CN116663139A - AIS data-based irregular wave ship parameter rolling motion probability evaluation method - Google Patents
AIS data-based irregular wave ship parameter rolling motion probability evaluation method Download PDFInfo
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Abstract
The invention discloses an AIS data-based irregular wave ship parameter rolling motion probability evaluation method, which comprises the following steps: establishing a probability density model of the ship density, and solving a probability density function to obtain a ship density map; according to the ship density map, selecting test points from the representative route area at medium distance, and calculating the irregular wave occurrence probability of the target sea area at a specific test point as P 1 The method comprises the steps of carrying out a first treatment on the surface of the Establishing a parameter roll motion differential equation, and calculating the occurrence parameter roll motion probability P of the ship in irregular waves 2 The method comprises the steps of carrying out a first treatment on the surface of the According to P 1 、P 2 Calculating a particularAnd the probability P of the parameter rolling motion of the ship under the test point in the irregular wave is obtained, so that the total occurrence probability of the parameter rolling motion of the ship on the representative route is obtained. The invention can improve the design of the ship, reduce the parameter roll of the ship, provide reasonable advice for how to avoid the parameter roll in the actual sailing of the ship, and improve the sailing safety in the stormy waves of the ship.
Description
Technical Field
The invention belongs to the field of ship nonlinear motion analysis, and particularly relates to an AIS data-based ship parameter rolling motion probability evaluation method in irregular waves.
Background
The parameter rolling is one of five failure modes in the second generation of the complete stability of the ship, and is the most comprehensive stability failure mode which is researched at home and abroad at present. The parameter rolling is a relatively unique rolling phenomenon, and the principle is that when a ship sails in a longitudinal wave or a diagonal wave, the wave surface shape and the area of a water plane of the ship in the wave change along with the change of time, so that the recovery characteristic of the ship body is periodically changed, the wet surface areas of the ship at the crest and the trough of the wave are different, and the area of the water plane is greatly changed. The ship motion parameter time history is essentially a strong nonlinear phenomenon caused by the time history change of the ship motion parameter, is not directly generated by external excitation, and has great damage to the marine navigation and transportation of the ship.
Early studies showed that the ship occurrence parameter roll motion needs to meet certain trigger conditions: the ship sails in the waves; the natural frequency of the rolling motion of the ship subjected to a wave frequency approximately 2 times; the wave encountered by the ship is approximately equal to the ship length; the wave height encountered by the ship is larger than a critical value; the rolling damping of the ship is smaller. Three main research methods are provided for the research of the current phase of the parameter rolling motion in irregular waves: nonlinear dynamics method, direct numerical simulation method, model test method. The non-dynamic rolling motion method has the characteristics of small calculated amount, high efficiency and low cost in the research of the parameter rolling motion, has irreplaceable effect on the research, and can be used as a main research means for researching the parameter rolling mechanism.
At present, most of researches on ship navigation safety are based on a certain point on the existing navigation line, but from the viewpoint of ship operation, the influence of the ship navigation track wave-up angle, sea state distribution and the like on the nonlinear motion of the ship is remarkable. Therefore, how to comprehensively consider the influence of the course and the random sea condition distribution on the rolling motion of the ship parameters is necessary for fully evaluating and analyzing the stability and the wave resistance of the ship under the complex sea conditions.
Disclosure of Invention
The invention aims to: the invention aims to provide an AIS data-based ship parameter rolling motion probability evaluation method in irregular waves. A ship density calculation model is established based on AIS data and a standard ship, a probability density function of the ship density is estimated by adopting a non-parameter nuclear density method, a ship density diagram obtained through the probability density function is used for finding a certain sea area representative route, the probability of parameter rolling motion in ship navigation on the representative route is researched, and in order to solve the limitation of the ship nonlinear parameter rolling motion in a non-regular wave by a numerical method, the provided non-linear dynamics probability analysis method for the ship parameter rolling motion in the non-regular wave is used for analyzing the probability of parameter rolling motion in the non-regular wave by adopting a nonlinear dynamics analysis method, so that an analysis method and a safe navigation basis are provided for the safe navigation of the ship.
The technical scheme is as follows: the invention relates to an AIS data-based ship parameter rolling motion probability evaluation method in irregular waves, which comprises the following steps:
step 1: AIS data of a target sea area in a target time period is selected, a probability density model of ship density is established, and a probability density function of the ship density is solved based on a non-parameter kernel density estimation method to obtain a ship density map;
step 2: presetting a ship critical density threshold value according to a ship density map, defining a region with larger traffic flow, identifying and representing a representative route region from the region, selecting a plurality of test points from the representative route region at medium distance, and calculating to obtain the irregular wave occurrence probability P of a target sea area at a specific test point according to wave scattering data of sea areas near each test point 1 ；
Step 3: establishing a ship motion coordinate system and a ship parameter rolling motion differential equation in irregular waves, determining a ship recovery moment, and solving a probability density function of a parameter rolling motion response in the irregular waves by a path integration method to obtain a ship occurrence parameter rolling motion probability P in the irregular waves 2 ；
Step 4: according to the probability of occurrence of irregular waves of a target sea area at a specific test point, the probability is P 1 And the probability of occurrence of parametric roll motions P of the vessel in irregular waves 2 Calculating to obtain the probability P of the ship under the specific test point to generate parameter rolling motion in irregular waves;
step 5: and (2) repeating the steps (2) to (4), and calculating the probability of the ship in each test point for generating the parameter rolling motion in the irregular waves to obtain the total generation probability of the ship in the representative route for generating the parameter rolling motion.
