A Nonlinear Dynamics Analysis Method for the Fluid Conduit of an Aircraft

Abstract

The present invention discloses a method for nonlinear dynamics analysis of an aircraft fluid conveying pipe, belonging to the technical field of aircraft fluid conveying pipe dynamics. According to the Hamilton variational principle, the control equations of the stability and nonlinear dynamics model are derived in the present invention. The quadratic perturbation method is extended to the nonlinear analysis of the aircraft fluid conveying pipe, a perturbation scheme is proposed, and by solving the nonlinear dynamics model, the initial deflection of the aircraft fluid conveying pipe is calculated. Based on the obtained relationship between the initial deflection and flow velocity of the aircraft fluid conveying pipe, approximate solutions of the natural frequency and the nonlinear frequency-amplitude relationship of the aircraft fluid conveying pipe are calculated. Finally, the stability and nonlinear dynamics analysis of the aircraft fluid conveying pipe are carried out, and a nonlinear dynamics response database of the aircraft fluid conveying pipe is established. The analysis method provided by the present invention has high calculation accuracy, comprehensively considers the nonlinear effects of the aircraft fluid conveying pipe, and is of great significance in the nonlinear dynamics analysis of the aircraft fluid conveying pipe.

Description

Technical Field

[0001] This application belongs to the technical field of aircraft fluid transfer pipe dynamics, and specifically relates to a method for nonlinear dynamics analysis of aircraft fluid transfer pipes. Background Technique

[0002] In the modern aviation field, the aircraft fluid transfer pipe system plays a crucial role. These carefully designed aircraft fluid transfer pipes undertake the important task of transporting various vital fluids, including fuel, hydraulic oil, coolant and other key fluids, and are the core elements to ensure the normal operation of each subsystem of the aircraft. With the continuous development of aviation technology, the performance requirements of aircraft are increasing day by day. In the process of pursuing higher speed, stronger maneuverability and lower energy consumption for the new generation of aircraft, the aircraft fluid transfer pipe system has become more complex and precise. However, in actual operation, the aircraft fluid transfer pipes face a variety of complex mechanical environments and working conditions. Factors such as strong airflow impact, huge inertial forces generated when the aircraft performs difficult maneuvering actions, and drastic fluctuations in temperature and pressure pose a severe test to the normal operation of the aircraft fluid transfer pipes.

[0003] Traditional linear dynamics analysis methods have certain limitations in dealing with aircraft fluid transfer pipe problems. Linear models usually describe the behavior of the system based on a series of simplified assumptions and linear relationships, and often cannot accurately describe the dynamic behavior of aircraft fluid transfer pipes in complex situations such as large deformations, strong nonlinear vibrations and fluid-structure interactions. When the aircraft fluid transfer pipe undergoes nonlinear vibrations, the linear analysis method may not be able to predict key parameters such as the amplitude, frequency and mode of the vibration, thus affecting the assessment of the structural reliability and safety of the aircraft fluid transfer pipe.

[0004] Based on this, it is necessary to provide a method for nonlinear dynamics analysis of aircraft fluid transfer pipes to provide technical support for accurately analyzing the dynamic characteristics of aircraft fluid transfer pipes and ensuring the safe and reliable operation of aircraft. Summary of the Invention

[0005] The present invention provides a method for nonlinear dynamics analysis of aircraft fluid transfer pipes, which can determine the flow velocity - initial deflection relationship and provide an initial research configuration for the nonlinear dynamics analysis of aircraft fluid transfer pipes. On this basis, the flow velocity - natural frequency curve and the nonlinear amplitude-frequency curve are determined, laying a foundation for the structural design of aircraft fluid transfer pipes, and thus can solve at least one of the technical problems involved in the background technique.

[0006] To solve the above technical problems, the present invention is implemented as follows:

[0007] A method for nonlinear dynamics analysis of aircraft fluid transfer pipes includes the following steps:

[0008] Step S1, construct a structural model based on the structural parameters of the aircraft fluid conveyance pipe, conduct an internal source excitation and external load analysis on the structural model, and construct a displacement field model, a geometric relationship model, and a constitutive relationship model of the aircraft fluid conveyance pipe;

[0009] Step S2, construct a relationship formula for the strain energy, kinetic energy, and external work of the aircraft fluid conveyance pipe based on the Hamilton variational principle, and substitute the displacement field model, constitutive model, and nonlinear geometric relationship model constructed in Step S1 into the expression to obtain a nonlinear dynamic model of the aircraft fluid conveyance pipe for transmitting supercritical high-speed pulsating flow represented by generalized displacements;

[0010] Step S3, use the second-order perturbation method to solve the nonlinear dynamic model, obtain the second-order perturbation format of the generalized displacement and flow velocity of the aircraft fluid conveyance pipe, and discretely solve the nonlinear stability model with the second-order perturbation format of the generalized displacement and flow velocity of the aircraft fluid conveyance pipe to obtain a relationship formula between the flow velocity and the initial deflection of the aircraft fluid conveyance pipe;

