A Method for Analyzing Nonlinear Bending and Forced Vibration of a Post-Buckling Aircraft Fluid-Conveying Pipe

Abstract

The present invention discloses a method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid conveying pipe, belonging to the technical field of mechanical property analysis of aircraft fluid conveying pipes. The method includes: establishing a static-dynamic model of a fluid-induced post-buckling aircraft fluid conveying pipe; extending the two-step perturbation technique to the post-buckling equilibrium path to provide an initial configuration for bending and resonance analysis; according to two symmetric initial bifurcation paths, again using the two-step perturbation method to obtain an explicit relationship between the bending load and deflection; using a method combining the two-step perturbation method and the improved Lindstedt-Poincaré method to obtain an approximate analytical solution for strongly nonlinear forced vibration. The present invention can guide the nonlinear static-dynamic structural design of aircraft fluid conveying pipes under the combined action of internal fluid flow-induced deformation and external loads.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical property analysis of aircraft fluid conveying pipes, and particularly relates to a method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid conveying pipe. Background Art

[0002] In the modern aviation field, the improvement of aircraft performance has always been the goal pursued. As a key component in the aircraft system, the aircraft fluid conveying pipe undertakes the important task of conveying various important fluids, such as fuel, coolant, etc. However, due to the complex dynamic environment that the aircraft experiences during flight, the mechanical properties of the fluid conveying pipe face severe challenges.

[0003] Traditional analysis methods for aircraft fluid conveying pipes are mostly based on linear theory and are effective to a certain extent in dealing with fluid conveying pipe problems under general working conditions. On the one hand, there are internal flow-induced static and dynamic instability modes such as buckling and self-excited vibration under different support conditions. On the other hand, the combined action of flow-induced deformation and external loads on the pipe increases the complexity of the problem. For an aircraft fluid conveying pipe in the post-buckling state, its mechanical behavior exhibits strong nonlinear characteristics. Such nonlinear bending and forced vibration phenomena will not only affect the structural integrity of the fluid conveying pipe itself, but may also have a significant impact on the overall performance and safety of the aircraft.

[0004] Currently, although there is some research on aircraft fluid conveying pipes in the field of aviation engineering, there is a lack of analysis methods specifically for the nonlinear bending and forced vibration of post-buckling aircraft fluid conveying pipes. Existing analysis means are difficult to accurately describe and predict the behavior of post-buckling fluid conveying pipes in a complex mechanical environment, which to a certain extent restricts the optimal design of aircraft fluid conveying pipes and the reliability of aircraft operation. Considering problems such as high experimental difficulty and high simulation cost, it is necessary to establish a comprehensive aircraft fluid conveying pipe model and propose corresponding analysis methods to predict the nonlinear static and dynamic behaviors of aircraft fluid conveying pipes, which is of great significance for the static and dynamic stiffness design of aircraft fluid conveying pipes.

[0005] In summary, developing an effective method for analyzing the nonlinear bending and forced vibration of post-buckling aircraft fluid conveying pipes has become an urgent problem to be solved in the field of aviation engineering, providing technical support for the nonlinear static and dynamic structural design of aircraft fluid conveying pipes under the combined action of internal flow-induced deformation and external loads. Summary of the Invention

[0006] The purpose of the embodiment of the present invention is to provide a method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid conveying pipe, which determines the nonlinear bending behavior and strongly nonlinear forced vibration of a flow-induced post-buckling aircraft fluid conveying pipe, laying a foundation for the static and dynamic design of a flow-induced post-buckling aircraft fluid conveying pipe, and thus can solve at least one technical problem involved in the background art.

[0007] To solve the above technical problems, the present invention is implemented as follows:

[0008] An embodiment of the present invention provides a method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid-conveying pipe, including the following steps:

[0009] Step S1, establish a static and dynamic model of the fluid-induced post-buckling aircraft fluid-conveying pipe;

[0010] Step S2, use the second-order perturbation method-Galerkin method to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe and the nonlinear bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe;

[0011] Step S3, expand the second-order perturbation method and the improved Lindstedt-Poincaré method to obtain an approximate analytical solution of the main resonance amplitude-frequency bifurcation equation;

[0012] Step S4, construct a database of the key parameters of the aircraft fluid-conveying pipe, the nonlinear bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe, and the main resonance.

