Method for analyzing stability of large-deformation cantilever pipe for conveying multiphase flow based on reduced-order model

Abstract

The invention discloses a stability analysis method of a large-deformation cantilever pipe for conveying multiphase flow based on a reduced-order model, and relates to the technical field of fluid-structure interaction dynamics and structural stability analys.The method comprises the steps that a reduced-order multiphase flow cantilever pipe motion differential equation is established, and boundary conditions are determined; introducing multiphase flow parameters; carrying out dimensionless processing on the multiphase flow cantilever pipeline motion differential equation; deriving a characteristic equation, a characteristic function and a corresponding characteristic value under a cantilever boundary condition, and defining an integral transformation pair; the original partial differential equation is converted into an ordinary differential equation set, matrix solving is conducted on the ordinary differential equation set, and pipeline stability analysis parameters are obtained. According to the method, the calculation efficiency and the numerical stability are improved while main dynamic characteristics are kept; the multiphase flow is introduced, so that the influence of the multiphase flow on the pipeline stability is described more accurately; and introducing dimensionless parameters to obtain a stability analysis result of the pipeline system. The safety and reliability of the pipeline system under different working conditions can be effectively evaluated.

Description

Technical Field

[0001] This invention relates to the field of fluid-structure interaction dynamics and structural stability analysis technology, and in particular to a stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model. Background Technology

[0002] In the field of oil and gas engineering, pipeline systems are a crucial component of the production process, and their stability significantly impacts the safe and efficient transportation of oil and gas. During transportation, complex interactions exist between the internal fluid and the pipe wall, easily leading to dynamic problems such as fluid-structure interaction (FSI). When multiphase flow is transported within the pipe, the differences in density and velocity among the phases further enhance FSI, resulting in a more complex system vibration response. Long-term multiphase flow-induced vibration can not only cause fatigue damage and crack propagation in the pipeline but may also lead to flutter instability in the entire system, and in severe cases, even equipment failure or leakage accidents. Therefore, accurately calculating and analyzing the dynamic response and stability of pipelines under complex multiphase flow conditions is a fundamental task for predicting critical instability conditions of the transport structure, optimizing pipeline design parameters, and ensuring the safe operation of oil and gas transportation systems, possessing significant theoretical and engineering application value.

[0003] Current methods for analyzing the vibration characteristics of multiphase flow pipelines are mostly based on the Euler-Bernoulli beam theory, establishing fourth-order partial differential equations of motion. The main analytical parameter is the lateral displacement of the pipeline. This model can accurately describe the structural dynamic response under small deflection conditions. However, the fourth-order model suffers from high computational complexity and poor convergence when solving complex boundary conditions. Therefore, there is an urgent need for an efficient and accurate computational analysis method for the dynamic response and stability analysis of multiphase flow transport systems. Summary of the Invention

[0004] To address the aforementioned issues, this invention discloses a stability analysis method for a large deformation cantilever pipe conveying multiphase flow based on a reduced-order model. This method, grounded in Hamilton's principle and the Euler-Bernoulli beam model, reduces the original fourth-order partial differential equation to a second-order form and changes the pipe's analysis parameters from vibration displacement to rotation angle. This significantly simplifies the mathematical model while preserving the main dynamic characteristics. The reduced-order equation form is more suitable for analytical derivation and parameter calculation using the Generalized Integral Transform (GITT) method, improving computational efficiency and numerical stability.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model includes the following steps:

[0007] S1. Establish the reduced-order multiphase flow cantilever pipe motion differential equations and define the boundary conditions;

[0008] S2. Introduce multiphase flow parameters;

[0009] S3. Introduce dimensionless parameters to make the kinematic differential equations of the multiphase flow cantilever pipe dimensionless.

[0010] S4. Derive the characteristic equations, characteristic functions, and corresponding eigenvalues ​​under cantilever boundary conditions, and define integral transform pairs;

[0011] S5. Transform the original partial differential equations into a system of ordinary differential equations, and solve the system of ordinary differential equations in a matrix manner to obtain the pipeline stability analysis parameters.

