张小明, 石玉龙, 田小广,等.树脂锚杆纵向导波特性理论计算与数值模拟研究[J].河南理工大学学报（自然科学版）,2026,45(1):118-127.

ZHANG X M, SHI Y L, TIAN X G,,et al.Theoretical calculation and numerical simulation of the characteristics of longitudinal guided wave in resin bolts[J].Journal of Henan Polytechnic University(Natural Science) ,2026,45(1):118-127.

树脂锚杆纵向导波特性理论计算与数值模拟研究

张小明1, 石玉龙1, 田小广2, 李智1

1.河南理工大学 机械与动力工程学院，河南 焦作  454000；2.中国平煤神马控股集团有限公司 平煤十一矿，河南 平顶山 467000

摘要: 目的 为了揭示树脂锚杆中的超声纵向导波传播和衰减特性，利用本征函数展开法求解树脂锚杆纵向导波频散方程。 方法 树脂锚杆是典型的双层柱状黏弹结构，基于Hysteretic黏弹性模型，考虑连续性边界条件，将其声场位移表示成Legendre多项式级数形式，经矩阵变换把复杂的波动控制偏微分方程组求解问题转化为特征值计算问题，特征值为波数，特征向量为位移场分布，得到树脂锚杆纵向导波频散关系，获得超越频散方程的解析解，进而绘制其纵向导波群速度频散曲线和衰减曲线。通过与已有文献计算的自由锚杆的群速度和衰减结果比较，验证所提方法的正确性。基于COMSOL软件模拟树脂锚杆纵向导波传播，揭示树脂层层厚和握裹段长度对树脂锚杆导波传播特性的影响。 结果 结果表明：（1）树脂层层厚对纵向导波的影响显著，随着层厚增大，回收信号波包弥散、波形畸变加剧；导波群速度和振幅比随着层厚增大而减小，衰减随着层厚增大而增大。（2）随着握裹段长度增加，树脂锚杆中的导波群速度减小，振幅比也逐渐减小，导波衰减增大。 结论 导波群速度与握裹段长度呈近似线性关系，通过导波群速度变化可对树脂锚杆握裹段长度进行量化判断。

关键词:树脂锚杆;纵向导波;本征函数展开法;群速度;衰减

doi:10.16186/j.cnki.1673-9787.2023040048

基金项目:国家自然科学基金资助项目（5197518）；河南省高校科技创新团队支持计划项目（23IRTSTHN016）；河南理工大学创新型科研团队研究项目（T2022-4）

收稿日期:2023/04/23

修回日期:2023/08/06

出版日期:2025-12-03

Theoretical calculation and numerical simulation of the characteristics of longitudinal guided wave in resin bolts

Zhang Xiaoming1, Shi Yulong1, Tian Xiaoguang2, Li Zhi1

1.School of Mechanical and Power Engineering， Henan Polytechnic University， Jiaozuo  454000， Henan， China；2.No. 11 Mine of Pingmei Coal Group ， China Pingmei Shenma Holding Group Co.， Ltd.， Pingdingshan  467000， Henan， China

Abstract: Objectives To illustrate the characteristics of longitudinal guided wave in resin bolts， the eigenfunction expansion method was used to solve the longitudinal guided wave dispersion equations of a resin bolt. Methods The resin anchor bolt was considered as a typical double-layer cylindrical viscoelastic structure. Based on the Hysteretic viscoelastic model， considering continuity boundary conditions， the acoustic field displacements were represented in the Legendre polynomial series form. Then the problem of solving the complex wave control partial differential equation system was transformed into the problem of eigenvalue calculation through matrix transformation. The eigenvalue was the wave number， and the eigenvector was the displacement field distribution， the longitudinal guided wave dispersion relationship was established， and the analytical solution of the transcendental dispersion equation was obtained. Then the group velocity dispersion curve and attenuation curve could be plotted. The correctness of the presented method was verified by comparing the obtained group velocity and attenuation results of a free bolt with those of existing literature. Based on the COMSOL software， the longitudinal guided waves in various resin bolts were simulated to illustrate the influences of resin layer thickness and bond length on the guided wave characteristics. Results The results showed that： （1） The thickness of the resin layer had a significant effect on the longitudinal guided waves. With the increase of layer thickness， the wave packet dispersion and waveform distortion of the recovered signal increased. The group velocity and amplitude ratio of guided waves decrease with the increase of layer thickness， but the attenuation increased. （2） With the increase of bond length， the group velocity of guided waves in the resin bolt decreased， the amplitude ratio decreased， and the guided wave attenuation increased. Conclusions The group velocity of guided wave was approximately linear with the bond length， and quantitative judgment of the length of the resin bolt bond section can be made through changes in guided wave group velocity.

Key words:resin bolt;longitudinal guided wave;eigenfunction expansion method;group velocity;attenuation