Suspension tunnel pipe-body-anchor cable three-dimensional coupling system vibration analysis method

Abstract

The present application belongs to the technical field of suspension tunnel, and proposes a kind of suspension tunnel pipe body-anchor three-dimensional coupling system vibration analysis method, three-dimensional nonlinear motion equation of anchor and three-dimensional motion equation of tunnel pipe body are established respectively using Hamilton principle, according to the geometric relationship between the displacement of the top node of anchor and the vertical displacement, lateral displacement and torsion angle of pipe body, three-dimensional deformation compatibility equation of anchor and pipe body is proposed, according to deformation compatibility equation, the motion equation of anchor and pipe body is fused, and the overall nonlinear motion equation of pipe body-anchor three-dimensional coupling system is established.Compared with prior art, the present application first proposes three-dimensional nonlinear motion equation of pipe body-anchor coupling system, simultaneously considers the dynamic characteristics of anchor and tunnel pipe body itself and the deformation compatibility relationship of connecting point, can more accurately clarify the coupling mechanism between anchor and tunnel pipe body, and can more comprehensively and objectively reflect the actual motion of pipe body-anchor coupling system under complex marine environment.

Description

Technical Field

[0001] This invention belongs to the field of suspended tunnel engineering technology, and specifically relates to a vibration analysis method for a three-dimensional coupled system of suspended tunnel tube and anchor cable. Background Technology

[0002] Suspended tunnels are a new type of transportation structure for solving major transportation projects such as crossing fjords and deep-sea passages. Figure 1 As shown, it consists of four parts: a tube structure suspended at a certain depth below the water surface, an anchor cable device anchored to an underwater foundation, connecting devices between the tunnel sections, and structures connecting the tunnel to the banks. The high flexibility and low damping of the anchor cables cause it to exhibit strong geometric nonlinear characteristics, and the dynamic behavior of the suspended tunnel under the influence of complex marine environments (such as waves, currents, and earthquakes) is extremely complex. Therefore, conducting research on the dynamic response of the suspended tunnel under marine environmental loads is of great significance for the initial design of the structure.

[0003] The vibration of a suspended tunnel under external loads will cause the tunnel body to be continuously affected by the dynamic load at the top of the anchor cable. The magnitude of the dynamic load at the top of the anchor cable is related to the displacement of the element node and the nonlinear total stiffness matrix of the element at the top of the anchor cable. The movement of the tunnel body will be affected by the nonlinear dynamic load of the anchor cable. Current research on the dynamic analysis of suspended tunnels has several limitations: some studies only analyze the dynamics of individual components such as the tunnel body or anchor cable, simplifying the motion of the tunnel body to parametric excitation at the top of the anchor cable or reducing the tunnel body to a concentrated mass, neglecting the dynamic characteristics of the anchor cable or tunnel body itself, leading to inaccurate analysis results; most studies on the dynamic response of suspended tunnel bodies simplify the constraint of the anchor cable to a massless spring or elastic foundation, neglecting the nonlinear dynamic characteristics of the anchor cable itself, resulting in inaccurate analysis results; some existing studies have established anchor cable-tunnel coupling systems that only consider the two-dimensional dynamic interaction between the anchor cable and the tunnel body, that is, only consider the coupling effect of the vertical displacement of the tunnel body and the lateral displacement of the anchor cable, neglecting the lateral displacement and torsional angle of the tunnel body and the axial and out-of-plane displacement of the anchor cable, lacking research on the three-dimensional coupling effect between the tunnel body and the anchor cable, making it difficult to comprehensively and objectively reflect the actual motion of the tunnel body-anchor cable coupling system in a complex three-dimensional marine environment. Summary of the Invention

[0004] To overcome the problems existing in the prior art, this invention focuses on suspended tunnels and establishes three-dimensional motion equations for the anchor cable and the tunnel tube based on the finite element method. Considering the geometric relationship between the node displacement at the top of the anchor cable and the vertical displacement, lateral displacement and torsional angle of the tunnel tube, a corresponding three-dimensional coordination equation is proposed, and the overall nonlinear motion equation of the suspended tunnel tube-anchor cable three-dimensional coupled system is established.

[0005] To establish the three-dimensional nonlinear motion equations of the suspended tunnel tube-anchor coupling system, it is first necessary to establish the three-dimensional motion equations of the anchor cable and the tunnel tube separately using the finite element method (such as Hamilton's principle). Then, using the three-dimensional deformation coordination equations of the connection points between the anchor cable and the tube, the three-dimensional nonlinear motion equations of the anchor cable and the three-dimensional linear motion equations of the tunnel tube are combined and integrated. Finally, the overall nonlinear motion equations of the anchor cable-tube three-dimensional coupling system are obtained.

[0006] To achieve the above objectives, this invention discloses a vibration analysis method for a three-dimensional coupled system of a suspended tunnel tube and anchor cable, the specific technical solution of which is as follows:

[0007] Step 1: Using the pipe's axial direction as the X-axis and the geocentric direction as the positive Y-axis, determine the Z-axis using the left-hand rule, and establish the overall coordinate system XYZ for the pipe-anchor cable three-dimensional coupling system; using the pipe's transverse direction as the x-axis and the geocentric direction as the positive y-axis, determine the z-axis using the right-hand rule, and establish the local coordinate system xyz for the anchor cable.

[0008] Step 2: Based on the nonlinear total stiffness matrix and mass matrix of the anchor cable element, the three-dimensional nonlinear motion equation of the anchor cable element in the local coordinate system xyz is established using Hamilton's principle. The three-dimensional nonlinear motion equation of a single anchor cable in the global coordinate system XYZ is obtained through element assembly.

[0009] It should be noted that in step 2, since the anchor cable has a large slenderness ratio and strong flexibility, the bending stiffness and shear stiffness of the anchor cable element can be ignored; based on the nonlinear total stiffness matrix and mass matrix of the anchor cable element, the three-dimensional nonlinear motion equation of the i-th anchor cable element in the local coordinate system xyz of the anchor cable is established according to Hamilton's principle as Equation (2.1):

[0010]

[0011] In the formula: d ic , and Let m be the displacement, velocity, and acceleration vector of the i-th anchor element; ic Let c be the mass matrix of the i-th anchor cable element; ic Let k be the damping matrix of the i-th anchor element; ic (d ic ) represents the nonlinear total stiffness matrix of the i-th anchor element; m ica f is the additional mass matrix of the i-th anchor cable element; icd Let i be the hydrodynamic resistance vector of the i-th anchor cable element;

[0012] Where, m ic It can be expressed as follows according to formula (2.2):

[0013]

[0014] In the formula: ρ c For the density of the anchor cable material, A c l is the cross-sectional area of ​​the anchor cable. ic It is the length of the i-th anchor cable unit;

[0015] Among them, due to k ic (d ic ) is the nonlinear total stiffness matrix, and a specific expression cannot be provided, but k ic (d ic The three-dimensional partial differential equations of motion governing the anchor cable element, derived from Hamilton's principle and based on Lagrange linear shape functions, can be obtained, i.e., k ic (d ic According to equation (2.3), it can be expressed as:

[0016]

[0017] In the formula: E c A represents the elastic modulus of the anchor cable; c The cross-sectional area of ​​the anchor cable; x i ={x′,y′,0} T ; Ni is a Lagrange linear shape function for the i-th anchor cable element;

[0018] Where, m ica It can be expressed as follows according to formula (2.4):

[0019]

[0020] In the formula: ρ f D is the density of the fluid. c The diameter of the anchor cable; l ic T1 is the length of the i-th anchor element; T1 is the transformation matrix from the anchor element coordinate system to the anchor local coordinate system xyz; C Mnc Let be the inertial force coefficient of the i-th anchor element along the transverse direction in the element coordinate system;

