CN116663139A - AIS data-based irregular wave ship parameter rolling motion probability evaluation method - Google Patents

Abstract

The invention discloses a method for evaluating the probability of ship parameter roll motion in irregular waves based on AIS data, comprising: establishing a probability density model of ship density, solving the probability density function to obtain a ship density map; according to the ship density map, from representative Select test points at equidistant distances in the route area, and calculate the occurrence probability of irregular waves in the target sea area at specific test points as P ₁ ; establish a differential equation of parametric roll motion, and calculate the probability of parametric roll motion of the ship in irregular waves P ₂ ; According to P ₁ and P ₂ , the probability P of the parametric rolling motion of the ship in irregular waves is calculated at a specific test point, and then the total occurrence probability of the parametric rolling motion of the ship on the representative route is obtained. The invention can improve the design of the ship, reduce the parametric roll of the ship, provide reasonable suggestions for how to avoid the parametric roll during the actual navigation of the ship, and improve the safety of the ship's navigation in wind and waves.

Description

Probabilistic Evaluation Method of Ship Parameters Rolling Motion in Irregular Waves Based on AIS Data

technical field

The invention belongs to the field of ship nonlinear motion analysis, in particular to a method for evaluating the probability of ship parameter roll motion in irregular waves based on AIS data.

Background technique

Parametric roll is one of the five failure modes in the second-generation intact stability of ships, and it is the most studied and comprehensive stability failure mode at home and abroad. Parametric roll is a relatively unique rolling phenomenon. Its principle is that when a ship sails in longitudinal or oblique waves, the shape of the ship's wave surface and the area of the water plane in the waves change with time, causing the hull The recovery characteristics of the ship change periodically, the wet surface area of the ship is different when the wave crest is in the midship and the trough is in the midship, and the area of the water plane changes greatly. In essence, it is a strong nonlinear phenomenon caused by the time-historical changes of ship motion parameters, not directly produced by external excitation, and it is extremely harmful to ship navigation and transportation at sea.

Early studies have shown that the parametric rolling motion of a ship needs to meet certain trigger conditions: the ship is sailing in head-on or following waves; the frequency of the ship encountering waves is approximately twice the natural frequency of the ship's rolling motion; the wavelength of the ship's encountering waves is approximately the same as the length of the ship equal; the wave height encountered by the ship is greater than the critical value; the roll damping of the ship itself is relatively small. There are mainly three research methods for the current research on parametric rolling motion in irregular waves: nonlinear dynamics method, direct numerical simulation method, and model test method. Among them, non-linear dynamics has the characteristics of small amount of calculation, high efficiency and low cost in the study of parametric rolling motion, which plays an irreplaceable role in the research and can be used as the main research method for studying the mechanism of parametric rolling.

At present, most of the research on the safety of ship navigation is based on a certain point on the existing route, but from the perspective of ship operation, the wave angle and sea state distribution on the ship's navigation track have a significant impact on the nonlinear motion of the ship. Therefore, it is necessary to fully evaluate and analyze the stability and seakeeping of ships under complex sea conditions, how to fully consider the influence of route and random sea state distribution on the rolling motion of ship parameters.

Contents of the invention

Purpose of the invention: The purpose of the present invention is to provide a method for evaluating the probability of rolling motion of ship parameters in irregular waves based on AIS data. Establish a ship density calculation model based on AIS data and standard ships, use the non-parametric kernel density method to estimate the probability density function of ship density, find a representative route in a certain sea area through the ship density map obtained by the probability density function, and study the ship navigation on the representative route In order to solve the limitation of the numerical method in the nonlinear parametric roll motion of the ship in the irregular wave, the probabilistic analysis method of the nonlinear dynamics of the ship parametric roll in the irregular wave is proposed. The nonlinear dynamics analysis method is used to analyze the probability of the parametric rolling motion of the ship in regular waves, which provides an analysis method and a safe navigation basis for the safe navigation of the ship.

Technical solution: The method for evaluating the probability of rolling motion of ship parameters in irregular waves based on AIS data of the present invention includes the following steps:

Step 1: Select the AIS data of the target sea area within the target time period, establish the probability density model of the ship density, and solve the probability density function of the ship density based on the non-parametric kernel density estimation method to obtain the ship density map;

Step 2: According to the ship density map, preset the critical density threshold of the ship and define the area with large traffic flow, identify and characterize the representative route area from this area, and then select several test points at equal distances from the representative route area, according to each The wave dispersal data of the sea area near the test point, the calculated irregular wave occurrence probability of the target sea area at the specific test point is _P1 ;

Step 3: Establish the ship motion coordinate system and differential equation of ship parametric roll motion in irregular waves, determine the restoring moment of the ship, solve the probability density function of parametric roll motion response in irregular waves by path integral method, and obtain the ship in The occurrence parameter rolling motion probability P ₂ in irregular waves;

Step 4: According to the occurrence probability of irregular waves in the target sea area at a specific test point is P ₁ and the probability of the ship's occurrence parameter rolling motion in irregular waves P ₂ , calculate the occurrence parameters of the ship in irregular waves at a specific test point The probability of rolling motion P;

Step 5: Repeat step 2-step 4 to calculate the probability of parametric rolling motion of the ship in irregular waves at each test point, and obtain the total occurrence probability of parametric rolling motion of the ship on the representative route.

