Structural testing and numerical simulation of a 34 m composite wind turbine blade
一类具有阻尼项的分数阶偏微分方程解的振动性
一类具有阻尼项的分数阶偏微分方程解的振动性李伟年【摘要】通过建立分数阶微分不等式,研究了一类具有阻尼项的分数阶偏微分方程解的振动性,并举例说明了主要结果的应用.【期刊名称】《滨州学院学报》【年(卷),期】2016(032)004【总页数】7页(P32-38)【关键词】分数阶偏微分方程;阻尼项;振动性【作者】李伟年【作者单位】滨州学院数学系,山东滨州256603【正文语种】中文【中图分类】O175众所周知,分数阶微分方程在工程、金融、应用数学以及非线性控制等领域都有着非常重要的应用。
近年来,分数阶微分方程理论研究取得了很大的进展,详细情况可参见专著[1-5]。
最近,关于分数阶偏微分方程的研究非常活跃,有关这方面的结果,可参见文献[6-10]及其相应的参考文献。
但是,涉及分数阶偏微分方程解的振动性的研究结果却不多,目前仅有文献[11-17]。
本文考虑以下具有阻尼项的分数阶偏微分方程解的振动性。
其中,Ω是Rn中具有逐片光滑区域∂Ω的有界区域,R+=(0,+∞),α∈(0,1)是一常数,(x,t)是u(x,t)关于t的α阶Liouville分数阶右导数,Δ是Rn上的Laplace变换。
边值条件为或者其中N是∂Ω的单位外法向量,g(x,t)是∂Ω×R+上的非负连续函数。
对于方程(1),约定下列条件总是成立:(A1) a∈C([0,∞);[0,∞)),p∈C([0,∞);[0,∞));(A2) q(x,t)。
问题(1)~(2)(或者(1)~(3))的解是指函数u(x,t)在上满足方程(1)和边界条件(2)(或者(3))。
问题(1)~(2)(或者(1)~(3))的解u(x,t)称为在上是振动的,如果它既不最终为正也不最终为负。
否则,就称为是非振动的。
定义1[2] 称为函数y(t):R+→R的α阶Liouville分数阶右积分,如果(4)式的右端在R+上是逐点定义的。
这里,α>0为一常数,Γ是通常的Γ函数。
概率论与数理统计专业攻读硕士学位研究生学术型培养方案专业代码070103
概率论与数理统计专业攻读硕士学位研究生(学术型)培养方案(专业代码:070103)一、培养目标旨在培养我国概率统计领域从事应用研究的专门人才。
具体要求如下:1、掌握当代社会主义优秀理论成果,热爱祖国,遵纪守法,品德高尚,有志于投身社会主义建设事业。
2、具有比较扎实的概率统计学理论基础,在概率统计学某研究方向上有系统的学习与研究,并能够利用概率统计理论解决实际应用问题。
3、掌握一门外国语,并能运用该门外国语比较熟练的阅读本专业的外文资料。
4、具有健康的体格和心理素质。
二、研究方向1、应用概率统计2、倒向随机微分方程与非线性期望3、量化金融与风险度量4、参数与非参数统计三、学习年限全日制硕士研究生的学制为3年,在校学习期限为2-3年。
原则上不提前毕业,对于特别优秀者,最多可提前一年。
提前毕业的硕士研究生除完成培养方案规定的课程外,必须有一篇以上SCI/CSSCI论文发表,并须经学位委员会审核通过。
所取得的科研成果均要求研究生为第一作者,作者单位需为山东大学。
四、培养方式根据宽口径、厚基础的原则,提倡按一级学科培养硕士研究生;充分利用校内外优质教育资源,鼓励研究生进行“三种经历”,实行双导师合作培养。
五、应修满的总学分数应修总学分:30,其中必修24学分(含前沿讲座与社会实践),选修6学分。
六、课程的类别及设置硕士研究生课程分为必修课与选修课两大类。
1.必修课是为达到培养目标要求,保证研究生培养质量而必须学习的课程。
必修课分学位公共课、学位基础课和学位专业课。
学位基础课一般按一级学科进行设置,学位专业课一般按二级学科设置。
选修课必须含有2门专业课。
经学校批准建设的全英语教学课程要纳入培养方案的课程体系中。
(1)思想政治理论,计3学分;(2)第一外国语,计3学分。
由学科开设的专业必修课包括:(1)专业外语,计2学分,学院考核。
重在培养研究生的学术论文外语写作和国际学术交流能力。
如学习2门及以上全英语专业必修课程(各专业培养计划课程表中所标注的全英语课程,是我院全英语系列课程项目建设中的全英语课程,但是否已开始全英语授课,需经学院审核批准后,以授课语言全英语为准),可免修专业外语,并通过申请可直接获得相应学分;(2)学位基础课2门,不少于6学分(测度与概率、高等数理统计引论、泛函分析基础、近世代数引论,四选二);(3)学位专业课2门,不少于6学分(随机过程及其应用、倒向随机微分方程及其应用、线性统计模型理论及其应用,三选二);(4)前沿讲座,计2学分;前沿讲座旨在使研究生熟悉本学科的重要学术理论和前沿性成果,提高硕士研究生参与学术活动的兴趣和学术交流能力。
水处理絮凝动力学及其效果的数值模拟
模拟全年原水水质, 考察 2 种不同几何尺寸的 搅拌桨对絮凝过程的影响。桨底到容器底部的距离
为12 mm ( 装置标准配置) 。桨1: 50 m m×40 m m× 0. 5 mm ; 桨2: 55 mm×20 m m×2 mm 。取静止沉淀 后的上清液浊度作为混凝沉淀效果的评价指标。试
验结果, 如表 1 所示。
1. 07
1. 96
25 0
1. 63
1. 00
1. 83
45 0
1. 39
0. 86
1. 59
2 容器内流场的数值模拟计算
2. 1 数值计算采用的软件与方法
采用FL UENT 6. 0 软件对杯罐内流场进行三维 数值模拟, 计算基于有限体积法, 对控制方程中的源 项作线性化处理, 在控制体内积分各控制方程, 可得 到变量在结点处的差分方程。经过差分离散后得到 复杂的方程组, 采用 T DM A 法求解, 压力与速度耦 合采用SIM PL E 算法。
28 0
解 放 军 理 工 大 学 学 报 ( 自 然 科 学 版)
第 9 卷
关数据等特点。本文尝试通过杯罐试验和对其内水 流流态的数值模拟, 研究絮凝搅拌桨的几何尺寸及 其置于容器中高度这两个水动力学因素对絮凝效果 的影响, 提出相应的合理工况。
1 混凝试验
1. 1 试验方案
1. 1. 1 试验设备、药剂与水样 主要试验设备: ZR4-6 智能型混凝试验搅拌机
液面会产生一个较小的下凹空穴, 该实验现象与计 算 流场云图( 如图 2 所示, 颜色反映水流流速的大 小) 中黑线下方的空穴相一致, N 为转速。容器内液 体流场, 如图3 所示。流体在桨叶边缘处产生强烈的 径向运动, 遇到容器壁分别向上和向下运动, 在桨的 上下方都呈现出一个循环状( 循环区) 。在桨叶区( 桨 旋转所划过的圆周区域) , 桨叶旋转推动液体, 液流 具有射流特征, 卷吸周围流体。表明计算结果能较可 靠地反映搅拌桨旋转时流体的流动状态。
涡流室直径对涡流管性能的影响
涡流室直径对涡流管性能的影响胡卓焕;杨欣【摘要】In this paper,the pressured air is used as the working fluid,and the numerical simulation was carried out for the energy separation effect of the vortex tube with different vortex chambers. The cooling,heating effects and the isen-tropic efficiency curves of the vortex tube with different structures were obtained. The research shows that:with the in-crease of the ratio of the cold flow,the cooling temperature difference and the entropy efficiency decrease gradually,while the thermal temperature difference increases gradually. When the cold flow ratio is about 0.28,the vortex tube has the max-imum unit cooling capacity. And the vortex tube with larger vortex chamber usually has better energy separation effects.%以高压气体为工质,对配有不同涡流室内径的涡流管冷热分离效应进行数值模拟研究,并获得了不同结构的涡流管随冷流比变化的制冷制热结果,以及等熵效率曲线.研究表明,随着冷流比的增加,涡流管制冷温差逐渐减小,制热温差逐渐增大,而等熵效率逐渐减小.当冷流比为0.28左右时,涡流管能获得最大的单位制冷量.通常装有较大涡流室的涡流管能获得更好能量分离效果.【期刊名称】《真空与低温》【年(卷),期】2017(023)001【总页数】6页(P52-57)【关键词】涡流室;数值仿真;能量分离;湍流强度【作者】胡卓焕;杨欣【作者单位】上海理工大学能源与动力工程学院,上海 200093;上海理工大学能源与动力工程学院,上海 200093【正文语种】中文【中图分类】TB61+9.1涡流管是一种精巧的能量分离装置,其技术起源于法国冶金师Ranque在1933年发现的“涡流管效应”[1]。
结构方程模型 交叉效度
结构方程模型交叉效度英文回答:Structural equation modeling (SEM) is a statistical method used to test and estimate the relationships between observed and latent variables. Cross-validity is an important aspect of SEM, as it helps to ensure that the measures used to assess latent variables are valid and reliable.There are two main types of cross-validity:Convergent validity assesses the extent to which different measures of the same latent variable are correlated with each other.Discriminant validity assesses the extent to which different measures of different latent variables are not correlated with each other.Cross-validity is essential for ensuring that the results of an SEM analysis are valid and reliable. If the measures used to assess latent variables are not valid or reliable, then the results of the analysis will be biased and inaccurate.There are a number of ways to assess cross-validity. One