[0011] Step S4, use the second-order perturbation method to propose the second-order perturbation format of the external load and the generalized displacement of the aircraft fluid conveyance pipe under the external load, discretize the nonlinear dynamic model of the aircraft fluid conveyance pipe, and discretely solve the nonlinear dynamic model with the second-order perturbation format of the external load and the generalized displacement of the aircraft fluid conveyance pipe under the external load to obtain an expression of the relationship between the load and displacement of the aircraft fluid conveyance pipe, and then further use the harmonic balance method to solve the nonlinear dynamic model to obtain approximate solutions of the natural frequency and nonlinear frequency;

[0012] Step S5, conduct a stability and nonlinear dynamics analysis of the aircraft fluid conveyance pipe, and establish a nonlinear dynamic response database of the aircraft fluid conveyance pipe.

[0013] As a preferred improvement, the structural parameters of the aircraft fluid conveyance pipe include: inner diameter, outer diameter, and length; in the structural model, both ends of the aircraft fluid conveyance pipe are modeled as simply supported states, and the interior is hollow for fluid conveyance.

[0014] As a preferred improvement, the displacement field model is used to describe the bending deformation of the aircraft fluid conveyance pipe and is selected from one of the Euler-Bernoulli beam theory, Timoshenko first-order shear beam theory, and Zhang-Fu deformed beam theory; the geometric relationship model is a nonlinear strain-displacement relationship considering the initial deflection under the assumption of large deformation and small strain.

[0015] As a preferred improvement, the displacement field model is selected as the Zhang-Fu deformed beam theory and is expressed as:

[0016]

[0017] In the formula, and respectively represent the displacements of the aircraft's fluid conveying pipe in the x and z directions. Here, the x and z directions respectively represent the directions indicated by the x-axis and z-axis in the Cartesian coordinate system with the center of the cross-section of the aircraft's fluid conveying pipe as the coordinate origin, the axis of the aircraft's fluid conveying pipe as the x-axis, the axis perpendicular to the ground as the z-axis, and the axis perpendicular to both the x-axis and z-axis as the y-axis; u x0 and u z0 represent the displacements of each point on the middle plane (z = 0 plane) of the aircraft's fluid conveying pipe in the x and z directions; θ0 represents the rotation angle of the middle plane, expressed by the angle between the tangent of the bent cross-section of the aircraft's fluid conveying pipe in the middle plane and the positive z-axis; f and g represent high-order shape functions; represents the partial derivative calculation;

[0018] Among them:

[0019] u x0 = u x * + u x ; u z0 = u z * + u z ; and θ0 = θ * + θ;

[0020] In the formula, u x * 、u z * and θ * all represent the initial deflections caused by the initial flow; u x and u z respectively represent the displacements of the points on the aircraft's fluid conveying pipe in the x and z directions under the action of vibration; θ represents the rotation angle of the middle plane under the action of vibration;

[0021]

[0022] In the formula, R0, R i respectively represent the outer diameter and inner diameter of the aircraft's fluid conveying pipe; r represents the pipe radius at the coordinate point (x, y, z) on the aircraft's fluid conveying pipe,

[0023] The geometric relationship model is expressed as:

[0024]

[0025] In the formula, ε xx represents the generalized plane normal strain; γ xy and γ zx respectively represent the shear strain in the xy plane and the shear strain in the xz plane;

[0026] The constitutive relation model is expressed as:

[0027]

[0028]

[0029] In the formula, σ xx , τ xy , τ zx respectively represent the normal stress in the x - direction, the shear stress in the xy - direction, and the shear stress in the xz - direction; E and ν respectively represent the elastic modulus and Poisson's ratio of the fluid - conveying pipe of the aircraft.

[0030] As a preferred improvement, Hamilton's variational principle is expressed as follows:

[0031]

[0032] In the formula, t1 and t2 respectively represent the upper and lower limits of the integral of the time domain t; δ represents variational calculation; δK P and K P respectively represent the virtual kinetic energy and kinetic energy of the fluid - conveying pipe of the aircraft; δK f and K f respectively represent the virtual kinetic energy and kinetic energy of the fluid in the fluid - conveying pipe of the aircraft; δU and U respectively represent the virtual strain energy and strain energy of the fluid - conveying pipe of the aircraft; δW and W respectively represent the virtual external work and external work, where:

[0033]

[0034] In the formula, Ω represents the region of the fluid - conveying pipe of the aircraft; ρ and ρ f respectively represent the density of the fluid - conveying pipe of the aircraft and the density of the fluid in the fluid - conveying pipe of the aircraft; v f represents the flow velocity of the fluid in the fluid - conveying pipe of the aircraft; A represents the cross - sectional area of the fluid - conveying pipe of the aircraft; A f represents the cross - sectional area of the fluid in the fluid - conveying pipe of the aircraft; q represents the external load, when the external load does not exist, q = 0; L represents the length of the fluid - conveying pipe of the aircraft;