[0021] Among them, f icd It can be expressed as follows according to formula (2.5):

[0022]

[0023] In the formula: F icu1 F icv1 F icw1 These represent the hydrodynamic resistance along the axial direction, transverse direction, and out-of-plane direction of the i-th anchor cable element in the element coordinate system;

[0024] Among them, the hydrodynamic resistance F acting on the i-th anchor cable unit icu1 F icv1 F icw1 According to the Morison equation, we obtain equation (2.6):

[0025]

[0026] In the formula: F icu2 F icv2 F icw2 C represents the additional mass forces along the axial direction, transverse direction, and out-of-plane direction of the i-th anchor element in the element coordinate system; Dlc C Dnc C represents the drag force coefficients of the i-th anchor cable element along the axial and lateral directions in the element coordinate system, respectively; Mlc Let be the inertial force coefficient of the i-th anchor element along the axial direction in the element coordinate system; These are the axial, lateral, and out-of-plane velocities of the midpoint of the i-th anchor cable element in the element coordinate system, respectively. These are the accelerations of the midpoint of the i-th anchor cable element along the axial direction, the transverse direction, and out-of-plane direction in the element coordinate system, respectively. These represent the axial, lateral, and out-of-plane velocities of the fluid in the anchor cable element coordinate system, respectively.

[0027] It should be noted that, in step 2, the three-dimensional nonlinear motion equation of the single anchor cable in the global coordinate system XYZ is obtained as follows:

[0028] The three-dimensional nonlinear motion equations of the anchor cable element in the local coordinate system xyz, as shown in equation (2.1), are transformed into the three-dimensional nonlinear motion equations in the global coordinate system XYZ, as shown in equation (2.7):

[0029]

[0030] Where: m icg c icg k icg f icg Let X represent the mass matrix, damping matrix, stiffness matrix, and force vector of the i-th anchor element in the global coordinate system XYZ. T2 is the transformation matrix from the local coordinate system xyz to the global coordinate system XYZ of the anchor cable, satisfying d ic =T2d icg ;d icg Let d be the displacement vector of the i-th anchor element in the global coordinate system XYZ. icg ={u iag ,v iag ,w iag,u ibg ,v ibg ,w ibg} T a and b are the node numbers at both ends of the i-th anchor cable element;

[0031] Based on the three-dimensional nonlinear motion equations of the anchor cable element in the global coordinate system XYZ, the three-dimensional nonlinear motion equations of a single anchor cable in the global coordinate system XYZ are obtained by assembling the anchor cable elements sequentially, as shown in equation (2.8):

[0032]

[0033] In the formula: U mc , and Let M be the displacement, velocity, and acceleration vectors of the m-th anchor cable, respectively; mc Let C be the mass matrix of the m-th anchor cable; mc K is the damping matrix of the m-th anchor cable; mc (U mc F is the nonlinear stiffness matrix of the m-th anchor cable; mc Let M be the force vector applied to the m-th anchor cable; where M is the force vector of a single anchor cable. mc C mc K mc (U mc ), F mc Based on the m of the anchor cable unit respectively icg c icg k icg f icg Determined according to the "matching rule".

[0034] Step 3: Simulate the tunnel tube body using a three-dimensional hollow circular cross-section beam. Based on the stiffness matrix and mass matrix of the three-dimensional beam element, use Hamilton's principle to establish the three-dimensional motion equations of the tunnel tube beam element in the global coordinate system XYZ. Then, obtain the three-dimensional motion equations of the tunnel tube body in the global coordinate system XYZ through element assembly.

[0035] It should be noted that in step 3, based on the stiffness matrix and mass matrix of the three-dimensional beam element, the three-dimensional motion equation of the j-th tube beam element in the global coordinate system XYZ is established using Hamilton's principle according to equation (3.1):

[0036]

[0037] In the formula: d jb , and Let m be the displacement, velocity, and acceleration vectors of the j-th tubular beam element; jbLet c be the mass matrix of the j-th tubular beam element; jb Let k be the damping matrix of the j-th tubular beam element; jb Let m be the stiffness matrix of the j-th tubular beam element; jba f is the additional mass matrix of the j-th tubular beam element; jbd Let be the hydrodynamic resistance vector of the j-th tube beam element;

[0038] Where, m jb It can be expressed as follows according to formula (3.2):

[0039]

[0040] In the formula: ρ b It is the density of the pipe material; A b It is the cross-sectional area of ​​the tube; l jb It is the length of the j-th tubular beam element; I P The polar moment of inertia of the tunnel pipe;

[0041] Where, k jb It can be expressed as follows according to formula (3.3):

[0042]

[0043] In the formula: E b I is the elastic modulus of the tunnel pipe. Y and I Z , respectively, represent the moments of inertia of the tunnel pipe about the Y and Z axes; G is the shear modulus of the tunnel pipe material; J X Let be the moment of inertia of the tunnel tube about the X-axis.

[0044] Where, m jba It can be expressed as follows according to formula (3.4):

[0045]

[0046] In the formula: C Mnb The inertial force coefficient of the tubular beam element along the transverse direction in the element coordinate system;

[0047] Among them, f jbd Equation (3.5) is used to express this as follows:

[0048]

[0049] In the formula: F jbu1 F jbv1 F jbw1 These represent the hydrodynamic resistances of the j-th beam element along the axial, transverse, and out-of-plane directions in the element coordinate system, respectively.

[0050] Among them, the hydrodynamic resistance F acting on the j-th tube beam elementjbu1 F jbv1 F jbw1 According to the Morison equation, we obtain equation (3.6):

[0051]

[0052] In the formula: D b The outer diameter of the tube; l jb F is the length of the j-th tubular beam element; jbu2 F jbv2 F jbw2 C represents the additional mass forces along the axial direction, transverse direction, and out-of-plane direction of the j-th tubular beam element in the element coordinate system; Dlb C Dnb C represents the drag force coefficients of the tubular beam element along the axial and transverse directions in the element coordinate system, respectively; Mlb The inertial force coefficient of the tubular beam element along the axial direction in the element coordinate system; and These are the axial, transverse, and out-of-plane velocities of the midpoint of the j-th tubular beam element in the element coordinate system, respectively. and These are the accelerations of the midpoint of the j-th tubular beam element along the axial direction, the transverse direction, and out-of-plane direction in the element coordinate system, respectively. and These represent the fluid velocities along the axial direction, transverse direction, and out-of-plane velocity in the tube element coordinate system, respectively.

[0053] It should be noted that in step 3, based on the three-dimensional motion equations of the tube beam element in the global coordinate system XYZ, the three-dimensional motion equations of the tunnel tube in the global coordinate system XYZ are obtained by assembling the tube beam element in sequence, as shown in equation (3.7):

[0054]

[0055] In the formula: U t , and These are the displacement, velocity, and acceleration vectors of the tunnel pipe, respectively; M t C is the mass matrix of the tunnel pipe; t K is the damping matrix of the tunnel pipe; t F is the stiffness matrix of the tunnel duct; t Let M be the force vector applied to the tunnel body; where M is the force vector of the tunnel body. t C t K t F t Based on the m of the tubular beam element respectively jb +m jba c jb kjb f jbd Determined according to the "matching rule".

