Multi-field coupling nonlinear vibration prediction method for deep-sea mining hydraulic hoisting pipe

Abstract

The application discloses a deep-sea mining hydraulic lifting pipe multi-field coupling nonlinear vibration prediction method and relates to the technical field of deep-sea mining. The method comprises the following steps: based on energy analysis of a system, a pipe column three-degree-of-freedom vibration control equation is established according to a Hamilton variation principle; fluid force adopts a van der Pol nonlinear vibration equation, and vortex-induced vibration time-history responses of each point of the lifting hard pipe are obtained through coupling iteration solution. The method is highly consistent with actual simulation experiment data, and the accuracy of the method is proved, thereby laying a method foundation for revealing vibration responses of the deep-sea mining lifting hard pipe.

Description

Technical Field

[0001] The present invention relates to the technical field of deep-sea mining, and in particular to a method for predicting multi-field coupled nonlinear vibration of a deep-sea mining hydraulic riser. Background Art

[0002] With the rapid development of science and technology and the global economy, demand for the quantity and variety of mineral resources is increasing. Land-based metal mineral resources are becoming increasingly depleted, and the shortage of these resources is becoming increasingly prominent. To alleviate this problem, humanity is gradually turning its attention to the underdeveloped deep sea. It is estimated that the guaranteed reserves of terrestrial metal mineral resources are limited, while the seabed's mineral deposits are extremely rich. Therefore, large-scale research, exploration, and exploitation of the mineral resources hidden in the seabed have become a key strategic goal for countries around the world.

[0003] However, during deep-sea mining, thousands of meters of pipe strings are subject to vortex-induced vibrations (VIVs), a complex nonlinear response of fluid-structure interaction, and one of the most detrimental factors in the dynamics of deep-sea lifting pipes. Pipes must match the operating water depth and can reach lengths of thousands of meters, resulting in a large aspect ratio. When faced with the threat of fluid-induced vibrations, the nonlinear vibrations caused by longitudinal and transverse coupling become more pronounced and intense. Furthermore, complex dynamic issues such as large deformations caused by the relatively free boundary conditions at the lower end cannot be ignored. The combined effects of these nonlinear factors make pipes highly susceptible to fatigue failure and buckling. Failure can lead to leakage accidents, resulting in significant economic losses, severe environmental pollution, and ecological safety concerns. Existing research on deep-sea mining lifting pipes primarily focuses on unidirectional vibrations and fluid-induced displacements, with insufficient research on three-dimensional vibrations. Furthermore, the combined effects of nonlinear factors such as fluid-structure interaction, longitudinal and transverse coupling, and large deformations are not considered. In addition, current research mainly focuses on water depths of 1,000 to 3,000 meters, but metallic nodules are mainly distributed on the seabed surface at depths of 4,000 to 6,000 meters. Existing research content is insufficient to support mining in deeper waters. Summary of the Invention

[0004] In view of the above technical problems, the purpose of the present invention is to provide a method for predicting multi-field coupled nonlinear vibration of deep-sea mining hydraulic riser to address the defects of the existing technology.

[0005] To achieve the above technical objectives, the present invention adopts the following technical solution: a method for studying the nonlinear vibration of a pipe string for deep-sea mining, comprising the following steps:

[0006] The top of the pipe string is taken as the coordinate origin, the vertical axis of the pipe string when it is not deformed is taken as the z-axis, and the downward direction is the positive direction; the direction parallel to the water flow is taken as the x-axis, and the direction perpendicular to the water flow is taken as the y-axis. A three-degree-of-freedom vibration control model of the pipe string is established:

[0007]

[0008] Where m υ is the unit mass of the string, m a is the additional fluid mass per unit string, m i is the additional mass of the internal fluid, υ x ,υ y ,u respectively represent the displacement of any point on the string in the x, y, and z directions, the superscript ′ represents the first-order derivative with respect to z, the superscript ″ represents the second-order derivative with respect to z, the superscript ″″ represents the fourth-order derivative with respect to z, and the superscript · Indicates the first-order derivative with respect to time t, the superscript ·· Indicates the second-order derivative with respect to time t, ρ υ is the string density, I is the polar moment of inertia, U is the fluid velocity inside the string, EI is the bending stiffness of the string, D i is the inner diameter of the tubing, f is the friction coefficient caused by fluid viscosity, ρ i is the density of the fluid inside the string, A is the cross-sectional area of ​​the string, E is the elastic modulus of the string material, M C is the mass of the intermediate chamber, M is the mass of the pipe string, and L is the length of the pipe string;

