CN109408899A - A kind of deep water hanger rope nonlinear Motion Response calculation method - Google Patents
A kind of deep water hanger rope nonlinear Motion Response calculation method Download PDFInfo
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Abstract
The invention discloses a kind of deep water hanger rope nonlinear Motion Response calculation methods, based on elastic wave theory, from the angle of energy, the dynamic model of hanger rope is established according to Hamilton principle, derive hanger rope in the three dimensional non-linear equation of motion in normal direction, tangential and secondary three directions of normal direction, consider plane motion, respective non-linear expression is obtained by Taylor expansion, the characteristics of according to equation, it is solved using finite difference method, the accuracy and reliability of method for solving is demonstrated, while analyzing and the reason of error occurs；The analysis for being loaded in nonlinear response rule under external disturbance for deep water hanger rope and hanging provides a method, also can be used for analyzing the horizontal and vertical movement of hanger rope.
Description
Technical field
The present invention relates to a kind of deep water hanger rope field, specifically a kind of deep water hanger rope nonlinear Motion Response calculation method.
Background technique
Deep water hanger rope nonlinear Motion Response is the new problem that ocean engineering technology occurs into deep-sea development process, is one
The nonlinear time-varying process of a complexity, non-linear, external drive non-linear and side caused by hanger rope natural resiliency deforms in addition
The non-linear of boundary's condition to become abnormal difficult to the solution of the problem, and its engineer application field is only limitted to hang under water and is
It unites, research achievement is relatively fewer inside and outside native land.
Summary of the invention
The purpose of the present invention is to provide a kind of deep water hanger rope nonlinear Motion Response calculation methods, to solve above-mentioned background
The problem of being proposed in technology.
To achieve the above object, the invention provides the following technical scheme:
A kind of deep water hanger rope nonlinear Motion Response calculation method, the specific steps are that:
S1, progress deep water hanger rope Dynamic Modeling obtain deep water hanger rope Nonlinear Equations of Motion: using elastic wave theory as base
Plinth considers the design feature and Elasticity performance of hanger rope, ignores its bending, shearing and torsion stiffness, uses S0Indicate hanger rope not
Geometry when being stretched, SiIndicate static balancing position, SfIndicate hanger rope dynamic geometry configuration, air as shown in Figs. 1-2
The middle dynamic geometric configuration schematic diagram of hanger rope；Arc coordinate s is established, hanger rope one end connects lash ship, and the other end is connected with load is hung, in figure
Given Ri(s) and Rf(s, t) respectively indicates position vector of the certain point on static balancing position and performance graph on hanger rope,
Then hanger rope can be indicated relative to the three-D displacement of equilbrium position are as follows:
R (s, t)=Rf(s,t)-Ri(s) (1-1)
By R (s, t) respectively along normal directionTangentiallyWith secondary normal directionIt is divided into three components Rs1(s,t)、R2(s,t)、R3(s,
T) it can obtain:
In view of the continuity and non-linear of hanger rope movement, it is continuously non-that classical force theory is no longer desirable for analysis hanger rope
Linear movement, therefore from the angle of energy, according to Hamilton principle, it is believed that the gross energy of hanger rope is by its own strain
A few part compositions of energy, kinetic energy, gravitional force, external work, and then derive the three dimensional non-linear movement side of deep water hanger rope
Journey.
The expression formula of hanger rope transient strain energy is then described are as follows:
Wherein, efIndicate the strain energy of hanger rope transient state, sfIndicate the arc length coordinate of transient state configuration after hanger rope extends, s0It indicates
Arc length coordinate when hanger rope does not extend.
Then hanger rope is in transient state configuration χfWhen strain energy are as follows:
WhereinFor equilbrium position χiWhen hanger rope strain energy, LiIndicate the length of hanger rope when equilbrium position, LiFor balance position
Set χiWhen hanger rope cross-sectional area, E be hanger rope elasticity modulus, Pi(si, t) and the static tensile of hanger rope when being balance, centered on ε
The dynamic component of Lagrange strain after line stretching, expression formula are；
When not considering hanger rope surrounding fluid, gravitional force be may be expressed as:
WhereinIt indicates in equilbrium position χiWhen hanger rope gravitional force, ρ be hanger rope density, lτAnd lnIt respectively indicates and cuts
To the direction cosines with normal direction.
At this point for the hanger rope in water, due to the effect by buoyancy, direction is with gravity direction on the contrary, then being produced by buoyancy
Raw potential energy can be expressed as follows:
WhereinIt indicates in equilbrium position χiWhen buoyancy generate potential energy, ρwIndicate the density of water, other meanings respectively measured
It is identical as aforementioned formula.
Then hanger rope is in transient state configuration χfKinetic energy may be expressed as:
Wherein VfFor the absolute velocity of particle on dynamic hanger rope configuration, expression formula are as follows:
The external force F work done then acted on hanger rope may be expressed as:
Wherein external force F can be divided into the component F in three directions along direction of displacement1、F2And F3。
Then according to Hamilton principle, have:
Expression formula (1-4), (1-7), (1-8), (1-9), (1-10) and (1-119) is substituted into formula (1-12), three can be obtained
The direction hanger rope three dimensional non-linear equation of motion.