[0035] Substitute the constitutive relation model and the non - linear geometric relation of the fluid - conveying pipe of the aircraft into the above formula to obtain the non - linear dynamic model of the fluid - conveying pipe of the aircraft, which is expressed as:

[0036]

[0037] In the formula, m f represents the fluid mass per unit length; P i represents the stiffness coefficient; ρ i represents the generalized density of the fluid - conveying pipe of the aircraft, i = 0, 1, 2, 3, 4; where:

[0038]

[0039] As a preferred improvement, the second-order perturbation format of the generalized displacement and flow velocity of the aircraft fluid conveyance pipe is:

[0040]

[0041]

[0042] In the formula, represents the small perturbation parameter; k represents the perturbation order; (v f 2 ) (k) , respectively represent the k-th order perturbation flow velocity, the mid-plane displacement, and the plane rotation angle;

[0043] To meet the requirements of the simply supported boundary, the deflection of the aircraft fluid conveyance pipe after the first-order buckling and the corresponding rotation angle θ1 * are expressed as:

[0044]

[0045] In the formula, represents the mid-plane displacement amplitude, represents the mid-plane rotation angle amplitude;

[0046] The asymptotic solution of the initial deflection of the aircraft fluid conveyance pipe is expressed as:

[0047]

[0048] In the formula, O(ε 4 ) represents the fourth-order truncation, and the flow velocity-initial deflection relationship of the aircraft fluid conveyance pipe is expressed as:

[0049]

[0050] Among them, v f represents the flow velocity of the fluid in the aircraft fluid conveyance pipe; v f (0) represents the critical flow velocity of the fluid in the aircraft fluid conveyance pipe; represents the maximum initial deflection of the aircraft fluid conveyance pipe; m f represents the mass of the fluid per unit length of the pipe ρ f represents the density of the fluid in the pipe, A f represents the fluid area in the pipe;

[0051] Among them:

[0052]

[0053] As a preferred improvement, the second-order perturbation format q of the external load is expressed as:

[0054]

[0055] In the formula, represents the dynamic term for delaying the high-order equation; q (k) represents the k-th order perturbation load;

[0056] The generalized displacement of the fluid conveying pipe of the aircraft under the action of the external load is expressed as:

[0057]

[0058]

[0059] In the formula, u z represents the displacement of the fluid conveying pipe of the aircraft under the action of the external load; θ represents the rotation angle of the fluid conveying pipe of the aircraft under the action of the external load; u zk and θ k respectively represent the displacement and rotation angle of the fluid conveying pipe of the aircraft under the k-th order perturbation load;

[0060] u z and the first-order form of θ are expressed as follows:

[0061] u z1 = A 10 sin(πx / L);

[0062] θ1 = B 10 cos(πx / L);

[0063] In the formula, A 10 and B 10 are respectively the amplitudes of u z1 and θ1;

[0064] Substitute the second-order perturbation format of the external load and the generalized displacement of the fluid conveying pipe of the aircraft under the action of the external load into the nonlinear dynamic model to obtain an approximate solution of the load of the fluid conveying pipe of the aircraft;

[0065] The approximate solution of the load of the fluid conveying pipe of the aircraft is expressed by the relationship between the load and the displacement, and the expression is as follows:

[0066]

[0067] Let the external load be: λ q = 0, then the nonlinear dynamic time-domain equation of the fluid conveying pipe of the aircraft is expressed as follows:

[0068]

[0069] In the formula, represent the dynamic coefficients of each order, i = 0, 1, 2, 3; u zm represent the lateral displacement amplitude of the fluid conveying pipe of the aircraft;

[0070] The approximate solution of the nonlinear frequency is expressed by the relationship between the nonlinear frequency and the amplitude, and the expression is:

[0071]

[0072] Then the natural frequency of the fluid conveying pipe of the aircraft is expressed as:

[0073]

[0074] As a preferred improvement, step S5 specifically includes the following process:

[0075] Step S51, numerically calculate the relationship between the initial deflection of the fluid conveying pipe of the aircraft and the fluid velocity in the fluid conveying pipe of the aircraft;

[0076] Step S52, numerically calculate the relationship between the natural frequency of the fluid conveying pipe of the aircraft and the fluid velocity; the analysis range of the relationship between the natural frequency and the fluid velocity is from the pre-buckling state - the critical buckling state - the post-buckling state;

[0077] Step S53, numerically calculate the relationship between the nonlinear frequency and the amplitude of the fluid conveying pipe of the aircraft;

[0078] Step S54, establish databases of the core variables of the fluid conveying pipe of the aircraft respectively related to stability and nonlinear dynamic response, where the core variables of the fluid conveying pipe of the aircraft include the material parameters, dimension parameters and boundary conditions of the fluid conveying pipe of the aircraft; the stability and nonlinear dynamic response of the fluid conveying pipe of the aircraft include the critical fluid velocity in the fluid conveying pipe of the aircraft, the relationship between the natural frequency of the fluid conveying pipe of the aircraft and the fluid velocity in the fluid conveying pipe of the aircraft, and the nonlinear frequency-amplitude relationship of the fluid conveying pipe of the aircraft.

