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Article(id=1281326731378279125, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281326672238580175, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2026.01.001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1750867200000, receivedDateStr=2025-06-26, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1783421698660, onlineDateStr=2026-07-07, pubDate=1768406400000, pubDateStr=2026-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1783421698660, onlineIssueDateStr=2026-07-07, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1783421698659, creator=13701087609, updateTime=1783421698659, updator=13701087609, issue=Issue{id=1281326672238580175, tenantId=1146029695717560320, journalId=1240685776644648972, year='2026', volume='30', issue='1', pageStart='1', pageEnd='176', issueExtLink='null', onlineDate='null', pubDate='1768406400000', pubDateStr='2026-01-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1783421684559, creator='13701087609', updateTime=1783422118948, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1281328494261026863, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281326672238580175, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1281328494261026864, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281326672238580175, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=1, endPage=10, ext={EN=ArticleExt(id=1281326731608965846, articleId=1281326731378279125, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Extreme value prediction of ship roll motion response and capsizing probability evaluation under combined effect of wind and waves, columnId=1241023037940748650, journalTitle=Journal of Ship Mechanics, columnName=Hydrodynamics, runingTitle=null, highlight=null, articleAbstract=

A single-degree-of-freedom model was established to describe the roll motion of ships under combined effect of wind and wave, then the accuracy of the numerical model was verified by model tests. The time history of roll motion responses in various sea conditions was calculated by the numerical method. The Gumbel method and average conditional exceedance rate (ACER) method were used to predict the extreme value of ship roll motion response. Then based on the extreme value prediction theory, an efficient method for evaluating the ship capsizing probabilities was established. The Monte Carlo method and Level 2 vulnerability criteria for dead ship stability were employed to validate the feasibility of the evaluation method. The research demonstrates the suitability of both the Gumbel method and the average conditional exceedance rate method in predicting the extreme value distribution of ship roll motion in various sea condition, while the average conditional exceedance rate method has a higher accuracy. Compared to the traditional Monte Carlo method, the ship capsizing probability evaluation method proposed in this paper could ensure the accuracy and improve the calculation efficiency. The research in this work could provide references for the dead ship stability evaluation as well as for the safety of ships at seas.

, authors=Xiang-peng YU, Xiao-fei MAO, Wei CHAI, Xing-yu ZHAN, Guo-rui LAI, authorsList=Xiang-peng YU, Xiao-fei MAO, Wei CHAI, Xing-yu ZHAN, Guo-rui LAI, authorCompany=null, correspAuthors=Xiao-fei MAO, authorNote=null, correspAuthorsNote=null, copyrightStatement=Copyright ©2026 Journal of Ship Mechanics. All rights reserved., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1281326741176173357, articleId=1281326731378279125, tenantId=1146029695717560320, journalId=1240685776644648972, language=CN, title=风浪联合作用下船舶横摇响应极值预报及倾覆概率评估, columnId=1241023038087549292, journalTitle=船舶力学, columnName=流体力学, runingTitle=null, highlight=null, articleAbstract=

本文建立风浪联合作用下船舶单自由度横摇运动响应模型，并通过模型试验验证其可靠性。基于该数值模型计算不同海况下船舶横摇运动响应时历，结合Gumbel方法和平均条件穿越率法（Average Conditional Exceedance Rate Method, ACER）开展船舶横摇运动响应极值预报研究。随后，基于极值预报理论建立船舶倾覆概率评估方法，通过蒙特卡洛法和瘫船稳性第二层薄弱性衡准验证该方法的可行性。研究表明，Gumbel方法和平均条件穿越率法均适用于预报不同风浪作用下船舶横摇运动响应极值分布，且平均条件穿越率法的精度更高。本文所提出的船舶倾覆概率评估方法相较于传统的蒙特卡洛法，既能够保证准确率，也能大幅提升计算效率。本研究可为船舶瘫船稳性评估和船舶安全性评估提供参考。

, authors=余象鹏, 毛筱菲, 柴威, 詹星宇, 赖国锐, authorsList=余象鹏, 毛筱菲, 柴威, 詹星宇, 赖国锐, authorCompany=null, correspAuthors=毛筱菲, authorNote=

余象鹏（1998–），男，硕士研究生

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毛筱菲（1962–），女，教授，硕士生导师，通讯作者，E-mail：
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余象鹏（1998–），男，硕士研究生

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余象鹏（1998–），男，硕士研究生

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Oslo, Norway: Det Norske Veritas, 2000., articleTitle=null, refAbstract=null)], funds=null, companyList=[AuthorCompany(id=1281326741448803118, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, xref=null, ext=[AuthorCompanyExt(id=1281326741465580335, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, companyId=1281326741448803118, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China), AuthorCompanyExt(id=1281326741478163248, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, companyId=1281326741448803118, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=武汉理工大学 船海与能源动力工程学院，武汉 430063)])], 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journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Fig.9, caption=Long-term capsizing probability evaluation by different methods, figureFileSmall=89LIIy8+YWAHjYTvGBADGg==, figureFileBig=SLcI6269CkVEFaqzOXZFaA==, tableContent=null), ArticleFig(id=1281326747903837027, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=CN, label=图9, caption=基于不同方法的长期倾覆概率评估, figureFileSmall=89LIIy8+YWAHjYTvGBADGg==, figureFileBig=SLcI6269CkVEFaqzOXZFaA==, tableContent=null), ArticleFig(id=1281326747991917412, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.1, caption=