Tangential motion equation:
The normal direction equation of motion:
The secondary normal direction equation of motion:
-ρAiU3, tt=[(Pi+EAiε)U3, s], s+F3 (1-15)
Assuming that the constitutive relation of hanger rope is linear, and only consider in-plane moving, ignores item related with secondary normal direction, in conjunction with
The fundamental equation of elastic wave can obtain hanger rope Nonlinear Equations of Motion are as follows:
S2, deep water hanger rope nonlinear Motion Response numerical solution:
More applicable difference scheme, the matrix of the obtained difference scheme are derived by according to Taylor expansion are as follows:
The difference scheme of displacement and speed with time correlation uses the result of study of scholar Mohammad:
Wherein U, v, a, displacement, velocity and acceleration, α are respectively indicated1、α2、β1、β2Respectively integral parameter takes respectively
0.5、1.0、0.5、1.0。
Normal direction and tangential acceleration and hanger rope Nonlinear Equations of Motion (2-16) at different nodes in equation (3-49)
The relationship between external force F in (2-17) can be obtained by defining discrete kinetics equation:
MAi+1+C|Vi|Vi+KUi=(Fexcit)i (3-50)
Wherein M is the unit length hanger rope quality including additional mass, and A indicates that acceleration, V indicate speed, and U is indicated
Displacement, FexcitIndicate external drive.
As a further solution of the present invention: the spread speed of nonlinear elasticity wave is sat with position in the step S1
The relevant function of the parameters such as mark, Horizontal Tension and hanger rope weight ratio λ.
As further scheme of the invention: the essence of finite difference method partial differential equation is in the step S2
Continuous problem discretization, the system of linear equations for being converted to finite form is solved, and main solution procedure includes: first pair and asks
It solves domain and carries out grid dividing, replace continuous function with the numerical value of grid intersection point；Second construction difference scheme appropriate, by differential side
Journey discretization exports system of linear equations；Third carries out interpolation to the approximation on discrete point and approaches, and obtains the approximation for solving domain
Solution.
Compared with prior art, the beneficial effects of the present invention are: based on elastic wave theory, go out from the angle of energy
Hair, the dynamic model of hanger rope is established according to Hamilton principle, derives hanger rope in normal direction, tangential and secondary three directions of normal direction
The three dimensional non-linear equation of motion, consider plane motion, respective non-linear expression is obtained by Taylor expansion, according to equation
Feature is solved using finite difference method, demonstrates the accuracy and reliability of method for solving, while analyzing and missing
The reason of difference；The analysis for being loaded in nonlinear response rule under external disturbance for deep water hanger rope and hanging provides a method, also may be used
For analyzing the horizontal and vertical movement of hanger rope.
Detailed description of the invention
Fig. 1 is the dynamic geometric configuration schematic diagram of hanger rope in air.
Fig. 2 is the dynamic geometric configuration schematic diagram of hanger rope in air.
Fig. 3 is a kind of stress diagram of hanger rope in water in deep water hanger rope nonlinear Motion Response calculation method.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts every other
Embodiment shall fall within the protection scope of the present invention.
Please refer to Fig. 1~3, in the embodiment of the present invention, a kind of deep water hanger rope nonlinear Motion Response calculation method is specific
Steps are as follows:
S1, progress deep water hanger rope Dynamic Modeling obtain deep water hanger rope Nonlinear Equations of Motion: using elastic wave theory as base
Plinth considers the design feature and Elasticity performance of hanger rope, ignores its bending, shearing and torsion stiffness, uses S0Indicate hanger rope not
Geometry when being stretched, SiIndicate static balancing position, SfIndicate hanger rope dynamic geometry configuration, air as shown in Figs. 1-2
The middle dynamic geometric configuration schematic diagram of hanger rope；Arc coordinate s is established, hanger rope one end connects lash ship, and the other end is connected with load is hung, in figure
Given Ri(s) and Rf(s, t) respectively indicates position vector of the certain point on static balancing position and performance graph on hanger rope,
Then hanger rope can be indicated relative to the three-D displacement of equilbrium position are as follows:
R (s, t)=Rf(s,t)-Ri(s) (1-1)
By R (s, t) respectively along normal directionTangentiallyWith secondary normal directionIt is divided into three components Rs1(s,t)、R2(s,t)、R3(s,
T) it can obtain:
In view of the continuity and non-linear of hanger rope movement, it is continuously non-that classical force theory is no longer desirable for analysis hanger rope
Linear movement, therefore from the angle of energy, according to Hamilton principle, it is believed that the gross energy of hanger rope is by its own strain
A few part compositions of energy, kinetic energy, gravitional force, external work, and then derive the three dimensional non-linear movement side of deep water hanger rope
Journey.
The expression formula of hanger rope transient strain energy is then described are as follows:
Wherein, efIndicate the strain energy of hanger rope transient state, sfIndicate the arc length coordinate of transient state configuration after hanger rope extends, s0It indicates
Arc length coordinate when hanger rope does not extend.
