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
本发明公开了一种基于AIS数据的非规则波中船舶参数横摇运动概率评估方法,包括:建立船舶密度的概率密度模型,求解概率密度函数得到船舶密度图;据船舶密度图,从代表性航线区域中等距离选择测试点,计算目标海域在特定测试点的非规则波出现概率为P1;建立参数横摇运动微分方程,计算船舶在非规则波浪中的发生参数横摇运动概率P2;根据P1、P2计算特定测试点下船舶在非规则波浪中发生参数横摇运动的概率P,进而得到代表性航线上船舶发生参数横摇运动总的发生概率。本发明可改进船舶设计,减小船舶的参数横摇,为船舶实际航行中如何规避参数横摇提出合理建议,提高船舶风浪中航行的安全性。
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
技术领域technical field
本发明属于船舶非线性运动分析领域,尤其涉及一种基于AIS数据的非规则波中船舶参数横摇运动概率评估方法。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.
背景技术Background technique
参数横摇是船舶第二代完整稳性中的五种失效模式之一,是目前国内外开展研究最多、最全面的稳性失效模式。参数横摇作为一种相对独特的横摇现象,其原理是船舶在纵浪或斜浪中航行时,船舶在波浪中的波面形状、水线面的面积随时间变化随之发生改变,引发船体的复原特性发生周期性的改变,波峰在船舯以及波谷在船舯时船舶湿表面积不同,同时水线面面积发生大幅改变。其本质上是由于船舶运动参数时历变化所导致的一种强非线性现象,并不是直接由外界激励作用产生的,其对于船舶海上航行与运输危害极大。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.
早期研究表明,船舶发生参数横摇运动需要满足一定的触发条件:船舶在迎浪或随浪中航行;船舶遭遇波浪频率近似2倍船舶的横摇运动的固有频率;船舶遭遇波波长与船长近似相等;船舶遭遇波波高大于临界值;船舶自身横摇阻尼比较小。针对非规则波中参数横摇运动现阶段研究主要有三种研究方法:非线性动力学法、直接数值模拟方法、模型试验方法。其中非性动力学在研究参数横摇运动中,具有计算量小,效率高,成本低的特点,对研究有着不可替代的作用,可以作为研究参数横摇机理的主要研究手段。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.
目前,关于船舶航行安全性的研究大多数基于现有航线上的某一个点,但从船舶运营角度来看,船舶航行轨迹上浪向角和海况分布等对船舶的非线性运动影响显著。因此,如何全面考虑航线、随机海况分布对船舶参数横摇运动的影响,对船舶在复杂海况下的稳性和耐波性有充分评估和分析的必要。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.
发明内容Contents of the invention
发明目的:本发明的目的在于提供一种基于AIS数据的非规则波中船舶参数横摇运动概率评估方法。基于AIS数据与标准船建立船舶密度计算模型,采用非参数核密度方法估计船舶密度的概率密度函数,通过概率密度函数得到的船舶密度图找到某一海域代表性航线,研究代表性航线上船舶航行中发生参数横摇运动的概率并为了解决数值方法在非规则波中船舶非线性参数横摇运动的局限性,而提出的非规则波中船舶参数横摇非线性动力学概率分析方法,对非规则波中船舶参数横摇运动采用非线性动力学分析方法分析发生参数横摇运动的概率,为船舶的安全航行提供分析方法和安全航行依据。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.
技术方案:本发明的基于AIS数据的非规则波中船舶参数横摇运动概率评估方法,包括如下步骤: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:
步骤1:选取目标海域在目标时间段内的AIS数据,建立船舶密度的概率密度模型,并基于非参数核密度估计法对船舶密度的概率密度函数进行求解,得到船舶密度图;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;
步骤2:根据船舶密度图,预设船舶临界密度阈值并定义交通流量较大的区域,从该区域中识别并表征代表性航线区域,再从代表性航线区域中等距离选择若干测试点,根据每个测试点附近海域的波浪散布数据,计算得到目标海域在特定测试点的非规则波出现概率为P1;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 ;
步骤3:在非规则波浪中建立船舶运动坐标系和船舶参数横摇运动微分方程,确定船舶的恢复力矩,通过路径积分法求解非规则波浪中参数横摇运动响应的概率密度函数,得到船舶在非规则波浪中的发生参数横摇运动概率P2;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;
步骤4:根据目标海域在特定测试点的非规则波出现概率为P1和船舶在非规则波浪中的发生参数横摇运动概率P2,计算得到特定测试点下船舶在非规则波浪中发生参数横摇运动的概率P;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;
步骤5:重复步骤2-步骤4,计算每一个测试点下船舶在非规则波浪中发生参数横摇运动的概率,得到代表性航线上船舶发生参数横摇运动总的发生概率。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.
