CN113033122B - Flexible pipeline nonlinear response prediction method, system and device under action of internal flow - Google Patents

Flexible pipeline nonlinear response prediction method, system and device under action of internal flow Download PDF

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CN113033122B
CN113033122B CN202110418298.8A CN202110418298A CN113033122B CN 113033122 B CN113033122 B CN 113033122B CN 202110418298 A CN202110418298 A CN 202110418298A CN 113033122 B CN113033122 B CN 113033122B
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高云
张壮壮
潘港辉
刘磊
姜泽成
柴盛林
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Harbin Institute of Technology Weihai
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Abstract

A method, a system and a device for predicting nonlinear response of a flexible pipeline under the action of internal flow belong to the technical field of pipeline design and prediction. The method aims to solve the problem that at present, stability analysis of the pipeline is based on linear theory development analysis, so that a plurality of critical information of structural vibration response cannot be obtained. The numerical prediction model can well simulate the influence of different internal flow rates, different axial forces, different fluid pressures, different gravity coefficients and different axial flexibilities on the vibration response of the flexible structure; and structural instability dynamic characteristics can be well revealed, and problems existing in practical engineering can be well described. The method is mainly applied to the nonlinear response prediction of the flexible pipeline.

Description

Flexible pipeline nonlinear response prediction method, system and device under action of internal flow
Technical Field
The invention relates to a method, a system and a device for predicting nonlinear response of a flexible pipeline, belonging to the technical field of pipeline design and prediction.
Background
In modern industry, a pipeline is the most common current-carrying device, and with the rapid development of modern science and technology, the application of a fluid transmission pipeline in the fields of aerospace, petroleum, chemical engineering, oceans, biological engineering, nuclear engineering and the like is more and more extensive. When the fluid conveying pipeline conveys internal fluid, the pipeline can generate a coupling effect with the internal fluid, and when the fluid flows in the pipe along the axial direction, the pipeline can vibrate under the action of the fluid. When the internal flow velocity is small, the resulting effect is weak and the influence on the pipe structure is negligible. As the fluid flow velocity within the pipe increases, the impact of the pipe with the internal fluid becomes more pronounced, and as the flow rate exceeds a certain level, the piping system structure becomes unstable and even severely damaged.
Therefore, the stability analysis of the pipeline is particularly important. At present, most of researches on the vibration response of the flexible cylinder are developed based on a linear theory, and more key information such as specific vibration response amplitude of the structure cannot be obtained. The structure vibration response characteristic can be well revealed by analyzing the structure by adopting a nonlinear theory. The advantages of the non-linear model compared to the linear model are as follows: (1) the structural motion equation obtained by nonlinear theory derivation can better describe the problems existing in the actual engineering; (2) can be used to reveal the dynamics of structural destabilization; (3) and more critical information of structural vibration response can be obtained by carrying out research based on nonlinear dynamics. Therefore, to more accurately and reliably estimate the vibrational response of an in-band flow riser, the internal fluid flow induced non-linearity factors must be considered.
Disclosure of Invention
The method aims to solve the problem that at present, stability analysis of the pipeline is based on linear theory development analysis, so that a plurality of critical information of structural vibration response cannot be obtained.
A flexible pipeline nonlinear response prediction method under the action of internal flow is characterized in that firstly, a flexible pipeline structure vibration partial differential equation under the action of internal flow excitation is established based on a Newton method; and then converting the structural vibration partial differential equation into an ordinary differential equation based on a Galerkin method, and calculating the structural vibration response of the flexible pipeline with the internal flow motion based on a Houbolt finite difference method and a Newton-Raphson iteration method.
Further, the process of establishing the vibration partial differential equation of the flexible pipeline structure under the action of the internal flow excitation based on the newton method in the step 1 includes the following steps:
establishing a coordinate system aiming at the flexible pipeline structure of the cylinder, wherein z is the flow direction of the inner flow; y is the transverse vibration direction; carrying out stress analysis on the fluid infinitesimal and the pipeline structure infinitesimal according to a infinitesimal method, and establishing a stress balance equation of the fluid in the z and y directions and a stress balance equation of the pipeline in the z and y directions;
according to the mechanics of materials, the expressions of transverse shear force Q and bending moment M inside the pipeline structure are shown as follows:
Figure BDA0003026871130000021
wherein E and E*The modulus of elasticity of the material and the dissipation factor of the pipe material, respectively.