[0079] Compared with the related technology, the beneficial effects of the present invention are as follows:

[0080] (1) The method proposed by the present invention can accurately carry out the nonlinear dynamics and stability modeling of the fluid conveying pipe of the aircraft, taking into account both the calculation accuracy and the solution efficiency;

[0081] (2) The method proposed by the present invention has strong applicability and can accurately obtain the natural frequency and the nonlinear frequency-amplitude relationship from the pre-buckling state - the critical buckling state - the post-buckling state, meeting the requirements of engineering applications;

[0082] (3) Through the analysis of core variables, the present invention establishes databases of core variables related to stability and nonlinear dynamic response respectively, and appropriate design parameters can be selected according to the actual engineering design requirements, which can be widely promoted in the fields such as the dynamics of fluid-conveying pipes of aircraft. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 FIG. is a schematic structural diagram of a fluid-conveying pipe of an aircraft provided by an embodiment of the present invention;

[0084] Figure 2 FIG. is a relationship curve between the initial deflection and flow velocity of a fluid-conveying pipe of an aircraft in Embodiment 1;

[0085] Figure 3 FIG. is the natural frequency curve of a fluid-conveying pipe of an aircraft in Embodiment 1;

[0086] Figure 4 FIG. is a nonlinear amplitude-frequency relationship diagram of a fluid-conveying pipe of an aircraft in Embodiment 1. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0087] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0088] Please refer to Figures 1-4 , the present invention provides a method for analyzing the nonlinear dynamics of a fluid-conveying pipe of an aircraft, including the following steps:

[0089] Step S1, constructing a structural model based on the structural parameters of the fluid-conveying pipe of the aircraft, performing a force analysis on the structural model, and constructing a displacement field model, a geometric relationship model, and a constitutive relationship model of the fluid-conveying pipe of the aircraft.

[0090] The structural parameters of the fluid-conveying pipe of the aircraft include: inner diameter, outer diameter, and length.

[0091] In the structural model, both ends of the fluid-conveying pipe of the aircraft are modeled as simply supported states, and the interior is hollow for fluid conveyance.

[0092] The displacement field model is used to describe the bending deformation of the fluid-conveying pipe of the aircraft, and is selected from one of the Euler-Bernoulli beam theory, the Timoshenko first-order shear beam theory, and the zhang-fu deformed beam theory. In this embodiment, the zhang-fu deformed beam theory is specifically selected, and the specific displacement field model is as follows:

[0093]

[0094] In the formula, and respectively represent the displacements of the aircraft's fluid conveying pipe in the x and z directions. Here, the x and z directions respectively represent the directions indicated by the x-axis and z-axis in the Cartesian coordinate system with the center of the cross-section of the aircraft's fluid conveying pipe as the coordinate origin, the axis of the aircraft's fluid conveying pipe as the x-axis, the axis perpendicular to the ground as the z-axis, and the axis perpendicular to both the x-axis and z-axis as the y-axis; u x0 and u z0 represent the displacements of each point on the middle plane (z = 0 plane) of the aircraft's fluid conveying pipe in the x and z directions; θ0 represents the rotation angle of the middle plane, expressed as the angle between the tangent of the curved cross-section of the aircraft's fluid conveying pipe on the middle plane and the positive z-axis; f and g represent high-order shape functions; represents the partial derivative calculation;

[0095] Where:

[0096] u x0 = u x * + u x ; u z0 = u z * + u z ; and θ0 = θ * + θ;

[0097] In the formula, u x * 、u z * and θ * all represent the initial deflections caused by the initial flow; u x and u z respectively represent the displacements of the points on the aircraft's fluid conveying pipe in the x and z directions under the action of vibration; θ represents the rotation angle of the middle plane under the action of vibration;

[0098]

[0099] In the formula, R0, R i respectively represent the outer diameter and inner diameter of the aircraft's fluid conveying pipe; r represents the pipe radius at the coordinate point (x, y, z) on the aircraft's fluid conveying pipe,

[0100] The geometric relationship model is a non-linear strain-displacement relationship considering the initial deflection under the assumption of large deformation and small strain, specifically expressed as:

[0101]

[0102] In the formula, ε xx represents the generalized plane normal strain; γ xy and γzx They are respectively represented as the shear strain in the xy plane and the shear strain in the xz plane.

[0103] The constitutive relation model is expressed as:

[0104]

[0105] In the formula, σ xx , τ xy , τ zx respectively represent the normal stress in the x direction, the shear stress in the xy direction, and the shear stress in the xz direction; E and ν respectively represent the elastic modulus and Poisson's ratio of the fluid-conveying pipe of the aircraft.

