Marine anchor cable dynamic stress analysis method

Abstract

The method for analyzing the dynamic stress of the ocean anchor cable, provided by the invention, is a semi-analytic and semi-numerical method, and the solving process of the method is in a closed form, so that the method has very high calculation efficiency and precision and is expected to lay a foundation for health monitoring of the service period of the ocean anchor cable. The method is simple in process and comprises the steps that firstly, a relational expression between the nonlinear modal frequency and the linear undamped modal frequency of the ocean anchor cable is given; then, a closed solution of a frequency equation of the inhaul cable is given according to a dynamic stiffness method, the undamped modal frequency of the system can be obtained by solving the frequency equation, then the nonlinear system frequency is obtained, and finally the dynamic strain of the inhaul cable is obtained according to the modal shape.

Description

Technical Field

[0001] This invention belongs to the field of civil engineering, focusing on the dynamic characteristics of marine anchor cables, and specifically relates to an analysis method for the dynamic stress of marine anchor cables. Background Technology

[0002] Immersive floating tunnels are an innovative transportation and display concept that integrates underwater suspension tunnel technology with immersive experience systems. They aim to solve the problem of deep-sea cross-sea passage while providing a multi-dimensional sensory interactive experience. Currently, they are one of the most promising marine crossing projects. As a core component of the integrity and functionality of immersive floating tunnels, the dynamic response of anchor cables under the action of waves and ocean currents is crucial to the overall safety and feasibility of the system.

[0003] To predict the lifespan of anchor cables and prevent safety issues caused by anchor chain breakage, anchor dragging, or excessive slack during use, dynamic analysis of the anchor cables is necessary. Previous dynamic analysis studies have often been achieved by building models and solving them. However, in the process of modeling and solving, simplification and approximation methods are often used for the sake of convenience, which to some extent affects the accuracy of the results. Therefore, in addition to adopting a more reasonable dynamic model, a unified analysis method with high computational accuracy and efficiency is also needed to make the dynamic analysis of anchor cables more accurate and efficient.

[0004] In their paper "Non-linear reduced-order model for parametric excitation analysis of an immersed vertical slender rod," Franzini et al. outlined the stability challenges faced by immersed slender structures. The main analytical methods can be categorized into time-domain and frequency-domain methods. Time-domain methods include lumped mass methods, finite difference methods, and finite element methods. The accuracy of these methods depends on the degree of discretization of the structure and the accuracy of the shape function. Therefore, although time-domain methods have a wide range of applications, a trade-off often needs to be made between accuracy and computational efficiency. Frequency-domain methods can clearly understand the system's response to different frequency components, facilitating the analysis of resonance phenomena and frequency sensitivity. However, for nonlinear systems, the application of frequency-domain methods is more complex and may lose accuracy.

[0005] To comprehensively consider the influence of design parameters while ensuring both accuracy and efficiency in calculation results, the invention patent application No. 202210505091.9, entitled "A Refined Analysis Method for the Dynamic Characteristics of Cables with Small Sag," proposes a semi-analytical, semi-numerical method. First, it presents the relationship between the nonlinear modal frequencies and the linear undamped modal frequencies of cables with small sag. Then, based on the dynamic stiffness method, it provides a closed-form solution to the cable frequency equation. Solving this equation yields the undamped modal frequencies of the system, and subsequently, the frequencies of the nonlinear system. However, for the dynamic analysis of anchor cables in deep-sea environments, this method does not fully consider the fatigue life of the anchor cable and the displacement of the trisection points in practical applications, leading to inaccurate and incomplete results.