Expressing the velocity in vector form, yielding a fluid and pipe acceleration expression:
Figure BDA0003026871130000022
according to the stress balance equation of the fluid in the z and y directions and the stress balance equation of the pipeline in the z and y directions, equations (3) and (4) are combined, and after a tangential force term of the pipeline wall to the fluid is eliminated, a transverse motion equation of the pipeline is obtained and is as follows:
Figure BDA0003026871130000023
wherein t represents time;
based on the force balance equation of the fluid in the z and y directions and the force balance equation of the pipeline in the z and y directions, the simultaneous formula (4) is obtained:
Figure BDA0003026871130000024
integrating z through L for equation (6):
Figure BDA0003026871130000025
assuming that at the point where z is equal to L, the axial force applied to the pipe is
Figure BDA0003026871130000026
The internal fluid is subjected to a pressure of
Figure BDA0003026871130000027
Equation (7) is written as:
Figure BDA0003026871130000028
when δ is 0, the pipeline structure has no axial motion constraint at the end point, and when δ is 1, the pipeline structure has axial motion constraint at the end point;
when the two ends of the pipeline are fixed, the pipeline can be contracted in the axial direction due to transverse movement, so that additional axial force is generated; aiming at the pipe infinitesimal, according to the relation between stress and strain, an additional axial force is obtained:
Figure BDA0003026871130000031
where y' represents the derivative of the y-direction displacement with respect to the axial position z.
Substituting the formulas (6), (8) and (9) into the formula (5) to obtain a differential equation of the transverse vibration of the internal flow coupling fluid transmission pipeline, wherein the differential equation is as follows:
Figure BDA0003026871130000032
neglecting the pipeline dissipation, the transverse vibration differential equation is simplified as:
Figure BDA0003026871130000033
wherein the content of the first and second substances,
Figure BDA0003026871130000034
the axial force and the fluid pressure applied to the pipeline at the point where the end point z is equal to L are respectively.
Further, the force balance equation of the fluid in the z and y directions is as follows:
Figure BDA0003026871130000035
Figure BDA0003026871130000036
wherein the length of the pipeline is L, and the inner diameter is DiOuter diameter of DoThe internal cross-sectional area is A, the rigidity is EI, and the internal fluid velocity of the pipeline is UiThe internal perimeter is S, and δ z is the length of the fluid cell; p is the internal fluid pressure to which the fluid element is subjected; f is the normal force of the unit length of the pipe wall to the fluid, q is the tangential force of the unit length of the pipe wall to the fluid, MfMass of fluid element per unit length, g is gravitational acceleration, afzAcceleration of fluid infinitesimal in z direction, afyIs the acceleration of the fluid infinitesimal in the y-direction, and w is the pipe lateral displacement.
Further, the stress balance equation of the pipeline in the z and y directions is as follows:
Figure BDA0003026871130000041
Figure BDA0003026871130000048
wherein T is the axial force of the pipe infinitesimal, and Q is the transverse shearing force in the pipe infinitesimalM is a bending moment, apyIs the acceleration of the pipe infinitesimal in the y direction, and m is the mass of the pipe infinitesimal per unit length.
Further, the step 2 of converting the structural vibration partial differential equation into an ordinary differential equation based on the galois method includes the following steps:
converting equation (11) to a dimensionless form:
Figure BDA0003026871130000042
wherein eta, xi and tau are dimensionless vibration displacement, dimensionless coordinate position and dimensionless time, uiBeta is a dimensionless internal flow velocity and mass ratio, gamma, pi, and pi0Respectively are a dimensionless gravity coefficient, a dimensionless axial force, a dimensionless fluid pressure and an axial compliance;
substituting the formula (12) into the formula (11) to obtain a flexible pipeline structure vibration dimensionless equation with the in-band flow motion as follows:
Figure BDA0003026871130000043
converting the high-order differential equation system of the formula (13) into a low-order differential equation system by using a Galerkin method to obtain the following formula:
Figure BDA0003026871130000044
wherein:
Figure BDA0003026871130000045
Figure BDA0003026871130000046
Figure BDA0003026871130000047
Figure BDA0003026871130000051
φj iv、φj″、φj' are the fourth derivative, the second derivative and the first derivative of the jth order transverse displacement mode function with respect to xi respectively.
