CN113111420B - Method for rapidly predicting unstable interval of boundary excitation elongated tension beam - Google Patents

Method for rapidly predicting unstable interval of boundary excitation elongated tension beam Download PDF

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CN113111420B
CN113111420B CN202110418299.2A CN202110418299A CN113111420B CN 113111420 B CN113111420 B CN 113111420B CN 202110418299 A CN202110418299 A CN 202110418299A CN 113111420 B CN113111420 B CN 113111420B
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高云
潘港辉
张壮壮
刘磊
姜泽成
柴盛林
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Harbin Institute of Technology Weihai
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Abstract

A method for rapidly predicting an unstable interval of a boundary excitation elongated tension beam. The marine riser generates fatigue damage due to resonance, and the important influence factor is the time-varying tension applied to the riser by the platform under the action of environmental load, but a related direct and accurate prediction mode is lacked. The invention establishes a complete vibration model with boundary excitation and a riser structure mutually coupled, forms a vibration control equation according to the vibration model, discretizes the vibration control equation by taking the first four-order vibration mode based on the Galerkin method, judges the unstable region of the vibration model by combining the Floquet theory, forms a minimum unstable region of the vibration model by changing the damping performance of the vibration model, and ensures the process that the vibration model is in a stable state by regulating and controlling the variable tension amplitude and the variable tension frequency. The invention is used in the field of ocean engineering.

Description

Method for rapidly predicting unstable interval of boundary excitation elongated tension beam
Technical Field
The invention relates to a rapid prediction method, and belongs to the technical field of electric digital data processing.
Background
For a deep sea platform, the length of a riser is generally longer, sea currents can act in a larger length range, the riser is caused to vibrate, incoming currents act on two sides of a cylinder to form alternately-falling vortexes, the vortexes which are periodically distributed generate vortex-induced lift forces perpendicular to the flow direction of the riser, and therefore the structure is induced to vibrate, namely vortex-induced vibration, and the riser is prone to fatigue damage under the action of the vortex-induced vibration. In a deep sea environment, an ocean platform generates heave motion under the action of environmental loads such as waves and ocean currents, periodic response is applied to the top of a riser, and time-varying tension is generated in the axial direction of the riser, so that the riser generates excitation vibration, and the vibration and fatigue damage of the riser are aggravated. On one hand, the heave motion of the platform structure can cause the axial tension and compression of the riser, the hydrodynamic response of the riser structure is caused, the instability of the system is caused, the parametric vibration system is provided with a plurality of external excitation resonance regions, and the external excitation resonance phenomenon also occurs when the external excitation frequency is equal to the combined value of the system natural frequency and the parametric excitation frequency except that the external excitation frequency is equal to the system natural frequency and the resonance occurs. On the other hand, when the frequency of the platform vibration is close to the natural frequency of the structure, resonance is generated, and the micro-amplitude platform vibration can amplify the amplitude of the vertical pipe by several times or even tens of times, so that the structure is damaged.
At present, most of researches on the fatigue characteristics of the marine riser simplify the tension applied by the marine platform into constant tension, only the displacement fixation of the platform structure is considered, and the time-varying tension generated at the top end of the riser caused by the sinking and floating of the platform is not considered. In fact, the parametric vibration is a special vibration form, and the special characteristic lies in that the external excitation of the parametric vibration is not applied to the system in the form of external force, but is indirectly realized through the periodic variation of the internal parameters of the system, and belongs to the research category of the nonlinear vibration theory. Because of the time-varying property of the parameters, the parametric excitation vibration system is a non-autonomous system, and the response of the system under the parametric excitation sometimes may be weak, but a severe resonance phenomenon may also occur, which depends on the stability of the parametric excitation vibration system. Under the action of axial parameter excitation, the inherent vibration characteristics of the vertical pipe structure are different in each moment compared with the previous moment, the dynamic characteristics of the vertical pipe are more complex, and an unstable area where parametric resonance occurs is a continuous parameter area, so that the avoidance of the parametric resonance is more difficult than the avoidance of common resonance generated due to forced vibration. Therefore, in order to better avoid fatigue damage to the marine riser due to resonance, the time-varying tension exerted by the platform on the riser under environmental loads must be considered. However, at present, a relevant prediction mode of the time-varying tension applied to the vertical tube by the platform under the action of environmental load is not considered.
The invention content is as follows:
aiming at the problems, the invention discloses a method for quickly predicting an unstable interval of a boundary excitation elongated tension beam.
