Method for predicting stability parameters of multi-suspended-span elastic bearing fluid conveying pipe

Abstract

The invention discloses a multi-suspended-span elastic bearing fluid conveying pipe stability parameter prediction method, and relates to the technical field of fluid conveying pipe vibration characteristic analysis. A multi-suspended-span pipeline is divided into a plurality of independent pipe sections at the supporting position, vibration control equations for describing all the suspended-span pipelines are established respectively, and the vibration control equations only consider the coupling effect of bending force, centrifugal force, gravity, Coriolis force and inertia force of the pipeline and do not contain any middle supporting constraint term; the intermediate support constraint is completely written into the cross-coupling boundary condition, so that the defects that the traditional Dirac function method is fuzzy in significance and the numerical calculation is difficult to converge are avoided; carrying out dimensionless processing on an oscillatory differential equation by introducing a dimensionless parameter, deducing to obtain a characteristic equation, a characteristic function and a characteristic value under the complex boundary condition by combining the end elastic boundary and the cross coupling constraint, converting a partial differential equation into an ordinary differential equation set, and carrying out matrix solution, so as to obtain an oscillatory differential equation set; and finally, an accurate prediction value of the inherent frequency can be obtained.

Description

Technical Field

[0001] This invention relates to the field of vibration characteristic analysis technology for transmission pipelines, and specifically to a method for predicting stability parameters of multi-span elastically supported transmission pipelines. Background Technology

[0002] Fluid transport pipelines are widely used in engineering fields such as aerospace, petrochemicals, and hydraulic transmission, and are core components of fluid transport systems. When the fluid velocity inside the pipe exceeds a critical value, the pipeline is prone to fluid-structure interaction vibration instability, manifested as buckling deformation and resonance, which seriously threatens system safety and structural lifespan. To suppress vibration and improve stability, intermediate supports are often used in engineering to locally constrain the pipeline, while elastic support boundaries composed of a combination of linear springs and torsion springs are used at the ends to simulate actual installation conditions.

[0003] like Figure 1 As shown, existing vibration modeling of transmission pipelines mostly adopts a monolithic pipe model, with intermediate support constraints implemented using the Dirac delta method. While directly embedding the function into the vibration control equation is simple to derive, it has several drawbacks. The deep coupling between the constraint terms and the main equation leads to ambiguity in physical meaning, making it difficult to intuitively reflect the influence of the support on the mechanical behavior of the pipe segments. The introduction of singular functions into the main equation can easily introduce calculation errors when numerical discretization occurs. Furthermore, it is difficult to clearly distinguish between "pipe mechanical behavior" and "support constraint effect," and numerical calculations are difficult to converge.

[0004] Studies have shown that the parameter combination of intermediate supports and end elastic constraints significantly affects the critical flow velocity and vibration characteristics of pipelines, while traditional monolithic pipe models have limitations in characterizing such multi-constraint effects. Therefore, there is an urgent need for a multi-span pipe modeling method that incorporates support constraints into the boundary conditions to predict the stability parameters of flow transmission pipelines with intermediate supports and end elastic constraints, providing a reliable basis for vibration reduction design in engineering projects. Summary of the Invention

[0005] The purpose of this invention is to address the aforementioned shortcomings by proposing a method for predicting the stability parameters of multi-span elastically supported transmission pipes. This method involves splitting the pipe into multiple independent structural segments at the support locations and fully incorporating the intermediate support constraints into the inter-span coupling boundary conditions, thus avoiding the Dirac delta problem found in traditional monolithic pipe models. Embedding functions into the master equation leads to drawbacks such as ambiguous physical meaning and difficulty in achieving convergence in numerical calculations.

[0006] The present invention specifically adopts the following technical solution:

[0007] A method for predicting stability parameters of multi-span elastically supported transmission pipes includes:

[0008] S1. Establish the vibration differential equation for a multi-span transmission pipe with N-1 spring supports in the middle and both ends constrained by linear and torsional springs; the vibration differential equation for the nth segment of a multi-span suspended transmission pipe with N-1 spring supports in the middle, N pipe segments, and both ends constrained by linear and torsional springs is:

[0009] (1);

[0010] in, For the bending stiffness of the transmission pipe, This refers to the lateral displacement of the pipeline. The coordinates are along the length of the conveying pipe. Mass of liquid per unit length Mass per unit length of pipe The fluid velocity inside the delivery pipe. For time, This represents the total number of spans in the transmission pipe. Numbering of axial sections of the conveying pipe;

[0011] S2. Introduce dimensionless parameters to make the vibration differential equation dimensionless, and establish the end elastic boundary and the span coupling boundary conditions.