Further, the process of converting the high-order differential equation system of formula (13) into a low-order differential equation system using the Galerkin method includes the steps of:
the following expression of the Galerkin method was introduced:
Figure BDA0003026871130000052
wherein: n is the Galerkin modal cutoff number; phi is ai(ξ) is the ith order lateral displacement mode function, the specific expression of which is determined by the boundary conditions;
Figure BDA0003026871130000053
is the ith generalized coordinate;
by substituting equation (14) for equation (13), both sides of the equation are multiplied by phii(xi) and integrating over 0-1, and obtaining by utilizing the orthogonality of the mode shape function
Figure BDA0003026871130000054
Further, the step 3 of calculating the vibration response of the flexible pipeline structure with the internal flow motion based on the Houbolt finite difference method and the Newton-Raphson iteration method comprises the following steps:
processing the equation (15) by using a Houbolt finite difference format, and solving by using a Newton-Raphson iteration method, wherein the Houbolt finite difference format of 4 orders is as follows:
Figure BDA0003026871130000055
Figure BDA0003026871130000056
wherein j represents the jth unknown quantity, delta tau represents the time step length, and n represents the time step number; converting the equation into N nonlinear ordinary differential equations by a Galerkin method; substituting a differential format into an equation (15);
at time step n +1
f(ηn+1)＝0 (18)
f is a non-linear function of dimension N, etan+1For the unknowns, η, to be solved forn，ηn-1And ηn-2Unknown quantity in the previous steps;
determination of etanAs etan+1The initial value of (a) is solved by a Newton-Raphson iterative method, and the iterative form is as follows:
ηn+1＝ηn-Jc[f(ηn+1)]-1f(ηn) (19)
wherein, Jc[f(ηn+1)]Is a system of equations f (η)n+1) When a Houbolt finite difference format is adopted to solve a nonlinear differential equation system, a Taylor expansion equation is needed to be used for calculating the value of the t-0 moment to obtain the known quantity of the previous steps; eta when given an initial time0And
Figure BDA0003026871130000062
is provided with
Figure BDA0003026871130000063
k is 1,2, 3; when higher calculation accuracy is required, the equation (18) and the initial value η are required0And
Figure BDA0003026871130000064
find out
Figure BDA0003026871130000065
Then according to
Figure BDA0003026871130000066
Calculating; when the accuracy of the initial condition and the numerical difference are different orders in the cellular mode, the deviation of the equation (18) is obtained0The higher order derivatives of (a) are then calculated using taylor expansion to obtain the values of the previous steps.
Further, after the flexible pipeline structure vibration response with the internal flow motion is calculated based on a Houbolt finite difference method and a Newton-Raphson iteration method, the pipeline example is calculated and analyzed based on analysis data; the process of performing computational analysis on the pipeline example comprises the influence of axial force, fluid pressure, axial flexibility and gravity coefficient on the buckling instability of the structure.
The system for predicting the nonlinear response of the flexible pipeline under the action of the internal flow is used for executing the method for predicting the nonlinear response of the flexible pipeline under the action of the internal flow.
And the device is used for storing and/or operating the flexible pipeline nonlinear response prediction system under the action of the internal flow.
Has the advantages that:
the numerical prediction model can well simulate the influence of different internal flow rates, different axial forces, different fluid pressures, different gravity coefficients and different axial flexibilities on the vibration response of the flexible structure; and structural instability dynamic characteristics can be well revealed, and problems existing in practical engineering can be well described.
Drawings
FIG. 1 is a schematic view of a flexible pipe under internal flow;
FIG. 2 is a schematic diagram of the force applied to the fluid micro-elements and the pipeline structure micro-elements;
FIG. 3 is a bifurcation diagram of point displacement along with internal flow velocity in a pipeline structure under different axial forces;
FIG. 4 is a bifurcation diagram of point displacement with internal flow velocity for a piping structure at different fluid pressures;
FIG. 5 is a bifurcation diagram of point displacement along with internal flow velocity in a pipeline structure under different gravity coefficients;
FIG. 6 is a bifurcation diagram of point displacement with internal flow velocity for lower pipe structures of different axial compliances.
Detailed Description
The first embodiment is as follows:
the embodiment is a flexible pipeline nonlinear response prediction method under the action of internal flow, and is a flexible pipeline nonlinear response prediction method under the action of internal flow under the condition of hinged boundaries at two ends.