[0006] Therefore, there is an urgent need to propose a new, refined, and universal method for anchor cable modeling and detailed analysis. Summary of the Invention

[0007] To address the shortcomings of existing technologies in the detailed analysis of marine anchor cables, this invention provides a more realistic and reliable method for the detailed analysis of the dynamic characteristics of marine anchor cables. This method first proposes a refined and universal anchor cable modeling approach, discovering that the maximum stress occurs near the upper anchor point. Therefore, during the engineering design phase, it is essential to prioritize the assessment of the anchor cable's fatigue life, especially at the connection with the tunnel body, while simultaneously monitoring the displacement of the anchor cable's trisection points to prevent excessive displacement. Next, a closed-form solution to the cable's frequency equation is derived using the dynamic stiffness method. Solving this equation yields the undamped modal frequencies of the system, and subsequently, the frequencies of the nonlinear system. Finally, the perturbation method is used to solve for the structure's modal functions, and the dynamic strain of the cable is calculated based on the mode shapes.

[0008] The technical solution of this invention is as follows:

[0009] A method for analyzing the dynamic stress of marine anchor cables, characterized by comprising the following steps:

[0010] Step 1: Establish the finite element model of the anchor cable and the vibration differential equation of the nonlinear system of the anchor cable, and linearize the nonlinear vibration differential equation to obtain the linear system;

[0011] Step 2: For the linear system obtained in Step 1, calculate the mode shape function of the anchor cable. ;

[0012] Step 3: Based on the finite element model of the anchor cable, calculate the stiffness matrix of each element in the finite element model. And assemble to obtain the overall stiffness matrix. ;

[0013] Step 4: Solve the frequency equations of the linear system using a numerical iterative algorithm. Obtain the natural frequency of the linear system structure Then, the frequency of the anchor cable nonlinear system can be solved. ;

[0014] Step 5: Based on the mode shape function of the anchor cable obtained in Step 2 The frequency of the anchor cable nonlinear system obtained in step four The perturbation method is used to solve the mode functions. ;

[0015] Step 6: Solve for the dynamic stress of the marine anchor cable using the stress calculation formula.

[0016] Furthermore, the vibration differential equation of the anchor cable nonlinear system in step one is as follows:

[0017] (1)

[0018] in:

[0019] It is Young's modulus; Represents the bending stiffness of the main beam; Represents the linear mass per unit length of the anchor cable; Represents the tension borne by the cable; Represents the viscous damping coefficient of the system; Represents the system's vibration displacement function; This represents the additional cable force caused by elastic elongation during cable vibration, and it is equal to the dynamic strain of the cable segment. and axial stiffness The product of, i.e.:

[0020] (2)

[0021] in, For the vertical-to-span ratio, It is the acceleration due to gravity. For the angle of inclination of the cable, The length of the cable; This is the effective length of the cable under its own weight.

[0022] Furthermore, the linearization process of the vibration differential equation of the nonlinear system is as follows:

[0023] According to the method of separation of variables, Substituting into equation (1), we obtain the linearized vibration differential equation:

[0024] (3)

[0025] in:

[0026] ; It is the mode shape function; It is a modal function.

[0027] Furthermore, the mode shape function of the anchor cable in step two... The calculation process is as follows:

[0028] Introducing coefficients , , , The mode shape function of the anchor cable Represented as:

[0029] (5)

[0030] in:

[0031] , ;

[0032] Based on the boundary conditions:

[0033] (6)

[0034] Seeking , , ;

[0035] make Then the mode shape function of the anchor cable is expressed as:

[0036] (7).

[0037] Furthermore, the stiffness matrix of each element in step three... as follows:

[0038] K e = [ n c o t n EA l 0 0 − n c s c n EA l 0 0 T NO l 3 Q NO l 2 0 − R NO l 3 H NO l 2 S NO l 0 − H NO l 2 C NO l − n c s c n EA l 0 0 n c o t n EA l 0 0 0 − R NO l 3 − H NO l 2 T NO l 3 − Q NO l 2 0 H NO l 2 C NO l S NO l ] (9)

[0039] in:

[0040] , ,in It is the structure's natural frequency. , It is the diameter of the marine anchor cable. It is an additional quality coefficient. It is the density of water; It is the cross-sectional area of ​​the cable; , , , , , The notation introduced here has no special meaning, among which:

[0041] (8).

[0042] Furthermore, the frequency of the anchor cable nonlinear system in step four... With the natural frequency of the linear system structure The relationship is as follows:

[0043] (10)

[0044] in:

[0045] .