进一步的,步骤1具体包括如下步骤:Further, Step 1 specifically includes the following steps:
步骤1-1:假设x1、x2…xn为随机变量x的样本,随机变量x的概率密度函数为则船舶密度的概率密度函数如下式所示: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:
其中,x是随机变量,表示船舶密度,xi表示第i个船舶密度样本,h表示带宽长度,n为样本数量,K代表核函数;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;
步骤1-2:选用Gaussian核函数作为非参数核密度估计方法中的核函数,Gaussian核函数公式如下: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:
步骤1-3:选择取为h作为带宽;Step 1-3: Select h as the bandwidth;
步骤1-4:采用非参数核密度估计法得到的船舶密度的概率密度函数如下所示:Steps 1-4: The probability density function of the ship density obtained by using the non-parametric kernel density estimation method is as follows:
得到船舶密度图。Get the ship density map.
进一步的,步骤2中,从代表性航线区域中等距离选择6-8个测试点。Further, in step 2, 6-8 test points are selected equidistantly from the representative route area.
进一步的,步骤2中,根据每个测试点附近海域的波浪散布数据,以不同特定有义波高和特征周期组合的不规则波出现的次数与波浪散布数据总的观测次数的比值,得到波高周期联合概率密度数据中不规则波出现的概率分布图,进而得到目标海域在特定测试点的非规则波出现概率为P1,其中: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:
P1=特定有义波高和特征周期组合的不规则波出现的次数/波浪散布数据总的观测次数。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.
进一步的,步骤3具体包括如下步骤:Further, step 3 specifically includes the following steps:
步骤3-1:假设船舶在波浪中的运动是准静态的,根据力的平衡原理,建立船舶参数横摇运动微分方程,如下式所示: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:
其中,Iφφ为横摇转动惯量,δIφφ为附加横摇转动惯量,φ为船舶横摇角,为横摇角速度,/>为横摇角加速度,/>代表船舶所受到的阻尼力,B1和B3分别为线性阻尼系数和三次非线性阻尼系数,/>代表横摇恢复力矩,Fwave代表船舶所受到的波浪力;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;
步骤3-2:确定的船舶的恢复力矩:Step 3-2: Determine the restoring moment of the ship:
求解船舶的恢复力矩是求解上述方程和研究船舶的参数横摇运动的重点。最主要的方法有两种:一种方法为直接求解法,将波浪中的初稳性高分为静水中的初稳性高和波浪中的变化量两部分,经过详细的推导,得到船舶在波浪中的初稳性高并可以直接推导求出初稳性高关于波面升高的变化;第二种方法为间接求横摇阻尼法,依据Dostal的研究,可以将船舶的恢复力矩表示成横倾角、波面升高、波面相位角和时间的函数,表示如下: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:
其中,Δ代表船舶的排水量,η代表波面升高,ψ代表相位角,范围为[0-2π],数值模型船舶的复原力矩函数后,再拟合成关于横摇角的函数;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;
在本发明中,不考虑升沉和纵摇运动对复原力臂的影响,并以间接求横摇阻尼为基础,采用切片理论,经过数值求解得到船舶的复原力臂,然后模拟得到横摇力臂函数。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.