[0056] Step 4: Assuming the connection between the anchor cable and the tunnel body is hinged, establish a three-dimensional deformation coordination equation for the anchor cable and the tunnel body based on the geometric deformation relationship at the connection point. Combine the deformation coordination equation, the three-dimensional nonlinear motion equation of the anchor cable, and the three-dimensional motion equation of the tunnel body to obtain the overall nonlinear motion equation of the suspended tunnel anchor cable-tunnel body three-dimensional coupled system. Solve the overall nonlinear motion equation to obtain the dynamic response, i.e., the vibration displacement U, of the suspended tunnel body-anchor cable three-dimensional coupled system.

[0057] It should be noted that in step 4, the inventors believe that the dynamic tension of the anchor cables on both sides of the tunnel tube is symmetrical under the influence of the vertical displacement of the tube. However, the lateral displacement and torsional angle of the tunnel tube will lead to the asymmetry of the dynamic tension of the anchor cables on both sides, which will generate a certain torque in the tube. The existing two-dimensional motion analysis method is difficult to describe the complex motion of the tunnel tube under three-dimensional load. Therefore, it is necessary to comprehensively consider the influence of the vertical and lateral displacements and torsional angles of the tunnel tube and establish the dynamic equation of the tunnel tube considering the influence of the nonlinear dynamic tension at the top of the anchor cable. Based on the geometric deformation relationship between the anchor cable and the left and right ends of the tunnel tube, the three-dimensional deformation coordination equation between the anchor cable and the tunnel tube is established as shown in equation (4.1):

[0058]

[0059] In the formula: u cL v cL and w cL These represent the displacements of the anchor cable connection point at the left end of the tunnel body along the X, Y, and Z axes, respectively; u cR v cR and w cR These represent the displacements of the right-end anchor cable connection point of the tunnel body along the X, Y, and Z axes, respectively; u t v t and w t These represent the displacements of the tunnel pipe connection points along the X and Y axes, respectively; D bo θ is the outer diameter of the tunnel pipe; θ is the anchoring angle of the anchor cable; α is the rotation angle of the pipe connection point along the X-axis.

[0060] It should be noted that in step 4, the three-dimensional deformation coordination equation, the three-dimensional nonlinear motion equation of the anchor cable, and the three-dimensional motion equation of the tunnel body are combined (for example, the motion equation of the anchor cable anchor point is removed, and the displacement of the anchor cable anchor point is substituted, etc.), and the overall nonlinear motion equation of the suspended tunnel body-anchor cable three-dimensional coupled system is obtained as shown in equation (4.2):

[0061]

[0062] In the formula: U, and Let M be the displacement, velocity, and acceleration vectors of the three-dimensional coupling system of the suspended tunnel tube-anchor cable; C be the mass matrix of the three-dimensional coupling system of the tube-anchor cable; K(U) be the overall nonlinear stiffness matrix of the three-dimensional coupling system of the tube-anchor cable; F(t) be the external force vector of the three-dimensional coupling system of the tube-anchor cable; and C be the damping matrix of the three-dimensional coupling system of the tube-anchor cable. The M, C, K(U), and F(t) of the three-dimensional coupling system of the tube-anchor cable are determined based on the M of the tunnel tube. t C t K t F t M and a single anchor cable mc C mc K mc (U mc ), F mc Determined according to the "matching rule";

[0063] Where C is the Rayleigh damping matrix, expressed as in equation (4.3):

[0064] C = a0M + a1K L (4.3)

[0065] In the formula: a0 and a1 are Rayleigh damping coefficients, i.e. Where ξ is the damping ratio, ω r and ω s These are the r-th and s-th natural frequencies of the corresponding linear system, respectively; K L This represents the linear part of the overall nonlinear stiffness matrix K(U).

[0066] It should be noted that in step 4, the overall nonlinear motion equation of the three-dimensional coupled system of the suspended tunnel tube-anchor cable is solved by numerical integration method, and the preferred numerical integration method is the fourth-order Runge-Kutta method.

[0067] Compared with the prior art, the present invention has the following beneficial technical effects:

[0068] 1) This invention uses the finite element method to establish the three-dimensional motion equations of the anchor cable and the tunnel pipe in the overall coordinate system XYZ. By analyzing the geometric deformation relationship of the connection point between the anchor cable and the tunnel pipe, the three-dimensional deformation coordination equations of the anchor cable and the tunnel pipe are constructed. Based on the deformation coordination equations, the three-dimensional motion equations of the anchor cable and the tunnel pipe are fused together, and the overall nonlinear motion equations of the pipe-anchor cable three-dimensional coupling system are obtained for the first time.

[0069] 2) Based on the asymmetry of dynamic tension of the anchor cables on both sides caused by the lateral displacement and torsion angle of the tunnel tube, this invention establishes a dynamic equation for the tunnel tube considering the influence of nonlinear dynamic tension at the top of the anchor cable. Based on the geometric relationship between the nodal displacement at the top of the anchor cable and the vertical, lateral, and torsion angles of the tunnel tube, a deformation coordination equation between the anchor cable and the tunnel tube is proposed, thereby obtaining the overall nonlinear motion equation of the three-dimensional coupled system of the suspended tunnel tube and anchor cable. Compared with the existing coupled dynamic model that only considers the two-dimensional dynamic interaction, i.e. only the vertical displacement of the tunnel tube and the lateral displacement of the anchor cable, this invention can more accurately explain the coupling mechanism between the anchor cable and the tunnel tube, and more comprehensively and objectively reflect the actual motion of the tube-anchor cable coupled system in a complex three-dimensional marine environment.

[0070] 3) Based on the dynamic characteristics of the anchor cable itself, this invention establishes the partial differential control equations of the anchor cable element motion considering the influence of geometric nonlinearity, and obtains the nonlinear total stiffness matrix and mass matrix of the anchor cable element. Based on these two matrices, a three-dimensional nonlinear motion equation of the anchor cable in the global coordinate system XYZ is established. Compared with the existing model that simulates the anchor cable as a spring or elastic foundation, it can more accurately reflect its nonlinear dynamic characteristics and the analysis results are more accurate.

[0071] 4) The method described in this invention will further enrich and improve the theoretical research foundation of suspended tunnels in my country, and provide a certain theoretical basis and technical support for the future construction and development of suspended tunnels. Attached Figure Description

[0072] Figure 1 This is a schematic diagram of an anchored cable-stayed suspension tunnel structure.

[0073] Figure 2 This is a schematic diagram of the dynamic modeling of the three-dimensional coupled system of the suspended tunnel tube and anchor cable.

[0074] Figure 3 This is a schematic diagram of the differential cable element before and after deformation of the three-dimensional anchor cable.

[0075] Figure 4 This is a schematic diagram showing the geometric deformation relationship between the anchor cable and the left and right ends of the pipe.

[0076] Figure 5 Figure 1 is a schematic diagram of the three-dimensional coupling system of the suspended tunnel tube and anchor cable; Figure (a) is a schematic diagram of the elevation structure of the suspended tunnel, and Figure (b) is a schematic diagram of the cross-sectional structure of the suspended tunnel.

[0077] Figure 6 This is a time history diagram of the displacement along the Y direction at the mid-span of the tunnel pipe in Example 1.

[0078] Figure 7 This is the displacement time history diagram along the Y-axis direction at the mid-span of the fifth group of anchor cables in Example 1.