Then hanger rope is in transient state configuration χfWhen strain energy are as follows:
WhereinFor equilbrium position χiWhen hanger rope strain energy, LiIndicate the length of hanger rope when equilbrium position, LiFor balance
Position χiWhen hanger rope cross-sectional area, E be hanger rope elasticity modulus, Pi(si, t) be balance when hanger rope static tensile, expression
Formula are as follows:
Pi(si, t) and=EAiei (1-5)
ε is the dynamic component of the Lagrange strain after center line stretches, expression formula are as follows:
When not considering hanger rope surrounding fluid, gravitional force be may be expressed as:
WhereinIt indicates in equilbrium position χiWhen hanger rope gravitional force, ρ be hanger rope density, lτAnd lnIt respectively indicates and cuts
To the direction cosines with normal direction.
At this point for the hanger rope in water, as shown in figure 3, due to the effect by buoyancy, direction and gravity direction on the contrary,
It can be then expressed as follows by the potential energy that buoyancy generates:
WhereinIt indicates in equilbrium position χiWhen buoyancy generate potential energy, ρwIndicate the density of water, other meanings respectively measured
It is identical as aforementioned formula.
Then hanger rope is in transient state configuration χfKinetic energy may be expressed as:
Wherein VfFor the absolute velocity of particle on dynamic hanger rope configuration, expression formula are as follows:
The external force F work done then acted on hanger rope may be expressed as:
Wherein external force F can be divided into the component F in three directions along direction of displacement1、F2And F3。
Then according to Hamilton principle, have:
Expression formula (1-4), (1-7), (1-8), (1-9), (1-10) and (1-119) is substituted into formula (1-12), three can be obtained
The direction hanger rope three dimensional non-linear equation of motion.
Tangential motion equation:
The normal direction equation of motion:
The secondary normal direction equation of motion:
-ρAiU3, tt=[(Pi+EAiε)U3, s], s+F3 (1-15)
Tension P when each equation includes hanger rope force balance state it can be seen from the equation being derived by above
With two unknown quantitys of curvature κ.Since tension has a major impact the nonlinear motion of hanger rope, while curvature and hanger rope configuration are direct
Correlation, it is closely related with hanger rope totality stress and movement, therefore the two amounts need to be analyzed when hanger rope is in equilibrium state.
Inquire into configuration of the hanger rope in equilibrium state, ignore displacement and the effect of external force, it is believed that the equilibrium state of hanger rope be it is instantaneous, can
It is zero to enable all and time correlation parameter, then can obtains the tension and song that calculate hanger rope equilibrium state by above-mentioned equation
The equation of rate are as follows:
Piκi=(ρ-ρw)Aigln (1-17)
Two above equation gives the equilibrium configuration of hanger rope, when meter and flow field when influence to hanger rope, that is, ρwWhen ≠ 0, side
Journey considers the effect of buoyancy, expression be hanger rope in water equilibrium configuration.Introduce φi, indicateWith the angle of vertical direction,
Then curvature can be respectively indicated with direction cosines are as follows:
κi=φi ,s (1-18)
lτ=sin φi (1-19)
ln=cos φi (1-20)
Integral transformation is carried out to equation (1-16) and (1-17), (1-18), (1-19) and (1-20) is substituted into, can must be hung
The tension of cable and the expression formula of curvature:
Wherein P0For the Horizontal Tension of hanger rope, two above equation is suitable for the hanger rope of relaxed state.
By above two formula it is found that the tension and curvature of hanger rope and arc length coordinate s are non-linear relation, equation cannot be parsed
Solution carries out Taylor series expansions to quadravalence to two equations, can obtain simultaneously to consider the non-linear of hanger rope simultaneously convenient for calculating:
It is convenient for expression, enable λ=P0/ ρ Ag, indicates the ratio of Horizontal Tension and hanger rope unit length gravity, and dimension is
1/m.As s=L,Indicate the ratio of Horizontal Tension and hanger rope self weight.The meaning for introducing λ is, opens when ignoring level
When Horizontal Tension of the power in other words in hanger rope is smaller,It is able to maintain to be a small amount of, then no matter hanger rope is in relaxation or tensioning
State, equation (1-23) and (1-24) can keep the convergence of series.
Ignore a small amount of of quadravalence and quadravalence or more, (1-23), (1-24) write out again, can be obtained:
κ (s, t)=λ (1- λ2s2) (1-26)
One of an important factor for λ is influence hanger rope tension and curvature (configuration) it can be seen from formula (1-25), (1-26),
Its size reflect hanger rope configuration and elastic state.When λ is larger, the higher order term in formula be can not ignore, and expression is hanger rope
In relaxed state, it is on the contrary then be tensioning state.
