2020年1月1日下午3时，课题组博士研究生王喆在北京理工大学中关村校区宇航楼509室进行学位论文答辩。答辩主席为北京航空航天大学航空科学与工程学院固体力学研究所邱志平教授，答辩委员有北京大学工学院航空航天工程系刘才山教授、清华大学航天航空学院王天舒教授、北京信息科技大学理学院戈新生教授、北京理工大学龙腾教授。答辩主持人为北京理工大学宇航学院罗凯讲师。
王喆博士师从胡海岩教授，主要研究方向为不确定性多体系统动力学，博士论文题目为《含区间不确定性参数的多体系统动力学研究》。经过一个半小时的汇报和提问环节，王喆博士获得了答辩委员会的一致认可，期间也对多个该领域前沿问题进行了热烈的讨论。最后，经过答辩委员会投票表决，一致通过王喆同学的博士学位论文答辩，并建议授予工学博士学位。
论文摘要：
近些年来，国内外学者围绕多体动力学开展了大量研究。现有研究大多针对仅含确定性参数的多体系统动力学，较少涉及含不确定性参数的多体系统动力学。而在工程中，多体系统不可避免地会含有不确定性参数，例如：不确定性密度、不确定性杨氏模量以及不确定性几何参数等等。忽略这些参数不确定性的多体系统动力学分析可能无法反映系统的真实动力学。
含不确定性参数的多体系统动力学研究仍存在很大挑战，如不确定性参数的描述方式以及求解大规模不确定性动力学方程的高效数值算法等。区间方法作为一种非概率方法，可处理含不确定性有界的区间参数问题，且获取区间参数的上下界比获取随机参数的概率密度函数更容易。
本文以问题为导向，基于Chebyshev代理模型对含区间不确定性参数的多体系统动力学进行求解与分析。本文主要学术贡献如下：
1) 提出了求解含区间不确定性参数的多体系统动力学方程组的非嵌入式计算方法。用Chebyshev代理模型，将区间非线性方程组转化为若干组含确定性采样参数的非线性方程组。为避免区间爆炸问题以及保持计算效率，利用扫描方法来获得Chebyshev代理模型的结果响应区间的上下界。
2) 研究了含区间间隙尺寸运动副的柔性多体系统动力学。对间隙运动副模型使用ANCF-RN建模。对比研究了修正的Coulomb摩擦模型以及LuGre摩擦模型。通过分析使用Chebyshev代理模型获取的不确定性动力学的区间响应，研究了其可能存在的分岔现象。
3) 提出了含不确定性区间参数的柔性多体系统动力学一阶灵敏度分析方案。推导了直接微分法以及伴随变量法，得到的动力学方程以及灵敏度方程均由区间微分-代数方程组描述。通过Chebyshev代理模型获得了其区间灵敏度。
4) 对于很多工程系统，随机参数的均值与方差本质上是区间参数。因此，将含区间均值与区间方差的随机参数作为一种混合不确定性参数，并研究该混合不确定性参数对系统动力学的影响。将摄动法与Chebyshev代理模型方法相结合，发展了一种计算含上述混合不确定性参数的多体系统动力学方程的方法。
5) 某些含不确定性参数的系统可能含多种动响应模式，仅用单一代理模型无法描述该类系统的所有动响应模式，因此提出了一种计算不确定性多体系统多种动响应模式的方法。研究结果表明，该方法可正确描述某些不确定性系统可能存在的多种动响应模式。
Abstract:
The recent decades have witnessed numerous studies on the dynamics of multibody systems. Most previous studies have focused on the dynamics of multibody systems with deterministic parameters so far, while very few studies have dealt with the dynamics of multibody systems with uncertain parameters. However, the practical multibody systems inevitably contain uncertain parameters, such as the uncertain material density, uncertain Young's modulus, uncertain geometrical parameters, and so on. Therefore, the dynamic analysis of multibody systems without taking these uncertain parameters into account cannot reflect their true dynamic behaviors.
To study the dynamics of the multibody systems with uncertain parameters, the main challenges are the description of the uncertain parameters and the solution of the huge sets of uncertain dynamic equations. The interval algorithm is one of the non-probabilistic methods and mainly used to study the problems with uncertain, but bounded interval parameters. It is usually easier to get the bounds of the interval parameters than to obtain the probability density functions of random parameters.