[0079] Figure 3-5 In the diagram, xyz represents the local coordinate system of the anchor cable, and XYZ represents the global coordinate system; u c v c and w c ds and ds1 represent the dynamic displacements of the 3D anchor cable element along the x, y, and z directions, respectively; ds and ds1 represent the lengths of the anchor cable element in static and dynamic states, respectively; du c dv c and dw c u c v c and w c The differential; v t w t These represent the dynamic displacements of the tunnel body along the Y and Z directions, respectively; O and O′ represent the positions of the tunnel body's cross-section center before and after deformation, respectively; D bo L is the outer diameter of the tunnel pipe. b Let L1, L2, ..., L be the lengths of the tunnel pipes. n L n+1 Here, n represents the length of the tunnel segment, θ represents the number of anchor cable sets, α represents the anchoring angle of the anchor cable, and α represents the rotation angle of the tunnel connection point along the X-axis. c The length of the anchor cable; The anchor cable inclination angle. Detailed Implementation

[0080] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0081] To establish the three-dimensional nonlinear motion equations of the suspended tunnel tube-anchor coupling system, it is first necessary to use the finite element method (such as Hamilton's principle) to establish the three-dimensional motion equations of the anchor cable and the tunnel tube separately. Then, using the three-dimensional deformation coordination equations of the connection point between the anchor cable and the tube, the three-dimensional nonlinear motion equations of the anchor cable and the tunnel tube are combined and integrated to finally obtain the overall nonlinear motion equations of the anchor cable-tube three-dimensional coupling system.

[0082] See Figure 1-5 , Figure 1 A schematic diagram of an anchored cable-stayed suspension tunnel structure; Figure 2 A schematic diagram of the dynamic modeling of a three-dimensional coupled system of a suspended tunnel tube and anchor cable; Figure 3 This is a schematic diagram of the differential cable element before and after deformation of a three-dimensional anchor cable; Figure 4A schematic diagram showing the geometric deformation relationship between the anchor cable and the left and right ends of the pipe body; Figure 5 This is a schematic diagram of a three-dimensional coupling system of a suspended tunnel tube and anchor cable. Figure 5 (a) and (b) are structural schematic diagrams of the elevation and cross-section, respectively.

[0083] This invention discloses a vibration analysis method for a three-dimensional coupled system of a suspended tunnel tube and anchor cable, the specific technical solution of which is as follows:

[0084] Step 1: Using the pipe's axial direction as the X-axis and the geocentric direction as the positive Y-axis, determine the Z-axis using the left-hand rule, and establish the overall coordinate system XYZ for the pipe-anchor cable three-dimensional coupling system; using the pipe's transverse direction as the x-axis and the geocentric direction as the positive y-axis, determine the z-axis using the right-hand rule, and establish the local coordinate system xyz for the anchor cable.

[0085] Step 2: Based on the nonlinear total stiffness matrix and mass matrix of the anchor cable element, the three-dimensional nonlinear motion equation of the anchor cable element in the local coordinate system xyz is established using Hamilton's principle. The three-dimensional nonlinear motion equation of a single anchor cable in the global coordinate system XYZ is obtained through element assembly.

[0086] Because anchor cables have a large slenderness ratio and strong flexibility, the bending stiffness and shear stiffness of the anchor cable elements can be ignored. Based on the nonlinear total stiffness matrix and mass matrix of the anchor cable elements, the three-dimensional nonlinear motion equation of the i-th anchor cable element in the local coordinate system xyz of the anchor cable is established according to Hamilton's principle as Equation (2.1):

[0087]

[0088] In the formula: d ic , and Let m be the displacement, velocity, and acceleration vector of the i-th anchor element; ic Let c be the mass matrix of the i-th anchor cable element; ic Let k be the damping matrix of the i-th anchor element; ic (d ic ) represents the nonlinear total stiffness matrix of the i-th anchor element; m ica f is the additional mass matrix of the i-th anchor cable element; icd Let i be the hydrodynamic resistance vector of the i-th anchor cable element;

[0089] Where, m ic It can be expressed as follows according to formula (2.2):

[0090]

[0091] In the formula: ρ c For the density of the anchor cable material, Ac l is the cross-sectional area of ​​the anchor cable. ic It is the length of the i-th anchor cable unit;

[0092] Among them, due to k ic (d ic ) is the nonlinear total stiffness matrix, and a specific expression cannot be provided, but k ic (d ic The three-dimensional partial differential equations of motion governing the anchor cable element, derived from Hamilton's principle and based on Lagrange linear shape functions, can be obtained, i.e., k ic (d ic According to equation (2.3), it can be expressed as:

[0093]

[0094] In the formula: E c A represents the elastic modulus of the anchor cable; c The cross-sectional area of ​​the anchor cable; x i ={x′,y′,0} T ; Ni is a Lagrange linear shape function for the i-th anchor cable element;

[0095] Where, m ica It can be expressed as follows according to formula (2.4):

[0096]

[0097] In the formula: ρ f D is the density of the fluid. c The diameter of the anchor cable; l ic T1 is the length of the i-th anchor element; T1 is the transformation matrix from the anchor element coordinate system to the anchor local coordinate system xyz; C Mnc Let be the inertial force coefficient of the i-th anchor element along the transverse direction in the element coordinate system;

[0098] Among them, f icd It can be expressed as follows according to formula (2.5):

[0099]

[0100] In the formula: F icu1 F icv1 F icw1 These represent the hydrodynamic resistance along the axial direction, transverse direction, and out-of-plane direction of the i-th anchor cable element in the element coordinate system;

[0101] Among them, the hydrodynamic resistance F acting on the i-th anchor cable unit icu1 F icv1F icw1 According to the Morison equation, we obtain equation (2.6):

[0102]

[0103] In the formula: F icu2 F icv2 F icw2 C represents the additional mass forces along the axial direction, transverse direction, and out-of-plane direction of the i-th anchor element in the element coordinate system; Dlc C Dnc C represents the drag force coefficients of the i-th anchor cable element along the axial and lateral directions in the element coordinate system, respectively; Mlc Let be the inertial force coefficient of the i-th anchor element along the axial direction in the element coordinate system; These are the axial, lateral, and out-of-plane velocities of the midpoint of the i-th anchor cable element in the element coordinate system, respectively. These are the accelerations of the midpoint of the i-th anchor cable element along the axial direction, the transverse direction, and out-of-plane direction in the element coordinate system, respectively. These represent the axial, lateral, and out-of-plane velocities of the fluid in the anchor cable element coordinate system, respectively.

[0104] The three-dimensional nonlinear motion equation of the single anchor cable in the global coordinate system XYZ is obtained by the following method:

[0105] The three-dimensional nonlinear motion equations of the anchor cable element in the local coordinate system xyz, as shown in equation (2.1), are transformed into the three-dimensional nonlinear motion equations in the global coordinate system XYZ, as shown in equation (2.7):

[0106]

[0107] Where: m icg c icg k icg f icg Let X represent the mass matrix, damping matrix, stiffness matrix, and force vector of the i-th anchor element in the global coordinate system XYZ. T2 is the transformation matrix from the local coordinate system xyz to the global coordinate system XYZ of the anchor cable, satisfying d ic =T2d icg ;d icg Let d be the displacement vector of the i-th anchor element in the global coordinate system XYZ. icg ={u iag ,v iag ,w iag ,u ibg ,v ibg ,w ibg} T a and b are the node numbers at both ends of the i-th anchor cable element;

[0108] Based on the three-dimensional nonlinear motion equations of the anchor cable element in the global coordinate system XYZ, the three-dimensional nonlinear motion equations of a single anchor cable in the global coordinate system XYZ are obtained by assembling the anchor cable elements sequentially, as shown in equation (2.8):

[0109]

[0110] In the formula: U mc , and Let M be the displacement, velocity, and acceleration vectors of the m-th anchor cable, respectively; mc Let C be the mass matrix of the m-th anchor cable; mc K is the damping matrix of the m-th anchor cable; mc (U mc F is the nonlinear stiffness matrix of the m-th anchor cable; mc Let M be the force vector applied to the m-th anchor cable; where M is the force vector of a single anchor cable. mc C mc K mc (U mc ), F mc Based on the m of the anchor cable unit respectively icg c icg k icg f icg Determined according to the "matching rule".