The simplification of S2, deep water hanger rope Nonlinear Equations of Motion: three maintenance and operations when according to the hanger rope equilibrium state being derived by
Dynamic equation, writes out the equation of motion again are as follows:
-ρAiR1,tt=[(P+EA ε) (1+R1,s-κR2)],s-κ(P+EAε)(R2,s-κR1)+F1 (2-1)
-ρAR2,tt=[(P+EA ε) (R2,s-κR1)],s-κ(P+EAε)(1+R1,s-κR2)+F2 (2-2)
-ρAR3,tt=[(P+EA ε) R3,s],s+F3 (2-3)
Wherein F1,F2,F3Indicating the external force in three directions, P (s, t) and κ (s, t) respectively indicate the tension and curvature of hanger rope,
ε is dynamic strain, expression formula are as follows:
Assuming that the constitutive relation of hanger rope is linear, and only consider in-plane moving, ignores item related with secondary normal direction, then move
State strain may be expressed as:
ε=R1,s-κR2 (2-5)
By P (s, t), κ (s, t) and equation (2-1) and (2-2) is substituted into, only considers in-plane moving, it can be deduced that hanger rope is non-thread
The property equation of motion are as follows:
After above-mentioned equation (2-6) and (2-7) are arranged, write out again:
By Such analysis it is recognised that for parameter lambda, the expression formula of tension and curvature when hanger rope equilibrium state is had ignored
Therefore item more than its 4 rank and 4 ranks is cast out 4 rank of parameter lambda in equation (2-8) and (2-9) and items more than 4 ranks, abbreviation
After obtain:
Nonlinear terms in equation (2-10) and (2-11) are merged, it can be deduced that:
Above-mentioned two equation (2-12) and (2-13) are extremely complex, and the multiple nonlinear terms for including in equation, which to solve, to be become
It is abnormal difficult, analytic solutions cannot be obtained.Therefore above-mentioned two equation need to be further simplified.The characteristics of passing through observation equation, knot
Close the fundamental equation of elastic wave:
As can be seen from the above equation, the parameter for influencing entire elastic wave propagation characteristic is spread speed, Linear Viscoelastic Constitutive Relation
What is influenced is also, that is to say, that it is for equation (2-12) and (2-13) major concern is contributed to elastic wave propagation speed
Item, other outliers can be ignored, in this way, continuing simplification to equation (2-12) and (2-13) can obtain:
Enable C1 2=a1+a2U1,s+a3U2, C2 2=b1+b2U1,s+b3U2Respectively indicate that nonlinear elasticity wave is tangential and the biography of normal direction
Broadcast speed, each coefficient are as follows:
In this way, just having obtained the non-linear plane differential equations of motion of deep water hanger rope (2-16) and (2-17).It can be seen that
It is that the spread speed of nonlinear elasticity wave is letter relevant to parameters such as position coordinates, Horizontal Tension and hanger rope weight ratio λ
Number.
S3, deep water hanger rope nonlinear Motion Response numerical solution: it is solved using finite difference calculus, finite difference calculus is asked
The essence for solving partial differential equation is continuous problem discretization, and the system of linear equations for being converted to finite form is solved, mainly
Solution procedure includes: that first pair of solution domain carries out grid dividing, replaces continuous function with the numerical value of grid intersection point；Second construction is suitable
When difference scheme differential equation discretization is exported into system of linear equations；Third carries out interpolation to the approximation on discrete point and forces
Closely, obtain solving the approximate solution in domain.
To established equation (2-16) and (2-17), it is discrete to the progress of partial differential item that suitable difference scheme is constructed first
It approaches.Variable is introduced, Taylor expansion was carried out to it, can be obtained:
Different point x can be chosen according to different equation and solving precisioniAnd order.Correlative study the result shows that, if
Using the difference scheme of 3 second orders, precision is limited during solving equation, using 5 quadravalences difference scheme when calculating
Precision is much better.
The quadravalence format of first-order partial derivative is constructed first, and finding out 5 points is respectively xi-2、xi-1、xi、xi+1And xi+2, then
It writes out except xiTaylor series of other outer four points of point to dependent variable are as follows:
Equation (3-2), (3-3), (3-4) and (3-5) is transformed, multiplication by constants a, b, c, d can be obtained respectively:
It sums to equation (3-2), (3-3), (3-4) and (3-5), retains first derivative item du (xi)/dx, is omitted
Other higher order terms introduce solving condition -2a-b+c+2d=1 (3-10)
In equation (3-10) " 1 " indicate in order to determine tetra- values of undetermined coefficient a, b, c, d, (3-2), (3-3), (3-
4) and (3-5) superimposed result retains first derivative item.Same method enables: 4a+b+c+ to eliminate Derivative Terms