Main parameters of DTMB-5415 standard model

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参数单位数值参数单位数值
垂线间长m142.190重心纵向位置m70.137
型宽m20.540重心垂向位置m7.555
吃水m6.150初稳性高m1.938
排水量t8635.000横向受风面积m23168.910
方形系数/0.505风倾作用力臂m13.940
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DTMB-5415标模主要船型参数

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参数单位数值参数单位数值
垂线间长m142.190重心纵向位置m70.137
型宽m20.540重心垂向位置m7.555
吃水m6.150初稳性高m1.938
排水量t8635.000横向受风面积m23168.910
方形系数/0.505风倾作用力臂m13.940
), ArticleFig(id=1281326748147106662, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.2, caption=

Sea conditions in model tests

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算例模型过零周期/s实型过零周期/s模型有义波高/cm实型有义波高/m海况等级
11.309.2810.805.506级
21.5411.0010.805.506级
), ArticleFig(id=1281326748226798439, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=CN, label=表2, caption=

模型试验海况

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算例模型过零周期/s实型过零周期/s模型有义波高/cm实型有义波高/m海况等级
11.309.2810.805.506级
21.5411.0010.805.506级
), ArticleFig(id=1281326748298101608, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.3, caption=

Analysis of statistical values of model tests and numerical simulations

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算例三一幅值/°平均值/°均方差/°
试验模拟误差试验模拟误差试验模拟误差
127.7728.783.64%17.8018.805.62%8.958.870.93%
230.6229.025.23%19.3319.390.31%9.368.746.62%
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模型试验及数值模拟统计值分析

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算例三一幅值/°平均值/°均方差/°
试验模拟误差试验模拟误差试验模拟误差
127.7728.783.64%17.8018.805.62%8.958.870.93%
230.6229.025.23%19.3319.390.31%9.368.746.62%
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Sea state parameters in numerical simulations

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算例过零周期 /s有义波高 /m定常风速 /(m·s−1)海况 等级
39.003.5013.955级
49.004.5016.506级
511.003.5013.955级
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数值模拟海况参数

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算例过零周期 /s有义波高 /m定常风速 /(m·s−1)海况 等级
39.003.5013.955级
49.004.5016.506级
511.003.5013.955级
), ArticleFig(id=1281326748629451628, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.5, caption=

Predicted extreme value of roll motion by ACER method with different k and η0 (Case 3)

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阶数kη0 = 15°η0 = 16°η0 = 17°η0 = 18°η0 = 19°η0 = 20°
140.38°39.72°40.13°40.03°39.79°39.42°
239.37°39.23°39.31°39.74°39.27°39.17°
339.39°39.24°39.13°39.30°39.16°39.15°
439.38°39.25°39.14°39.11°39.46°39.09°
538.71°38.96°39.05°38.93°38.83°38.65°
638.64°38.89°39.04°38.92°38.82°38.64°
), ArticleFig(id=1281326748717532013, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=CN, label=表5, caption=

不同k和η0时ACER方法预报的横摇极值 （算例3）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数kη0 = 15°η0 = 16°η0 = 17°η0 = 18°η0 = 19°η0 = 20°
140.38°39.72°40.13°40.03°39.79°39.42°
239.37°39.23°39.31°39.74°39.27°39.17°
339.39°39.24°39.13°39.30°39.16°39.15°
439.38°39.25°39.14°39.11°39.46°39.09°
538.71°38.96°39.05°38.93°38.83°38.65°
638.64°38.89°39.04°38.92°38.82°38.64°
), ArticleFig(id=1281326748801418094, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.6, caption=

Results of extreme value prediction of roll motion response of ship in different sea conditions

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算例TZ /sHS /mUw /（m·s−1）Gumbel方法ACER方法
η95% 置信区间不确定度η95% 置信区间不确定度
39.003.5013.9539.34°（37.63°, 41.41°）9.61%39.13°（38.02°, 39.93°）4.90%
49.004.5016.5049.57°（47.22°, 52.41°）10.47%48.42°（46.99°, 49.44°）5.06%
511.003.5013.9537.83°（36.04°, 40.04°）10.60%37.21°（36.34°, 38.05°）4.60%
), ArticleFig(id=1281326748910469999, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=CN, label=表6, caption=

不同海况下船舶横摇运动响应极值预报结果

, figureFileSmall=null, figureFileBig=null, tableContent=
算例TZ /sHS /mUw /（m·s−1）Gumbel方法ACER方法
η95% 置信区间不确定度η95% 置信区间不确定度
39.003.5013.9539.34°（37.63°, 41.41°）9.61%39.13°（38.02°, 39.93°）4.90%
49.004.5016.5049.57°（47.22°, 52.41°）10.47%48.42°（46.99°, 49.44°）5.06%
511.003.5013.9537.83°（36.04°, 40.04°）10.60%37.21°（36.34°, 38.05°）4.60%
), ArticleFig(id=1281326749002744688, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=EN, label=Tab.7, caption=