Further, Step 1 specifically includes the following steps:

Step 1-1: Suppose x ₁ , x ₂ ... x _n are samples of random variable x, and the probability density function of random variable x is Then the probability density function of ship density is shown in the following formula:

where x is a random variable, represents the ship density, x _i represents the i-th ship density sample, h represents the bandwidth length, n represents the number of samples, and K represents the kernel function;

Step 1-2: Select the Gaussian kernel function as the kernel function in the non-parametric kernel density estimation method. The Gaussian kernel function formula is as follows:

Step 1-3: Select h as the bandwidth;

Steps 1-4: The probability density function of the ship density obtained by using the non-parametric kernel density estimation method is as follows:

Get the ship density map.

Further, in step 2, 6-8 test points are selected equidistantly from the representative route area.

Further, in step 2, according to the wave dispersal data in the sea area near each test point, the wave height period is obtained by the ratio of the number of occurrences of irregular waves with different combinations of specific meaningful wave heights and characteristic periods to the total number of observations of the wave dispersive data The probability distribution map of the occurrence of irregular waves in the joint probability density data, and then the probability of occurrence of irregular waves in the target sea area at a specific test point is P ₁ , where:

P ₁ = the number of occurrences of irregular waves for a specific combination of significant wave height and characteristic period/the total number of observations of wave dispersion data.

Further, step 3 specifically includes the following steps:

Step 3-1: Assuming that the motion of the ship in the waves is quasi-static, according to the principle of force balance, the differential equation of the motion of the ship’s parameter roll is established, as shown in the following formula:

Among them, I _φφ is the moment of inertia of roll, δI _φφ is the moment of inertia of additional roll, φ is the roll angle of the ship, is the rolling angular velocity, /> is the rolling angular acceleration, /> Represents the damping force on the ship, B ₁ and B ₃ are the linear damping coefficient and the cubic nonlinear damping coefficient respectively, /> Represents the roll restoring moment, F _wave represents the wave force on the ship;

Step 3-2: Determine the restoring moment of the ship:

Solving the restoring moment of the ship is the focus of solving the above equations and studying the parametric rolling motion of the ship. There are two main methods: one is the direct solution method, which divides the initial stability height in waves into two parts: the initial stability height in still water and the variation in waves. The initial stability in waves can be directly derived to obtain the change of the initial stability with respect to the rise of the wave surface; the second method is the indirect roll damping method. According to Dostal’s research, the restoring moment of the ship can be expressed as function of inclination, wavefront elevation, wavefront phase angle, and time, expressed as follows:

Among them, Δ represents the displacement of the ship, η represents the rise of the wave surface, ψ represents the phase angle, and the range is [0-2π]. After the restoration moment function of the numerical model ship, it is then fitted into a function about the roll angle;

In the present invention, the influence of heave and pitch motion on the righting arm is not considered, and based on the indirect calculation of roll damping, the slice theory is used to obtain the ship’s righting arm through numerical solution, and then the rolling force is obtained by simulation arm function.

According to the Froude-Krylov assumption, the wave force and wave moment are obtained by integrating the underwater pressure with the underwater wet surface area:

F＝ _-∫∫Ω pndΩ (6)

Among them, Ω represents the underwater wet surface area of the hull, r is the position vector, n represents the unit normal vector of the wet surface of the hull, and the underwater pressure p includes two parts, which are composed of hydrostatic pressure and hydrodynamic pressure generated by the rise of the wave surface, as follows Shown:

Among them, ρ is the seawater density, g is the acceleration of gravity, z is the water depth, η is the wave amplitude, k is the wave number, and c is the wave velocity;

Integrating equation (8), the restoring moment on the ship is obtained:

Among them, β is the heading angle, S(x′) is the cross-sectional area of each submerged water, with /> is the coordinate of the centroid B ₀ of each submerged cross-section in the reference coordinate system, and F(x′) is the pressure gradient coefficient of each cross-section, calculated according to the following formula:

Wherein, d(x′) represents the draft of each cross-section when the ship is in an upright state in still water, and B(x′) represents the waterline width of each cross-section when the ship is in an upright state in still water;

In order to determine the restoring moment of the ship, it is necessary to obtain the approximate analytical expression of GZ(φ,η,Ψ), and expand GZ(φ,η,Ψ) into a combination of polynomial and Fourier series about the roll angle φ As shown in the following formula:

Among them, N _φ represents the highest power of φ in the numerical fitting process, which is 2N _φ +1; N _k is the number of polynomial expansion items; N _S and N _C are the number of Fourier series expansion items; the above variables are determined by The value is determined by the least square method; Q, Q ^S , Q ^C , Represents the coefficients of the expansion terms, respectively determined by the results of numerical fitting;

The difference between the expansion of GZ _app (φ,η,ψ) and the calculation result GZ(φ,η,ψ) of the numerical simulation needs to be the smallest, even if the following formula E reaches the minimum:

After comprehensively considering the accuracy of numerical fitting and the complexity of subsequent calculations, finally determine N _φ = 2, N _k = 0, N _S = 1, N _C = 1, and set 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, the expansion of GZ _app (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 )

Due to the random process _ηc = ηcos(ψ) obtained by the decomposition of irregular waves, 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)

Among them, q _i (i=1, 2, 3, 4) represents the coefficient of the expansion term, which is determined by the numerical fitting results of GZ _app (φ, η, ψ) and GZ (φ, η, ψ);

Step 3-3: Solve the steady-state probability density function:

In the present invention, in order to simplify the calculation, the colored property of the wave is not considered, that is, the wave is assumed to be white noise. Based on the path integral method, the corresponding Fokker-Planck equation is solved in the time domain to obtain the transition probability density of the ship's rolling motion over time, and then solve the steady-state probability density of the parametric rolling motion response of the ship in irregular waves function. The present invention takes a two-dimensional calculation example, and the specific steps are as follows:

Considering the effect of random wave external excitation under oblique waves, substituting Equation (14) into Equation (4), the differential equation of motion of ship degrees of freedom considering parametric roll motion is obtained:

Divide both sides of the above equation by I _φφ +δI _φφ :

In the formula,

Then formula (16) is transformed into the following expression:

make ξ ₁ (t)=p(t), ξ ₂ (t)=f _wave (t), then:

Let x ₁ = φ; Transform the stochastic differential equation (18) into the following state equations:

The parameter excitation and wave load ξ _k (t) are regarded as colored noise sequences, which are output after passing through the linear filter respectively, as shown in formula (20):

Among them, W _k (t) is a Wiener function; it is combined with formula (19) to form a six-dimensional stochastic motion differential equation:

Write the above equations as the following Ito stochastic differential equation:

dX=m(X,t)dt+Q(X,t)dW(t) (22)

Among them: m(X,t) and Q(X,t) are the offset coefficient moment and diffusion coefficient matrix respectively;

When the external excitation is a white noise, the response of the dynamic system is a Markov process, and the transition probability density of the process satisfies the initial condition: P(X,t′∣X′,t′)=δ(X-X′) and the boundary Condition: P(-∞, t|X′, t′)=P(+∞, t|X′, t′)=0 the solution of the Fokker-Planck equation; the Fokker-Planck equation is shown in the following formula:

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)

Among them, t' is the initial moment; X' is the roll angle and roll angular velocity matrix at the initial moment; X is the roll angle and roll angular velocity matrix at time t; x ₁ is the roll angle at time t; x ₂ is Rolling angular velocity at time t ₁ ;

Solve the Fokker-Planck equation by the path integral method to obtain the transition probability density of the ship's rolling motion in the time domain, and then obtain the steady-state probability density function p(x ₁ ) of the ship's parametric rolling motion, and obtain the The occurrence parameter rolling motion probability P ₂ , where:

φ ₀ is the criterion angle for parametric rolling motion of the ship.

Further, in step 3-3, the present invention solves the Fokker-Planck equation through the path integral method (PIS) to obtain the evolution of the probability distribution of the ship's rolling motion in the time domain. The basic idea of the path integral method is to discretize the space and time respectively, and replace the integral with the path, that is, the global transition probability density is formed by connecting the short-term probability density, and the joint probability density of the state vector is obtained. The transition probability density function of the system in a small time period τ(τ=t-t′) is written as:

In the formula: δ is the Direck function; r _i (x′) is the drift coefficient m _i (x′) obtained by the Longo-Kutta method; γ ² is the diffusion coefficient; x ₁ ′ is the roll at time t′ angle; x ₂ ′ is the roll angular velocity at time t′, and the formula (26) is accurate to τ ^2. After obtaining the transition probability density in a short period of time τ, the transition probability density function of the system is expressed as:

In the formula, t _k =t′+kτ; t=t _S ; X=X ^(S) ; X′=X ⁽⁰⁾ ; S is the number of time intervals at the initial moment t′ at the moment t=t _s ; R ^{n (S-1)} represents the number of integrations. After setting the initial distribution ω(X ⁽⁰⁾ ), the probability density function at time t is obtained as:

The steady-state probability density function of the ship's rolling motion can be obtained by applying the path integral method, and the probability P ₂ of the ship's parametric rolling motion in irregular waves can be obtained.

Further, step 4 is specifically as follows: according to the occurrence probability _P1 of the irregular wave in the target sea area and the parameter roll probability _P2 of the ship in the irregular wave, the ship parameter roll of the specific irregular wave in the test point area representing the route is calculated. The probability P of shaking motion is shown in the following formula:

P=P ₁ P ₂ (29).