[0046] Furthermore, the specific implementation process of step five is as follows:

[0047] For the equation of motion described by equation (3), let The equations of motion for the derived system are:

[0048] (4)

[0049] The equation of motion for the anchor cable is a fourth-order partial differential equation with second-order nonlinear damping, namely:

[0050] (11)

[0051] in:

[0052] ; It is the average viscous damping coefficient of the system within the period;

[0053] Due to equation coupling, we linearize the nonlinear damping. The basic principle is that the energy consumed by the nonlinear damping in one cycle is equal to the energy consumed by the linearized damping in one cycle, that is:

[0054] (12)

[0055] in:

[0056] It is the energy consumed by the nonlinear damping within one cycle. It is the energy consumed by linear damping within one cycle;

[0057] Therefore, we can conclude that:

[0058] (15)

[0059] in:

[0060] It is the force of nonlinear damping of the mode, and its calculation formula is:

[0061] (13)

[0062] It is the linearized damping force for each mode, and its calculation formula is:

[0063] (14)

[0064] in These are equivalent parameters;

[0065] Similarly, Cable Power and the force of each order They are the same, and ,but:

[0066] (17)

[0067] Therefore, we can conclude that:

[0068] κ = Ψ ∫ 0 l Φ ( x ) d x ∫ 0 l Φ 2 ( x ) d x = e 2 l p ( 1 − 2 e − l p − l p + e − 2 l p ( 1 + l p ) ) 2 p ( − 4 l p + 2 l p c o s h [ 2 l p ] + 8 s i n h [ l p ] − 3 s i n h [ 2 l p ] ) (18)

[0069] Combining formula (4) with formulas (11) to (18), we get:

[0070] (19)

[0071] Multiply by each term of equation (19), and The integral yields:

[0072] (20)

[0073] in:

[0074] ; ; ζ = c ¯ m ¯ ∫ 0 l Φ 2 | Φ | d x ∫ 0 l Φ 2 d x = c ¯ m ¯ e − l p ( − 5 2 + 2 7 c o s h [ l p ] + 3 6 c o s h [ 2 l p ] − 1 1 c o s h [ 3 l p ] + 6 l p ( − 9 s i n h [ l p ] + s i n h [ 3 l p ] ) ) 3 ( − 4 l p + 2 l p c o s h [ 2 l p ] + 8 s i n h [ l p ] − 3 s i n h [ 2 l p ] ) ;

[0075] Equation (20) can be written as the standard differential equation of motion with viscous damping and quadratic nonlinear damping:

[0076] (twenty one)

[0077] in:

[0078] ,in , ; ;

[0079] Using a multi-scale method:

[0080] (twenty two)

[0081] in:

[0082] , , These represent disturbance terms at different time scales, and , , ; It is a tiny amount;

[0083] Introducing the partial derivative operator:

[0084] (twenty three)

[0085] Compare The coefficients of the same power yield a series of linear differential equations:

[0086] (twenty four)

[0087] in:

[0088] It represents the conjugate of the preceding terms;

[0089] The solution form of formula (24) is as follows:

[0090] Y 0 = α ( T 1 )cos[ ω 0 T 0 + β ( T 1 )] (25)

[0091] To facilitate the solution Formula (25) can be written in complex form:

[0092] (26)

[0093] in:

[0094] The imaginary unit;

[0095] Substitution get:

[0096] (27)

[0097] in:

[0098] It is an imaginary number;

[0099] To avoid the presence of a perpetual term, the right-hand side of the above equation cannot contain... or Such a term requires that the Fourier coefficient on the right-hand side of the above equation be zero:

[0100] (28)

[0101] remember:

[0102] (29)

[0103] Substituting it into equation (28), we obtain the trigonometric function form of this condition:

[0104] (30)

[0105] Separating the real and imaginary parts of formula (30) yields:

[0106] (31)

[0107] (32)

[0108] (33)

[0109] Based on the solutions of equations (32) and (33):

[0110] (34)

[0111] α = α 0 [ e εμ 1 t ( 1 + α 0 4 μ 2 ω 0 3 πμ 1 ) − α 0 ] − 1 (35)

[0112] in, and These are the initial amplitude and initial phase, determined by the initial displacement and initial velocity, respectively.