The technical scheme adopted by the invention is as follows:
a method for rapidly predicting an unstable interval of a boundary excitation elongated tension beam includes the steps of establishing a complete vibration model with the boundary excitation and a vertical tube structure coupled with each other, forming a vibration control equation according to the vibration model, obtaining a first four-order vibration mode based on the Galerkin method to disperse the vibration control equation, judging the unstable interval of the vibration model by combining the Floquet theory, forming a minimum unstable area of the vibration model by changing damping performance of the vibration model, and ensuring that the vibration model is in a stable state by regulating and controlling variable tension amplitude and frequency.
As a preferable scheme: the prediction method comprises the following steps:
the method comprises the following steps: establishing a vibration control equation with the boundary excitation and the riser structure coupled with each other:
taking a flow transmission vertical pipe with the length of L and the diameter of D as a variable-tension flexible cylinder, taking the variable-tension flexible cylinder as a vibration model, hinging the upper end of the flow transmission vertical pipe with a floating platform and the lower end of the flow transmission vertical pipe with the seabed to establish a right-hand rectangular coordinate system, and further establishing a vibration control equation as follows:
Figure BDA0003026871260000021
in the above formula, m is the mass of the vibration system per unit length; t is time; EI is bending stiffness; r is structural damping RsAnd fluid damping RfSumming;
variable tension T (T) T of vibration model given by floating platform0-kasin(ωet)，T0Is the initial tension, and k is the stiffness compensation coefficient;
Wafor the immersion weight of the lifter per unit length, acIs the critical amplitude, k ═ LWa/acA is the amplitude of the variable tension, omegaeTo a variable tension frequency;
fL(z, t) is the lift per unit length and is denoted as fL(z,t)＝ρU2DCL0q(z,t)/4，CL0Is the transverse lift coefficient, q (z, t) is the dimensionless wake variable related to the lift coefficient;
converting vibration control equation (1) into a dimensionless form, let:
Figure BDA0003026871260000031
in the above formula, τ is dimensionless time, η is dimensionless vibration displacement, ξ is dimensionless coordinate position, the above formula (2) is substituted into the vibration control equation (1), the unstable interval of the vibration model is analyzed at the same time, the lift force term at the right end of the vibration control equation (1) is ignored, and the following dimensionless equations are obtained by sorting:
Figure BDA0003026871260000032
in the above formula (3), c is dimensionless damping expressed as
Figure BDA0003026871260000033
b is a dimensionless variable tension expressed as:
Figure BDA0003026871260000034
as a preferable scheme: the prediction method further comprises the following steps: judging an unstable interval of a coupled vibration control equation based on a Floquet theory, wherein the judging process is as follows:
the expression for the system of first order differential equations expressed in matrix form is:
Figure BDA0003026871260000035
wherein Y is (Y)1,y2,…y2N)T，
Figure BDA0003026871260000036
Representing the first derivative with respect to time, A (t) being a periodic matrix of 2N x 2N, in particular
Figure BDA0003026871260000037
The stability for expression (4) was analyzed using the Floquet theory, i.e.:
the expression for the basic solution u (t) for expression (4) is:
U(t)＝P(t)exp(tF) (5)
in the above formula, U (t) is a periodic matrix of 2N × 2N with a period of 2 π/ω; f is a 2 Nx 2N periodic matrix whose fundamental matrix solution U (t) grows over time and only if at least one eigenvalue of F has a positive real part; to numerically evaluate the eigenvalues of F, a state transition matrix Φ (t, t) is introduced0)：
Φ(t,t0)＝U(t)U-1(t0) (6)
Substituting the basic matrix solution into the expression, and taking T as T; t is t0Obtaining when the yield is 0:
Φ(T,0)＝P(T)exp(TF)P-1(0)＝P(0)exp(TF)P-1(0) (7)
the stability of the matrix F is independent of the initial conditions, and as can be seen from equation (7), the initial conditions are U (0) ═ P (0) ═ I, where I is the identity matrix, so that equation (7) is simplified as follows:
Φ(T,0)＝exp(TF) (8)
therefore, when the initial condition U (0) is P (0) is I, the state transition matrix expression of the matrix F through the period T is:
F＝T-1ln(Φ(T,0)) (9)