Beneficial effect: compared with the prior art, the present invention has the following significant advantages:

1. A method for calculating and analyzing nonlinear dynamics probability based on AIS data in irregular waves of the present invention to evaluate ships navigating under the excitation of irregular waves, which can predict the parametric roll motion of ships navigating in irregular waves probability.

2. The present invention is based on the modern nonlinear dynamics theory method, considers the effect of irregular waves, and adopts the path integral method to calculate the probability density function of the ship parameter rolling motion, the calculation amount is small, the solution accuracy is high, the efficiency is high, and the cost is low .

3. The nonlinear dynamics calculation and analysis method of a ship under the action of irregular wave excitation of the present invention is applicable to various ship types and has strong versatility.

4. The present invention is used for the analysis theory and method of ship motion in waves, can be used in two aspects of ship design and ship navigation, can improve ship design according to the method, reduce the parameter roll of ship, and provide a practical solution for ship navigation. How to avoid parametric roll in the paper puts forward reasonable suggestions to improve the safety of ships navigating in wind and waves.

5. Based on the hydrodynamics of ships and ocean engineering, the basic conditions of ship navigation and the theory of nonlinear dynamics, the present invention has invented a method for analyzing the nonlinear dynamics of ship parameter rolling in irregular waves. According to the motion characteristics of parametric rolling of ships in irregular waves, a physical and mathematical model is established by combining ship motion theory, probability theory and nonlinear dynamics theory, and based on the path integral method, it is solved in the time domain According to the corresponding Fokker-Planck equation, the transition probability density function is obtained. According to the boundary conditions and initial conditions of the differential equation, the steady-state probability density function of the parametric roll motion response of the ship is obtained, and then the parametric roll of the ship in irregular waves is determined. The law of motion, from which the risk of possible instability of the ship's rolling motion can be predicted.

Description of drawings

Figure 1 Hull motion coordinate system;

Figure 2. Ship density map in the ideal route area;

Figure 3 shows a schematic diagram of the seven test points A, B, C, D, E, F, and G at equidistant distances on the route;

Figure 4 is the wave distribution data map of one point on the route;

Figure 5 is a schematic diagram of the stability of a ship in water;

Fig. 6 is a flow chart of the method for evaluating the probability of parametric rolling motion of a ship based on AIS data in irregular waves of the present invention.

Detailed ways

The technical solution of the present invention will be further described below in conjunction with the accompanying drawings.

The invention combines the theory of ship motion with the theory of nonlinear dynamics, proposes a new measurement index and new concept of ship motion instability, and establishes an analysis method and flow of nonlinear dynamics.

The nonlinear dynamics calculation and analysis method of the ship under the action of irregular wave excitation, the coordinate system of the ship’s motion is shown in Figure 1. In Fig. 1, O(x,y,z) is the geodetic fixed coordinate system, O′(x′,y′,z′) is the ship’s coordinate system, O″(x″,y″,z″) is the wave coordinate system, ξ _G is the position of the center of gravity of the ship in the geodetic coordinate system, and ζ _G is the position of the center of gravity of the ship in the wave coordinate system.

The present invention is a non-linear dynamics calculation and analysis method based on AIS data to evaluate the probability of ship parameter roll under irregular wave excitation. The description of the navigation method for specifically analyzing ship parameter roll motion is as follows:

Step 1: Select the AIS data in any period of time in a sea area, such as the AIS data of ships from January 23 to March 1, 2022, establish a probability density model of ship density, and use the non-parametric kernel density estimation method to estimate the ship density. The probability density function of the density is solved.

Suppose x ₁ , x ₂ ... x _n are samples of random variable x, the probability density function of random variable x is Then the probability density function of ship density is shown in the following formula:

where x is a random variable, Indicates the ship density, _xi represents the i-th ship density sample, h represents the bandwidth length, n represents the number of samples, and K represents the kernel function.

The 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: In this paper, the bandwidth is directly taken as h;

Finally, the probability density function of ship density obtained by non-parametric kernel density estimation method is as follows:

The probability density function of ship density is calculated by non-parametric kernel density estimation method, and the ship density map is obtained. The ship density map of the selected route area is shown in Figure 2.

Step 2: According to the ship density map in Figure 2, select the critical density threshold of ships and define the area with large traffic flow, identify and characterize the representative route area, and then select A, B, Seven test points C, D, E, F, and G (for example, choose the route from Porto, Portugal in the North Atlantic Ocean to Southampton, England, as shown in the black line in Figure 3).