[0009] Establish a fluid force analysis model:

[0010]

[0011]

[0012] Where, F D is the drag force, F L is the lift, ρ w is the external fluid density, U C is the external fluid flow rate, D o is the outer diameter of the pipe string, is the steady-state drag coefficient, q x ,q y are the dimensionless wake oscillator variables in the downstream and cross-stream directions, respectively, and C l is the pulsating lift coefficient;

[0013] Establish the control equation of the wake oscillator:

[0014]

[0015]

[0016] Where, ε x ,ε y ,A x ,A y All are dimensionless parameters determined experimentally;

[0017] Combining the three-degree-of-freedom vibration control model, the fluid force analysis model, and the wake oscillator control equation, the nonlinear vibration control model of the string can be obtained:

[0018]

[0019] Create boundary conditions:

[0020] u(0,t)=u boat (t)

[0021]

[0022] υ x (0,t)=0

[0023]

[0024] υ y (0,t)=0

[0025]

[0026] EIυ″ x | (0,t) =0,EIυ″ x | (L,t) =0,EIυ″ Y | (0,t) =0,EI″υ y | (L,t) =0

[0027] The pipe string is discretized using the cubic Hermit interpolation function and matrix assembly is performed to obtain the pipe string vibration control matrix equation. The Newmark-β method and Newton Raphson method are then used to jointly solve the pipe string vibration control matrix equation. At the same time, the fourth-order Runge-Kutta method is used to solve the wake oscillator model.

[0028] The beneficial effects of the present invention are:

[0029] The present invention addresses the vibration issues of deep-sea hydraulic lifting pipes for hydraulic lifting applications. Using the finite element method and the Hamiltonian variational principle, a three-dimensional nonlinear vibration model for deep-sea hydraulic lifting pipes under multi-field and multi-scale coupling is established. This model takes into account nonlinear factors such as longitudinal and transverse coupling effects, vortex-induced effects, and large deformations of the lifting pipes. The longitudinal and transverse coupling nonlinear vibration model of the pipe string is discretely solved using the finite element method, the Newmark-β method, and the Newton-Raphson iterative method. The wake oscillator model is solved using the fourth-order Runge-Kutta method. The theoretical model is validated based on experimental parameters and test data from literature, laying the foundation for a model that reveals the vibration response of deep-sea hydraulic lifting pipes. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 Schematic diagram of the lifting string and its coordinate system structure;

[0031] Figure 2 Schematic diagram of the external fluid force applied to the lifting string;

[0032] Figure 3 Solve the flow chart for the computational model;

[0033] Figure 4 This is the solution result and verification diagram of the present invention. DETAILED DESCRIPTION

[0034] In order to make the technical solutions and technical advantages of the present invention more clear, the technical solutions in the implementation process of the present invention will be clearly and completely described below using embodiments and drawings.

[0035] The setting method of this embodiment includes the following steps:

[0036] S1. Establish a three-degree-of-freedom vibration control model for the string:

[0037] like Figure 1 As shown, the top of the pipe is taken as the origin of coordinates, the vertical axis of the pipe without deformation is taken as the z axis, and the downward direction is the positive direction; the direction parallel to the water flow is taken as the x axis, and the direction perpendicular to the water flow is taken as the y axis, and υ is used as the coordinate system. x ,υ y ,u represent the displacement of any point on the string in the x, y, and z directions respectively;

[0038] Assuming that the string has a small deformation, the displacement field function is obtained according to Kirchhoff's hypothesis as shown in Equation 1:

[0039]

[0040] Where, u1 is the displacement field function in the x direction, u2 is the displacement field function in the y direction, and u3 is the displacement field function in the z direction; vx is the displacement of the pipe in the x direction, v y is the displacement of the string in the y direction, u(z,t) is the displacement of the string in the z direction, and t is time.