根据Froude-Krylov假设,通过水下压力对水下湿表面积积分获得波浪力和波浪力矩: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=-∫∫ΩpndΩ (6)F= -∫∫Ω pndΩ (6)
其中,Ω代表船体的水下湿表面积,r是位置向量,n代表船体湿表面的单位法向量,水下压力p包括两部分,由静水压力和波面升高产生的动水压力组成,如下式所示: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:
其中,ρ是海水密度,g是重力加速度,z是水深,η是波幅,k是波数,c是波速;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;
对式(8)积分,得到船舶所受到的恢复力矩:Integrating equation (8), the restoring moment on the ship is obtained:
其中,β为航向角,S(x′)为各浸水横剖面面积,与/>为各浸水横剖面形心B0在参考坐标系下的坐标,F(x′)为各个横剖面的压力梯度系数,按如下式计算: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:
其中,d(x′)表示为船舶在静水中为直立状态时各个横剖面的吃水,B(x′)表示为船舶在静水中为直立状态时各个横剖面的水线宽;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;
为了确定船舶的复原力矩,需要得到GZ(φ,η,Ψ)的近似解析表达式,将GZ(φ,η,Ψ)展开为关于横摇角φ的多项式和傅里叶级数的组合形式如下式所示: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:
其中,Nφ代表数值拟合过程中φ所取的最高幂次,为2Nφ+1;Nk为多项式展开项数;NS、NC为傅里叶级数展开项数;上述变量由最小二乘法确定取值;Q、QS、QC、代表展开项的系数,分别由数值拟合的结果确定;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;
GZapp(φ,η,ψ)的展开式与数值模拟的计算结果GZ(φ,η,ψ)的差值需要最小,即使下式E达到最小: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:
在综合考虑数值拟合的精度和后面计算的复杂程度后,最后确定Nφ=2、Nk=0、NS=1、NC=1,并令QS 0,1,1=0、QS 1,1,1=0、QS 2,1,1=0,QC 1,1,1=0、, 那么,GZapp(φ,η,ψ)的展开式表示为: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:
GZapp(φ,η,ψ)=Q0,0,0φ+Q1,0,0φ3+Q2,0,0φ5+QC 0,1,1ηcos(ψ)φ (13)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 )
由于非规则波浪分解得到的随机过程ηc=ηcos(ψ),GZapp(φ,η,ψ)写为:Due to the random process ηc = ηcos(ψ) obtained by the decomposition of irregular waves, GZ app (φ,η,ψ) is written as:
GZapp(φ,η,ψ)=q1φ+q2φ3+q3φ5+q4ηcφ (14)GZ app (φ,η,ψ)=q 1 φ+q 2 φ 3 +q 3 φ 5 +q 4 η c φ (14)
其中,qi(i=1,2,3,4)代表展开项的系数,通过GZapp(φ,η,ψ)与GZ(φ,η,ψ)的数值拟合的结果确定;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);
步骤3-3:求解稳态概率密度函数:Step 3-3: Solve the steady-state probability density function:
在本发明中,为了简化计算,暂不考虑波浪的有色性质,即假定波浪为白噪声。以路径积分法为基础,在时间域内求解相应的Fokker-Planck方程,得到随时间演变的船舶横摇运动的转移概率密度,进而求解船舶在不规则波浪中参数横摇运动响应的稳态概率密度函数。本发明取二维的算例,具体步骤如下所示: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:
考虑斜浪下随机波浪外激励的作用,将式(14)代入式(4),得到考虑参数横摇运动的船舶自由度运动微分方程: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:
上述方程两边同时除以Iφφ+δIφφ:Divide both sides of the above equation by I φφ +δI φφ :
式中, In the formula,
则式(16)转化成如下表达式:Then formula (16) is transformed into the following expression:
令ξ1(t)=p(t),ξ2(t)=fwave(t),则有:make ξ 1 (t)=p(t), ξ 2 (t)=f wave (t), then:
令x1=φ;将随机微分方程式(18)转化为如下状态方程组:Let x 1 = φ; Transform the stochastic differential equation (18) into the following state equations:
将参数激励和波浪载荷ξk(t)作为有色噪声序列,分别通过线性滤波器后输出,如式(20)所示: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):
其中,Wk(t)为Wiener函数;与式(19)联立形成六维随机运动微分方程:Among them, W k (t) is a Wiener function; it is combined with formula (19) to form a six-dimensional stochastic motion differential equation:
将上述方程组写成如下Ito随机微分方程:Write the above equations as the following Ito stochastic differential equation:
dX=m(X,t)dt+Q(X,t)dW(t) (22)dX=m(X,t)dt+Q(X,t)dW(t) (22)
其中:m(X,t)和Q(X,t)分别为偏移系数矩和扩散系数矩阵;Among them: m(X,t) and Q(X,t) are the offset coefficient moment and diffusion coefficient matrix respectively;
当外部激励为一白噪声时,动态系统的响应是一个Markov过程,该过程的转移概率密度是满足初始条件:P(X,t′∣X′,t′)=δ(X-X′)与边界条件:P(-∞,t|X′,t′)=P(+∞,t|X′,t′)=0的Fokker-Planck方程的解;Fokker-Planck方程如下式所示: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:
同时转移概率密度P(X,t∣X′,t′)满足归一化条件:At the same time, the transition probability density P(X,t∣X′,t′) satisfies the normalization condition:
∫∫P(X,t∣X′,t′)dx1dx2dx3dx4dx5dx6=1 (24)∫∫P(X,t∣X′,t′)dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 =1 (24)
其中,t′为初始时刻;X′为初始时刻的横摇角与横摇角速度矩阵;X为t时刻的横摇角与横摇角速度矩阵;x1为时刻t的横摇角;x2为时刻t1的横摇角速度;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 ;
通过路径积分法求解Fokker-Planck方程得到船舶横摇运动的在时间域内的转移概率密度,进而求得船舶参数横摇运动的稳态概率密度函数p(x1),得到船舶在非规则波浪中的发生参数横摇运动概率P2,其中: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:
φ0为船舶发生参数横摇运动的衡准角。φ 0 is the criterion angle for parametric rolling motion of the ship.
