基于比例边界有限元的复合梁自由振动频率计算

李文武; 王为

doi:10.21656/1000-0887.440208

基于比例边界有限元的复合梁自由振动频率计算

doi: 10.21656/1000-0887.440208

李文武^1, ,,
王为^2,

1.
湖南省交通规划勘察设计院有限公司, 长沙 410219
2.
湖南省建设投资集团有限责任公司, 长沙 410009

基金项目:

国家自然科学基金 51875159

详细信息

作者简介:
王为(1973—)，男，高级工程师(E-mail: tian3316625@163.com)

通讯作者:
李文武(1980—)，男，博士，教授级高级工程师(通讯作者. E-mail: m19818937339@163.com)

中图分类号: TB332
计量
- 文章访问数: 154
- HTML全文浏览量: 69
- PDF下载量: 24
- 被引次数: 0
出版历程
- 收稿日期: 2023-07-10
- 修回日期: 2024-03-04
- 刊出日期: 2024-07-01

Natural Vibration Frequencies of Laminated Composite Beams Based on the Scaled Boundary Finite Element Method

LI Wenwu^{1
, ,},
WANG Wei^2
,

1.
Hunan Provincial Communications Planning, Survey & Design Insititute Co., Ltd., Changsha 410219, P.R.China
2.
Hunan Construction Investment Group, Changsha 410009, P.R.China

摘要

摘要: 将比例边界有限元方法(SBFEM)拓展用于计算复合梁的自由振动频率. 该方法将梁简化为一维模型，并且仅选用x和z方向的弹性线位移作为基本未知量. 从弹性力学基本方程出发，通过比例边界坐标、虚功原理和对偶变量技术推导得到了复合梁的一阶常微分比例边界有限元动力控制方程，其通解为解析的矩阵指数函数. 利用Padé级数求解矩阵指数函数可得各个梁层的动力刚度矩阵，根据自由度匹配原则组装得到复合梁的整体刚度和质量矩阵. 求解特征值方程，最终可得复合梁的自由振动频率. 该方法对复合梁的层数和边界条件均无限制，具有广泛的适用性. 将该文的解与三层、四层和十层复合梁振动频率的数值参考解以及阶梯型悬臂梁固有频率的实验实测值进行对比，验证了比例边界有限元算法的准确性、高效性和快速收敛性.
- 复合梁 /
- 自由振动频率 /
- 比例边界有限元 /
- Padé级数
Abstract: The scaled boundary finite element method (SBFEM) was extended to calculate the natural frequencies of laminated composite beams. With this method, the beam was simplified as a 1D model. Only the displacement components along the x and z directions were selected as the fundamental unknowns. Based on the fundamental equations of elasticity and the scaled boundary coordinates, under the principle of virtual work and with the dual vector technique, the 1st-order ordinary differential scaled boundary finite element dynamic equation for composite beams was obtained, with its general solution in the form of the analytical matrix exponential function. The Padé expansion was utilized to solve the matrix exponential function and the dynamic stiffness matrix for each beam layer was acquired. According to the principle of matching degrees of freedom, the global stiffness and mass matrices of the laminated beam were gained. The eigenvalue equation was solved to give the natural vibration frequencies of the laminated composite beam. The results show that, the proposed method is widely applicable without limitation on the layer number and boundary conditions. Comparisons between the numerical natural frequencies and the experiment results of 3-, 4- and 10-layered step-shaped cantilever beams, validate the accuracy, high efficiency and fast convergence of the SBFEM.
- laminate composite beam /
- natural frequency /
- scaled boundary finite element method (SBFEM) /
- Padé expansion

HTML全文

图 1 四层复合梁示意图

Figure 1. The 4 layered composite beam

下载: 全尺寸图片幻灯片

图 2 四阶谱单元

Figure 2. The 4th order spectral element

下载: 全尺寸图片幻灯片

图 3 梁层模型图

Figure 3. The model for the beam layer

下载: 全尺寸图片幻灯片

图 4 (0°/90°/0°/90°)四层复合梁

Figure 4. The 4-layered (0°/90°/0°/90°) beam

下载: 全尺寸图片幻灯片

图 5 材料纤维坐标系(1-2-3)与直角坐标系

Figure 5. The fiber-matrix coordinate system (1-2-3) and the Cartesian coordinate system