Further, the step 1 specifically includes the following steps:
step 1-1: let x be 1 、x 2 …x n For a sample of the random variable x, the probability density function of the random variable x isThe probability density function of the ship density is as follows:
where x is a random variable,representing ship density x i Representing an ith ship density sample, h representing bandwidth length, n representing sample number, and K representing a kernel function;
step 1-2: the Gaussian kernel function is selected as the kernel function in the non-parametric kernel density estimation method, and the Gaussian kernel function has the following formula:
step 1-3: selecting h as bandwidth;
step 1-4: the probability density function of the ship density obtained by adopting the non-parameter nuclear density estimation method is as follows:
and obtaining a ship density map.
Further, in step 2, 6-8 test points are selected from the representative airline area at a medium distance.
Further, in step 2, according to the wave scattering data of the sea area near each test point, the probability distribution map of the occurrence of the irregular wave in the wave height period combined probability density data is obtained according to the ratio of the occurrence times of the irregular wave of different specific sense wave heights and characteristic period combinations to the total observation times of the wave scattering data, and the probability of the occurrence of the irregular wave of the target sea area at the specific test point is obtained as P 1 Wherein:
P 1 number of occurrences of irregular waves of specific sense wave height and characteristic period combination/total number of observations of wave spread data.
Further, the step 3 specifically includes the following steps:
step 3-1: assuming that the motion of the vessel in the wave is quasi-static, according to the force balance principle, a differential equation of the vessel parameter roll motion is established, as shown in the following formula:
wherein I is φφ For moment of inertia, δI φφ For additional roll moment of inertia, phi is the vessel roll angle,for roll angular velocity +.>For roll angular acceleration +.>Represents the damping force to which the ship is subjected, B 1 And B 3 A linear damping coefficient and a cubic nonlinear damping coefficient, respectively,/->Representing roll recovery moment, F wave Representing the wave forces to which the vessel is subjected;
step 3-2: the determined recovery moment of the ship:
solving the recovery moment of the vessel is the focus of solving the above equations and studying the parametric roll motions of the vessel. There are two main approaches: the method is a direct solution method, wherein the primary stability in the wave is divided into a primary stability high in still water and a variable quantity in the wave, and the primary stability high in the wave of the ship is obtained through detailed deduction, and the change of the primary stability high relative to the elevation of the wave surface can be directly deduced; the second method is an indirect roll damping method, and according to the study of Dostal, the recovery moment of the ship can be expressed as a function of roll angle, wave surface rise, wave surface phase angle and time as follows:
wherein delta represents the water displacement of the ship, eta represents the wave surface rise, phi represents the phase angle, the range is [0-2 pi ], and the function of the roll angle is fitted after the recovery moment function of the numerical model ship;
in the invention, the influence of heave and pitch movements on the recovery moment arm is not considered, the recovery moment arm of the ship is obtained through numerical solution by adopting a slicing theory based on indirect roll damping solving, and then the roll moment arm function is obtained through simulation.
According to the Froude-Krylov assumption, wave force and wave moment are obtained by integrating underwater pressure on underwater wet surface area:
F＝-∫∫ Ω pndΩ (6)
wherein Ω represents the underwater wet surface area of the hull, r is a position vector, n represents a unit normal vector of the wet surface of the hull, and the underwater pressure p comprises two parts, namely hydrostatic pressure and hydrodynamic pressure generated by wave surface elevation, as shown in the following formula:
wherein ρ is the sea water density, g is the gravitational acceleration, z is the water depth, η is the amplitude, k is the wave number, and c is the wave velocity;
integrating the formula (8) to obtain the recovery moment to which the ship is subjected:
wherein beta is the course angle, S (x') is the cross-sectional area of each immersion,and->For each submerged cross-section centroid B 0 The coordinates in the reference coordinate system, F (x'), are the pressure gradient coefficients of the respective cross sections, calculated as follows:
wherein d (x ') represents the draft of each cross section of the ship in the standing state in still water, and B (x') represents the line width of each cross section of the ship in the standing state in still water;
in order to determine the restoring moment of the ship, it is necessary to obtain an approximate analytical expression of GZ (Φ, η, ψ), which is developed as a combination of a polynomial and fourier series with respect to the roll angle Φ as follows:
wherein N is φ Represents the highest power of phi in the numerical fitting process, which is 2N φ +1；N k Expanding the number of terms for the polynomial; n (N) S 、N C Expanding the number of terms for the Fourier series; the variable is determined to be valued by a least square method; q, Q S 、Q C 、Coefficients representing the expansion term are respectively determined by the results of numerical fitting;
GZ app the difference between the expansion of (Φ, η, ψ) and the calculation result GZ (Φ, η, ψ) of the numerical simulation needs to be minimum even if the following formula E is minimized:
after comprehensively considering the accuracy of numerical fitting and the complexity of the subsequent calculation, finally determining N φ ＝2、N k ＝0、N S ＝1、N C =1, and let Q S 0,1,1 ＝0、Q S 1,1,1 ＝0、Q S 2,1,1 ＝0，Q C 1,1,1 ＝0、， Then GZ app The expansion of (φ, η, ψ) is expressed as:
GZ app (φ,η,ψ)＝Q 0,0,0 φ+Q 1,0,0 φ 3 +Q 2,0,0 φ 5 +Q C 0,1,1 ηcos(ψ)φ (13)
random process eta due to irregular wave decomposition c ＝ηcos(ψ)，GZ app (φ, η, ψ) is written as:
GZ app (φ,η,ψ)＝q 1 φ+q 2 φ 3 +q 3 φ 5 +q 4 η c φ (14)
wherein q i (i=1, 2,3, 4) represents the coefficient of the expansion term by GZ app Results of numerical fits of (phi, eta, phi) to GZ (phi, eta, phi) are determined;
step 3-3: solving a steady-state probability density function:
in the present invention, in order to simplify the calculation, the colored nature of the waves, i.e., the waves are assumed to be white noise, is temporarily not considered. Based on a path integration method, a corresponding Fokker-Planck equation is solved in a time domain to obtain a transition probability density of ship rolling motion evolving along with time, and then a steady probability density function of the ship in response to parameter rolling motion in irregular waves is solved. The invention takes a two-dimensional calculation example, and the specific steps are as follows:
substituting the formula (14) into the formula (4) by considering the effect of random wave external excitation under oblique waves to obtain a ship freedom degree motion differential equation considering parameter rolling motion:
dividing both sides of the above equation by I φφ +δI φφ ：
In the method, in the process of the invention,
then formula (16) translates into the following expression:
order theξ 1 (t)＝p(t)，ξ 2 (t)＝f wave (t) then there is:
let x 1 ＝φ；The stochastic differential equation (18) is converted into the following set of state equations:
exciting the parameters and wave load ζ k (t) as a colored noise sequence, respectively passing through a linear filter and outputting, as shown in the formula (20):
wherein W is k (t) is a Wiener function; in conjunction with equation (19), a six-dimensional stochastic motion differential equation is formed:
the above equation set is written as the following Ito random differential equation:
dX＝m(X,t)dt+Q(X,t)dW(t) (22)
wherein: m (X, t) and Q (X, t) are the offset coefficient moment and the diffusion coefficient matrix, respectively;
when the external stimulus is a white noise, the response of the dynamic system is a Markov process whose transition probability density is such that the initial condition is satisfied: p (X, t '|x', t ')=δ (X-X') and boundary conditions: p (- ≡t|X ', t') =p (+infinity, t|x ', solution of Fokker-Planck equation with t')=0; the Fokker-Planck equation is shown below:
at the same time, the transition probability density P (X, t|x ', t') satisfies the normalization condition:
∫∫P(X,t∣X′,t′)dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 ＝1 (24)
wherein t' is the initial time; x' is a matrix of roll angle and roll angular velocity at the initial moment; x is a matrix of roll angle and roll angular velocity at time t; x is x 1 The roll angle at time t; x is x 2 For time t 1 Is used for the roll angular velocity;
solving Fokker-Planck equation by path integration method to obtain transfer probability density of ship rolling motion in time domain, and further obtaining steady probability density function p (x) of ship parameter rolling motion 1 ) Obtaining the shipProbability of occurrence of parametric roll motion P in irregular waves 2 Wherein:
φ 0 is the angle of balance at which the parameter roll motion of the vessel occurs.
Further, in step 3-3, the evolution of the probability distribution of the ship rolling motion in the time domain can be obtained by solving the Fokker-Planck equation through a path integration method (PIS). The basic idea of the path integration method is to discretize in space and time respectively, and replace the integration with the path, that is, the global transition probability density is formed by connecting the short-time probability densities, so as to obtain the joint probability density of the state vector. The transition probability density function of the system over a small period of time τ (τ=t-t') is written as:
wherein: delta is a direx function; r is (r) i (x') obtaining the drift coefficient m by the Long Geku column method i (x′)；γ 2 Is the diffusion coefficient; x is x 1 Roll angle at time t; x is x 2 Roll angular velocity at time t, equation (26) is accurate to τ 2 After the transition probability density over a small period of time τ is found, the transition probability density function of the system is expressed as:
wherein t is k ＝t′+kτ；t＝t S ；X＝X (S) ；X′＝X (0) The method comprises the steps of carrying out a first treatment on the surface of the S is the initial time t' time t=t s Is a time period interval number of (a); r is R n(S-1) Represents the integration times, and the initial distribution omega (X (0) ) Then, the probability density function at time t is obtained as:
the steady probability density function of the ship rolling motion can be obtained by applying the path integration method, and the occurrence parameter rolling motion probability P of the ship in irregular waves can be obtained 2 。
Further, the step 4 specifically includes: according to probability P of irregular wave occurrence in target sea area 1 And the probability of roll P of the parameter of the ship in irregular waves 2 The probability P of the ship occurrence parameter rolling motion of the specific irregular wave in the test point area representing the route is calculated as follows:
P＝P 1 P 2 (29)。
the beneficial effects are that: compared with the prior art, the invention has the following remarkable advantages:
1. according to the nonlinear dynamics probability calculation analysis method for estimating the ship sailing under the excitation action of the irregular wave based on AIS data in the irregular wave, the probability of parameter rolling motion of the ship sailing in the irregular wave can be predicted.