[0012] Optionally, in step S1, based on Hamilton's principle and the Euler-Bernoulli beam model, the differential equations of motion and boundary conditions for the pipe are established, and the forces generated by each phase of the transported fluid are taken into account in the differential equations of motion, resulting in the following differential equations of motion and boundary conditions for the multiphase flow cantilever pipe: (1);

[0013] (2);

[0014] in, Mass per unit length of pipe The mass of each phase per unit length, Let the flow rate of each phase be , Represents the gas phase. Represents the liquid phase. The rotation angle of the pipe centerline, subscript These are the parameter values ​​corresponding to the end of the pipe. For a point whose position is described in deformed coordinates, For the length of the pipe, Let g be the bending stiffness of the pipe, and g be the acceleration due to gravity. Represents the derivative with respect to time. It represents the derivative with respect to space.

[0015] Optionally, in step S2, the following multiphase flow parameters are introduced:

[0016] (3);

[0017] in, This is the gas phase volumetric flow rate. The volumetric flow rate is the liquid phase. The flow area occupied by the gas phase. The flow area occupied by the liquid phase. This refers to the gas phase volume fraction. The gas content of the cross section, It is the slip factor;

[0018] The cantilever pipe is oriented downwards, and the liquid velocity is higher than the gas velocity. The slip factor should be selected as follows:

[0019] (4).

[0020] Optionally, step S3 specifically includes:

[0021] Introduce the following dimensionless parameters:

[0022] (5);

[0023] Substituting equation (5) into equation (1), we obtain the dimensionless form of the equation of motion as follows:

[0024] (6);

[0025] To further linearize, equation (6) introduces the Taylor expansion of trigonometric functions as follows:

[0026] (7);

[0027] (8);

[0028] (9);

[0029] in, Indicates all that satisfy Higher-order terms;

[0030] Assuming the internal fluid velocity is constant, substituting equations (7), (8), and (9) into equation (6) yields the dimensionless differential equation of motion as follows:

[0031] (10);

[0032] in, The positions of various points in the dimensionless pipeline. For dimensionless time, Let be the dimensionless flow velocity of each phase. This represents the ratio of the dimensionless fluid mass of each phase to the total mass. This is a dimensionless gravity parameter;

[0033] The corresponding boundary conditions and initial conditions are as follows:

[0034] (11);

[0035] (12);

[0036] Based on equations (3) and (4), the gas phase mass ratio and gas phase velocity are derived as follows:

[0037] (13);

[0038] (14);

[0039] in, The mass ratio of the gas phase. The density is the gas phase density. The liquid phase mass ratio The density of the liquid phase is... The dimensionless gas phase velocity, The value is the dimensionless liquid flow rate.

[0040] Optionally, in step S4, based on the basic transformation form of the generalized integral transform, the corresponding characteristic equation, eigenvalues, and characteristic functions are derived, specifically including:

[0041] Definition of solution The separated variable form is:

[0042] (15);

[0043] in, For space-dependent functions, A time-dependent function;

[0044] Substituting equation (15) into equation (10), we get: (16);

[0045] Further organized as follows:

[0046] (17);

[0047] Divide both sides of equation (17) by Write it in the following form:

[0048] (18);

[0049] Where, assuming To ensure that this equation has non-zero solutions, then Must meet <0, assuming ;

[0050] Based on equation (18), the characteristic equation of equation (10) is constructed as follows:

[0051] (19);

[0052] By solving the characteristic equation, we can obtain the characteristic function and characteristic roots that satisfy the given boundary conditions;

[0053] The general solution of equation (19) is as follows:

[0054] (20);

[0055] In this reduced-order model, the cantilever boundary conditions are as follows:

[0056] (twenty one);

[0057] Substituting equation (21) into equation (20), we obtain the following matrix:

[0058] (twenty two);

[0059] Seeking ,but The non-zero solutions are as follows:

[0060] (twenty three);

[0061] Equation (23) is the characteristic value of equation (10);

[0062] Characteristic function The orthogonal normalization conditions that need to be met are as follows:

[0063] (twenty four);

[0064] in, For the Kronecker delta function, when hour ,when hour ;

[0065] Based on equations (20), (23), and (24), the characteristic function of equation (10) is obtained as follows:

[0066] (25);

[0067] The integral transform pair for equation (10) is defined as follows:

[0068] (26);

[0069] (27).