[0041] Based on the assumption of small deformation of the pipe string, the Green strain is shown in Equation 2:

[0042]

[0043] Where, ε xx is the axial tensile strain in the x direction, ε yy is the axial tensile strain in the y direction, ε zz is the axial tensile strain in the z direction, ε xz is the shear strain in the xz plane, ε yz is the shear strain in the yz plane, ε xy is the shear strain in the xy plane.

[0044] Substituting Equation 1 into Equation 2, we can obtain:

[0045]

[0046] Calculate the speed according to formula 4 and formula 5:

[0047]

[0048]

[0049] Where, ρ i is the density of the fluid inside the string, U is the flow rate of the fluid inside the string, is the mass source, ΔV is the volume of the control volume; p is the internal pressure; is the pipe inclination; h is the fluid height of the mixture; and τ is the shear stress. The terms on the right side of Equation 5 represent the pressure gradient, water level gradient, interfacial and wall friction, and gravity, respectively.

[0050] According to Equation 3, the strain energy and kinetic energy of the pipe string can be obtained as shown in Equations 6 and 7 respectively:

[0051]

[0052]

[0053] Where L is the pipe length, m; E is the elastic modulus of the pipe material, Pa; A is the cross-sectional area of ​​the pipe, m 2 ρ υ is the string density, EI is the bending stiffness of the string; M C is the mass of the intermediate warehouse; Represents the internal flow velocity; the superscript ' represents the first-order derivative of the parameter on the z coordinate, the superscript ″ represents the second-order derivative of the parameter on the z coordinate, and the superscript · Indicates that the first derivative of the parameter at time t is calculated. The superscript apostrophe and a dot indicate that the derivative of the parameter at z coordinate is calculated first and then at time t. For example, u′ represents the first derivative of u in the z coordinate system, and u″ is the second derivative of u in the z coordinate system. represents the first-order derivative of u with respect to time t, Represents υ x After taking the first derivative of the z coordinate, take the first derivative of the time t, m υ is the unit mass of the string, A i is the cross-sectional area of ​​the flow channel in the string, and I is the polar moment of inertia.

[0054] Since the fluid inside the string has a certain viscosity, the energy loss caused by this viscosity is shown in Equation 8:

[0055]

[0056] Where f is the friction coefficient caused by fluid viscosity, D i is the inner diameter of the tubing string.

[0057] The structural damping, the external hydrodynamic force and the work done by the damping are shown in Equation 9:

[0058]

[0059] Where, F cx ,F cy ,F cz is the structural damping force, ω s is the natural angular frequency of the simply supported pipe column, F D ,F L They represent the drag force of the external fluid in the downstream direction and the lift in the lateral direction, w g is the longitudinal force of the string.

[0060]

[0061] M is the total mass of the string, D o is the outer diameter of the pipe, m a is the additional fluid mass per unit string, F ax is the inertial force in the x direction, F ay is the inertial force in the y direction.

[0062] According to Hamilton's principle and variational principle Substituting Equations 7 and 8 into equations 8, we can obtain the vibration equations of the pipe string in the x, y, and z directions, as shown in Equation 11:

[0063]

[0064] m i is the additional mass of the internal fluid, ρ υ is the column density, and the superscript “” indicates the fourth-order derivative with respect to z.

[0065] Equation 11 is the three-degree-of-freedom vibration control equation of the lifting rigid pipe established based on the energy analysis of the system and the Hamiltonian variation principle.

[0066] S2. After establishing the three-degree-of-freedom vibration control equation, the van der Pol nonlinear vibration equation is used to analyze the fluid force of the string. The specific steps are as follows:

[0067] Assume the relative velocity between the fluid and the string is V r , the outflow velocity of the column is U c Based on the steady flow, the steady drag and lift acting on the string are as follows: Figure 2 shown.

[0068] According to the Morison equation, the steady-state drag force and lift force acting on the string are:

[0069]

[0070] Where, is the steady-state drag coefficient, is the steady-state lift coefficient, ρ w is the external fluid density, V r is the relative velocity between the fluid outside the string and the string.

[0071] Depend on Figure 2 It can be seen that:

[0072]

[0073] U c is the external fluid flow rate.