2. According to the method, the probability density function of the ship parameter rolling motion is calculated by adopting a path integration method according to the modern nonlinear dynamics theory method and considering the action of irregular waves, so that the calculated amount is small, the solving precision is high, the efficiency is high, and the cost is low.
3. The nonlinear dynamics calculation and analysis method of the ship under the excitation action of the irregular wave is applicable to various ship types and has strong universality.
4. The method is used for analyzing theory and method of motion in ship wave, can be used for two aspects of ship design and ship navigation, can improve the ship design according to the method, reduces the parameter roll of the ship, provides reasonable advice for how to avoid the parameter roll in the actual navigation of the ship, and improves the navigation safety in the wind wave of the ship.
5. The invention discloses a ship parameter rolling nonlinear dynamics probability analysis method in irregular waves based on basic conditions of ship and ocean engineering hydrodynamics and ship navigation and nonlinear dynamics theory. According to the motion characteristics of the ship in the irregular wave, a physical mathematical model is established by combining a ship motion theory, a probability theory and a nonlinear dynamics theory, a corresponding Fokker-Planck equation is solved in a time domain based on a path integration method to obtain a transition probability density function, and a steady probability density function of the ship in the parameter rolling motion response is obtained according to the boundary condition and the initial condition of a differential equation, so that the parameter rolling motion law of the ship in the irregular wave is determined, and the possible rolling motion instability risk of the ship can be predicted.
Drawings
FIG. 1 hull motion coordinate system;
FIG. 2 is a diagram of the density of a ship in an ideal course area;
FIG. 3 is a schematic diagram illustrating an equidistant A, B, C, D, E, F, G seven test points on a route;
FIG. 4 is a graph of wave spread data for one of the points along the course of the route;
FIG. 5 is a schematic view of the stability of the vessel in water;
fig. 6 is a flow chart of a method for evaluating the probability of ship parameter roll motion based on AIS data in irregular waves.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings.
The invention combines the ship motion theory with the nonlinear dynamics theory, provides a new measurement index and a new concept of ship motion instability, and establishes a nonlinear dynamics analysis method and an analysis flow.
The ship motion coordinate system of the nonlinear dynamics calculation analysis method under the excitation action of irregular waves is shown in figure 1. In FIG. 1, O (x, y, z) is the geodetic fixed coordinate system, O '(x', y ', z') is the onboard coordinate system, O "(x", y ", z") is the wave coordinate system, ζ G Zeta is the position of the center of gravity of the ship under the geodetic coordinate system G Is the position of the center of gravity of the ship under the wave coordinate system.
The invention relates to a nonlinear dynamics calculation analysis method for estimating ship parameter rolling probability based on AIS data under the excitation action of irregular waves, which is described as follows:
step 1: AIS data in any time period of a sea area, such as ship AIS data in 2022, 1 month, 23 days to 3 months, 1 day, are selected, a probability density model of ship density is established, and a non-parameter nuclear density estimation method is used for solving a probability density function of the ship density.
Let x be 1 、x 2 …x n For a sample of the random variable x, the probability density function of the random variable x isThe probability density function of the ship density is as follows:
where x is a random variable,representing ship density x i Represents the ith ship density sample, h represents the bandwidth length, n is the number of samples, and K represents the kernel function.
A Gaussian kernel function is selected as the kernel function in the non-parametric kernel density estimation method. The Gaussian kernel function formula is as follows:
selection of bandwidth: the bandwidth is taken directly herein as h;
finally, the probability density function of the ship density obtained by the non-parameter nuclear density estimation method is as follows:
and calculating a probability density function of the ship density by adopting a non-parameter kernel density estimation method to obtain a ship density map. Wherein the selected course area ship density map is shown in fig. 2.
Step 2: from the ship density map of fig. 2, a ship critical density threshold is selected and a region of greater traffic flow is defined, from which a representative course region is identified and characterized, and seven test points are selected A, B, C, D, E, F, G at intermediate distances in the selected representative course region (e.g., selecting a north atlantic grape dental bohr map to a south ampton course in the united kingdom as shown in the black line of fig. 3).
According to the wave scattering data of the sea area near the A, B, C, D, E, F, G seven places in the covered area of the route, the probability distribution map of the occurrence of the irregular wave in the wave height period combined probability density data is obtained according to the ratio of the occurrence times of the irregular wave of different specific sense wave heights and characteristic period combinations to the total observation times of the wave scattering map, and the wave scattering data around one point is shown as figure 4, so that the probability of the occurrence of the irregular wave of the target sea area at a specific test point is obtained 1 Wherein:
P 1 number of occurrences of irregular waves of specific sense wave height and characteristic period combination/total number of observations of wave spread data.
