大挠度欧拉梁三参数曲率模型及其在平面柔顺机构中的应用

doi:10.3901/JME.2021.13.144

机械工程学报 ›› 2021, Vol. 57 ›› Issue (13): 144-152.doi: 10.3901/JME.2021.13.144

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大挠度欧拉梁三参数曲率模型及其在平面柔顺机构中的应用

谢丹¹, 黄勇刚²

1. 西南大学工程技术学院重庆 400715;
2. 重庆工商大学制造装备机构设计与控制重庆市重点实验室重庆 400067

收稿日期:2020-07-28 修回日期:2020-11-09 出版日期:2021-08-31 发布日期:2021-08-31
通讯作者: 谢丹(通信作者),女,1987年出生,博士,讲师。主要研究方向为刚柔耦合多体系统理论及应用。E-mail:danxie2017@swu.edu.cn
作者简介:黄勇刚,男,1976年出生,博士,副教授。主要研究方向为机构学与机器人及机械系统CAE。E-mail:hyg@ctbu.edu.cn
基金资助:
国家自然科学基金资助项目（51805448）

Novel Model with Three Curvature Variables for Euler Beam under Large Deflection and Its Application in Planar Compliant Mechanisms

XIE Dan¹, HUANG Yonggang²

1. College of Engineering and Technology, Southwest University, Chongqing 400715;
2. Chongqing Municipal Key Laboratory of Mechanism Design and Control for Manufacturing Equipment, Chongqing Technology and Business University, Chongqing 400067

Received:2020-07-28 Revised:2020-11-09 Online:2021-08-31 Published:2021-08-31

摘要/Abstract

摘要： 提出了一种基于二次Bernstein多项式的平面大挠度欧拉梁三参数曲率模型。三个曲率参数视为描述梁平衡位形的广义坐标，应用微分几何曲线论基本定理建立梁弯曲的转角方程和积分形式的挠曲线方程，基于虚功原理导出了以三个曲率参数为基本未知量的平面欧拉梁大挠度弯曲的几何非线性平衡方程，并给出了采用高斯积分近似以及牛顿-拉弗森迭代法的数值计算格式。通过末端载荷状态下的悬臂直梁、曲梁的大挠度变形计算，以及对一种典型的平面曲柄摇杆柔顺机构的详细建模分析，验证了三参数曲率模型方法的高精度、高计算效率和对曲梁的通用性以及变形度量呈现的直观性和丰富性，结果表明该方法在柔顺机构设计和分析中具有明显优势和广阔的应用前景。

关键词: 欧拉梁, 大挠度, 曲率, 参数化模型, 平面柔顺机构

Abstract: A novel model with three curvature variables for Euler beam under large deflection is proposed based on the quadratic Bernstein polynomials. The three curvature variables are considered as generalized coordinates for bending equilibrium configuration of the beam. Then, the rotation equation and the deflection curve equation in integral form are formulated using the curve theory of differential geometry. By the virtual work principle, the nonlinear geometric equilibrium equation of Euler beam is derived in which three curvature parameters are unknown variables. At the same time the numerical formulation of the equilibrium equation is shown using Gaussian quadrature and Newton-Raphson iteration method. Finally, through the typical numerical examples of straight and curved cantilever beams and the partially compliant crank rocker mechanism, the curvature model proposed above is fully proved to have high accuracy, high calculation efficiency, general applicability to curved beams and intuitive and rich expressions for deformation measurements. The calculation and analysis results show the obvious advantages and application prospects of the proposed method in the design and analysis of compliant mechanisms.

Key words: Euler beam, large deflection, curvature, parametric model, planar compliant mechanisms

中图分类号:

TH112

谢丹, 黄勇刚. 大挠度欧拉梁三参数曲率模型及其在平面柔顺机构中的应用[J]. 机械工程学报, 2021, 57(13): 144-152.

XIE Dan, HUANG Yonggang. Novel Model with Three Curvature Variables for Euler Beam under Large Deflection and Its Application in Planar Compliant Mechanisms[J]. Journal of Mechanical Engineering, 2021, 57(13): 144-152.

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大挠度欧拉梁三参数曲率模型及其在平面柔顺机构中的应用

Novel Model with Three Curvature Variables for Euler Beam under Large Deflection and Its Application in Planar Compliant Mechanisms

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