A method for predicting parametric vibration stability of underwater suspended tunnel anchor cable

Abstract

The application discloses a kind of underwater suspended tunnel anchor cable parametric vibration stability prediction method, belongs to underwater suspended tunnel technical field.To solve the problem that the existing prediction method for anchor cable parametric vibration stability is greatly deviated from the actual situation, the sag effect of anchor cable is determined by introducing equivalent elastic modulus, and a numerical prediction model of suspended tunnel anchor cable vibration under parametric excitation is established based on D'Alembert principle.Then, the damping effect of fluid on the structure is considered by introducing a linearized damping model, and a series of ordinary differential equations containing only time items are obtained by reducing the high-order partial differential equations varying with time and space based on Galerkin method.Finally, based on Floquet theory, a matrix eigenvalue problem is established, and the stability of anchor cable is determined by solving the eigenvalue.

Description

Technical Field

[0001] This invention belongs to the field of underwater suspended tunnel technology, specifically relating to a method for predicting the stability of anchor cables in underwater suspended tunnels under parametric vibration. Background Technology

[0002] A submerged floating tunnel (SFT), also known as an Archimedes bridge, is a tubular underwater structure completely submerged in water, relying on its own buoyancy to bear a portion of the external load and maintaining its stability through specific connection methods. Compared to traditional cross-sea structures, in deep water and other special water areas, floating tunnels are less affected by seabed geological conditions, have strong resistance to natural disasters and war damage, and offer considerable cost-effectiveness. Anchor cables are the ideal support structure for floating tunnels, generally made of steel strands or fiber materials. Anchor cables have advantages such as low flexibility and light weight. Under the axial dynamic tension load caused by the dynamic displacement of the upper tube, they will undergo para-excited vibration. Although para-excited vibration is generated by external excitation, its excitation mode does not appear in the vibration equation as a non-homogeneous term, but rather in the vibration equation in the form of variable coefficients with periodic characteristics. Therefore, mathematically speaking, the para-excited vibration equation of the structure is a differential equation with time-varying coefficients. From both a theoretical analysis and engineering application perspective, any mechanical system must first ensure the stability of its operating state before it is necessary to discuss the dynamic response characteristics of the system under stable operating conditions. Therefore, establishing an accurate and reliable method for predicting the parametric vibration stability of anchor cables in suspended tunnels, and analyzing the parametric stability of anchor cables based on this method, is a key prerequisite for the feasibility study of suspended tunnels.

[0003] Currently, most scholars disregard the bending stiffness of anchor cables, simplifying them into a tension beam model and only considering the first-order modal contribution. They then analyze the para-excited vibration stability of anchor cables based on the Lyapunov exponent method. However, this simplified approach, neglecting bending stiffness and considering only the first-order modal contribution, while achieving a certain level of computational accuracy in most cases, is flawed in its entire prediction process, both in terms of the mechanical model and mathematical derivation. This simplified method for predicting anchor cable para-excited vibration stability yields predictions that deviate significantly from reality, and in some cases, even result in the loss of higher-order modes. Therefore, to further refine the mechanical and mathematical models to better reflect reality, a theoretical analysis and numerical model of the para-excited vibration stability of anchor cables in suspended tunnels is established, comprehensively considering bending stiffness and contributions from second-order and higher modes. Based on this, the study investigates the variation of structural vibration stability characteristics with para-excitation amplitude, para-excitation frequency, and fluid damping, delving into the para-excited vibration stability characteristics of anchor cables. This aims to provide theoretical basis and technical support for the safe design and construction of key infrastructure equipment in suspended tunnels in my country. Summary of the Invention

[0004] The present invention aims to address the problem that existing methods for predicting the stability of anchor cable parametric vibrations deviate significantly from reality.

[0005] A method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration includes the following steps:

[0006] Step 1: Establish a numerical prediction model for the vibration of anchor cables in suspended tunnels under parametric excitation.

