A wind tunnel test vibration analysis method based on vibration control equation

Abstract

The present invention belongs to the technical field of flutter detection of aircraft model support systems, and discloses a vibration analysis method for wind tunnel experiments based on vibration control equations. According to the elastic body theory, the engineering model is simplified into a beam with non-uniform cross-section, and the constraints are that one end is fixed, the other end is attached with a mass block, and the surface is distributed in the form of non-uniform dynamic excitation loads. The vibration control equations of the micro-units are listed according to the D'Alembert principle; according to the Euler-Bernoulli theory, the relationship equation between the bending moment and the deflection of the elastic body is constructed; the system vibration differential equation is obtained, and the number of required measurement points is determined according to the properties of the equation and the equation is solved to obtain the vibration analysis model of the system, and further analyze the vibration relationship between the local parts of the vibration system. The wind tunnel experiment vibration analysis method based on the vibration control equation proposed in the present invention lays a theoretical foundation for engineering application scenarios such as vibration control, vibration decoupling, and vibration testing of wind tunnel experiments, and is a simple and efficient analytical calculation method.

Description

Technical Field

[0001] The invention belongs to the technical field of buffeting detection of aircraft model support systems, and relates to a wind tunnel experiment vibration analysis method based on vibration control equations. Background Art

[0002] As modern flight missions place higher demands on the ultra-high speed and high maneuverability of aircraft, wind tunnel tests are increasingly becoming an indispensable part of the aircraft design process. Due to the limitations of the computational accuracy of simulation models such as computational fluid dynamics, in order to ensure the service performance of modern high-end aerospace equipment such as supersonic fighters, ultra-high-speed aircraft, and rockets, their development mostly relies on wind tunnel experiments to accurately evaluate technical indicators such as aerodynamic design. In actual wind tunnel experiments, due to the advantages of small interference with the flow field surrounding the experimental model, simple structure and easy installation, the support method of fixing the target model with the tail support rod is currently widely used. For this type of configuration, under the conditions of complex and changeable aerodynamic loads and high-speed and high-pressure flow fields, it is very easy to cause safety hazards caused by large-amplitude vibrations. Analytical model vibration is of great significance for vibration control and real-time monitoring of the system. In order to accurately and reliably obtain the motion data of high-speed wind tunnel models, high-precision and high-reliability measurements under strong disturbances in complex flow fields must be guaranteed. In addition, due to the limited test space, the arrangement of measurement points and the types of measured physical quantities are restricted. The traditional wind tunnel test vibration analysis method is dominated by modal method and vibration type analysis method, which has the advantages of fast response, accurate analysis and high precision, but has the disadvantage of high sensitivity to the arrangement of measuring points. At the same time, modal method and vibration type analysis method lack effective solutions for solving problems such as phase difference between parts of the vibration system, which involve the analysis of the vibration relationship between local parts of the system. It is challenging to achieve the purpose of analyzing the vibration model by arranging measuring points or directly analyzing and calculating, reversely decoupling the vibration of the system, and reconstructing the vibration equation of the system.

[0003] The patent "Array antenna vibration deformation prediction method and device based on principal mode method and strain" by Wang Zhihai et al., patent number CN202010172598.8, introduces a method for predicting the vibration deformation of an array antenna based on the principal mode method and strain. This method constructs a differential equation for a multi-degree-of-freedom system under the action of ground motion under the action of external high-dynamic force excitation, solves multi-order vibration modes, and thus derives multi-order modes and modal matrices. Finally, by arranging multiple measuring points and solving the high-order modal matrix, the vibration modeling is completed. The limitation of this method is that the number of measuring points required to be set is the same as the number of solvable vibration modes. Therefore, it is highly sensitive to the arrangement of measuring points, and cannot efficiently deal with the problem of vibration relationship between local parts of the vibration system.

[0004] In the article "Analysis of Vibration and Mechanical Characteristics of Flexible Beams Based on the Nodal Coordinate Method", Qu Hanlong et al. established a three-dimensional two-node high-order fully parametric flexible beam unit model based on the nodal coordinate method. Through the coordinate matrix function and the dynamic equation of the flexible beam, the Jacobian matrix and the generalized elastic mechanics strain energy theory were cited, and the vibration equation of the flexible beam was obtained by combining the mass matrix of the physical model. This method performs a complete analytical calculation on the required vibration model. Therefore, in actual engineering applications, if you want to reconstruct the system vibration equation, you only need to arrange any two measurement points to obtain the boundary conditions of the full analytical equation. It has the advantages of clear physical images and low sensitivity to the arrangement of measurement points. However, this method only theoretically models the research object and cannot be applied to working conditions with complex excitation loads, so it is not widely applicable.