[0111] According to existing technology (Kun Wang, Guo-Kang Er, Vai Pan Iu. Nonlinear random vibrations of 3D cable-moored floating structures under seismic and wave excitations. Journal of Sound and Vibration. 452 (2019), 58-81.), the nonlinear total stiffness matrix and mass matrix of the anchor cable element are obtained by the following method:

[0112] 1) Based on the deformation characteristics of the three-dimensional anchor cable element before and after motion in the local coordinate system xyz of the anchor cable (e.g. Figure 3 The axial strain differential equation of the three-dimensional anchor cable element is established as shown in equation (2.9):

[0113]

[0114] In the formula: ε is the axial strain of the three-dimensional anchor cable element; Represents the derivative with respect to s; ds and ds1 are the lengths of the anchor cable element in static and dynamic conditions, respectively; u cv c and w c These represent the dynamic displacements of the three-dimensional anchor cable element along the x, y, and z directions, respectively.

[0115] 2) The partial differential equation governing the three-dimensional motion of the anchor cable under unstressed conditions is expressed by equation (2.10) according to Hamilton's principle:

[0116]

[0117] In the formula: δK is the variation of the kinetic energy of the anchor cable element relative to the unstressed state; δΠ is the variation of the potential energy of the anchor cable relative to the unstressed state; δW is the variation of the virtual work of the anchor cable; t1 and t2 are the upper and lower limits of the integration, which are any two times, t1≤t2; the unstressed state refers to the state without considering the weight of the anchor cable.

[0118] Wherein, δΠ is expressed according to equation (2.11):

[0119]

[0120] In the formula: l c T is the length of the anchor cable; E is the static tension of the anchor cable; c A represents the elastic modulus of the anchor cable; c δ represents the cross-sectional area of ​​the anchor cable; δε represents the variation of the axial strain of the anchor cable element.

[0121] Wherein, δK is expressed according to equation (2.12):

[0122]

[0123] In the formula: ρ c The density of the anchor cable material; and These represent the dynamic accelerations of the anchor cable element along the x, y, and z axes, respectively; δu c δv c and δw c These are the variational values ​​of the dynamic displacement of the anchor cable element along the x, y, and z axes, respectively.

[0124] Wherein, δW is expressed according to equation (2.13):

[0125]

[0126] In the formula: g is the acceleration due to gravity; δu c δv c and δw c These are the variational values ​​of the dynamic displacement of the anchor cable element along the x, y, and z axes, respectively. and The values ​​represent the velocities of the anchor cable element along the x, y, and z axes; f1, f2, and f3 are the external distributed loads per unit length along the x, y, and z axes, respectively; c1, c2, and c3 are the damping coefficients per unit length along the x, y, and z axes, respectively.

[0127] Substituting the axial strain differential equation (2.9) into the three-dimensional partial differential governing equation of the anchor cable under unstressed state (2.10), we obtain the three-dimensional partial differential governing equation of the anchor cable under the initial state (2.14):

[0128]

[0129] In the formula: δΠ′ is the potential energy variation of the anchor cable element relative to the initial state; δW′ is the virtual work variation of the anchor cable element relative to the initial state; the initial state refers to the stress state when only the weight of the anchor cable is considered.

[0130] Wherein, δΠ′ is expressed according to equation (2.15):

[0131]

[0132] Wherein, δW′ is expressed according to equation (2.16):

[0133]

[0134] 3) Based on the three-dimensional partial differential equations of motion of the anchor cable element in the initial state, a Lagrange shape function is used to simulate the axial strain of the anchor cable element, letting u ic ={u ic ,v ic ,w ic} T =N i d ic Let d be the displacement vector of the i-th anchor element in the local coordinate system xyz, where d ic ={u iac ,v iac ,w iac ,u ibc ,v ibc ,w ibc} T a and b are the node numbers of the two ends of the i-th anchor cable element, N i It is the Lagrange linear shape function for the i-th anchor cable element;

[0135] The potential energy variation δΠ′ of the anchor cable element relative to the initial state, as shown in equation (2.15), can be transformed into equation (2.17), and the kinetic energy variation relative to the unstressed state, as shown in equation (2.12), can be transformed into equation (2.18).

[0136]

[0137]

[0138] In the formula: T i This represents the tension of the i-th anchor cable element; xi={x′,y′,0} T ; l ic It is the length of the i-th anchor cable unit;

[0139] Based on equations (2.17) and (2.18), the three-dimensional motion partial differential control equations based on shape functions are obtained simultaneously;

[0140] Where, N i According to formula (2.19):

[0141]

[0142] In the formula, N1 = 1 - s / l ic N2 = s / l ic , l ic It is the length of the i-th anchor cable unit;

[0143] Based on the shape function-based three-dimensional motion partial differential control equations, δΠ′ is as shown in equation (2.17), δK is as shown in equation (2.18), and N... i As shown in (2.19), the nonlinear total stiffness matrix of the three-dimensional anchor cable element is obtained as shown in equation (2.20) and the mass matrix is ​​obtained as shown in equation (2.21):

[0144]

[0145]

[0146] In the formula: k ic Let m be the nonlinear total stiffness matrix of the i-th anchor element; i Let be the mass matrix of the i-th anchor cable element;

[0147] Equation (2.3) can be obtained from equation (2.20), and equation (2.2) can be obtained from equation (2.21).

[0148] Step 3: Simulate the tunnel tube body using a three-dimensional hollow circular cross-section beam. Based on the stiffness matrix and mass matrix of the three-dimensional beam element, use Hamilton's principle to establish the three-dimensional motion equations of the tunnel tube beam element in the global coordinate system XYZ. Then, obtain the three-dimensional motion equations of the tunnel tube body in the global coordinate system XYZ through element assembly.

[0149] Based on the stiffness matrix and mass matrix of the three-dimensional beam element, the three-dimensional motion equation of the j-th tubular beam element in the global coordinate system XYZ is established according to Hamilton's principle as Equation (3.1):

[0150]

[0151] In the formula: d jb , and Let m be the displacement, velocity, and acceleration vectors of the j-th tubular beam element; jb Let c be the mass matrix of the j-th tubular beam element; jb Let k be the damping matrix of the j-th tubular beam element; jb Let m be the stiffness matrix of the j-th tubular beam element; jba f is the additional mass matrix of the j-th tubular beam element; jbd Let be the hydrodynamic resistance vector of the j-th tube beam element;

[0152] Where, m jb It can be expressed as follows according to formula (3.2):

[0153]

[0154] In the formula: ρ b It is the density of the pipe material; A b It is the cross-sectional area of ​​the tube; l jb It is the length of the j-th tubular beam element; I P Let k be the polar moment of inertia of the tunnel duct; where k jb It can be expressed as follows according to formula (3.3):

[0155]

[0156] In the formula: E b I is the elastic modulus of the tunnel pipe. Y and I Z , respectively, represent the moments of inertia of the tunnel pipe about the Y and Z axes; G is the shear modulus of the tunnel pipe material; J X Let be the moment of inertia of the tunnel tube about the X-axis.