[0013] Optionally, in step S1, to establish a static and dynamic model of the fluid-induced post-buckling aircraft fluid-conveying pipe, specifically including:

[0014] Step S11, according to the structural characteristics of the aircraft fluid-conveying pipe, establish a post-buckling model of the aircraft fluid-conveying pipe, which is expressed by the following formula:

[0015]

[0016] Among them, Ξ i (i = 1, 2, 3, 4) is the stiffness coefficient, v is the internal flow velocity, m f is the generalized density, w * is the post-buckling lateral displacement, θ * is the post-buckling rotation angle, and x is the x-direction coordinate;

[0017] Step S12: Establish a nonlinear dynamic model of the aircraft fluid-conveying pipe considering the post-buckling configuration, which is expressed by the following formula:

[0018]

[0019] Among them, m f , ρ0, ρ1, ρ2, ρ3 are the generalized densities, q is the external load, is the lateral vibration displacement, is the vibration rotation angle, is the lateral acceleration, is the rotational acceleration, is the lateral vibration velocity;

[0020] Step S13: Establish a nonlinear bending model of the aircraft fluid-conveying pipe considering the post-buckling configuration, which is expressed by the following formula:

[0021]

[0022] Optionally, in step S2, the quadratic perturbation method - Galerkin method is used to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe and the nonlinear bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe, specifically including:

[0023] Step S21, using the quadratic perturbation method, propose the perturbation formats of the generalized displacement and the internal flow velocity to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe;

[0024] Step S22, based on the quadratic perturbation method, propose the perturbation formats of the lateral load and the generalized displacement, and combine the Galerkin integral method to obtain the nonlinear bending load-deflection relationship.

[0025] Optionally, in step S21, using the quadratic perturbation method, propose the perturbation formats of the generalized displacement and the internal flow velocity to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe, specifically including:

[0026] In order to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe, that is, the relationship between the flow velocity and the deflection, establish a perturbation scheme for the flow velocity and the generalized displacement:

[0027]

[0028] where ε is a small perturbation parameter without physical meaning; k is the k-th term in the perturbation expansion; (v 2 ) (k) , are the k-th order internal flow velocity and the generalized post-buckling displacement of the perturbation expansion respectively;

[0029] To satisfy the boundary conditions of simply supported at both ends, assume the first-order post-buckling displacements and are as follows:

[0030]

[0031] where, represents the dimensionless maximum post-buckling deflection;

[0032] The post-buckling equilibrium path of the aircraft fluid-conveying pipe is:

[0033]

[0034] where, (v 2 ) (0) represents the square of the dimensionless critical buckling flow velocity, which is expressed as:

[0035]

[0036] Optionally, in step S22, based on the second-order perturbation method, the perturbation format of the transverse load and the generalized displacement is proposed, and the nonlinear bending load-deflection relationship is obtained by combining the Galerkin integral method, specifically including:

[0037] In order to obtain the nonlinear bending load-deflection relationship, for the nonlinear bending control equations of the post-buckling aircraft fluid-conveying pipe, the perturbation format of the transverse load and the generalized displacement is proposed to carry out discrete solution, specifically as follows:

[0038]

[0039] In order to satisfy the boundary conditions, the trial function can be written as:

[0040]

[0041] The high-order asymptotic analytical solution of the load is obtained:

[0042]

[0043] Among them, the expressions of the coefficients g i (i = 1, 2, 3) are:

[0044]

[0045] The Galerkin integral method can be used to obtain the nonlinear bending load-deflection relationship, and the Galerkin integral method is:

[0046]

[0047] The specific expression of the nonlinear bending load-deflection relationship is:

[0048]

[0049] Optionally, in step S3, the second-order perturbation method and the improved Lindstedt-Poincaré method are extended to obtain an approximate analytical solution of the primary resonance amplitude-frequency bifurcation equation, specifically including:

[0050] Step S31, in the framework of the second-order perturbation method, the high-order asymptotic analytical solution of the load is obtained, which is expressed by the following formula:

[0051]

[0052] Among them, W m is the midpoint displacement of the aircraft fluid-conveying pipe, is the second-order derivative of the midpoint displacement with respect to time;

[0053] Step S32: Obtain an approximate analytical solution of the frequency-amplitude curve of the strongly nonlinear system by using the modified Lindstedt-Poincaré method. The approximate analytical solution of the frequency-amplitude curve is expressed by the following formula:

[0054]

[0055] where, Ω and ω L are the nonlinear frequency and the linear frequency respectively, is the amplitude of the external load; is the discriminant for judging whether the system exhibits hardening or softening spring characteristics. g1, g2, and g3 are the dynamic coefficients of each order.

