微分求积法求解悬臂L梁固有振动特性研究

李智超; 郝育新

doi:10.21656/1000-0887.430382

微分求积法求解悬臂L梁固有振动特性研究

doi: 10.21656/1000-0887.430382

李智超,
郝育新^,

北京信息科技大学机电工程学院，北京 100192

基金项目:

国家自然科学基金项目 11872127

详细信息

作者简介:
李智超(1998—)，男，硕士生(E-mail: lzc19980510@163.com)

通讯作者:
郝育新(1972—)，男，教授(通讯作者. E-mail: bimhao@163.com)

中图分类号: O31; O32
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- 被引次数: 0
出版历程
- 收稿日期: 2022-11-29
- 修回日期: 2023-02-17
- 刊出日期: 2023-05-01

Study on Natural Vibration Characteristics of L-Shaped Cantilever Beams With the Differential Quadrature Method

LI Zhichao,
HAO Yuxin^,

Mechanical Electrical Engineering School, Beijing Information Science and Technology University, Beijing 100192, P.R.China

摘要

摘要: 悬臂L梁结构由于具有柔性大、可设计性强、空间利用充分，振动过程中变形方式多样等独特优势而受到了广泛的关注与研究. 该文提出了一种基于微分求积法求解末端附加质量块的矩形等截面均质悬臂细长L梁的各阶固有频率和模态的方法. 在双坐标系下，基于Euler-Bernoulli梁理论建立了悬臂L梁的动力学方程，然后通过选取Chebyshev多项式的根作为节点坐标、选取Lagrange插值基函数、求解各阶权系数、处理边界条件等步骤，最终利用求解矩阵广义特征值问题的方法求得结构各阶固有频率及模态. 在边界条件的处理上，直接将边界条件施加于边界点上，通过对比研究验证了该文固有频率理论解的正确性. 最后分析了末端质量、内外梁的长度比、宽度、厚度对各阶固有振动特性的影响. 该方法可以进一步应用推广到相关结构振动的研究中.
- 微分求积法 /
- 悬臂L梁 /
- 固有振动特性
Abstract: The L-shaped cantilever beam structure has many unique advantages such as large flexibility, strong designability, full utilization of space and various deformation modes during vibration, and is widely regarded and studied. A differential quadrature method was proposed to solve the natural frequencies and modes of rectangular-section homogeneous slender L-shaped cantilever beams with additional end masses. In the double coordinate systems, the dynamic equations for the L-shaped cantilever beam based on the Euler-Bernoulli beam theory were established. With selected roots of the Chebyshev polynomial as the node coordinates, the Lagrange interpolation basis function was employed, the weight coefficients of each order were solved, and the boundary conditions were considered, to obtain the natural frequencies and modes of all orders of the structure through resolution of the generalized matrix eigenvalue problem. The theoretical solution of the natural frequencies was verified in comparison with the previous theoretical results and the finite element results. Finally, the effects of the end mass, the length ratio, the width and the thickness of the inner and outer beams on the natural vibration characteristics of all orders were discussed. This method can be further applied to the study of related structural vibrations.
- differential quadrature method /
- L-shaped cantilever beam /
- natural vibration characteristic

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图 1 悬臂L梁示意图

Figure 1. Schematic diagram of the L-shaped cantilever beam

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图 2 结构前五阶模态图

Figure 2. Diagram of the first five modes of the structure

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图 3 内、外梁长度比对结构各阶固有频率的影响

Figure 3. The effect of the length ratio of the outer beam to the inner beam on natural frequencies of the structure

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图 4 梁宽度对结构各阶固有频率的影响

Figure 4. The effect of the beam width on natural frequencies of the structure

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图 5 梁厚度对结构各阶固有频率的影响

Figure 5. The effect of the beam thickness on natural frequencies of the structure

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图 6 末端质量对结构各阶固有频率的影响

Figure 6. The effect of the end mass on natural frequencies of the structure

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表 1 悬臂L梁的几何与材料参数表

Table 1. Geometric and material parameters of the L-shaped cantilever beam

parameter	value^[20]	parameter	value^[20]
length l₁/m, l₂/m	0.75, 0.75	elasticity modulus E/Pa	7×10¹⁰
width b/m	0.04	polar moment of inertia I_p/m⁴	1.535×10^-9
thickness h/m	0.005	end mass m/kg	0.5
density ρ/(kg/m³)	2 700	Poisson’s ratio υ	0.3

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表 2 不同节点数下的结构前五阶固有频率表(单位：Hz)

Table 2. First five-order natural frequencies of the structure with different number of nodes (unit: Hz)

node	mode
node	1	2	3	4	5
N_1，2=9	1.377 7	5.514 0	27.671 8	42.532 3	94.087 5
N_1，2=10	1.377 7	5.513 6	27.661 4	42.342 1	93.287 7
N_1，2=11	1.377 7	5.513 6	27.659 9	42.259 0	93.070 6
N_1，2=12	1.377 7	5.513 6	27.660 2	42.261 4	93.072 7
N_1，2=13	1.377 7	5.513 6	27.660 2	42.263 6	93.088 2
N_1，2=14	1.377 7	5.513 6	27.660 2	42.263 5	93.089 0
N_1，2=15	1.377 7	5.513 6	27.660 2	42.263 4	93.088 4

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表 3 结构前五阶固有频率对比表(单位：Hz)

Table 3. Comparison of the structure's first five-order natural frequencies (unit: Hz)

mode	1	2	3	4	5
present N_1，2=13	1.377 7	5.513 6	27.660 2	42.263 6	93.088 2
ref. [20] (error δ/%)	1.377 2(0.04)	5.531 6(-0.33)	27.761 9(-0.37)	42.456 2(-0.45)	93.500 7(-0.44)
PATRAN (error δ/%)	1.370 1(0.55)	5.486 1(0.50)	27.582 0(0.28)	41.738 0(1.26)	92.665 0(0.46)
COMSOL (error δ/%)	1.405 3(-1.96)	5.554 0(-0.73)	27.779 0(-0.43)	41.556 0(1.70)	92.551 0(0.58)

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