用Mindlin板理论精确求解中厚双导边矩形板自由振动的新方法

IF 6.6 1区工程技术 Q1 ENGINEERING, CIVIL

Thin-Walled Structures Pub Date : 2025-09-06 DOI:10.1016/j.tws.2025.113866

Fei Long, Ya-Wei Wang, Xian-Fang Li

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引用次数: 0

摘要

这项研究有两个目的。一是为推导Reissner-Mindlin板理论或Mindlin板理论（MPT）的控制方程提供了一种新的方法；二是获得了厚度-扭转-锁定（TTL）中厚矩形板的自由振动的精确解。首先，本文介绍了一种简化的解析方法来推导控制方程，该方程保留了截面的剪切变形和旋转惯量的基本特征。通过引入两个未知函数，一个满足控制板挠度的四阶偏微分方程，另一个满足控制厚度方向扭转的二阶偏微分方程。对于TTL板，只需一个函数即可满足四阶偏微分方程，并引入有效剪切力以包括扭矩的作用。对于有两个相对边导向的板，TTL板的自由振动可以用lsamv型解精确求解。确定了其他边（包括夹紧边、铰接边、导向边和自由边）的十种组合情况的精确特征方程。给出了精确的固有频率参数和相应的模态振型。详细讨论了宽高比、厚边长比和边界条件对板振动特性的影响。只有当剪切刚度足够大时，具有相反导向支承的薄板的精确闭型特征方程及其相关特征函数才能减小。

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A novel approach for Mindlin plate theory and application to exact solution of free vibration of moderately thick rectangular plates with two opposite guided sides

This study has two aims. One is to provide a novel approach for deriving the governing equation of the Reissner–Mindlin plate theory or Mindlin plate theory (MPT) and the other is to obtain exact solutions for the free vibration of thickness-torsion-locking (TTL) moderately thick rectangular plates with two opposite guided sides. First, the paper introduces a simplified analytical method to deduce the governing equations that retain the essential features of shear deformation and rotary inertia of the section. By introducing two unknown functions, one satisfies a fourth-order partial differential equation (PDE) which dominates the flexure of plates and the other meets a second-order PDE which dominates the torsion along the thickness direction. For TTL plates, only a single function is required to satisfy a fourth-order PDE and effective shear force is introduced to include the role of the twisting moment. For plates with two opposite sides guided, the free vibration of TTL plates is exactly solved by a Lévy-type solution. The exact characteristic equations are determined for ten combination cases of the other sides, including clamped, hinged, guided, and free edges. Accurate natural frequency parameters and the corresponding mode shapes are given. The effects of aspect ratio, thickness-to-side-length ratio, and boundary conditions on the vibration characteristics of plates are discussed in detail. The exact closed-form characteristic equations and their associated characteristic functions for thin plates with opposite guided supports can be reduced only if the shear stiffness is sufficiently large.

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来源期刊

Thin-Walled Structures 工程技术-工程：土木

CiteScore

9.60

自引率

20.30%

发文量

801

审稿时长

66 days

期刊介绍： Thin-walled structures comprises an important and growing proportion of engineering construction with areas of application becoming increasingly diverse, ranging from aircraft, bridges, ships and oil rigs to storage vessels, industrial buildings and warehouses. Many factors, including cost and weight economy, new materials and processes and the growth of powerful methods of analysis have contributed to this growth, and led to the need for a journal which concentrates specifically on structures in which problems arise due to the thinness of the walls. This field includes cold– formed sections, plate and shell structures, reinforced plastics structures and aluminium structures, and is of importance in many branches of engineering. The primary criterion for consideration of papers in Thin–Walled Structures is that they must be concerned with thin–walled structures or the basic problems inherent in thin–walled structures. Provided this criterion is satisfied no restriction is placed on the type of construction, material or field of application. Papers on theory, experiment, design, etc., are published and it is expected that many papers will contain aspects of all three.