[0070] Optionally, step S5 specifically includes:

[0071] Transforming a partial differential equation into a system of ordinary differential equations includes the following steps:

[0072] Multiply each term of equation (10) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term:

[0073] (28);

[0074] (29);

[0075] (30);

[0076] (31);

[0077] (32);

[0078] Adding equations (28) to (32), we obtain the following set of ordinary differential transformation equations:

[0079] (33);

[0080] (34);

[0081] In the formula, the coefficients of the system of ordinary differential equations as follows:

[0082] (35);

[0083] Equation (33) is the final form of the dimensionless differential equation of motion after transformation, which can be rewritten in matrix form as follows:

[0084] (36);

[0085] (37);

[0086] In the formula, matrices [A], [B], [D], and [F] represent the elements respectively. [E] indicates that it is made by ( A diagonal matrix consisting of numbers from 1 to the truncation number;

[0087] By solving equation (36), we obtain And the angle of the pipe can be obtained according to equation (27). Changes;

[0088] Further assume that the solution to equation (37) can be written in the following form:

[0089] (38);

[0090] in, It is a constant magnitude vector. It is the natural frequency;

[0091] Substituting equation (38) into equation (36), we get:

[0092] (39);

[0093] If equation (39) has a non-zero solution, then it must satisfy:

[0094] (40);

[0095] The natural frequency of the pipe is obtained by solving equation (40). The time history variation and stability of the pipeline are analyzed by comparing the obtained angle changes with the natural frequency.

[0096] The following assumptions are made in the above steps: (1) the fluid is incompressible; (2) the fluid velocity distribution is uniform; (3) the pipe characteristics are similar to those of an Euler-Bernoulli beam; (4) the motion is planar; (5) the pipe has large deflection and small strain, and rotational inertia and shear deformation are ignored; (6) the pipe centerline is not expandable.

[0097] The beneficial effects of this invention are that it simplifies the mathematical model structure by establishing a reduced-order differential equation of motion for a multiphase flow cantilever pipe based on Hamilton's principle and the Euler-Bernoulli beam model, while maintaining the main dynamic characteristics. This method comprehensively considers the effects of mass, density, and velocity differences of each phase, introducing parameters such as gas volume fraction, cross-sectional gas holdup, and slip factor, enabling the model to more realistically reflect the pipe dynamics under multiphase flow. By introducing dimensionless parameters and performing analytical solutions based on the Generalized Integral Transform (GITT) method, the calculation process is clear, numerically stable, and converges quickly, achieving high-precision prediction of stability parameters such as pipe rotation angle and critical velocity. This invention's method is not only applicable to vibration and stability analysis of riser systems but can also be extended to the dynamic response calculation and critical instability parameter evaluation of other multiphase flow transport structures, providing a reliable theoretical basis and technical support for vibration control and safety design of engineering structures. Attached Figure Description

[0098] Figure 1 This is a flowchart of the stability analysis method for a large deformation cantilever pipe for multiphase flow based on a reduced-order model according to the present invention.

[0099] Figure 2 This is a frequency distribution diagram of the first to fourth order of the transmission pipeline shown in an embodiment of the present invention;

[0100] Figure 3 This is a schematic diagram illustrating the change of the pipe end rotation angle over time, as shown in an embodiment of the present invention.

[0101] Figure 4 This is an envelope diagram shown in an embodiment of the present invention. Detailed Implementation

[0102] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0103] A stability analysis method for a large deformation cantilever pipe transporting multiphase flow based on a reduced-order model, such as... Figure 1 As shown, it includes the following steps:

[0104] S1. Based on Hamilton's principle and the Euler-Bernoulli beam model, the differential equations of motion and boundary conditions for the pipe are established. The forces generated by each phase of the transported fluid are taken into account in the differential equations of motion, resulting in the following differential equations of motion and boundary conditions for the multiphase flow cantilever pipe:

[0105] (1);

[0106] (2);

[0107] in, Mass per unit length of pipe The mass of each phase per unit length, Let the flow rate of each phase be , Represents the gas phase. Represents the liquid phase. The rotation angle of the pipe centerline, subscript These are the parameter values ​​corresponding to the end of the pipe. For a point whose position is described in deformed coordinates, For the length of the pipe, Let g be the bending stiffness of the pipe, and g be the acceleration due to gravity. Represents the derivative with respect to time. It represents the derivative with respect to space.