[0074] Then the fluid force components acting on the pipe string in the x and y directions are:

[0075]

[0076] In addition to the steady-state fluid component, the forces acting on the pipe string also include the pulsating drag force and pulsating lift force in a harmonic form.

[0077]

[0078] Where, F D ′,C D , F L ′,C Lare the pulsating drag force, the pulsating drag force coefficient, the pulsating lift force, and the pulsating lift coefficient, respectively. Combining equations (12), (13), (14) and (15), the external fluid force acting on the string can be obtained:

[0079] F D =F x +F D ′,F L =F y +F L ′ Equation 16

[0080] Since the outer cross section of the pipe column is circular, the steady-state lift coefficient It is usually 0. At the same time, simplify the equation, that is, Also ignore the effects of higher-order terms:

[0081]

[0082] The van der Pol nonlinear vibration equation is used to describe the shedding characteristics of the fluid vortex, and the control equation of the wake oscillator is:

[0083] Where q x ,q y is the dimensionless wake oscillator variable in the downstream and cross-stream directions, ω s is the vortex shedding frequency, S t is the Sterthal number. x ,ε y ,A x ,A y All are dimensionless parameters determined experimentally.

[0084] Dimensionless wake oscillator variables q in the downstream and transverse directions x and q y Introduced into the drag coefficient C D (=C′ d q x / 2) and the pulsating lift coefficient C L (=C′ l q y / 2), the final form of the external fluid force acting on the tube can be obtained:

[0085]

[0086] make The nonlinear vibration control equation of the lifting rigid pipe can be obtained:

[0087]

[0088] According to the conclusions of Facchinetti and Violette, the parameters of the van der pol equation are as follows: The experimentally determined values ​​are obtained, A x =48,A y =12,ε x =1.2,ε y =0.3,C′ d =0.3,C′ l =0.4.

[0089] This study assumes that the upper end of the deep-sea mining lifting pipe is connected to a ball joint and the lower end is suspended with an intermediate chamber. Its boundary conditions can be expressed as:

[0090]

[0091] The above equations 20 and 21 are the nonlinear vibration control model of the string. After the model is established, it needs to be solved.

[0092] S3. When solving the above model, first use the cubic Hermit difference function to discretize the pipe string into N units (a total of N+1 nodes), with a unit length of l. The displacements of the lifting pipe in the x, y, and z directions are respectively expressed as Specifically, see Equation 20, Equation 21, and Equation 22:

[0093]

[0094]

[0095]

[0096] By assembling the unit string control equations into a matrix, we can obtain the string vibration control matrix equation shown in Equation 25:

[0097]

[0098] Where M is the mass matrix of the system; C is the damping matrix of the system; K is the stiffness matrix of the system, and the specific composition is as follows:

[0099]

[0100]

[0101]

[0102]

[0103] Then the Newmark-β method and Newton Raphson method are used to solve the riser control equation. The wake oscillator model is solved using the fourth-order Runge-Kutta method. The solution process is as follows: Figure 3 shown.

[0104] In order to further verify the accuracy of this method, a specific example is used to demonstrate it below.

[0105] In their paper, Experimental investigation on vortex-induced vibrations of ahang-off evacuated drilling riser[J]. Nonlinear Dynamics, 2020, 102(3):1499-1516, Mao et al. conducted a model verification of a hang-off evacuated riser in a 2.3 m water tank. The pipe string was made of polyethylene, and the boundary conditions of the experimental model were a fixed top end and a free bottom end. The parameters are shown in Table 1.

[0106] Table 1 Parameters

[0107] parameter Numerical Tube length L 2.7m <![CDATA[Outer diameter D o > 0.020m <![CDATA[Inner diameter D i > 0.016m Elastic modulus E 1.9GPa <![CDATA[Seawater density ρ w > <![CDATA[1025kg·m -3 ]]> String density <![CDATA[0.9kg·m -3 ]]> Outflow velocity <![CDATA[0.1-0.8m·s -1 ]]> <![CDATA[Strouhal number S t > 0.18

[0108] In the numerical calculation, the string is discretized into 64 units, the time step is 0.001s, and the final measured results are as follows: Figure 4 As shown, Figure 4 a is the verification diagram of the RMS distribution in the cross flow direction, Figure 4 b is a verification diagram of the RMS distribution in the downstream direction. As can be seen from the figure, the calculated results of the present invention are highly consistent with the actual experimental results, which confirms the accuracy of the present invention.