下载: 全尺寸图片幻灯片

图 6 梁各阶谱单元

Figure 6. The different orders of spectral elements

下载: 全尺寸图片幻灯片

图 7 (0°/90°/0°/90°)四层复合梁的振动频率

Figure 7. The vibration frequencies of the 4-layered (0°/90°/0°/90°) composite beam

下载: 全尺寸图片幻灯片

图 8 三层夹层梁

Figure 8. The 3-layered sandwich beam

下载: 全尺寸图片幻灯片

图 9 三层简支夹层梁振动频率

Figure 9. The eigenfrequencies of the 3-layered SS sandwich beam

下载: 全尺寸图片幻灯片

图 10 (0°/90°/0°)三层复合梁

Figure 10. The 3-layered (0°/90°/0°) composite beam

下载: 全尺寸图片幻灯片

图 11 阶梯型悬臂梁(单位: cm)

Figure 11. The step-shaped cantilever beam (unit: mm)

下载: 全尺寸图片幻灯片

表 1 梁两端处的约束情况

Table 1. The constraint conditions at ends of the beam

constraint condition	x=0	x=l
SS	u_z(0, z)=0	u_z(l, z)=0
CS	u_z(0, z)=u_x(0, z)=0	u_z(l, z)=0
CC	u_z(0, z)=u_x(0, z)=0	u_z(l, z)=u_x(l, z)=0
CF	u_z(0, z)=u_x(0, z)=0	free

下载: 导出CSV

表 2 l/t=100时(0°/90°/0°/90°)四层梁的振动频率

Table 2. The natural frequencies of the 4-layered (0°/90°/0°/90°) beam with l/t=100

element order	1st frequency	2nd frequency	3rd frequency	4th frequency	5th frequency	6th frequency
2nd order	12.309 2	44.215 7	505.889 6	956.705 8	2 118.235 8	2 708.157 8
3rd order	11.252 5	48.945 4	115.329 1	184.133 7	958.889 6	1 005.616 1
4th order	11.212 9	44.560 2	101.600 6	211.074 6	328.678 5	431.467 9
5th order	11.212 9	44.571 8	99.420 0	173.882 4	280.551 6	498.252 5
6th order	11.212 9	44.561 7	99.217 6	174.345 3	269.073 7	380.223 2
7th order	11.212 9	44.561 7	99.206 1	173.834 0	267.054 3	377.217 1
2D^[7]	11.193 0	44.477 0	98.988 0	173.390 0	266.010 0	374.910 0
error δ/%	0.177 8	0.190 5	0.220 4	0.256 1	0.392 6	0.615 4

下载: 导出CSV

表 3 三层梁(0°/90°/0°)振动频率

Table 3. Eigensolutions of the (0°/90°/0°) beam

l/t		SS	CF	CC
50	SBFEM	17.480	6.270	37.808
	ref. [6]	17.462	6.267	37.679
	error δ/%	0.107	0.053	0.343
30	SBFEM	17.103	6.205	34.511
	ref. [6]	17.055	6.198	34.268
	error δ/%	0.281	0.119	0.709
20	SBFEM	16.432	6.084	29.982
	ref. [6]	16.338	6.070	29.695
	error δ/%	0.5780	0.234	0.967

下载: 导出CSV

表 4 十层梁的振动频率

Table 4. The eigenvalues of the 10-layered beam

l/t		SS	CS	CC	CF
10	SBFEM	10.862	14.058	17.245	4.184
	ref. [29]	10.893	14.221	17.580	4.197
	error δ/%	-0.288	-1.147	-1.905	-0.316
5	SBFEM	8.118	9.199	10.541	3.497
	ref. [29]	8.156	9.356	10.784	3.527
	error δ/%	-0.467	-1.675	-2.250	-0.856

下载: 导出CSV

表 5 阶梯型悬臂梁的振动频率

Table 5. Vibration frequencies of the step-shaped cantilever beam

	SBFEM	experiment^[30]	CEM^[30]
1st frequency	36.224 9	39.06	36.595
2nd frequency	146.580 7	138.67	148.13
3rd frequency	428.683 8	417.48	433.75
4th frequency	788.257 6	776.37	798.28