The method for predicting the nonlinear response of the flexible pipeline under the action of the internal flow comprises the following steps:
step 1, establishing a vibration partial differential equation of a flexible pipeline structure under the action of internal flow excitation based on a Newton method;
step 2, converting the structural vibration partial differential equation into an ordinary differential equation based on the Galerkin method;
step 3, calculating the vibration response of the flexible pipeline structure with the internal flow motion based on a Houbolt finite difference method and a Newton-Raphson iteration method;
and 4, performing calculation analysis on the example based on the analysis data.
More specifically, the process of establishing the vibration partial differential equation of the flexible pipeline structure under the action of internal flow excitation based on the newton method in step 1 includes the following steps:
the flexible pipe under the action of the internal flow is shown in fig. 1, and the problem of structural stability caused by the action of stabilizing the internal flow of the slender flexible cylinder is researched. Taking the hinged boundary conditions adopted at the two ends of the cylinder as an example, the structural length is considered to be L (m), and the inner diameter is considered to be Di(m) outer diameter Do(m) internal cross-sectional area A (m)2) Stiffness is EI (N.m)2) The velocity of fluid in the pipeline is Ui(m/s). Establishing a coordinate system shown in FIG. 1; z is the inner flow direction; y isThe direction of the lateral vibration.
According to the force analysis of the fluid micro-elements and the pipeline structure micro-elements by the micro-element method, as shown in fig. 2, the cross-sectional area of the interior of the pipeline is A, the perimeter of the interior is S, and δ z is the length of the fluid unit. Through analysis, the stress of the fluid infinitesimal comprises the static pressure P of the fluid, the vertical acting force F delta z and the tangential acting force qS delta z of the fluid on the pipe wall, and the gravity M of the fluid infinitesimalfg δ z. And (3) applying Newton's second law, neglecting high-order small terms, and establishing a stress balance equation of the fluid in the z and y directions:
Figure BDA0003026871130000071
wherein F is the normal force of the unit length of the pipe wall to the fluid, q is the tangential force of the unit length of the pipe wall to the fluid, MfMass of fluid element per unit length, g is gravitational acceleration, afzAcceleration of fluid infinitesimal in z direction, afyThe acceleration of the fluid micro element in the y direction is shown, and w is the transverse displacement of the pipeline;
through analysis, the stress of the pipe infinitesimal comprises tangential reaction force qS delta z and vertical reaction force F delta z of the internal fluid on the pipe, axial force T, internal transverse shear force Q, bending moment M and pipe infinitesimal gravity mg delta z. Applying Newton's second law, neglecting high-order small terms, and establishing a stress balance equation of the pipeline in the z and y directions:
Figure BDA0003026871130000081
wherein, apyThe acceleration of the pipe infinitesimal in the y direction is shown, and m is the mass of the pipe infinitesimal in unit length;
according to the mechanics of materials, the expressions of transverse shear force Q and bending moment M inside the pipeline structure are shown as follows:
Figure BDA0003026871130000082
m-bending moment (kN · M); e and E*The modulus of elasticity of the material and the dissipation factor of the pipe material, respectively.
And expressing the velocity in a vector form, and obtaining fluid and pipeline acceleration expressions according to the definition of a material derivative and omitting high-order small quantity:
Figure BDA0003026871130000083
the joint type (1), (2), (3) and (4) eliminates the tangential force term of the pipe wall to the fluid, and the transverse motion equation of the pipeline can be obtained as follows:
Figure BDA0003026871130000084
wherein t represents time;
both the pipeline axial force T and the fluid pressure P appear in the above equation as functions of the coordinate z, so the above equation cannot be solved directly and needs to be transformed. The united type (1), (2) and (4) can obtain:
Figure BDA0003026871130000085
integrating equation (6) from z to L yields:
Figure BDA0003026871130000086
assuming that at the point where z is equal to L, the axial force applied to the pipe is
Figure BDA0003026871130000096
The internal fluid is subjected to a pressure of
Figure BDA0003026871130000097
Then the above formula can be written as:
Figure BDA0003026871130000091
when δ is 0, the pipe structure has no axial motion constraint at the end point, and when δ is 1, the pipe structure has axial motion constraint at the end point.
When the two ends of the tube are fixed, the tube may contract in the axial direction due to the lateral movement, thereby generating an additional axial force. As shown in fig. 2, assuming that the pipe has no initial strain, the expression of the additional axial force obtained from the relationship between stress and strain is:
Figure BDA0003026871130000092
where y' represents the derivative of the y-direction displacement with respect to the axial position z.