4d=0 (3-11)
What " 0 " in above formula indicated is that tetra- values of determining a, b, c, d eliminate Derivative Terms.Similarly, in order to eliminate three ranks
With Fourth-Derivative item, can obtain:
- 8a-b+c+8d=0 (3-12)
16a+b+c+16d=0 (3-13)
(3-10), (3-11), (3-12) and (3-31) four is thus obtained for solving the linear equation of a, b, c, d,
A=2/4 can be obtained after solution！, b=-16/4！, c=16/4！, d=-2/4！.Later by a, b, c, d rewind equation (3-6),
Inside (3-7), (3-8) and (3-9), first derivative du (x is obtainediThe expression formula of)/dx:
When u (x) is about u (xi) it is symmetrical, it is noted that formula (3-14) is the centered difference form of quadravalence, to obtain
du(x1The approximate solution of)/dx can use u (x2)、u(x3)、u(x4) and u (x5) Taylor series expansion form indicate:
Equally, first derivative du (x in order to obtain1)/dx is introduced condition a+2b+3c+4d=1 (3-19)
The higher order term of second order or more is eliminated simultaneously, is enabled:
A+4b+9c+16d=0 (3-20)
A+8b+27c+64d=0 (3-21)
A+16b+81c+256d=0 (3-22)
It is solved by simultaneous equations (3-19), (3-20), (3-21) and (3-22), available a=96/4！, b=-72/
4！, c=32/4！, d=-6/4！, take back equation (3-15), (3-16), (3-17) and (3-18), available du (x1)/dx's
Expression formula are as follows:
Same method, du (x2The approximate solution of)/dx u (x1)、u(x3)、u(x4) and u (x5) Taylor series form
It indicates, such undetermined coefficient equation group are as follows:
- a+b+2c+3d=1 (3-24)
A+b+4c+9d=0 (3-25)
- a+b+8c+27d=0 (3-26)
A+b+16c+81d=0 (3-27)
Then obtain du (x2The expression formula of)/dx are as follows:
Then du (xN-1)/dx and du (xNThe expression formula of)/dx are as follows:
In this way by the matrix of equation (3-14), (3-23), (3-28), (3-29) and (3-30) available difference scheme
Are as follows:
Continue the quadravalence format that second order partial differential is derived according to preceding method, for equation (3-2), (3-3), (3-
4) and (3-5) retains Derivative Terms, enables 4a+b+c+4d=2 (3-32)
The higher order term for omitting three ranks and three ranks or more, can obtain coefficient equation are as follows:
- 8a-b+c+8d=0 (3-33)
16a+b+c+16d=0 (3-34)
- 32a-b+c+32d=0 (3-35)
A=-2/4 can be obtained by solving equation (3-32), (3-33), (3-34) and (3-35)！, b=32/4！, c=32/4！, d
=-2/4！, four coefficients are taken back into available du in original equation2(xi)/dx2Expression formula:
For du2(x2)/dx2Approximate expression, can be using working as x=x1,x3,x4,x5,x6When u (x) linear combination come
It indicates:
au(x1)+bu(x3)+cu(x4)+du(x5)+eu(x6) (3-37)
In order to omit du (x2)/dx, order-a+b+2c+3d+4e=0 (3-38)
Likewise, being enabled to omit the item of three ranks and three ranks or more:
- a+b+8c+27d+64e=0 (3-39)
A+b+16c+81d+256e=0 (3-40)
- a+b+32c+243d+1024e=0 (3-41)
Retain d2u(x2)/dx2, introduce condition a+b+4c+9d+16e=2 (3-42)
Equation (3-38), (3-39), (3-40), (3-41) and (3-42) is solved it can be concluded that a=-2/4！, b=32/
4！, c=32/4！, d=-2/4！, substitute into original equation, available du2(x2)/dx2Expression formula:
D is obtained using same method later2u(xN-1)/dx2Expression formula:
Finally find out d2u(x1)/dx2And d2u(xN)/dx2Approximate expression, due to d2u(x1)/dx2And d2u(xN)/dx2Packet
Boundary condition is contained, therefore has chosen and work as x=x2,x3,x4,x5When u (x) and du (x1The linear combination of)/dx, it may be assumed that
Solve system of equation (3-45), substitution original equation can obtain d after obtaining coefficient2u(x1)/dx2Expression formula:
D is obtained with same method later2u(xN)/dx2Expression formula:
Equation (3-31) and (3-48) are to establish for solving partial differential equation (2-16) and the space (2-17) is micro-
The difference scheme divided.
The difference scheme of displacement and speed with time correlation uses the result of study of scholar Mohammad:
Wherein U, v, a, displacement, velocity and acceleration, α are respectively indicated1、α2、β1、β2Respectively integral parameter takes respectively
0.5、1.0、0.5、1.0。
Normal direction and tangential acceleration and hanger rope Nonlinear Equations of Motion (2-16) at different nodes in equation (3-49)
The relationship between external force F in (2-17) can be obtained by defining discrete kinetics equation:
MAi+1+C|Vi|Vi+KUi=(Fexcit)i (3-50)
Wherein M is the unit length hanger rope quality including additional mass, and A indicates that acceleration, V indicate speed, and U is indicated
Displacement, FexcitIndicate external drive.
Although the present invention is described in detail referring to the foregoing embodiments, for those skilled in the art,
It is still possible to modify the technical solutions described in the foregoing embodiments, or part of technical characteristic is carried out etc.
With replacement, all within the spirits and principles of the present invention, any modification, equivalent replacement, improvement and so on should be included in this
Within the protection scope of invention.