within the time interval T is more than or equal to 0 and less than or equal to T, the state transition matrix phi (T,0) is obtained by numerically solving the expression (4), the expression (4) needs to be solved for 2N times, a unit initial value of one component of the vector y is given each time, all other initial values are kept to be zero, namely when a first row solution of the basic matrix is obtained, the first y is taken to be 1, and the rest are 0; when solving the second row solution, the second y is 1, the rest are 0, and the calculation is carried out by analogy, and when the state transition matrix phi (T,0) is formed, the characteristic values are lambda respectivelyiWhere i is 1,2, … 2N, then 2N eigenvalues of this matrix need to be found; judging the relation between the absolute values of all the characteristic values and 1, and when the absolute values of the characteristic values are less than 1, indicating that the vibration model is unstable; when the absolute values of all the characteristic values are greater than or equal to 1, the vibration model is stable;
firstly, discretizing the formula (3) based on a Galerkin method, and introducing:
Figure BDA0003026871260000041
in the above-mentioned formula (10),
Figure BDA0003026871260000042
is a basis function, and an ith order mode function phi is introduced due to the hinged connection of two ends of the vibration modeli(ξ)，
Figure BDA0003026871260000043
The reason that only the first four-order mode is adopted is that the second-order mode is not enough to satisfy the discrete solving precision of the vibration control equation, and the eighth-order mode does not greatly improve the overall precision;
according to the mode function, there are:
Figure BDA0003026871260000044
the formula (11) is used for discretizing the formula (3), and the two ends of the discretized formula are simultaneously multiplied by phii(xi) and integrating over 0-1 to obtain a discrete form, wherein phi is1(xi) as an example, equation (12) is obtained:
Figure BDA0003026871260000045
in the same way will phi2(ξ)、φ3ξ and φ4(xi) is expanded and merged into a matrix form, i.e. equation (13), equation (13) is as follows:
Figure RE-RE-GDA0003077152940000051
equation (13) is further written in matrix form as:
Figure BDA0003026871260000052
in the formula (14), the first and second groups,
Figure BDA0003026871260000053
is a column vector of a 4 × 1 matrix, namely expressed as:
Figure BDA0003026871260000054
[L]、[B]and [ N]Are 4 × 4 matrices, and the expression of each element in the matrix is:
Figure BDA0003026871260000055
the second order ordinary differential equation of the formula (15) is rewritten as a first order ordinary differential equation, that is:
Figure BDA0003026871260000056
wherein:
Figure BDA0003026871260000057
firstly, the solution of a first-order ordinary differential equation is set as:
z＝Aeλτ (18)
back in equation (16):
[D]λAeλτ+[Y]Aeλτ＝0 (19)
about eλτAnd multiplied by [ D ] at the left end]-1Then, the following results are obtained:
(λI-(-D-1Y))A＝0 (20)
let H be-D-1Y, according to the Floquet theory, when the absolute value of the characteristic value of H is larger than 1, the variable-tension flexible cylinder is indicated to be in an unstable state, only the coefficient b of the variable-tension term influences the value of the characteristic value in the formula (3), and each group of different omegaeAnd a corresponds to a group of different characteristic values, the unstable interval of the variable-tension flexible cylinder is determined by judging the absolute value of the characteristic value, namely, each group of variable-tension frequency and amplitude are brought into a formula (20) to obtain the absolute value of the characteristic value, when the absolute value of the characteristic value corresponding to the group of data is determined to be more than 1, the group of data is indicated to belong to the unstable interval, and when the absolute value of the characteristic value corresponding to the group of data is determined to be less than or equal to 1, the group of data is indicated to belong to the stable interval.
As a preferable scheme: the calculation process of the mass m of the vibration system per unit length is as follows:
m＝ms+mf (21)
in the above formula, msIs the mass of the structure per unit length; m isfExternal fluid added mass per unit length; wherein the additional mass m of the external fluid per unit lengthfThe calculation formula of (2) is as follows:
mf＝CMρD2π/4 (22)
in the above formula, ρ is the fluid density, CMIs added with a mass coefficient, and CM＝1.0；
And calculating the mass m of the vibration system in unit length according to the two formulas.
As a preferable scheme: fluid damping RfThe calculation process of (2) is as follows:
Rf＝γΩfρD2 (23)
in the above formula, omegafThe local Steehar vortex release frequency is calculated according to the Steehar relational expression, and gamma is a viscous force coefficient; ρ is the fluid density; calculating the fluid damping R according to the formulaf。
As a preferable scheme: the process of establishing the right-hand rectangular coordinate system comprises the following steps: the lower end of the flow transmission vertical pipe is a coordinate origin O, the downstream direction is an X direction, the vertical direction is a Z direction, and the transverse vibration direction of the flow transmission vertical pipe is a Y direction.