[0007] For the suspended tunnel anchor cable, take the midpoint of the anchor cable as point B, and the hinge point between the anchor cable and the seabed as point b; take segment Bb as the isolated body, and the tension at the midpoint of the anchor cable as T0. Take the moment about point b, the hinge point between the anchor cable and the seabed, and write the equilibrium equations:

[0008]

[0009] Where f is the anchor cable mid-span sag; T0 is the anchor cable tension at static equilibrium; α is the angle between the anchor cable and the bottom surface; L is the anchor cable length in the stress-free state; m = γ f A is the buoyant self-weight of the anchor cable per unit length, A' is the cross-sectional area of ​​the anchor cable, and γ' is the buoyant self-weight of the anchor cable per unit length. f The buoyant unit weight of the anchor cable;

[0010] The sag of the anchor cable is approximated as a parabola. Based on the formulas for anchor cable mid-span sag and parabolic length, the length S at which the sag occurs and the anchor cable elongation ΔL = SL are determined. Furthermore, the elastic modulus E of the anchor cable during sag is then determined. f Equivalent elastic modulus E of anchor cable eq ;

[0011] Establish a coordinate system with the z-axis along the length of the anchor cable, the x-axis perpendicular to the z-axis in the plane along the depth direction, and the y-axis perpendicular to the xz plane. The excitation of the end of the anchor cable by the pipe is Z(t) = Z c cos(ω T t), Z c For the excitation amplitude, ω T The excitation frequency is given by t, and the time is given by t; the equivalent elastic modulus E of the anchor cable during sag is also given. eq The corresponding sag effect is considered by treating the suspended tunnel anchor cable as a simply supported beam. A stress analysis of the anchor cable is performed using the infinitesimal element method. A infinitesimal element dz is taken at coordinate point z along the z-direction. The length of the infinitesimal element during vibration is... u represents the vibration displacement of the anchor cable along the x-direction; the upper end of the anchor cable element is subjected to a clockwise bending moment M, tension T, and shear force V; the lower end of the anchor cable element is subjected to a counterclockwise bending moment. tension Shear force The excitation force F at the center of the anchor cable infinitesimal segment is D dz, damping is Where C is the structural damping coefficient of the anchor cable, and F D This is the sum of the drag damping force per unit length and the additional inertial force caused by anchor cable vibration;

[0012] The equilibrium equations in the x-direction are established using D'Alembert's principle, neglecting higher-order minor quantities, and combined with the infinitesimal shear force V and bending moment M to obtain:

[0013]

[0014] Under parametric excitation, the axial tension T of the anchor cable undergoes periodic changes:

[0015] T = T0 + E eq Aε t (10)

[0016] Where T0 is the tension of the anchor cable in static equilibrium, and ε t For the dynamic strain of the anchor cable;

[0017] Furthermore, the differential equation for anchor cable vibration was determined as follows:

[0018]

[0019] Where I is the moment of inertia of the anchor cable section;

[0020] Furthermore, a numerical prediction model for the vibration of anchor cables in suspended tunnels under parametric excitation was determined:

[0021]

[0022] in,

[0023] Step 2: The vibration mode of the suspended tunnel anchor cable is taken as the standard chord vibration mode, that is:

[0024]

[0025]

[0026] Where N is the modal order; φ n (z) represents the nth order vibration mode; q n (t) represents the coordinates of the nth mode shape;

[0027] Based on fluid damping, the fluid damping force of the nth mode, the work done by the actual damping in each mode, the damping after linearization of the nth mode, and the work done by the damping after linearization of each mode, the following can be obtained:

[0028]

[0029] Where, ξ n These are the equivalent parameters for the nth mode; It is the natural frequency of the nth mode;

[0030] Solving equation (22) yields ξ n ,make Where α n It is the modal proportionality coefficient. If the maximum amplitude of the mode is given, then the damping after linearization of the nth mode is:

[0031]

[0032] in, The equivalent modal damping coefficients;

[0033] The first four modes are selected based on the Galerkin method, i.e. Multiply both sides of equation (15) by any m-order vibration mode φ m After (z), integrate along the z-direction of the anchor cable from 0 to L; we get:

[0034]

[0035] Divide each term in the above formula by And write it in matrix form:

[0036]

[0037] In the above formula, q is a 4×1 matrix column vector, represented as: q=[q1,q2,q3,q4] T M, C, and K(t) are 4×4 matrices, and the expressions for each element in the matrices are as follows:

[0038]

[0039] Equation (25) can be transformed into a first-order differential equation:

[0040]

[0041]

[0042] Further writing equation (27):

[0043]

[0044]

[0045] Substituting (17) into (26), we obtain the expressions for each element in the M, C and K(t) matrices. Then, we substitute the M, C, K(t) matrices and the natural frequency into equation (30) to obtain the expression for A(t).