[0012] Introduce the following dimensionless parameters:

[0013] (2);

[0014] Substituting equation (2) into equation (1), we obtain the dimensionless form of the vibration differential equation:

[0015] (3);

[0016] in, The positions of various points in the dimensionless pipeline. This refers to the dimensionless lateral displacement of the pipeline. For dimensionless time, The velocity is dimensionless. This is the ratio of the dimensionless liquid mass to the total mass of the pipeline. For the first The dimensionless location of the end point of the suspended cross-flow pipe. For the length of the pipe, , These are the spring constants of the torsion springs at the left and right ends, respectively. , These are the spring constants of the left and right end springs, respectively. The spring constant of the intermediate spring is denoted as . , These are the dimensionless forms of the torsional spring stiffness at the left and right ends, respectively. , These are the dimensionless forms of the spring stiffness at the left and right ends, respectively. This is the dimensionless form of the stiffness of the centerline spring;

[0017] The dimensionless end elastic boundary conditions and the span coupling boundary conditions are shown in the corresponding continuity conditions of equations (4)-(11):

[0018] At both ends of the pipe, the elastic support constraint of the combination of linear spring and torsion spring is satisfied:

[0019] (4);

[0020] (5);

[0021] (6);

[0022] (7);

[0023] At each span support location, the conditions of displacement continuity, rotation continuity, bending moment continuity, and shear force equilibrium are satisfied:

[0024] , (8);

[0025] , (9);

[0026] , (10);

[0027] , (11);

[0028] S3. Derive the characteristic equation, eigenvalues ​​and characteristic functions, and define the integral transform pairs;

[0029] S4. After converting the partial differential equation into an ordinary differential equation, solve for the natural frequency using matrix transformation.

[0030] Preferably, in S3, the characteristic equation, eigenvalues, and eigenfunctions are derived based on the elastic supports at both ends and the multi-span coupling constraints, and the integral transformation pairs are defined, including the following steps:

[0031] Based on the basic transformation form of the generalized integral transform, the Euler-Bernoulli beam vibration equation is selected as an approximation of equation (3). The steps for deriving the corresponding characteristic equation, eigenvalues, and characteristic functions are as follows:

[0032] (12);

[0033] in, The mass per unit length of the beam;

[0034] Introduce the following dimensionless parameters:

[0035] (13);

[0036] in, For the dimensionless time in equation (12);

[0037] The dimensionless form of the beam vibration equation is:

[0038] (14);

[0039] Definition of solution The separated variable form is:

[0040] (15);

[0041] Substituting equation (15) into equation (14), we get:

[0042] (16);

[0043] make The characteristic equation of equation (3) can be written as:

[0044] (17);

[0045] The general solution of the characteristic equation is in the form of:

[0046] (18);

[0047] The integral constant is determined by the boundary conditions;

[0048] The end elastic boundary conditions and the span coupling boundary conditions are as follows:

[0049] (19);

[0050] (20);

[0051] (twenty one);

[0052] (twenty two);

[0053] , (twenty three);

[0054] , (twenty four);

[0055] , (25);

[0056] , (26);

[0057] Substituting equations (19)-(26) into equation (18), we obtain the following matrix equation:

[0058] (27);

[0059] in It is about eigenvalues 1-th order matrix It concerns the integral constant. Column vectors of order;

[0060] Further solving equation (27) yields the equation satisfied by the eigenvalues:

[0061] (28);

[0062] The integral constant is obtained by using equations (27) and (28);

[0063] Substituting the integral constant into equation (18) yields the characteristic function of the multi-span elastic support condition;

[0064] Normalize the characteristic function:

[0065] (29);

[0066] The normalized characteristic function is obtained from the normalization condition:

[0067] (30);

[0068] After normalization, the orthogonal normalization conditions for the characteristic functions are as follows:

[0069] (31);

[0070] The integral transform pair is defined for equation (3) as follows:

[0071] (32);

[0072] (33)

[0073] Preferably, in S4, the partial differential equation is transformed into a system of ordinary differential equations through integral transformation and then solved in matrix form, including the following steps:

[0074] Multiply each term of equation (3) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term:

[0075] (34);

[0076] (35);

[0077] (36);

[0078] (37);

[0079] Adding equations (34) to (37) together, we obtain the following set of ordinary differential transformation equations:

[0080] (38);

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

[0082] (39);

[0083] (40);

[0084] Equation (38) is the final form of the vibration differential equation after transformation;

[0085] Truncate equation (38) to a finite number of truncations. And rewrite it in matrix form:

[0086] (41);

[0087] In the formula, the matrix The elements they represent are , Indicates by The diagonal matrix formed Equal to 1 to the cutoff number;

[0088] The standard form of equation (38) is as follows:

[0089] (42);

[0090] (43);

[0091] In the formula, , For the quality matrix, Here is the damping matrix. Here is the stiffness matrix;

[0092] The solution to equation (42) can be written in the following form:

[0093] (44);

[0094] In the formula, It is a constant magnitude vector. Angular frequency;

[0095] Substituting equation (44) into equation (42) yields:

[0096] (45);

[0097] If equation (45) has a non-zero solution, then it must satisfy:

[0098] (46);

[0099] This means that the determinant of the matrix within the brackets is calculated, and the natural frequency of the transmission tube is obtained by solving formula (46).

[0100] The present invention has the following beneficial effects:

[0101] By adopting a multi-span pipe modeling scheme, the intermediate support constraints are fully incorporated into the inter-span coupled boundary conditions rather than the master equation. Based on the Euler-Bernoulli beam model and momentum theorem, the vibration control equation is established and solved using the generalized integral transform method (GITT). This effectively considers the effects of intermediate support stiffness, position, and end constraints on the dynamic characteristics of the conveying pipe, and achieves high-precision prediction of the natural frequency.

[0102] This method has clear physical meaning, good numerical stability, and wide applicability. It is not only applicable to the prediction of stability parameters such as natural frequency of flow transport pipeline systems with intermediate supports and elastic end constraints, but can also be extended to the prediction of critical parameters of other multi-span and multi-support constraints fluid transport systems, providing technical support for engineering vibration reduction design and structural safety assessment. Attached Figure Description

[0103] Figure 1 A schematic diagram for solving a traditional monolithic pipeline model;

[0104] Figure 2 A schematic diagram for solving a multi-span pipe segmented model;

[0105] Figure 3 This is a scatter plot of the first four natural frequencies based on the multi-span tube segmentation model. Detailed Implementation

[0106] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and specific examples:

[0107] This scheme, under the assumption of small deformation, divides the pipeline into multiple independent structural segments at the intermediate support location. Based on Euler-Bernoulli beam theory and momentum theorem, the vibration control equations for multi-span conveying pipelines are derived. These equations only reflect the coupling effects of bending force, centrifugal force, gravity, Coriolis force, and inertial force, without including any intermediate support constraint terms. Dimensionless parameters are then defined and the vibration control equations are made dimensionless. Combining the elastic support boundary conditions of the end-spring-torsion spring combination with the continuous boundary conditions of continuous displacement, rotation angle, bending moment, and shear force balance between spans, the characteristic equations, characteristic functions, and eigenvalues ​​under these complex boundary conditions are derived. Orthogonality conditions of the characteristic functions are obtained through orthogonal normalization. Then, a generalized integral transformation is applied to each term of the vibration differential equations, converting the partial differential equations into a system of ordinary differential equations. Finally, the system of ordinary differential equations obtained by the integral transformation is solved in a matrix manner, allowing numerical calculation to obtain the natural frequency.