The invention has the beneficial effects that:
the invention researches the unstable interval of the slender tension beam by considering boundary excitation, establishes a variable-tension flexible cylinder which is a complete boundary excitation-structure coupling vibration model, takes the first four-order vibration mode to disperse a vibration control equation based on the Galerkin method, and judges the unstable interval by combining the Floquet theory. The influence between the change of the tension of the top end of the riser and the structure is considered in real time by the model, and the unstable region of the variable-tension flexible cylinder can be stably and comprehensively predicted, so that the unstable region of the cylinder system under the boundary excitation is obtained, and the fatigue damage of the marine riser due to resonance is avoided.
Description of the drawings:
for ease of illustration, the invention is described in detail by the following detailed description and the accompanying drawings.
FIG. 1 is a schematic structural diagram of a vibration model with two hinged ends under uniform incoming flow, in which a vertically arranged double-headed arrow indicates a sinking and floating direction of a floating platform, and a plurality of horizontally arranged arrows indicate uniform incoming flow directions; the arrows arranged along the longitudinal direction of the vibration model indicate the acting direction of the variable tension t (t);
FIG. 2 is a first schematic diagram of the variation of the outflow velocity with a varying tension amplitude;
FIG. 3 is a second schematic diagram of the variation of the outflow velocity with a varying tension amplitude;
FIG. 4 is a third schematic view of the variation of the outflow velocity with respect to the variable tension amplitude;
fig. 5 is a fourth schematic diagram showing the variation relationship between the outflow velocity and the variable tension amplitude.
In the figure, 30-floating platform; 40-vibration model; 50-sea level.
The specific implementation mode is as follows:
to make the objects, technical solutions and advantages of the present invention more clear, the basic principles of the shaking table mixing test using the method of the present invention are explained based on the shaking table mixing test principle, but it should be understood that the descriptions are only exemplary and are not intended to limit the scope of the present invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
The first embodiment is as follows: the embodiment is described with reference to fig. 1, fig. 2, fig. 3, fig. 4, and fig. 5, in the embodiment, a vibration model 40 in which complete boundary excitation and a riser structure are coupled to each other is established, a vibration control equation is formed according to the vibration model 40, a first four-order vibration mode is taken based on the galois method to discretize the vibration control equation, an unstable region of a variable-tension flexible cylinder is determined with reference to the Floquet theory, a minimum unstable region of the variable-tension flexible cylinder is formed by changing the damping performance of the variable-tension flexible cylinder, and the variable-tension flexible cylinder is in a stable state by regulating and controlling the amplitude and frequency of the variable-tension flexible cylinder.
The second embodiment is as follows: in this embodiment, which is a further limitation of the first embodiment, the prediction method includes the steps of: the prediction method comprises the following steps:
the method comprises the following steps: establishing a vibration control equation with the boundary excitation and the riser structure coupled with each other:
taking a flow transmission vertical pipe with the length of L and the diameter of D as a variable-tension flexible cylinder, taking the variable-tension flexible cylinder as a vibration model 40, and respectively hinging two ends of the flow transmission vertical pipe with a floating platform 30 to establish a right-hand rectangular coordinate system so as to establish a vibration control equation as follows:
Figure BDA0003026871260000081
in the above formula, m is the mass of the vibration system per unit length; r is structural damping RsAnd fluid damping RfIn sum, the variable tension T (T) T ═ T of the vibration model 40 given by the floating platform 300-kasin(ωet)，T0For initial tension, k is the stiffness compensation coefficient, WaFor the immersion weight of the lifter per unit length, acIs the critical amplitude, k ═ LWa/acA is the variable tension amplitude, omegaeFor variable tension frequency, t is time; f. ofL(z, t) is the lift per unit length and is denoted as fL(z,t)＝ρU2DCL0q(z,t)/4，CL0Is the transverse lift coefficient, q (z, t) is the dimensionless tail flow variation related to the lift coefficient; EI is bending stiffness;
converting vibration control equation (1) into a dimensionless form, let:
Figure BDA0003026871260000082
in the above formula, τ is dimensionless time, η is dimensionless vibration displacement, ξ is dimensionless coordinate position, the above formula (2) is substituted into the vibration control equation (1), the unstable interval of the vibration model 40 itself is analyzed, the lift force term at the right end of the vibration control equation (1) is ignored, and the following dimensionless equations are obtained by sorting:
Figure BDA0003026871260000083
in the above formula (3), c is a dimensionless resistanceNi, is represented as
Figure BDA0003026871260000084
b is a dimensionless variable tension expressed as:
Figure BDA0003026871260000085
in the present embodiment, a variable tension flexible cylinder is used as the vibration model 40, which is a complete vibration model 40 with boundary excitation coupled to the riser structure. The floating platform 30 is located above sea level 50, the floating platform 30 is connected to the seabed through a vibration model 40, the upper end of the flow transfer riser is hinged to the floating platform 30, and the lower end of the flow transfer riser is hinged to the seabed. Boundary excitations are applied to the vibrational model 40 by the sea level 50 acting on the floating platform 30.