Step 3: a nonlinear dynamics analysis method of ship occurrence parameter rolling motion in irregular waves is as follows:
the ship sails in the wave, and the ship parameter roll motion differential equation is established according to the force balance principle on the assumption that the motion of the ship in the wave is quasi-static, wherein the equation is shown as follows:
wherein I is φφ For moment of inertia, δI φφ For additional roll moment of inertia, phi is the vessel roll angle,for roll angular velocity +.>For roll angular acceleration +.>Represents the damping force to which the ship is subjected, B 1 And B 3 A linear damping coefficient and a cubic nonlinear damping coefficient, respectively,/->Representing roll recovery moment, F as shown in FIG. 6 wave Representing the wave forces to which the vessel is subjected.
Determined restoring moment of ship
Solving the recovery moment of the vessel is the focus of solving the above equations and studying the parametric roll motions of the vessel. There are two main approaches: the method is a direct solution method, wherein the primary stability in the wave is divided into a primary stability high in still water and a variable quantity in the wave, and the primary stability high in the wave of the ship is obtained through detailed deduction, and the change of the primary stability high relative to the elevation of the wave surface can be directly deduced; the second method is an indirect roll damping method, and according to the study of Dostal, the recovery moment of the ship can be expressed as a function of roll angle, wave surface rise, wave surface phase angle and time as follows:
wherein, delta represents the water displacement of the ship, eta represents the wave surface elevation, and psi represents the phase angle, and the range is [0-2 pi ]. After the restoring moment function of the numerical model ship, the function about the roll angle is fitted again.
In the invention, the influence of heave and pitch movements on the recovery moment arm is not considered, the recovery moment arm of the ship is obtained through numerical solution by adopting a slicing theory based on indirect roll damping solving, and then the roll moment arm function is obtained through simulation.
According to the Froude-Krylov assumption, froude-Krylov force is the main part of wave force, and wave force and wave moment can be obtained by integrating underwater pressure on underwater wet surface area.
F＝-∫∫ Ω pndΩ (6)
Wherein omega represents the underwater wet surface area of the hull,is a position vector, +.>The underwater pressure p, which represents the unit normal vector of the wet surface of the hull, consists of two parts, namely hydrostatic pressure and hydrodynamic pressure generated by wave surface elevation, and is shown in the following formula:
wherein ρ is the sea water density, g is the gravitational acceleration, z is the water depth,is the amplitude, k is the wavenumber, and c is the wavespeed.
Integrating equation (8) can obtain the recovery moment to which the ship is subjected:
wherein beta is the course angle, S (x') is the cross-sectional area of each immersion,and->For each submerged cross-section centroid B 0 The coordinates in the reference coordinate system, F (x'), are the pressure gradient coefficients of the respective cross sections, calculated as follows:
wherein d (x') is represented by draft of each cross section when the ship is in an upright state in still water,expressed as the water line width of each cross section when the ship is standing in still water.
In order to determine the restoring moment of the ship, it is necessary to obtain an approximate analytical expression of GZ (Φ, η, ψ), which is developed as a combination of a polynomial and fourier series with respect to the roll angle Φ as follows:
wherein N is φ Represents the highest power of phi in the numerical fitting process, which is 2N φ +1；N k Expanding the number of terms for the polynomial; n (N) S 、N C Expanding the number of terms for the Fourier series; the variables are determined to be valued by a least square method. Q, Q S 、Q C 、Q C* Coefficients representing the expansion term are determined from the results of the numerical fit, respectively.
GZ app The difference between the expansion of (Φ, η, ψ) and the calculation result GZ (Φ, η, ψ) of the numerical simulation needs to be minimum even if the following formula E is minimized:
after comprehensively considering the accuracy of numerical fitting and the complexity of the subsequent calculation, finally determining N φ ＝2、N k ＝0、N S ＝1、N C =1, and let Q S 0,1,1 ＝0、Q S 1,1,1 ＝0、Q S 2,1,1 ＝0，Q C 1,1,1 ＝0、， Then GZ app The expansion of (φ, η, ψ) can be expressed as:
GZ app (φ,η,ψ)＝Q 0,0,0 φ+Q 1,0,0 φ 3 +Q 2,0,0 φ 5 +Q C 0,1,1 ηcos(ψ)φ (13)
random process eta due to irregular wave decomposition c ＝ηcos(ψ)，GZ app (φ, η, ψ) can be further written as:
GZ app (φ,η,ψ)＝q 1 φ+q 2 φ 3 +q 3 φ 5 +q 4 η c φ (14)
wherein q i (i=1, 2,3, 4) represents the coefficient of the expansion term, and can be obtained by GZ app The result of the numerical fit of (phi, eta, phi) to GZ (phi, eta, phi) is determined.