[0109] The embodiments described above are only some embodiments of the present invention, which are used to describe the basic principles, implementation purposes and detailed processes of the present invention, and do not limit the scope of use of the present invention. Any modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention all fall within the scope of the technical solution of the present invention. The present invention has been disclosed in the above preferred embodiments, but it should be understood by those skilled in the art that these embodiments are only used to describe the present invention and should not be construed as limiting the scope of the present invention. Without departing from the principles of the present invention, further improvements to the present invention should also be considered to be within the scope of protection of the present invention.

Claims

1. A method for predicting multi-field coupled nonlinear vibration of a deep-sea mining hydraulic riser, characterized in that: The following steps are involved: The top of the pipe string is taken as the coordinate origin, the vertical axis of the pipe string when it is not deformed is taken as the z-axis, and the downward direction is the positive direction; the direction parallel to the water flow is taken as the x-axis, and the direction perpendicular to the water flow is taken as the y-axis. A three-degree-of-freedom vibration control model of the pipe string is established: Where m υ is the unit mass of the string, m a is the additional fluid mass per unit string, m i is the additional mass of the internal fluid, υ x ,υ y , u represents the displacement of any point on the string in the x, y, and z directions respectively, the superscript ′ represents the first-order derivative with respect to z, the superscript ″ represents the second-order derivative with respect to z, the superscript ″″ represents the fourth-order derivative with respect to z, the superscript · represents the first-order derivative with respect to time t, the superscript represents the second-order derivative with respect to time t, ρ υ is the string density, I is the polar moment of inertia, U is the fluid velocity inside the string, EI is the bending stiffness of the string, D i is the inner diameter of the tubing, f is the friction coefficient caused by fluid viscosity, ρ i is the density of the fluid inside the string, A is the cross-sectional area of ​​the string, E is the elastic modulus of the string material, c = 2M c ω s , M C is the mass of the intermediate chamber, M is the mass of the pipe string, and L is the length of the pipe string; Establish a fluid force analysis model: Where, F D is the drag force, F L is the lift, ρ w is the external fluid density, U C is the external fluid flow rate, D o is the outer diameter of the pipe string, is the steady-state drag coefficient, q x ,q y are the dimensionless wake oscillator variables in the downstream and cross-stream directions, respectively, and C l is the pulsating lift coefficient; Establish the control equation of the wake oscillator: Where, ε x ,ε y ,A x ,A y All are dimensionless parameters determined experimentally; Combining the three-degree-of-freedom vibration control model, the fluid force analysis model, and the wake oscillator control equation, the nonlinear vibration control model of the string can be obtained: Create boundary conditions: u(0,t)=u boat (t) u x (0,t)=0 u y (0,t)=0 EIυ″ x | (0,t) =0,EIυ″ x | (L,t) =0,EIυ″ Y | (0,t) =0,EIυ″ y | (L,t) =0 The tubing string is discretized using the cubic Hermit interpolation function and matrix assembly is performed to obtain the tubing string vibration control matrix equation. The Newmark-β method and Newton-Raphson method are then used to jointly solve the tubing string vibration control matrix equation. At the same time, the fourth-order Runge-Kutta method is used to solve the wake oscillator model.

2. The method according to claim 1, characterized in that In the wake oscillator control model, A x =48,A y =12,ε x =1.2,ε y =0.3,C′ d =0.3,C′ l =0.

4.

3. The method according to claim 1, characterized in that The string vibration control matrix equation is established through the following process: The string is discretized into N units using the cubic Hermit interpolation function, with a unit length of l, and the following is obtained: Where, To express the displacement of the pipe string in the x, y, and z directions, Where l is the length of the string unit; Assemble the unit string into a matrix and obtain the string vibration control matrix equation: Where M is the mass matrix of the system, C is the damping matrix of the system, K is the stiffness matrix of the system, and D is the displacement matrix. is the velocity matrix, is the acceleration matrix.

4. The method according to claim 3, characterized in that The specific composition of M, C, and K is as follows:

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