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Article(id=1228048673725154145, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.03.014, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1660406400000, receivedDateStr=2022-08-14, revisedDate=1672848000000, revisedDateStr=2023-01-05, acceptedDate=null, acceptedDateStr=null, onlineDate=1770719219857, onlineDateStr=2026-02-10, pubDate=1711555200000, pubDateStr=2024-03-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770719219857, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770719219857, creator=13701087609, updateTime=1770719219857, updator=13701087609, issue=Issue{id=1228048667874095618, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='3', pageStart='365', pageEnd='538', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770719218462, creator=13701087609, updateTime=1770795476854, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368518803030940, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368518803030941, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=497, endPage=504, ext={EN=ArticleExt(id=1228048675440624492, articleId=1228048673725154145, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Random response and reliability analysis of aircraft landing gear under uneven runway excitation, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

A nonlinear two-degree-of-freedom system is used to construct an aircraft landing gear model. The stochastic excitation of the uneven runway to the system is described by time-domain noise，and the road roughness coefficient is used to describe the roughness of runway. Based on the probability density function and the statistics of system response，the influence of uneven runway on aircraft landing gear system is investigated. The reliability of the landing gear model and the passenger comfort under different road roughness coefficients are analyzed by establishing the relationship between the safety zones，comfort zones and the system response. The results show that the larger the road roughness coefficient is，the more fluctuation of the system state variable will be. The reliability and comfort of the system are negatively correlated with the road roughness coefficient. In addition，when the road roughness coefficient is small，the mean first-passage time and comfort of the system are more significantly affected by random disturbance. The present paper provides a theoretical basis for aircraft riding comfort and landing gear development and design.

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采用非线性二自由度系统构建了飞机起落架模型，通过时域噪声刻画了不平整跑道对飞机起落架系统的随机激励，并引入路面不平度系数描述了跑道的不平整程度；基于飞机起落架模型响应的概率密度函数及其统计量，开展了不平整跑道对飞机起落架系统影响规律的研究。应用随机可靠性分析方法以及人体振动舒适性理论，通过建立安全域和舒适域与系统响应之间的关系，分析了起落架模型在不同路面不平度系数下的可靠性，并对乘客舒适性进行了评价。研究结果表明：路面不平度系数越大，系统状态变量的波动越剧烈，系统的可靠性和舒适性与路面不平度系数呈一定的负相关性；在路面不平度系数较小的情况下，系统平均首次穿越时间以及舒适性受随机扰动的影响更加显著。

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许勇（1977—），男，博士，教授。电话：（029）88431637；E-mail：。
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张莹（1981—），女，博士，教授。电话：（029）88431637；E-mail：。

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张莹（1981—），女，博士，教授。电话：（029）88431637；E-mail：。

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张莹（1981—），女，博士，教授。电话：（029）88431637；E-mail：。

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路面等级/
A16
B64
C256
D1024
E4096
F16384
G65536
H262144
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不同路面等级对应的路面不平度系数

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路面等级/
A16
B64
C256
D1024
E4096
F16384
G65536
H262144
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Landing gear structure parameters

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参数数值参数数值
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起落架结构参数

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参数数值参数数值
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总加权加速度均方根值人的主观感受
小于0.315没有不舒服
0.315~0.63有些不舒服
0.5~1比较不舒服
0.8~1.6不舒服
1.25~2.5很不舒服
大于2极不舒服
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总加权加速度均方根值与人的主观感受之间的关系

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总加权加速度均方根值人的主观感受
小于0.315没有不舒服
0.315~0.63有些不舒服
0.5~1比较不舒服
0.8~1.6不舒服
1.25~2.5很不舒服
大于2极不舒服
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不平整跑道激励下飞机起落架随机响应及可靠性分析
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张莹 1 , 金峥嵘 1 , 贾万涛 1 , 刘小川 2 , 许勇 1
振动工程学报 | 2024,37(3): 497-504
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振动工程学报 | 2024, 37(3): 497-504
不平整跑道激励下飞机起落架随机响应及可靠性分析
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张莹1 , 金峥嵘1, 贾万涛1, 刘小川2, 许勇1
作者信息
  • 1西北工业大学数学与统计学院，陕西　西安　710129
  • 2中国飞机强度研究所，陕西　西安　710065

    • 张莹（1981—），女，博士，教授。电话：（029）88431637；E-mail：。

通讯作者:

