Vibration of nonlocal strain gradient functionally graded nonlinear nanobeams using a novel locally adaptive strong quadrature element method

M. Trabelssi, S. El-Borgi
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Abstract

The primary objective of this paper is to propose a novel method to derive Differential Quadrature Method matrices with several degrees of freedom at the boundaries that can be used to build Strong Quadrature Elements to solve fourth and higher-order equations of motion. The proposed method, referred to as Locally adaptive Strong Quadrature Element Method, is applied to higher-order equations of motion for nonlinear graded Timoshenko and Euler-Bernoulli nanobeams formulated using the Second Strain Gradient Theory or the Nonlocal Strain Gradient Theory. To limit the formulation complexity, the proposed approach is based on the regular formulation of the differential quadrature method combined with custom-built transfer matrices. Moreover, it does not require a different formulation for fourth and sixth-order equations and can be extended beyond sixth-order equations. Validation was carried out using examples from the literature as well as data obtained using the classical Locally adaptive Quadrature Element Method. Both linear and nonlinear frequencies were evaluated for a large number of configurations and boundary conditions. The proposed approach resulted in good accuracy and a convergence speed comparable to the conventional Locally adaptive Quadrature Element Method.
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基于局部自适应强正交元法的非局部应变梯度功能梯度非线性纳米梁振动研究
本文的主要目的是提出一种新的方法来导出在边界处具有多个自由度的微分正交方法矩阵,这些矩阵可用于构建强正交单元来求解四阶和高阶运动方程。该方法被称为局部自适应强正交单元法,应用于用第二应变梯度理论或非局部应变梯度理论表述的非线性梯度Timoshenko和Euler-Bernoulli纳米梁的高阶运动方程。为了限制公式的复杂性,本文提出的方法是基于微分正交法的正则公式,并结合定制的传递矩阵。此外,对于四阶和六阶方程不需要不同的公式,并且可以推广到六阶方程之外。利用文献中的算例和经典的局部自适应正交元法得到的数据进行了验证。对大量的结构和边界条件进行了线性和非线性频率的计算。与传统的局部自适应正交元法相比,该方法具有较高的精度和收敛速度。
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来源期刊
CiteScore
6.00
自引率
1.70%
发文量
24
期刊介绍: Proceedings of the Institution of Mechanical Engineers Part N-Journal of Nanomaterials Nanoengineering and Nanosystems is a peer-reviewed scientific journal published since 2004 by SAGE Publications on behalf of the Institution of Mechanical Engineers. The journal focuses on research in the field of nanoengineering, nanoscience and nanotechnology and aims to publish high quality academic papers in this field. In addition, the journal is indexed in several reputable academic databases and abstracting services, including Scopus, Compendex, and CSA's Advanced Polymers Abstracts, Composites Industry Abstracts, and Earthquake Engineering Abstracts.
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