Capsizing probability under different combined effects of wind and wave

, figureFileSmall=null, figureFileBig=null, tableContent=
算例TZ /sHS /mUw /（m·s−1）倾覆概率
Gumbel方法ACER方法蒙特卡洛法
39.003.5013.950.622%0.38%0.16%
49.004.5016.5078.24%79.50%79.00%
511.003.5013.950.22%0.05%0.08%
), ArticleFig(id=1281326749074047857, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281326731378279125, language=CN, label=表7, caption=

不同风浪作用下船舶倾覆概率

, figureFileSmall=null, figureFileBig=null, tableContent=
算例TZ /sHS /mUw /（m·s−1）倾覆概率
Gumbel方法ACER方法蒙特卡洛法
39.003.5013.950.622%0.38%0.16%
49.004.5016.5078.24%79.50%79.00%
511.003.5013.950.22%0.05%0.08%
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风浪联合作用下船舶横摇响应极值预报及倾覆概率评估
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余象鹏 , 毛筱菲 , 柴威 , 詹星宇 , 赖国锐
船舶力学 | 流体力学 2026,30(1): 1-10
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船舶力学 |流体力学 2026 , 30 (1) : 1 -10
风浪联合作用下船舶横摇响应极值预报及倾覆概率评估
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余象鹏, 毛筱菲 , 柴威, 詹星宇, 赖国锐
作者信息
  • 武汉理工大学 船海与能源动力工程学院，武汉 430063
通讯作者:
毛筱菲（1962–），女，教授，硕士生导师，通讯作者，E-mail：
作者简介:

余象鹏（1998–），男，硕士研究生

Extreme value prediction of ship roll motion response and capsizing probability evaluation under combined effect of wind and waves
Xiang-peng YU, Xiao-fei MAO , Wei CHAI, Xing-yu ZHAN, Guo-rui LAI
Affiliations
  • School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China
出版时间: 2026-01-15 doi: 10.3969/j.issn.1007-7294.2026.01.001
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摘要
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本文建立风浪联合作用下船舶单自由度横摇运动响应模型，并通过模型试验验证其可靠性。基于该数值模型计算不同海况下船舶横摇运动响应时历，结合Gumbel方法和平均条件穿越率法（Average Conditional Exceedance Rate Method, ACER）开展船舶横摇运动响应极值预报研究。随后，基于极值预报理论建立船舶倾覆概率评估方法，通过蒙特卡洛法和瘫船稳性第二层薄弱性衡准验证该方法的可行性。研究表明，Gumbel方法和平均条件穿越率法均适用于预报不同风浪作用下船舶横摇运动响应极值分布，且平均条件穿越率法的精度更高。本文所提出的船舶倾覆概率评估方法相较于传统的蒙特卡洛法，既能够保证准确率，也能大幅提升计算效率。本研究可为船舶瘫船稳性评估和船舶安全性评估提供参考。

横摇运动响应  /  极值预报  /  瘫船稳性  /  倾覆概率

A single-degree-of-freedom model was established to describe the roll motion of ships under combined effect of wind and wave, then the accuracy of the numerical model was verified by model tests. The time history of roll motion responses in various sea conditions was calculated by the numerical method. The Gumbel method and average conditional exceedance rate (ACER) method were used to predict the extreme value of ship roll motion response. Then based on the extreme value prediction theory, an efficient method for evaluating the ship capsizing probabilities was established. The Monte Carlo method and Level 2 vulnerability criteria for dead ship stability were employed to validate the feasibility of the evaluation method. The research demonstrates the suitability of both the Gumbel method and the average conditional exceedance rate method in predicting the extreme value distribution of ship roll motion in various sea condition, while the average conditional exceedance rate method has a higher accuracy. Compared to the traditional Monte Carlo method, the ship capsizing probability evaluation method proposed in this paper could ensure the accuracy and improve the calculation efficiency. The research in this work could provide references for the dead ship stability evaluation as well as for the safety of ships at seas.