[0108] S2. Introduce multiphase flow parameters as follows:

[0109] (3);

[0110] in, This is the gas phase volumetric flow rate. The volumetric flow rate is the liquid phase. The flow area occupied by the gas phase. The flow area occupied by the liquid phase. This refers to the gas phase volume fraction. The gas content of the cross section, It is the slip factor;

[0111] Since the cantilever pipe in this method is oriented downwards, the liquid velocity is generally higher than the gas velocity. The slip factor is selected as follows:

[0112] (4).

[0113] S3. Introduce dimensionless parameters to make the differential equations of motion for the multiphase flow cantilever pipe dimensionless; specifically including:

[0114] Introduce the following dimensionless parameters:

[0115] (5);

[0116] Substituting equation (5) into equation (1), we obtain the dimensionless form of the equation of motion as follows:

[0117] (6);

[0118] To further linearize, equation (6) introduces the Taylor expansion of trigonometric functions as follows:

[0119] (7);

[0120] (8);

[0121] (9);

[0122] in, Indicates all that satisfy Higher-order terms;

[0123] Assuming the internal fluid velocity is constant, substituting equations (7), (8), and (9) into equation (6) yields the dimensionless differential equation of motion as follows:

[0124] (10);

[0125] in, The positions of various points in the dimensionless pipeline. For dimensionless time, Let be the dimensionless flow velocity of each phase. This represents the ratio of the dimensionless fluid mass of each phase to the total mass. This is a dimensionless gravity parameter;

[0126] The corresponding boundary conditions and initial conditions are as follows:

[0127] (11);

[0128] (12);

[0129] Based on equations (3) and (4), the gas phase mass ratio and gas phase velocity are derived as follows:

[0130] (13);

[0131] (14);

[0132] in, The mass ratio of the gas phase. The density is the gas phase density. The liquid phase mass ratio The density of the liquid phase is... The dimensionless gas phase velocity, The value is the dimensionless liquid flow rate.

[0133] S4. Based on the basic transformation form of the generalized integral transform, derive the corresponding characteristic equation, eigenvalues, and characteristic functions, specifically including:

[0134] Definition of solution The separated variable form is:

[0135] (15);

[0136] in, For space-dependent functions, A time-dependent function;

[0137] Substituting equation (15) into equation (10), we get: (16);

[0138] Further organized as follows:

[0139] (17);

[0140] Divide both sides of equation (17) by Write it in the following form:

[0141] (18);

[0142] Where, assuming To ensure that this equation has non-zero solutions, then Must meet <0, assuming ;

[0143] Based on equation (18), the characteristic equation of equation (10) is constructed as follows:

[0144] (19);

[0145] By solving the characteristic equation, we can obtain the characteristic function and characteristic roots that satisfy the given boundary conditions;

[0146] The general solution of equation (19) is as follows:

[0147] (20);

[0148] The integral constant is determined by the boundary conditions;

[0149] In this reduced-order model, the cantilever boundary conditions are as follows:

[0150] (twenty one);

[0151] Substituting equation (21) into equation (20), we obtain the following matrix:

[0152] (twenty two);

[0153] Seeking ,but The non-zero solutions are as follows:

[0154] (twenty three);

[0155] Equation (23) is the characteristic value of equation (10);

[0156] Characteristic function The orthogonal normalization conditions that need to be met are as follows:

[0157] (twenty four);

[0158] in, For the Kronecker delta function, when hour ,when hour ;

[0159] Based on equations (20), (23), and (24), the characteristic function of equation (10) is obtained as follows:

[0160] (25);

[0161] The integral transform pair for equation (10) is defined as follows:

[0162] (26);

[0163] (27).

[0164] S5. Transform the original partial differential equations into a system of ordinary differential equations, and solve the system of ordinary differential equations using matrix transformation to obtain the pipeline stability analysis parameters, specifically including:

[0165] Multiply each term of equation (10) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term:

[0166] (28);

[0167] (29);

[0168] (30);

[0169] (31);

[0170] (32);

[0171] Adding equations (28) to (32), we obtain the following set of ordinary differential transformation equations:

[0172] (33);

[0173] (34);

[0174] In the formula, the coefficients of the system of ordinary differential equations as follows:

[0175] (35);

[0176] Equation (33) is the final form of the dimensionless differential equation of motion after transformation, which can be rewritten in matrix form as follows:

[0177] (36);

[0178] (37);