[0056] Optionally, in step S31, the dynamic coefficients g i (i = 0, 1, 2, 3) are expressed by the following formula:

[0057]

[0058] Optionally, in step S4, construct a database of the key parameters of the aircraft fluid conveying pipe and the nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid conveying pipe, specifically including:

[0059] The key parameters of the aircraft fluid conveying pipe include the internal flow velocity, material parameters, and size parameters;

[0060] The nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid conveying pipe include the load-deflection relationship and the amplitude-frequency relationship.

[0061] Optionally, in step S1, the two ends of the aircraft fluid conveying pipe are immovable and simply supported. For the nonlinear bending analysis, the aircraft fluid conveying pipe is subjected to a uniformly distributed lateral static load; for the forced vibration analysis, the aircraft fluid conveying pipe is subjected to a lateral harmonic load; the aircraft fluid conveying pipe conveys an incompressible, inviscid, and steady constant flow inside; where:

[0062] The form of the uniformly distributed lateral static load is q(t) = q s , where, q s is a constant;

[0063] The form of the lateral harmonic excitation load is q(t) = q d cos(ωt), where, q d is the load amplitude and ω is the excitation frequency.

[0064] Compared with the related technology, the beneficial effects of the present invention are as follows:

[0065] The method proposed by the present invention precisely establishes the non - linear bending and forced vibration models of aircraft fluid - conveying pipes in the post - buckling state, expands the quadratic perturbation method and the improved Lindstedt - Poincaré method, and effectively overcomes the problems of fluid - solid coupling, strong non - linearity, high experimental difficulty, and high simulation cost in the aircraft fluid - conveying pipe system in the post - buckling state. It has strong applicability in revealing the non - linear bending behavior and primary resonance behavior of fluid - induced post - buckling aircraft fluid - conveying pipes, and effectively guides the non - linear static and dynamic design of aircraft fluid - conveying pipes. For the above reasons, the present invention can be widely promoted in the fields of non - linear static and dynamic design of aircraft fluid - conveying pipes, etc. Description of the Drawings

[0066] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings, where:

[0067] Figure 1 is the flowchart of the method for analyzing the non - linear bending and forced vibration of a post - buckling aircraft fluid - conveying pipe provided by the present invention;

[0068] Figure 2 (a) and Figure 2 (b) are the structural schematic diagrams of the fluid - induced post - buckling aircraft fluid - conveying pipe provided by the present invention;

[0069] Figure 3 (a) and Figure 3 (b) are the bending load - deflection curves corresponding to different flow velocities provided by the present invention;

[0070] Figure 4 is the non - linear amplitude - frequency relationship diagram of the post - buckling aircraft fluid - conveying pipe provided by the present invention. Detailed Embodiments

[0071] The following will clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of them. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0072] Please refer to Figure 1 、 Figure 2 (a), Figure 2 (b) as shown. The embodiments of the present invention provide a method for analyzing the non - linear bending and forced vibration of a post - buckling aircraft fluid - conveying pipe, including the following steps:

[0073] Step S1: Establish the static and dynamic model of the fluid-induced post-buckling aircraft pipe conveying fluid;

[0074] Step S2: Use the second-order perturbation method - Galerkin method to obtain the post-buckling equilibrium path of the aircraft pipe conveying fluid and the nonlinear bending behavior of the fluid-induced post-buckling aircraft pipe conveying fluid;

[0075] Step S3: Expand the second-order perturbation method and the improved Lindstedt-Poincaré method to obtain the approximate analytical solution of the primary resonance amplitude-frequency bifurcation equation;

[0076] Step S4: Construct the key parameters of the aircraft pipe conveying fluid and the database of the nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft pipe conveying fluid.

[0077] In Step S1, both ends of the aircraft pipe conveying fluid are immovable and simply supported. For the nonlinear bending analysis, the aircraft pipe conveying fluid is subjected to a uniformly distributed static load in the transverse direction; for the forced vibration analysis, the aircraft pipe conveying fluid is subjected to a transverse harmonic load; the aircraft pipe conveying fluid conveys an incompressible, inviscid and stable steady flow inside.

[0078] The form of the uniformly distributed static load is q(t) = q s , where q s is a constant.

[0079] The form of the transverse harmonic excitation load is q(t) = q d cos(ωt), where q d is the load amplitude and ω is the excitation frequency.