许勇（1977—），男，博士，教授。电话：（029）88431637；E-mail：。
Random response and reliability analysis of aircraft landing gear under uneven runway excitation
Ying ZHANG1 , Zheng-rong JIN1, Wan-tao JIA1, Xiao-chuan LIU2, Yong XU1
Affiliations
  • 1School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China
  • 2Aircraft Strength Research Institute of China, Xi'an 710065, China
出版时间: 2024-03-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.03.014
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摘要
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采用非线性二自由度系统构建了飞机起落架模型，通过时域噪声刻画了不平整跑道对飞机起落架系统的随机激励，并引入路面不平度系数描述了跑道的不平整程度；基于飞机起落架模型响应的概率密度函数及其统计量，开展了不平整跑道对飞机起落架系统影响规律的研究。应用随机可靠性分析方法以及人体振动舒适性理论，通过建立安全域和舒适域与系统响应之间的关系，分析了起落架模型在不同路面不平度系数下的可靠性，并对乘客舒适性进行了评价。研究结果表明：路面不平度系数越大，系统状态变量的波动越剧烈，系统的可靠性和舒适性与路面不平度系数呈一定的负相关性；在路面不平度系数较小的情况下，系统平均首次穿越时间以及舒适性受随机扰动的影响更加显著。

飞机起落架  /  随机响应  /  可靠性  /  跑道不平整

A nonlinear two-degree-of-freedom system is used to construct an aircraft landing gear model. The stochastic excitation of the uneven runway to the system is described by time-domain noise，and the road roughness coefficient is used to describe the roughness of runway. Based on the probability density function and the statistics of system response，the influence of uneven runway on aircraft landing gear system is investigated. The reliability of the landing gear model and the passenger comfort under different road roughness coefficients are analyzed by establishing the relationship between the safety zones，comfort zones and the system response. The results show that the larger the road roughness coefficient is，the more fluctuation of the system state variable will be. The reliability and comfort of the system are negatively correlated with the road roughness coefficient. In addition，when the road roughness coefficient is small，the mean first-passage time and comfort of the system are more significantly affected by random disturbance. The present paper provides a theoretical basis for aircraft riding comfort and landing gear development and design.