[0179] In the formula, matrices [A], [B], [D], and [F] represent the elements respectively. [E] indicates that it is made by ( A diagonal matrix consisting of numbers from 1 to the truncation number;

[0180] By solving equation (36), we obtain And the angle of the pipe can be obtained according to equation (27). Changes;

[0181] Further assume that the solution to equation (37) can be written in the following form:

[0182] (38);

[0183] in, It is a constant magnitude vector. It is the natural frequency;

[0184] Substituting equation (38) into equation (36), we get:

[0185] (39);

[0186] If equation (39) has a non-zero solution, then it must satisfy:

[0187] (40);

[0188] The natural frequency of the pipe is obtained by solving equation (40). The time history variation and stability of the pipeline are analyzed by comparing the obtained angle changes with the natural frequency.

[0189] The following assumptions are made in the above steps: (1) the fluid is incompressible; (2) the fluid velocity distribution is uniform; (3) the pipe characteristics are similar to those of an Euler-Bernoulli beam; (4) the motion is planar; (5) the pipe has large deflection and small strain, and rotational inertia and shear deformation are ignored; (6) the pipe centerline is not expandable.

[0190] Example

[0191] Substitute the pipe parameters from Table 1 into the model and solve for the desired results. Figure 2 Frequency distribution diagrams of first to fourth order current transmission channels. Figure 3 Schematic diagram of rotation angle changing over time Figure 4 Envelope diagram.

[0192] Table 1. Parameters of a multiphase flow cantilever pipe

[0193]

[0194] Figure 2 Based on the frequency distribution corresponding to the first to fourth modes of the pipeline, the critical flow velocity for this condition can be determined to be 9.182. Selecting flow velocities of 9.18 and 9.19, the rotation angle variation at the pipeline end within the dimensionless time range of 0–20 is calculated, yielding the following results: Figure 3 The results shown are from Figure 3 It can be seen that when the flow velocity is 9.18, the maximum amplitude of the pipe rotation angle is 0.000336 rad, which is in a stable state; when the flow velocity is 9.19, the pipe rotation angle increases with the increase of dimensionless time, which is in an unstable state. Figure 4 The figure shows the change in the overall rotation angle of the pipeline during dimensionless time intervals of 0 to 20 when the flow velocities are 9.18 and 9.19.

[0195] Although the method and derivation process of the present invention have been described in detail above with reference to specific embodiments, it should be understood that the above description is only used to illustrate the principles and preferred embodiments of the present invention, and is not intended to limit the scope of protection of the present invention. Guided by the core concept of the present invention, those skilled in the art can reasonably adjust, replace, or equivalently improve the setting methods of relevant parameters, dimensionless processing methods, characteristic function construction strategies, implementation processes of generalized integral transforms, and specific steps of matrix-based solutions according to actual application scenarios. Furthermore, based on the established auxiliary characteristic equations and combined with different boundary conditions, the dynamic response of the transport pipeline under various boundary constraint forms can also be solved. Any modifications and improvements made within the scope of the claims of the present invention and their equivalent technical solutions without departing from the concept and spirit of the present invention should be considered to fall within the patent protection scope of the present invention.

Claims

1. A stability analysis method for a large deformation cantilever pipe conveying multiphase flow based on a reduced-order model, characterized in that, Includes the following steps: S1. Establish the reduced-order multiphase flow cantilever pipe motion differential equations and define the boundary conditions; S2. Introduce multiphase flow parameters; S3. Introduce dimensionless parameters to make the kinematic differential equations of the multiphase flow cantilever pipe dimensionless. S4. Derive the characteristic equations, characteristic functions, and corresponding eigenvalues ​​under cantilever boundary conditions, and define integral transform pairs; S5. Transform the original partial differential equations into a system of ordinary differential equations, and solve the system of ordinary differential equations in a matrix manner to obtain the pipeline stability analysis parameters.

2. The stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model as described in claim 1, characterized in that, In step S1, the differential equations of motion and boundary conditions for the multiphase flow cantilever pipe are as follows: (1); (2)； in, Mass per unit length of pipe The mass of each phase per unit length, Let the flow rate of each phase be , Represents the gas phase. Represents the liquid phase. The rotation angle of the pipe centerline, subscript These are the parameter values ​​corresponding to the end of the pipe. For a point whose position is described in deformed coordinates, For the length of the pipe, Let g be the bending stiffness of the pipe, and g be the acceleration due to gravity. Represents the derivative with respect to time. It represents the derivative with respect to space.

