Kirchhoff板和Mindlin板上动态分布载荷的辨识问题的研究
摘要
众所周知,载荷辨识问题一直是工程及科学上极具研究价值的问题之一,历来受到众多研究学者的关注。在诸如航空航天、船舶工程、汽车、建筑、桥梁等实际工程领域中,对于实体结构的承重承载能力的研究无疑是至关重要的。准确获知结构表面的载荷分布对于改善及优化器械结构和制造材料都有重大意义。然而现实情况及经验表明并非所有结构上的任意性质的载荷都能通过相关仪器设备测量得到,此时我们希望通过某些容易测量得到的相关量如应变或位移等来反演载荷,问题的关键在于如何建立测量信息和载荷之间的联系。以往的研究大多集中在结构表面的冲击载荷的辨识问题,而对于动态分布载荷的辨识问题的研究则较少见。本文采用反分析技巧研究了Kirchhoff板和Mindlin板上动态分布载荷的辨识问题。我们知道板是一种非常重要的结构元件,其中Kirchhoff板是古典意义下的薄板,而Mindlin板是厚板理论的基础,选择这两种板作为研究对象具有一定的代表性和普遍性。求解一个一般的反问题,如何选取和得到合适的附加信息甚为关键,本文选择板表面的应变作为载荷反演的附加信息,因为实验中应变通过应变测量仪容易测量得到。但本文所使用的附加信息数据事实上是通过计算而非测量得到的,因此在载荷反演之前需要解一个板的强迫振动问题(正问题)。Rayleigll-Ritz法是目前最常用、较为成熟的解决振动问题的近似方法之一。本文在使用Rayleigh-Ritz法求解Kirchhoff矩形板和Mindlin矩形板的振动问题过程中,分别选取了正交的Bernouilli-Euler梁函数和Timoshenko梁函数作为未知变量展开的基函数。利用Rayleigh-Ritz法求解这一一问题的一大优势是最后能够得到作为附加信息的应变的解析表达式,这无疑是极为方便的。Kirchhoff板的振动控制系统在空间上是关于一个自变量的四阶偏微分方程,而Mindlin板的振动控制系统在空间上是一个关于三个未知变量的二阶偏微分方程组。为了反演右端载荷项,需要对这样的微分系统在空间上进行离散。本文分别采用有限元法和拟谱方法将Kirchhoff板和Mindlin板的连续微分控制系统均离散为一个易解的代数系统。根据板理论中位移-应变关系,得到了联系未知载荷和已知应变的转换函数的显示表达式,然而我们发现这是一个病态矩阵,通过高斯消元法或矩阵求逆而直接得到的解的精确性和稳定性是无法保证的,因此本文借助传统的Tikhonov正则化方法求解这一病态系统,其间使用了L-curve准则选择正则化参数。另外,本文的研究对象是一个时间及空间域上的三维问题,右端未知载荷亦是动态的,为此引入Laplace变换及数值Lapalce逆变换处理系统的动态行为。最后对应于上述分析分别作了两种方板上动态载荷辨识的数值例子,结果表明使用正则化方法得到的解比直接通过矩阵求逆得到的解更加准确,抗噪能力更强,验证了本文提出的动态载荷反演策略是成功的。
It is well-known that identification of load distributions has always been a worthy problem in science and engineering which catches researchers' much attention. Researches on loading capacity of structures are very important in the fields of aviation, spaceflight, watercraft, automobile, building and bridges et al. So it is significant for improving and optimizing structures and its materials to make clear exactly of load distributions applied to the structures. However, not any kind of load applied to any structure can be measured through some special instruments according to actual experiences. In the circumstances we hope to identify load distributions through correlative quantities like strains and displacements etc that can be measured conveniently. And the key point is how to set up the relationship between the measured information and the unknown load distributions. The former researches on this problem mostly focused on identification of impact forces applied to structures, while the problem of identification of dynamic load distributions has been studied singularly as far as know. The problem of identification of dynamic load distributions applied to Kirchhoff plate and Mindlin plate by inverse analysis is studied in this paper. Kirchhoff plate is also called classic thin plate and Mindlin plate is one basic kind of thick plate, and it is representative and universal to select the two typical kinds of plates as research objects. How to select and obtain suitable additional information is very important to a common inverse problem. The strains on the surface of plate, which are easily measured by strain-gauges during experiments, are selected as additional information for load identification in this study. Yet the additional information data used in the paper are obtained through calculation rather than direct measuring, which means necessary to solve a forced vibration problem of plate (direct problem) before load identification. Rayleigh-Ritz method is a very popular and efficient approximate approach for vibration problems. In the procedure of solving the vibration problems of Kirchhoff rectangular plate and Mindlin rectangular plate by means of Rayleigh- Ritz method, the orthogonal Bernouilli-Euler beam functions and Timoshenko beam functions are employed as the base functions to expand the unknown variables. Moreover another advantage of applying Rayleigh-Ritz method to the direct problem is that we could get the exact analysis expressions of strains, which is no doubt much convenient. The control vibration system of Kirchhoff plate is a four-order partial differential equation about one independent variable in space, while that of Mindlin plate is a two-order system of partial differential equations about three independent variables in space. For identifying the load distributions in right hand, it is necessary to discretize such differential systems in space. The finite element method and pseudospectral method are applied respectively to discretize the continuous differential control systems of Kirchhoff plate and Mindlin plate into a simple algebraic system. According to the relationship between displacement-strain in plate theories, we obtain the transform function connecting the unknown load vector