[0157] k jb and m jb All of these can be obtained from existing technology records (Lei Hujun. Study on coupled vibration of vehicle-track-bridge under non-uniform seismic excitation and driving safety [D]. Sichuan: Southwest Jiaotong University, 2014.), and will not be repeated here;

[0158] Where, m jba It can be expressed as follows according to formula (3.4):

[0159]

[0160] In the formula: C Mnb The inertial force coefficient of the tubular beam element along the transverse direction in the element coordinate system;

[0161] Among them, f jbd Equation (3.5) is used to express this as follows:

[0162]

[0163] In the formula: F jbu1 F jbv1 F jbw1 These represent the hydrodynamic resistances of the j-th tube beam element along the axial, transverse, and out-of-plane directions in the element coordinate system, respectively.

[0164] Among them, the hydrodynamic resistance F acting on the j-th tube beam element jbu1 F jbv1 F jbw1 According to the Morison equation, we obtain equation (3.6):

[0165]

[0166] In the formula: D b The outer diameter of the tube; l jb F is the length of the j-th tubular beam element; jbu2 F jbv2 F jbw2 Let C represent the additional mass forces along the axial direction, transverse direction, and out-of-plane direction of the j-th tunnel beam element in the element coordinate system; Dlb C Dnb C represents the drag force coefficients of the tubular beam element along the axial and transverse directions in the element coordinate system, respectively; Mlb The inertial force coefficient of the tubular beam element along the axial direction in the element coordinate system; and These are the axial, transverse, and out-of-plane velocities of the midpoint of the j-th tubular beam element in the element coordinate system, respectively. and These are the accelerations of the midpoint of the j-th tubular beam element along the axial direction, the transverse direction, and out-of-plane direction in the element coordinate system, respectively. and These represent the fluid velocities along the axial direction, transverse direction, and out-of-plane velocity in the tube element coordinate system, respectively.

[0167] Based on the three-dimensional motion equations of the tube beam element in the global coordinate system XYZ, the three-dimensional motion equations of the tunnel tube in the global coordinate system XYZ are obtained by assembling the tube beam element in sequence, as shown in equation (3.7):

[0168]

[0169] In the formula: U t , and These are the displacement, velocity, and acceleration vectors of the tunnel pipe, respectively; M t C is the mass matrix of the tunnel pipe; t K is the damping matrix of the tunnel pipe; t F is the stiffness matrix of the tunnel duct; t Let M be the force vector applied to the tunnel body; where M is the force vector of the tunnel body. t C t K t F t Based on the m of the tubular beam element respectively jb +m jba c jb k jb f jbd The determination is made according to the "matching rule"; the "matching rule" is a conventional technical method in this field and will not be elaborated here.

[0170] Step 4: Assuming the connection between the anchor cable and the tunnel body is hinged, establish a three-dimensional deformation coordination equation for the anchor cable and the tunnel body based on the geometric deformation relationship at the connection point. Combine the three-dimensional deformation coordination equation, the three-dimensional nonlinear motion equation of the anchor cable, and the three-dimensional motion equation of the tunnel body to obtain the overall nonlinear motion equation of the suspended tunnel body-anchor cable three-dimensional coupled system. Solve the overall nonlinear motion equation to obtain the dynamic response, i.e., the vibration displacement U, of the suspended tunnel body-anchor cable three-dimensional coupled system.

[0171] The inventors believe that the vibration of a suspended tunnel under external loads will cause the tunnel body to be continuously affected by the dynamic load at the top of the anchor cable. The magnitude of the dynamic load at the top of the anchor cable is related to the displacement of the anchor cable element node and the nonlinear total stiffness matrix of the element. Therefore, the motion of the tunnel body will be affected by the nonlinear dynamic load of the anchor cable. The dynamic tension of the anchor cables on both sides of the tunnel body is symmetrical under the influence of the vertical displacement of the tunnel body. However, the lateral displacement and torsional angle of the tunnel body will cause the dynamic tension of the anchor cables on both sides to be asymmetrical, which will generate a certain torque in the tunnel body. The two-dimensional motion analysis method in the existing technology is difficult to describe the complex motion of the tunnel body under three-dimensional loads.

[0172] Therefore, it is necessary to comprehensively consider the effects of vertical and lateral displacements and torsional angles of the tunnel duct, and establish a dynamic equation for the tunnel duct that takes into account the influence of nonlinear dynamic tension at the top of the anchor cable; based on the geometric deformation relationship between the anchor cable and the left and right ends of the tunnel duct (e.g. Figure 4 As shown in the figure, the three-dimensional deformation compatibility equation between the anchor cable and the tunnel duct is established as shown in equation (4.1):

[0173]

[0174] In the formula: u cL v cL and w cL These represent the displacements of the anchor cable connection point at the left end of the tunnel body along the X, Y, and Z axes, respectively; u cR v cR and w cR These represent the displacements of the right-end anchor cable connection point of the tunnel body along the X, Y, and Z axes, respectively; u t v t and w t These represent the displacements of the tunnel pipe connection points along the X and Y axes, respectively; D bo θ is the outer diameter of the tunnel pipe; θ is the anchoring angle of the anchor cable; α is the rotation angle of the tunnel pipe connection point along the X-axis.

[0175] By combining the three-dimensional deformation coordination equations of the anchor cable and the tunnel body, the three-dimensional nonlinear motion equations of the anchor cable system, and the three-dimensional motion equations of the tunnel body (e.g., removing the motion equations of the anchor cable anchor points, substituting the displacements of the anchor cable anchor points, etc.), the overall nonlinear motion equations of the suspended tunnel body-anchor cable three-dimensional coupled system are obtained as shown in equation (4.2):

[0176]

[0177] In the formula: U, and Let M be the displacement, velocity, and acceleration vectors of the three-dimensional coupling system of the suspended tunnel tube-anchor cable; C be the mass matrix of the three-dimensional coupling system of the tube-anchor cable; K(U) be the overall nonlinear stiffness matrix of the three-dimensional coupling system of the tube-anchor cable; F(t) be the external force vector of the three-dimensional coupling system of the tube-anchor cable; and C be the damping matrix of the three-dimensional coupling system of the tube-anchor cable. The M, C, K(U), and F(t) of the three-dimensional coupling system of the tube-anchor cable are determined based on the M of the tunnel tube. t C t K t F t M and a single anchor cable mc C mc K mc (U mc ), F mc Determined according to the "matching rule";

[0178] Where C is the Rayleigh damping matrix, expressed as in equation (4.3):

[0179] C = a0M + a1K L (4.3)

[0180] In the formula: a0 and a1 are Rayleigh damping coefficients, i.e. Where ξ is the damping ratio, ωr and ω s These are the r-th and s-th natural frequencies of the corresponding linear system, respectively; K L This represents the linear part of the overall nonlinear stiffness matrix K(U).

[0181] The overall nonlinear motion equation of the suspended tunnel tube-anchor cable three-dimensional coupled system is solved by numerical integration method to obtain the dynamic response, i.e., vibration displacement U, of the suspended tunnel tube-anchor cable three-dimensional coupled system; the preferred numerical integration method is the fourth-order Runge-Kutta method.

[0182] The establishment of the anchor cable unit coordinate system and the tube beam unit coordinate system is a conventional technical means in this field. The anchor cable unit coordinate system refers to the unit coordinate system established based on the anchor cable unit obtained after discretization of a single anchor cable. The tube beam unit coordinate system refers to the unit coordinate system established based on the tube beam unit obtained after discretization of the tunnel tube. Usually, the direction along the anchor cable unit (direction of the tube beam unit) is taken as the axis, and the other two directions perpendicular to the axis are taken as the transverse and out-of-plane directions.

