Curvature Parameterization Model for Variable Cross-section Euler Beams under Large Deflection

doi:10.3969/j.issn.1004-132X.2025.08.011

Abstract

Abstract:

In order to achieve optimization of structures and performances， variable cross-section beams were increasingly valued in compliant mechanisms. A parametric model was proposed to address the challenge of accuracy and efficiency of large deflection analysis based on curvature approximation. The proposed approach represented the curvature of deflection curve by Bernstein polynomials. Then the static equilibrium equation was established based on the principle of minimum total potential energy， and the numerical solution was obtained via Gaussian quadrature and Newton-Raphson iteration. The advantages are that the expression of bending strain energy and stress are simple and accurate， and the generalized stiffness matrix is a constant symmetric square matrix related to the variation of the cross-sectional moment of inertia. Therefore， both modeling accuracy and computational efficiency are significantly improved. Furthermore， the post-processing is simple and fast based on the solution of curvature parameters. Numerical examples fully demonstrate the effectiveness and superiority of the proposed model.

Key words: compliant mechanism, variable cross-section beam, large deflection, parameterization model

摘要：

为了实现结构和性能的优化，变截面梁在柔顺机构中愈来愈受到重视，针对其大挠度分析难以兼顾精度和效率的挑战，提出了一种基于曲率逼近的参数化建模方法。采用Bernstein多项式表示挠曲线曲率，基于最小总势能原理建立静力平衡方程，应用高斯积分和牛顿-拉弗森迭代法进行数值求解。其优点是弯曲应变能及应力的计算简单而精确，且广义刚度矩阵是与截面惯性矩变化相关的常值对称方阵，因而建模精度高、数值计算收敛速度快。同时，基于曲率参数解，结果后处理简便快捷。多个算例的数值仿真结果充分证明了所提方法的有效性和优越性。

关键词: 柔顺机构, 变截面梁, 大挠度, 参数化模型

CLC Number:

TH112

Yonggang HUANG, Dan XIE. Curvature Parameterization Model for Variable Cross-section Euler Beams under Large Deflection[J]. China Mechanical Engineering, 2025, 36(8): 1757-1766.

黄勇刚, 谢丹. 大挠度变截面欧拉梁曲率参数化模型[J]. 中国机械工程, 2025, 36(8): 1757-1766.

Figures/Tables 20

Fig.1 Planar curve and body-fixed frame

Fig.2 Straight cantilever beam under tip load

Fig.3 Straight cantilever beam with parabolic thickness

Fig.4 Comparison of curvature solutions

Tab.1 The results of curvature model and exact solution

	精确解	3参数模型		5参数模型
	精确解	值	误差/%	值	误差/%
应变能/J	0.041 251	0.041 035	0.52	0.041 245	0.0146
角刚度/（N·m·rad^-1）	0.019 393	0.019 496	0.53	0.019 396	0.0145

Fig.5 Angle error of deflection curve

Fig.6 Position error of deflection curve

Fig.7 Bending stress nephogram

Fig.8 Displacement nephogram

Tab.2 Beam tip coordinates and angle under two load cases

	参数类别	载荷1	载荷2
载荷	F_x /N	-45.0	-22.5
	F_y /N	15.0	7.5
	M（N·m）	2	1
有限元 ^［17］	x（L）/m	0.1599	0.1880
	y（L）/m	0.062 59	0.038 11
	θ（L）/rad	2.5974	1.3492
7参数模型	x（L）/m	0.1597	0.1879
	y（L）/m	0.062 03	0.038 22
	θ（L）/rad	2.6765	1.3737
相对误差/%	x（L）/m	0.15	0.07
	y（L）/m	0.90	0.28
	θ（L）/rad	3.05	1.18

Fig.9 Deflection curves under two load cases

Fig.10 Curvature curves under two load cases

Fig.11 Geometry of cross-axis flexural pivot

Fig.12 Coordinates and forces of components

Fig.13 Relative error of curvature of different models

Fig.14 Curvature curves under 1.13 rad rotation

Fig.15 Configuration under 1.13 rad rotation

Tab.3 The results of curvature model and FEM

	扭转刚度/（N·m·rad^-1）	最大应力/MPa
有限元^［5］	1.220	63.30
10参数模型	1.248	66.27
相对误差/%	2.28	4.70

Fig.16 Moment-deflection relation

Fig.17 Equivalent stress nephogram under 1.13 rad rotation

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