进一步的,步骤3-3中,本发明通过路径积分法(PIS)求解Fokker-Planck方程可以得到船舶横摇运动的概率分布在时间域内的演变。路径积分法的基本思想是在空间和时间上分别离散化,以路径代替积分,即通过连接短时概率密度形成全局转移概率密度,得到状态矢量的联合概率密度。系统在一个小的时间段τ(τ=t-t′)上的转移概率密度函数写成: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:
式中:δ为狄雷克函数;ri(x′)为通过龙哥库塔法求出漂移系数mi(x′);γ2为扩散系数;x1′为t′时刻的横摇角;x2′为t′时刻的横摇角速度,式(26)精确至τ2在求得了一小时间段τ上的转移概率密度后,系统的转移概率密度函数表示为: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:
式中,tk=t′+kτ;t=tS;X=X(S);X′=X(0);S为初始时刻t'时刻t=ts的时间段间隔数;Rn(S-1)表示积分次数,在定初始分布ω(X(0))之后,得到时刻t的概率密度函数为: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:
应用路径积分法可求船舶横摇运动的稳态概率密度函数,并得到船舶在非规则波浪中的发生参数横摇运动概率P2。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.
进一步的,步骤4具体为:根据目标海域非规则波出现的概率P1和船舶在非规则波中的参数横摇概率P2计算得到代表航线的测试点区域特定非规则波的船舶发生参数横摇运动的概率P如下式所示: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=P1P2 (29)。P=P 1 P 2 (29).
有益效果:与现有技术相比,本发明具有如下显著优点:Beneficial effect: compared with the prior art, the present invention has the following significant advantages:
1.本发明的一种基于非规则波中基于AIS数据评估船舶在非规则波激励作用下航行的非线性动力学概率计算分析方法,可以预报船舶在非规则波中航行发生参数横摇运动的概率。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.本发明依据现代非线性动力学理论方法,考虑非规则波浪的作用,采用路径积分法计算了船舶参数横摇运动的概率密度函数,计算量较小,求解精度高,效率高,成本低。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.本发明的一种船舶在非规则波激励作用下的非线性动力学计算分析方法,适用于各种船型,通用性强。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.本发明用于属于舰船波浪中运动的分析理论与方法,可以用于船舶设计和船舶航行两个方面,可以根据本方法改进船舶设计,减小船舶的参数横摇,为船舶实际航行中如何规避参数横摇提出合理建议,提高船舶风浪中航行的安全性。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.本发明基于船舶与海洋工程水动力学、船舶航行的基本条件以及非线性动力学理论,发明了一种非规则波中船舶参数横摇非线性动力学概率分析方法。根据船舶在非规则波中发生参数横摇的运动特点,通过船舶运动理论、概率学理论和非线性动力学理论相结合的方式,建立物理数学模型,以路径积分法为基础,在时间域内求解相应的Fokker-Planck方程,得到转移概率密度函数,根据微分方程的边界条件和初始条件,得到船舶的参数横摇运动响应的稳态概率密度函数,进而确定船舶在不规则波浪中的参数横摇运动规律,由此可以预报船舶可能的横摇运动失稳风险。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.