According to the wave distribution data in the sea area near the seven locations A, B, C, D, E, F, and G in the area covered by the route, the occurrence times and wave distribution of irregular waves with different combinations of specific meaningful wave heights and characteristic periods The ratio of the total number of observations in the graph can be used to obtain the probability distribution graph of the occurrence of irregular waves in the joint probability density data of the wave height cycle, as shown in Figure 4 for the wave distribution data around one of the points, and then obtain the non-regular waves of the target sea area at a specific test point The probability of regular wave appearance is P ₁ , where:

P ₁ = the number of occurrences of irregular waves for a specific combination of significant wave height and characteristic period/the total number of observations of wave dispersion data.

Step 3: A nonlinear dynamic analysis method for parametric rolling motion of a ship in irregular waves is as follows:

The ship sails in the waves, assuming that the motion of the ship in the waves is quasi-static, according to the principle of force balance, the differential equation of the ship parameter roll motion is established, as shown in the following formula:

Among them, I _φφ is the moment of inertia of roll, δI _φφ is the moment of inertia of additional roll, φ is the roll angle of the ship, is the rolling angular velocity, /> is the rolling angular acceleration, /> Represents the damping force on the ship, B ₁ and B ₃ are the linear damping coefficient and the cubic nonlinear damping coefficient respectively, /> Represents the roll restoring moment, as shown in Fig. 6, F _wave represents the wave force on the ship.

The determined restoring moment of the ship

Solving the restoring moment of the ship is the focus of solving the above equations and studying the parametric rolling motion of the ship. There are two main methods: one is the direct solution method, which divides the initial stability height in waves into two parts: the initial stability height in still water and the variation in waves. The initial stability in waves can be directly derived to obtain the change of the initial stability with respect to the rise of the wave surface; the second method is the indirect roll damping method. According to Dostal’s research, the restoring moment of the ship can be expressed as function of inclination, wavefront elevation, wavefront phase angle, and time, expressed as follows:

Among them, Δ represents the displacement of the ship, η represents the rise of the wave surface, and ψ represents the phase angle, and the range is [0-2π]. After the restoring moment function of the numerical model ship, it is fitted into a function about the roll angle.

In the present invention, the influence of heave and pitch motion on the righting arm is not considered, and based on the indirect calculation of roll damping, the slice theory is used to obtain the ship’s righting arm through numerical solution, and then the rolling force is obtained by simulation arm function.

According to the Froude-Krylov hypothesis, the Froude-Krylov force is the main part of the wave force, and the wave force and wave moment can be obtained by integrating the underwater pressure with the underwater wet surface area:

F＝ _-∫∫Ω pndΩ (6)

Among them, Ω represents the underwater wet surface area of the hull, is the position vector, /> Represents the unit normal vector of the wet surface of the hull, and the underwater pressure p includes two parts, which are composed of hydrostatic pressure and hydrodynamic pressure generated by the rise of the wave surface, as shown in the following formula:

Among them, ρ is the seawater density, g is the acceleration due to gravity, z is the water depth, is the wave amplitude, k is the wave number, and c is the wave speed.

Integrating equation (8), the restoring moment on the ship can be obtained:

Among them, β is the heading angle, S(x′) is the cross-sectional area of each submerged water, with /> is the coordinate of the centroid B ₀ of each submerged cross-section in the reference coordinate system, and F(x′) is the pressure gradient coefficient of each cross-section, calculated according to the following formula:

where, d(x′) represents the draft of each transverse section when the ship is in an upright state in still water, Expressed as the waterline width of each transverse section when the ship is upright in still water.

In order to determine the restoring moment of the ship, it is necessary to obtain the approximate analytical expression of GZ(φ,η,Ψ), and expand GZ(φ,η,Ψ) into a combination of polynomial and Fourier series about the roll angle φ As shown in the following formula:

Among them, N _φ represents the highest power of φ in the numerical fitting process, which is 2N _φ +1; N _k is the number of polynomial expansion items; N _S and N _C are the number of Fourier series expansion items; the above variables are determined by The value is determined by the method of least squares. Q, Q ^S , Q ^C , and Q ^C* represent the coefficients of the expansion items, which are respectively determined by the results of numerical fitting.

The difference between the expansion of GZ _app (φ,η,ψ) and the calculation result GZ(φ,η,ψ) of the numerical simulation needs to be the smallest, even if the following formula E reaches the minimum:

After comprehensively considering the accuracy of numerical fitting and the complexity of subsequent calculations, finally determine N _φ = 2, N _k = 0, N _S = 1, N _C = 1, and set 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, the expansion of GZ _app (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 )

Due to the random process _ηc = ηcos(ψ) obtained by the decomposition of irregular waves, 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)

Among them, q _i (i=1, 2, 3, 4) represents the coefficient of the expansion term, which can be determined by the numerical fitting results of GZ _app (φ, η, ψ) and GZ (φ, η, ψ).