[0183] Based on the asymmetry of dynamic tension of the anchor cables on both sides caused by the lateral displacement and torsion angle of the tunnel tube, this invention establishes the dynamic equation of the tunnel tube under the influence of nonlinear dynamic tension at the top of the anchor cable. Taking into account the geometric relationship between the nodal displacement of the anchor cable top and the vertical, lateral displacement and torsion angle of the tunnel tube, a deformation coordination equation between the anchor cable and the tunnel tube is proposed. Thus, the overall nonlinear motion equation of the three-dimensional coupled system of the suspended tunnel tube-anchor cable is obtained. This equation is proposed for the first time in this field. Compared with the existing coupled dynamic model (motion equation) that only considers the two-dimensional dynamic interaction, that is, only the vertical displacement of the tunnel tube and the lateral displacement of the anchor cable, it can more accurately explain the coupling mechanism between the anchor cable and the tunnel tube and more comprehensively and objectively reflect the actual motion of the tube-anchor cable coupled system in a three-dimensional complex marine environment.

[0184] Example 1;

[0185] See Figure 1-7 In this embodiment, a three-dimensional coupled vibration analysis model of the suspended tunnel tube-anchor cable system is established using Visual Studio C++ software. The tube is 500m long and has 9 sets of anchor cables, with two cables in each set. The spacing between the anchor cables in each set is 50m, and the distance between the end anchor cable and the end of the tube is 50m. At the same time, this embodiment uses Abaqus software to establish a vibration analysis model of the suspended tunnel for comparative analysis. Table 1 shows the basic parameters of the suspended tunnel.

[0186] Table 1 Basic Parameters of Suspended Tunnels

[0187]

[0188]

[0189] At the initial equilibrium position, the static buoyancy of the hollow tunnel tube is balanced with the vertical static tension of the anchor cables, that is:

[0190]

[0191] In the formula: n is the number of anchor cable sets, T cv For the vertical static tension of each anchor cable, ρ f ρ is the density of the fluid. b L is the density of the tunnel pipe. b D is the length of the tunnel pipe. bo D is the outer diameter of the tunnel pipe. bi This refers to the inner diameter of the tunnel pipe.

[0192] This embodiment uses the vibration analysis method of the three-dimensional coupled system of suspended tunnel pipe-anchor cable described in this invention for dynamic analysis. The specific steps are described in the above specific implementation method and will not be repeated here.

[0193] Assume that the suspended tunnel tube is subjected to a concentrated harmonic load F in the Y direction at mid-span (250m). y =F A sin(ωt), where F A =1×10 7 N, ω = 2 rad / s. Using numerical integration methods (such as the fourth-order Runge-Kutta method), with a time step of Δt = 0.0008 s, the dynamic response, i.e., the vibration displacement U, of the three-dimensional coupled system of the suspended tunnel tube and anchor cable is obtained. The velocity can also be obtained simultaneously. and acceleration (Not described in this embodiment.) Figure 6 This is a time history diagram of the displacement along the Y direction at the mid-span of the pipe. Figure 7 This is the displacement time history diagram along the Y direction at the mid-span of the fifth group of anchor cables. Figure 6 , 7 In the diagram, the solid line represents the numerical simulation results of the proposed method (implemented in C++ software), and the dotted line represents the simulation results of Abaqus software with the same data. It can be seen that the data match well, verifying the effectiveness of the method. Figure 6 , 7 It can also be seen that the mid-span displacement of the pipe and the mid-span displacement of the fifth group of anchor cables reach steady state after t=70s, and the frequency of their steady-state response is related to the external force F. y The frequency ω is consistent.

[0194] Matters not covered in this invention are existing technologies.

[0195] The above embodiments are only for illustrating the technical concept and features of the present invention. Their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. They should not be used to limit the scope of protection of the present invention. All equivalent changes or modifications made in accordance with the spirit and essence of the present invention should be covered within the scope of protection of the present invention.

Claims

1. A vibration analysis method for a three-dimensional coupled system of a suspended tunnel pipe-anchor cable, characterized in that: Includes the following steps: Step 1: Taking the axial direction of the pipe body as... The axis, with the Earth's center facing the direction of... Positive axis, determined using the left-hand rule. Axis, establish the global coordinate system of the pipe-anchor cable three-dimensional coupled system. ; with the transverse direction of the pipe body as The axis, with the Earth's center facing the direction of... Positive axis, determined using the right-hand rule. Establish the local coordinate system for the anchor cable. ; Step 2: Based on the nonlinear total stiffness matrix and mass matrix of the anchor cable element, establish the anchor cable element in the local coordinate system of the anchor cable using Hamilton's principle. The three-dimensional nonlinear motion equations in the model are obtained by assembling individual elements to obtain the motion of a single anchor cable in the global coordinate system. The three-dimensional nonlinear motion equations in the model; Step 3: Simulate the tunnel tube using a three-dimensional hollow circular cross-section beam. Based on the stiffness and mass matrices of the three-dimensional beam element, establish the tunnel tube beam element in the global coordinate system using Hamilton's principle. The three-dimensional motion equations are obtained, and the tunnel body in the global coordinate system is obtained through unit assembly. The three-dimensional equations of motion are as follows; Step 4: The connection between the anchor cable and the tunnel tube is hinged. Based on the geometric deformation relationship of the connection point between the anchor cable and the tunnel tube, a three-dimensional deformation coordination equation for the anchor cable and the tunnel tube is established. The three-dimensional deformation coordination equation, the three-dimensional nonlinear motion equation of the anchor cable system, and the three-dimensional motion equation of the tunnel tube are combined to obtain the overall nonlinear motion equation of the suspended tunnel anchor cable-tube three-dimensional coupled system. The dynamic response of the suspended tunnel tube-anchor cable three-dimensional coupled system is obtained by solving the overall nonlinear motion equation.

2. The vibration analysis method for the three-dimensional coupled system of suspended tunnel pipe-anchor cable as described in claim 1, characterized in that: In step 2, based on the nonlinear total stiffness matrix and mass matrix of the anchor cable element, the first... An anchor cable element in the local coordinate system of the anchor cable The three-dimensional nonlinear motion equations are established using Hamilton's principle according to equation (2.1): (2.1) In the formula: , and For the first Displacement, velocity, and acceleration vectors of each anchor cable element; For the first Mass matrix of each anchor cable unit; For the first Damping matrix of each anchor cable element; For the first Nonlinear full stiffness matrix of each anchor cable element; For the first Additional mass matrix of each anchor cable element; For the first The hydrodynamic resistance vector of each anchor cable unit; in, It can be expressed as follows according to formula (2.2): (2.2) In the formula: The density of the anchor cable material, Let be the cross-sectional area of ​​the anchor cable. It is the first The length of each anchor cable unit; in, The anchor cable element, derived from Hamilton's principle, is obtained based on the three-dimensional partial differential equations of motion of Lagrange linear shape functions, and is expressed as equation (2.3): (2.3) In the formula: The elastic modulus of the anchor cable; The cross-sectional area of ​​the anchor cable; ; ; ; ; This is the first Lagrange linear shape function for each anchor cable element; in, It can be expressed as follows according to formula (2.4): (2.4) In the formula: The density of the fluid; The diameter of the anchor cable; For the first The length of each anchor cable unit; From the anchor cable element coordinate system to the anchor cable local coordinate system The transformation matrix of the transformation; For the first The inertial force coefficient of each anchor cable element in the element coordinate system along the lateral direction.