[0113] (36).

[0114] Furthermore, the stress calculation formula in step six is ​​as follows:

[0115] (37)

[0116] in:

[0117] It is the dynamic stress of marine anchor cables; It is the diameter of the cable; It is the moment of inertia.

[0118] The beneficial effects of this invention are as follows:

[0119] This invention presents a semi-analytical, semi-numerical method for analyzing the dynamic stress of marine anchor cables. This method combines the advantages of analytical and numerical methods and is simple in process. First, it gives the relationship between the modal frequencies of marine anchor cables and the linear undamped modal frequencies, and discusses the nonlinear behavior of marine anchor cables, including their vibration displacement and the bending stress of the anchor cable, to observe the mechanical behavior of the structure under initial excitation. Once the vibration displacement of the anchor cable is obtained, the bending stress of the cable can be further calculated using the stress solution formula. This method has high computational efficiency and accuracy, and can comprehensively consider the influence of cable design parameters, which is expected to lay the foundation for the optimized design and service life health monitoring of marine anchor cables.

[0120] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0121] Figure 1 Simplified dynamic model diagram of marine anchor cables. Detailed Implementation

[0122] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0123] See Figure 1 A method for analyzing the dynamic stress of marine anchor cables, the method comprising the following:

[0124] Step 1: According to Figure 1 The finite element model of the anchor cable and the vibration differential equation of the nonlinear system of the anchor cable are established as shown, and the nonlinear vibration differential equation is linearized to obtain the linear system.

[0125] Step 2: For a linear system, calculate the mode shape function of the anchor cable. ;

[0126] Step 3: Based on the finite element model of the anchor cable, calculate the stiffness matrix of each element in the finite element model. And assemble to obtain the overall stiffness matrix. ;

[0127] Step 4: Solve the system frequency equations using a numerical iterative algorithm (or the Wittrick-Williams algorithm). Obtain the natural frequency of the linear system structure Then, the frequency of the anchor cable nonlinear system can be solved. ;

[0128] Step 5: Based on the mode shape function of the anchor cable and the frequency of the anchor cable nonlinear system The perturbation method is used to solve the mode functions. ;

[0129] Step 6: Solve for the dynamic stress of the marine anchor cable using the stress calculation formula.

[0130] The dynamic stress calculation and analysis of anchor cables requires vibration displacement. After dynamic response calculation of the structure, it was found that the first-order mode, i.e., the minimum frequency, has the main influence. The following is a detailed explanation of each step above:

[0131] A dynamic model of the anchor cable is established, and the vibration differential equation of the nonlinear system is derived from this model as follows:

[0132] (1)

[0133] in:

[0134] It is Young's modulus; Represents the bending stiffness of the main beam; Represents the linear mass per unit length of the anchor cable; Represents the tension borne by the cable; Represents the viscous damping coefficient of the system; Represents the system's vibration displacement function; This represents the additional cable force caused by elastic elongation during cable vibration, and it is equal to the dynamic strain of the cable segment. and axial stiffness The product of, i.e.:

[0135] (2)

[0136] in, For the vertical-to-span ratio, It is the acceleration due to gravity. For the angle of inclination of the cable, The length of the cable; This is the effective length of the cable under its own weight.