common method is to use a confirmatory factor analysis (CFA). A CFA is a statistical model that tests the relationships between observed and latent variables. By examining the fit of the CFA model, researchers can assess the convergent and discriminant validity of the measures used to assess latent variables.Another method for assessing cross-validity is to use a multitrait-multimethod (MTMM) analysis. A MTMM analysis compares the correlations between different measures of the same latent variable with the correlations betweendifferent measures of different latent variables. By comparing these correlations, researchers can assess the convergent and discriminant validity of the measures used to assess latent variables.Cross-validity is an important aspect of SEM, and it is essential for ensuring the validity and reliability of the results of an SEM analysis. Researchers should carefully assess the cross-validity of the measures used in their SEM analyses to ensure that the results are accurate and reliable.中文回答:结构方程模型(SEM)是一种用于检验和估计观测变量和潜在变量之间关系的统计方法。
基于蒙特卡罗数值模拟的大跨桥梁状态评估
公 路 交 通 科 技 Journal of Highway and Transportation Research and Development
Vol122 No18 Aug12005
文章编号 : 1002Ο0268 (2005) 08Ο0026Ο05
对模型进行随机模拟或统计抽样 , 最终利用所得结果
求出这些特征的统计估计值作为原来的数学计算问题
的近似解 。
具体进行蒙特卡罗模拟时 , 通常是利用计算机按
人们所关心和讨论的随机变量的某种分布方式产生足
够多的随机变数 , 即进行足够多的试验 , 然后对这些
随机变数进行统计推断 , 将所得的统计估计值作为随
图 3 等间隔蒙特卡罗抽样
更多的机会落入失效域内 , 改变随机变量的抽样重
心 , 这就是蒙特卡罗法的重要抽样法 , 此时的失效概
率为
∫ Pf =
+∞ -∞
I[ g ( v) ] f x ( v) Pv ( v)
Pv ( v) d v
式中 , f x ( x) 为随机变量 V 的概率密度函数 ; Pv ( v) 为
(3) 重要抽样 ( Important Sampling) 对于小概率事件的结构失效问题 , 采用直接蒙特 卡罗模拟方法将导致很大的计算量 , 为了提高抽样效 率的途径 , 需增加失效的机会 , 即令抽取的样本点有
28 公 路 交 通 科 技 第 22 卷
( - ∞, + ∞) 上的随机数可由下列关系式产生
X = m +σ·<- 1 [ rand (0 ,1) ]
(1)
在子区间 ( a , b) 上的随机数可由下列关系式产生
土木工程结构试验2章1
第2章土木工程结构试验的设计Chapter 2 Design for Civil Structural Test 目录Contents2.1 研究性试验(research oriented testing) (12)2.1.1 设计阶段(planning phase) (12)1.试件设计(specimens design) (12)2.加载方案设计(design of loading scheme) (13)3.量测方案设计(design of measurement scheme) (14)4.安全措施(safety measures) (15)2.1.2 准备阶段(preparation phase) (16)1.试件的制作(specimens fabrication techniques) (16)2.试件质量检查(specimens quality inspection) (16)3.试件及加载设备安装(specimens and loading setup assembling) (16)4.仪器的率定(calibration of instrument) (17)5.仪表的安装(instrument assembling) (17)6.设计记录表格(design of record sheet) (17)2.1.3 实施阶段(loading phase) (17)1.加载(loading) (17)2.观测与记录(observation and record) (17)2.1.4 总结阶段(data analysis and report writing phase) (17)1.数据处理(data processing) (17)2.试验分析(test result analysis) (17)3.试验报告(test report) (18)2.2 检验性试验(verification testing) (18)2.2.1 试验对象(test structure) (18)2.2.2 试验荷载的取值(determination of testing load) (18)2.2.3 检验报告(inspection report) (19)本章小结(summary) (19)思考题(problems) (19)结构试验设计是一项细致而又复杂的工作,必须严格认真地对待,任何疏忽都会影响试验结果,甚至导致试验失败或危及人身安全。
基于OpenFOAM的5×5棒束流动数值模拟
反应堆燃料组件是堆芯的主要部件,是堆 芯热工水力性能的重要影响因素之一。与研究
周期长、费用高的试验研究相比,CFD模拟已 成为安全、快速研究手段之一,因此利用CFD
收稿日期:2020-08-17;修回日期:2020-10-20 基金项目:国家自然科学基金(11675057)广东省自然科学基金(2020A1515010373) "通信作者:蔡杰进
" 2. 2 SST k 模型模拟结果与讨论
截面1位于格架下方15 mm处,截面2位 于格架中部位置,距离格架入口 15 mm处,截 面3位于格架上方搅混翼下游出口,截面4位 于远离格架出口 60 mm位置°各截面轴向速度 u示于图5°
图3横截面选取和通道A示意图 Fig. 3 Selection of plane and channel A
参数 入口流速,m/s 参考值压力,MPa 岀口平均静压,MPa 密度,kg/m3
收敛精度
夕十
数值
1 0 0
994 10"6 1.2 〜2.6
图2 3级精度网格划分 Fig. 2 Level 3 mesh
表1 网格敏感性分析
Table 1 Sensitivity analysis of mesh
网格 精度等级
截面4
由图5可见,位于格架中部的截面2,由于 单弹簧和双弹簧实体的阻碍作用,弹簧附近的 轴向流动速度较小,形成角通道流动速度大、棒 束边界流动速度小,受实体挤压导致局部流速 达到1.80 m/s0截面3位于搅混翼下游,轴向 速度在搅混翼的作用下出现规则对称的扰流 , 如图6所示。在搅混翼下游出现对称的轴向速 度分布,中心通道的流速较角通道流速大,从流 线图可看出,在搅混翼下游出现涡流,这种涡流 结构之间相互干涉,在搅混翼分布密集的中心 通道,涡流之间干涉最为明显。
靖边气田水平井适应性评价及其应用
表 3 2010 年完钻水平井开发效果 统计表
井号
水平井 含气层 有效储层
邻近直井 水平井与
无阻流量 日产气量
段长度 长度 ( m) ( m)
钻遇率 (%)
( 104m3/d)
(
104 m 3 /
平均产气量 直井
d)
(
104
3 m
/
d)
产气倍数
靖平 X- 8 1301. 0 781
60. 0
113. 96
[ 参考文献]
[ 1] 杨洪志, 陈友莲, 陈伟等. 气藏水平井开发条件 探讨[ J] . 天然气工业, 2005, 25( 增刊 A) : 95~ 99.
[ 2] 周英杰. 胜利油田稠油油藏水平井技术经济政 策界限[ J] . 新疆石油地质, 2008, 25( 1) : 76~ 78.
[ 3] 孙玉平, 陆家亮, 巩玉政等. 我国气藏水平井技 术应用 综述[ J] . 天然气技 术与经济, 2011, 5 ( 1) : 24~27.
马五1+ 2储层参数对比表
项目
水平井适应性条件 靖边气田马五1+ 2储层参数
气藏深度( m) 有效厚度( m)
气层系数 有效厚度与气层系数乘积
气层渗透率( 10- 3Lm2) 单井可采储量( 108m3) 千米深产气量( 104m3/ d)
1 00 0~4 30 0 > 6. 0 < 4. 0 < 100. 0 > 0. 2 > 2. 0 > 1. 0
关键词: 靖边气田; 水平井; 适应性; 经济界限渗透率 中图分类号: T E355. 6 文献标识码: A 文章编号: 1006—7981( 2012) 13—0124—03
simulink仿真教程计算机软件及应用it计算机专业资料课件
Block Parabmeters
Ports d Subsystems Signal Attributes Signal Routing
Open the 'Sinks'Library Sourees
User-Defined Functions
Library: siulink/Sinks
Eile Edit Yiew Tarmt Help
Parameters Numerator;
[25]
Denominator:
1347]
Absolute tolerance;
auto
0K
Cancel
Melp
spply
仿真技术
9
第九章 Simulink动态仿真
9.1.2建立Simulink仿真模型
f) 模块的连接
■ 模块之间的连接是用连接线将一个模块的输出端与另一模块 的输入端连接起来;也可用分支线把一个模块的输出端与几 个模块的输入端连接起来。
存(s)即可。
第九章 Simulink动态仿真
Eile Edit Yiew Simulation Format Iools Help
No
保存在T):
untitled adl
2s+1
3s2+s+1 Transfer Fcn
tt_et_rtw nonlinesrSystem ndl
[说明]
文件名q): 保存类型①): Simulink Models(.ma)
Initial step size
Output options
Refine oupu
Refine factor
[说明]若不设置仿真参数,则采用Simulink缺省设置.