[0106] Step S2: Based on Hamilton's variational principle, construct the relationships of the strain energy, kinetic energy, and external work of the fluid-conveying pipe of the aircraft. Substitute the displacement field model, constitutive model, and nonlinear geometric relationship model constructed in Step S1 into the expression to obtain the nonlinear dynamic model of the fluid-conveying pipe of the aircraft expressed by the generalized displacement for transmitting supercritical high-speed pulsating flow.

[0107] Hamilton's variational principle is expressed as follows:

[0108]

[0109] In the formula, t1 and t2 respectively represent the upper and lower limits of integration of the time domain t; δ represents variational calculation; δK P and K P respectively represent the virtual kinetic energy and kinetic energy of the fluid-conveying pipe of the aircraft; δK f and K f respectively represent the virtual kinetic energy and kinetic energy of the fluid in the fluid-conveying pipe of the aircraft; δU and U respectively represent the virtual strain energy and strain energy of the fluid-conveying pipe of the aircraft; δW and W respectively represent the virtual external work and external work, where:

[0110]

[0111] In the formula, Ω represents the region of the fluid-conveying pipe of the aircraft; ρ and ρ f respectively represent the density of the fluid-conveying pipe of the aircraft and the density of the fluid in the fluid-conveying pipe of the aircraft; v f represents the flow velocity of the fluid in the fluid-conveying pipe of the aircraft; A represents the cross-sectional area of the fluid-conveying pipe of the aircraft; A f represents the cross-sectional area of the fluid in the fluid-conveying pipe of the aircraft; q represents the external load, and when the external load does not exist, q = 0; L represents the length of the fluid-conveying pipe of the aircraft;

[0112] Substitute the constitutive relation model and nonlinear geometric relation of the fluid-conveying pipe of the aircraft into the above formula, and the nonlinear dynamic model of the fluid-conveying pipe of the aircraft can be obtained, which is expressed as:

[0113]

[0114] where m f represents the fluid mass per unit length; P i represents the stiffness coefficient; ρ i represents the generalized density of the fluid-conveying pipe of the aircraft, i = 0, 1, 2, 3, 4; where:

[0115]

[0116] Step S3, the quadratic perturbation method is used to solve the non-linear dynamic model to obtain the quadratic perturbation format of the generalized displacement and flow velocity of the fluid-conveying pipe of the aircraft, and the non-linear stability model is discretely solved with the quadratic perturbation format of the generalized displacement and flow velocity of the fluid-conveying pipe of the aircraft to obtain the relationship between the flow velocity and the initial deflection of the fluid-conveying pipe of the aircraft.

[0117] The quadratic perturbation format of the generalized displacement and flow velocity is:

[0118]

[0119] where represents the small perturbation parameter; k represents the perturbation order; (v f 2 ) (k) , respectively represent the k-th order perturbation flow velocity, the mid-plane displacement and the plane rotation angle;

[0120] To meet the requirements of the simply supported boundary, the deflection after the first buckling of the fluid-conveying pipe of the aircraft and the corresponding rotation angle θ1 * are expressed as:

[0121]

[0122] where represents the mid-plane displacement amplitude, represents the mid-plane rotation angle amplitude

[0123] The asymptotic solution of the initial deflection of the fluid-conveying pipe of the aircraft is expressed as:

[0124]

[0125] where O(ε 4 ) represents the fourth-order truncation, and the relationship between the flow velocity and the initial deflection of the fluid-conveying pipe of the aircraft is expressed as:

[0126]

[0127] where v f represents the flow velocity of the fluid in the fluid-conveying pipe of the aircraft; v f (0)Represents the critical flow velocity of the fluid in the fluid conveyance pipe of the aircraft; Represents the maximum initial deflection of the fluid conveyance pipe of the aircraft; m f Represents the mass of the fluid per unit length of the pipe ρ f Represents the density of the fluid in the pipe, A f Represents the fluid region in the pipe.

[0128] Where:

[0129]

[0130] Step S4, using the second-order perturbation method to propose the second-order perturbation format of the external load and the generalized displacement of the fluid conveyance pipe of the aircraft under the action of the external load, discretizing the nonlinear dynamic model of the fluid conveyance pipe of the aircraft, and discretely solving the nonlinear dynamic model in the second-order perturbation format of the external load and the generalized displacement of the fluid conveyance pipe of the aircraft under the action of the external load to obtain the expression of the relationship between the load and displacement of the fluid conveyance pipe of the aircraft, and then further using the harmonic balance method to solve the nonlinear dynamic model to obtain the approximate solutions of the natural frequency and the nonlinear frequency.