[0080] Establish the static and dynamic model of the fluid-induced post-buckling aircraft pipe conveying fluid, specifically including:

[0081] Step S11, establish the post-buckling model of the aircraft pipe conveying fluid according to the structural characteristics of the aircraft pipe conveying fluid;

[0082] In a specific embodiment, considering the influence of the fluid-induced initial post-buckling deformation on the bending and vibration deformation of the aircraft pipe conveying fluid, the displacement field components also consist of two parts. The three mid-plane generalized displacements of the initial post-buckling are defined as u * (x), w * (x), θ * (x), and the additional bending or vibration displacements of the mid-plane are defined as u(x,t), w(x,t), θ(x,t). Therefore, the mid-plane displacement composed of these two processes can be written as

[0083] The post-buckling model of the aircraft pipe conveying fluid is expressed by the following formula:

[0084]

[0085] where Ξi (i = 1, 2, 3, 4) are stiffness coefficients, v is the internal flow velocity, m f is the generalized density, w * is the post - buckling lateral displacement, θ * is the post - buckling rotation angle, and x is the coordinate in the x - direction. The specific expressions are as follows:

[0086]

[0087] Among them, f and g are high - order displacement shape functions, which are defined as R1 and R2 are the outer diameter and inner diameter of the aircraft fluid - conveying pipe respectively; E is the elastic modulus of the fluid - conveying pipe, A is the area of the fluid - conveying pipe, and Z is the coordinate in the z - direction.

[0088] Step S12: Establish a nonlinear dynamic model of the aircraft fluid - conveying pipe considering the post - buckling configuration;

[0089] Based on the Hamilton variational principle, taking the post - buckling configuration as the initial loading configuration, a nonlinear static - dynamic model of the aircraft fluid - conveying pipe considering the post - buckling configuration is obtained.

[0090] The nonlinear dynamic model of the aircraft fluid - conveying pipe considering the post - buckling configuration is expressed by the following formula:

[0091]

[0092] Among them, m f , ρ0, ρ1, ρ2, ρ3 are generalized densities, q is the external load, is the lateral vibration displacement, is the vibration rotation angle, is the lateral acceleration, is the rotational acceleration, is the lateral vibration velocity.

[0093] In the nonlinear dynamic model, the load form of the lateral harmonic excitation is q(t)=q d cos(ωt), where q d is the load amplitude and ω is the excitation frequency.

[0094] Step S13: Establish a nonlinear bending model of the aircraft fluid - conveying pipe considering the post - buckling configuration;

[0095] Based on the principle of minimum potential energy, on the basis of considering the post - buckling configuration, a nonlinear bending model of the aircraft fluid - conveying pipe considering the post - buckling configuration is established.

[0096] The nonlinear bending model of the aircraft fluid - conveying pipe considering the post - buckling configuration is expressed by the following formula:

[0097]

[0098] In step S2, the secondary perturbation method - Galerkin method is used to obtain the post - buckling equilibrium path of the aircraft fluid - conveying pipe and the nonlinear bending behavior of the fluid - induced post - buckling aircraft fluid - conveying pipe, specifically including:

[0099] Step S21: Using the secondary perturbation method, a perturbation format of the generalized displacement and the internal flow velocity is proposed to obtain the post - buckling equilibrium path of the aircraft fluid - conveying pipe;

[0100] To obtain the post - buckling equilibrium path of the aircraft fluid - conveying pipe, that is, the relationship between the flow velocity and the deflection, a perturbation scheme for the flow velocity and the generalized displacement is established as follows:

[0101]

[0102] where ε is a small perturbation parameter without physical meaning; k is the k - th term in the perturbation expansion; (v 2 ) (k) , are the k - th order internal flow velocity and the generalized post - buckling displacement of the perturbation expansion, respectively.

[0103] To satisfy the boundary conditions of simply supported at both ends, assume the first - order post - buckling displacements and as follows:

[0104]

[0105] where represents the dimensionless maximum post - buckling deflection, is the amplitude of the first - order post - buckling transverse deflection, is the amplitude of the first - order post - buckling rotation angle; by gradually solving Equation (1) using Equations (6) - (7), the asymptotic solution of the aircraft fluid - conveying pipe can be obtained.