[0005] Based on the problems existing in the above technologies, it is necessary to propose a wind tunnel experiment vibration analysis method based on the vibration control equation, which can reduce the sensitivity of the vibration model reconstruction to the measurement point arrangement while ensuring the accuracy of the vibration analysis reconstruction calculation, and provide a method for decoupling and analyzing the vibration relationship between local parts of the vibration system. Summary of the invention

[0006] The purpose of the present invention is to provide a wind tunnel experiment vibration analysis method based on the vibration control equation, which overcomes the shortcomings of the prior art and is used to solve the problems of high sensitivity of measuring point arrangement, difficulty in modeling complex excitation load conditions, and difficulty in analyzing the vibration relationship between local parts of the vibration system in the prior art, and can accurately and quickly perform model vibration analysis calculations.

[0007] The technical solution of the present invention:

[0008] A wind tunnel experimental vibration analysis method based on vibration control equations, firstly, according to the elastic body theory, the engineering model is simplified into a beam with non-uniform cross-section, the constraint condition is that one end is fixed, the other end is attached with a mass block, and the surface is distributed in the form of non-uniform dynamic excitation load, and the vibration control equation of the micro unit is listed according to the D'Alembert principle; secondly, according to the Euler-Bernoulli theory, the relationship equation between the bending moment and deflection of the elastic body is constructed; finally, the system vibration differential equation is obtained, the number of required measurement points is determined according to the properties of the equation, and the equation is solved to obtain the vibration analytical model of the system, and further analyze the vibration relationship between local parts of the vibration system;

[0009] The specific steps are as follows:

[0010] Step 1: Simplify the engineering model and construct the differential vibration control equation

[0011] Consider the support method of using the tail support rod to fix the model to be tested. In actual working conditions, the system consists of a support rod 1 and a model to be tested 2. One end of the support rod is fixed and the other end is connected to the model to be tested. There is a non-uniform and discontinuous dynamic load in the tangential direction. Establish a three-dimensional rectangular coordinate system and set the coordinate origin to be located on the plane where the cross section corresponding to the fixed end of the support rod is located. The axis direction is parallel to the axis direction of the support rod, and the positive direction points to the model to be tested. The origin position is taken as the centroid of the cross section of the fixed end of the support rod. Since the three-dimensional geometric figure of the support rod is about The axis is rotationally symmetric, so Axis and The direction of the axis can be based on The orientation of the axis can be selected arbitrarily. Since the support rod has uniform mass distribution and good rotational symmetry, it can be equivalent to a uniform elastic body, and its physical properties conform to the basic assumptions of elastic mechanics. The scale of the model to be tested can be considered as a small amount compared to the support rod, so the model to be tested can be equivalent to a mass block attached to one end of the support rod.

[0012] Since vibration can be superimposed, only one perpendicular to The coordinate components of the axis can describe the vibration of the system in three-dimensional space. Differential of the original coordinate system on the plane , the inertia force of the support rod unit corresponding to this microelement can be expressed as

[0013]

[0014] Where: For the support rod unit The displacement in the axial direction, is the support rod density, is the partial differential operator, is the cross-sectional area of ​​the support rod Function of

[0015] Since the lateral vibration of the support rod has a greater impact on the stability and safety of the system, only the lateral dynamic equation is considered. According to the D'Alembert principle, we get

[0016]

[0017] Where: is the tangential force element, It is a tangential non-uniform discontinuous dynamic load;

[0018] According to the moment balance equation, we get

[0019]

[0020] Where: is the bending moment microelement;

[0021] Tangential force and bending moment Perform variational operations and obtain

[0022]

[0023]

[0024] Substituting equations (4) and (5) into equations (2) and (3) together, we can obtain

[0025]

[0026] Where: is the bending moment function;

[0027] At this point, the differential vibration control equation is established.