[0131] The second-order perturbation format q of the external load is expressed as:

[0132]

[0133] In the formula, Represents the dynamic term for delaying the high-order equation; q (k) Represents the k-th order perturbation load;

[0134] The generalized displacement of the fluid conveyance pipe of the aircraft under the action of the external load is expressed as:

[0135]

[0136] In the formula, u z Represents the displacement of the fluid conveyance pipe of the aircraft under the action of the external load; θ represents the rotation angle of the fluid conveyance pipe of the aircraft under the action of the external load; u zk and θ k respectively represent the displacement and rotation angle of the fluid conveyance pipe of the aircraft under the k-th order perturbation load;

[0137] u z and the first-order form of θ are expressed as follows:

[0138] u z1 =A 10 sin(πx / L);

[0139] θ1 = B 10 cos(πx / L);

[0140] In the formula, A 10 and B10 They are the amplitudes of u z1 and θ1 respectively.

[0141] Substitute the external load and the second-order perturbation format of the generalized displacement of the fluid-conveying pipe of the aircraft under the external load into the non-linear dynamic model to obtain an approximate solution of the load of the fluid-conveying pipe of the aircraft.

[0142] The approximate solution of the load of the fluid-conveying pipe of the aircraft is expressed by the relationship between the load and the displacement, and the expression is as follows:

[0143]

[0144] Let the external load be: λ q = 0, then the non-linear dynamic time-domain equation of the fluid-conveying pipe of the aircraft is expressed as follows:

[0145]

[0146] In the formula, represents each order of dynamic coefficient, i = 0, 1, 2, 3; u zm represents the lateral displacement amplitude of the fluid-conveying pipe of the aircraft.

[0147] The approximate solution of the non-linear frequency is expressed by the relationship between the non-linear frequency and the amplitude, and the expression is:

[0148]

[0149] Then the natural frequency of the fluid-conveying pipe of the aircraft is expressed as:

[0150]

[0151] Step S5: Conduct the stability and non-linear dynamics analysis of the fluid-conveying pipe of the aircraft, and establish a non-linear dynamics response database of the fluid-conveying pipe of the aircraft.

[0152] Step S5 specifically includes the following process:

[0153] Step S51: Numerically calculate the relationship between the initial deflection of the fluid-conveying pipe of the aircraft and the fluid velocity in the fluid-conveying pipe of the aircraft;

[0154] Step S52: Numerically calculate the relationship between the natural frequency of the fluid-conveying pipe of the aircraft and the fluid velocity; the analysis range of the relationship between the natural frequency and the fluid velocity is from the pre-buckling state - the critical buckling state - the post-buckling state;

[0155] Step S53: Numerically calculate the relationship between the non-linear frequency of the fluid-conveying pipe of the aircraft and the amplitude;

[0156] Step S54: Establish the core variables of the aircraft's fluid conveyance pipe respectively with the stability and nonlinear dynamics response database, where the core variables of the aircraft's fluid conveyance pipe include the material parameters, dimension parameters, and boundary conditions of the aircraft's fluid conveyance pipe; the stability and nonlinear dynamics response of the aircraft's fluid conveyance pipe include the critical flow velocity of the fluid inside the aircraft's fluid conveyance pipe, the relationship between the natural frequency of the aircraft's fluid conveyance pipe and the flow velocity of the fluid inside the aircraft's fluid conveyance pipe, and the nonlinear frequency-amplitude relationship of the aircraft's fluid conveyance pipe.

[0157] Embodiment 1

[0158] In this embodiment, the structural parameters of the aircraft's fluid conveyance pipe are as follows: inner diameter R i = 0.09 m, outer diameter R o = 0.1 m, length L = 40Ro = 4 m. The post-buckling, natural frequency, and frequency-amplitude relationship of the aircraft's fluid conveyance pipe are analyzed by using the nonlinear dynamics analysis method of an aircraft's fluid conveyance pipe provided by the present invention.

[0159] The analysis results are as Figures 2-4 shown, where Figure 2 represents the curve between the initial deflection and the flow velocity of the aircraft's fluid conveyance pipe; Figure 3 is the natural frequency curve of the aircraft's fluid conveyance pipe; Figure 4 is the nonlinear frequency-amplitude relationship diagram of the aircraft's fluid conveyance pipe. It can be seen from Figure 2 that the aircraft's fluid conveyance pipe will undergo bifurcation instability. It can be seen from Figure 3 that as the flow velocity increases, the natural frequency of the aircraft's fluid conveyance pipe first decreases and then increases. Since the vibration of the aircraft's fluid conveyance pipe (including linear and nonlinear vibrations) is based on the instability behavior induced by the initial flow, two-step analysis is required. It can be seen from Figure 4 that when the aircraft's fluid conveyance pipe is in the post-buckling state, the aircraft's fluid conveyance pipe will show softening characteristics. Therefore, it can be concluded that opposite conclusions may be obtained for linear and nonlinear vibrations. Thus, it can be concluded that the analysis method of the present invention is effective for the nonlinear dynamics of the aircraft's fluid conveyance pipe in the unstable state, and the key parameters of the aircraft's fluid conveyance pipe provided by the present invention and the stability and nonlinear dynamics response database have reference value for the nonlinear dynamics design of the aircraft's fluid conveyance pipe.