基于SRR结构的多频带频率选择表面
第32卷 第1期2009年2月电子器件Chinese J ournal Of Elect ron DevicesVol.32 No.1Feb.2009Multi 2B and Frequency Selective Surfaces B ased on SRR Metamaterials 3J I A N G J un 1,X I E Yong 2j un13,W A N G R ui 1,YA N G R ui 1,L I D an 1,L I U Zhi 2gang21.N ational Key L aboratory of A ntennas and Microw ave Technolog y ,X i dian Universit y ,X i ’an 710071,China;2.S ichuan J i uz hou Elect ri Grou p Co.,L t d.,Miany ang ,S ichuan 621000,ChinaAbstract :Multi 2band f requency selective surfaces (FSSs )are realized by utilizing t he split 2ring resonators (SRRs )as t he basic unit cells.The magnetic resonance of SRRs ,which makes an additional dip in t he re 2flection and t ransmission spectrums of FSSs ,is discussed based on t he equivalent circuit met hod and t he simulation models.Finally ,ideal infinitely large Multi 2band FSS has been simulated by H FSS ,and t he de 2pendence of t he geometrical parameters of SRRs on t he resonance frequencies of FSSs is investigated.K ey w ords :split 2ring resonators (SRR );multi 2band ;f requency selective surface (FSS )EEACC :5270基于SRR 结构的多频带频率选择表面3蒋 俊1,谢拥军13,王 瑞1,杨 锐,李 丹,刘志刚21.西安电子科技大学天线与微波技术国家重点实验室,西安710071;2.四川九洲电器集团有限责任公司,四川绵阳621000收稿日期:2008208225基金项目:国家自然基金资助项目资助(60771040);国家重点实验室基金资助项目资助(9140C5306010604);教育部新世纪优秀人才资助项目(NCET 20420950)作者简介:蒋 俊(19822)男,西安电子科技大学博士生,主要研究方向为左手媒质、频率选择表面等,Jiangjun8273@ ;谢拥军(19682)男,教授,博士生导师,主要研究方向为左手媒质、计算微波与计算电磁学,微波通信,电磁兼容及电波传播特性等,yjxie @ ;王 瑞(19852),女,博士研究生,主要研究方向为左手媒质、频率选择表面等摘 要:为实现频率选择表面(FSS )的多频带特性,将开口谐振环(SRR )结构引入多频带FSS 的设计中。
analytical method和numerical method
analytical method和numerical method Analytical MethodIntroductionAnalytical methods are mathematical methods that are used to solve equations or problems in a closed form. Analytical methods are used when exact solutions to problems can be obtained using mathematical equations. These methods are widely used in various fields of science and engineering.Advantages of Analytical MethodsOne of the main advantages of analytical methods is that they provide exact solutions to problems. This means that the results obtained using analytical methods are highly accurate and reliable. Furthermore, analytical methods can be used to derive general formulas that can be applied to a wide range of problems.Disadvantages of Analytical MethodsOne of the main disadvantages of analytical methods is that they can only be applied to simple problems with well-definedboundary conditions. Furthermore, analytical methods may not always provide practical solutions to real-world problems due to their complexity.Examples of Analytical MethodsSome examples of analytical methods include:1. Differential Equations: Differential equations are used to describe the behavior of physical systems such as heat transfer, fluid flow, and electromagnetic fields.2. Fourier Analysis: Fourier analysis is used to decompose complex signals into simple sinusoidal components, which can then be analyzed more easily.3. Laplace Transform: The Laplace transform is used to solve differential equations by transforming them into algebraic equations.Numerical MethodIntroductionNumerical methods are mathematical techniques that are usedto obtain approximate solutions to complex mathematical problems. Numerical methods involve the use of computers and algorithms to perform calculations on large datasets and complex systems.Advantages of Numerical MethodsOne of the main advantages of numerical methods is that they can be applied to complex problems with uncertain boundary conditions, which cannot be solved using analytical methods. Furthermore, numerical methods can provide accurate results even for large datasets and complex systems.Disadvantages of Numerical MethodsOne major disadvantage of numerical methods is that they involve a degree of approximation and uncertainty in their results. Furthermore, numerical methods can be computationally expensive and time-consuming.Examples of Numerical MethodsSome examples of numerical methods include:1. Finite Element Method: The finite element method is used to solve complex problems in engineering and physics by dividingthe problem into smaller, simpler elements.2. Monte Carlo Method: The Monte Carlo method is used to simulate complex systems by generating random numbers and analyzing their behavior.3. Numerical Integration: Numerical integration is used to approximate the value of integrals that cannot be solved analytically.ConclusionIn conclusion, both analytical and numerical methods have their advantages and disadvantages, depending on the problem being solved. Analytical methods are best suited for simple problems with well-defined boundary conditions, while numerical methods are best suited for complex problems with uncertain boundary conditions. Both methods are important tools for scientists and engineers in various fields of study.。
随机扰动下统一混沌系统的有限时间同步
随机扰动下统一混沌系统的有限时间同步王娇;涂俐兰;朱泽飞【摘要】本文研究了具有随机扰动的统一混沌系统的有限时间同步问题,其中随机扰动是一维标准的维纳随机过程。
利用了有限时间随机李雅普诺夫稳定性理论、伊藤公式,本文分三个步骤设立了三个控制器获得了驱动–响应系统在有限时间内的均方渐近同步。
最后进行的数值模拟验证了理论结果的正确性和方法的有效性。
%In this paper, finite-time synchronization of the unified chaotic system with stochastic perturbation is investigated, in which the perturbation is a Wiener process of one-dimensional standards. Based on finite-time stochastic Lyapunov stability theory and Ito formula, three steps are presented to consecutively design three controllers to guarantee the finite-time mean-square asymptotical synchronization of the drive-response systems. Finally, numerical simulations are provided to illustrate the correctness and effectiveness of the theoretical results.【期刊名称】《数学杂志》【年(卷),期】2017(037)001【总页数】8页(P193-200)【关键词】随机扰动;统一混沌系统;有限时间同步;伊藤公式;李雅普诺夫稳定性理论【作者】王娇;涂俐兰;朱泽飞【作者单位】武汉科技大学冶金工业过程系统科学湖北省重点实验室,湖北武汉430065;武汉科技大学冶金工业过程系统科学湖北省重点实验室,湖北武汉430065;武汉科技大学冶金工业过程系统科学湖北省重点实验室,湖北武汉430065【正文语种】中文【中图分类】O231.3混沌现象是指发生在确定的非线性系统中的随机现象,是一种非常普遍的自然现象.例如,在流体对流实验中观测到倍周期分叉和混沌运动,在化学反应、非线性电路、光学双稳态、非线性声学、激光振荡等系统中能观测到混沌现象,甚至社会、经济领域中也存在混沌现象[1].1963年,Lorenz提出了第一个混沌模型,即Lorenz系统[2],从此开启了混沌研究的热潮.各种混沌系统相继被人们发现,如Rossler系统、Chua电路系统、Chen系统、L系统和统一混沌系统等等[3].混沌的同步问题是混沌研究中的一个重要方向.自从Pecoro和Carroll于1990年提出了混沌的反馈控制同步之后[4],越来越多的科学家、学者们开始致力于这一方面的研究并提出了各种各样的混沌控制同步的方法,如自适应同步[5]、非线性同步[6]、耦合同步[7]、投影同步[8]和H∞同步[9]等等.到现在为止,对于混沌同步,研究的主流方向是使得系统在无限时间内达到同步[4-10].但是,在处理实际情况下的混沌系统问题时,不仅要使得系统达到同步,而且要在有限时间之内达到同步,即如何在有限的时间之内使驱动–响应系统达到同步.赵等[11]研究了分数阶超混沌Lorenz系统的有限时间同步问题,Wang Liyang等[12]研究了不确定参数下两个不同的混沌系统的有限时间同步,Wang Hua等[13]研究了不确定参数下的统一混沌系统的有限时间同步.与同步时间为无穷的情况下相比,有限时间更加切合人们的实际应用,这也使得有限时间同步的研究更加受到学者们的青睐.同时,实际生活中,混沌系统不是孤立地存在于社会之中的,当然也就存在扰动,比如由于随机波动所带来的信号传输就是一个有扰动的过程.在具体的函数形式下的随机干扰项具有丰富的内容,随机扰动的产生会使得系统不稳定并难以控制,减少扰动的影响才能保持系统良好的稳定性.Salarieh等[14]研究了带有随机扰动的两个混沌系统在不确定参数下的自适应同步,Sun Yonghui等[15]研究了有时滞的情况下带有随机扰动的混沌系统的指数同步.基于以上所述,本文将研究随机扰动下的统一混沌系统的有限时间同步问题,其中随机扰动是一维标准的维纳随机过程.