【作者】 江湘清；

【导师】 胡海岩；

【作者基本信息】 南京航空航天大学 ， 固体力学， 2009， 博士

【摘要】 本文研究Euler-Bernoulli梁和弹性薄板上稳态和瞬态的分布动载荷的识别问题,期望找到简单却重要的性质或规律,并谋求建立新颖、简洁、有效的识别理论和方法。由于实际测量所得数据总是空间上极为有限的离散信息,远比时间上获得的信息要少,通常达不到有限元法划分单元所需要的基本的节点数,因此应用有限元的方法不太合适;同时,线弹性系统上的分布动载荷识别问题本质上是无限维问题。因此,为近似识别出整个空间上动载荷的分布情况,需要应用模态方法进行空间坐标的变换,并应用有限维近似。由于测量数据为有限的离散信息,无法直接应用Galerkin投影法进行有限维近似,而需要将Galerkin投影法和配置投影法结合进行有限维近似,本文将其命名为“近似投影法”。线弹性系统上的分布动载荷识别是不适定问题,而不适定问题存在信息不足的困扰,通常需要寻找附加信息。作者认为响应的时间信息和空间信息之间一定隐含着重要的关联信息,即使所获得的响应信息是部分而离散的。当Euler-Bernoulli梁承受单模态分布的简谐动载荷时,其动载荷和响应间呈现出简单而重要的时空关系,由此本文从中提出“缩放因子”的概念。自然界生物的各种感知器官本质上都是识别系统,而其敏感区域总是有限却各不相同。结合这种自然而合理的有限性识别和“重要的事物一定会产生重要的影响”的想法,本文提出“识别有限性”的假设。融合这种假设和缩放因子概念,从而提出“模态选择法”并建立线弹性系统上稳态分布动载荷的识别理论。“近似投影法”和“模态选择法”实质上是对线弹性系统的分布动载荷识别这个无限维不适定问题,进行物理概念上的策略化的正则化处理;如有必要,之后还可以运用其它正则化方法。在线弹性系统的固定边界附近,分布动载荷识别有很大难度。如果采用系统固有模态或其修改型为基函数来表达动载荷的空间分布,那么固定边界处的动载荷将无法识别。为此,作者提出动载荷空间分布的“一致性表述”概念,并初步尝试运用Legendre多项式进行表达,在固定边界处获得很好的效果。小波变换具有分析局部信息的优势,但是对于动力学系统的输入、输出信息的变换没有简单好用的性质,因此应用起来很困难。作者提出“小波近似法”识别瞬态动载荷,获得不错的效果。同时,从非零初始条件的响应中识别瞬态动载荷一般都有困难,作者提出“分段识别”的想法,在小波近似法中获得很好的成功。运用上述方法和概念,本文建立了Euler-Bernoulli梁和弹性薄板上稳态分布动载荷的识别理论,以及Euler-Bernoulli梁上的瞬态分布动载荷的识别理论。数值模拟显示新理论可给出很好的识别结果,并可揭示出若干新问题。更多还原

【Abstract】 This dissertation deals with the reconstruction of distributed dynamic loads on Euler-Bernoulli beam and elastic thin plate, to find simple but important properties or principles underlying the problem, and to propose new, concise and effective reconstruction theory.The finite element method cannot find its application in the situation, since the discrete data from engineering measurement is usually on only a small number of spatial points, which is much less than the data on time domain. However, the reconstruction of distributed dynamic loads on a linear elastic system is originally a problem involving infinite dimensions. Therefore, to approximately reconstruct the spatial distribution of the dynamic loads, both modal transformation and finite dimensional approximation are employed. Both Galerkin method and collocation method are applied together as projection methods to transform the original infinite dimensional problem into a finite dimensional one, and this coupling application is named approximate projection method.The reconstruction of distributed dynamic loads on a linear elastic system is an ill-posed problem, which always needs additional information due to lack of information. The authors believe that some important connection lies between the spatial data and the temporal data in the partial discrete response data. When an Euler-Bernoulli beam is excited by a harmonic load with single mode-shape distribution, the simple and important relationship finally emerges between the spatial data and the temporal data during the transformation from load to response, and the concept of scale factor is thus proposed.Animals’perceptual organs are naturally identification system, all of which are sensitive in finite but different bandwidth. Combing the naturally reasonable limit and the idea that important things will cause important effect, it is naturally to infer that finite reconstruction is also sensible. Based on the recognition and the concept of scale factor, a new reconstruction theory is found with the proposition of the mode selection method. The approximate projection method and the mode selection method are applied basically as physical regularization strategy to deal with the infinite-dimensional and ill-posed reconstruction problem, before applying other purely mathematical regularization methods.Load reconstruction near the fixed spatial boundaries is usually hard. The loads near the fixed spatial boundaries are impossible to be correctly reconstructed, if the natural modal functions or their modified forms are applied as base functions to express the spatial distribution of the load as generalized Fourier series. To tackle this problem, the concept of consistent spatial expression is proposed, and Legendre polynomials are applied as the consistent spatial base functions, which result in good effect in numerical simulations.Wavelet transform is not conveniently applied on the load reconstruction of a dynamic system, since it has no good and simple properties on the relationship between the input and output of a dynamic system, though it dose have profits on local analysis of signals. Nevertheless, the authors proposed the method of wavelet approximation to reconstruct the distributed dynamic loads on linear elastic system, and relatively good results are obtained. Meanwhile, as difficulties usually confront researchers identifying transient force from response with non-zero initial conditions, the method of fragment analysis is proposed to deal with this problem, and the method succeeds cooperating with the wavelet approximation method.Based on abovementioned methods and concepts, the reconstruction theories of distributed dynamic loads on both Euler-Bernoulli beam and elastic thin plate are proposed. The numerical simulations show good accordance with the theories, and many new phenomena are disclosed.更多还原