CN102507115A - Large deformation bending vibration analysis method of deep-water top-tensioned riser - Google Patents

Abstract

The invention relates to a research method of an ocean deep-water riser, in particular to a large deformation bending vibration analysis method of a deep-water top-tensioned riser. The method improves the existing large deformation bending vibration analysis technique of the deep-water top-tensioned riser, the problem of large deformation of deep-water top-tensioned riser bending vibration is considered, a large deformation bending vibration equation of the deep-water top-tensioned riser is proposed and the large deformation bending vibration analysis method of the deep-water top-tensioned riser is created based on the equation. Since the method considers the influences of large deformation to the bending vibration of the deep-water top-tensioned riser, the bending vibration analysis of the deep-water top-tensioned riser is enabled to be more accordant with the actual stress and the deformation status of the riser.

Description

The analytical approach of a kind of deep water top tension-type vertical pipe large deformation flexural vibrations

Technical field

The present invention relates to the research method of ocean deepwater standpipe, be specifically related to the analytical approach of a kind of deep water top tension-type vertical pipe large deformation flexural vibrations.

Background technology

Deep water top tension-type vertical pipe is the important equipment of deep-sea oil gas exploitation, and (1000～3000m) are far longer than its sectional dimension, and (0.3～0.5m), therefore, its bending stiffness is less, belongs to big flexible structure owing to standpipe length.Particularly novel compliant type standpipe, its flexibility be the scope of traditional girder construction head and shoulders above especially, belongs to the super large flexible structure.At present; Main complicated bend (considering the bending of the axial force) theory that adopts traditional Euler's beam is analyzed in the flexural vibrations of deep water top tension-type vertical pipe; This theory supposes based on small deformation, ignore because the gravitional force that flexural deformation causes changes and curvature to deep water top tension-type vertical pipe geometric stiffness and The Effect of Inertia Force.This scheme is set up top tension-type vertical pipe flexural vibrations equation based on mechanical model shown in Figure 1.Among Fig. 1, the gravity of infinitesimal section is included in the tension force T.

Based on this model, prior art adopts the standpipe flexural vibrations equation of formula (1):

$m\frac{{\∂}^{2}y}{\∂t^2}+c\frac{\∂y}{\∂t}+\mathrm{EI}\frac{{\∂}^{4}y}{{\∂x}^4}-\frac{\∂}{\∂x}\left(T\frac{\∂y}{\∂x}\right)=q---\left(1\right)$

In the formula: the quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structural damping coefficient of c--standpipe unit length;

Y--standpipe sag;

T--standpipe tension force;

The t--time;

The axial coordinate of x--standpipe;

Q--acts on the fluid load on the standpipe.

The major defect of prior art is following:

1, do not consider that the standpipe large deformation causes the vertical component of flexural vibrations acceleration

During small deformation, the acceleration vertical component of standpipe flexural vibrations is less, therefore, is left in the basket.And during large deformation, the vertical component of standpipe flexural vibrations acceleration is bigger, should not be left in the basket.

2, do not consider the influence of gravity that the standpipe large deformation causes to flexural vibrations

During small deformation, because the distortion of hypothesis standpipe occurs over just laterally, therefore, gravitional force does not change.But during large deformation,, cause gravity that effect of contraction has been played in flexural deformation because rotate in the cross section that flexural deformation causes.Therefore, during large deformation considered gravity to diastrophic influence.

Summary of the invention

The objective of the invention is to defective to prior art; Consider that the flexural vibrations acceleration of the vertical direction that deep water top tension-type vertical pipe large deformation causes and the gravitional force that large deformation causes change the influence to flexural vibrations, set up big flexible deep water standpipe flexural vibrations analysis on Large Deformation method.