This dissertation presents a problem-oriented research, based on the Chebyshev surrogate models to preform the compution and analysis of dynamics of multibody systems with uncertain interval parameters. The major contributions of the dissertation can be summarized as follows.
1) A non-intrusive computation methodology is proposed to study the dynamics of multibody systems with a large number of uncertain interval parameters. The Chebyshev surrogate models are used to transform the nonlinear interval algebraic equations into sets of nonlinear algebraic equations with deterministic sampling parameters. To circumvent the interval explosion problem and maintain computation efficiency, the scanning method is used to determine the upper and lower bounds of the deduced Chebyshev surrogate models.
2) The nonlinear dynamics of flexible multibody systems with interval clearance size joint is studied. The kinetics model of the revolute clearance joints is formulated by using ANCF-RN coordinates. The influence of the LuGre and the modified Coulomb’s friction models on the system dynamic responses is comparatively studied. Furthermore, by analyzing the bounds of dynamic response obtained by scanning the Chebyshev surrogate model, the bifurcation diagrams are also observed.
3) A computation method is proposed to perform the first order sensitivity analysis of multibody systems with uncertain interval parameters. The direct differentiation method and adjoint variable method are performed to do sensitivity analysis, and the dynamics and sensitivity equations with uncertain interval parameters are both governed by Interval Differential Algebraic Equations (IDAEs). The Chebyshev surrogate models are used to obtain the interval sentivities.
4) For many engineering systems, the mean values and variances of those random parameters are inherently interval parameters. A computation method is proposed to study the dynamics of multibody systems with hybrid uncertain parameters, i.e., the random parameters with interval mean values and interval variances. The perturbation-based method together with the Chebyshev surrogate model is developed in order to incorporate hybrid uncertain parameters in the dynamic analysis of flexible multibody systems.
5) The mechanisms with uncertain parameters may exhibit multiple dynamic response patterns. As a single surrogate model can hardly describe all the dynamic response patterns of mechanism dynamics, a computation method is proposed to study multiple dynamic response patterns of a multibody system with uncertain parameters. The results computed by using the proposed computation method can correctly reflect multiple dynamic response patterns of the uncertain system.
博士期间发表论文：
[1] Zhe Wang, Qiang Tian, Haiyan Hu. Multiple dynamic response patterns of flexible multibody systems with random uncertain parameters[J]. Journal of Computational and Nonlinear Dynamics, (2019) 14(2): 021008.
[2] Zhe Wang, Qiang Tian, Haiyan Hu. Dynamics of flexible multibody systems with hybrid uncertain parameters[J]. Mechanism and Machine Theory, (2018) 121: 128-147.
[3] Zhe Wang, Qiang Tian, Haiyan Hu, Paulo Flores. Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance[J]. Nonlinear Dynamics, (2016) 86:1571-1597.
[4] Zhe Wang, Qiang Tian, Haiyan Hu. Dynamics of spatial rigid-flexible multibody systems with uncertain interval parameters[J]. Nonlinear Dynamics, (2016) 84:527-548.
[5] Zhe Wang, Qiang Tian, Haiyan Hu. Dynamics of spatial flexible multibody systems with interval probabilities[C]. 8th ECCOMAS Thematic conference on MULTIBODY DYNAMICS, Prague, June 19-22, 2017.
[6] Zhe Wang, Qiang Tian, Haiyan Hu. Dynamics study and sensitivity analysis of flexible multibody systems with interval parameters[C]. ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. 2016:V006T09A052.
[7] 王喆, 田强, 胡海岩. 含混合不确定性参数的柔性多体系统动力学研究[C]. 第十届全国多体动力学与控制暨第五届全国航天动力学与控制学术会议. 成都, 2016.
[8] 王喆, 田强, 胡海岩. 含有区间参数的柔性机械臂动力学仿真与灵敏度分析[C]. 第二届可展开空间结构学术会议. 北京, 2016.
[9] 王喆, 田强, 胡海岩. 含有区间不确定性参数的柔性多体系统动力学析[C]. 第九届全国多体系统动力学暨第四届全国航天动力学与控制学术会议. 武汉, 2015.
毕业博士简介：
王喆，男，1991年4月生于吉林省白山市。2010年9月考入北京理工大学飞行器设计与工程专业，2014年7月毕业并获得学士学位。2014年9月起，进入该校力学专业攻读博士学位，从事多体系统动力学研究，导师胡海岩教授。