3. The vibration analysis method for the three-dimensional coupled system of suspended tunnel pipe-anchor cable as described in claim 2, characterized in that: The first Hydrodynamic resistance vector of each anchor cable unit It can be expressed as follows according to formula (2.5): (2.5) In the formula: , , They were respectively in the second The hydrodynamic resistance of an anchor cable element along the axial, lateral, and out-of-plane directions in the element coordinate system; in, , , According to the Morison equation, we obtain equation (2.6): (2.6) In the formula: , , The first Additional mass forces of an anchor cable element along the axial, transverse, and out-of-plane directions in the element coordinate system; , The first The drag force coefficients of each anchor cable element along the axial and lateral directions in the element coordinate system; For the first The inertial force coefficient of an anchor cable element along the axial direction in the element coordinate system; , , The first The velocity of the midpoint of each anchor cable element along the axial direction, transverse direction, and out-of-plane direction in the element coordinate system; , , The first The acceleration of the midpoint of an anchor cable element along the axial direction, transverse direction, and out-of-plane direction in the element coordinate system; , , These represent the fluid velocities along the axial direction, transverse direction, and out-of-plane direction of the anchor element in the anchor element coordinate system, respectively.

4. The vibration analysis method for the three-dimensional coupled system of suspended tunnel pipe-anchor cable as described in claim 1, characterized in that: In step 2, the single anchor cable is in the global coordinate system The three-dimensional nonlinear motion equations in the figure are obtained as follows: Place the anchor cable element in the local coordinate system of the anchor cable. The three-dimensional nonlinear motion equations in equation (2.1) are transformed into the global coordinate system. The three-dimensional nonlinear motion equations are shown in equation (2.7): (2.7) In the formula: , , , The first Each anchor cable element in the global coordinate system The mass matrix, damping matrix, stiffness matrix, and force vector in the equation. , , , ; For the local coordinate system of the anchor cable Towards the global coordinate system The transformation matrix of the transformation satisfies ; For the first Each anchor cable element in the global coordinate system The displacement vector in; Based on the anchor cable element in the global coordinate system The three-dimensional nonlinear motion equations in the model are obtained by assembling the anchor cable elements sequentially, thus obtaining the equations of motion for a single anchor cable in the global coordinate system. The three-dimensional nonlinear motion equations are shown in equation (2.8): (2.8) In the formula: , and The first The displacement, velocity, and acceleration vectors of the anchor cable; For the first The mass matrix of the anchor cable; For the first Damping matrix of the anchor cable; For the first The nonlinear stiffness matrix of the anchor cable; To apply in the Force vector on the root anchor cable.

5. The vibration analysis method for a three-dimensional coupled system of a suspended tunnel pipe-anchor cable as described in claim 1, characterized in that: In step 3, based on the stiffness matrix and mass matrix of the three-dimensional beam element, the tunnel tube beam element in the global coordinate system... The three-dimensional equations of motion are established using Hamilton's principle according to equation (3.1): (3.1) In the formula: , and The first Displacement, velocity, and acceleration vectors of a single tubular beam element; For the first Mass matrix of each tubular beam element; For the first Damping matrix of a single tubular beam element; For the first Stiffness matrix of a single tubular beam element; For the first Additional mass matrix of each tubular beam element; For the first The hydrodynamic resistance vector of a single tube beam element; in, It can be expressed as follows according to formula (3.2): (3.2) In the formula: It is the density of the pipe material; It is the cross-sectional area of ​​the tube; It is the first The length of each tubular beam element; The polar moment of inertia of the tunnel pipe; in, It can be expressed as follows according to formula (3.3): (3.3) In the formula: The elastic modulus of the tunnel pipe; and These are the tunnel pipe body relative to the global coordinate system. middle shaft and Moment of inertia of the axis; Shear modulus of the pipe material; For tunnel pipe body Moment of inertia of the shaft; in, It can be expressed as follows according to formula (3.4): (3.4) In the formula: The density of the fluid; Let be the inertial force coefficient of the tubular beam element along the transverse direction in the element coordinate system.

6. The vibration analysis method for the three-dimensional coupled system of suspended tunnel pipe-anchor cable as described in claim 5, characterized in that: The first Hydrodynamic resistance vector of each tube beam element It can be expressed as follows according to formula (3.5): (3.5) In the formula: , , The first Hydrodynamic resistance of a single tube beam element along the axial, transverse, and out-of-plane directions in the element coordinate system; in, , , According to the Morison equation, we obtain equation (3.6): (3.6) In the formula: The outer diameter of the tunnel pipe; For the first The length of each tubular beam element; , , The first Additional mass forces along the axial, transverse, and out-of-plane directions of a tube beam element in element coordinates; , These are the drag force coefficients of the tubular beam element along the axial and transverse directions in the element coordinate system, respectively. The inertial force coefficient of the tubular beam element along the axial direction in the element coordinate system; , and The first The velocity of the midpoint of a tube beam element along the axial direction, transverse direction, and out-of-plane direction in the element coordinate system; , and The first The acceleration of the midpoint of a tube beam element along the axial direction, transverse direction, and out-of-plane direction in the element coordinate system; , and These represent the axial, lateral, and out-of-plane velocities of the fluid in the tube beam element coordinate system, respectively.

7. The vibration analysis method for a three-dimensional coupled system of a suspended tunnel pipe-anchor cable as described in claim 1, characterized in that: In step 3, based on the three-dimensional linear motion equation of the tube beam element in the global coordinate system XYZ, the three-dimensional linear motion equation of the tunnel tube in the global coordinate system XYZ is obtained by assembling the tube beam elements in sequence, as shown in equation (3.7): (3.7) In the formula: , and These are the displacement, velocity, and acceleration vectors of the tunnel pipe, respectively. The mass matrix of the tunnel pipe; The damping matrix of the tunnel pipe; Here is the stiffness matrix of the tunnel pipe; This is the force vector applied to the tunnel pipe.

8. The vibration analysis method for a three-dimensional coupled system of a suspended tunnel pipe-anchor cable as described in claim 1, characterized in that: In step 4, based on the geometric deformation relationship between the anchor cable and the left and right ends of the tunnel pipe, the three-dimensional deformation compatibility equation between the anchor cable and the tunnel pipe is established as shown in equation (4.1): (4.1) In the formula: , and The anchor cable connection points at the left end of the tunnel body are respectively along axis, shaft and Displacement in the axial direction; , and These represent the displacements of the right-end anchor cable connection point of the tunnel body along the X, Y, and Z axes, respectively. , and The tunnel pipe connection points are respectively along and Displacement in the axial direction; The outer diameter of the tunnel pipe; The anchorage angle of the anchor cable; For the pipe connection point along Angle of rotation along the axis.

9. The vibration analysis method for a three-dimensional coupled system of a suspended tunnel tube and anchor cable as described in claim 1, characterized in that: In step 4, the three-dimensional deformation coordination equations of the anchor cable and the tunnel body, the three-dimensional nonlinear motion equations of the anchor cable system, and the three-dimensional motion equations of the tunnel body are combined and rearranged to obtain the overall nonlinear motion equations of the suspended tunnel body-anchor cable three-dimensional coupling system as shown in equation (4.2): (4.2) In the formula: , and These are the displacement, velocity, and acceleration vectors of the three-dimensional coupled system of the suspended tunnel tube-anchor cable; The mass matrix of the pipe-anchor cable three-dimensional coupled system; The damping matrix of the tube-anchor cable three-dimensional coupling system; The overall nonlinear stiffness matrix of the three-dimensional coupled pipe-anchor system; The external force vector of the pipe-anchor cable three-dimensional coupled system; The damping matrix is ​​the three-dimensional coupling system of the tube and anchor cable.

10. The vibration analysis method for a three-dimensional coupled system of a suspended tunnel tube-anchor cable as described in claim 1, characterized in that: In step 4, the numerical integration method is used to solve the overall nonlinear motion equation of the three-dimensional coupled system of the anchor cable-tube body in the suspended tunnel.

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