Technical scheme of the present invention is following: the analytical approach of a kind of deep water top tension-type vertical pipe large deformation flexural vibrations, and the deep water top tension-type vertical pipe flexural vibrations analytical model of foundation is following:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}$

In the formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structural damping coefficient of c--standpipe unit length;

T--standpipe tension force;

X--standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on the standpipe;

Based on above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model, calculate the large deformation flexural vibrations response of deep water top tension-type vertical pipe as follows:

1) set Cartesian coordinates, the summit of establishing deep water top tension-type vertical pipe is a true origin, and the x axle is the vertical direction coordinate axis, and the y axle is the horizontal direction coordinate axis;

2) equation of above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model can be obtained the finite element equation of deep water top tension-type vertical pipe large deformation flexural vibrations after discrete with Finite Element Method:

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_x\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+\frac{\∂T}{\∂x}{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y]

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]-β[K _x]

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient:

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

The l--element length;

The n--element number:

E--makes the unit of accumulating operation;

3) riser is carried out dividing elements;

4) will divide the parameter substitution step 2 of each unit, back) in each matrix and the formula of vector calculate the matrix and the vector of all unit;

5) adopt time-histories integral method solution procedure 2) in the finite element equation of deep water top tension-type vertical pipe large deformation flexural vibrations, can obtain the large deformation flexural vibrations response of deep water top tension-type vertical pipe, vibratory response comprises displacement, speed, acceleration.

Beneficial effect of the present invention is following: the present invention improves the analytical approach of existing deep water top tension-type vertical pipe large deformation flexural vibrations; Considered the large deformation problem of deep water top tension-type vertical pipe flexural vibrations; Proposed deep water top tension-type vertical pipe large deformation flexural vibrations equation, and set up the large deformation flexural vibrations analytical approach of deep water top tension-type vertical pipe based on this equation.This method has been considered the influence of large deformation to the tension-type vertical pipe flexural vibrations of deep water top, makes the flexural vibrations of deep water top tension-type vertical pipe analyze actual loading and the deformation state that more meets standpipe.

Description of drawings

Fig. 1 is the small deformation mechanical model synoptic diagram of prior art neutral tube infinitesimal section;

Fig. 2 is a deep water of the present invention top tension-type vertical pipe large deformation mechanical model synoptic diagram;

Fig. 3 is the coordinate system synoptic diagram of top tension-type vertical pipe;

Fig. 4 is a method flow diagram of the present invention.

Embodiment

Describe the present invention below in conjunction with accompanying drawing and embodiment.

Deep water top tension-type vertical pipe is the right cylinder of a high-fineness ratio, and its slenderness ratio can reach 5000～6000.Therefore, its bending stiffness is less, is a structure with big flexibility.Under the marine environment effect, the displacement of its flexural vibrations is bigger, belongs to the large deformation structure.If adopt the small deformation hypothesis of prior art, then the actual loading and the deformation state of result of calculation and deep water top tension-type vertical pipe will have bigger error.For this reason, the present invention proposes the deep water top tension-type vertical pipe flexural vibrations analytical approach of considering large deformation, be intended to solve the flexural vibrations analysis of deep water top tension-type vertical pipe large deformation.

The present invention considers that the tension-type vertical pipe large deformation of deep water top changes the influence to flexural vibrations to the gravitional force that the influence and the large deformation of flexural vibrations acceleration causes; Proposed to consider the deep water top tension-type vertical pipe flexural vibrations analytical model of large deformation, and based on this modelling the analytical approach of deep water top tension-type vertical pipe large deformation flexural vibrations.