Substituting the expressions (6), (8) and (9) into the expression (5) can obtain a differential equation of the transverse vibration of the internal flow coupling fluid transmission pipeline, wherein the differential equation is as follows:
Figure BDA0003026871130000093
if the pipeline dissipation is neglected, the structural transverse vibration differential equation can be simplified as follows:
Figure BDA0003026871130000094
wherein the content of the first and second substances,
Figure BDA0003026871130000095
the axial force and the fluid pressure applied to the pipeline at the point where the end point z is equal to L are respectively.
Step 2, the process of converting the structural vibration partial differential equation into the ordinary differential equation based on the Galerkin method comprises the following steps:
first, equation (11) is transformed into a dimensionless form, requiring the use of the following expression:
Figure BDA0003026871130000101
in the above formula, eta, xi and tau are dimensionless vibration displacement, dimensionless coordinate position and dimensionless time, uiBeta is a dimensionless internal flow velocity and mass ratio, gamma, pi, and pi0Dimensionless gravity coefficient, dimensionless axial force, dimensionless fluid pressure, and axial compliance, respectively. Substituting the formula (12) into the formula (11) to obtain a flexible pipeline structure vibration dimensionless equation with the in-band flow motion as follows:
Figure BDA0003026871130000102
in order to facilitate the analysis of the motion equation, the high-order differential equation set of the formula (13) is converted into a low-order differential equation set by using a Galerkin method, and the high calculation accuracy can be achieved by generally taking a fourth-order form. The following expression of the Galerkin method was introduced:
Figure BDA0003026871130000103
wherein N is the Galerkin modal cutoff number; phi is ai(ξ) is the ith order lateral displacement mode function, the specific expression of which is determined by the boundary conditions;
Figure BDA0003026871130000104
is the ith generalized coordinate.
By substituting equation (14) for equation (13), both sides of the equation are multiplied by phii(xi) and integrating over 0-1, and obtaining the following formula by utilizing the orthogonality of the mode shape function:
Figure BDA0003026871130000105
wherein:
Figure BDA0003026871130000106
Figure BDA0003026871130000107
Figure BDA0003026871130000108
Figure BDA0003026871130000111
φj iv、φj"and phij' are the fourth derivative, the second derivative and the first derivative of the jth order transverse displacement mode function with respect to xi respectively.
Step 3, the process for calculating the vibration response of the flexible pipeline structure with the internal flow motion based on the Houbolt finite difference method and the Newton-Raphson iteration method comprises the following steps:
in the above, the Galerkin method has been adopted to discretize the formula (15) into a coupled nonlinear differential equation set, and in order to solve the equation set, the invention firstly adopts a Houbolt finite difference format to process the equation, and then utilizes a Newton-Raphson iteration method to solve, wherein the 4-order Houbolt finite difference format is:
Figure BDA0003026871130000112
Figure BDA0003026871130000113
where j represents the jth unknown, Δ τ represents the time step, and n represents the number of time steps. And converting the equation into N nonlinear ordinary differential equations by a Galerkin method. The differential format is substituted into equation (15).
At time step n +1
f(ηn+1)＝0 (18)
f is a non-linear function of dimension N，ηn+1For the unknowns, η, to be solved forn，ηn-1And ηn-2Unknown for the first few steps. When the time step is small, η can be assumed to be consideredn+1And ηnApproximately equal, so η can be calculatednAs etan+1The initial guess value of. Then, a Newton-Raphson iteration method is used for solving, and the iteration form is as follows:
ηn+1＝ηn-Jc[f(ηn+1)]-1f(ηn) (19)
wherein Jc[f(ηn+1)]Is a system of equations f (η)n+1) When the Houbolt finite difference format is adopted to solve the nonlinear differential equation system, the known quantities of the previous steps need to be obtained by calculating the value at the time t being 0 by using the Taylor expansion. Eta when given an initial time0And
Figure BDA0003026871130000114
is provided with
Figure BDA0003026871130000115
k＝1,2,3。
When higher calculation accuracy is required, the equation (18) and the initial value η are required0And
Figure BDA0003026871130000116
find out
Figure BDA0003026871130000117
Then according to
Figure BDA0003026871130000118
And (6) performing calculation. When the accuracy and value of the initial conditions differ by different orders in a cellular manner, a large global error results. In this case, a higher order approximation is also required, and the derivation with respect to η is obtained by the derivation of the equation (18)0The values of the previous steps are obtained by using Taylor expansion.