研究这种扰动下的混沌系统更具有一般性,更加符合实际生活.基于有限时间随机李雅普诺夫稳定性理论、伊藤公式,本文将分三个步骤设立三个控制器使得驱动–响应系统在有限时间内达到均方渐近同步.本节首先介绍随机扰动下的统一混沌系统的驱动系统和响应系统的数学模型,之后再阐述文章中所需要的预备知识.考虑随机扰动下的统一混沌系统的驱动系统为其中x1,x2,x3是状态向量,α∈[0,1]是系统参数,H(t,xi)(i=1,2,3)是非线性函数,并满足Lipschitz条件,即其中Li(i=1,2,3)是Lipschitz系数,w(t)是随机扰动,是一维标准的维纳随机过程,且满足对于系统(2.1),如果去掉随机项,则该系统是一个统一混沌系统,其中当α取不同数值时,对应的混沌系统将相应发生改变,如当0≤α<0.8时,系统(2.1)为一般Lorenz混沌系统;当α=0.8时,为L系统;当0.8<α≤1时,系统为Chen系统.再设响应系统为其中y1,y2,y3是状态向量.设控制器为u1,u2,u3,则加入控制器后的响应系统为定义响应系统(2.4)和驱动系统(2.1)之间的状态误差为本文的目标是通过设置合适的控制器u1,u2,u3,使得系统(2.1)和(2.4)达到有限时间内的均方渐近同步,即如果存在正数T>0,使得且当t≥T时,有|yi-xi|≡0,则称系统(2.1)和(2.4)在T时刻内达到有限时间内的均方渐近同步.为了获得系统(2.1)和(2.4)在有限时间内达到同步的条件,需要用到以下两个引理. 引理1(有限时间稳定性定理[16])假设存在连续的,正定的函数V(t)满足下面的微分不等式其中c>0,0<η<1,那么对于任意给定的t0,V(t)满足下面的微分不等式且当∀t≥t1,V(t)≡0,引理2(伊藤公式[17])设f(t,x(t))是关于t和随机过程{x(t),t∈T}的二次微分函数,若x(t)的随机微分是则Y(t)=f(t,x(t)),有其中由系统(2.1)和系统(2.5)以及状态误差(2.6),获得系统的误差方程为为了获得误差系统在有限时间内达到稳定,本节下面将分三步分别设置控制器u1,u2,u3.第1步设置控制器这里是一个合适的有理数,p和q都是正的奇数并且有p>q,将(3.2)式代入(3.1)式有构造李雅普诺夫函数应用引理2得由(2.3)式可知(3.5)式的均方(即数学期望)只有第一部分不为零,所以后面只需考虑第一部分,可设其中LV1是应用引理2后,去掉公式中的随机微分项获得的.由(2.2)式得则所以根据引理1,在某个T1时刻内,误差e1渐近趋于零,且当t≥T1时,e1恒等于零.第2步类似地,在第一步的基础上,设置控制器将(3.8)式代入到状态误差方程(3.1)式中,有同样地,构造李雅普诺夫函数应用引理2得到由公式(3.7)得所以根据引理1,在某个T2时刻内,误差e2也渐近趋于零,且当t≥T2时,e2恒等于零.第3步在第一步和第二步的基础上,设置控制器构造李雅普诺夫函数由引理2得由公式(3.7)得同样地,根据引理1,可知在某个T3时刻内,误差e3是渐近稳定的,且当t≥T3时,e3恒等于0.由此得到状态误差方程(3.1)在T3时刻内渐近趋于零,即驱动系统和响应系统在T3时刻内达到有限时间同步.注在施加控制器u2的时候,e1已经恒为0,本文的证明是在过了时间T1之后再添加的u2,然后过了时间T2之后施加控制器u3,这样就有T3≥T2≥T1.当然,实际操作的时候,也不一定非得这样做,可以一次性地把这三个控制器同时加上去,只求最后的时间T3.为了说明第3节中所提出的理论结果的正确性和方法的有效性,本节将运用数值模拟来验证驱动–响应系统的均方渐近同步情况.在本节中α分别取0,0.8和1.为了简单起见,设则可取Lipschitz系数L1=L2=L3=6,再设在模拟中,我们始终设驱动系统和响应系统的初始值分别为当t0=0时,,由有限时间引理1,获得理论有限时间T≈3.64秒,要注意的是这个理论值依赖于初值的选取情况.利用Matlab,下面给出驱动系统、响应系统和误差系统相对应的数值模拟图.图1、图4和图7是随机扰动下的统一混沌系统在参数α分别取0、0.8和1时驱动系统的图形,图2、图5和图8是对应的响应系统的图形,这些图形说明了系统的运动轨迹是杂乱无章的,且驱动系统和响应系统的对应轨迹不一致.图3、图6和图9是在控制器u1,u2,u3下误差系统的图形,可以看出在0.1秒左右的时间内,驱动–响应系统都达到了有限时间同步,而且比理论上计算出来的有限时间更小,体现了良好的同步性能.混沌系统的有限时间同步控制是控制理论界的热门研究方向.本文对带有随机扰动的两个统一混沌系统的有限时间均方渐近同步问题进行了研究.与同步时间在无穷的情况下相比,本文中有限时间同步问题的研究更加符合现实生活,而且考虑了随机扰动的影响,具有一定的现实意义.文中分三个步骤分别设置了三个控制器,相应地构造了三个李雅普诺夫函数,其中控制器和李雅普诺夫函数的形式相对简单,计算过程也不繁琐.文章最后进行的Matlab数值模拟表明当α取不同的数值时,对应的各个系统均体现出了良好的同步性能,验证了理论结果的正确性和方法的有效性.【相关文献】[1]陈士华,陆君安.混沌动力学初步[M].武汉:武汉水利电力大学出版社,1998.[2]Lorenz E N.Deterministic non periodic flow[J].J.Atmos.Sci.,1963,20(2):130–141.[3]陈关荣,吕金虎.Lorenz系统族的动力学分析,控制与同步[M].北京:科学出版社,2003.[4]Pecora L M,Carroll T L.Synchronization in chaoticsystems[J].Phys.Rev.Lett.,1990,64(8):821–824.[5]祝大伟,涂俐兰.随机扰动下Lorenz混沌系统的自适应同步与参数识别[J].物理学报,2013,62(5): 98–103.[6]Wu Xiaoqun,Zheng Weixing,Zhou Jin.Generalized outer synchronization between complex dynamical networks[J].Chaos,2009,19(1):013109.[7]Simonovi J.Synchronization in coupled systems with different type of couplingelements[J].Diff. Equ.Dyn.Sys.,2013,21(1-2):141–148.[8]Ma Shaojuan,Shen Qiong,Hou Jing.Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters[J].Nonl.Dyn.,2013,73(1-2):93–100.[9]涂俐兰,柯超,丁咏梅.随机扰动下一般混沌系统的H∞同步[J].物理学报,2011,60(5):584–591.[10]王建根.混沌同步研究进展综述[J].计算机工程与应用,2005,41(22):1–4.[11]赵灵冬,胡建兵,包志华.分数阶系统有限时间稳定性理论及分数阶超混沌Lorenz系统有限时间同步[J].物理学报,2011,60(10):100507–100507.[12]Wang Liyang,Xia Xiaodong,Yu Caidong,et al.Finite time synchronization between two different chaotic systems with uncertain parameters[J]rm.Sci.,2010,3(3):174–179.[13]Wang Hua,Han Zhengzhi,Xie Qiyue,et al.Finite-time chaos synchronization of unified chaotic system with uncertainparameters[J].Commun.Nonli.Sci.Numer.Simul.,2009,14(5):2239–2247.[14]Salarieh H,Alasty A.Adaptive synchronization of two chaotic systems with stochastic unknown parameters[J].Communi.Nonli.Sci.Numer.Simul.,2009,14(2):508–519.[15]Sun Yonghui,Cao Jinde,Wang Zidong.Exponential synchronization of stochastic perturbed chaotic delayed neural networks[J]p.,2007,70(13):2477–2485. [16]Chen Weisheng,Jiao Licheng.Finite-time stability theorem of stochastic nonlinear systems[J].Automatica,2010,46(12):2105–2108.[17]龚光鲁,钱敏平.应用随机过程教程及在算法和智能计算中的随机模型[M].北京:清华大学出版社, 2004.。
侏罗系弱胶含水层特征及首采区防治水方案研究
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三维自由面湍流场数值模拟及其在水利工程中的应用
河海大学博士学位论文三维自由面湍流场数值模拟及其在水利工程中的应用姓名:王志东申请学位级别:博士专业:水力学及河流动力学指导教师:汪德爟20040322摘要自然界中的流体流动问题一般都具有三维性和湍动性,对于最常见的流体一水则同时还拥有自由表面,水利工程中绕闸墩的泄洪水流、船舶工程中的船体绕流等均具有上述流动的三种特性,这三种特性是计算流体力学(CFD领域非常重要的研究内容和方向。
.本文针对粘流场数值模拟技术的研究现状和发展动态作了比较全面的回顾与展望,重点研究了网格生成技术、数值求解技术、湍流模型技术和动边界模拟技术,在此基础上建立了模拟自由面湍流场的数学模型,并成功地应用N-维溢流坝和带闸墩的三维溢流坝过坝水流的数值计算。
本文完成的具体工作和主要结论如下:1以代数网格生成方法为基础提出了一种简单的、可独立于网格生成方法之外的边界正交化技术;针对分区结构网格系统建立了分区交界面处的数据结构与计算模型: 2利用有限体积方法在非正交同位网格系统中建立了SIMPLE求解算法,对非正交网格系统中的边界条件、延迟修正技术及计算节点的梯度计算等专题进行了深入讨论。
通过对倾斜方腔驱动流、后台阶绕流、圆柱绕流等典型流场的数值模拟,表明本文建立的非正交同位网格系统中的SIMPLE算法是合理的;3研究比较了各种湍流模型的特点及应用范围,以水翼绕流场为例完成了RNGk 一£模型与标准k—s模型的计算比较,结果表明RNGk—s模型能够更好地模拟流场的湍流特性:4研究了运动边界模拟技术中的VOF方法,详细建立了一种精度更高的自由面重构模型,通过对典型运动界面的数值模拟表明,该方法能够更加准确地确定运动界面的形状和位置:5利用本文建立的非正交网格系统中自由面湍流场数学模型,对二维溢流坝过坝水流进行了数值计算,分析了不同坝顶水头下的水面线位置、坝面压力及流量、流速等水流特性,通过与典型实验资料的对比,水面线高度及坝面负压极值均与实验值具有非常好的一致性。
一种修正的Tikhonov方法求解拉普拉斯方程的柯西问题
一种修正的Tikhonov方法求解拉普拉斯方程的柯西问题熊向团;任丽婷【摘要】考虑矩形区域上拉普拉斯方程的柯西问题.使用修正的Tikhonov方法给出了该问题基于分离变量的近似解,并通过不同的参数选择规则得到了H(o)1der类型的误差估计.【期刊名称】《西北师范大学学报(自然科学版)》【年(卷),期】2018(054)005【总页数】4页(P1-4)【关键词】不适定问题;拉普拉斯方程的柯西问题;Tikhonov方法;参数选取规则;误差估计【作者】熊向团;任丽婷【作者单位】西北师范大学数学与统计学院,甘肃兰州 730070;西北师范大学数学与统计学院,甘肃兰州 730070【正文语种】中文【中图分类】O241.820 引言拉普拉斯方程的柯西问题出现在许多工程领域,如稳定状态的逆热传导[1]、心脏病学[2]等.然而这个问题是不适定的,为了解决其稳定性,需要一种有效且易于使用的数值近似方法.在过去的几十年里,已经提出了许多稳定数值近似方法来解决这类问题,如傅里叶正则化方法[3-5]、吉洪诺夫正则化方法[6-8]、截断方法[9]、扰动方法[10]、变分方法[11]、小波方法[12]、逆拟方法[13]等等.但是,这些结果主要集中于正则化参数的先验选取,进而得到误差估计,然而在正则化参数选取规则中,先验界往往难以估计,针对这种缺陷,文中同时考虑正则化参数的后验选取问题. 考虑矩形区域上拉普拉斯方程的柯西问题,从输入数据g(·):=u(·,0)处确定解u(x,y)(0<y<1).这里u(x,y)满足利用分离变量法得到问题(1)~(4)的解为(5)其中cn=g(x)sin(nx)dx.(6)从解u(x,y)可以看出这个问题的不适定性,文中用修正的Tikhonov方法在矩形区域上求解拉普拉斯方程的柯西问题,通过不同的参数选择规则得到了Hölder类型的误差估计.1 先验参数选择和误差估计假设精确数据g(x)和带噪音数据gδ(x)都属于L2(R)且满足||g(·)-gδ(·)||≤δ,(7)其中||·||为L2(R)上的范数,δ>0表示噪音水平.对于不适定问题,在精确解上做一些先验假设是有必要的,否则,正则化近似解的收敛速度可能任意慢[13].作先验假设如下:||u(·,1)||≤E,(8)这里E>0是固定的常数,则不等式等价于(9)带有噪音数据的近似解定义为其中ρn(y)是过滤函数,即这里α是正则化参数,且引理1[6] (a) 如果0≤ω≤1,0≤s≤1,α>0,则有(b)如果1≤ω≤2,0≤s≤1,α>0,则定理1 设u(x,y)是问题(1)~(4)的精确解,且是问题(1)~(4)的正则化解,假设噪音水平(7)和先验条件(8)成立,取则有如下误差估计:证明由(5),(10)式及三角不等式可得令s=e-n,则0<s<1.使用引理1可得由于因此2 后验参数选择和误差估计下面通过Morozov偏差原理选择后验正则化参数,令α满足如下方程: 这里τ>1是一个常数,α是正则化参数.进一步,有引理2 设并且0<τδ<||gδ||,则下列性质成立:(a)φ(α)为连续函数;(d)φ(α)为严格递增函数.结论是明显的,略去证明.注1:由引理2可知方程(18)存在唯一的解α.引理3 如果α是方程(18)的解,则有(20)证明由(7),(9),(13)式和三角不等式可得进而不等式(20)成立. 】引理4 如果α是方程(18)的解,则有证明由(7),(9),(14)式可得将(20)式代入(22)式可得(21)式成立. 】定理2 如下不等式成立:证明由(7)式和三角不等式可得定理3 设u(x,y)是问题(1)~(4)的精确解,且是问题(1)~(4)的正则化解,假设先验界和噪音先验水平成立,正则化参数通过偏差原理选取,则有如下估计:其中证明联合(21),(23)式及Hölder不等式可得3 结束语众所周知,每一种正则化方法都有各自优缺点,提出解决所有问题的通用方法是非常困难的,对于不同的问题,它需要给出一个合适的解,这就促使我们去研究和构建正则化方法.针对本文具体内容我们使用修正的Tikhonov 方法求解拉普拉斯方程的柯西问题,通过先验和后验参数选择得到了Hölder类型的误差估计.修正的Tikhonov 方法也可能适用于Helmholtz方程柯西问题及其他不适定问题,这也有待于我们进一步探究.