The third concrete implementation mode: the present embodiment is further limited to the first or second embodiment, wherein the prediction method further includes a second step of: judging an unstable interval of a coupled vibration control equation based on a Floquet theory, wherein the judging process is as follows:
the expression for the system of first order differential equations expressed in matrix form is:
Figure BDA0003026871260000091
wherein Y is (Y)1,y2,…y2N)T，
Figure BDA0003026871260000092
Representing the first derivative with respect to time, A (t) being a periodic matrix of 2N x 2N, in particular
Figure BDA0003026871260000093
The stability for expression (4) was analyzed using the Floquet theory, i.e.:
the expression for the basic solution u (t) for expression (4) is:
U(t)＝P(t)exp(tF) (5)
in the above formula, the first and second carbon atoms are,u (t) is a 2N × 2N periodic matrix with a period of 2 π/ω; f is a 2 Nx 2N periodic matrix whose fundamental matrix solution U (t) grows over time and only if at least one eigenvalue of F has a positive real part; to numerically evaluate the eigenvalues of F, a state transition matrix Φ (t, t) is introduced0)：
Φ(t,t0)＝U(t)U-1(t0) (6)
Substituting the basic matrix solution into the expression, and taking T as T; t is t0Obtaining when the yield is 0:
Φ(T,0)＝P(T)exp(TF)P-1(0)＝P(0)exp(TF)P-1(0) (7)
the stability of the matrix F is independent of the initial conditions, and as can be seen from equation (7), the initial conditions are U (0) ═ P (0) ═ I, where I is the identity matrix, so that equation (7) is simplified as follows:
Φ(T,0)＝exp(TF) (8)
therefore, when the initial condition U (0) is P (0) is I, the state transition matrix expression of the matrix F through the period T is:
F＝T-1ln(Φ(T,0)) (9)
within the time interval T is more than or equal to 0 and less than or equal to T, the state transition matrix phi (T,0) is obtained by numerically solving the expression (4), the expression (4) needs to be solved for 2N times, a unit initial value of one component of the vector y is given each time, all other initial values are kept to be zero, namely when a first row solution of the basic matrix is obtained, the first y is taken to be 1, and the rest are 0; when solving the second row solution, the second y is 1, the rest are 0, and the calculation is carried out by analogy, and when the state transition matrix phi (T,0) is formed, the characteristic values are lambda respectivelyiWhere i is 1,2, … 2N, 2N eigenvalues of this matrix are found. Judging the relation between the absolute values of all the characteristic values and 1, and when the absolute values of all the characteristic values are less than 1, indicating that the system is stable; when the absolute values of all the characteristic values are greater than or equal to 1, the system is stable;
firstly, discretizing the formula (3) based on a Galerkin method, and introducing:
Figure BDA0003026871260000101
in the above-mentioned formula (10),
Figure BDA0003026871260000102
is a basis function, and introduces an ith order mode function phi due to the hinged connection of the two ends of the vibration model 40i(ξ)，
Figure BDA0003026871260000103
The reason that only the first four-order mode is adopted is that the second-order mode is insufficient to meet the requirement of discrete solving precision of the vibration control equation, and the eighth-order mode does not greatly improve the overall precision;
according to the mode function, there are:
Figure BDA0003026871260000104
the formula (11) is used for discretizing the formula (3), and the two ends of the discretized formula are simultaneously multiplied by phii(xi) and integrating over 0-1 to obtain a discrete form, wherein phi is1(xi) as an example, equation (12) is obtained:
Figure BDA0003026871260000105
in the same way will phi2(ξ)、φ3ξ and φ4(xi) is expanded and combined in a matrix form, i.e. equation (13):
Figure BDA0003026871260000106
equation (13) is further written in matrix form as:
Figure BDA0003026871260000111
in the formula (14), the first and second groups,
Figure BDA0003026871260000112
is a column vector of a 4 × 1 matrix, namely expressed as:
Figure BDA0003026871260000113
[L]、[B]and [ N]Are 4 × 4 matrices, and the expression of each element in the matrix is:
Figure BDA0003026871260000114
the second order ordinary differential equation is rewritten into the first order ordinary differential equation for the above three equations (15), that is:
Figure BDA0003026871260000115
wherein:
Figure BDA0003026871260000116
firstly, the solution of a first-order ordinary differential equation is set as:
z＝Aeλτ (18)
back in equation (16):
[D]λAeλτ+[Y]Aeλτ＝0 (19)
about eλτAnd multiplied by [ D ] at the left end]-1Then, the following results are obtained:
(λI-(-D-1Y))A＝0 (20)
let H be-D-1Y, according to the Floquet theory, when the absolute value of the characteristic value of H is larger than 1, the variable-tension flexible cylinder is indicated to be in an unstable state, only the coefficient b of the variable-tension term influences the value of the characteristic value in the formula (3), and each group of different omegaeAnd a corresponds to a group of different characteristic values, and the unstable interval of the variable-tension flexible cylinder is determined by judging the absolute value of the characteristic value, namely each group of variable-tension frequency and amplitudeAnd the absolute value of the characteristic value is obtained by converting the absolute value into a formula (20), when the absolute value of the characteristic value corresponding to the group of data is determined to be more than 1, the group of data is indicated to belong to an unstable interval, and when the absolute value of the characteristic value corresponding to the group of data is determined to be less than or equal to 1, the group of data is indicated to belong to a stable interval.