The process of performing a computational analysis of the instance based on the analysis data as described in step 4 comprises the steps of:
a4-order Houbolt finite difference method is adopted to analyze the nonlinear dynamic response of the structural system under the constant inflow effect, and the problem of buckling instability of the pipeline structure caused by different system parameters is researched. Because the model established by the invention has symmetry, when the parameters of the system change, the result can converge to different steady-state solutions for different initial values and time step lengths. The invention takes a pipeline with two hinged ends as an example, and researches the influence of axial force, fluid pressure, axial flexibility and gravity coefficient on buckling instability of a structure.
Fig. 3 shows a bifurcation diagram of point displacement along with internal flow velocity in a pipeline structure under different axial forces, and parameters adopted in the calculation process are as follows: β is 0.3 ═ Π0500, γ is 0, and ii is 0. When the axial force is greater than 0, the action of the tension of the pipeline is indicated; when the axial force is less than 0, it indicates that the pipe is under the action of pressure. When the dimensionless axial force is kept unchanged, the critical speed of buckling instability of the structure is higher than the condition without external tension. The critical speed of buckling instability of the structure is higher with the increase of the applied tensile force. When the velocity of the fluid inside the pipeline is kept constant, the larger the applied tension is, the smaller the buckling amplitude of the pipeline structure is. When the axial force of the pipeline is negative, the axial force of the pipeline indicates that the pipeline is under the action of compressive force at the moment, and the vibration response of the pipeline is opposite to the action of tensile force, namely, the larger the pressure borne by the pipeline structure is, the larger the corresponding buckling amplitude is, and the lower the corresponding critical speed is.
Fig. 4 shows a bifurcation diagram of point displacement along with internal flow velocity in a pipeline structure under different fluid pressures, and parameters adopted in the calculation process are as follows: β is 0.3 ═ Π0500, γ ═ Γ ═ 0. As can be seen from fig. 4, when the pressure of the dimensionless fluid increases, the critical flow rate of buckling instability of the pipeline is correspondingly reduced; as the velocity of the internal fluid remains constant, the buckling amplitude of the tubular structure increases as the dimensionless fluid pressure increases. In contrast to the effect of axial forces on the structure, it can be seen that the effect of fluid pressure on the pipe structure is exactly opposite to axial forces, since the fluid pressure acts on the pipe to create an additional axial pressure.
FIG. 5 showsAnd (3) generating a bifurcation diagram of the point displacement of the pipeline structure along with the internal flow velocity under different gravity coefficients, wherein the parameters adopted in the calculation process are as follows: β is 0.3 ═ Π0500, Γ ═ Π ═ 0. As can be seen from fig. 5, as the dimensionless gravity coefficient increases, the critical flow rate of buckling instability of the pipeline also increases, and when the velocity of the internal fluid is kept constant, as the dimensionless gravity coefficient increases, the buckling amplitude of the pipeline structure decreases. In contrast to the effect of axial forces on the structure, it can be seen that the effect of the gravity coefficient on the pipe structure is similar to axial forces, since when the gravity coefficient γ is greater than 0, the pipe structure is subjected to axial pulling forces caused by the action of gravity.
Fig. 6 shows a bifurcation diagram of point displacement along with internal flow velocity in a pipeline structure under different axial flexibilities, and parameters adopted in the calculation process are as follows: β is 0.3, γ is Γ is 0. As can be seen from fig. 6, the critical flow rate of buckling instability of the pipeline does not change with the change of axial compliance. When the velocity of the internal fluid is kept constant, the buckling amplitude of the pipe structure decreases with increasing axial compliance.
The second embodiment is as follows:
the present embodiment is a system for predicting a nonlinear response of a flexible pipe under an internal flow effect, where the system is configured to execute the method for predicting a nonlinear response of a flexible pipe under an internal flow effect, that is, the system is a software product corresponding to the method for predicting a nonlinear response of a flexible pipe under an internal flow effect, and may execute the method for predicting a nonlinear response of a flexible pipe under an internal flow effect to complete a process such as predicting a nonlinear response of a flexible pipe.
The third concrete implementation mode:
the embodiment is a device for predicting the nonlinear response of a flexible pipeline under the action of an internal flow, wherein the device is used for storing and/or operating a system for predicting the nonlinear response of the flexible pipeline under the action of the internal flow, namely the device is storage equipment or operating equipment corresponding to the system for predicting the nonlinear response of the flexible pipeline under the action of the internal flow, the storage equipment comprises but is not limited to a hard disk, and the operating equipment comprises but is not limited to a PC (personal computer), a workstation, mobile equipment, a purposefully developed singlechip and the like.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope of the present invention.