[0106] The post - buckling equilibrium path of the aircraft fluid - conveying pipe is:

[0107]

[0108] where (v 2 ) (0) represents the square of the dimensionless critical buckling flow velocity, which is expressed as:

[0109]

[0110] Step S22: Further based on the secondary perturbation method, a perturbation format of the transverse load and the generalized displacement is proposed, and the nonlinear bending load - deflection relationship is obtained by combining the Galerkin integral method;

[0111] To obtain the nonlinear bending load-deflection relationship, for the nonlinear bending control equations of the post-buckling aircraft fluid-conveying pipe, the perturbation formats of the transverse load and the generalized displacement are proposed for discrete solution, as follows:

[0112]

[0113] To satisfy the boundary conditions, the trial function can be written as:

[0114]

[0115] Substituting Eqs. (10)-(11) into Eq. (4), the high-order asymptotic analytical solution of the load is obtained:

[0116]

[0117] where the coefficients g i (i = 1, 2, 3) are expressed as:

[0118]

[0119] The Galerkin integration method is used for Eq. (12) to obtain the nonlinear bending load-deflection relationship, and the Galerkin integration method is:

[0120]

[0121] The specific expression of the nonlinear bending load-deflection relationship is:

[0122]

[0123] In step S3, the extended second-order perturbation method and the improved Lindstedt-Poincaré method are used to obtain the approximate analytical solution of the primary resonance amplitude-frequency bifurcation equation, specifically including:

[0124] Step S31: In the framework of the second-order perturbation method, the high-order asymptotic analytical solution of the load is obtained;

[0125] For the nonlinear dynamic control equations of the fluid-induced pre-buckling and post-buckling aircraft fluid-conveying pipes, the perturbation forms of the time-domain generalized displacement and the transverse load are shown as follows:

[0126]

[0127] where is introduced to delay the dynamic terms from the low-order perturbation equations to the high-order equations. Considering the boundary conditions, the trial functions of each order can be written as:

[0128]

[0129] Substitute Equation (17) into each perturbation equation and use t instead of The high-order asymptotic analytical solution of the load is obtained as follows:

[0130]

[0131] Similarly, by combining the Galerkin integration method, the high-order asymptotic analytical solution of the load can be obtained, and its specific form is:

[0132]

[0133]

[0134] Therefore, the high-order asymptotic analytical solution of the load in the time domain can be expressed as:

[0135]

[0136] Step S32: Adopt the modified Lindstedt-Poincaré method to obtain the approximate analytical solution of the frequency-amplitude curve of the strongly nonlinear system.

[0137] For undamped nonlinear forced vibration, the lowest-order mode is selected for primary resonance analysis. Equation (20) can be rewritten in the following form:

[0138]

[0139] The modified Lindstedt-Poincaré (MLP) method can effectively obtain the approximate analytical solution of the strongly nonlinear Duffing system. By combining Equation (21), the second-order approximate closed solutions of the two frequency-amplitude curves (in-phase and out-of-phase) can be obtained:

[0140]

[0141] where, Ω and ω L are the nonlinear frequency and the linear frequency respectively, is the amplitude of the external load, g1, g2, g3 are the dynamic coefficients of each order, is the discriminant to judge whether the system shows the characteristics of a hardening or softening spring, and also determines what parameter transformation to use in the MLP method.

[0142] In Step S4, the key parameters of the aircraft fluid conveying pipe and the nonlinear bending behavior and primary resonance database of the fluid-induced post-buckling aircraft fluid conveying pipe are constructed, specifically including:

[0143] The key parameters of the aircraft fluid conveying pipe include the internal flow velocity, material parameters, dimension parameters, etc.

[0144] The nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid-conveying pipe include the load-deflection relationship and the amplitude-frequency relationship.

[0145] The following takes Specific Embodiment 1 to detail a method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid-conveying pipe provided by the present invention.

[0146] Embodiment 1

[0147] In this Embodiment 1, the influence of the flow velocity on the nonlinear bending and forced vibration of the fluid-induced post-buckling aircraft fluid-conveying pipe is considered. The dimensional parameters are set as R i = 9 mm, R o = 10 mm, L = 40R o , the fluid density is ρ f = 1000 kg / m 3 . During forced vibration, the excitation amplitude is q0 = 3 N.