roll motion response  /  extreme value prediction  /  dead ship stability  /  capsizing probability
余象鹏, 毛筱菲, 柴威, 詹星宇, 赖国锐. 风浪联合作用下船舶横摇响应极值预报及倾覆概率评估. 船舶力学, 2026 , 30 (1) : 1 -10 . DOI: 10.3969/j.issn.1007-7294.2026.01.001
Xiang-peng YU, Xiao-fei MAO, Wei CHAI, Xing-yu ZHAN, Guo-rui LAI. Extreme value prediction of ship roll motion response and capsizing probability evaluation under combined effect of wind and waves[J]. Journal of Ship Mechanics, 2026 , 30 (1) : 1 -10 . DOI: 10.3969/j.issn.1007-7294.2026.01.001
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0　引言
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瘫船稳性作为国际海事组织颁布的第二代完整稳性衡准暂行指南中的稳性失效模式之一，指的是船舶由于机械故障或操舵问题而失去动力，在零航速漂移状态下受横风横浪联合作用产生大幅横摇甚至倾覆的现象。瘫船稳性是船舶在波浪中的稳性研究领域最具代表性的问题之一，也是船舶设计及检验阶段的重要内容，深入研究瘫船稳性可最大限度地保障船舶在波浪中航行的安全性。
围绕瘫船稳性问题，国内外学者们主要采用模型试验和数值模拟等方法开展相关研究[1−5]。曾柯等[1]基于单自由度模型计算船舶在不规则风浪中的横摇运动，并通过模型试验验证其可靠性，结合蒙特卡洛法求解船舶倾覆概率，并与瘫船稳性第二层薄弱性衡准计算结果进行对比。Kubo等[2]建立横荡–垂荡−横摇–纵摇耦合的四自由度模型并与模型试验结果对比验证其准确性，继而通过蒙特卡洛法评估船舶倾覆概率。胡丽芬等[3]基于单自由度模型，通过蒙特卡洛法结合北大西洋波浪长期分布进行瘫船稳性直接评估，并分析了初稳性高和有效波倾系数对倾覆概率的影响。然而，通过模型试验和数值模拟方法进行倾覆概率评估普遍存在成本高、耗时长等弊端，因此需引入更为高效、准确的倾覆概率评估方法用于船舶设计及检验过程中的稳性评估。
而从另一个角度出发，瘫船稳性可以理解为失去动力的船舶在横风横浪作用下，横摇运动响应超过一定阈值而导致的极限失稳事件。因此，基于船舶稳性失效和横摇角极值分布之间的密切联系，部分学者运用极值预报方法评估横摇角极值分布以反映船舶稳性，并通过计算船舶浸水角或稳性消失角对应的超越概率近似预报船舶的倾覆概率[6−9]。如此可大幅减小时域模拟样本数目，从而提高计算效率。Taggart[6]基于FREDYN软件计算得到了船舶横摇运动响应时历，结合Gumbel方法对横摇运动响应极值分布进行了描述。Campbell等[7]运用LAMP软件计算得到了船舶横摇运动响应，结合波高阈值法对横摇运动响应极值进行了预报。然而，目前大部分极值预报方法在瘫船稳性的应用研究中仅考虑了随机横浪作用，忽略了横风的影响，因此，与船舶实际航行过程中更为相符的风浪联合作用下的船舶横摇运动响应极值预报值得关注。
本文首先建立风浪联合作用下船舶单自由度横摇运动响应模型，并根据横浪中的模型试验验证数值模型可靠性；其次，通过时域模拟得到风浪联合作用下的船舶横摇运动响应时历，结合Gumbel方法和ACER方法预报船舶横摇运动极值分布并开展不确定性分析，探究极值预报方法适用性；最后，基于极值预报方法进行船舶短期及长期倾覆概率评估，并通过与蒙特卡洛法和第二层薄弱性衡准的对比，验证所提方法的高效性及准确性。
1　数学模型与研究方法
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1.1　风浪联合作用下船舶横摇运动单自由度模型
船舶在风浪联合作用下的横摇运动方程可以表示为如下形式[9]
$ \left( {{I_{44}} + {A_{44}}} \right)\ddot \varphi (t) + {B_{44}}\dot \varphi (t) + {B_{44q}}\dot \varphi (t)\left| {\dot \varphi (t)} \right| + \Delta \cdot {l_{\mathrm{R}}}(\varphi (t)) = {M_{{\mathrm{wind}}}}(t) + {M_{{\mathrm{wave}}}}(t) $
式中：$\varphi $、$\dot \varphi $、$\ddot \varphi $分别为船舶横摇角、横摇角速度和横摇角加速度，I44为船舶横摇惯性矩，A44为船舶横摇附加惯性矩，B44、B44q分别为船舶线性和平方横摇阻尼系数，Δ为船舶排水量，lR为船舶复原力臂曲线，Mwind （t）为风作用力矩，Mwave （t）为波浪作用力矩。
船舶的复原力臂曲线可以由以下公式表示
$ {l_{\mathrm{R}}}(\varphi ) = \sum\limits_{i = 1}^{2r + 1} {{C_i}{\varphi ^i}} $