Claims (3)

1. The method for predicting the nonlinear response of the flexible pipeline under the action of the internal flow is characterized by firstly establishing a vibration partial differential equation of the flexible pipeline structure under the action of the internal flow excitation based on a Newton method; then converting the structural vibration partial differential equation into an ordinary differential equation based on a Galerkin method, and calculating the structural vibration response of the flexible pipeline with the internal flow motion based on a Houbolt finite difference method and a Newton-Raphson iteration method;
the process for establishing the vibration partial differential equation of the flexible pipeline structure under the action of internal flow excitation based on the Newton method comprises the following steps:
establishing a coordinate system aiming at the flexible pipeline structure of the cylinder, wherein z is the flow direction of the inner flow; y is the transverse vibration direction; carrying out stress analysis on the fluid infinitesimal and the pipeline structure infinitesimal according to a infinitesimal method, and establishing a stress balance equation of the fluid in the z and y directions and a stress balance equation of the pipeline in the z and y directions;
according to the mechanics of materials, the expressions of transverse shear force Q and bending moment M inside the pipeline structure are shown as follows:
Figure FDA0003549030830000011
wherein, E, E*Respectively the elastic modulus of the material and the dissipation coefficient of the pipeline material;
expressing the velocity in vector form, yielding a fluid and pipe acceleration expression:
Figure FDA0003549030830000012
according to the stress balance equation of the fluid in the z and y directions and the stress balance equation of the pipeline in the z and y directions, equations (3) and (4) are combined, and after a tangential force term of the pipeline wall to the fluid is eliminated, a transverse motion equation of the pipeline is obtained and is as follows:
Figure FDA0003549030830000013
wherein t represents time;
based on the force balance equation of the fluid in the z and y directions and the force balance equation of the pipeline in the z and y directions, the simultaneous formula (4) is obtained:
Figure FDA0003549030830000014
integrating z through L for equation (6):
Figure FDA0003549030830000015
assuming that at the point where z is equal to L, the axial force applied to the pipe is
Figure FDA0003549030830000016
The internal fluid is subjected to a pressure of
Figure FDA0003549030830000017
Equation (7) is written as:
Figure FDA0003549030830000021
when δ is 0, the pipeline structure has no axial motion constraint at the end point, and when δ is 1, the pipeline structure has axial motion constraint at the end point;
when the two ends of the pipeline are fixed, the pipeline can be contracted in the axial direction due to transverse movement, so that additional axial force is generated; aiming at the pipe infinitesimal, according to the relation between stress and strain, an additional axial force is obtained:
Figure FDA0003549030830000022
wherein y' represents the derivative of the y-direction displacement with respect to the axial position z;
substituting the formulas (6), (8) and (9) into the formula (5) to obtain a differential equation of the transverse vibration of the internal flow coupling fluid transmission pipeline, wherein the differential equation is as follows:
Figure FDA0003549030830000023
neglecting the pipeline dissipation, the transverse vibration differential equation is simplified as:
Figure FDA0003549030830000024
wherein the content of the first and second substances,
Figure FDA0003549030830000025
axial force and fluid pressure applied to the pipeline at the end point z-L are respectively;
the stress balance equation of the fluid in the z and y directions is as follows:
Figure FDA0003549030830000026
wherein the length of the pipeline is L, and the inner diameter is DiOuter diameter of DoThe internal cross-sectional area is A, the rigidity is EI, and the internal fluid velocity of the pipeline is UiThe inner perimeter is S, and P is the inner fluid pressure of the fluid infinitesimal; f is the normal force of the unit length of the pipe wall to the fluid, q is the tangential force of the unit length of the pipe wall to the fluid, MfMass of fluid element per unit length, g is gravitational acceleration, afzAcceleration of fluid infinitesimal in z direction, afyThe acceleration of the fluid micro element in the y direction is shown, and w is the transverse displacement of the pipeline;
the stress balance equation of the pipeline in the z direction and the y direction is as follows:
Figure FDA0003549030830000031
wherein T is the axial force of the pipe infinitesimal, Q is the transverse shearing force in the pipe infinitesimal, M is the bending moment, apyThe acceleration of the pipe infinitesimal in the y direction is shown, and m is the mass of the pipe infinitesimal in unit length;