[0148] Embodiment 1 studies the influence of the flow velocity on the nonlinear bending and forced vibration of the fluid-induced post-buckling aircraft fluid-conveying pipe. Figure 3 (a) Analyzed the bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe at different flow velocities. Obviously, when the lateral load acts on the convex or concave surface of the post-buckled pipe, the results are very different. When the load acts on the concave surface, the bending stiffness exhibits non-linear hardening characteristics. In addition, as the flow velocity increases, the lateral deflection becomes larger and the bending stiffness becomes larger. On the other hand, when the load acts on the convex surface, a pole-type instability phenomenon may occur, and the pipe body exhibits bistable characteristics. It should be noted that these two steady-state positions are symmetric with the horizontal position, which is different from the prestress-free arch. Therefore, when the pipe reaches the second steady-state position and continues to be loaded, the load-deflection curve is consistent with the case of loading on the concave surface. Further obtained, as the flow velocity increases, the post-buckling deflection of the pipe increases, the initial stiffness increases, and the critical over-break point also increases.

[0149] Figure 4 For the amplitude-frequency characteristic curves of the aircraft fluid-conveying pipe at different flow velocities, it can be found that when the velocity is less than the critical buckling velocity, the curve exhibits hardening characteristics, that is, when the external excitation frequency sweeps from high frequency to low frequency, there is a saddle-node bifurcation point with an amplitude jump. On the contrary, when the pipe reaches the post-buckling state, the curve exhibits softening characteristics. As the velocity increases, the critical jump frequency first decreases and then increases, which is consistent with the change of the pipe material stiffness.

[0150] It can be concluded that the method proposed by the present invention accurately establishes the non-linear bending and forced vibration models of the aircraft fluid conveying pipe in the post-buckling state, expands the secondary perturbation method and the improved Lindstedt-Poincaré method, and effectively overcomes the problems of fluid-structure interaction, strong non-linearity, high experimental difficulty, and high simulation cost in the aircraft fluid conveying pipe system in the post-buckling state. It has strong applicability in revealing the non-linear bending behavior and primary resonance behavior of the fluid-induced post-buckling aircraft fluid conveying pipe, and effectively guides the non-linear static and dynamic design of the aircraft fluid conveying pipe system.

[0151] It should be noted that in this article, the term "including", "comprising" or any other variant thereof is intended to cover non-exclusive inclusion, so that a process, method, article or equipment including a series of elements not only includes those elements, but also includes other elements not expressly listed, or further includes elements inherent to such process, method, article or equipment. Without more limitations, an element defined by the statement "including a..." does not exclude the existence of additional identical elements in the process, method, article or equipment including that element. In addition, it should be pointed out that the scope of the methods and equipment in the embodiments of the present invention is not limited to performing functions in the order shown or discussed, and may also include performing functions in a substantially simultaneous manner or in the reverse order according to the functions involved. For example, the described methods may be performed in an order different from that described, and various steps may be added, omitted, or combined. Additionally, the features described with reference to certain examples may be combined in other examples.

[0152] The embodiments of the present invention have been described above in conjunction with the accompanying drawings. However, the present invention is not limited to the above specific embodiments. The above specific embodiments are merely illustrative and not restrictive. Under the inspiration of the present invention, those of ordinary skill in the art can also make many forms without departing from the spirit of the present invention and the scope protected by the claims, and all of them belong to the protection scope of the present invention.

Claims

1. A method for analyzing the nonlinear bending and forced vibration of a post-buckling aircraft fluid conveying pipe, characterized in that, It includes the following steps: Step S1, establish the static and dynamic model of the fluid-induced post-buckling aircraft fluid-conveying pipe, specifically including: Step S11, according to the structural characteristics of the aircraft fluid-conveying pipe, establish the post-buckling model of the aircraft fluid-conveying pipe, which is expressed by the following formula: Among them, is the stiffness coefficient, ; v is the internal flow velocity, is the generalized density, is the post-buckling transverse displacement, is the post-buckling rotation angle, is the x-direction coordinate; Step S12: Establish the nonlinear dynamic model of the aircraft fluid-conveying pipe considering the post-buckling configuration, which is expressed by the following formula: Among them, is the generalized density, q is the external load, is the lateral vibration displacement, is the lateral acceleration, is the rotational acceleration, is the lateral vibration velocity; Step S13: Establish the nonlinear bending model of the aircraft fluid-conveying pipe considering the post-buckling configuration, which is expressed by the following formula: ； Step S2, adopt the second-order perturbation method - Galerkin method to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe and the nonlinear bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe; Step S3, expand the second-order perturbation method and the improved Lindstedt-Poincaré method to obtain the approximate analytical solution of the primary resonance amplitude-frequency bifurcation equation, specifically including: Step S31, in the framework of the second-order perturbation method, obtain the high-order asymptotic analytical solution of the load, which is expressed by the following formula: In the formula, is the second derivative of the midpoint displacement with respect to time; Step S32: Adopt the modified Lindstedt-Poincaré method to obtain the approximate analytical solution of the frequency-amplitude curve of the strongly nonlinear system, and the approximate analytical solution of the frequency-amplitude curve is expressed by the following formula: wherein, and are the non-linear frequency and the linear frequency respectively, is the amplitude of the external load; is the discriminant for judging whether the system exhibits hardening or softening spring characteristics, , , , are the dynamic coefficients of each order; Step S4, construct the key parameters of the aircraft fluid-conveying pipe and the database of the nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid-conveying pipe.