下载: 导出CSV

参考文献(30)

[1]	SAYYAD A S, GHUGAL Y M. Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature[J]. Composite Structures, 2017, 171: 486-504. doi: 10.1016/j.compstruct.2017.03.053
[2]	杨坤, 张玮, 杜度. 复合材料夹层结构动力学特性研究进展[J]. 玻璃钢/复合材料, 2019, 9: 110-118. https://www.cnki.com.cn/Article/CJFDTOTAL-BLGF201909020.htm YANG Kun, ZHANG Wei, DU Du. The research progress of dynamic characteristics of the composite sandwich structure[J]. Fibre Reinforced Plastics/Composites, 2019, 9: 110-118. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-BLGF201909020.htm
[3]	宋丽红, 陈殿云, 张传敏. 层合梁自由振动的微分求积分析[J]. 河南科技大学学报(自然科学版), 2005, 26 (2): 89-92. https://www.cnki.com.cn/Article/CJFDTOTAL-LYGX200502025.htm SONG Lihong, CHEN Dianyun, ZHANG Chuanmin. Free vibration analysis of laminated beam by differential quadrature[J]. Journal of Henan University of Science and Technology (Natural Science), 2005, 26 (2): 89-92. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LYGX200502025.htm
[4]	贺丹, 杨万里. 基于广义变分原理和锯齿理论的高精度层合梁模型[J]. 宇航总体技术, 2017, 1 (2): 26-32. https://www.cnki.com.cn/Article/CJFDTOTAL-YHZJ201702005.htm HE Dan, YANG Wanli. A high-accuracy composite laminated beam model based on generalized variational principle and Zigzag theory[J]. Astronautical Systems Engineering Technology, 2017, 1 (2): 26-32. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YHZJ201702005.htm
[5]	惠维维, 韩宾, 张钱城, 等. 基于一种简化剪切变形理论的层合梁自由振动分析[J]. 应用力学学报, 2017, 34 (6): 1067-1071. https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201706010.htm HUI Weiwei, HAN Bin, ZHANG Qiancheng, et al. Free vibration analysis of laminated composite beams based on a simplified shear deformation theory[J]. Chinese Journal of Applied Mechanics, 2017, 34 (6): 1067-1071. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201706010.htm
[6]	NGUYEN T K, NGUYEN N D, VO T P, et al. Trigonometric-series solution for analysis of laminated composite beams[J]. Composite Structures, 2017, 160: 142-151. doi: 10.1016/j.compstruct.2016.10.033
[7]	KAPURIA S, DUMIR P C, JAIN N K. Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams[J]. Composite Structures, 2004, 64 (3/4): 317-327.
[8]	杨胜奇, 张永存, 刘书田. 一种准确预测层合梁结构层间剪应力的新锯齿理论[J]. 航空学报, 2019, 40 (11): 223028. YANG Shengqi, ZHANG Yongcun, LIU Shutian. A new zig-zag theory for accurately predicting interlaminar shear stress of laminated beam structures[J]. Acta Aeronautica et Astronautica Sinica, 2019, 40 (11): 223028. (in Chinese)
[9]	陈玲俐, 赵晓昱, 张鑫, 等. 复合材料简支梁的模态分析[J]. 农业装备与车辆工程, 2020, 58 (4): 128-130. https://www.cnki.com.cn/Article/CJFDTOTAL-SDLG202004030.htm CHEN Lingli, ZHAO Xiaoyu, ZHANG Xin, et al. Modal analysis of composite simply supported beam[J]. Agricultural Equipment & Vehicle Engineering, 2020, 58 (4): 128-130. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SDLG202004030.htm
[10]	GARG A, CHALAK H D. Novel higher-order zigzag theory for analysis of laminated sandwich beams[J]. Proceedings of the Institution of Mechanical Engineers (Part L): Journal of Materials: Design and Applications, 2021, 235 (1): 176-194. doi: 10.1177/1464420720957045