landing gear  /  random response  /  reliability  /  unevenness of the runway
张莹, 金峥嵘, 贾万涛, 刘小川, 许勇. 不平整跑道激励下飞机起落架随机响应及可靠性分析. 振动工程学报, 2024 , 37 (3) : 497 -504 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.014
Ying ZHANG, Zheng-rong JIN, Wan-tao JIA, Xiao-chuan LIU, Yong XU. Random response and reliability analysis of aircraft landing gear under uneven runway excitation[J]. Journal of Vibration Engineering, 2024 , 37 (3) : 497 -504 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.014
正文
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引言
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飞机起落架在飞机着陆和滑跑过程中起着举足轻重的作用，关系着飞机的滑跑安全，因此起落架结构的动力学分析问题一直是研究者们关注的重点。20世纪30年代起，国际上就开展了关于飞机起落架滑跑动力学建模的研究。学者们将起落架结构简化为一个线性阻尼弹簧振子，得到二质量块模型，基于此模型对起落架的动力学特性开展了研究［1］。随后，为了对起落架进行更加全面和精确的建模和分析，能有效反映起落架非线性特征的因素被引入到该模型中，构建了二质量块非线性模型。如Milwitzky等［2］将起落架缓冲器的空气弹簧力、油液阻尼力以及摩擦力等非线性力考虑到模型中，并对其相关非线性动力学特性进行了研究。
由于飞机在地面滑跑阶段通常会受到不平整跑道激励产生随机振动，极易造成飞机起落架结构的疲劳损伤，缩短飞机寿命。因此，开展飞机在地面滑跑时起落架系统的随机响应研究并进行可靠性分析十分重要。通过将跑道激励简化为平稳高斯过程，Tung［3］采用摄动法和等效线性化法对非线性二自由度飞机滑跑模型的随机响应进行了数值模拟；贾玉红等［4］基于主动控制起落架的非线性二质量块模型建立目标函数，对主动控制起落架在地面滑跑的随机响应进行了分析。
随着研究的不断深入，研究人员开始使用更符合实际的跑道谱来描述跑道不平度。张冠超等［5］基于非线性二质量块起落架模型，将实测数据作为路面激励，采用时域确定性方法，对飞机在不平整跑道上滑行的响应进行了探究。然而实测跑道方法虽准确度高，但不具有普遍性且成本较高。因此，刘莉等［6-7］针对二质量块起落架模型，借助跑道功率谱密度描述路面不平度，并采用谱密度法研究了飞机滑跑时的平稳响应。Tung等［8］利用确定性数值分析和统计分析两种方法对飞机的跑道激励振动问题进行了讨论。此外，刘莉等［9］、赵莹芳等［10］利用等效线性化方法计算了非线性系统在不平稳随机激励下的响应。以上研究都是利用功率谱密度函数来模拟路面不平度，但谱分析方法对于非线性系统并不适用。
为了能将频域噪声引入到非线性系统，有学者采用谐波叠加法，将一系列相位不同的正弦波进行叠加，使路面不平度表达形式从功率谱转化到了时域的时间序列上，进而研究了飞机在不平整跑道滑跑时的随机响应，并对机场道面平整度进行了评价［11-13］。同时，由于铺砌平整程度决定了跑道的等级，为了更好地描述不同类型跑道的不平整度，颜光锋［14］采用不同的功率谱密度函数表征了不同等级路面的不平整度，并通过时域随机微分方程将其转化为时域噪声，刻画了不同平整程度的跑道，使非线性起落架系统复杂的动态响应分析更贴近实际。
因飞机在受到不平整跑道激励时产生的颠簸会直接影响滑跑的安全性、驾驶的平稳性以及乘坐的舒适度，故国内外学者对滑跑系统开展了可靠性分析，并将确定性系统的研究成果应用到了飞机的设计和优化中［15-17］。随着飞机设计的飞速发展，舒适性指标在飞机设计的评价中所占比重也越来越大［18-19］，现有研究通常使用GB/T 13441.1—2007标准［20］和GJB 67.8A—2008标准［21］来讨论飞机客舱人体的振动舒适性问题。周福强等［22］依据系统加速度均方根值对飞行中飞机座椅舒适度进行了评价，但针对不平整跑道作用下飞机客舱振动环境舒适性的研究仍有待开展。
本文构建了不平整跑道激励下的非线性二自由度起落架模型，研究了飞机滑跑过程中跑道不平度对飞机起落架系统随机响应的影响规律，分析了起落架系统在不平整跑道扰动下的可靠性以及乘客舒适性，为飞机乘坐舒适性及起落架研制和设计提供了理论依据。
1　飞机起落架动力学模型
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为更准确地研究系统的动力学响应，Wang等［23］在不考虑机身弹性和机体俯仰运动的情况下，建立了如图1所示的非线性二质量块模型。此模型重点考虑起落架缓冲器的非线性特征和轮胎的参数特征，注重起落架的整体结构和局部参数特征。其中，机体、机翼、缓冲器外筒等被简化为弹性质量，缓冲器活塞杆、刹车装置、机轮及车架等被简化为非弹性质量。表示上质量块的加速度，表示下质量块的加速度。设模型中缓冲器空气弹簧力、油液阻尼力和缓冲器活塞杆与外筒间的摩擦力均为非线性的，为跑道对轮胎的支撑力，机轮被视作一个具有线性阻尼和线性刚度的轮胎。