Solve the steady-state probability density function

In the present invention, in order to simplify the calculation, the colored property of the wave is not considered, that is, the wave is assumed to be white noise. Based on the path integral method, the corresponding Fokker-Planck equation is solved in the time domain to obtain the transition probability density of the ship's rolling motion over time, and then solve the steady-state probability density of the parametric rolling motion response of the ship in irregular waves function. The present invention takes a two-dimensional calculation example, and the specific steps are as follows:

Considering the effect of random wave external excitation under oblique waves, substituting Equation (14) into Equation (4), the ship-degree-of-freedom motion differential equation considering parametric roll motion is obtained:

Divide both sides of the above equation by I _φφ +δI _φφ :

In the formula,

Then formula (16) can be transformed into the following expression:

make ξ ₁ (t)=p(t), ξ ₂ (t)=f _wave (t), then:

Let x ₁ = φ; Transform the stochastic differential equation (18) into the following state equations:

The parameter excitation and wave load ξ _k (t) are regarded as colored noise sequences, which are output after passing through the linear filter respectively, as shown in formula (20):

Among them, W _k (t) is a Wiener function. Combined with formula (19) to form a six-dimensional random motion differential equation:

The above equations can be written as the following Ito stochastic differential equation:

dX=m(X,t)dt+Q(X,t)dW(t) (22)

Among them: m(X,t) and Q(X,t) are the offset coefficient moment and diffusion coefficient matrix respectively.

When the external excitation is a white noise, the response of the dynamic system is a Markov process, and the transition probability density of the process satisfies the initial condition: P(X,t′∣X′,t′)=δ(X-X′) and the boundary Condition: Solution of the Fokker-Planck equation for P(-∞, t|X', t')=P(+∞, t|X', t')=0. The Fokker-Planck equation is shown in the following formula:

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)

Among them, t' is the initial moment; X' is the roll angle and roll angular velocity matrix at the initial moment; X is the roll angle and roll angular velocity matrix at time t; x ₁ is the roll angle at time t; x ₂ is Rolling angular velocity at time _t1 .

The present invention solves the Fokker-Planck equation through the path integral method (PIS) to obtain the transition probability density of the ship's rolling motion in the time domain, and then obtains the steady-state probability density function p(x ₁ ) of the ship's parameter rolling motion, and obtains Probability of parametric rolling motion P ₂ of the ship in irregular waves, where:

φ ₀ is the criterion angle for parametric rolling motion of the ship.

The basic idea of the path integral method is to discretize the space and time respectively, and replace the integral with the path, that is, the global transition probability density is formed by connecting the short-term probability density, and the joint probability density of the state vector is obtained. The transition probability density function of the system in a small time period τ(τ=t-t′) can be written as:

In the formula: δ is the Direck function; r _i (x′) is the drift coefficient m _i (x′) obtained by the Longo-Kutta method; γ ² is the diffusion coefficient; x ₁ ′ is the roll at time t′ angle; x ₂ ′ is the roll angular velocity at time t′. Equation (26) is accurate to τ ² After obtaining the transition probability density in a short period of time τ, the transition probability density function of the system can be expressed as:

In the formula, t _k =t′+kτ; t=t _S ; X=X ^(S) ; X′=X ⁽⁰⁾ ; S is the number of time intervals at the initial moment t′ at the moment t=t _s ; R ^{n (S-1)} represents the number of times of integration. After determining the initial distribution ω(X ⁽⁰⁾ ), the probability density function at time t can be obtained as:

Therefore, the steady-state probability density function of the parametric rolling motion of the ship can be obtained by applying the path integral method, and the probability P ₂ of the parametric rolling motion of the ship in irregular waves can be obtained.

Step 4: According to the probability P ₁ of irregular waves in the target sea area and the probability P ₂ of the parametric roll of the ship in the irregular waves, the probability P of the parametric roll motion of the ship in the specific irregular wave of the test point area representing the route is calculated As shown in the following formula:

P=P ₁ P ₂ (29)

By analogy, according to the wave scatter diagram in the area covered by the ideal route, the probability of parametric rolling motion of the ship under each random sea condition at the test point is calculated.

Step 5: Repeat steps 2-4 to calculate the probability of parametric rolling motion of the ship in different irregular waves at other test points, and solve the total probability of parametric rolling motion of the ship on the representative route.

To sum up, the flow chart of the method for evaluating the probability of parametric rolling motion of a ship based on AIS data in irregular waves according to the present invention is shown in FIG. 6 .

Claims (7)