Fig. 2 is the deep water top tension-type vertical pipe infinitesimal section mechanical model that the present invention adopts, and can obtain considering the deep water top tension-type vertical pipe flexural vibrations differential equation of large deformation based on the dynamic analysis of Fig. 2:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q---\left(2\right)$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}---\left(3\right)$

In the formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structural damping coefficient of c--standpipe unit length;

T--standpipe tension force;

X-standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on the standpipe;

Based on above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model, calculate the large deformation flexural vibrations response of deep water top tension-type vertical pipe as follows, as shown in Figure 4:

1) set Cartesian coordinates, the summit of establishing deep water top tension-type vertical pipe is a true origin, and the x axle is the vertical direction coordinate axis, and the y axle is the horizontal direction coordinate axis, and is as shown in Figure 3;

2) equation of above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model can be obtained the finite element equation of deep water top tension-type vertical pipe large deformation flexural vibrations after discrete with Finite Element Method (known technology):

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}---\left(4\right)$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}---\left(5\right)$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}---\left(6\right)$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]---\left(7\right)$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$ (8)

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$ (9)

$+\frac{\∂T}{\∂x}-{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y] (10)

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]+β[K _x] (11)

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}---\left(12\right)$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}---\left(13\right)$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient;

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

The l--element length;

The n--element number;

E--makes the unit of accumulating operation;

3) riser is carried out dividing elements (known technology);

Dividing elements can be divided according to the method for API RP 2RD specification recommends, also can be partial to conservative division according to the ability of computing machine and the length of standpipe, and is long like 1m.

4) will divide the matrix and the vector of all unit of formula calculating of each matrix and vector in parameter (comprising element length and section modulus EI, EA) substitution formula (6)～(13) of each unit, back;

5) adopt time-histories integral method (known technology), like Newmark-β method or Wilson-θ method, solution procedure 2) in equation (4) and (5), the large deformation flexural vibrations that can obtain deep water top tension-type vertical pipe respond.

The mass matrix of standpipe and stiffness matrix adopt Finite Element Method to calculate; The vertical inertial force system matrix number wherein and the stiffness matrix of both direction are different with prior art; It is the content that the present invention proposes; Damping matrix then adopts auspicious thunder damping matrix (known technology), and the lateral load vector of standpipe also calculates according to fluid dynamics and Wave Theory, and these calculate and all belong to known technology for a person skilled in the art.

Obviously, those skilled in the art can carry out various changes and modification to the present invention and not break away from the spirit and scope of the present invention.Like this, belong within the scope of claim of the present invention and equivalent technology thereof if of the present invention these are revised with modification, then the present invention also is intended to comprise these changes and modification interior.

Claims (1)

1. the analytical approach of deep water top tension-type vertical pipe large deformation flexural vibrations, it is characterized in that: the deep water top tension-type vertical pipe flexural vibrations analytical model of foundation is following:

$\mathrm{EI}\frac{{\∂}^{4}v}{\∂x^4}+\mathrm{EI}{\mathrm{\κ}}^2\frac{{\∂}^{2}v}{\∂x^2}-\frac{\∂}{\∂x}\left(T\frac{\∂v}{\∂x}\right)-\mathrm{mg}\frac{\∂v}{\∂x}+m\frac{{\∂}^{2}v}{\∂t^2}+c\frac{\∂v}{\∂t}=q$

$\mathrm{EI}\frac{{\∂}^{4}u}{\∂x^4}+(\mathrm{EI}{\mathrm{\κ}}^2-T)\frac{{\∂}^{2}u}{\∂x^2}-\frac{\∂T}{\∂x}\frac{\∂u}{\∂x}-m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\frac{{\∂}^{2}u}{{\∂t}^2}+c\frac{\∂u}{\∂t}=\frac{\∂T}{\∂x}-\mathrm{mg}$

In the formula: v--standpipe horizontal direction flexural vibrations displacement;

U--standpipe vertical direction flexural vibrations displacement;

The quality of m--standpipe unit length;

EI--standpipe flexural rigidity of section;

The structural damping coefficient of c--standpipe unit length;

T--standpipe tension force;

X-standpipe axial coordinate;

The t-time;

The curvature of κ--standpipe;

G-acceleration of gravity;

Q--acts on the fluid load on the standpipe;

Based on above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model, calculate the large deformation flexural vibrations response of deep water top tension-type vertical pipe as follows:

1) set Cartesian coordinates, the summit of establishing deep water top tension-type vertical pipe is a true origin, and the x axle is the vertical direction coordinate axis, and the y axle is the horizontal direction coordinate axis;

2) equation of above-mentioned deep water top tension-type vertical pipe flexural vibrations analytical model can be obtained the finite element equation of deep water top tension-type vertical pipe large deformation flexural vibrations after discrete with Finite Element Method:

$\left[M_y\right]\left\{\stackrel{..}{v}\right\}+\left[C_y\right]\left\{\stackrel{.}{v}\right\}+\left[K_y\right]\left\{v\right\}=\left\{F_y\right\}$

$\left[M_x\right]\left\{\stackrel{..}{u}\right\}+\left[C_x\right]\left\{\stackrel{.}{u}\right\}+\left[K_x\right]\left\{u\right\}=\left\{F_x\right\}$

Wherein, [M _y]--the transverse mass matrix of standpipe;

$\left[M_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}$

[M _x]--the vertical inertial coefficient matrix of standpipe;

$\left[M_x\right]=-\underset{e=1}{\overset{n}{\mathrm{\Σ}}}m\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}{\∫}_0^l{\left[N\right]}^T\left[N\right]\mathrm{dx}=-\frac{\mathrm{\κ}}{\left|\mathrm{\κ}\right|}\left[M_y\right]$

[K _y]--the lateral stiffness matrix of standpipe;

$\left[K_y\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+(\frac{\∂T}{\∂x}+\mathrm{mg}){\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[K _x]--the vertical stiffness matrix of standpipe;

$\left[K_x\right]=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}\left\{\mathrm{EI}{\∫}_0^l{\left[N^{\′\′}\right]}^T\right[N^{\′\′}]\mathrm{dx}+(\mathrm{EI}{\mathrm{\κ}}^2-T){\∫}_0^l{\left[N^{\′}\right]}^T[N^{\′}]\mathrm{dx}$

$+\frac{\∂T}{\∂x}{\∫}_0^l{\left[N^{\′}\right]}^T\left[N\right]\mathrm{dx}\}$

[C _y]--the horizontal damping matrix of standpipe;

[C _y]＝α[M _y]+β[K _y]

[C _x]--standpipe vertical damping matrix;

[C _x]＝α[M _x]+β[K _x]

--standpipe transverse acceleration vector;

--standpipe transverse velocity vector;

{ v}--standpipe transversal displacement vector;

{ F _y--standpipe lateral load vector:

$\left\{F_y\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_y{\}}_e\mathrm{dx}$

{ ü }--the vertical vector acceleration of standpipe;

--standpipe vertical velocity vector;

{ u}--standpipe vertical displacement vector;

{ F _x--standpipe vertical load vector:

$\left\{F_x\right\}=\underset{e=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_0^l{\left[N\right]}_e^T\{q_x{\}}_e\mathrm{dx}$

[N]--standpipe unit interpolating function matrix;

[N ']--standpipe unit interpolating function matrix is to the first order derivative of x;

[N ＂]--standpipe unit interpolating function matrix is to the second derivative of x;

{ q _y--standpipe lateral load vector;

{ q _x--standpipe vertical load vector:

$\left\{q_x\right\}=\{\frac{\∂T}{\∂x}-\mathrm{mg}\}$

α, β--Rui Leizuni coefficient;

X-standpipe axial coordinate;

The curvature of κ--standpipe;

The quality of m-standpipe unit length;

G-acceleration of gravity;

The l--element length;

The n--element number;

E--makes the unit of accumulating operation;

3) riser is carried out dividing elements;

4) will divide the parameter substitution step 2 of each unit, back) in each matrix and the formula of vector calculate the matrix and the vector of all unit;

5) adopt time-histories integral method solution procedure 2) in the finite element equation of deep water top tension-type vertical pipe large deformation flexural vibrations, can obtain the large deformation flexural vibrations response of deep water top tension-type vertical pipe.

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