论文标题
控制和稳定几何精确的光束
Control and stabilization of geometrically exact beams
论文作者
论文摘要
我们研究了良好的稳定度，稳定和控制问题，这些问题涉及可能会发生大幅度运动的光束 - 即参考线的大型位移和横截面的大旋转。这种光束可剪切且非常灵活，通常称为几何精确的光束，在现代高度灵活的轻质重量结构中尤其需要，在那里，人们无法忽略这些大型运动。我们从两个角度查看这些光束。第一个观点是，其中梁是根据其参考线的位置和横截面的方向描述（在某些固定坐标系中表达）的。由于Eric Reissner和Juan C. Simo，这是通常遇到的模型。在时间和空间上的二阶顺序，它是六个方程式的准线性系统。第二个视角是，梁被固有变量（速度，应变或内部力量和力矩）描述为一种视角，这些变量在附着在光束上的移动坐标系中表示。该系统由Dewey H. Hodges以最通用的形式提出，由两倍的方程组成，但在时空，双曲线和仅半线性（二次）（二次）。从后一个模型的状态的定义来看，可以看到这两种观点都是通过非线性转换链接的。由固有模型控制的梁解决了良好的问题，稳定和控制的问题，而通过使用转换，我们还证明了对内在模型的经典解决方案的存在和独特性意味着以位置和旋转方式编写的模型的经典解决方案。特别是，这使我们能够推断后一种模型的相应结果。我们还通过刚性接头彼此相连的光束网络解决了这些问题。
We study well-posedness, stabilization and control problems involving freely vibrating beams that may undergo motions of large magnitude -- i.e. large displacements of the reference line and large rotations of the cross sections. Such beams, shearable and very flexible, are often called geometrically exact beams and are especially needed in modern highly flexible light-weight structures, where one cannot neglect these large motions. We view these beams from two perspectives. The first perspective is one in which the beam is described in terms of the position of its reference line and the orientation of its cross sections (expressed in some fixed coordinate system). This is the generally encountered model, due to Eric Reissner and Juan C. Simo. Of second order in time and space, it is a quasilinear system of six equations. The second perspective is one in which the beam is rather described by intrinsic variables -- velocities and strains or internal forces and moments -- which are moreover expressed in a moving coordinate system attached to the beam. This system, proposed in its most general form by Dewey H. Hodges, consists of twice as many equations, but is of first order in time and space, hyperbolic and only semilinear (quadratic). From the definition of the state of the latter model, one can see that both perspectives are linked by a nonlinear transformation. The questions of well-posedness, stabilization and control are addressed for beams governed by the intrinsic model, while by using the transformation we also prove that the existence and uniqueness of a classical solution to the intrinsic model implies that of a classical solution to the model written in terms of positions and rotations. In particular, this enables us to deduce corresponding results for the latter model. We also also address these questions for networks of beams attached to each other by means of rigid joints.