3. The stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model as described in claim 1, characterized in that, In step S2, the following multiphase flow parameters are introduced: (3)； in, This is the gas phase volumetric flow rate. The volumetric flow rate is the liquid phase. The flow area occupied by the gas phase. The flow area occupied by the liquid phase. This refers to the gas phase volume fraction. The gas content of the cross section, It is the slip factor; The cantilever pipe is oriented downwards, and the liquid velocity is higher than the gas velocity. The slip factor should be selected as follows: (4)。 4. The stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model as described in claim 1, characterized in that, Step S3 specifically includes: Introduce the following dimensionless parameters: (5)； Substituting equation (5) into equation (1), we obtain the dimensionless form of the equation of motion as follows: (6)； To further linearize, equation (6) introduces the Taylor expansion of trigonometric functions as follows: (7)； (8)； (9)； in, Indicates all that satisfy Higher-order terms; Assuming the internal fluid velocity is constant, substituting equations (7), (8), and (9) into equation (6) yields the dimensionless differential equation of motion as follows: (10)； in, The positions of various points in the dimensionless pipeline. For dimensionless time, Let be the dimensionless flow velocity of each phase. This represents the ratio of the dimensionless fluid mass of each phase to the total mass. This is a dimensionless gravity parameter; The corresponding boundary conditions and initial conditions are as follows: (11)； (12)； Based on equations (3) and (4), the gas phase mass ratio and gas phase velocity are derived as follows: (13)； (14)； in, The mass ratio of the gas phase. The density is the gas phase density. The liquid phase mass ratio The density of the liquid phase is... The dimensionless gas phase velocity, The value is the dimensionless liquid flow rate.

5. The stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model as described in claim 1, characterized in that, Step S4 specifically includes: Definition of solution The separated variable form is: (15)； in, For space-dependent functions, It is a time-dependent function; substituting equation (15) into equation (10), we get: (16); Further organized as follows: (17)； Divide both sides of equation (17) by Write it in the following form: (18)； Where, assuming To ensure that this equation has non-zero solutions, then Must meet <0, assuming ; Based on equation (18), the characteristic equation of equation (10) is constructed as follows: (19)； By solving the characteristic equation, we can obtain the characteristic function and characteristic roots that satisfy the given boundary conditions; The general solution of equation (19) is as follows: (20)； The integral constant is determined by the boundary conditions; In this reduced-order model, the cantilever boundary conditions are as follows: (21)； Substituting equation (21) into equation (20), we obtain the following matrix: (22)； Seeking ,but The non-zero solutions are as follows: (23)； Equation (23) is the characteristic value of equation (10); Characteristic function The orthogonal normalization conditions that need to be met are as follows: (24)； in, For the Kronecker delta function, when hour ,when hour ; Based on equations (20), (23), and (24), the characteristic function of equation (10) is obtained as follows: (25)； The integral transform pair for equation (10) is defined as follows: (26)； (27)。 6. The stability analysis method for a large deformation cantilever pipe for transporting multiphase flow based on a reduced-order model as described in claim 1, characterized in that, Step S5 specifically includes: Transforming a partial differential equation into a system of ordinary differential equations includes the following steps: Multiply each term of equation (10) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term: (28)； (29)； (30)； (31)； (32)； Adding equations (28) to (32), we obtain the following set of ordinary differential transformation equations: (33)； (34)； In the formula, the coefficients of the system of ordinary differential equations as follows: (35)； Equation (33) is the final form of the dimensionless differential equation of motion after transformation, which can be rewritten in matrix form as follows: (36)； (37)； In the formula, matrices [A], [B], [D], and [F] represent the elements respectively. [E] indicates that it is made by The diagonal matrix formed; By solving equation (36), we obtain And the angle of the pipe can be obtained according to equation (27). Changes; Further assume that the solution to equation (37) can be written in the following form: (38)； in, It is a constant magnitude vector. It is the natural frequency; Substituting equation (38) into equation (36), we get: (39)； If equation (39) has a non-zero solution, then it must satisfy: (40)； The natural frequency of the pipe is obtained by solving equation (40). ; The time history variation and stability of the pipeline were analyzed by examining the angle changes and natural frequency.

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