[0046] Step 3: For formula (29), establish the matrix eigenvalue problem based on Floquet theory, and determine the stability of the anchor cable by solving the eigenvalues.

[0047] Furthermore, the length at which the anchor cable sag is generated. The anchor cable elongation

[0048] Furthermore, the elastic modulus E of the anchor cable when it sags f Equivalent elastic modulus E of anchor cable eq as follows:

[0049]

[0050]

[0051] Where σ=T0 / A is the anchor cable stress, ε is the anchor cable elastic strain, and ε f The strain that causes anchor cable sag, where A is the cross-sectional area of ​​the anchor cable, γ f Let E be the buoyant unit weight of the anchor cable, and E be the elastic modulus of the anchor cable.

[0052] Furthermore, the equilibrium equations in the x-direction established by D'Alembert's principle are as follows:

[0053]

[0054] Furthermore, the shear force V and bending moment M of the cable element are as follows:

[0055]

[0056] Furthermore, the dynamic strain of the anchor cable

[0057] Furthermore, the aforementioned Where, ρ w D is the density of water; C is the diameter of the anchor cable; D is the density of water. D C is the drag coefficient; m This is an additional quality coefficient.

[0058] Furthermore, the process of obtaining the linear damping model based on fluid damping, the fluid damping force of the nth mode, the work done by the actual damping in each mode, the linearized damping of the nth mode, and the work done by the linearized damping in each mode is as follows:

[0059] Fluid damping is:

[0060] The fluid damping force of the nth mode is:

[0061] Based on formula (20), the work done by the equivalent viscous damping of each mode in one vibration cycle is equal to the work done by the actual damping of each mode, and the nonlinear damping is equivalent to viscous damping.

[0062] W drag.n =W drag.n.linear (20)

[0063] Let the damping after linearization of the nth mode be:

[0064]

[0065] Based on equations (16) to (21), we finally obtain:

[0066]

[0067] Furthermore, regarding formula (29), based on Floquet theory, a matrix eigenvalue problem is established, and the stability of the anchor cable is determined by solving the eigenvalues. The specific determination process is as follows:

[0068] For formula (29), based on Floquet theory, the stability problem of the system is directly addressed:

[0069]

[0070] Matrix A(t) is an 8×8 matrix with period T = 2π / ω. T Matrix;

[0071] Introducing the state transition matrix Φ(t), and expressing the state transition matrix as the product of the periodic component and the exponential component Φ(t)=P(t)etF Where P(t) is a matrix with period T, and F is a constant matrix; the general solution is expressed as a linear combination of characteristic solutions, and the characteristic constant is λ, and its characteristic solution is:

[0072] Q(t) = e λt P(t) (32)

[0073] P(t) is a periodic matrix. Expand P(t) using a Fourier series.

[0074] Then, A(t) in equation (30) is further expressed as:

[0075]

[0076] in,

[0077] Stiffness matrix Representing the form of formula (36):

[0078]

[0079] In the formula, and If A is a constant matrix, then A(t) can be expressed as:

[0080]

[0081]

[0082] Substituting the Fourier series expansion of P(t), and equations (32) and (37) into equation (29), and comparing the coefficients of each harmonic term on both sides of the equation, we obtain:

[0083]

[0084] When i=1:

[0085]

[0086]

[0087] When i>1:

[0088]

[0089]

[0090] Rewriting equations (39) to (43) in matrix form completes the problem of establishing matrix eigenvalues ​​based on Floquet theory:

[0091] H(λ)Y=0 (44)

[0092]

[0093] H(λ)=λI-H1 (46)

[0094]

[0095] Determine system stability by finding the eigenvalues ​​of matrix H1: the system is stable when the real parts of all eigenvalues ​​are negative; the system is unstable when the real part of any eigenvalue is greater than zero.