[0028] Step 2: Construct the relationship equation between the bending moment and deflection of the elastic body

[0029] According to the Euler-Bernoulli theory, the relationship between the bending moment and deflection of an elastic body can be expressed as

[0030]

[0031] Where: is Young's modulus, is the moment of inertia of the cross section;

[0032] This equation relates the bending moment and deflection in an elastic body. For the research background of the present invention, since the model to be tested is attached to one end of the support rod and the other end is a fixed constraint, it is only necessary to transform the infinitesimal elements in the support rod using the Euler-Bernoulli theory. The states of the two endpoints exist only as boundary conditions and do not need to be taken into account in the dynamic equation, thereby simplifying some calculations.

[0033] At this point, the relationship equation between the bending moment and deflection of the elastic body has been constructed.

[0034] Step 3: Get the system vibration differential equation and evaluate the measurement points

[0035] Substituting equation (7) into equation (6), we get

[0036]

[0037] Expanding and simplifying the equation, we can get

[0038]

[0039] Where: for right The second derivative of

[0040] Equation (9) is the vibration differential equation of the system. Obviously, this equation is a fourth-order partial differential equation, which requires two initial conditions and four boundary conditions. The boundary conditions can be taken as The displacement and velocity at the moment, assuming that the system does not vibrate in the initial state, then there are boundary conditions

[0041]

[0042]

[0043] Where: is the trajectory equation of the edge of the support rod;

[0044] For the four boundary conditions, the fixed constraint at one end of the support rod can contribute two boundary conditions, namely

[0045]

[0046]

[0047] For the other two boundary conditions, consider the end points of the support rod attached to the equivalent mass block. If the damping and equivalent stiffness of the model to be tested are ignored, the boundary conditions are:

[0048]

[0049]

[0050] Where: is the coordinate of the contact point between the support rod and the model to be tested;

[0051] It can be seen that if the damping and equivalent stiffness provided by the model to be tested are not considered, the four boundary conditions can be uniquely determined. In the case of small vibrations, the fitting effect is better in theory and the calculation cost is saved. When the vibration amplitude is large, the damping and equivalent stiffness provided by the model to be tested cannot be ignored. At this time, the boundary conditions become

[0052]

[0053] Where: is the damping coefficient, is the equivalent stiffness, all of which are unknown and need to be measured;

[0054] It can be seen that, considering the damping and equivalent stiffness provided by the model to be tested, it is meaningless to list the boundary condition equation for one end of the equivalent attached mass block of the support rod, but it increases the number of unknowns. Therefore, for more general cases, in actual engineering applications, it is necessary to arrange at least two auxiliary measuring points at different positions to determine the two boundary conditions, and there is no additional requirement for the location of the measuring points.

[0055] At this point, the analytical calculation of the wind tunnel experiment vibration based on the vibration control equation is completed.

[0056] The beneficial effect of the present invention is that a vibration analysis method for wind tunnel experiments based on vibration control equations is proposed. Different from the traditional modal method and vibration type analysis method, the vibration reconstruction analytical calculation using this method is less sensitive to the distribution of measuring points, and requires fewer measuring points, which is conducive to the layout of test equipment for narrow wind tunnel experiments and reduces the standard for selecting test equipment. At the same time, since the full response equation of the system deflection is obtained, it is convenient to analyze and quantify the vibration relationship between local parts of the system. An analytical calculation method for the vibration system under quasi-stable conditions is proposed, which saves computing resources and has the advantages of high response speed, high reconstruction quality, and high calculation accuracy. In summary, the vibration analysis method for wind tunnel experiments based on vibration control equations proposed in the present invention has laid a theoretical foundation for engineering application scenarios such as vibration control, vibration decoupling, and vibration testing of wind tunnel experiments, and is a simple and efficient analytical calculation method. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 is a schematic diagram of the vibration system experimental device of the support rod and the model to be tested; is the location of the center of gravity of the model to be tested, is the distance from the centroid to the connection point, is the gravity vector;

[0058] Figure 2 It is a simplified schematic diagram of the vibration system;

[0059] Figure 3 Schematic diagram of the phase difference of vibration test response; (a) is the sampling diagram of lift test, (b) is the sampling diagram of pitch moment test, (c) is the Lissajous figure corresponding to the phase difference of the acceleration measurement point of lift test, and (d) is the Lissajous figure corresponding to the phase difference of the acceleration measurement point of pitch moment test;

[0060] Figure 4 It is a flow chart of the vibration analysis method of wind tunnel experiment based on the vibration control equation;

[0061] In the figure: 1-support rod, 2-model to be tested. DETAILED DESCRIPTION

[0062] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0063] This embodiment uses a vibration system composed of a support rod 1 and a model to be tested 2 for modeling and calculation. To simplify the theoretical calculation process, this embodiment ignores the relationship between the cross-sectional area of ​​the support rod and the coordinates. The goal is to solve the vibration phase difference between the end of the model to be tested and the fixed end under quasi-stable conditions. This is a commonly used mathematical relationship in actual engineering applications. This example is used to illustrate the wide application of this method in wind tunnel experiment vibration analysis calculation. All similar problems can be solved using the ideas provided by this method.