[0108] Combination Figure 2 A method for predicting stability parameters of multi-span elastically supported transmission pipes, comprising:

[0109] S1. Establish the vibration differential equation for a multi-span transmission pipe with N-1 spring supports in the middle and both ends constrained by linear and torsional springs; the vibration differential equation for the nth segment of a multi-span suspended transmission pipe with N-1 spring supports in the middle, N pipe segments, and both ends constrained by linear and torsional springs is:

[0110] (1);

[0111] in, For the bending stiffness of the transmission pipe, This refers to the lateral displacement of the pipeline. The coordinates are along the length of the conveying pipe. Mass of liquid per unit length Mass per unit length of pipe The fluid velocity inside the delivery pipe. For time, This represents the total number of spans in the transmission pipe. The axial segment numbering of the transmission pipe is used.

[0112] S2. Introduce dimensionless parameters to make the vibration differential equation dimensionless, and establish the end elastic boundary and the span coupling boundary conditions.

[0113] Introduce the following dimensionless parameters:

[0114] (2);

[0115] Substituting equation (2) into equation (1), we obtain the dimensionless form of the vibration differential equation:

[0116] (3);

[0117] in, The positions of various points in the dimensionless pipeline. This refers to the dimensionless lateral displacement of the pipeline. For dimensionless time, The velocity is dimensionless. This is the ratio of the dimensionless liquid mass to the total mass of the pipeline. For the first The dimensionless location of the end point of the suspended cross-flow pipe. For the length of the pipe, , These are the spring constants of the torsion springs at the left and right ends, respectively. , These are the spring constants of the left and right end springs, respectively. The spring constant of the intermediate spring is denoted as . , These are the dimensionless forms of the torsional spring stiffness at the left and right ends, respectively. , These are the dimensionless forms of the spring stiffness at the left and right ends, respectively. This is the dimensionless form of the stiffness of the centerline spring;

[0118] The dimensionless end elastic boundary conditions and the span coupling boundary conditions are shown in the corresponding continuity conditions of equations (4)-(11):

[0119] At both ends of the pipe, the elastic support constraint of the combination of linear spring and torsion spring is satisfied:

[0120] (4);

[0121] (5);

[0122] (6);

[0123] (7);

[0124] At each span support location, the conditions of displacement continuity, rotation continuity, bending moment continuity, and shear force equilibrium are satisfied:

[0125] , (8);

[0126] , (9);

[0127] , (10);

[0128] , (11).

[0129] S3. Derive the characteristic equation, eigenvalues ​​and characteristic functions, and define the integral transform pairs;

[0130] Based on the elastic supports at both ends and the multi-span coupling constraints, the characteristic equation, eigenvalues, and characteristic functions are derived, and the integral transformation pairs are defined, including the following steps:

[0131] Based on the basic transformation form of the generalized integral transform, the Euler-Bernoulli beam vibration equation is selected as an approximation of equation (3). The steps for deriving the corresponding characteristic equation, eigenvalues, and characteristic functions are as follows:

[0132] (12);

[0133] in, The mass per unit length of the beam;

[0134] Introduce the following dimensionless parameters:

[0135] (13);

[0136] in, For the dimensionless time in equation (12);

[0137] The dimensionless form of the beam vibration equation is:

[0138] (14);

[0139] Definition of solution The separated variable form is:

[0140] (15);

[0141] Substituting equation (15) into equation (14), we get:

[0142] (16);

[0143] make The characteristic equation of equation (3) can be written as:

[0144] (17);

[0145] The general solution of the characteristic equation is in the form of:

[0146] (18);

[0147] The integral constant is determined by the boundary conditions;

[0148] The end elastic boundary conditions and the span coupling boundary conditions are as follows:

[0149] (19);

[0150] (20);

[0151] (twenty one);

[0152] (twenty two);

[0153] , (twenty three);

[0154] , (twenty four);

[0155] , (25);

[0156] , (26);

[0157] Substituting equations (19)-(26) into equation (18), we obtain the following matrix equation:

[0158] (27);

[0159] in It is about eigenvalues 1-th order matrix It concerns the integral constant. Column vectors of order;

[0160] Further solving equation (27) yields the equation satisfied by the eigenvalues:

[0161] (28);

[0162] The integral constant is obtained by using equations (27) and (28);

[0163] Substituting the integral constant into equation (18) yields the characteristic function of the multi-span elastic support condition;

[0164] Normalize the characteristic function:

[0165] (29);

[0166] The normalized characteristic function is obtained from the normalization condition:

[0167] (30);

[0168] After normalization, the orthogonal normalization conditions for the characteristic functions are as follows:

[0169] (31);

[0170] The integral transform pair is defined for equation (3) as follows:

[0171] (32);

[0172] (33)

[0173] S4. After converting the partial differential equations into ordinary differential equations, solve for the natural frequencies using matrix transformation. This involves the following steps:

[0174] Multiply each term of equation (3) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term:

[0175] (34);

[0176] (35);

[0177] (36);

[0178] (37);

[0179] Adding equations (34) to (37) together, we obtain the following set of ordinary differential transformation equations:

[0180] (38);

[0181] In the formula, the coefficients of the system of ordinary differential equations are as follows:

[0182] (39);

[0183] (40);

[0184] Equation (38) is the final form of the vibration differential equation after transformation;

[0185] Truncate equation (38) to a finite number of truncations. And rewrite it in matrix form:

[0186] (41);

[0187] In the formula, the matrix The elements they represent are , Indicates by The diagonal matrix formed Equal to 1 to the cutoff number;

[0188] The standard form of equation (38) is as follows:

[0189] (42);

[0190] (43);

[0191] In the formula, , For the quality matrix, Here is the damping matrix. Here is the stiffness matrix;

[0192] The solution to equation (42) can be written in the following form:

[0193] (44);

[0194] In the formula, It is a constant magnitude vector. Angular frequency;

[0195] Substituting equation (44) into equation (42) yields:

[0196] (45);

[0197] If equation (45) has a non-zero solution, then it must satisfy:

[0198] (46);

[0199] This means that the determinant of the matrix within the brackets is calculated, and the natural frequency of the transmission tube is obtained by solving formula (46).

[0200] The selected example data is and In this case, the schematic diagram of the transmission pipe is as follows: Figure 2Substituting the pipe data from Table 1 into formula (46) and solving, we obtain... Figure 3 Scatter plot of the first four natural frequencies of the transmission tube.

[0201] Table 1

[0202]

[0203] Figure 3 The figure shows the variation of the real and imaginary parts of the first four natural frequencies of a multi-span pipeline with the dimensionless flow velocity Γ. The real part reflects the attenuation characteristics and stability level of the system's vibration amplitude, while the imaginary part characterizes the magnitude of the angular frequency of the vibration. As shown in the figure, with the continuous increase of the dimensionless flow velocity Γ, the real part of each natural frequency generally shows a monotonically decreasing trend, indicating that the system is gradually transitioning from a strongly stable state to a critically stable state and even an unstable state. The imaginary part shows obvious steps and fluctuations in the high-velocity range, reflecting the modal transition characteristics under strong fluid-structure interaction. That is, the coupling effect of fluid inertial force and pipeline elastic force excites the participation of higher-order modes, resulting in abrupt changes in vibration frequency.

[0204] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A method for predicting stability parameters of a multi-span elastically supported transmission pipe, characterized in that, include: S1. Establish the vibration differential equation for a multi-span transmission pipe with N-1 spring supports in the middle and both ends constrained by linear and torsional springs; the vibration differential equation for the nth segment of a multi-span suspended transmission pipe with N-1 spring supports in the middle, N pipe segments, and both ends constrained by linear and torsional springs is: （1）； in, For the bending stiffness of the transmission pipe, For the first Lateral displacement of the pipeline section The coordinates are along the length of the conveying pipe. For the first The coordinates of the location of the end point of the suspended cross-flow pipe. Mass of liquid per unit length Mass per unit length of pipe The fluid velocity inside the delivery pipe. For time, This represents the total number of spans in the transmission pipe. Numbering of axial sections of the conveying pipe; S2. Introduce dimensionless parameters to make the vibration differential equation dimensionless, and establish the end elastic boundary and the span coupling boundary conditions. Introduce the following dimensionless parameters: （2）； Substituting equation (2) into equation (1), we obtain the dimensionless form of the vibration differential equation: （3）； in, The positions of various points in the dimensionless pipeline. For the first Dimensionless lateral displacement of a section of pipeline For dimensionless time, The velocity is dimensionless. This is the ratio of the dimensionless liquid mass to the total mass of the pipeline. For the first The dimensionless location of the end point of the suspended cross-flow pipe. For the length of the pipe, , These are the spring constants of the torsion springs at the left and right ends, respectively. , These are the spring constants of the left and right end springs, respectively. For the first The spring constant of a center-line spring. , These are the dimensionless forms of the torsional spring stiffness at the left and right ends, respectively. , These are the dimensionless forms of the spring stiffness at the left and right ends, respectively. For the first The dimensionless form of the stiffness of a midline spring; The dimensionless end elastic boundary conditions and the span coupling boundary conditions are shown in the corresponding continuity conditions of equations (4)-(11): At both ends of the pipe, the elastic support constraint of the combination of linear spring and torsion spring is satisfied: （4）； （5）； （6）； （7）； At each span support location, the conditions of displacement continuity, rotation continuity, bending moment continuity, and shear force equilibrium are satisfied: ， （8）； ， （9）； ， （10）； ， (11)； S3. Derive the characteristic equation, eigenvalues ​​and characteristic functions, and define the integral transform pairs; S4. After converting the partial differential equation into an ordinary differential equation, solve for the natural frequency using matrix transformation.