附图说明Description of drawings
图1船体运动坐标系;Figure 1 Hull motion coordinate system;
图2理想航线区域船舶密度图;Figure 2. Ship density map in the ideal route area;
图3示意航线上等距离取A、B、C、D、E、F、G七个测试点示意图;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;
图4航线上其中一点的波浪散布数据图;Figure 4 is the wave distribution data map of one point on the route;
图5船舶在水中的稳性示意图;Figure 5 is a schematic diagram of the stability of a ship in water;
图6本发明非规则波中基于AIS数据评估船舶发生参数横摇运动概率的方法思路流程图。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.
具体实施方式Detailed ways
下面结合附图对本发明的技术方案作进一步说明。The technical solution of the present invention will be further described below in conjunction with the accompanying drawings.
本发明是将船舶运动理论与非线性动力学理论相结合,提出新的船舶运动失稳的衡量指标和新概念,建立了非线性动力学的分析方法,分析流程。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.
船舶在非规则波激励作用下的非线性动力学计算分析方法,船体运动坐标系如图1所示。图1中,O(x,y,z)为大地固定坐标系,O′(x′,y′,z′)为随船坐标系,O″(x″,y″,z″)为波浪坐标系,ξG为大地坐标系下船舶重心的位置,ζG为波浪坐标系下船舶重心的位置。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.
本发明是一种非规则波激励作用下基于AIS数据评估船舶参数横摇概率的非线性动力学计算分析方法,具体分析船舶参数横摇运动航行方法的描述如下所示: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:
步骤1:选取某一海域任意一时间段内的AIS数据,例如2022年1月23日至3月1日内的船舶AIS数据,建立船舶密度的概率密度模型,使用非参数核密度估计法对船舶密度的概率密度函数进行求解。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.
假设x1、x2…xn为随机变量x的样本,随机变量x的概率密度函数为则船舶密度的概率密度函数如下式所示: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:
其中,x是随机变量,表示船舶密度,xi表示第i个船舶密度样本,h表示带宽长度,n为样本数量,K代表核函数。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.
选用Gaussian核函数作为非参数核密度估计方法中的核函数。Gaussian核函数公式如下: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:
带宽的选择:本文中带宽直接取为h;Selection of bandwidth: In this paper, the bandwidth is directly taken as h;
最后用非参数核密度估计法得到的船舶密度的概率密度函数如下所示:Finally, the probability density function of ship density obtained by non-parametric kernel density estimation method is as follows:
采用非参数核密度估计方法计算船舶密度的概率密度函数,得到船舶密度图。其中所选航线区域船舶密度图如图2所示。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.
步骤2:根据图2的船舶密度图,选择船舶临界密度阈值并定义交通流量较大的区域,从中识别并表征代表性航线区域,再在选择的代表性航线区域中等距离选择选择A、B、C、D、E、F、G七个测试点(例如选择北大西洋葡萄牙波尔图至英国南安普顿航线如图3黑色线路所示)。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).
根据该航线所覆盖区域中A、B、C、D、E、F、G七个地点附近海域的波浪散布数据,以不同特定有义波高和特征周期组合的不规则波出现的次数与波浪散布图总的观测次数的比值,得到该波高周期联合概率密度数据中不规则波出现的概率分布图,如其中一点周围的波浪散布数据如图4所示,进而得到目标海域在特定测试点的非规则波出现概率为P1,其中: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:
P1=特定有义波高和特征周期组合的不规则波出现的次数/波浪散布数据总的观测次数。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.
步骤3:一种非规则波中船舶发生参数横摇运动的非线性动力学分析方法如下所示:Step 3: A nonlinear dynamic analysis method for parametric rolling motion of a ship in irregular waves is as follows:
船舶在波浪中航行,假设船舶在波浪中的运动是准静态的,根据力的平衡原理,建立船舶参数横摇运动微分方程,如下式所示: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:
其中,Iφφ为横摇转动惯量,δIφφ为附加横摇转动惯量,φ为船舶横摇角,为横摇角速度,/>为横摇角加速度,/>代表船舶所受到的阻尼力,B1和B3分别为线性阻尼系数和三次非线性阻尼系数,/>代表横摇恢复力矩,如图6所示,Fwave代表船舶所受到的波浪力。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.