参考文献:【相关文献】[1] QIAN Zhi,FU Chu-li,XIONG Xiang-tuan.Fourth order modified method for the Cauchy problem for the Laplace equation[J].J Comput and Appl Math,2006,192(2):205.[2] HALL W S,MAO X Q.Boundary element investigation of irregular frequencies inelectromagnetic scattering[J].Eng Anal Bound Elem,1995,16(3):245.[3] FU Chu-li,LI Hong-fang,XIONG Xiang-tuan.Fourier regularization method for solving a Cauchy problem for the Laplace equation[J].Inverse Problems in Science and Engineering,2008,16(2):159.[4] HO D N.A mollification method for ill-posed problems[J].Numer Math,1994,68(4):469.[5] XIONG Xiang-tuan,SHI Wan-xia,FAN Xiao-fan.Two numerical methods for a Cauchy problem for modified Helmholtz equation[J].Applied MathematicalModelling,2011,35(10):4915.[6] QIAN Zhi,FENG Xiao-li.A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation[J].Appl Anal,2017,96(10):1656.[7] 刘继军.不适定问题的正则化方法及其应用[M].北京:科学出版社,2005.[8] QIAN Ai-lin,WU Yu-jiang.A computational method for a Cauchy problem of Laplace’s equaiont[J].Appl Math and Comput,2009,207(2):478.[9] QIAN Zhi,FU Chu-li,LI Zhen-ping.Two regularization methods for a Cauchy problem for the Laplace equation[J].J Math Anal Appl,2008,338(1):479.[10] WEI Ting,CHEN Yong-Gang.A regularization methods for a Cauchy problem for the Laplace equation in an annular domain[J].Math Comput Simul,2012,82(11):2129.[11] VANI C,AVUDAINAYAGAM A.Regulared solution of the Cauchy problem for the Laplace equation using Meyer wavelets[J].Math and Comput Modelling,2002,36(9/10):1151.[12] LI Zhen-ping,FU Chu-li.A mollification method for a Cauchy problem for the Laplace equation[J].Appl Math Comput,2011,217(22):9209.[13] CHENG Hao,FENG Xiao-li.A new filtering method for the Cauchy problem of the Laplace equation[J].International Journal of Computer Mathmatics,2014,91(12):2621.。
多维正态分布的数值计算
多维正态分布的数值计算岳晓鹏;李慧慧【摘要】为了克服多维正态分布积分计算量过大的问题,引入多维正态分布积分的一种降维的变换形式,对变换后的积分运用蒙特卡洛以及拟蒙特卡洛算法进行计算求值.在此基础上,提出一种改进的方法,并做了相应的数值试验.【期刊名称】《西安文理学院学报(自然科学版)》【年(卷),期】2013(016)004【总页数】4页(P29-32)【关键词】多维正态分布;降维;蒙特卡洛;拟蒙特卡洛【作者】岳晓鹏;李慧慧【作者单位】西安电子科技大学理学院数学系,西安710071;西安电子科技大学理学院数学系,西安710071【正文语种】中文【中图分类】O242.2近年来,在科学与工程计算、统计分析和信号处理等领域经常会遇到多维正态分布的求值问题.对一维、二维的情况,大多数可以使用一些传统的数值计算方法[1-2]得到一个比较满意的值.但是对于三维甚至更高维的情况,传统算法的计算量会随着维数的增加呈几何倍的增大.为了解决这一难题,本文首先运用Genz的思想[3-4],通过一系列的转换,把积分区域从原始的积分区域转换成超立方体[0,1]m,从而将原问题转换成m-1维的问题,运用拟蒙特卡洛算法[5]进行求值.1 变换形式多维正态分布的形式其中,∑为m × m 的协方差矩阵,积分下限,积分上限为了把一般的多维正态分布转换成标准正态分布,也就是需要把积分区域从原始的积分区域转换成超立方体[0,1]m,我们引入Genz提出的一种变换方法.首先对A的协方差矩阵∑进行Cholesky分解得令X=Cy,转换成如下的形式:其中令,其中从而得出其中令,则其中由(2)可见,最内层的积分等于1,从而使原积分减少到了m-1维.因此多维标准正态分布可以变换成如下形式:其中这样,问题(1)的求解就转换为问题(3)的求解.运用蒙特卡洛或者拟蒙特卡洛算法进行进一步的求解.2 蒙特卡洛和拟蒙特卡洛算法2.1 蒙特卡洛算法对于选取区间[0,1]的一个概率分布函数 g(w1,w2,…,wm-1),使它满足当时令则即P(b)是随机变量函数的数学期望.抽选服从的N个随机样本点,计算这N个样本点处的函数值则利用中心极限定理,可以得出蒙特卡洛算法的误差为O(N-1/2).这里,O(N-1/2)只是一个概率意义上的统计结果,并不是真正意义上的绝对误差的上界.2.2 拟蒙特卡洛算法由蒙特卡洛算法可知,用N个随机点,其误差阶的平均数量级为O(N-1/2).显然,必存在这样的N个点,使其误差的绝对值不大于平均值,由此引出拟蒙特卡洛算法.该算法的主要目标是构造积分误差比平均误差显著好的点集,其积分形式与蒙特卡洛算法是一致的,只是所选取的随机数不同.对于其拟蒙特卡洛估计可表示为:虽然该式形式看起来像蒙特卡洛估计,但是此处使用的是确定性的点(xi1,xi2,…,xim-1),i∈[1,N].因此,拟蒙特卡洛算法的核心思想就是使用确定性的点代替蒙特卡洛算法中随机选取的点,从而减少误差.这里,我们采用文献[6]中提供的序列以及误差估计进行拟蒙特卡洛算法的求解.3 数值试验由文献[7]的理论可知,当相关系数较大时,协方差矩阵接近奇异时会产生较大的精度损失.为了解决这一问题,本文通过先对积分区域进行排序,把最大的积分区域放在最内层,减少最内层的误差,从而可以很大程度上减少积分值的误差,再进行cholesy分解及拟蒙特卡洛算法的求值.为将该方法与拟蒙特卡洛算法进行比较,分别对多组六维、十维的数据进行计算,仿真结果如下表所示:序号估计值误差时间估计值误差时间1 2.62312349e -002 8.07439194e -006 0.297 2.62333209e-002 2.57031605e-006 0.297 2 4.97496759e -001 4.76955889e -007 0.250 4.97480981e -001 7.03207325e -005 3.985 3 5.48123747e -001 1.64104098e -006 0.235 5.48102347e -001 7.78144094e -005 3.437 4 5.12168700e -002 6.94379142e -008 0.250 5.12167946e -002 6.69679258e -008 0.250 5 2.51149184e -001 4.61793785e -010 0.187 2.51149639e -001 5.28955459e -006 0.218序号估计值误差时间估计值误差时间1 2.49254838e -002 4.36894733e -007 0.453 2.49277454e -002 2.05852323e -005 15.719 2 1.54580431e -001 1.33293074e -005 0.4531.54595088e -001 9.91384169e -005 2.578 3 3.55658621e -001 5.18584498e -005 0.438 3.55709492e -001 8.34338008e -005 0.437 4 4.35359491e -001 5.59966862e -006 0.328 4.35396913e -001 8.10297581e -005 9.500 5 7.47189221e -001 3.02017468e -005 5.500 7.47199897e-001 7.29217114e-005 4.5164 结论通过以上数值试验,可以看出Genz的变换方法对多维正态分布的计算有非常明显的效果.在此基础上对积分区域进行排序,对排序后的计算结果与未排序的计算结果进行比较,说明在大多数情况下排序后的结果在时间以及精度上都优于未排序的计算结果.[参考文献][1] DREZNER Z,WESOLOWSKY G O.On the computation of the bivariate normal integral[J].J.Statist.Comput.Simul.,1989,35:101 -107.[2] DREZNER Z.Computation of the trivariate normal integral [J].Mathematics of Computation,1994,63:289 -294.[3] GENZ A.numerical computation of themultivariate normal probabilities[J].J.Comp.Graph.Stat,1992,1:141 -150.[4] GENZ A.Comparison ofmethods for the computation ofmultivariate normal probabilities[J].Computing Science and Statistics,1993,25,400 -405.[5]雷佳媛.关于蒙特卡罗及拟蒙特卡罗的若干研究[博士毕业论文][D].浙江大学,2003.[6] GENZ A,BRETZ F.numerical computation ofmultivariate t probabilitieswith application to power calculation ofmultiple contrasts [J].J.Statist.Comput.Simul,1999,63:361 -378.[7] GENZ A.numerical computation of rectangular bivariate and trivariate normal and t probabilities[J].Statistics and Computing,2004,14(3):251 -260.。
旋翼系统模态阻尼数值计算方法[1]
收稿日期: 2010-11-16; 修订日期: 2011-03-22 通讯作者: 王华明, 男, 教授, E-mail: hm- wang @ nuaa . edu. cn。
第3 期
王华明, 等: 旋翼系统模态阻尼数值计算方法
30 9
稳定性试验数据, 得到了模型旋翼在不同转速下的 阻尼比。
然而, 旋翼动稳定性试验风险大、成本高、耗时 长, 而且噪声、旋翼的不平衡信号、旋翼台传动系统 的固有模态等, 都可能对阻尼识别精度产生影 响[ 4] 。为了得到相应模态的阻尼, 需要在旋翼或机 体上施加激振[ 5] , 物理试验中激励点相位不一致也 会造成阻尼识别的误差。
利用坐标系之间的增广转换矩阵, 采用一种基 于递推格式的方法, 得到桨叶截面各点的位置坐标
RP S7 = ( RP 1) = S1H1 H2 H3H4H5H6H7 ( 2)
式中: Hk( k = 1, …, 6) 为第 k 个与第 k+ 1 个坐标系 之间的增广转换矩阵。
桨叶单元动能的变分为
Tb =
Numerical Method for Computing Modal Damping of Rotor System
W ang H uaming1 , W ang A nw u1 , H e Rui heng1, W ang H aow en2
( 1. Science and T echnolog y o n Rot or cr aft A er omechanics Labo rat or y, Nanjing U niv er sity o f A er onautics & A str onautics, N anjing , 210016, China ;