[0137] Linearization of the vibration differential equation:

[0138] According to the method of separation of variables, Substituting into equation (1), we get:

[0139] (3)

[0140] in:

[0141] ; It is the mode shape function; It is a mode function;

[0142] For the equation of motion described by equation (3), let The equations of motion for the derived system can be obtained as follows:

[0143] (4)

[0144] in:

[0145] ,in It is the diameter of the marine anchor cable. It is an additional quality coefficient. It is the density of water; It is the inherent frequency of the structure;

[0146] Calculate the mode shape function of the anchor cable :

[0147] Introducing coefficients , , , The mode shape function of the anchor cable It can be represented as:

[0148] (5)

[0149] in:

[0150] , ;

[0151] Based on the boundary conditions:

[0152] (6)

[0153] Seeking , , ;

[0154] make Then the mode shape function of the anchor cable can be expressed as:

[0155] (7)

[0156] Dynamic stiffness matrix Solving for:

[0157] Introduction notation:

[0158] (8)

[0159] Then the element stiffness matrix for:

[0160] K e = [ n c o t n EA l 0 0 − n c s c n EA l 0 0 T NO l 3 Q NO l 2 0 − R NO l 3 H NO l 2 S NO l 0 − H NO l 2 C NO l − n c s c n EA l 0 0 n c o t n EA l 0 0 0 − R NO l 3 − H NO l 2 T NO l 3 − Q NO l 2 0 H NO l 2 C NO l S NO l ] (9)

[0161] in:

[0162] ; , It is the cross-sectional area of ​​the cable;

[0163] Finally, the element stiffness matrix is... Placed in the global stiffness matrix In this context, for each element, the position of its nodal degrees of freedom in the global stiffness is determined, and the corresponding values ​​of the element stiffness matrix are added to these positions.

[0164] Undamped modal frequency Solving for:

[0165] element stiffness matrix The overall stiffness matrix is ​​obtained by assembly. Then, the characteristic equations of the system can be solved using a numerical iterative algorithm (or the Wittrick-Williams algorithm). Thus, the undamped modal frequencies of the system are obtained, which are the natural frequencies of the linear system structure described in formula (4). .

[0166] Calculate the frequency of the original system:

[0167] Determine the undamped modal frequencies of the system. Then, through the following formula:

[0168] (10)

[0169] in:

[0170] ;

[0171] Obtain the frequency of the nonlinear system .

[0172] The perturbation solution of the structural dynamic response is derived using exact functions and natural frequencies. :

[0173] The equation of motion for the anchor cable is a fourth-order partial differential equation with second-order nonlinear damping, namely:

[0174] (11)

[0175] in:

[0176] ; It is the average viscous damping coefficient of the system within the period;

[0177] Due to equation coupling, we linearize the nonlinear damping. The basic principle is that the energy consumed by the nonlinear damping in one cycle is equal to the energy consumed by the linearized damping in one cycle, that is:

[0178] (12)

[0179] in:

[0180] It is the energy consumed by the nonlinear damping within one cycle. It is the energy consumed by linear damping within one cycle;

[0181] The nonlinear damping force of the mode is:

[0182] (13)

[0183] Let the linearized damping force for each mode be:

[0184] (14)

[0185] in:

[0186] These are equivalent parameters;

[0187] From equation (12), we get:

[0188] (15)

[0189] Similarly, let the force for each mode at each order be:

[0190] (16)

[0191] According to cable power and the force of each order They are the same; we have:

[0192] (17)

[0193] have to:

[0194] κ = Ψ ∫ 0 l Φ ( x ) d x ∫ 0 l Φ 2 ( x ) d x = e 2 l p ( 1 − 2 e − l p − l p + e − 2 l p ( 1 + l p ) ) 2 p ( − 4 l p + 2 l p c o s h [ 2 l p ] + 8 s i n h [ l p ] − 3 s i n h [ 2 l p ] ) (18)

[0195] Based on the above calculations and substituting equation (4) into equation (11), we obtain:

[0196] (19)

[0197] Multiply by each term of equation (19), and The integral yields:

[0198] (20)

[0199] in:

[0200] ; ; ζ = c ¯ m ¯ ∫ 0 l Φ 2 | Φ | d x ∫ 0 l Φ 2 d x = c ¯ m ¯ e − l p ( − 5 2 + 2 7 c o s h [ l p ] + 3 6 c o s h [ 2 l p ] − 1 1 c o s h [ 3 l p ] + 6 l p ( − 9 s i n h [ l p ] + s i n h [ 3 l p ] ) ) 3 ( − 4 l p + 2 l p c o s h [ 2 l p ] + 8 s i n h [ l p ] − 3 s i n h [ 2 l p ] ) ;