The fourth concrete implementation mode: the embodiment is further limited to the first, second or third embodiment, and the calculation process of the mass m of the vibration system per unit length is as follows:
m＝ms+mf (21)
in the above formula, msIs the mass of the structure per unit length; m isfExternal fluid added mass per unit length; wherein the additional mass m of the external fluid per unit lengthfThe calculation formula of (2) is as follows:
mf＝CMρD2π/4 (22)
in the above formula, ρ is the fluid density, CMIs added with a mass coefficient, and CM＝1.0；
And calculating the mass m of the vibration system in unit length according to the two formulas.
The fifth concrete implementation mode: this embodiment is a further limitation of the first, second, third or fourth embodiment, the fluid damping RfThe calculation process of (2) is as follows:
Rf＝γΩfρD2 (23)
in the above formula, omegafThe local Steehar vortex release frequency is calculated according to the Steehar relational expression, and gamma is a viscous force coefficient; ρ is the fluid density; calculating the fluid damping R according to the formulaf。
The sixth specific implementation mode: the present embodiment is further limited to the first, second, third, fourth, or fifth embodiment, and the process of establishing the right-hand rectangular coordinate system is as follows: the lower end of the flow transmission vertical pipe is a coordinate origin O, the downstream direction is an X direction, the vertical direction is a Z direction, and the transverse vibration direction of the flow transmission vertical pipe is a Y direction.
The seventh embodiment: the embodiment is further limited to the first, second, third, fourth, fifth or sixth specific embodiments, and the rapid prediction method further includes a third step of performing calculation analysis on the example based on the analysis data, and performing specific analysis according to the specific example data, wherein the process is as follows:
the specific data are as follows:
Figure BDA0003026871260000121
Figure BDA0003026871260000131
as can be seen from fig. 2, 3, 4 and 5, the black area in each figure represents the unstable interval of the vibration model 40, and the black area in each figure has a change rule, so that the unstable interval of the vibration model 40 always appears periodically and alternately, and as the flow rate of the external uniform incoming flow continuously increases, the unstable interval of the vibration model 40 also presents a rapid reduction mode, which not only represents the reduction of the area of the whole black area, but also gradually reduces the width of the black area in each period. As can be seen from equation (1), the variation of the outflow velocity directly affects the damping term of the system, and the variation of the entire unstable region substantially decreases as the damping of the vibration model 40 increases.