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Structural testing and numerical simulation of a 34m compositewind turbine bladeF.M.Jensena,*,B.G.Falzon b ,J.Ankersen b ,H.StangcaDepartment of Wind Energy,Risoe National Laboratory,P.O.Box 49,Roskilde,DenmarkbDepartment of Aeronautics,Imperial College London,South Kensington Campus,London SW72AZ,UKcDepartment of Civil Engineering,Technical University of Denmark,Lyngby,DenmarkAvailable online 12July 2006AbstractA full-scale 34m composite wind turbine blade was tested to failure under flap-wise loading.Local displacement measurement equip-ment was developed and displacements were recorded throughout the loading history.Ovalization of the load carrying box girder was measured in the full-scale test and simulated in non-linear FE-calculations.The non-linear Brazier effect is characterized by a crushing pressure which causes the ovalization.To capture this effect,non-linear FE-analyses at different scales were employed.A global non-linear FE-model of the entire blade was prepared and the boundaries to a more detailed sub-model were extracted.The FE-model was calibrated based on full-scale test measurements.Local displacement measurements helped identify the location of failure initiation which lead to catastrophic parisons between measurements and FE-simulations showed that delamination of the outer skin was the initial failure mechanism followed by delamination buckling which then led to collapse.Ó2006Elsevier Ltd.All rights reserved.Keywords:Structural testing;Wind turbine blades;Non-linear finite element analysis;Sub-modelling;Failure mechanism;Brazier effect;Anticlastic effect1.IntroductionThe size of wind turbine blades are expected to increase considerably in the future,demanding a better understand-ing of the structural behaviour on a different scale.Detailed structural behaviour must be investigated beyond the elastic range.This may include the identification of fail-ure modes which lead to ultimate collapse.Wind turbine blades have typically been optimized by experimental tests and simplified analytical methods.However,numerical simulation tools are gaining wider acceptance as they become more sophisticated in their predictive capability as well as offering a route to significantly lower develop-mental costs.2.Numerical methodsThe finite element method (FEM)has traditionally been used in the development of wind turbine blades mainly to investigate the global behaviour in terms of,for example,eigenfrequencies,tip deflections,and global stress/strain levels.This type of FE-simulation usually predicts the glo-bal stiffness and stresses with a good accuracy.Local defor-mations and stresses are often more difficult to predict and little has been published in this area.One reason is that the highly localised deformations and stresses can be non-lin-ear,while the global response appears linear for relatively small deflections.Another factor is that a relatively simple shell model can be used for representing the global behav-iour,while a computationally more expensive 3D-solid model may be necessary to predict this localised behaviour.Even with a highly detailed 3D solid model it would rarely be possible to predict deformations or stresses accurately without calibration of the FE-model.This0263-8223/$-see front matter Ó2006Elsevier Ltd.All rights reserved.doi:10.1016/pstruct.2006.06.008*Corresponding author.E-mail address:find.moelholt.jensen@risoe.dk (F.M.Jensen)./locate/compstructComposite Structures 76(2006)52–61calibration is necessary due to large manufacturing toler-ances.Features such as box girder corners and adhesive joints often differ from specifications.Geometric imperfec-tions are often seen and can cause unexpected behaviour,especially relating to the strength predictions but also the local deformations can be affected.In this paper,box girder corners were not modelled in detail using solids.Instead,the rotational stiffness of the corners was calibrated with deflections measured in the full-scale test.A big advantage of using FEM is that,once the model is set up and calibrated,complex load cases representing actual wind conditions can be analyzed.Only idealised loads can be imposed in a full-scale test and in this paper the critical flap-wise load case is evaluated.3.Full-scale test of 34m SSP-bladeA full-scale 34m wind turbine blade,manufactured by SSP-Technology A/S,was tested to failure under flap-wise loading as shown in Fig.1.The blade had passed all static and dynamic (correspond to 20years life time)tests required by the classification bodies.The SSP-blade is made of glass-epoxy pre-preg material and is designed with a load carrying box girder,see Fig.2.Earlier experience from another full-scale test [1]and FE-simulations [2]has shown that the cap deflect non-lin-early.The conclusion from these investigations is that the load carrying box girder ovalizes when the blade bends flap-wise.New equipment has been developed to record the web deflection resulting from ovalization,see Fig.3.Each web sensor can independently measure in-and outward web deflection which occurs when the box girder ovalizes or is subjected to other types of deformation.In the 8m segment near the root,effects due to change in geometry interact with conventional ovalization effects resulting in a counterintuitive non-linear response.Mecha-nisms which explain these deformations and the relevance of including these effects are presented.A combination of measurements and FE-simulation has shown that delamination of the outer skin was the initial failure mechanism followed by delamination buckling which led to collapse.4.Sub-modelling techniquesA local model,using shell and brick elements,was devel-oped for a span-wise segment of the blade.The 0–13m seg-ment is found to be the most critical part and final failure also occurred in this section.The boundary conditions used in this sub-model were based on the displacement field taken from the global FE-model.Normally,the linear dis-placement field is used in sub-modelling techniques,but in this case this technique cannot be used,since non-linear effects dominate.The explanation as to why the use ofaFig.1.Full-scale test –flap-wiseloading.Fig.2.Blade with a load carrying main spar.F.M.Jensen et al./Composite Structures 76(2006)52–6153non-linear displacementfield is required,even at moderate non-linearity,is given in this paper.5.Experimental resultsDeflections at the centre of the upper spar cap have been measured at seven different sections.In the section10.3m from the root,three chord-wise locations were measured, as shown in Fig.3.New equipment for deflection measurement was devel-oped for use inside the box girder.The equipment is shown in Fig.3and measures local web deflections with afixed reference frame in the centre of the box girder.The central reference frame is telescopic to allow for change in section height during ovalization of the box girder.6.Non-linear geometric deformationsDifferent deformation patterns were observed during the full-scale test.The blade showed three distinct deformation patterns dependent on the span-wise location(all measured from the root):–Root segment(0–4m)–Transition segment(4–8m)from root to box girder segment–Box girder segment(8–34m)Thefirst two segments will only be treated very briefly in this paper,but are included to illustrate the complexity of local deformations.Deformations in the root segment also influence the boundary conditions on the box girder seg-ment,which starts8m from the root.It should be noted that the box girder actually runs along the full length of the blade