[0096] Furthermore, P(t) can be expanded into a Fourier series as follows:

[0097]

[0098]

[0099] Beneficial effects:

[0100] This invention investigates the parametric vibration stability of anchor cables in suspended tunnels under parametric excitation, proposing a rapid and accurate prediction method that considers the contributions of bending stiffness and multiple modes. The prediction method of this invention can effectively analyze the influence of different excitation frequencies, amplitudes, and fluid damping on anchor cable stability, taking into account bending stiffness and multiple modes, thus significantly improving the prediction accuracy of parametric vibration stability of anchor cables. Attached Figure Description

[0101] Figure 1 This is a simplified calculation diagram of the anchor cable structure.

[0102] Figure 2 This is a simplified schematic diagram of a suspended tunnel anchor cable structure.

[0103] Figure 3 This is a schematic diagram of the forces acting on a micro-element of an anchor cable.

[0104] Figure 4 This represents the unstable range for different excitation amplitudes and frequencies.

[0105] Figure 5 The unstable range of para-excited vibration of anchor cables with different fluid damping is defined. Detailed Implementation Specific implementation method one:

[0107] This embodiment is a method for predicting the stability of parametrically excited vibrations of anchor cables in underwater suspended tunnels. It is a method that considers the parametric excitation effect when predicting the stability of anchor cables in suspended tunnels, and includes the following steps:

[0108] Step 1: Introduce the equivalent elastic modulus to determine the sag effect of the anchor cable, and establish a numerical prediction model for the vibration of the anchor cable in the suspended tunnel under parametric excitation based on D'Alembert's principle.

[0109] Step 2: By introducing a linearized damping model, the damping effect of the fluid on the structure is considered. Based on the Galerkin method, the high-order partial differential equations that vary with time and space are reduced to a series of ordinary differential equations containing only time terms.

[0110] Step 3: Based on Floquet theory, establish the matrix eigenvalue problem, and determine the stability of the anchor cable by solving the eigenvalues.

[0111] Furthermore,

[0112] Step 1: Introduce the equivalent elastic modulus to determine the sag effect of the anchor cable. Based on D'Alembert's principle, establish a numerical prediction model for the vibration of the anchor cable in the suspended tunnel under parametric excitation. The specific process is as follows:

[0113] First, the equivalent elastic modulus is introduced to determine the sag effect of the anchor cable:

[0114] like Figure 1 As shown, take the midpoint of the anchor cable as point B, and the hinge point between the anchor cable and the seabed as point b. Cut the anchor cable at the midpoint B, and take segment Bb as the isolated body. The tension at the midpoint of the anchor cable is T0, and its direction is along the positive z-axis (the z-axis is along the length of the anchor cable). Take the moment about point b, the hinge point between the anchor cable and the seabed, and write the equilibrium equations:

[0115]

[0116] Where f is the anchor cable mid-span sag; T0 is the anchor cable tension at static equilibrium; α is the angle between the anchor cable and the bottom surface; L is the anchor cable length in the stress-free state; m = γ f A is the buoyant self-weight of the anchor cable per unit length, A' is the cross-sectional area of ​​the anchor cable, and γ' is the buoyant self-weight of the anchor cable per unit length. f The buoyant unit weight of the anchor cable;

[0117] Approximating the sag of the anchor cable as a parabola, and based on the formula for calculating the length of a parabola, the length S at which the anchor cable sag occurs is:

[0118]

[0119] The anchor cable elongation is:

[0120]

[0121] And obtain the elastic modulus E when the anchor cable is sagging. f Equivalent elastic modulus E of anchor cable eq :

[0122]

[0123]

[0124] Where E is the elastic modulus of the anchor cable, σ = T0 / A is the anchor cable stress, ε is the elastic strain of the anchor cable, and ε f Strain when anchor cable sag occurs;

[0125] Then, a numerical prediction model for the vibration of anchor cables in suspended tunnels under parametric excitation is established:

[0126] like Figure 2 As shown, establish a coordinate system with the z-axis along the length of the anchor cable and in the plane along the depth direction ( Figure 2 The direction parallel to the paper plane) is perpendicular to the z-axis, and the direction perpendicular to the xz plane is ( Figure 2 With the y-axis perpendicular to the plane of the paper, the excitation at the end of the pipe to the anchor cable is Z(t) = Z c cos(ω T t), Z c For the excitation amplitude, ω T Let t be the excitation frequency and t be the time. Consider the fluid damping force and additional inertial force caused by the anchor cable vibration. The sag effect of the anchor cable is considered using the equivalent elastic modulus introduced by equation (5). Treating the suspended tunnel anchor cable as a simply supported beam, the stress analysis of the anchor cable is performed using the infinitesimal element method, such as... Figure 3 As shown, consider a small element dz along the z-direction at coordinate point z. The length of the small element during vibration is... u represents the vibration displacement of the anchor cable along the x-direction. The upper end of the anchor cable element is subjected to a clockwise bending moment M, tension T, and shear force V; the lower end of the anchor cable element is subjected to a counterclockwise bending moment. tension Shear force The excitation force F at the center of the anchor cable infinitesimal segment is D dz, damping is Where C is the structural damping coefficient of the anchor cable, and F D This is the sum of the drag damping force per unit length and the additional inertial force caused by the vibration of the anchor cable.

[0127] Establish the equilibrium equations in the x-direction using D'Alembert's principle:

[0128]

[0129] Ignoring higher-order minor quantities, we get:

[0130]

[0131] Based on knowledge of mechanics of materials, the expressions for the shear force V and bending moment M of the anchor cable micro-element are as follows:

[0132]

[0133] Substituting equation (8) into equation (7), we get:

[0134]

[0135] Under parametric excitation, the axial tension T of the anchor cable undergoes periodic changes. Based on the relationship between stress and strain, the expression for the axial tension T of the anchor cable is as follows:

[0136] T = T0 + E eq Aε t (10)

[0137] T0 is the tension of the anchor cable when it is in static equilibrium, and ε t The dynamic strain of the anchor cable is calculated using the following formula:

[0138]

[0139]

[0140] Neglecting small quantities in the dynamic strain of the anchor cable, the differential equation for anchor cable vibration is:

[0141]

[0142] Where I is the moment of inertia of the anchor cable section;

[0143] F D Represented as:

[0144]

[0145] Where, ρ w D is the density of water; C is the diameter of the anchor cable; D is the density of water. D C is the drag coefficient; m The additional quality factor is set to 1.

[0146] Substituting (14) into (13) and rearranging, we obtain the numerical prediction model for the vibration of the anchor cable in the suspended tunnel under parametric excitation:

[0147]

[0148] in,

[0149] Step 2: Introduce a linear damping model to consider the damping effect of the fluid on the structure. Based on the Galerkin method, reduce the order of the high-order partial differential equations that vary with both time and space to obtain a series of ordinary differential equations containing only time terms, as follows:

[0150] Because the sag-to-span ratio of the anchor cables in a suspended tunnel is very small, their vibration mode can be approximated as a standard string vibration mode, i.e.:

[0151]

[0152] Where N is the modal order; φ n (z) represents the nth order vibration mode; q n (t) represents the coordinates of the nth mode shape.

[0153] (1) Establish a linear damping model:

[0154] Fluid damping is:

[0155] The fluid damping force of the nth mode is:

[0156] Based on the principle of energy equivalence, complex nonlinear damping can be equivalent to viscous damping in a certain sense. Specifically, this can be achieved by ensuring that the work done by the equivalent viscous damping in each mode is equal to the work done by the actual damping in each mode within one oscillation cycle.

[0157] W drag.n =W drag.n.linear (20)

[0158] Let the damping after linearization of the nth mode be:

[0159]

[0160] Where, ξ n The equivalent parameters for the nth mode.

[0161] According to equations (16) to (21), we can obtain:

[0162]

[0163] in, It is the natural frequency of the nth mode.

[0164] Solving equation (22) yields: ξ n It is a parameter related to the mode and natural frequency, let Where α n It is the modal proportionality coefficient. If the maximum amplitude of the mode is given, then the damping after linearization of the nth mode is:

[0165]

[0166] in, This represents the equivalent modal damping coefficient.