Claims (6)
1. a kind of deep water hanger rope nonlinear Motion Response calculation method, which is characterized in that it is dynamic to mainly comprise the following steps progress deep water hanger rope
Mechanical modeling obtains deep water hanger rope Nonlinear Equations of Motion: based on elastic wave theory, considering the design feature and bullet of hanger rope
Property mechanical property, ignore its bending, shearing and torsion stiffness, use S0Indicate geometry when hanger rope is not stretched, SiIndicate quiet
State equilbrium position, SfIndicate hanger rope dynamic geometry configuration；Arc coordinate s is established, hanger rope one end connects lash ship, and the other end connects with load is hung
It connects, the R given in figurei(s) and Rf(s, t) respectively indicates position of the certain point on static balancing position and performance graph on hanger rope
Vector is set, then hanger rope can be indicated relative to the three-D displacement of equilbrium position are as follows:
R (s, t)=Rf(s,t)-Ri(s) (1-1)
By R (s, t) respectively along normal directionTangentiallyWith secondary normal directionIt is divided into three components Rs1(s,t)、R2(s,t)、R3(s, t) can
:
From the angle of energy, according to Hamilton principle, it is believed that the gross energy of hanger rope by the strain energy of its own, kinetic energy,
A few part compositions of gravitional force, external work；
The expression formula of hanger rope transient strain energy is then described are as follows:
Wherein efIndicate the strain energy of hanger rope transient state, sfIndicate the arc length coordinate of transient state configuration after hanger rope extends, s0Indicate hanger rope not
Arc length coordinate when elongation；
Then hanger rope is in transient state configuration χfWhen strain energy are as follows:
WhereinFor equilbrium position χiWhen hanger rope strain energy, LiIndicate the length of hanger rope when equilbrium position, LiFor equilbrium position χi
When hanger rope cross-sectional area, E be hanger rope elasticity modulus, Pi(si, t) be balance when hanger rope static tensile, expression formula are as follows:
Pi(si, t) and=EAiei (1-5)
ε is the dynamic component of the Lagrange strain after center line stretches, expression formula are as follows:
When not considering hanger rope surrounding fluid, gravitional force be may be expressed as:
WhereinIt indicates in equilbrium position χiWhen hanger rope gravitional force, ρ be hanger rope density, lτAnd lnRespectively indicate tangentially with
The direction cosines of normal direction；
At this point for the hanger rope in water, due to the effect by buoyancy, direction and gravity direction are on the contrary, then by buoyancy generation
Potential energy can be expressed as follows:
WhereinIt indicates in equilbrium position χiWhen buoyancy generate potential energy, ρwIndicate water density, other meanings respectively measured with it is aforementioned
Formula is identical；
Then hanger rope is in transient state configuration χfKinetic energy may be expressed as:
Wherein VfFor the absolute velocity of particle on dynamic hanger rope configuration, expression formula are as follows:
The external force F work done then acted on hanger rope may be expressed as:
Wherein external force F can be divided into the component F in three directions along direction of displacement1、F2And F3；
Then according to Hamilton principle, have:
Expression formula (1-4), (1-7), (1-8), (1-9), (1-10) and (1-119) is substituted into formula (1-12), three sides can be obtained
To the hanger rope three dimensional non-linear equation of motion；
Tangential motion equation:
The normal direction equation of motion:
The secondary normal direction equation of motion:
Tension P and song when each equation includes hanger rope force balance state it can be seen from the equation being derived by above
Two unknown quantitys of rate κ；Configuration of the hanger rope in equilibrium state is inquired into, ignores displacement and the effect of external force, it is believed that the balance of hanger rope
State be it is instantaneous, enabling all with time correlation parameter is zero, then can be obtained calculating hanger rope equilibrium-like by above-mentioned equation
The tension of state and the equation of curvature are as follows:
Piκi=(ρ-ρw)Aigln (1-17)
Two above equation gives the equilibrium configuration of hanger rope, when meter and flow field when influence to hanger rope, that is, ρwWhen ≠ 0, equation is examined
Considered the effect of buoyancy, expression be hanger rope in water equilibrium configuration；Introduce φi, indicateIt is with the angle of vertical direction, then bent
Rate can be respectively indicated with direction cosines are as follows:
κi=φi ,s (1-18)
lτ=sin φi (1-19)
ln=cos φi (1-20)
Integral transformation is carried out to equation (1-16) and (1-17), (1-18), (1-19) and (1-20) is substituted into, of hanger rope can be obtained
The expression formula of power and curvature:
Wherein P0For the Horizontal Tension of hanger rope, two above equation is suitable for the hanger rope of relaxed state；
By above two formula it is found that the tension and curvature of hanger rope and arc length coordinate s are non-linear relation, equation cannot get analytic solutions,