the process of converting the structural vibration partial differential equation into the ordinary differential equation based on the Galerkin method comprises the following steps:
converting equation (11) to a dimensionless form:
Figure FDA0003549030830000032
wherein eta, xi and tau are dimensionless vibration displacement, dimensionless coordinate position and dimensionless time, uiBeta is a dimensionless internal flow velocity and mass ratio, gamma, pi, and pi0Respectively are a dimensionless gravity coefficient, a dimensionless axial force, a dimensionless fluid pressure and an axial compliance;
substituting the formula (12) into the formula (11) to obtain a flexible pipeline structure vibration dimensionless equation with the in-band flow motion as follows:
Figure FDA0003549030830000033
converting the high-order differential equation system of the formula (13) into a low-order differential equation system by using a Galerkin method to obtain the following formula:
Figure FDA0003549030830000034
wherein:
Figure FDA0003549030830000035
Figure FDA0003549030830000041
Figure FDA0003549030830000042
Figure FDA0003549030830000043
φj iv、φj″、φj' are respectively a fourth derivative, a second derivative and a first derivative of the jth order transverse displacement mode shape function relative to xi;
the process of converting the high order differential equation set of equation (13) to a low order differential equation set using the Galerkin method comprises the steps of:
the following expression of the Galerkin method was introduced:
Figure FDA0003549030830000044
wherein: n is the Galerkin modal cutoff number; phi is ai(ξ) is the ith order lateral displacement mode function, the specific expression of which is determined by the boundary conditions;
Figure FDA0003549030830000045
is the ith generalized coordinate;
by substituting equation (14) for equation (13), both sides of the equation are multiplied by phii(xi) and integrating over 0-1, and obtaining by utilizing the orthogonality of the mode shape function
Figure FDA0003549030830000046
The process for calculating the vibration response of the flexible pipeline structure with the internal flow motion based on the Houbolt finite difference method and the Newton-Raphson iteration method comprises the following steps:
processing the equation (15) by using a Houbolt finite difference format, and solving by using a Newton-Raphson iteration method, wherein the Houbolt finite difference format of 4 orders is as follows:
Figure FDA0003549030830000047
Figure FDA0003549030830000048
wherein j represents the jth unknown quantity, delta tau represents the time step length, and n represents the time step number; converting the equation into N nonlinear ordinary differential equations by a Galerkin method; substituting a differential format into an equation (15);
at time step n +1
f(ηn+1)＝0 (18)
f is a non-linear function of dimension N, etan+1For the unknowns, η, to be solved forn，ηn-1And ηn-2Unknown quantity in the previous steps;
determination of etanAs etan+1The initial value of (a) is solved by a Newton-Raphson iterative method, and the iterative form is as follows:
ηn+1＝ηn-Jc[f(ηn+1)]-1f(ηn) (19)
wherein, Jc[f(ηn+1)]Is a system of equations f (η)n+1) When a Houbolt finite difference format is adopted to solve a nonlinear differential equation system, a Taylor expansion equation is needed to be used for calculating the value of the t-0 moment to obtain the known quantity of the previous steps; when given an initial momentη0And
Figure FDA0003549030830000051
is provided with
Figure FDA0003549030830000052
Figure FDA0003549030830000053
When higher calculation accuracy is required, the equation (18) and the initial value η are required0And
Figure FDA0003549030830000054
find out
Figure FDA0003549030830000055
Then according to
Figure FDA0003549030830000056
Calculating; when the accuracy of the initial condition and the numerical difference are different orders in the cellular mode, the deviation of the equation (18) is obtained0The higher-order derivative is calculated by using the Taylor expansion to obtain the values of the previous steps;
the method comprises the steps of calculating the vibration response of a flexible pipeline structure with internal flow motion based on a Houbolt finite difference method and a Newton-Raphson iteration method, and then performing calculation analysis on a pipeline example based on analysis data; the process of performing computational analysis on the pipeline example comprises the influence of axial force, fluid pressure, axial flexibility and gravity coefficient on the buckling instability of the structure.
2. A system for predicting the non-linear response of a flexible pipe under the action of an internal flow, the system being configured to perform the method of claim 1.
3. An apparatus for predicting the non-linear response of a flexible pipe under internal flow conditions, the apparatus being configured to store and/or operate the system of claim 2.
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