[0160] The embodiments of the present invention have been described above in conjunction with the accompanying drawings, but the present invention is not limited to the above specific embodiments. The above specific embodiments are merely illustrative and not restrictive. Those of ordinary skill in the art, under the inspiration of the present invention, without departing from the spirit and scope protected by the present invention and the claims, can also make many forms, all of which belong to the protection scope of the present invention.

Claims

1. A nonlinear dynamic analysis method for an aircraft fluid delivery pipe, characterized in that: The steps include: Step S1, constructing a structural model based on structural parameters of the aircraft fluid delivery pipe, performing endogenous excitation and external load analysis on the structural model, and constructing a displacement field model, a geometric relationship model, and a constitutive relationship model of the aircraft fluid delivery pipe; Step S2, constructing a relationship between strain energy, kinetic energy and external work of the aircraft fluid delivery pipe based on the Hamiltonian variational principle, substituting the displacement field model, constitutive model and nonlinear geometric relationship model constructed in step S1 into the expression to obtain a nonlinear dynamic model of the aircraft fluid delivery pipe transmitting supercritical high-speed pulsating flow represented by generalized displacement; Step S3, using a quadratic perturbation method to propose a quadratic perturbation format of the generalized displacement and flow velocity of the aircraft fluid delivery pipe, and using the quadratic perturbation format of the generalized displacement and flow velocity of the aircraft fluid delivery pipe to perform a discrete solution on the nonlinear dynamic model, and obtain a relationship between the flow velocity and the initial deflection of the aircraft fluid delivery pipe; Step S4, using a quadratic perturbation method to propose a quadratic perturbation format of the external load and the generalized displacement of the aircraft fluid delivery pipe under the external load, performing a discrete solution to the nonlinear dynamic model, obtaining an expression of the relationship between the load and the displacement of the aircraft fluid delivery pipe, and then further using a harmonic balance method to solve the nonlinear dynamic model to obtain an approximate solution of the natural frequency and the nonlinear frequency; Step S5, conducting stability and nonlinear dynamics analysis of the aircraft fluid delivery pipe, and establishing a nonlinear dynamic response database of the aircraft fluid delivery pipe; The displacement field model is used to describe the bending deformation of the aircraft fluid delivery pipe, and the Zhang-Fu deformation beam theory is selected, which is expressed as: ； In the formula, and They represent the flow pipes of the aircraft in x and z The displacement in the direction, where , The directions are respectively represented by the center of the cross section of the aircraft fluid delivery pipe as the coordinate origin and the axis of the aircraft fluid delivery pipe as the coordinate origin. Axis, with the axis perpendicular to the ground as axis, perpendicular to Axis and The axis of the shaft is The Cartesian coordinate system of the axis axis, The direction indicated by the axis; and Indicates that each point on the plane of the aircraft flow pipe is x Direction and z Displacement in direction; The rotation angle of the midplane is expressed by the angle between the tangent line of the curved section of the aircraft flow pipe in the midplane and the normal z The angle between the axes is expressed as; f and g represents a higher-order shape function; represents partial derivative calculation; in: ; ;and ; In the formula, , and Both represent the initial deflection caused by the initial flow; and They represent the points on the aircraft flow pipe under vibration. x and z Displacement in direction; represents the rotation angle of the midplane under vibration; ； In the formula, , represent the outer diameter and inner diameter of the aircraft fluid delivery pipe respectively; r Indicates the coordinate point on the aircraft flow pipe ( x , y , z ), ; The geometric relationship model is based on the assumption of large deformation and small strain, considering the nonlinear strain-displacement relationship of the initial deflection. The geometric relationship model is expressed as: ； In the formula, represents the generalized plane normal strain; and Respectively expressed as xy The shear strain in the plane and xz Shear strain in the plane; The constitutive relationship model is expressed as: ； In the formula, They represent the normal stress in the x direction, the shear stress in the xy direction, and the shear stress in the xz direction respectively; , represent the elastic modulus and Poisson’s ratio of the aircraft fluid delivery pipe respectively; The Hamiltonian variational principle is expressed as follows: ； In the formula, , Represents the time domain t The upper and lower limits of the integral; represents variational calculation; and denote the virtual kinetic energy and kinetic energy of the aircraft flow pipe respectively; and They represent the virtual kinetic energy and kinetic energy of the fluid in the aircraft fluid delivery pipe respectively; and denote the virtual strain energy and strain energy of the aircraft fluid delivery pipe respectively; and represent the external virtual work and external work respectively, where: ； In the formula, represents the aircraft flow pipe area; and denote the density of the aircraft fluid delivery pipe and the density of the fluid in the aircraft fluid delivery pipe respectively; It indicates the flow rate of the fluid in the aircraft fluid delivery pipe; represents the cross-sectional area of ​​the aircraft fluid delivery pipe; represents the cross-sectional area of ​​the fluid in the aircraft fluid delivery pipe; represents external load. When external load does not exist, ; L Indicates the length of the aircraft fluid delivery pipe; Substituting the constitutive relationship model and nonlinear geometric relationship of the aircraft fluid delivery pipe into the above formula, the nonlinear dynamic model of the aircraft fluid delivery pipe is obtained, which is expressed as: ； In the formula, It represents the mass of fluid per unit length; represents the stiffness coefficient; represents the generalized density of the aircraft fluid pipe, ;in: ； 。 2. The nonlinear dynamic analysis method of an aircraft fluid delivery pipe according to claim 1, characterized in that: The structural parameters of the aircraft fluid delivery pipe include: inner diameter, outer diameter and length; in the structural model, both ends of the aircraft fluid delivery pipe are modeled as simple support states, and the interior is hollow for conveying fluid.