1. A method for assessing the probability of ship parameter roll motion in irregular waves based on AIS data is characterized in that, the probability density function of the AIS data of the target sea area is selected to solve the ship density in the target time period, and the ship density map is obtained, based on For the ship density map, by performing steps 2 to 5, the total occurrence probability of parameter rolling motion of the ship on the representative route in the ship density map is obtained; Step 1: Select the AIS data of the target sea area within the target time period, establish the probability density function of the ship density, and solve the probability density function of the ship density based on the non-parametric kernel density estimation method to obtain the ship density map; Step 2: According to the ship density map, preset the critical density threshold of ships and define the area with high traffic flow, identify and characterize the representative route area from this area, and then select several test points at equal distances from the representative route area, according to each The wave distribution data in the sea area near the test point is calculated to obtain the occurrence probability of irregular waves in the target sea area at a specific test point as P ₁ ; Step 3: Establish the ship motion coordinate system and differential equation of ship parametric roll motion in irregular waves, determine the restoring moment of the ship, solve the probability density function of parametric roll motion response in irregular waves by path integral method, and obtain the ship in The occurrence parameter rolling motion probability P ₂ in irregular waves; Step 4: According to the occurrence probability of irregular waves in the target sea area at a specific test point is P ₁ and the probability of the ship's occurrence parameter rolling motion in irregular waves P ₂ , calculate the occurrence parameters of the ship in irregular waves at a specific test point The probability of rolling motion P; Step 5: Repeat step 2-step 4 to calculate the probability of parametric rolling motion of the ship in irregular waves at each test point, and thus obtain the total occurrence probability of parametric rolling motion of the ship on the representative route. 2. a kind of ship parameter rolling motion probability assessment method in the irregular wave based on AIS data according to claim 1, is characterized in that, step 1 specifically comprises the steps: Step 1-1: Suppose x ₁ , x ₂ ... x _n are samples of random variable x, and the probability density function of random variable x is Then the probability density function of ship density is shown in the following formula: where x is a random variable, represents the ship density, x _i represents the i-th ship density sample, h represents the bandwidth length, n represents the number of samples, and K represents the kernel function; Step 1-2: Select the Gaussian kernel function as the kernel function in the non-parametric kernel density estimation method. The Gaussian kernel function formula is as follows: Step 1-3: Select h as the bandwidth; Steps 1-4: The probability density function of the ship density obtained by using the non-parametric kernel density estimation method is as follows: Get the ship density map. 3. a kind of ship parameter roll motion probability assessment method in the irregular wave based on AIS data according to claim 1, it is characterized in that, in step 2, select 6-8 test points equidistantly from representative route area . 4. a kind of ship parameter roll motion probability evaluation method in the irregular wave based on AIS data according to claim 1, it is characterized in that, in step 2, according to the wave dispersion data of each test point nearby sea area, with different The ratio of the number of occurrences of irregular waves in a specific combination of significant wave height and characteristic period to the total number of observations in wave dispersion data is used to obtain the probability distribution map of irregular waves in the combined probability density data of wave height and period, and then to obtain the target sea area in a specific test. The occurrence probability of the irregular wave of the point is P ₁ , where: P ₁ = the number of occurrences of irregular waves for a specific combination of significant wave height and characteristic period/the total number of observations of wave dispersion data. 5. a kind of ship parameter roll motion probability assessment method in the irregular wave based on AIS data according to claim 1, it is characterized in that, step 3 specifically comprises the following steps: Step 3-1: Assuming that the motion of the ship in the waves is quasi-static, according to the principle of force balance, the differential equation of the motion of the ship’s parameter roll is established, as shown in the following formula: Among them, I _φφ is the moment of inertia of roll, δI _φφ is the moment of inertia of additional roll, φ is the roll angle of the ship, is the rolling angular velocity, /> is the rolling angular acceleration, /> Represents the damping force on the ship, B ₁ and B ₃ are the linear damping coefficient and the cubic nonlinear damping coefficient respectively, /> Represents the roll restoring moment, F _wave represents the wave force on the ship; Step 3-2: Determine the restoring moment of the ship: The restoring moment of the ship is expressed as a function of heel angle, wave surface rise, wave surface phase angle and time, expressed as follows: Among them, Δ represents the displacement of the ship, η represents the rise of the wave surface, ψ represents the phase angle, and the range is [0-2π]. After the restoration moment function of the numerical model ship, it is then fitted into a function about the roll angle; According to the Froude-Krylov assumption, the wave force and wave moment are obtained by integrating the underwater pressure with the underwater wet surface area: F＝ _-∫∫Ω pndΩ (6) Among them, Ω represents the underwater wet surface area of the hull, is the position vector, /> Represents the unit normal vector of the wet surface of the hull, and the underwater pressure p includes two parts, which are composed of hydrostatic pressure and hydrodynamic pressure generated by the rise of the wave surface, as shown in the following formula: Among them, ρ is the seawater density, g is the acceleration due to gravity, z is