式中：$i = 1,3, \cdots ,2 r + 1$；$r \geqslant 1$且为正整数； Ci为复原力臂系数。文中选取$\varphi $、${\varphi ^3}$、${\varphi ^5}$三个基函数进行拟合，对应C1为线性复原力系数，C3、C5为非线性复原力系数。
由于随机海浪可以通过平稳的高斯过程描述，因此通过如下波浪作用力矩谱描述波浪的作用
$ {S_M}\left( \omega \right) = {\left[ {\Delta \cdot {l_{\mathrm{R}}} \cdot {f_{r,{\mathrm{waves}}}}\left( \omega \right)} \right]^2}\frac{{{\omega ^4}}}{{{g^2}}}{S_\zeta }\left( \omega \right) $
式中：fr, waves （ω）为有效波倾系数，可通过谐摇试验求得；Sζ （ω）为海浪谱，文中选用ITTC双参数谱，其一般形式表示为
$ {S_\zeta }(\omega ) = \frac{A}{{{\omega ^5}}}\exp \left( - \frac{B}{{{\omega ^4}}}\right) $
式中：$A = 173H_{\mathrm{S}}^2/T_{\mathrm{Z}}^4$，$B = 691/T_{\mathrm{Z}}^4$，HS为有义波高，TZ为过零周期，ω为波浪圆频率。
风的作用包括定常风和阵风两部分，其中定常风的风倾力矩可表示为
$ {M_{{\mathrm{wind}}}} = \frac{1}{2}{\rho _{{\mathrm{air}}}}U_{\mathrm{w}}^2{A_{\mathrm{L}}}{C_{\mathrm{y}}}\left( {{H_{\mathrm{w}}} + {H_{{\mathrm{yd}}}}} \right) $
式中：ρair为空气密度，其值取1.222 kg/m3；AL为水面以上船舶横向受风面积；Cy为风倾力矩系数，其值取1.22；Hw为水线面到风场作用中心的垂向距离；Hyd为从水线面到波浪力作用中心的垂向距离，Uw为定常风速，可根据不规则波的有义波高确定，Uw = （HS/0.06717）1/1.5。
阵风产生的力矩谱可由下式计算
$ {S_{{M_f}}}(\omega ) = {\left[ {{\rho _{{\mathrm{air}}}}{U_{\mathrm{w}}}{C_{\mathrm{y}}}{A_{\mathrm{L}}}\left( {{H_{\mathrm{w}}} + {H_{{\mathrm{yd}}}}} \right)} \right]^2}{\chi ^2}\left( \omega \right){S_U}\left( \omega \right) $
式中：χ（ω）为标准气动导纳函数，可通过下式得出
$ \chi \left( \omega \right) = \dfrac{1}{{1 + {{\left( {\dfrac{{\omega \sqrt {{A_{\mathrm{L}}}} }}{{{\text{π}} {U_{\mathrm{w}}}}}} \right)}^{4/3}}}}$
SU（ω）为阵风谱，文中选用Davenport阵风谱，其标准表达式如下
$ {S_U}\left( \omega \right) = 4K\frac{{U_{\mathrm{w}}^2}}{\omega }\frac{{X_D^2}}{{{{\left( {1 + X_D^2} \right)}^{{4 \mathord{\left/ {\vphantom {4 3}} \right. } 3}}}}} $
式中：K=0.003，XD = 600·ω·（π·Uw）−1。
1.2　Gumbel方法极值预报
极值预报指的是根据一个随机过程的样本数据推算其极端值的出现概率。对于一个随机过程X（t），X1, X2,…, Xn表示随机过程的n个序列样本，定义序列样本极值：${M_n} = \max \left\{ {{X_1},{X_2}, \cdots ,{X_n}} \right\}$。若各序列样本彼此独立且满足同一分布FX（x），则随机过程的极值η满足以下分布[10]
$ \begin{gathered} {F_{{M_n}}}(\eta ) = {P} ({M_n} \leqslant \eta ) = {P} ({X_1} \leqslant \eta , \cdots ,{X_n} \leqslant \eta ) = {P} ({X_1} \leqslant \eta ) \cdot \ldots \cdot {P} ({X_n} \leqslant \eta ) = {({F_X}(\eta ))^n} \\ \end{gathered} $
随着序列样本n的逐渐增大，不论FX（x）为何分布，随机过程的极值分布都趋近于某些渐近极值分布，如Gumbel分布等。Gumbel概率分布模型可表示为
$ {F_{{M_n}}}(\eta ) = \exp \left\{ { - \exp \left[ { - \alpha (x - \mu )} \right]} \right\} $
式中：α为比例参数，μ为位置参数，两参数通过在Gumbel概率图纸中将数据点拟合为一条直线确定。
根据Gumbel概率图纸拟合确定的Gumbel分布函数，进一步计算超越概率λ为
$ \lambda = 1 - {F_{{M_n}}}(\eta ) $
基于各海况下的短期极值预报，可结合海况发生频率加权的组合计算长期的超越概率，其形式可表示为