but behaves differently in the three segments describe in the following.6.1.Root segment(0–4m)Part of the web deformations are due to the gravity load, which causes the webs to show a non-symmetric behaviour before loading.This behaviour is specific to the SSP-blade, and is due to the change in box girder section geometry along the root segment.A sketch of the non-symmetric web deflection is shown in Fig.4a and measurements are shown in Fig.4b.The measurements exclude deflections due to pre-load(gravity load)because the measuring gauges were reset prior to loading of the blade.The unusual behaviour of the webs is not discussed further here but is most likely related to the constraints imposed by the stiffroot connection to the hub.In the current context,it is merely of interest to be able to apply the correct boundary conditions on the segments further away from the root.6.2.Transition segment(4–8m)A large change in cap curvature,and therefore cap height,takes place in the transition segment,see Fig.5a.Fig.4a.Sketch of non-symmetric.54 F.M.Jensen et al./Composite Structures76(2006)52–61Outward cap deformations were measured during the full-scale test as shown in Fig.5b .These outward deflections were seen on the compression side of the blade and are linked to the rapid change in cap profile in this segment.Box girder segment (8–34m).In this segment the expected ovalization caused by flap-wise bending was observed.This non-linear deformation or ‘‘flattening’’of the cross section is also known as the Brazier effect [1]and is most pronounced for long thin hollow beamssubjected to bending.The crushing pressure caused by the Brazier effect has a large effect on the web deflections and local deformation sensors have measured up to 20mm displacement at the trailing web and 11mm for leading web (Fig.6).Outer skin debonding from the box girder,see Fig.7a ,was observed after collapse,but it was not clear at which load this failure mode had occurred.A study of measured cap deflections has shown that the skin peeling starts at 92%of the ultimate load,see Fig.7c .The jump in mea-sured displacement is not due to sudden chord-wise bend-ing of the cap,instead it was caused by the skin debonding from the cap where the skin assumes more of the initial curvature.The reference frame for measuring cap deflection is attached to the outer skin rather than directly to the box girder corner.Fig.7b illustrates this observation.In Fig.7c another skin peeling jump can be observed at 97%of ultimate load.The actual maximum cap deformation of 4.3mm was obtained by extrapolation,see Fig.7c .This is the value that would have been observed if measurements were taken directly on the box girder or had debonding not occurred.The skin debonding is expected to have a major influ-ence on the final collapse.Buckling capacity of the blade is reduced dramatically once the outer skin and box girder separate.This buckling behaviour can be observed in Fig.7c at 100%load,just before failure.7.Final failure –buckling assumption 7.1.Cap and web deformationsThe cap centerline deflections were measured at four span-wise positions (8.5m,10.3m,12.5m and 15.2m)in the box girder segment from 8to 15m.A 8–15m segment of the blade is shown in Fig.8a with the upper cap centr-eline indicated.The measured cap deflections are plotted in Fig.8b .Unfortunately no measurement was taken atFig.5a.Changes in profileheight.F.M.Jensen et al./Composite Structures 76(2006)52–6155Fig.7a.Skin peeling from cap.Fig.8a.Cap centreline.56 F.M.Jensen et al./Composite Structures 76(2006)52–61Web and cap deflections in 8.5 m from root (Box girder segment)-8-6-4-20246810120%10%20%30%40%50%60%70%80%90%100%Load (% of ultimate)R e l a t i v e c a p a n d w e b d e f l e c t i on s [m m ]Web towards trailing edge Cap centerR e l a t i v e c a p a n d w e b d e f l e c t i o n s [m m ]Fig.9.Buckling collapse.F.M.Jensen et al./Composite Structures 76(2006)52–615711.5m where the final failure occurred,so cap deformation at this section is assumed.If the deformation is corrected from the skin peeling mentioned above,the cap in the lon-gitudinal direction behaves in a smooth manner until 98%load is reached.Above this load,most likely the cap starts to buckle as shown in Fig.8b (dark line –100%load).This assumption is based on two observations;pictures of the blade after collapse (as in Fig.13)and the cap measure-ment in 10.3m,12.5m and 15.2m which fit into this buck-ling pattern,see Fig.8b .The buckling assumption does not take the clamp at 13.2m into account,see Fig.9.This approximation may affect the buckling pattern since the cap is not able to buckle outwards at this location while inwards deforma-tion is unconstrained.The inwards displacement actually looks smooth and fits into the deformation pattern up to the ultimate buckling load,see the blue ()and pink ()line in Fig.8b .1The skin peeling assumption and the final buckling behaviour assumption can be confirmed by checking other measurements (strain gauges and web deflection sensors)in this segment of the wind turbine blade.Unfortunately,web sensors were only placed at 8.5m and 10.3m and no web measurements were taken at 12.5m and 15.2m.Measurements from both sections at 8.5m and 10.3m support the skin peeling assumption and the buckling assumption.At 92%load where skin peeling starts,there is no sign of sudden displacement jumps in the webs at either of the sections.Displacement jumps in the cap are seen at 10.3m but not at 8.5m.This ties in with what one would expect from skin peeling occurring at 11.5m,i.e.only cap deflection is affected at the nearest measuring station.The web towards the trailing edge shows a moder-ate increase in deflection at the 8.5m station and a much more pronounce increase at the 10.3m station as the load approaches ultimate.This does to some degree also con-firm that buckling occurs further outboard of the 10.3m station.Alternatively,the web may have failed locally lead-ing to buckling in the cap.Wing structures used in the aero-nautical industry (helicopter blades and fixed aircraft wings)are usually designed with the trailing edge web stronger than the web towards the leading edge.Perhapsthis should also be considered in the design of wind turbine blades.Another obvious difference between wind turbines blades and aircraft wings is the lack of internal ribs/bulk-heads in the wind turbine blades.This should also be con-sidered eventhough these ribs/bulkheads may add complexity to the manufacturing process.An example of how a web/bulkhead could be incorporated in the existing blade design is shown in Fig.10(Fig.11).8.Finite element resultsIn this section results from the numerical modelling (FEM)will be presented.The main