[0064] A wind tunnel test vibration analysis method based on the vibration control equation is as follows Figure 4 As shown, the specific calculation steps are as follows:

[0065] Step 1: Simplify the engineering model and construct the differential vibration control equation

[0066] Assume that support rod 1 is a homogeneous cylindrical long rod, and take its density as , take its Young's modulus as , take its cross-sectional radius as , its physical properties conform to various assumptions of classical elastic mechanics. Since the model 2 to be tested is fixed to the support rod 1 and has a small size, it can be equivalent to a mass block attached to one end of the support rod, and its mass is set to , take the distance between its center of gravity and the origin as Substituting each physical quantity into equation (6) we can get

[0067]

[0068] Construct the differential governing equations of the vibration system.

[0069] Step 2: Construct the relationship equation between the bending moment and deflection of the elastic body

[0070] Since the support rod has the same cross-sectional area, the moment of inertia of the support rod can be expressed as

[0071]

[0072] Substituting each physical quantity into equation (7), the relationship between the bending moment and deflection of the elastic body can be written as

[0073]

[0074] Step 3: Get the system vibration differential equation and evaluate the measurement points

[0075] In combination with this embodiment, equation (9) is appropriately simplified to obtain a complete analytical solution to the deflection response. The simplified equation (9) is:

[0076]

[0077] Considering that the phase difference at the two ends of the support rod can be ignored in the case of small vibration, the initial condition is obtained from equations (10) and (11):

[0078]

[0079]

[0080] The boundary conditions are obtained from equations (12), (13), (14), and (15):

[0081]

[0082]

[0083]

[0084]

[0085] Solved

[0086]

[0087] Where: is the angular frequency of the time factor term in the vibration equation, , , , , , They are 6 constants determined by initial conditions and boundary conditions;

[0088] make

[0089]

[0090] Where: is the wave number of the spatial factor term in the vibration equation;

[0091] Substituting the boundary conditions, we can get , , , The fourth-order linear equations of four parameters are as follows:

[0092]

[0093] in

[0094]

[0095]

[0096]

[0097] Where: is the coefficient matrix, is a column vector consisting of four parameters, is the length of the support rod;

[0098] To ensure that the equation has non-zero solutions, we have

[0099]

[0100] The phase difference between the two ends of the support rod can be expressed as

[0101]

[0102] Substituting the data, we get

[0103]

[0104]

[0105] Compared with the dynamic fitting results of the finite element method and the experimental results ( Figure 3 ), the phase difference between the two ends is of a small order of magnitude, which is close to the vibration time scale.

[0106] Since multiple steps of simplification are performed in this embodiment, according to the uniqueness theorem of partial differential equation solutions, the solution of the system vibration partial differential equation corresponding to this embodiment is uniquely determined by two initial conditions and four boundary conditions, so a full analytical solution can be obtained without arranging measurement points. If in actual engineering applications, there is a situation where a full analytical solution cannot be obtained through initial conditions and boundary conditions, it is necessary to determine the number of required measurement points based on the number of coefficients to be determined, and substitute the boundary conditions measured at the measurement points into the system vibration partial differential equation to determine the coefficients.

[0107] At this point, the calculation of this embodiment is completed.

[0108] This vibration analysis method for wind tunnel experiments based on vibration control equations is mainly aimed at a common vibration system in wind tunnel experiments. Starting from practical engineering problems, the vibration system is modeled using vibration control equations. By reasonably selecting boundary conditions and appropriate model simplification, the dynamic model of the vibration system is obtained from the Euler-Bernoulli theory. It also provides a theoretical basis for the selection of measurement points in practical engineering applications by combining vibration analysis theory, and proposes a solution to the vibration relationship between local structures of the vibration system.