Simultaneously to consider the non-linear of hanger rope simultaneously convenient for calculating, Taylor series expansions are carried out to quadravalence to two equations, can be obtained:
It is convenient for expression, enable λ=P0/ ρ Ag indicates the ratio of Horizontal Tension and hanger rope unit length gravity, dimension 1/m；When
When s=L,Indicate the ratio of Horizontal Tension and hanger rope self weight；
Ignore a small amount of of quadravalence and quadravalence or more, (1-23) and (1-24) write out again, can be obtained:
κ (s, t)=λ (1- λ2s2) (1-26)。
2. a kind of deep water hanger rope nonlinear Motion Response calculation method according to claim 1, which is characterized in that the depth
Water hanger rope Nonlinear Equations of Motion carries out subsequent simplification: three-dimensional motion equation when according to the hanger rope equilibrium state being derived by,
Again the equation of motion is write out are as follows:
-ρAiR1,tt=[(P+EA ε) (1+R1,s-κR2)],s-κ(P+EAε)(R2,s-κR1)+F1 (2-1)
-ρAR2,tt=[(P+EA ε) (R2,s-κR1)],s-κ(P+EAε)(1+R1,s-κR2)+F2 (2-2)
-ρAR3,tt=[(P+EA ε) R3,s],s+F3 (2-3)
Wherein F1,F2,F3Indicate the external force in three directions, P (s, t) and κ (s, t) respectively indicate the tension and curvature of hanger rope, and ε is
Dynamic strain, expression formula are as follows:
Assuming that the constitutive relation of hanger rope is linear, and only consider in-plane moving, ignores item related with secondary normal direction, then dynamically answer
Change may be expressed as:
ε=R1,s-κR2 (2-5)
By P (s, t), κ (s, t) and equation (2-1) and (2-2) is substituted into, only considers in-plane moving, it can be deduced that the non-linear fortune of hanger rope
Dynamic equation are as follows:
After above-mentioned equation (2-6) and (2-7) are arranged, write out again:
By Such analysis it is recognised that for parameter lambda, the expression formula of tension and curvature when hanger rope equilibrium state have ignored its 4
Therefore item more than rank and 4 ranks is cast out 4 rank of parameter lambda in equation (2-8) and (2-9) and items more than 4 ranks, after abbreviation
It arrives:
Nonlinear terms in equation (2-10) and (2-11) are merged, it can be deduced that:
Above-mentioned two equation (2-12) and (2-13) are extremely complex, and the multiple nonlinear terms for including in equation, which to solve, becomes abnormal
Difficulty cannot obtain analytic solutions；Therefore the characteristics of above-mentioned two equation need to being further simplified, pass through observation equation, in conjunction with bullet
The fundamental equation of property wave:
As can be seen from the above equation, the parameter for influencing entire elastic wave propagation characteristic is spread speed, and Linear Viscoelastic Constitutive Relation influences
Be also, that is to say, that for equation (2-12) and (2-13) major concern be to the contributive item of elastic wave propagation speed,
Other outliers can be ignored, in this way, continuing simplification to equation (2-12) and (2-13) can obtain:
Enable C1 2=a1+a2U1,s+a3U2, C2 2=b1+b2U1,s+b3U2It respectively indicates nonlinear elasticity wave tangentially and the propagation of normal direction is fast
Degree, each coefficient are as follows:
3. a kind of deep water hanger rope nonlinear Motion Response calculation method according to claim 2, which is characterized in that the depth
Water hanger rope Nonlinear Equations of Motion is solved after carrying out subsequent simplification, i.e. deep water hanger rope nonlinear Motion Response numerical solution:
To established equation (2-16) and (2-17), suitable difference scheme is constructed first, discrete force is carried out to partial differential item
Closely；Variable is introduced, Taylor expansion was carried out to it, can be obtained:
Different point x can be chosen according to different equation and solving precisioniAnd order；
The quadravalence format of first-order partial derivative is constructed first, and finding out 5 points is respectively xi-2、xi-1、xi、xi+1And xi+2, then write out and remove
xiTaylor series of other outer four points of point to dependent variable are as follows:
Equation (3-2), (3-3), (3-4) and (3-5) is transformed, multiplication by constants a, b, c, d can be obtained respectively:
It sums to equation (3-2), (3-3), (3-4) and (3-5), retains first derivative item du (xi)/dx, omits other
Higher order term introduces solving condition -2a-b+c+2d=1 (3-10)
In equation (3-10) " 1 " indicate in order to determine tetra- values of undetermined coefficient a, b, c, d, (3-2), (3-3), (3-4) and (3-
5) superimposed result retains first derivative item；Same method enables: 4a+b+c+4d=to eliminate Derivative Terms
0 (3-11)
What " 0 " in above formula indicated is that tetra- values of determining a, b, c, d eliminate Derivative Terms；Similarly, in order to eliminate three ranks and four
Order derivative item can obtain:
- 8a-b+c+8d=0 (3-12)
16a+b+c+16d=0 (3-13)
(3-10), (3-11), (3-12) and (3-31) four is thus obtained for solving the linear equation of a, b, c, d, is solved
After can obtain a=2/4！, b=-16/4！, c=16/4！, d=-2/4！；Later by a, b, c, d rewind equation (3-6), (3-7),
Inside (3-8) and (3-9), first derivative du (x is obtainediThe expression formula of)/dx:
When u (x) is about u (xi) it is symmetrical, it is noted that formula (3-14) is the centered difference form of quadravalence, to obtain du
(x1The approximate solution of)/dx can use u (x2)、u(x3)、u(x4) and u (x5) Taylor series expansion form indicate:
Equally, first derivative du (x in order to obtain1)/dx is introduced condition a+2b+3c+4d=1 (3-19)
The higher order term of second order or more is eliminated simultaneously, is enabled:
A+4b+9c+16d=0 (3-20)
A+8b+27c+64d=0 (3-21)
A+16b+81c+256d=0 (3-22)
It is solved by simultaneous equations (3-19), (3-20), (3-21) and (3-22), available a=96/4！, b=-72/4！, c
=32/4！, d=-6/4！, take back equation (3-15), (3-16), (3-17) and (3-18), available du (x1The expression of)/dx
Formula are as follows:
Same method, du (x2The approximate solution of)/dx u (x1)、u(x3)、u(x4) and u (x5) Taylor series form indicate,
Undetermined coefficient equation group in this way are as follows:
- a+b+2c+3d=1 (3-24)
A+b+4c+9d=0 (3-25)
- a+b+8c+27d=0 (3-26)
A+b+16c+81d=0 (3-27)
Then obtain du (x2The expression formula of)/dx are as follows:
Then du (xN-1)/dx and du (xNThe expression formula of)/dx are as follows:
In this way by the matrix of equation (3-14), (3-23), (3-28), (3-29) and (3-30) available difference scheme are as follows:
Continue to derive the quadravalence format of second order partial differential according to preceding method, for equation (3-2), (3-3), (3-4) and
(3-5) retains Derivative Terms, enables 4a+b+c+4d=2 (3-32)
The higher order term for omitting three ranks and three ranks or more, can obtain coefficient equation are as follows:
- 8a-b+c+8d=0 (3-33)
16a+b+c+16d=0 (3-34)
- 32a-b+c+32d=0 (3-35)
A=-2/4 can be obtained by solving equation (3-32), (3-33), (3-34) and (3-35)！, b=32/4！, c=32/4！, d=-2/
4！, four coefficients are taken back into available du in original equation2(xi)/dx2Expression formula:
For du2(x2)/dx2Approximate expression, can be using working as x=x1,x3,x4,x5,x6When u (x) linear combination indicate:
au(x1)+bu(x3)+cu(x4)+du(x5)+eu(x6) (3-37)
In order to omit du (x2)/dx, order-a+b+2c+3d+4e=0 (3-38)
Likewise, being enabled to omit the item of three ranks and three ranks or more:
- a+b+8c+27d+64e=0 (3-39)
A+b+16c+81d+256e=0 (3-40)
- a+b+32c+243d+1024e=0 (3-41)
Retain d2u(x2)/dx2, introduce condition a+b+4c+9d+16e=2 (3-42)
Equation (3-38), (3-39), (3-40), (3-41) and (3-42) is solved it can be concluded that a=-2/4！, b=32/4！, c
=32/4！, d=-2/4！, substitute into original equation, available du2(x2)/dx2Expression formula:
D is obtained using same method later2u(xN-1)/dx2Expression formula:
Finally find out d2u(x1)/dx2And d2u(xN)/dx2Approximate expression, due to d2u(x1)/dx2And d2u(xN)/dx2It contains
Boundary condition, therefore choose and work as x=x2,x3,x4,x5When u (x) and du (x1The linear combination of)/dx, it may be assumed that
Solve system of equation (3-45), substitution original equation can obtain d after obtaining coefficient2u(x1)/dx2Expression formula:
D is obtained with same method later2u(xN)/dx2Expression formula:
Equation (3-31) and (3-48) are to establish for solving partial differential equation (2-16) and (2-17) space differentiation
Difference scheme；
The difference scheme of displacement and speed with time correlation uses the result of study of scholar Mohammad:
Wherein U, v, a, displacement, velocity and acceleration, α are respectively indicated1、α2、β1、β2Respectively integral parameter, take 0.5 respectively,
1.0,0.5,1.0；
Normal direction and tangential acceleration and hanger rope Nonlinear Equations of Motion (2-16) and (2- at different nodes in equation (3-49)
17) relationship between external force F in can be obtained by defining discrete kinetics equation:
MAi+1+C|Vi|Vi+KUi=(Fexcit)i (3-50)
Wherein M is the unit length hanger rope quality including additional mass, and A indicates that acceleration, V indicate speed, and U indicates position
It moves, FexcitIndicate external drive.
4. a kind of deep water hanger rope nonlinear Motion Response calculation method according to claim 2, which is characterized in that described non-
The spread speed of linear elasticity wave is function relevant to parameters such as position coordinates, Horizontal Tension and hanger rope weight ratio λ.
5. a kind of deep water hanger rope nonlinear Motion Response calculation method according to claim 3, which is characterized in that described to have
The essence that limit calculus of finite differences solves partial differential equation is continuous problem discretization to be converted to the system of linear equations progress of finite form
It solves.
6. a kind of deep water hanger rope nonlinear Motion Response calculation method according to claim 5, which is characterized in that described to have
The main solution procedure of limit calculus of finite differences includes: that first pair of solution domain carries out grid dividing, is replaced continuously with the numerical value of grid intersection point
Function；Differential equation discretization is exported system of linear equations by the second construction difference scheme appropriate；Third is on discrete point
Approximation carries out interpolation and approaches, and obtains the approximate solution for solving domain.
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