focus will be on the 8–13m section,where the collapse occurred.1For interpretation of colour in Figs.1,4,5,7–10,and 12–15,the reader is referred to the web version of thisarticle.Fig.11.Rib/bulkhead to prevent crushingpressure.Fig.12a.Global 34mFE-model.Fig.12b.0–13m sub-model.58 F.M.Jensen et al./Composite Structures 76(2006)52–618.1.0–13m sub-modelCapturing the correct deformation pattern required a more detailed sub-model of the 0–13m section.The detailed sub-model includes solid elements in the webs and spar caps,see Fig.12b .The main purpose of a global FE-model of the whole blade is to provide representative boundaries for the 0–13m sub-model.Non-linear analysis is clearly necessary here due to the large deflections seen in the blade at higher loads.More interestingly,a combination of the anticlastic effect [3],which is a linear phenomenon,and the non-linear Brazier effect [4,5]seem to counteract one another at cer-tain locations along the blade.At small loads the anticlasticFig.14a.Section through a box corner.Fig.14b.Box corner used for calibration.F.M.Jensen et al./Composite Structures 76(2006)52–6159effect can cause outward cap deflection which then changes to inward deflection at higher loads as the Brazier effect (ovalization)takes over.This observation is illustrated in Figs.12a and12b.Linear analysis would simply not cap-ture deflections due to the Brazier effect(Fig.13).9.Calibration of corner stiffness in the FE-modelWorking with large wind turbine blades,made offibre composites,a large margin of production tolerances must be accepted.SSP-Technology A/S uses a relative high qual-ity production method(pre-preg without autoclave). Imperfections,such as voids,are most pronounced in areas where pre-preg layers terminate such as the corner of a box girder(Figs.14a and14b).In principle all variations could be identified by cutting the blade into small pieces,measure the local stiffness,and then make a detailed FE-model which includes all these variations.In practise this is prohibitively expensive and a simpler approach was applied.Box girder corners inthe FE-model were only represented with a coarse mesh of layered shell elements,which of course do not give the true picture of the corner stiffness.However,the modulus of an‘‘adjustable’’element was tuned to obtain the best fit with the actual deflections measured in the full-scale test.parison of experimental test and numerical modellingComparing experimentally measured cap deflections with FE predictions,the agreement is found to be accept-able following calibration of the corner stiffness,see Fig.15a.The web deflections,however,are not satisfactory above80%load,see Fig.15b.It was not possible to obtain better results by further adjustment of the corner stiffness and other improvements must be considered in the future work.It appears that a geometrical softening mechanism or material non-linearity exists in the actual blade.This is not accounted for in the FE-model hence good correlation is seen at lower loads while the FE-model is over-stiffat higher loads approaching ultimate strength.Also,a kink is evident on the FE deflection curves at60%load which is hardly noticeable on the curves obtained from the test. If certain features in the FE-model are too stiffit is conceiv-able that localized post buckling or snap-through has a more pronounced effect on the lateral deflections of the box girder.It should be mentioned that the overallflap-wise blade deflection shows exact agreement between test and FE-model.This is also the case for natural vibration modes,which are not included in this paper.If the accuracy of the local FE-results are to be improved,manufacturing imperfections must be included in the FE-model.For the blade in question,the section at 10.3m turned out to have a relatively large geometric imperfection in the load carrying cap which one would expect to influence local deformation behaviour.11.ConclusionNon-linearfinite element analyses of a34m wind tur-bine blade with a load carrying box girder and loaded in flap-wise bending have revealed a variety of behaviour along the span.The Brazier effect dominates outwards of the8m section.This non-linear effect was also found to dominate at the13m section where the boundary condi-tions from the global FE-model were applied to a local FE-model.This ties in with the requirement for a non-lin-ear global FE-model.Simulation of local behaviour relied on calibration of the FE-model,mainly due to the variations in production. Especially corners in the box girder show large variations, hence the sub-model did not includefine details,but was calibrated against measured local deformations from the full-scale test.60 F.M.Jensen et al./Composite Structures76(2006)52–61Measurements supported by FE-results also show that debonding of the outer skin was the initial failure mecha-nism followed by delamination buckling which led to col-lapse.When the skin debond reaches a certain size the buckling strength of the load carrying laminate becomes critical andfinal collapse occurred.12.Future workThe skin debonding failure,observed in the full-scale test,will be investigated further using computational meth-ods.It is intended to use fracture mechanics based cohesive interface elements in afinite element model.These elements are capable of predicting both initiation and propagation of a delamination.Extensive delamination was observed in the failed blade and this failure mode was due to interlaminar tensile stress caused by straightening of the curved caps.This deforma-tion is a result of the crushing pressure from the Brazier effect.This failure mode will be studied in more detail also by the use of cohesive elements in afinite element model.Finally,in this paper only the displacement measure-ments are presented but strain measurements were also recorded throughout the loading history and will be com-pared with the FE-results.References[1]Jørgensen ER,Borum KK,McGugan M,Thomsen CL,Jensen FM,Debel CP.Full scale testing of wind turbine blade to failure–flapwise loading.Risø-R-1392(EN)ISBN87-550-3184-6;ISBN87-550-3185-4 (Internet)ISSN0105-2840.[2]Jensen pression strength of afibre composite main spar in awind turbine blade.Risø-R-1391(EN)ISBN87-550-3184-6;ISBN87-550-3185-4(Internet)ISSN0105-2840.[3]Swanson SR.Anticlastic effects and the transition from narrow to widebehaviour in orthotropic p Struct2001;53:449–55.[4]Brazier LG.Theflexure of thin cylindrical shells and other‘thin’sections,Late of the Royal Aircraft Establishment.Reports and Memoranda,No.1081(M.49),1926,p.1–30.[5]Cecchini LS,Weaver PM.The Brazier effect in multi-bay aerofoilsections,University of Bristol,UK.F.M.Jensen et al./Composite Structures76(2006)52–6161。