It is well-known that identification of load distributions has always been a worthy problem in science and engineering which catches researchers' much attention. Researches on loading capacity of structures are very important in the fields of aviation, spaceflight, watercraft, automobile, building and bridges et al. So it is significant for improving and optimizing structures and its materials to make clear exactly of load distributions applied to the structures. However, not any kind of load applied to any structure can be measured through some special instruments according to actual experiences. In the circumstances we hope to identify load distributions through correlative quantities like strains and displacements etc that can be measured conveniently. And the key point is how to set up the relationship between the measured information and the unknown load distributions. The former researches on this problem mostly focused on identification of impact forces applied to structures, while the problem of identification of dynamic load distributions has been studied singularly as far as know. The problem of identification of dynamic load distributions applied to Kirchhoff plate and Mindlin plate by inverse analysis is studied in this paper. Kirchhoff plate is also called classic thin plate and Mindlin plate is one basic kind of thick plate, and it is representative and universal to select the two typical kinds of plates as research objects. How to select and obtain suitable additional information is very important to a common inverse problem. The strains on the surface of plate, which are easily measured by strain-gauges during experiments, are selected as additional information for load identification in this study. Yet the additional information data used in the paper are obtained through calculation rather than direct measuring, which means necessary to solve a forced vibration problem of plate (direct problem) before load identification. Rayleigh-Ritz method is a very popular and efficient approximate approach for vibration problems. In the procedure of solving the vibration problems of Kirchhoff rectangular plate and Mindlin rectangular plate by means of Rayleigh- Ritz method, the orthogonal Bernouilli-Euler beam functions and Timoshenko beam functions are employed as the base functions to expand the unknown variables. Moreover another advantage of applying Rayleigh-Ritz method to the direct problem is that we could get the exact analysis expressions of strains, which is no doubt much convenient. The control vibration system of Kirchhoff plate is a four-order partial differential equation about one independent variable in space, while that of Mindlin plate is a two-order system of partial differential equations about three independent variables in space. For identifying the load distributions in right hand, it is necessary to discretize such differential systems in space. The finite element method and pseudospectral method are applied respectively to discretize the continuous differential control systems of Kirchhoff plate and Mindlin plate into a simple algebraic system. According to the relationship between displacement-strain in plate theories, we obtain the transform function connecting the unknown load vector
引文
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[21] A.W.Leissa (1973). The free vibration of rectangular plates. J.Sound Vib.,31,257-293.
[22] R.D.Mindlin, A.Schacknow and H.Deresiewicz (1956). Flexural vibrations of rectangular plates. ASME J.Appl.Mech.,23,430-436.
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[25] H.M.Nelson (1978). High frequency flexural vibration of thick rectangular bars and plates. J. Sound Vib., 60,101-118.
[26] S.Wang and D.J.Dawe (1993). Vibration of shear-deformable rectangular plates using a spline-function Rayleigh-Ritz approach. Int.J.Num.Meth.Eng.,36,695-l\\.
[27] K.M.Liew, C.M.Wang, Y.Xiang and S.Kitipornchai (1998). Vibration of Mindlin plate, programming the p-version method. Elsevier Amsterdam.
[28] K.M.Liew, J.B.Han and Z.M.Xiao (1997). Vibration analysis of circular Mindlin plates using differential quadrature method. J.Sound Vib., 205(5),617-630.
[29] K.M.Liew, K.C.Hung and M.K.Lim (1994). Free vibration studies on stress free three-dimensional elastic solids. ASME J.Appl.Mech.
[30] K.M.Liew, Y.Xiang and S.Kitipornchai (1995). Research on thick plate vibration: a literature survey. J.Sound WA.,180(l),163-176.
[31] H.Inoue, R.Watanabe, T.Shibuya, and T.Koizumi (1989). Measurement of impact force by the deconvolution method. Trans. JSNDI, 2,63-73.
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[35] T.Hosono (1979). Numerical inversion of Laplace transform. Trans.Inst.Elect.Eng.Jpn.,(in Japanese),99-A,494.
[36] R.Piessens and R.Huysmans (1984). Algorithm 619: Automatic numerical inversion of the Laplace transform. ACM Trans.Math.Soft.,10,384.
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