针对图1所示的模型，根据牛顿第二定律，建立系统动力学方程：
下面详述，，以及的物理意义［23］。
（a） 空气弹簧力
空气弹簧力用于描述氮气在上腔室所产生的压力。假设该气体的压强和体积满足气体状态方程：
式中　 ，，，和分别表示初始状态气体的压强、体积、气瓶长度及当前状态下气体的压强和体积；为活塞横截面积；为缓冲器行程；为气体常数，通常取值为。显然，，联立方程（2）~（4）得：
（b） 油液阻尼力
油液阻尼力的大小取决于流过孔板的油液所耗散的能量。假设油液是不可压缩的，表示上下腔压力差。由质量守恒定律和伯努利方程可得：
式中　参数为孔板流量系数，由实验确定；为孔板面积，为孔板直径；为油液流过孔板的速度；为油液的质量密度。
根据起落架结构参数可知，联立方程（6）~（8）得：
则油液阻尼力为：
这里用代替，使得与速度的方向相同。表示油气缓冲器活塞相对于外缸的速度，当时，油液是静态的，则为零。
（c） 摩擦力
起落架受到的摩擦力主要由密封件的密封性引起，可表示为：
式中　 和代表摩擦系数。
（d） 支撑力
设地面通过轮胎传递给起落架的力为线性力［23］，则支撑力为：
式中　刚度和阻尼系数为常数。由于飞机在地面滑跑时总会受到跑道不平整的影响，故在模型中引入随机噪声来刻画不平整跑道对起落架模型产生的随机激励，则地面通过轮胎传递至系统的力可修正为：
将式（5），（10），（11）和（13）代入方程（1），可得如下随机扰动下的起落架动力学模型：
式中　跑道不平整所引起的随机激励的统计特性可由路面功率谱刻画［24］：
式中　 表示参考空间频率，即每米长度包含个波长；代表最低截止角频率；表示路面谱角频率，单位为rad/s；v为飞机滑跑的速度，单位为m/s；为路面不平度系数，《机械振动道路路面谱测量数据报告》［24］中将路面分为如表1所示的8个等级，不同路面等级对应的路面不平度系数即为，根据中国所修建跑道的实际情况，常用前三级来刻画其不平度。
式（14）中随机激励满足如下微分方程［25］：
式中　 为路面空间截止频率，通常取；表示单位功率谱密度的高斯白噪声。
2　跑道随机激励下起落架系统的响应分析
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为方便分析式（14）的动力学特性，令，，，，则式（14）可表示为：
式中　 ；。
下面将运用随机模拟方法，对式（17）的响应及其统计特性进行分析。
2.1　系统响应的时间历程及矩分析
式（17）中的起落架结构参数如表2所示。设飞机的滑跑速度为，初始条件为 。
不同路面等级情形下，随机激励以及飞机下质量块位移随时间变化的计算结果如图2所示。不难看出，随路面不平整系数的增加，随机扰动强度增加，下质量块位移的波动幅度增大。
针对不同，计算了从到 的随机激励以及系统随机响应的方差。由图3（a）可知，方差的振荡幅度随的增加而增大。由图3（b）~（e）可知，系统随机响应的方差随时间的推移先快速增加随后趋于稳定；且越大，系统响应的方差越大，系统响应的波动程度越大。
特别地，在A和B级路面上，系统响应的方差随的增大仅呈小幅增加；但是在C级路面上，系统响应的方差显著增加。可见系统在不平整跑道激励较小时，具有较好的抗干扰能力，但对于跑道不平度的持续增加，响应的波动程度会大幅增加；且在随机激励一致的情况下，相较于系统的缓冲器冲程以及下质量块位移，上、下质量块速度的波动程度更剧烈，并且在C级路面上，该现象更为显著。
2.2　系统响应的概率密度函数
为进一步研究系统随机响应的统计特性，计算了不同下系统状态变量的概率密度函数。结果如图4所示，随着的增加，系统状态变量的稳态概率密度函数峰值逐渐减小，即系统稳定在平衡位置的概率逐渐减小。不难看出，与B和C级路面相比，A级路面使系统稳定在平衡位置的概率更大；并且系统状态变量的概率密度函数峰值随的增加大致呈的比例增加。
图5给出了不同下上质量块的速度在不同时刻的瞬态概率密度函数。可以看出，当飞机在不同等级路面滑跑时，随着时间的增加，上质量块速度的概率密度函数峰值逐渐增大，即系统上质量块的速度在零附近波动的概率增大；随着的增加，上质量块速度的瞬态概率密度函数峰值逐渐减小，即系统上质量块的速度在零附近波动的概率降低。
3　随机激励下系统可靠性及乘客舒适性分析
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3.1　系统的可靠性
随机振动系统的可靠性描述了系统在指定安全域运行的概率，是衡量系统安全性、舒适性的重要指标。通常可用可靠性函数、首次穿越时间的概率密度函数以及平均首次穿越时间等来刻画［24，26-28］。
可靠性函数为系统状态在时间区间内处于安全域的概率，即：
首次穿越时间的概率密度函数可由式（18）导出，即：
式中　 为可靠性函数关于时间的导数。进而由式（19）可得系统平均首次穿越时间满足：
式中　首次穿越时间是一个依赖于系统、激励及初始状态的随机变量。
3.2　可靠性分析
在讨论不同对于系统可靠性函数的影响时，本文选取了几种不同的安全域，在实际问题中可根据具体情况自行设置安全域。针对飞机缓冲器冲程设定安全域 ，针对飞机缓冲器位移以及飞机上质量块位移设定安全域，针对飞机缓冲器位移以及上质量块速度、位移设定安全域，系统初始状态取为 。