By properly utilizing the stability graphs, one can derive a rapid assessment of the dynamic behavior of a given system under a range of realistic operating conditions. According to the above analysis, the unstable region can be minimized by changing the damping performance of the system. The system can be in a stable state by regulating and controlling the amplitude and the frequency of the variable tension, so that the rapid prediction process of the unstable interval of the boundary excitation long and thin tension beam is realized.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are given by way of illustration of the principles of the present invention, and that various changes and modifications may be made without departing from the spirit and scope of the invention as defined by the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (4)

1. A method for rapidly predicting an unstable interval of a boundary excitation elongated tension beam is characterized by comprising the following steps: establishing a complete vibration model (40) with boundary excitation and a riser structure mutually coupled, forming a vibration control equation according to the vibration model (40), taking the first four-order vibration mode based on the Galerkin method to disperse the vibration control equation, judging an unstable interval of the vibration model (40) by combining the Floquet theory, forming a minimum unstable area of the vibration model (40) by changing the damping performance of the vibration model (40), and ensuring the vibration model (40) to be in a stable state by regulating and controlling the variable tension amplitude and frequency;
the prediction method comprises the following steps:
the method comprises the following steps: establishing a vibration control equation with the boundary excitation and the riser structure coupled with each other:
taking a flow transmission vertical pipe with the length of L and the diameter of D as a variable-tension flexible cylinder, taking the variable-tension flexible cylinder as a vibration model (40), connecting the upper end of the flow transmission vertical pipe with a floating platform (30) and connecting the lower end of the flow transmission vertical pipe with the seabed in a hinged manner, establishing a right-hand rectangular coordinate system, and further establishing a vibration control equation as follows:
Figure FDA0003548955630000011
in the above formula, m is the mass of the vibration system per unit length; t is time; EI is bending stiffness; r is structural damping RsAnd fluid damping RfSumming;
the variable tension T (T) of the vibration model (40) given by the floating platform (30) is T0-kasin(ωet)，T0Is the initial tension, and k is the stiffness compensation coefficient;
Wafor the immersion weight of the lifter per unit length, acIs the critical amplitude, k ═ LWa/acA is the amplitude of the variable tension, omegaeTo a variable tension frequency;
fL(z, t) is the lift per unit length and is denoted as fL(z,t)＝ρU2DCL0q(z,t)/4，CL0Is the transverse lift coefficient, q (z, t) is the dimensionless wake variable related to the lift coefficient;
converting vibration control equation (1) into a dimensionless form, let:
Figure FDA0003548955630000012
in the above formula, τ is dimensionless time, η is dimensionless vibration displacement, ξ is dimensionless coordinate position, the above formula (2) is substituted into the vibration control equation (1), meanwhile, the unstable interval of the vibration model (40) is analyzed, the lift force term at the right end of the vibration control equation (1) is ignored, and the following dimensionless equations are obtained through arrangement:
Figure FDA0003548955630000013
in the above formula (3), c is dimensionless damping expressed as
Figure FDA0003548955630000021
CDThe coefficient of drag force is, U is the outflow velocity;
b is dimensionless variable tension expressed as
Figure FDA0003548955630000022
The prediction method further comprises the following steps: judging an unstable interval of a coupled vibration control equation based on a Floquet theory, wherein the judging process is as follows:
the expression for the system of first order differential equations expressed in matrix form is:
Figure FDA0003548955630000023
wherein Y is (Y)1,y2,…y2N)T，
Figure FDA0003548955630000024
Representing the first derivative with respect to time, A (t) being a periodic matrix of 2N x 2N, in particular
Figure FDA0003548955630000025
The stability for expression (4) was analyzed using the Floquet theory, i.e.:
the expression for the basic solution u (t) for expression (4) is:
U(t)＝P(t)exp(tF) (5)
in the above formula, U (t) is a periodic matrix of 2N × 2N with a period of 2 π/ω; f is a 2 Nx 2N periodic matrix whose fundamental matrix solution U (t) grows over time and only if at least one eigenvalue of F has a positive real part; to numerically evaluate the eigenvalues of F, a state transition matrix Φ (t, t) is introduced0)：
Φ(t,t0)＝U(t)U-1(t0) (6)
Substituting the basic matrix solution into the expression, and taking T as T; t is t0Obtaining when the yield is 0:
Φ(T,0)＝P(T)exp(TF)P-1(0)＝P(0)exp(TF)P-1(0) (7)
the stability of the matrix F is independent of the initial conditions, and as can be seen from equation (7), the initial conditions are U (0) ═ P (0) ═ I, where I is the identity matrix, so that equation (7) is simplified as follows:
Φ(T,0)＝exp(TF) (8)
therefore, when the initial condition U (0) is P (0) is I, the state transition matrix expression of the matrix F through the period T is:
F＝T-1ln(Φ(T,0)) (9)