[0167] (2) Based on the Galerkin method, the higher-order partial differential equations that vary with both time and space are reduced to a series of ordinary differential equations containing only time terms:

[0168] Substituting the linear damping model into equation (15), and taking the first four modes based on the Galerkin method, i.e. Multiply both sides of equation (15) by any m-order vibration mode φ m After (z), integrate along the z-direction of the anchor cable from 0 to L. The reason for only considering the first four modes here is that they are sufficient for the calculation requirements of most working conditions. We obtain:

[0169]

[0170] Divide each term in the above formula by And write it in matrix form:

[0171]

[0172] In the above formula, q is a 4×1 matrix column vector, which can be represented as: q=[q1,q2,q3,q4] T M, C, and K(t) are 4×4 matrices, and the expressions for each element in the matrices are as follows:

[0173]

[0174] Equation (25) can be transformed into a first-order differential equation:

[0175]

[0176]

[0177] Further writing equation (27):

[0178]

[0179] in,

[0180]

[0181] Substituting (17) into (26), we obtain the expressions for each element in the M, C, and K(t) matrices. Then, substituting the M, C, and K(t) matrices and the natural frequency into equation (30), we obtain the expression for A(t):

[0182]

[0183] Wherein, ω1-ω4 are the natural frequencies of the 1st to 4th order modes, respectively.

[0184] Step 3: Based on Floquet theory, establish the matrix eigenvalue problem, and determine the stability of the anchor cable by solving the eigenvalues. The specific determination method is as follows:

[0185] Equation (29) corresponding to the system in this paper is a system of linear ordinary differential equations with periodic coefficients, as shown below; based on Floquet theory, the stability problem of this system can be directly addressed:

[0186]

[0187] Matrix A(t) is an 8×8 matrix with period T = 2π / ω. T The matrix.

[0188] Introducing a new quantity needed to analyze the system: the state transition matrix or fundamental matrix Φ(t), the state transition matrix is ​​expressed as the product of the periodic and exponential components Φ(t) = P(t)e tF , where P(t) is a matrix with period T, and F is a constant matrix. The general solution can be expressed as a linear combination of characteristic solutions. Let the characteristic constant be λ, and its characteristic solution is:

[0189] Q(t) = e λt P(t) (32)

[0190] Since P(t) is a periodic matrix, it can be expanded using Fourier series as follows:

[0191]

[0192]

[0193] The A(t) in equation (30) can be further expressed as:

[0194]

[0195] in,

[0196] Stiffness matrix It can be expressed in the form of formula (36):

[0197]

[0198] In the formula, K c0 K s1 and K c1 If A is a constant matrix, then A(t) can be expressed as:

[0199]

[0200]

[0201] Substituting equations (32), (33), and (37) into equation (29), and comparing the coefficients of each harmonic term on both sides of the equation, we obtain:

[0202]

[0203] When i=1:

[0204]

[0205]

[0206] When i>1:

[0207]

[0208]

[0209] Rewriting equations (39) to (43) in matrix form completes the problem of establishing matrix eigenvalues ​​based on Floquet theory:

[0210] H(λ)Y=0 (44)

[0211]

[0212] H(λ)=λI-H1 (46)

[0213]

[0214] The stability of a system can be determined by finding the eigenvalues ​​of matrix H1. The system is stable when the real parts of all eigenvalues ​​are negative; the system is unstable when the real part of any eigenvalue is greater than zero.

[0215] Example

[0216] We selected the design parameters of a classic suspended tunnel proposed abroad and conducted calculations and analyses based on the above prediction methods:

[0217] To date, no suspended tunnel has been built in the world. Therefore, this invention selects the design parameters of a classic suspended tunnel proposed abroad as the basis for calculation, and the parameter values ​​are shown in Table 1:

[0218] Table 1 Anchor Cable Calculation Parameters

[0219]

[0220] Damping ratio

[0221] (1) The influence of excitation amplitude and frequency on the stability of anchor cable para-excited vibration:

[0222] The excitation amplitude ranges from 0 to 0.2 m, the excitation frequency ranges from 0 to 22 rad / s, and the equivalent modal damping coefficient c is taken as 0.3. Under different excitation amplitudes and frequencies, the unstable range of the anchor cable's para-excited vibration is as follows: Figure 4As shown in the figure, the black area represents the unstable region of the anchor cable's parametric vibration, and the white area represents the stable region. The inverted hump region corresponding to twice the natural frequency is the first-order unstable region, with modes 1-4 arranged from left to right. The inverted hump region corresponding to one natural frequency is the second-order unstable region. It can be seen from the figure that the first-order unstable region increases with the modal order; the range of the first-order unstable region for the same mode is much larger than the range of the second-order unstable region; when the excitation frequency is close to twice the natural frequency, the anchor cable is more likely to be in an unstable state. The unstable regions for different excitation amplitudes and frequencies are shown below. Figure 4 As shown.