由图6可见，对于三种不同的安全域，系统的可靠性函数均随时间的增加而减小，即系统停留在安全域的概率随时间的增加而减小。在同一安全域下，随着的增加，系统的可靠性函数急剧下降，系统跃出安全域的概率大幅增加；且当路面等级为A时，系统的可靠性函数减小的速率远小于B和C等级，说明在较小的情况下，系统具有较好的可靠性。
在不同安全域以及对系统首次穿越时间概率密度函数的影响机制分析中，系统初值以及安全域选取与图6设定保持一致。从图7可以看出，首次穿越时间的概率密度函数峰随着的增大向左移动，且其峰值也随之增大，即首次穿越最可能发生的时间前移且概率增大，系统更易被破坏。
图8给出了安全域为时，不同下系统的平均首次穿越时间的三维图像。可见，系统初始状态越靠近穿越边界，平均首次穿越时间越短，则系统可靠性越差；且越大，系统的平均首次穿越时间越短，即系统停留在安全域的平均时间越短。进一步观察不难发现，在较小的情况下，平均首次穿越时间下降更快，即跑道随机激励对系统平均首次穿越时间的影响更明显。
3.3　舒适性评价
飞机在地面滑跑的过程中，跑道不平整所引起的振动会影响乘客的舒适性，标准［20］将总加权加速度均方根值作为评价乘客舒适性的指标，其与人主观感受间的关系如表3所示。本文仅考虑垂直于飞机客舱地板方向的加速度均方根，其中为客舱地板处垂直于地面的加速度，为用于预测振动对舒适影响的频率计权，通常取值为0.4。
图9所示为固定时间 时，不同下系统的总加权加速度均方根值的稳态概率密度函数。不难看出，随着的增大，的概率密度函数峰值下降，且函数峰的坡度越来越平缓，即达到较大值的概率随的增大而增大。当飞机在A级路面滑跑时，取时的概率密度函数最大，对照表3可知，此时乘客在飞机滑跑阶段的大部分时间不会感到不舒适。而在B级路面滑跑时，时的概率密度函数达到峰值，且在滑跑阶段的取值范围为；这意味着大于的概率变大，即乘客在滑跑这一阶段感到很不舒适的概率较大。而当飞机滑跑在C级路面上时，的概率密度函数峰值为，且其值在之间波动；说明与A，B级路面相比，乘客在C级路面上处于极不舒适状态的时间最长。
为了直观地对乘客舒适性进行评价，定义舒适性函数来衡量在时间区间内乘客处在舒适域的概率。根据表3，当总加权加速度均方根值小于时，乘客处在舒适域，即舒适域选取为。图10给出了不同下乘客舒适性函数的计算结果，可见舒适性函数在初始时间会急剧下降并快速趋于稳定，且越大，舒适性下降得越快，并且稳定值越低。从图10可以观察到，路面等级从A级变化到B级时，乘客舒适性函数的变化程度大于从B级到C级时，即相对较小时，乘客舒适性受到不平整跑道的影响较显著。
4　结论
收起
本文基于不平整跑道下的非线性二自由度起落架模型，采用随机模拟方法，对飞机滑跑过程中系统的非线性振动及其统计特性进行研究，通过定义系统振动响应的安全域以及乘客舒适区间，着重分析了路面不平度系数对系统响应、可靠性以及乘客舒适性的影响规律。主要结论如下：
（1） 随路面不平度系数增加，系统稳定在平衡位置的概率减小，即飞机的上、下质量块的速度及位移波动范围变大，系统响应在路面不平度系数较大的情况下变化更剧烈。相较于系统的缓冲器冲程以及下质量块位移，不平整跑道对速度的影响更大。
（2） 路面不平度系数越大，系统可靠性函数下降越快，首次穿越时间的概率密度函数峰越早出现且峰值越大，即系统发生破坏的时间越早且在峰值对应的时刻发生破坏的概率越大；且系统的平均首次穿越时间越短，即系统损坏的概率越大。
（3） 乘客舒适性随时间的增大趋于稳定，且随路面不平度系数的增加而减小，即跑道不平度越高，乘客处于不舒适状态的时间越长。随机激励在路面不平度系数较小的情况下对乘客舒适性影响较为显著。
基金
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  • 国家自然科学基金资助项目(12172286)
  • 国家自然科学基金资助项目(11872306)
  • 航空科学基金资助项目(201941053004)
文献
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2024年第37卷第3期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.03.014
  • 接收时间：2022-08-14
  • 首发时间：2026-02-10
  • 出版时间：2024-03-28
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  • 收稿日期：2022-08-14
  • 修回日期：2023-01-05
基金
国家自然科学基金资助项目(12172286)
国家自然科学基金资助项目(11872306)
航空科学基金资助项目(201941053004)
作者信息
    1西北工业大学数学与统计学院，陕西　西安　710129
    2中国飞机强度研究所，陕西　西安　710065

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许勇（1977—），男，博士，教授。电话：（029）88431637；E-mail：。
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