within the time interval T is more than or equal to 0 and less than or equal to T, the state transition matrix phi (T,0) is obtained by numerically solving the expression (4), and the expression (4) needs to be solved for 2N times, and one component of the vector y is given each timeThe unit initial value of (1) and all other initial values are kept to be zero, that is, when the first row solution of the basic matrix is obtained, the first y is taken to be 1, and the rest are 0; when solving the second row solution, the second y is 1, the rest are 0, and the calculation is carried out by analogy, and when the state transition matrix phi (T,0) is formed, the characteristic values are lambda respectivelyiWhere i is 1,2, … 2N, then 2N eigenvalues of this matrix need to be found; judging the relation between the absolute values of all the characteristic values and 1, and when the absolute value of the characteristic value is less than 1, indicating that the vibration model (40) is unstable; when the absolute values of all the characteristic values are greater than or equal to 1, the vibration model (40) is stable;
firstly, discretizing the formula (3) based on a Galerkin method, and introducing:
Figure FDA0003548955630000031
in the above-mentioned formula (10),
Figure FDA0003548955630000032
is a basis function, and introduces an ith order mode function phi due to the hinged connection of two ends of the vibration model (40)i(ξ)，
Figure FDA0003548955630000033
The reason that only the first four-order mode is adopted is that the second-order mode is not enough to satisfy the discrete solving precision of the vibration control equation, and the eighth-order mode does not greatly improve the overall precision;
according to the mode function, there are:
Figure FDA0003548955630000034
the formula (11) is used for discretizing the formula (3), and the two ends of the discretized formula are simultaneously multiplied by phii(xi) and integrating over 0-1 to obtain a discrete form, wherein phi is1(xi) as an example, equation (12) is obtained:
Figure FDA0003548955630000035
in the same way will phi2(ξ)、φ3ξ and φ4(xi) is expanded and merged into a matrix form, i.e. equation (13), equation (13) is as follows:
Figure FDA0003548955630000041
equation (13) is further written in matrix form as:
Figure FDA0003548955630000042
in the formula (14), the first and second groups,
Figure FDA0003548955630000043
is a column vector of a 4 × 1 matrix, namely expressed as:
Figure FDA0003548955630000044
[L]、[B]and [ N]Are 4 × 4 matrices, and the expression of each element in the matrix is:
Figure FDA0003548955630000045
the second order ordinary differential equation of the formula (15) is rewritten as a first order ordinary differential equation, that is:
Figure FDA0003548955630000046
wherein:
Figure FDA0003548955630000047
firstly, the solution of a first-order ordinary differential equation is set as:
z＝Aeλτ (18)
back in equation (16):
[D]λAeλτ+[Y]Aeλτ＝0 (19)
about eλτAnd multiplied by [ D ] at the left end]-1Then, the following results are obtained:
(λI-(-D-1Y))A＝0 (20)
let H be-D-1Y, according to the Floquet theory, when the absolute value of the characteristic value of H is larger than 1, the variable-tension flexible cylinder is indicated to be in an unstable state, only the coefficient b of the variable-tension term influences the value of the characteristic value in the formula (3), and each group of different omegaeAnd a corresponds to a group of different characteristic values, the unstable interval of the variable-tension flexible cylinder is determined by judging the absolute value of the characteristic value, namely, each group of variable-tension frequency and amplitude is converted into a formula (20), the absolute value of the characteristic value is calculated, when the absolute value of the characteristic value corresponding to the group of data is determined to be more than 1, the group of data belongs to the unstable interval, and when the absolute value of the characteristic value corresponding to the group of data is determined to be less than or equal to 1, the group of data belongs to the stable interval.
2. The method for rapidly predicting the unstable interval of the boundary excitation elongated tension beam as claimed in claim 1, wherein: the calculation process of the mass m of the vibration system per unit length is as follows:
m＝ms+mf (21)
in the above formula, msIs the mass of the structure per unit length; m isfExternal fluid added mass per unit length; wherein the additional mass m of the external fluid per unit lengthfThe calculation formula of (2) is as follows:
mf＝CMρD2π/4 (22)
in the above formula, ρ is the fluid density, CMIs added with a mass coefficient, and CM＝1.0；
And calculating the mass m of the vibration system in unit length according to the two formulas.
3. The method for rapidly predicting the unstable interval of the boundary excitation elongated tension beam as claimed in claim 1, wherein: fluid damping RfThe calculation process of (2) is as follows:
Rf＝γΩfρD2 (23)
in the above formula, omegafThe local Steehar vortex release frequency is calculated according to the Steehar relational expression, and gamma is a viscous force coefficient; ρ is the fluid density; calculating the fluid damping R according to the formulaf。
4. The method for rapidly predicting the unstable interval of the boundary excitation elongated tension beam as claimed in claim 1, wherein: the process of establishing the right-hand rectangular coordinate system comprises the following steps: the lower end of the flow transmission vertical pipe is a coordinate origin O, the downstream direction is an X direction, the vertical direction is a Z direction, and the transverse vibration direction of the flow transmission vertical pipe is a Y direction.
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