$ {\tilde \lambda _{{\mathrm{longterm}}}}(\zeta ) = \int\limits_{{h_{\rm{s}}}} {\int\limits_{{t_{\rm{z}}}} {\lambda (\zeta |{h_{\rm{s}}},{t_{\rm{z}}}){f_{{H_{\rm{s}}}{T_{\rm{z}}}}}({h_{\rm{s}}},{t_{\rm{z}}})} } {\text{d}}{h_{\rm{s}}}{\text{d}}{t_{\rm{z}}} $
式中：$\tilde \lambda_{{\mathrm{longterm}}}(\zeta )$为长期预报的超越概率；$\lambda (\zeta |{h_{\rm{s}}},{t_{\rm{z}}})$为特定海况HS = hs，TZ= tz下短期预报的超越概率；${f_{H_{\mathrm{s}}T_{\mathrm{z}}}}({h_{\rm{s}}},{t_{\rm{z}}})$为HS和TZ的联合概率密度函数。
1.3　ACER方法极值预报
不同于Gumbel方法基于参数分布函数预报极值，在预报过程中舍弃了大部分数据，仅应用横摇最大幅值数据，ACER方法充分利用所有幅值数据构建不同阶的ACER函数用于预报极值，使其同样适用于非平稳数据集[11]。基于数值模拟得到了横摇运动响应时历，横摇极值η的累积分布可描述为以下形式
$ P(\eta ) \approx \exp \left[ { - \left( {N - k + 1} \right) \cdot {{\hat \varepsilon }_k}\left( \eta \right)} \right] $
式中：N为每个给定持续时长的横摇时历内幅值的平均数量；k为ACER函数的阶数，且满足k$\ll $N；${\hat \varepsilon _k}\left( \eta \right)$为k阶的经验ACER函数，通过模拟的横摇时历得到。上式的精度随ACER函数的阶数k的增加而提高，但同时用于极值预报的数据相对减少，因此当精度接近时应选择较小的阶数。
继而结合有效的外推方法用于极值预报，外推方法基于工程事实：即对于大部分的船舶和海洋结构物，不同响应极值对应的ACER函数均具有高度规律性。尤其在η ≥ η0的尾部区域，ACER函数类似于函数exp{−a（η−b）c}，其中a、b、c为合适的常数，且满足a＞0、b ≤ η0、c＞0。因此可以假设经验ACER函数为
$ {\hat \varepsilon _k}(\eta ) \approx {q_k}\exp \left\{ { - {a_k}{{\left( {\eta - {b_k}} \right)}^{{c_k}}}} \right\},\eta \geqslant {\eta _0} $
式中：qk、ak、bk和ck均为与阶数k相关的常数。这些常数通过对数级的优化拟合取值，可由均方误差函数确定，其表达式为
$ F\left( {{q_k},{a_k},{b_k},{c_k}} \right) = \sum_{i = 1}^M {{\rho _i}{{\left| {\ln {{\hat \varepsilon }_k}\left( {{\eta _i}} \right) - \ln {q_k} + {a_k}{{\left( {{\eta _i} - {b_k}} \right)}^{{c_k}}}} \right|}^2}} $
式中：ηi（i = 1, …, M）为ACER函数经验估计的极值水平，ρi为权重因子，可增加更可靠数据点的权重，通常权重因子随极值水平ηi的升高而降低。权重因子由经验ACER函数的95%置信区间（C, confidence interval）确定，其表达式为
$ {C^ \pm }({\eta _i}) = {\hat \varepsilon _k}({\eta _i}) \cdot \left(1 \pm \frac{{1.96}}{{\sqrt {(N - k + 1){{\hat \varepsilon }_k}({\eta _i})} }}\right) $
1.4　瘫船稳性第二层薄弱性衡准
在2020年国际海事组织正式颁布的第二代完整稳性衡准暂行指南中，第二层薄弱性衡准评估船舶在给定的短期环境条件下，暴露在特定时间Texp内的瘫船稳性失效倾覆指数为
$ {C_{EA}} = 1 - \exp \left\{ { - \frac{{{T_{\exp }}}}{{{T_Z}}} \cdot \left[ {\exp \left( { - \frac{1}{{2 \cdot R_{EA + }^2}}} \right) + \exp \left( { - \frac{1}{{2 \cdot R_{EA - }^2}}} \right)} \right]} \right\} $
式中：R为危险指数，具体计算公式详见参考文献[12]。
基于短期倾覆概率评估，结合具体海区中每种海况的发生概率取权重平均，可进行瘫船稳性长期倾覆概率P的评估，其计算公式如下
$ P = \int\limits_{{h_{\rm{s}}}} {\int\limits_{{t_{\rm{z}}}} {{C_{EA}}\left( {{T_{\exp }}\left| {{h_{\rm{s}}},{t_{\rm{z}}}} \right.} \right){f_{{H_{\rm{s}}}{T_{\rm{z}}}}}({h_{\rm{s}}},{t_{\rm{z}}})} } {\text{d}}{h_{\rm{s}}}{\text{d}}{t_{\rm{z}}} $