the water depth, is the amplitude, k is the wave number, and c is the wave speed; Integrating equation (8), the restoring moment on the ship is obtained: Among them, β is the heading angle, S(x′) is the cross-sectional area of each submerged water, with /> is the coordinate of the centroid B ₀ of each submerged cross-section in the reference coordinate system, and F(x′) is the pressure gradient coefficient of each cross-section, calculated according to the following formula: where, d(x′) represents the draft of each transverse section when the ship is in an upright state in still water, Expressed as the waterline width of each transverse section when the ship is upright in still water; In order to determine the restoring moment of the ship, it is necessary to obtain the approximate analytical expression of GZ(φ,η,Ψ), and expand GZ(φ,η,Ψ) into a combination of polynomial and Fourier series about the roll angle φ As shown in the following formula: Among them, N _φ represents the highest power of φ in the numerical fitting process, which is 2N _φ +1; N _k is the number of polynomial expansion items; N _S and N _C are the number of Fourier series expansion items; the above variables are determined by The value is determined by the least square method; Q, Q ^S , Q ^C , Represents the coefficients of the expansion terms, respectively determined by the results of numerical fitting; The difference between the expansion of GZ _app (φ,η,ψ) and the calculation result GZ(φ,η,ψ) of the numerical simulation needs to be the smallest, even if the following formula E reaches the minimum: After comprehensively considering the accuracy of numerical fitting and the complexity of subsequent calculations, finally determine N _φ = 2, N _k = 0, N _S = 1, N _C = 1, and let Then, the expansion of GZ _app (φ,η,ψ) is expressed as: Due to the random process _ηc = ηcos(ψ) obtained by the decomposition of irregular waves, 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) Among them, q _i (i=1, 2, 3, 4) represents the coefficient of the expansion term, which is determined by the numerical fitting results of GZ _app (φ, η, ψ) and GZ (φ, η, ψ); Step 3-3: Solve the steady-state probability density function: Considering the effect of random wave external excitation under oblique waves, substituting Equation (14) into Equation (4), the differential equation of motion of ship degrees of freedom considering parametric roll motion is obtained: Divide both sides of the above equation by I _φφ +δI _φφ : In the formula, Then formula (16) is transformed into the following expression: make ξ ₁ (t)=p(t), ξ ₂ (t)=f _wave (t), then: Let x ₁ = φ; Transform the stochastic differential equation (18) into the following state equations: The parameter excitation and wave load ξ _k (t) are regarded as colored noise sequences, which are output after passing through the linear filter respectively, as shown in formula (20): Among them, W _k (t) is a Wiener function; it is combined with formula (19) to form a six-dimensional stochastic motion differential equation: Write the above equations as the following Ito stochastic differential equation: dX=m(X,t)dt+Q(X,t)dW(t) (22) Among them: m(X,t) and Q(X,t) are the offset coefficient moment and diffusion coefficient matrix respectively; When the external excitation is a white noise, the response of the dynamic system is a Markov process, and the transition probability density of the process satisfies the initial condition: P(X,t′∣X′,t′)=δ(X-X′) and the boundary condition: The solution of the Fokker-Planck equation of P(-∞,t∣X′,t′)=P(+∞,t∣X′,t′)=0; the Fokker-Planck equation is shown in the following formula: 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) Among them, t' is the initial moment; X' is the roll angle and roll angular velocity matrix at the initial moment; X is the roll angle and roll angular velocity matrix at time t; x ₁ is the roll angle at time t; x ₂ is Rolling angular velocity at time t ₁ ; Solve the Fokker-Planck equation by the path integral method to obtain the transition probability density of the ship's rolling motion in the time domain, and then obtain the steady-state probability density function p(x ₁ ) of the ship's parametric rolling motion, and obtain the The occurrence parameter rolling motion probability P ₂ , where: φ ₀ is the criterion angle for parametric rolling motion of the ship. 6. The method for assessing the probability of ship parameter rolling motion in irregular waves based on AIS data according to claim 5, characterized in that, in step 3-3, the path integration method is space and time respectively Discretization, replacing the integral with a path, that is, the global transition probability density is formed by connecting the short-term probability densities, and the joint probability density of the state vector is obtained. The transition probability density function of the system in a small time period τ(τ=t-t′) is written as : In the formula: δ is the Direck function; r _i (x′) is the drift coefficient m _i (x′) obtained by the Longo-Kutta method; γ ² is the diffusion coefficient; x ₁ ′ is the roll at time t′ angle; x ₂ ′ is the roll angular velocity at time t′, and the formula (26) is accurate to τ ^2. After obtaining the transition probability density in a short period of time τ, the transition probability density function of the system is expressed as: In the formula, t _k =t′+kτ; t=t _S ; X=X ^(S) ; X′=X ⁽⁰⁾ ; S is the number of time intervals at the initial moment t′ at the moment t=t _s ; R ^{n (S-1)} represents the number of integrations. After setting the initial distribution ω(X ⁽⁰⁾ ), the probability density function at time t is obtained as: The steady-state probability density function of the parametric rolling motion of the ship can be obtained by applying the path integral method, and the probability P ₂ of the parametric rolling motion of the ship in irregular waves can be obtained. 7. a kind of ship parameter roll motion probability evaluation method in the irregular wave based on AIS data according to claim 1, it is characterized in that, step 4 is specifically: according to the probability _P that irregular wave occurs in the target sea area and ship The parametric rolling probability _P2 in the irregular wave is calculated to obtain the probability P of the parametric rolling motion of the ship in the specific irregular wave of the test point area representing the route, as shown in the following formula: P = P ₁ P ₂ (29).