确定的船舶的恢复力矩The determined restoring moment of the ship
求解船舶的恢复力矩是求解上述方程和研究船舶的参数横摇运动的重点。最主要的方法有两种:一种方法为直接求解法,将波浪中的初稳性高分为静水中的初稳性高和波浪中的变化量两部分,经过详细的推导,得到船舶在波浪中的初稳性高并可以直接推导求出初稳性高关于波面升高的变化;第二种方法为间接求横摇阻尼法,依据Dostal的研究,可以将船舶的恢复力矩表示成横倾角、波面升高、波面相位角和时间的函数,表示如下: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:
其中,Δ代表船舶的排水量,η代表波面升高,ψ代表相位角,范围为[0-2π]。数值模型船舶的复原力矩函数后,再拟合成关于横摇角的函数。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.
在本发明中,不考虑升沉和纵摇运动对复原力臂的影响,并以间接求横摇阻尼为基础,采用切片理论,经过数值求解得到船舶的复原力臂,然后模拟得到横摇力臂函数。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.
根据Froude-Krylov假设,Froude-Krylov力是波浪作用力的主要部分,可以通过水下压力对水下湿表面积积分获得波浪力和波浪力矩: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=-∫∫ΩpndΩ (6)F= -∫∫Ω pndΩ (6)
其中,Ω代表船体的水下湿表面积,是位置向量,/>代表船体湿表面的单位法向量,水下压力p包括两部分,由静水压力和波面升高产生的动水压力组成,如下式所示: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:
其中,ρ是海水密度,g是重力加速度,z是水深,是波幅,k是波数,c是波速。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.
对式(8)积分,可以得到船舶所受到的恢复力矩:Integrating equation (8), the restoring moment on the ship can be obtained:
其中,β为航向角,S(x′)为各浸水横剖面面积,与/>为各浸水横剖面形心B0在参考坐标系下的坐标,F(x′)为各个横剖面的压力梯度系数,按如下式计算: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:
其中,d(x′)表示为船舶在静水中为直立状态时各个横剖面的吃水,表示为船舶在静水中为直立状态时各个横剖面的水线宽。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.
为了确定船舶的复原力矩,需要得到GZ(φ,η,Ψ)的近似解析表达式,将GZ(φ,η,Ψ)展开为关于横摇角φ的多项式和傅里叶级数的组合形式如下式所示: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:
其中,Nφ代表数值拟合过程中φ所取的最高幂次,为2Nφ+1;Nk为多项式展开项数;NS、NC为傅里叶级数展开项数;上述变量由最小二乘法确定取值。Q、QS、QC、QC*代表展开项的系数,分别由数值拟合的结果确定。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.
GZapp(φ,η,ψ)的展开式与数值模拟的计算结果GZ(φ,η,ψ)的差值需要最小,即使下式E达到最小: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:
在综合考虑数值拟合的精度和后面计算的复杂程度后,最后确定Nφ=2、Nk=0、NS=1、NC=1,并令QS 0,1,1=0、QS 1,1,1=0、QS 2,1,1=0,QC 1,1,1=0、, 那么,GZapp(φ,η,ψ)的展开式可表示为: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:
GZapp(φ,η,ψ)=Q0,0,0φ+Q1,0,0φ3+Q2,0,0φ5+QC 0,1,1ηcos(ψ)φ (13)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 )
由于非规则波浪分解得到的随机过程ηc=ηcos(ψ),GZapp(φ,η,ψ)可进一步写为: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:
GZapp(φ,η,ψ)=q1φ+q2φ3+q3φ5+q4ηcφ (14)GZ app (φ,η,ψ)=q 1 φ+q 2 φ 3 +q 3 φ 5 +q 4 η c φ (14)
其中,qi(i=1,2,3,4)代表展开项的系数,可以通过GZapp(φ,η,ψ)与GZ(φ,η,ψ)的数值拟合的结果确定。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).
