This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including -dimensional elasticity models (where = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

中文翻译：

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多维柔性线性机械系统端口哈密尔顿建模的系统方法


本文介绍了一种新颖的系统方法，用于对一类多维线性机械系统进行建模，该系统可以直接获得其无限维端口哈密尔顿表示。虽然该方法是针对特定运动学假设控制的系统量身定制的，但它涵盖了当前文献中发现的各种模型，包括维弹性模型（其中 = 1、2、3）、振动弦、圆杆扭转、经典梁和板模型等。该方法涉及使用初级广义坐标通过线性代数关系来制定位移场。然后使用二次广义坐标计算和表达应变张量的非零分量，从而能够表征与端口哈密尔顿表示相关的斜伴随微分算子。通过应用哈密尔顿原理并针对所考虑的微分算子类别采用专门开发的分部积分公式，可以直接获得端口哈密尔顿模型以及边界输入和输出的定义。为了说明该方法，以基于 Reddy 三阶剪切变形理论的板块建模过程为例。据我们所知，这是文献中首次提出该系统的波特哈密尔顿表示。