2. The method for predicting stability parameters of a multi-span elastically supported transmission pipe as described in claim 1, characterized in that, In S3, based on the elastic supports at both ends and the multi-span coupling constraints, the characteristic equation, eigenvalues, and eigenfunctions are derived, and the integral transformation pairs are defined, including the following steps: Based on the basic transformation form of the generalized integral transform, the Euler-Bernoulli beam vibration equation is selected as an approximation of equation (3). The steps for deriving the corresponding characteristic equation, eigenvalues, and characteristic functions are as follows: (12)； in, The mass per unit length of the beam; Introduce the following dimensionless parameters: (13)； in, For the dimensionless time in equation (12); The dimensionless form of the beam vibration equation is: (14)； Definition of solution The separated variable form is: (15)； Substituting equation (15) into equation (14), we get: (16)； make The characteristic equation of equation (3) can be written as: (17)； The general solution of the characteristic equation is in the form of: (18)； The integral constant is determined by the boundary conditions; The end elastic boundary conditions and the span coupling boundary conditions are as follows: (19)； (20)； (21)； (22)； ， (23)； ， (24)； ， (25)； ， (26)； Substituting equations (19)-(26) into equation (18), we obtain the following matrix equation: (27)； in It is about eigenvalues 1-th order matrix It concerns the integral constant. Column vectors of order; Further solving equation (27) yields the equation satisfied by the eigenvalues: (28)； The integral constant is obtained by using equations (27) and (28); Substituting the integral constant into equation (18) yields the characteristic function of the multi-span elastic support condition; Normalize the characteristic function: (29)； The normalized characteristic function is obtained from the normalization condition: (30)； After normalization, the orthogonal normalization conditions for the characteristic functions are as follows: (31)； The integral transform pair is defined for equation (3) as follows: (32)； (33)。 3. The method for predicting stability parameters of a multi-span elastically supported transmission pipe as described in claim 1, characterized in that, In S4, the partial differential equations are transformed into a system of ordinary differential equations through integral transformation and then solved in matrix form, including the following steps: Multiply each term of equation (3) by the orthogonally normalized characteristic function. And in Integrating in the middle, we obtain the following ordinary differential transformations for each term: (34)； (35)； (36)； (37)； Adding equations (34) to (37) together, we obtain the following set of ordinary differential transformation equations: (38)； In the formula, the coefficients of the system of ordinary differential equations are as follows: (39)； (40)； Equation (38) is the final form of the vibration differential equation after transformation; Truncate equation (38) to a finite number of truncations. And rewrite it in matrix form: (41)； In the formula, the matrix The elements they represent are , Indicates by The diagonal matrix formed Equal to 1 to the cutoff number; The standard form of equation (38) is as follows: (42)； (43)； In the formula, , For the quality matrix, Here is the damping matrix. Here is the stiffness matrix; The solution to equation (42) can be written in the following form: (44)； In the formula, It is a constant magnitude vector. Angular frequency; Substituting equation (44) into equation (42) yields: (45)； If equation (45) has a non-zero solution, then it must satisfy: (46)； This means that the determinant of the matrix within the brackets is calculated, and the natural frequency of the transmission tube is obtained by solving formula (46).

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