[0223] (2) Influence of fluid damping on the stability of anchor cable para-excited vibration

[0224] The effect of fluid damping on the stability of anchor cable para-excited vibration, such as Figure 5 As shown. From Figure 5 It is evident that as fluid damping increases, the unstable region of anchor cable para-excited vibration gradually shifts upwards. An excitation amplitude and frequency that are unstable under low damping may become stable under high damping. In other words, the excitation amplitude required for anchor cable para-excited vibration to reach an unstable state under high damping is greater than that required under low damping. Therefore, increasing the fluid damping of the anchor cable makes it easier for the anchor cable to reach a stable state. Figure 5 The unstable range of para-excited vibration of anchor cables with different fluid damping is defined. Figure 5 (a) to (d) in the figure correspond to the unstable intervals of para-excited vibration of the anchor cable when c is 0, 0.1, 0.2 and 0.3, respectively.

[0225] By applying the method described in this invention, the stability of parametrically excited anchor cables in suspended tunnels can be rapidly predicted. Comparative studies under different working conditions provide a reference for further exploring structural stability under complex end excitations and reducing structural vibration, thereby improving the safety, economy, and comfort of suspended tunnels.

[0226] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration, characterized in that, Includes the following steps: Step 1: Establish a numerical prediction model for the vibration of anchor cables in suspended tunnels under parametric excitation. For the suspended tunnel anchor cable, take the midpoint of the anchor cable as point B, and the hinge point between the anchor cable and the seabed as point b; take segment Bb as the isolator, and the tension at the midpoint of the anchor cable is... Taking the moment at point b, the hinge point between the anchor cable and the seabed, and establishing the equilibrium equations: (1) in, For anchor cable mid-span sag; The tension of the anchor cable when in static equilibrium; The angle between the anchor cable and the bottom surface. L The length of the anchor cable in its stress-free state; The buoyancy of the anchor cable per unit length is the weight of the anchor cable itself. A The cross-sectional area of ​​the anchor cable. The buoyant unit weight of the anchor cable; The sag of the anchor cable is approximated as a parabola, and the length at which the anchor cable sag occurs is determined based on the formulas for anchor cable mid-span sag and parabolic length. S and anchor cable elongation This allows us to determine the elastic modulus of the anchor cable when it sags. Equivalent elastic modulus of anchor cable ; Establish a coordinate system along the length of the anchor cable. z The axis, in the plane along the depth direction, is intersected with... z The axis is perpendicular to The axis, the direction perpendicular to the xz plane is The shaft and tube body provide end excitation for the anchor cable. , To excite the amplitude, For the excitation frequency, For time; based on the equivalent elastic modulus of the anchor cable during sag. The corresponding sag effect is considered by treating the suspended tunnel anchor cable as a simply supported beam, and the stress analysis of the anchor cable is performed using the infinitesimal element method. z Direction at coordinate point z Take a small element dz The length of the infinitesimal element during vibration is , The anchor cable vibrates and displaces along the x-direction; the upper end of the anchor cable element is subjected to a clockwise bending moment. M ,tension T shear force V The lower end of the anchor cable element is subjected to a counterclockwise bending moment. ,tension shear force The excitation force at the center of the anchor cable micro-element segment is Damping is ;in, C The structural damping coefficient of the anchor cable. This is the sum of the drag damping force per unit length and the additional inertial force caused by anchor cable vibration; Established from D'Alembert's principle The directional equilibrium equations, neglecting higher-order small quantities, are combined with infinitesimal shear forces. V and bending moment M The result is: (9) The axial tension of the anchor cable under parametric excitation Periodic changes occur: (10) in, The tension of the anchor cable when it is in static equilibrium. For the dynamic strain of the anchor cable; Furthermore, the differential equation for anchor cable vibration was determined as follows: (13) in, The moment of inertia of the anchor cable section; Furthermore, a numerical prediction model for the vibration of anchor cables in suspended tunnels under parametric excitation was determined: (15) in, ; ; Step 2: The vibration mode of the suspended tunnel anchor cable is taken as the standard chord vibration mode, that is: (16) (17) in, N The modal order; It is the nth order vibration mode; These are the coordinates of the nth mode shape; Based on fluid damping, the first The fluid damping force of the first mode, the work done by the actual damping in each mode, and the first mode. n The damping after linearization of each mode and the work done by the damping after linearization of each mode are obtained as follows: (22) in, For the first Equivalent parameters of the first mode; It is the first The natural frequencies of the first mode; Solving equation (22) yields ,make ,in It is the modal proportionality coefficient. If it is the modal maximum amplitude, then the first... After linearization of the first mode, the damping is: (23) in, The equivalent modal damping coefficients; The first four modes are selected based on the Galerkin method, i.e. Multiply both sides of equation (15) by any... Mode shape Then, along the z-direction of the anchor cable from 0 to L Integral; obtained: (24) Divide each term in the above formula by And write it in matrix form: (25) In the above formula, A matrix column vector is represented as: , , as well as They are respectively A matrix, and the expressions for each element in the matrix are as follows: (26) Equation (25) can be transformed into a first-order differential equation: (27) (28) Further writing equation (27): (29) (30) Substituting (17) into (26), we get as well as The expressions for each element in the matrix, and then... Substituting the matrix and the natural frequency into equation (30), we obtain... expression; Step 3: For formula (29), establish the matrix eigenvalue problem based on Floquet theory, and determine the stability of the anchor cable by solving the eigenvalues.

2. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 1, characterized in that, The length at which the anchor cable sag occurs. The anchor cable elongation .

3. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 1 or 2, characterized in that, The elastic modulus of the anchor cable when it is sagging Equivalent elastic modulus of anchor cable as follows: ； in, For anchor cable stress, Elastic strain of anchor cable Strain when anchor cable sag occurs A The cross-sectional area of ​​the anchor cable. The buoyant density of the anchor cable. E This is the elastic modulus of the anchor cable.

4. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 3, characterized in that, Established by D'Alembert's principle The directional equilibrium equations are as follows: 。 5. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 4, characterized in that, The shear force of the cable element V and bending moment M as follows: 。 6. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 5, characterized in that, The dynamic strain of the anchor cable .

7. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 6, characterized in that, The ;in, The density of water; D The diameter of the anchor cable; This is the drag coefficient; This is an additional quality coefficient.

8. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 7, characterized in that, Based on fluid damping, the first n The fluid damping force of the first mode, the work done by the actual damping in each mode, and the first mode. n The process of obtaining the linear damping model by linearizing the first mode and the work done by the damping after linearization of each mode is as follows: Fluid damping is: (18) No. The fluid damping force for the first mode is: (19) Based on formula (20), the work done by the equivalent viscous damping of each mode in one vibration cycle is equal to the work done by the actual damping of each mode, and the nonlinear damping is equivalent to viscous damping. (20) Let the first After linearization of the first mode, the damping is: (21) Based on equations (16) to (21), we finally obtain: (22) 。 9. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 8, characterized in that, For formula (29), the matrix eigenvalue problem is established based on Floquet theory. The stability of the anchor cable is determined by solving the eigenvalues. The specific determination process is as follows: For formula (29), based on Floquet theory, the stability problem of the system is directly addressed: matrix It is The period is Matrix; Introducing the state transition matrix The state transition matrix is ​​represented as the product of the periodic component and the exponential component. ,in For the period is T The matrix, F Let be a constant matrix; the general solution is expressed as a linear combination of characteristic solutions, and let the characteristic constants be . Its characteristic solution is: (32) For a periodic matrix, Expand using Fourier series; Then the equation (30) A ( t This can be further expressed as: (35) in, ; Stiffness matrix Representing the form of formula (36): (36) In the formula, If it is a constant matrix, then Represented as: (37) (38) The formula Substituting the Fourier series expansion, and (32) and (37) into equation (29), and comparing the coefficients of each harmonic term on both sides of the equation, we obtain: (39) hour: (40) (41) hour: (42) (43) Rewriting equations (39) to (43) in matrix form completes the problem of establishing matrix eigenvalues ​​based on Floquet theory: (44) (45) (46) (47) Solve the matrix H 1. Eigenvalues ​​determine system stability: When the real parts of all eigenvalues ​​are negative, the system is stable; when the real part of any eigenvalue is greater than zero, the system is unstable.

10. The method for predicting the stability of underwater suspended tunnel anchor cables under parametric vibration according to claim 9, characterized in that, Will The Fourier series expansion is as follows: （33） (34)。

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