2. The method according to claim 1, wherein In step S2, the second-order perturbation method - Galerkin method is adopted to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe and the nonlinear bending behavior of the fluid-induced post-buckling aircraft fluid-conveying pipe, specifically including: Step S21, adopt the second-order perturbation method, propose the perturbation format of the generalized displacement and the internal flow velocity, and obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe; Step S22, based on the second-order perturbation method, propose the perturbation format of the lateral load and the generalized displacement, and combine the Galerkin integral method to obtain the nonlinear bending load-deflection relationship.

3. The method according to claim 2, characterized in that, In step S21, the second-order perturbation method is adopted, the perturbation format of the generalized displacement and the internal flow velocity is proposed, and the post-buckling equilibrium path of the aircraft fluid-conveying pipe is obtained, specifically including: In order to obtain the post-buckling equilibrium path of the aircraft fluid-conveying pipe, that is, the relationship between the flow velocity and the deflection, establish a perturbation scheme for the flow velocity and the generalized displacement: Among them, is a small perturbation parameter without physical meaning; k is the k -th term in the perturbation expansion; are respectively the k -th order internal flow velocity, generalized lateral post-buckling displacement, and post-buckling rotation angle of the perturbation expansion; To satisfy the boundary conditions of simply supported at both ends, assume the first-order post-buckling displacements and as follows: Among them, represents the non-dimensionalized maximum post-buckling deflection, is the amplitude of the first-order post-buckling lateral deflection, is the amplitude of the first-order post-buckling rotation angle; The post-buckling equilibrium path of the aircraft fluid-conveying pipe is: Among them, represents the square of the dimensionless critical buckling flow velocity and is expressed as follows: 。 4. The method according to claim 3, wherein In step S22, based on the second-order perturbation method, the perturbation format of the lateral load and the generalized displacement is proposed, and the nonlinear bending load-deflection relationship is obtained by combining the Galerkin integral method, specifically including: In order to obtain the nonlinear bending load-deflection relationship, for the nonlinear bending control equations of the post-buckling aircraft fluid-conveying pipe, the perturbation format of the lateral load and the generalized displacement is proposed to carry out discrete solution, specifically as follows: In order to satisfy the boundary conditions, the trial function is written as: Obtain the high-order asymptotic analytical solution of the load: Among them, the coefficients of each order , , , The expression is: The Galerkin integral method can be used to obtain the nonlinear bending load-deflection relationship, and the Galerkin integral method is: The nonlinear bending load-deflection relationship is specifically expressed as: 。 5. The method according to claim 1, characterized in that In step S4, construct the key parameters of the aircraft fluid-conveying pipe and the database of the nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid-conveying pipe, specifically including: The key parameters of the aircraft fluid-conveying pipe include the internal flow velocity, material parameters, and size parameters; The nonlinear bending behavior and primary resonance of the fluid-induced post-buckling aircraft fluid conveying pipe include the load-deflection relationship and the amplitude-frequency relationship.

6. The method according to claim 1, characterized in that, In step S1, both ends of the aircraft fluid conveying pipe are immovable and simply supported. For the nonlinear bending analysis, the aircraft fluid conveying pipe is subjected to a uniformly distributed static load in the transverse direction; for the forced vibration analysis, the aircraft fluid conveying pipe is subjected to a harmonic load in the transverse direction; an incompressible, inviscid and steady constant flow is conveyed inside the aircraft fluid conveying pipe; where: The form of the horizontally uniformly distributed static load is q ( t ) = qs , where qs is a constant; The load situation of the lateral harmonic excitation is , where is the load amplitude, is the excitation frequency.

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