[0201] Equation (20) can be written as the standard differential equation of motion with viscous damping and quadratic nonlinear damping:

[0202] (twenty one)

[0203] in:

[0204] ,in , ; ;

[0205] Using a multi-scale method:

[0206] (twenty two)

[0207] in:

[0208] , , These represent disturbance terms at different time scales, and , , ; It is a tiny amount;

[0209] Introducing the partial derivative operator:

[0210] (twenty three)

[0211] Compare The coefficients of the same power yield a series of linear differential equations:

[0212] (twenty four)

[0213] in:

[0214] It represents the conjugate of the preceding terms;

[0215] Obviously, this set of equations can be solved sequentially. First, it is easy to see that the solution form of equation (24) is as follows:

[0216] Y 0 = α ( T 1 )cos[ ω 0 T 0 + β ( T 1 )] (25)

[0217] To facilitate the solution Write the above expression in plural form:

[0218] (26)

[0219] in:

[0220] The imaginary unit;

[0221] Substitution get:

[0222] (27)

[0223] in:

[0224] It is an imaginary number;

[0225] This can be understood as an undamped system under periodic excitation;

[0226] To avoid the presence of a perpetual term, the right-hand side of the above equation cannot contain... or Such a term requires that the Fourier coefficient on the right-hand side of the above equation be zero:

[0227] (28)

[0228] remember:

[0229] (29)

[0230] Substituting it into equation (28), we obtain the trigonometric function form of this condition:

[0231] (30)

[0232] Separating the real and imaginary parts of the above equation, we get:

[0233] (31)

[0234] (32)

[0235] (33)

[0236] Based on the solutions of equations (32) and (33):

[0237] (34)

[0238] α = α 0 [ e εμ 1 t ( 1 + α 0 4 μ 2 ω 0 3 πμ 1 ) − α 0 ] − 1 (35)

[0239] in, and These are the initial amplitude and initial phase, which can be determined by the initial displacement and initial velocity, i.e.:

[0240] (36)

[0241] Solving for the dynamic stress in marine anchor cables:

[0242] Using the stress calculation formula:

[0243] (37)

[0244] in:

[0245] It is the diameter of the cable; It is the moment of inertia;

[0246] The dynamic stress of the marine anchor cable was calculated.

[0247] The following section verifies the accuracy of the calculation formula for the free vibration dynamic characteristics of marine anchor cables given above. The relevant design parameters for the anchor cables are as follows:

[0248] , , , , ,

[0249] This method yields the first-order modal frequencies of the structure as follows: Simulation results obtained using the finite element software Ansys Very close. Then, the maximum dynamic stress of the cable was calculated. It occurred near the fixed end ( ) place.

Claims

1. A method for analyzing the dynamic stress of marine anchor cables, characterized in that, Includes the following steps: Step 1: Establish the finite element model of the anchor cable and the vibration differential equation of the nonlinear system of the anchor cable, and linearize the nonlinear vibration differential equation to obtain the linear system; Step 2: For the linear system obtained in Step 1, calculate the mode shape function of the anchor cable. ; Step 3: Based on the finite element model of the anchor cable, calculate the stiffness matrix of each element in the finite element model. And assemble to obtain the overall stiffness matrix. ; Step 4: Solve the frequency equations of the linear system using a numerical iterative algorithm. Obtain the natural frequency of the linear system structure Then, the frequency of the anchor cable nonlinear system can be solved. ; Step 5: Based on the mode shape function of the anchor cable obtained in Step 2 The frequency of the anchor cable nonlinear system obtained in step four The perturbation method is used to solve the mode functions. ; Step 6: Solve for the dynamic stress of the marine anchor cable using the stress calculation formula.

2. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The vibration differential equation of the anchor cable nonlinear system in step one is as follows: （1） in: It is Young's modulus; Represents the bending stiffness of the main beam; Represents the linear mass per unit length of the anchor cable; Represents the tension borne by the cable; Represents the viscous damping coefficient of the system; Represents the system's vibration displacement function; This represents the additional cable force caused by elastic elongation during cable vibration, and it is equal to the dynamic strain of the cable segment. and axial stiffness The product of, i.e.: （2） in, For the vertical-to-span ratio, It is the acceleration due to gravity. For the angle of inclination of the cable, The length of the cable; This is the effective length of the cable under its own weight.

3. The method for analyzing the dynamic stress of marine anchor cables as described in claim 2, characterized in that, The linearization process of the vibration differential equation of the nonlinear system is as follows: According to the method of separation of variables, Substituting into equation (1), we obtain the linearized vibration differential equation: （3） in: ; It is the mode shape function; It is a modal function.

4. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The mode shape function of the anchor cable in step two The calculation process is as follows: Introducing coefficients , , , The mode shape function of the anchor cable Represented as: （5） in: ， ； Based on the boundary conditions: （6） Seeking , , ; make Then the mode shape function of the anchor cable is expressed as: （7）。 5. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The stiffness matrix of each element in step three as follows: （9） in: , ,in It is the structure's natural frequency. , It is the diameter of the marine anchor cable. It is an additional quality coefficient. It is the density of water; It is the cross-sectional area of ​​the cable; , , , , , The notation introduced here has no special meaning, among which: （8）。 6. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The frequency of the anchor cable nonlinear system in step four With the natural frequency of the linear system structure The relationship is as follows: （10） in: 。 7. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The specific implementation process of step five is as follows: For the equation of motion described by equation (3), let The equations of motion for the derived system are: （4） The equation of motion for the anchor cable is a fourth-order partial differential equation with second-order nonlinear damping, namely: （11） in: ; It is the average viscous damping coefficient of the system within the period; Due to equation coupling, we linearize the nonlinear damping. The basic principle is that the energy consumed by the nonlinear damping in one cycle is equal to the energy consumed by the linearized damping in one cycle, that is: （12） in: It is the energy consumed by the nonlinear damping within one cycle. It is the energy consumed by linear damping within one cycle; Therefore, we can conclude that: （15） in: It is the force of nonlinear damping of the mode, and its calculation formula is: （13） It is the linearized damping force for each mode, and its calculation formula is: （14） in These are equivalent parameters; Similarly, Cable Power and the force of each order They are the same, and ,but: （17） Therefore, we can conclude that: （18） Combining formula (4) with formulas (11) to (18), we get: （19） Multiply by each term of equation (19), and The integral yields: （20） in: ； ； ； Equation (20) can be written as the standard differential equation of motion with viscous damping and quadratic nonlinear damping: （21） in: ,in , ; ; Using a multi-scale method: （22） in: , , These represent disturbance terms at different time scales, and , , ; It is a tiny amount; Introducing the partial derivative operator: （23） Compare The coefficients of the same power yield a series of linear differential equations: （24） in: It represents the conjugate of the preceding terms; The solution form of formula (24) is as follows: （25） To facilitate the solution Formula (25) can be written in complex form: （26） in: The imaginary unit; Substitution get: （27） in: It is an imaginary number; To avoid the presence of a perpetual term, the right-hand side of the above equation cannot contain... or Such a term requires that the Fourier coefficient on the right-hand side of the above equation be zero: （28） remember: （29） Substituting it into equation (28), we obtain the trigonometric function form of this condition: （30） Separating the real and imaginary parts of formula (30) yields: （31） （32） （33） Based on the solutions of equations (32) and (33): （34） （35） in, and These are the initial amplitude and initial phase, determined by the initial displacement and initial velocity, respectively. （36）。 8. The method for analyzing the dynamic stress of marine anchor cables as described in claim 1, characterized in that, The stress calculation formula in step six is ​​as follows: （37） in: It is the dynamic stress of marine anchor cables; It is the diameter of the cable; It is the moment of inertia.

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