2　数值模拟及试验验证
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2.1　船型参数
选取DTMB-5415船型进行模型试验和数值计算研究，缩尺比为1∶51。其标模示意图、复原力臂曲线如图1和图2所示，主要船型参数见表1。
2.2　数值模型的建立及验证
模型试验在武汉理工大学船舶性能实验室进行[13]。基于模型试验测得的水动力系数建立船模横摇单自由度模型，并开展不规则波中的横摇试验以验证数值模型的可靠性，其中设置的试验海况如表2所示。
将随机横浪中各个海况下数值模拟所得的横摇运动响应时历的特征值和模型试验结果进行对比，以验证数值模型的准确性和可靠性。其中，船模在各海况下的试验、数值模拟结果对比如表3所示。由表可知，各海况下模型试验与数值模拟得到的船模横摇运动响应统计值误差基本控制在5%以内，存在最大误差约为7%，数值模型可靠性得到了验证。
随后，将数值模拟和模型试验横摇幅值的概率密度函数（Probability Density Function, PDF）进行对比。图3为船模在随机横浪中算例2条件下，分别通过数值模拟和模型试验得到的横摇幅值概率密度函数。由图3可知，两者结果吻合较好。因此，本文提出的数值模型准确性和可靠性符合要求，可用于开展后续研究与分析。
3　极值预报研究
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3.1　Gumbel方法极值预报
基于验证后的数值模型，根据相似理论将数值模型中通过试验得到的模型参数换算至实船参数，考虑风的影响，通过数值模拟得到实船在随机横风横浪中的横摇运动响应时历。设置算例海况如表4所示，各海况下模拟次数为50次，单次模拟时长为3600 s。对各海况下每组横摇运动响应时历取其最大幅值，将得到的50个横摇角绝对值的最大值在Gumbel概率图纸中通过最小二乘法拟合为直线。取超越概率λ = 0.01，延长拟合所得直线得到估计的极值η。船舶在算例3条件下的拟合效果与极值估计如图4所示。
在图4中，与横轴相平行的直线即为超越概率λ = 0.01所对应的值，其与拟合直线的交点所对应的横坐标为估计的横摇极值，该海况下为39.34°。由图可知，船舶在随机横风横浪下的横摇运动响应极值在Gumbel概率图纸中能够取得较好的拟合效果，仅在尾部区域的拟合效果稍有欠佳。表明Gumbel方法可用于预报船舶横摇运动响应极值分布。
为进一步验证Gumbel方法所预报横摇极值的可靠性，对其进行不确定度分析。根据Gumbel概率图纸中拟合所得的比例参数α和位置参数μ，利用参数自举法获得估计极值的95%置信区间，95%置信区间的宽度与估计极值的比值即为不确定度。
图5展示了随机横风横浪中船舶在算例3条件下，从104个参数样本中获得的Gumbel概率密度函数，星号表示95%置信区间的上下限。由图可知，该条件下所预报的横摇极值为39.34°时，其对应的95%置信区间为（37.63°, 41.41°），则不确定度为9.61%。
3.2　ACER方法极值预报
基于模拟得到了各海况下的50组横摇运动响应时历，利用横摇幅值数据绘制经验ACER函数，图6所示为各海况下不同阶的经验ACER函数。由图可知，当横摇极值较低时，不同阶的经验ACER函数之间有所差异，表明其数据点之间存在一定的依赖性。当k ≥ 3时，经验ACER函数在极值较高范围内不再随k的增加而产生显著变化，数据间的依赖性影响可忽略不计。因此，为充分运用数据可选择经验ACER3（下标为阶数，下同）函数以预报横摇极值。此外，从图中可以观察到，在极值较高的尾部区域，经验ACER函数均趋近于线性，而实际上Gumbel分布在ACER函数图中也将对应为一条直线，表明Gumbel分布模型可较好地描述极值数据。
同样选择超越概率λ = 0.01，结合式（13）展开极值预报。表5所示为随机横风横浪中算例3条件下基于不同阶ACER函数及不同尾部区域所估计的横摇极值，η0即为尾部区域阈值，可根据敏感性研究结果确定其最佳取值。
由表5可知，经验ACER1函数及经验ACER2函数在不同η0时所预报的横摇极值未出现明显的收敛，ACER3函数在η0 ≥ 17°时呈现出显著的收敛性。阶数k ≥ 3后预报的横摇极值亦展现出收敛性，但收敛所对应的η0随阶数的升高呈现增大的趋势，用于极值预报的尾部区域减小，所运用的数据量相对减少，故选择k = 3、η0 = 17°进行极值预报。
图7所示为随机横风横浪中算例3条件下运用ACER方法预报的横摇极值及其95%置信区间，与横轴相平行的直线即为超越概率λ = 0.01所对应的经验ACER3函数的值，与拟合曲线的交点的横坐标即为预报横摇极值。此时，预报的横摇极值为39.13°，其95%置信区间为（38.02°, 39.93°），计算得到的不确定度为4.90%。
3.3　极值预报结果分析
分别采用Gumbel方法和ACER方法对风浪联合作用下不同海况的船舶横摇运动响应极值分布进行预报，所得结果见表6。
由表6可知，分别通过Gumbel方法和ACER方法预报的不同海况下的横摇极值差异较小，证明了Gumbel方法和ACER方法用于预报风浪联合作用下的横摇运动响应极值分布的可行性。并且可以看出ACER方法预报的横摇极值的95%置信区间较Gumbel方法更窄，不确定度更小，即ACER方法的预报精度更高，故在数据充分的情况下，可优先选用ACER方法预报横摇极值分布。此外，对比不同海况下的横摇极值结果可以发现，当有义波高升高，或当波浪特征周期接近船舶横摇固有周期（即进入船舶横摇临界区域）时，横摇极值将发生显著增加。