3. The nonlinear dynamic analysis method of an aircraft fluid delivery pipe according to claim 1, characterized in that: The quadratic perturbation format of the generalized displacement and flow velocity of the aircraft fluid pipe is: ； ； In the formula, represents the small perturbation parameter; k represents the perturbation series; Respectively represent k perturbation velocity, mid-plane displacement and plane rotation angle; In order to meet the requirements of simply supported boundaries, the deflection of the aircraft fluid delivery pipe after first-order buckling is and the corresponding turning angle It is expressed as: ； ； In the formula, represents the mid-plane displacement amplitude, represents the mid-plane rotation angle amplitude; The asymptotic solution of the initial deflection of the aircraft fluid delivery pipe is expressed as: ； In the formula, represents the fourth-order truncation, and the velocity-initial deflection relationship of the aircraft flow pipe is expressed as: ； in, It indicates the flow rate of the fluid in the aircraft fluid delivery pipe; It represents the critical flow velocity of the fluid in the aircraft fluid delivery pipe; It represents the maximum initial deflection of the aircraft fluid delivery pipe; Indicates the mass of fluid per unit length of the tube , represents the density of the fluid in the tube, represents the fluid area in the tube; in: ； 。 4. The nonlinear dynamic analysis method of an aircraft fluid delivery pipe according to claim 3, characterized in that: Secondary perturbation format for external loads It is expressed as: In the formula, represents the dynamic terms used to delay higher-order equations; Indicates k perturbation load; The generalized displacement of the aircraft fluid delivery pipe under the action of external load is expressed as: ； ； In the formula, It represents the displacement of the aircraft fluid delivery pipe under the external load; It represents the rotation angle of the aircraft fluid delivery pipe under the action of external load; and Respectively k Displacement and rotation angle of the aircraft fluid delivery pipe under the order perturbation load; and The first-order form of is as follows: ； In the formula, and They are and Amplitude of Substituting the quadratic perturbation format of the external load and the generalized displacement of the aircraft fluid delivery pipe under the external load into the nonlinear dynamic model to obtain an approximate solution of the load of the aircraft fluid delivery pipe; The approximate solution of the load on the aircraft fluid delivery pipe is expressed by the relationship between load and displacement, as shown below: ； Let the external load be: , then the nonlinear dynamic time domain equation of the aircraft fluid delivery pipe is expressed as follows: ； In the formula, represents the kinetic coefficients of each order, ; It represents the lateral displacement amplitude of the aircraft fluid delivery pipe; The approximate solution of the nonlinear frequency is expressed by the relationship between the nonlinear frequency and the amplitude, which is expressed as follows: ； Then the natural frequency of the aircraft flow pipe is expressed as: 。 5. The nonlinear dynamic analysis method of an aircraft fluid delivery pipe according to claim 4, characterized in that: Step S5 specifically includes the following process: Step S51, numerically calculating the relationship between the initial deflection of the aircraft fluid delivery pipe and the flow velocity of the fluid in the aircraft fluid delivery pipe; Step S52, numerically calculating the relationship between the natural frequency and the flow velocity of the aircraft fluid delivery pipe; the analysis range of the relationship between the natural frequency and the flow velocity is from the pre-buckling state-critical buckling state-post-buckling state; Step S53, numerically calculating the relationship between the nonlinear frequency and amplitude of the aircraft fluid delivery pipe; Step S54, establishing a database of core variables and stability and nonlinear dynamic response of the aircraft fluid pipe, wherein the core variables of the aircraft fluid pipe include material parameters, dimensional parameters and boundary conditions of the aircraft fluid pipe; the stability and nonlinear dynamic response of the aircraft fluid pipe include the critical flow velocity of the fluid in the aircraft fluid pipe, the relationship between the natural frequency of the aircraft fluid pipe and the flow velocity of the fluid in the aircraft fluid pipe, and the nonlinear frequency-amplitude relationship of the aircraft fluid pipe.

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