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Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

机译：底部驱动的非线性柔性梁的哈密顿建模和屈曲分析

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摘要

The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson's effect can be accounted in modelling by the use of a two-dimensional stress-strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions.
机译：在许多领域，包括微和纳米技术设备中，越来越多地使用梁和类似的结构元件。为了进行屈曲分析和控制，在对这些挠性结构进行建模时，必须考虑应变中的非线性项。此外，可以通过使用二维应力-应变关系在模型中考虑泊松效应。本文研究了细长的垂直梁（在无夹持状态下）的屈曲效应，该梁在固定端具有水平致动，在自由端具有尖端质量。包括梁的不可伸缩性约束，可以得出运动方程。对该系统进行了初步的模态分析，以描述候选的屈曲后构型并研究这些平衡的稳定性。光束在重力作用下的垂直构型不失一般性，因为目标是对确定平衡的势场建模。忽略不可扩展性约束，然后将运动方程式以汉密尔顿形式移植，并适当定义流量和作用力，作为保留结构离散化和模拟的基础。最后，对有限元模型进行仿真，以获得不同载荷条件下叶尖质量的时间响应。

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