Solving steady state probability density functions
In the present invention, in order to simplify the calculation, the colored nature of the waves, i.e., the waves are assumed to be white noise, is temporarily not considered. Based on a path integration method, a corresponding Fokker-Planck equation is solved in a time domain to obtain a transition probability density of ship rolling motion evolving along with time, and then a steady probability density function of the ship in response to parameter rolling motion in irregular waves is solved. The invention takes a two-dimensional calculation example, and the specific steps are as follows:
substituting the formula (14) into the formula (4) under the action of random wave external excitation under oblique waves to obtain a ship one-degree-of-freedom motion differential equation considering parameter rolling motion:
dividing both sides of the above equation by I φφ +δI φφ ：
In the method, in the process of the invention,
then formula (16) can be converted into the following expression:
order theξ 1 (t)＝p(t)，ξ 2 (t)＝f wave (t) then there is:
let x 1 ＝φ；The stochastic differential equation (18) is converted into the following set of state equations:
exciting the parameters and wave load ζ k (t) as a colored noise sequence, respectively passing through a linear filter and outputting, as shown in the formula (20):
wherein W is k (t) is a Wiener function. In conjunction with equation (19), a six-dimensional stochastic motion differential equation is formed:
the above equation set can be written as the following Ito random differential equation:
dX＝m(X,t)dt+Q(X,t)dW(t) (22)
wherein: m (X, t) and Q (X, t) are the moment of the offset coefficient and the diffusion coefficient matrix, respectively.
When the external stimulus is a white noise, the response of the dynamic system is a Markov process whose transition probability density is such that the initial condition is satisfied: p (X, t '|x', t ')=δ (X-X') and boundary conditions: p (- ≡t|X ', t') =p (+infinity, t|x ', solution of Fokker-Planck equation with t')=0. The Fokker-Planck equation is shown below:
at the same time, the transition probability density P (X, t|x ', t') satisfies the normalization condition:
∫∫P(X,t∣X′,t′)dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 ＝1 (24)
wherein t' is the initial time; x' is a matrix of roll angle and roll angular velocity at the initial moment; x is a matrix of roll angle and roll angular velocity at time t; x is x 1 The roll angle at time t; x is x 2 For time t 1 Is used for the roll angular velocity of the roll.
The invention solves Fokker-plane equation by Path Integration (PIS) to obtain the transition probability density of ship rolling motion in time domain, and further obtains the steady probability density function p (x) of ship parameter rolling motion 1 ) Obtaining the occurrence parameter rolling motion probability P of the ship in irregular waves 2 Wherein:
φ 0 is the angle of balance at which the parameter roll motion of the vessel occurs.
The basic idea of the path integration method is to discretize in space and time respectively, and replace the integration with the path, that is, the global transition probability density is formed by connecting the short-time probability densities, so as to obtain the joint probability density of the state vector. The transition probability density function of the system over a small period of time τ (τ=t-t') can be written as:
wherein: delta is a direx function; r is (r) i (x') obtaining the drift coefficient m by the Long Geku column method i (x′)；γ 2 Is the diffusion coefficient; x is x 1 Roll angle at time t; x is x 2 'is the roll angular velocity at time t'. Equation (26) is accurate to τ 2 After the transition probability density over a small period of time τ is found, the transition probability density function of the system can be expressed as:
wherein t is k ＝t′+kτ；t＝t S ；X＝X (S) ；X′＝X (0) The method comprises the steps of carrying out a first treatment on the surface of the S is the initial time t' time t=t s Is a time period interval number of (a); r is R n(S-1) Indicating the number of integration. In determining an initial distribution omega (X (0) ) Then, the probability density function at time t can be obtained as:
therefore, a steady probability density function of the ship parameter rolling motion can be obtained by applying a path integration method, and the occurrence parameter rolling motion probability P of the ship in irregular waves is obtained 2 。
Step 4: according to probability P of irregular wave occurrence in target sea area 1 And the probability of roll P of the parameter of the ship in irregular waves 2 The probability P of the ship occurrence parameter rolling motion of the specific irregular wave in the test point area representing the route is calculated as follows:
P＝P 1 P 2 (29)
and by analogy, calculating the probability of the occurrence parameter rolling motion of the ship under each random sea condition of the test point according to the wave scattering diagram in the coverage area of the ideal route.
Step 5: and (2) repeating the steps (2-4) to calculate the ship occurrence parameter rolling motion probability in different irregular waves of other test points, and solving the total occurrence probability of the ship occurrence parameter rolling motion on a representative route.
In summary, a flow chart of a method for evaluating the probability of ship parameter rolling motion based on AIS data in irregular waves is shown in fig. 6.