求解稳态概率密度函数Solve the steady-state probability density function
在本发明中,为了简化计算,暂不考虑波浪的有色性质,即假定波浪为白噪声。以路径积分法为基础,在时间域内求解相应的Fokker-Planck方程,得到随时间演变的船舶横摇运动的转移概率密度,进而求解船舶在不规则波浪中参数横摇运动响应的稳态概率密度函数。本发明取二维的算例,具体步骤如下所示: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:
考虑斜浪下随机波浪外激励的作用,将式(14)代入式(4),得到考虑参数横摇运动的船舶一自由度运动微分方程: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:
上述方程两边同时除以Iφφ+δIφφ:Divide both sides of the above equation by I φφ +δI φφ :
式中, In the formula,
则式(16)可以转化成如下表达式:Then formula (16) can be transformed into the following expression:
令ξ1(t)=p(t),ξ2(t)=fwave(t),则有:make ξ 1 (t)=p(t), ξ 2 (t)=f wave (t), then:
令x1=φ;将随机微分方程式(18)转化为如下状态方程组:Let x 1 = φ; Transform the stochastic differential equation (18) into the following state equations:
将参数激励和波浪载荷ξk(t)作为有色噪声序列,分别通过线性滤波器后输出,如式(20)所示: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):
其中,Wk(t)为Wiener函数。与式(19)联立形成六维随机运动微分方程:Among them, W k (t) is a Wiener function. Combined with formula (19) to form a six-dimensional random motion differential equation:
可将上述方程组写成如下Ito随机微分方程:The above equations can be written as the following Ito stochastic differential equation:
dX=m(X,t)dt+Q(X,t)dW(t) (22)dX=m(X,t)dt+Q(X,t)dW(t) (22)
其中:m(X,t)和Q(X,t)分别为偏移系数矩和扩散系数矩阵。Among them: m(X,t) and Q(X,t) are the offset coefficient moment and diffusion coefficient matrix respectively.
当外部激励为一白噪声时,动态系统的响应是一个Markov过程,该过程的转移概率密度是满足初始条件:P(X,t′∣X′,t′)=δ(X-X′)与边界条件:P(-∞,t|X′,t′)=P(+∞,t|X′,t′)=0的Fokker-Planck方程的解。Fokker-Planck方程如下式所示: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:
同时转移概率密度P(X,t∣X′,t′)满足归一化条件:At the same time, the transition probability density P(X,t∣X′,t′) satisfies the normalization condition:
∫∫P(X,t∣X′,t′)dx1dx2dx3dx4dx5dx6=1 (24)∫∫P(X,t∣X′,t′)dx 1 dx 2 dx 3 dx 4 dx 5 dx 6 =1 (24)
其中,t′为初始时刻;X′为初始时刻的横摇角与横摇角速度矩阵;X为t时刻的横摇角与横摇角速度矩阵;x1为时刻t的横摇角;x2为时刻t1的横摇角速度。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 .
本发明通过路径积分法(PIS)求解Fokker-Planck方程可以得到船舶横摇运动的在时间域内的转移概率密度,进而求得船舶参数横摇运动的稳态概率密度函数p(x1),得到船舶在非规则波浪中的发生参数横摇运动概率P2,其中: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:
φ0为船舶发生参数横摇运动的衡准角。φ 0 is the criterion angle for parametric rolling motion of the ship.
路径积分法的基本思想是在空间和时间上分别离散化,以路径代替积分,即通过连接短时概率密度形成全局转移概率密度,得到状态矢量的联合概率密度。系统在一个小的时间段τ(τ=t-t′)上的转移概率密度函数可以写成: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:
式中:δ为狄雷克函数;ri(x′)为通过龙哥库塔法求出漂移系数mi(x′);γ2为扩散系数;x1′为t′时刻的横摇角;x2′为t′时刻的横摇角速度。式(26)精确至τ2在求得了一小时间段τ上的转移概率密度后,系统的转移概率密度函数可以表示为: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:
式中,tk=t′+kτ;t=tS;X=X(S);X′=X(0);S为初始时刻t'时刻t=ts的时间段间隔数;Rn(S-1)表示积分次数。在定初始分布ω(X(0))之后,可以得到时刻t的概率密度函数为: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:
由此,应用路径积分法可以求得船舶参数横摇运动的稳态概率密度函数,并得到船舶在非规则波浪中的发生参数横摇运动概率P2。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.
步骤4:根据目标海域非规则波出现的概率P1和船舶在非规则波中的参数横摇概率P2计算得到代表航线的测试点区域特定非规则波的船舶发生参数横摇运动的概率P如下式所示: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=P1P2 (29)P=P 1 P 2 (29)
由此类推,根据理想航线所覆盖区域内的波浪散布图,计算该测试点每一种随机海况下船舶的发生参数横摇运动概率。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.
步骤5:重复步骤2-4计算其它测试点的不同非规则波中船舶发生参数横摇运动概率,求解代表性航线上船舶发生参数横摇运动总的发生概率。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.
综上,本发明非规则波中基于AIS数据评估船舶发生参数横摇运动概率的方法思路流程图如图6所示。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 .
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