4　倾覆概率评估
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4.1　短期倾覆概率评估
由于极值预报中超越概率与横摇角水平的一一对应关系，因此可通过各横摇角水平分析其对应的超越概率。分别采用Gumbel方法和ACER方法计算得到随机横风横浪中算例3条件下各横摇角水平下的超越概率，同时引入蒙特卡洛方法与上述两种方法进行对比。蒙特卡洛模拟预报的超越概率的可靠性随模拟次数的增加而增强，本文选用5000次数值模拟的蒙特卡洛法进行验证，以得到统计意义上稳定的超越概率[14]。三种方法的对比结果如图8所示。
由图可知，采用极值预报方法和蒙特卡洛法计算得到风浪联合作用下船舶各横摇角水平对应的超越概率差异较小，这证明了通过极值预报方法计算超越概率的可行性。此外可以发现，由于基于参数分布的Gumbel方法存在尾部区域拟合效果欠佳的可能，因此在评估较高横摇角水平的超越概率时可能存在一定的误差，此时ACER方法的准确性更高。
进一步地，当船舶横摇角度超过船舶实际倾覆角时，即认为船舶会发生倾覆。通过计算船舶实际倾覆角对应的超越概率，可近似预报船舶的倾覆概率。在第二代完整稳性衡准暂行指南的稳性直接评估中，规定船舶实际倾覆角取以下三者中的最小值：40°，船舶在静水中的稳性消失角，以及船舶浸水角。因此，本文选择40°作为船舶实际倾覆角，进而计算得到各海况下船舶倾覆概率，结果如表7所示。
由表7可知，通过蒙特卡洛法计算的船舶各海况下的倾覆概率，证明了极值预报方法评估船舶短期倾覆概率的可靠性。并且由于船舶倾覆在大多数海况下为小概率事件，文中蒙特卡洛法基于5000次时域模拟才能得到统计意义上稳定的倾覆概率，而极值预报方法进行时域模拟的次数仅为50次，大幅提高了倾覆概率评估效率。
4.2　长期倾覆概率评估
基于极值预报方法和蒙特卡洛法计算得到的随机横风横浪中的短期倾覆概率评估结果，再结合北大西洋长期海况概率分布[15]可进行长期倾覆概率评估，同时引入第二代完整稳性衡准中的瘫船稳性第二层薄弱性衡准[12]进一步进行对比分析。此外，还通过极值预报方法预报了随机横浪（不考虑横风）中的长期倾覆概率，与风浪联合作用下的倾覆概率展开对比，所得结果如图9所示。
由图9可知，基于极值预报方法预报的风浪联合作用下各横摇角水平的长期超越概率、倾覆概率和蒙特卡洛法、第二层薄弱性衡准计算结果吻合程度较高；而在不考虑横风影响时，随机横浪作用下所评估的倾覆概率较第二层薄弱性衡准评估、蒙特卡洛法的计算结果显著偏小。该结果证明了极值预报方法评估长期超越概率及倾覆概率的可靠性，以及进行倾覆概率评估时考虑横风作用的重要性。此外，通过蒙特卡洛法、极值预报方法和第二层薄弱性衡准评估的随机横风横浪下船舶长期倾覆概率分别为26.27%、26.02%和23.64%，第二层薄弱性衡准的计算结果相较于极值预报方法和蒙特卡洛法更为保守。因此，鉴于极值预报方法评估倾覆概率的高效性和准确性，相较于传统的蒙特卡洛法，对于未能通过第二层薄弱性衡准评估的船舶，可考虑优先采用极值预报方法进行进一步的瘫船稳性直接评估。
5　结论
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本文建立了风浪联合作用下船舶单自由度横摇运动模型，通过极值预报方法评估了风浪联合作用下船舶横摇极值分布和倾覆概率。主要得到如下结论：
（1）随机横浪下船舶单自由度横摇运动响应模型计算得到的横摇运动响应统计特性与模型试验结果误差基本控制在5%以内，验证了本文数值模型的可靠性。
（2）Gumbel方法和ACER方法均适用于预报风浪作用下的船舶横摇运动极值分布，且ACER方法预报的横摇极值精度更高，计算结果的不确定度较Gumbel方法小5%左右。
（3）通过与蒙特卡洛法和第二层薄弱性衡准的对比，验证了极值预报方法评估船舶短期及长期倾覆概率的可行性，且极值预报方法较蒙特卡洛法减少约100倍计算时长，大幅提升了倾覆概率评估效率，因此可将极值预报方法用于未通过第二层薄弱性衡准后的进一步瘫船稳性评估。

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doi: 10.3969/j.issn.1007-7294.2026.01.001
  • 接收时间：2025-06-26
  • 首发时间：2026-07-07
  • 出版时间：2026-01-15
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  • 收稿日期：2025-06-26
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    武汉理工大学 船海与能源动力工程学院，武汉 430063

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毛筱菲（1962–），女，教授，硕士生导师，通讯作者，E-mail：
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2种不同金属材料的力学参数
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多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
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