the modal steady state amplitude; is the phase difference of the response to the excitation, which has the value 90° in the case of resonance; The modal superposition is performed to obtain the general solution of two modal coordinates Substitute the coordinate transformation formula: The final general solution of the physical coordinates is obtained: （30）。

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