Claims (7)
1. The AIS data-based irregular wave ship parameter rolling motion probability evaluation method is characterized in that AIS data of a target sea area in a target time period is selected to solve a probability density function of ship density to obtain a ship density map, and based on the ship density map, the total occurrence probability of ship occurrence parameter rolling motion on a representative route in the ship density map is obtained by executing steps 2-5;
step 1: AIS data of a target sea area in a target time period is selected, a probability density function of ship density is established, and the probability density function of the ship density is solved based on a non-parameter kernel density estimation method to obtain a ship density map;
step 2: presetting a ship critical density threshold value according to a ship density map, defining a region with high traffic flow, identifying and characterizing a representative route region from the region,selecting a plurality of test points from the representative route area at medium distance, and calculating to obtain the irregular wave occurrence probability P of the target sea area at the specific test point according to the wave scattering data of the sea area near each test point 1 ；
Step 3: establishing a ship motion coordinate system and a ship parameter rolling motion differential equation in irregular waves, determining a ship recovery moment, and solving a probability density function of a parameter rolling motion response in the irregular waves by a path integration method to obtain a ship occurrence parameter rolling motion probability P in the irregular waves 2 ；
Step 4: according to the probability of occurrence of irregular waves of a target sea area at a specific test point, the probability is P 1 And the probability of occurrence of parametric roll motions P of the vessel in irregular waves 2 Calculating to obtain the probability P of the ship under the specific test point to generate parameter rolling motion in irregular waves;
step 5: and (2) repeating the steps (2) to (4), and calculating the probability of the parameter rolling motion of the ship under each test point in the irregular wave, thereby obtaining the total occurrence probability of the parameter rolling motion of the ship on the representative route.
2. The method for estimating the rolling motion probability of the ship parameter in the irregular wave based on the AIS data according to claim 1, wherein the step 1 specifically comprises the following steps:
step 1-1: let x be 1 、x 2 …x n For a sample of the random variable x, the probability density function of the random variable x isThe probability density function of the ship density is as follows:
where x is a random variable,representing ship density x i Representing an ith ship density sample, h representing bandwidth length, n representing sample number, and K representing a kernel function;
step 1-2: the Gaussian kernel function is selected as the kernel function in the non-parametric kernel density estimation method, and the Gaussian kernel function has the following formula:
step 1-3: selecting h as bandwidth;
step 1-4: the probability density function of the ship density obtained by adopting the non-parameter nuclear density estimation method is as follows:
and obtaining a ship density map.
3. The method for estimating the rolling motion probability of the ship parameter in the irregular wave based on the AIS data according to claim 1, wherein in the step 2, 6-8 test points are selected from the representative route area at a medium distance.
4. The AIS data-based ship parameter rolling motion probability evaluation method in irregular waves according to claim 1, wherein in step 2, according to wave dispersion data of sea areas near each test point, the probability distribution map of irregular wave occurrence in the wave height period joint probability density data is obtained by using the ratio of the number of irregular wave occurrences of different specific sense wave height and characteristic period combinations to the total observation number of wave dispersion data, and the probability of irregular wave occurrence of a target sea area at a specific test point is obtained 1 Wherein:
P 1 irregular wave number/wave dispersion of occurrence of combination of specific sense wave height and characteristic periodThe total number of observations was counted.
5. The method for estimating the rolling motion probability of the ship parameter in the irregular wave based on the AIS data according to claim 1, wherein the step 3 specifically comprises the following steps:
step 3-1: assuming that the motion of the vessel in the wave is quasi-static, according to the force balance principle, a differential equation of the vessel parameter roll motion is established, as shown in the following formula:
wherein I is φφ For moment of inertia, δI φφ For additional roll moment of inertia, phi is the vessel roll angle,for roll angular velocity +.>For roll angular acceleration +.>Represents the damping force to which the ship is subjected, B 1 And B 3 A linear damping coefficient and a cubic nonlinear damping coefficient, respectively,/->Representing roll recovery moment, F wave Representing the wave forces to which the vessel is subjected;
step 3-2: the determined recovery moment of the ship:
the recovery moment of the vessel is expressed as a function of transverse inclination, wave surface rise, wave surface phase angle and time as follows:
wherein delta represents the water displacement of the ship, eta represents the wave surface rise, phi represents the phase angle, the range is [0-2 pi ], and the function of the roll angle is fitted after the recovery moment function of the numerical model ship;
according to the Froude-Krylov assumption, wave force and wave moment are obtained by integrating underwater pressure on underwater wet surface area:
F＝-∫∫ Ω pndΩ (6)
wherein omega represents the underwater wet surface area of the hull,is a position vector, +.>The underwater pressure p, which represents the unit normal vector of the wet surface of the hull, consists of two parts, namely hydrostatic pressure and hydrodynamic pressure generated by wave surface elevation, and is shown in the following formula:
wherein ρ is the sea water density, g is the gravitational acceleration, z is the water depth,is the amplitude, k is the wave number, c is the wave velocity;
integrating the formula (8) to obtain the recovery moment to which the ship is subjected:
wherein beta is the course angle, S (x') is the cross-sectional area of each immersion,and->For each submerged cross-section centroid B 0 The coordinates in the reference coordinate system, F (x'), are the pressure gradient coefficients of the respective cross sections, calculated as follows:
wherein d (x') is represented by draft of each cross section when the ship is in an upright state in still water,the water line width of each cross section when the ship stands upright in still water;
in order to determine the restoring moment of the ship, it is necessary to obtain an approximate analytical expression of GZ (Φ, η, ψ), which is developed as a combination of a polynomial and fourier series with respect to the roll angle Φ as follows:
wherein N is φ Represents the highest power of phi in the numerical fitting process, which is 2N φ +1；N k Expanding the number of terms for the polynomial; n (N) S 、N C Expanding the number of terms for the Fourier series; the variable is determined to be valued by a least square method; q, Q S 、Q C 、Coefficients representing the expansion term are respectively determined by the results of numerical fitting;
GZ app the difference between the expansion of (Φ, η, ψ) and the calculation result GZ (Φ, η, ψ) of the numerical simulation needs to be minimum even if the following formula E is minimized:
after comprehensively considering the accuracy of numerical fitting and the complexity of the subsequent calculation, finally determining N φ ＝2、N k ＝0、N S ＝1、N C =1, and letThen GZ app The expansion of (φ, η, ψ) is expressed as:
random process eta due to irregular wave decomposition c ＝ηcos(ψ)，GZ app (φ, η, ψ) is written as:
GZ app (φ,η,ψ)＝q 1 φ+q 2 φ 3 +q 3 φ 5 +q 4 η c φ (14)
wherein q i (i=1, 2,3, 4) represents the coefficient of the expansion term by GZ app Results of numerical fits of (phi, eta, phi) to GZ (phi, eta, phi) are determined;
step 3-3: solving a steady-state probability density function:
substituting the formula (14) into the formula (4) by considering the effect of random wave external excitation under oblique waves to obtain a ship freedom degree motion differential equation considering parameter rolling motion:
dividing both sides of the above equation by I φφ +δI φφ ：
In the method, in the process of the invention,
then formula (16) translates into the following expression:
order theξ 1 (t)＝p(t)，ξ 2 (t)＝f wave (t) then there is:
let x 1 ＝φ；The stochastic differential equation (18) is converted into the following set of state equations:
exciting the parameters and wave load ζ k (t) as a colored noise sequence, respectively passing through a linear filter and outputting, as shown in the formula (20):
wherein W is k (t) is a Wiener function; in conjunction with equation (19), a six-dimensional stochastic motion differential equation is formed:
the above equation set is written as the following Ito random differential equation:
dX＝m(X,t)dt+Q(X,t)dW(t) (22)
wherein: m (X, t) and Q (X, t) are the offset coefficient moment and the diffusion coefficient matrix, respectively;
when the external stimulus is a white noise, the response of the dynamic system is a Markov process whose transition probability density is such that the initial condition is satisfied: p (X, t '|x', t ')=δ (X-X') and boundary conditions:
p (- ≡t|X ', t') =p (+infinity, t|x ', solution of Fokker-Planck equation with t')=0; the Fokker-Planck equation is shown below:
at the same time, the transition probability density P (X, t|x ', t') satisfies the normalization condition:
∫∫P(X,t∣X′,t′)dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 ＝1 (24)
wherein t' is the initial time; x' is a matrix of roll angle and roll angular velocity at the initial moment; x is a matrix of roll angle and roll angular velocity at time t; x is x 1 The roll angle at time t; x is x 2 For time t 1 Is used for the roll angular velocity;
solving Fokker-Planck equation by path integration method to obtain transfer probability density of ship rolling motion in time domain, and further obtaining steady probability density function p (x) of ship parameter rolling motion 1 ) Obtaining the occurrence parameter transverse of the ship in irregular wavesProbability of shaking motion P 2 Wherein:
φ 0 is the angle of balance at which the parameter roll motion of the vessel occurs.
6. The method for estimating the rolling motion probability of the ship parameter in the irregular wave based on AIS data according to claim 5, wherein in the step 3-3, the path integration method is that the path is discretized in space and time respectively, and the path is used for replacing integration, namely, global transition probability density is formed by connecting short-time probability density, so as to obtain the joint probability density of the state vector, and the transition probability density function of the system in a small time period tau (tau=t-t') is written as follows:
wherein: delta is a direx function; r is (r) i (x') obtaining the drift coefficient m by the Long Geku column method i (x′)；γ 2 Is the diffusion coefficient; x is x 1 Roll angle at time t; x is x 2 Roll angular velocity at time t, equation (26) is accurate to τ 2 After the transition probability density over a small period of time τ is found, the transition probability density function of the system is expressed as:
wherein t is k ＝t′+kτ；t＝t S ；X＝X (S) ；X′＝X (0) The method comprises the steps of carrying out a first treatment on the surface of the S is the initial time t' time t=t s Is a time period interval number of (a); r is R n (S-1) Represents the integration times, and the initial distribution omega (X (0) ) Then, the probability density function at time t is obtained as:
the steady probability density function of the ship parameter rolling motion can be obtained by applying the path integration method, and the occurrence parameter rolling motion probability P of the ship in irregular waves is obtained 2 。
7. The method for estimating the rolling motion probability of the ship parameter in the irregular wave based on the AIS data according to claim 1, wherein the step 4 is specifically: according to probability P of irregular wave occurrence in target sea area 1 And the probability of roll P of the parameter of the ship in irregular waves 2 The probability P of the ship occurrence parameter rolling motion of the specific irregular wave in the test point area representing the route is calculated as follows:
P＝P 1 P 2 (29)。
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