[0109] This method is not limited to the vibration system and calculation examples to be solved, but is applicable to the solution of the support system of wind tunnel aircraft models with any geometric shape and any configuration. This calculation method solves the shortcomings of traditional methods that are not conducive to solving the problem of vibration relationship analysis between local parts of the vibration system, and theoretically can reduce the sensitivity of the test system to the arrangement of measurement points. It expands the vibration detection method in wind tunnel experiments.

Claims

1. A wind tunnel test vibration analysis method based on vibration control equation, characterized in that: The specific steps are as follows: Step 1, simplify the model to be tested and construct the differential vibration control equation; In actual working conditions, the system consists of a support rod and a model to be tested. The support method of fixing the model to be tested with a support rod is used. One end of the support rod is fixed, and the other end is connected to the model to be tested. There is a non-uniform and discontinuous dynamic load loading in the tangential direction. A three-dimensional rectangular coordinate system is established, and the coordinate origin is set to be located in the plane where the corresponding cross section of the fixed end of the support rod is located. The x-axis direction is parallel to the axis direction of the support rod, and the positive direction points to the model to be tested. The origin position is taken as the centroid of the cross section of the fixed end of the support rod. The three-dimensional geometric figure of the support rod is rotationally symmetric about the x-axis, and it is concluded that the directions of the y-axis and the z-axis can be arbitrarily selected based on the orientation of the x-axis. Based on the uniform mass distribution and rotational symmetry of the support rod, it is equivalent to a uniform elastic body, and the physical properties conform to the assumptions of elastic mechanics. The scale of the model to be tested is a small amount compared to the support rod, so the model to be tested is equivalent to a mass block attached to one end of the support rod. Based on the vibration, superposition operation can be performed. It is only necessary to analyze a coordinate component perpendicular to the x-axis to describe the vibration of the system in three-dimensional space. Select the microelement dx of the original coordinate system on the x, y plane, and the inertial force of the support rod unit corresponding to the microelement is expressed as: Where: u(x,t) is the displacement of the support rod unit in the y-axis direction, ρ is the support rod density, is the partial differential operator, S(x) is the function of the cross-sectional area of ​​the support rod with respect to x; The lateral vibration of the support rod has a great influence on the stability and safety of the system, so only the lateral dynamic equation is considered. According to the D'Alembert principle, we get: Where: dF is the tangential force element, f(x, t) is the tangential non-uniform discontinuous dynamic load; According to the moment balance equation, we get: dM-Fdx=0 (3) Where: dM is the bending moment; F is the tangential force; Performing variational operations on the tangential force F and the bending moment M, we obtain: Substituting equations (4) and (5) into equations (2) and (3) in parallel, we get: Where: M(x,t) is the bending moment function; At this point, the differential vibration control equation is established; The second step is to construct the relationship equation between the bending moment and deflection of the elastic body; According to the Euler-Bernoulli theory, the relationship between the bending moment and deflection of an elastic body is expressed as: Where: E is Young's modulus, I(x) is the moment of inertia of the cross section; At this point, the relationship equation between the bending moment and deflection of the elastic body has been constructed; The third step is to obtain the system vibration differential equation and evaluate the measurement points; Substituting formula (7) into formula (6), we get: Expanding and simplifying the equation, we get: Where: g(x) is the second-order derivative of I(x) with respect to x; Formula (9) is the vibration differential equation of the system, which requires two initial conditions and four boundary conditions. The boundary conditions are the displacement and velocity at time t = 0. If the system is set to have no vibration in the initial state, then the boundary conditions are: u(x,t=0)=u0(x) (10) Where: u0(x) is the trajectory equation of the edge of the support rod; For the four boundary conditions, the fixed constraint at one end of the support rod contributes two boundary conditions, namely: u(x=0,t)=0 (12) For the other two boundary conditions, consider the end points of the support rod attached to the equivalent mass block. If the damping and equivalent stiffness of the model to be tested are ignored, the boundary conditions are: Where: x0 is the coordinate of the contact point between the support rod and the model to be tested; If the damping and equivalent stiffness provided by the model to be tested are not considered, the four boundary conditions are uniquely determined. In theory, the fitting effect is better in the case of small vibrations. When the vibration amplitude is large, the damping and equivalent stiffness provided by the model to be tested cannot be ignored. At this time, the boundary conditions become: In the formula: c is the damping coefficient, k is the equivalent stiffness, both are unknown to be measured; At this point, the analytical calculation of the wind tunnel experiment vibration based on the vibration control equation is completed.

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