基于动力刚度理论和Wittrick-Williams算法的多段梁点阵色散研究

彭长清; 刘金兴

doi:10.21656/1000-0887.450043

基于动力刚度理论和Wittrick-Williams算法的多段梁点阵色散研究

doi: 10.21656/1000-0887.450043

彭长清,
刘金兴^,

江苏大学土木工程与力学学院，江苏镇江 212013

基金项目:

国家自然科学基金 11972174

详细信息

作者简介:
彭长清(1998—)，男，硕士生(E-mail: changqing9810@163.com)

通讯作者:
刘金兴(1979—)，男，教授(通讯作者. E-mail: jxliu@mails.ucas.ac.cn)

中图分类号: O32
计量
- 文章访问数: 63
- HTML全文浏览量: 29
- PDF下载量: 14
- 被引次数: 0
出版历程
- 收稿日期: 2024-02-22
- 修回日期: 2024-03-18
- 刊出日期: 2025-02-01

Research on Dispersion Characteristics of Multi-Segment Beam Lattices Based on the Dynamic Stiffness Theory and the Wittrick-Williams Algorithm

PENG Changqing,
LIU Jinxing^,

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R.China

摘要

摘要: 基于动力刚度法(DSM)表述了多段梁(MSB)点阵的动态响应，进而利用Wittrick-Williams(WW)算法计算其各阶固有频率. 首先，根据MSB内部节点处的位移与应力连续条件，以分块矩阵组装的形式得到了MSB的动力刚度矩阵. 利用这种方法获得的MSB动力刚度矩阵仍然是两节点式，并不会增大刚度矩阵的维度. WW算法与动力刚度矩阵的结合可以精确求解点阵的各阶固有频率. 针对MSB点阵的周期性胞元，在原始动力刚度矩阵中考虑Floquet边界条件，再采取WW算法并最终得到了相应点阵的色散曲线. 在不可约Brillouin区内，利用该法算得的色散曲线与基于COMSOL仿真的结果相比，两者误差在6%以内，验证了所建方法的可行性. 进而系统研究了微观几何和材料参数对色散曲线的影响规律，结果表明使用MSB搭建点阵是调节点阵色散特性的有效方法.
- 动力刚度法 /
- Wittrick-Williams算法 /
- Floquet边界条件 /
- 多段梁点阵 /
- 色散曲线
Abstract: The dynamic stiffness method (DSM) was employed to describe dynamic responses of periodic lattices composed of multi-segment beams (MSBs), and the dispersion characteristics were examined based on the Wittrick-Williams algorithm (WWA). First, the dynamic stiffness matrix of the MSB was obtained with the continuity conditions in terms of displacements and stresses at inner joints. The obtained dynamic stiffness matrix in nature remains a 2-node type element, and has the same dimensions as those of a 2-node homogeneous beam. The combination of the DSM and the WWA enables the accurate calculation of natural frequencies of the lattice. As for a periodic unit cell of the MSB lattice, the Floquet boundary condition was introduced into the initial DSM, and then dispersion curves and natural frequencies can were obtained with the WWA. In the irreducible Brillouin zone, results obtained with the proposed method agree reasonably well with those by software COMSOL, with errors no larger than 6%, which verifies the effectiveness of the proposed method. Furthermore, the effects of microscopic geometric and material parameters on lattice dispersion curves were studied. The results show that, the MSB makes an effective way to adjust dispersion characteristics of lattices by building periodic lattices.
- dynamic stiffness method /
- Wittrick-Williams algorithm /
- Floquet boundary condition /
- multi-segment beam lattice /
- dispersion curves

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图 1 三段梁模型以及各个分段的受力情况

Figure 1. The 3-segment beam and the forces on each segment

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图 2 模型Ⅰ和模型Ⅱ

Figure 2. Modle Ⅰ and modle Ⅱ

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图 3 分别考虑为普通梁和三段梁，使用本文方法以及FEM得到的两个模型的色散曲线

Figure 3. The dispersion curves of the 2 models considered respectively for the 1-segment beam and the 3-segment beam, in comparison with the results of the finite element model

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图 4 两个模型色散曲线受到中间段宽度t₂的影响(t₁=0.003 m, l₂/l=1/3)

Figure 4. The dispersion curves of the 2 models with different width t₂ values of the middle segment (t₁=0.003 m, l₂/l=1/3)

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图 5 两个模型色散曲线受到中间段长度l₂的影响(t₁=0.003 m)

Figure 5. The dispersion curves of the 2 models with different length l₂ values of the middle segment (t₁=0.003 m)

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图 6 两个模型色散曲线受到中间段密度ρ₂的影响(ρ₁=2 500 kg/m³, t₁=0.003 m, t₂=0.003 m, l₂/l=1/3)

Figure 6. The dispersion curves of the 2 models with different density ρ₂ values of the middle segment (ρ₁=2 500 kg/m³, t₁=0.003 m, t₂=0.003 m, l₂/l=1/3)

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图 7 两个模型色散曲线受到中间段弹性模量E₂的影响(E₁=2.1×10¹⁰ Pa, t₁=0.003 m, t₂=0.003 m, l₂/l=1/3)

Figure 7. The dispersion curves of the 2 models with different Young's modulus E₂ values of the middle segment (E₁=2.1×10¹⁰ Pa, t₁=0.003 m, t₂=0.003 m, l₂/l=1/3)

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