[11]	鲍四元, 周静, 陆健炜. 任意弹性边界的多段梁自由振动研究[J]. 应用数学和力学, 2020, 41 (9): 985-993. doi: 10.21656/1000-0887.410045 BAO Siyuan, ZHOU Jing, LU Jianwei. Free vibration of multi-segment beams with arbitrary boundary conditions[J]. Applied Mathematics and Mechanics, 2020, 41 (9): 985-993. (in Chinese) doi: 10.21656/1000-0887.410045
[12]	李智超, 郝育新. 微分求积法求解悬臂L梁固有振动特性研究[J]. 应用数学和力学, 2023, 44 (5): 525-534. doi: 10.21656/1000-0887.430382 LI Zhichao, HAO Yuxin. Study on natural vibration characteristics of L-shaped cantilever beams with the differential quadrature method[J]. Applied Mathematics and Mechanics, 2023, 44 (5): 525-534. (in Chinese) doi: 10.21656/1000-0887.430382
[13]	赵翔, 孟诗瑶. 基于Green函数分析Euler-Bernoulli双曲梁系统的受迫振动[J]. 应用数学和力学, 2023, 44 (2): 168-177. 10.21656/1000-0887.430298 ZHAO Xiang, MENG Shiyao. Forced vibration analysis of Euler-Bernoulli double-beam systems by means of Green's functions[J]. Applied Mathematics and Mechanics, 2023, 44 (2): 168-177. (in Chinese) 10.21656/1000-0887.430298
[14]	韩泽军, 林皋. 三维层状地基动力刚度矩阵连分式算法[J]. 大连理工大学学报, 2012, 52 (6): 862-869. https://www.cnki.com.cn/Article/CJFDTOTAL-DLLG201206016.htm HAN Zejun, LIN Gao. A continued-fraction algorithm for dynamic stiffness matrix of foundation located or embedded in three-dimensional layered subgrade[J]. Journal of Dalian University of Technology, 2012, 52 (6): 862-869. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-DLLG201206016.htm
[15]	韩泽军, 林皋, 钟红. 改进的比例边界有限元法求解层状地基动力刚度矩阵[J]. 水电能源科学, 2012, 30 (7): 100-104. https://www.cnki.com.cn/Article/CJFDTOTAL-SDNY201207029.htm HAN Zejun, LIN Gao, ZHONG Hong. Modified scaled boundary finite element solution for dynamic stiffness matrix of laminar foundation[J]. Water Resources and Power, 2012, 30 (7): 100-104. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SDNY201207029.htm
[16]	张海廷, 杨林青, 郭芳. 基于SBFEM的层状地基埋置管道动力响应求解与分析[J]. 岩土力学, 2019, 40 (7): 2713-2722. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201907025.htm ZHANG Haiting, YANG Linqing, GUO Fang. Solution and analysis of dynamic response for rigid pipe in multi-layered soil based on SBFEM[J]. Rock and Soil Mechanics, 2019, 40 (7): 2713-2722. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201907025.htm
[17]	MAN H, SONG C, GAO W, et al. A unified 3D-based technique for plate bending analysis using scaled boundary finite element method[J]. International Journal for Numerical Methods in Engineering, 2012, 91 (5): 491-515. doi: 10.1002/nme.4280
[18]	MAN H, SONG C, XIANG T, et al. High-order plate bending analysis based on the scaled boundary finite element method[J]. International Journal for Numerical Methods in Engineering, 2013, 95 (4): 331-360. doi: 10.1002/nme.4519
[19]	林皋, 张鹏冲. 板结构计算模型的新发展[J]. 计算力学学报, 2019, 36 (4): 429-440. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG201904001.htm LIN Gao, ZHANG Pengchong. New development of the computational model for the analysis of laminated plate structures[J]. Chinese Journal of Computational Mechanics, 2019, 36 (4): 429-440. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG201904001.htm
[20]	何宜谦, 王霄腾, 祝雪峰, 等. 求解粘弹性问题的时域自适应等几何比例边界有限元法[J]. 工程力学, 2020, 37 (2): 23-33. https://www.cnki.com.cn/Article/CJFDTOTAL-GCLX202002005.htm HE Yiqian, WANG Xiaoteng, ZHU Xuefeng, et al. A temporally piecewise adaptive isogeometric SBFEM for viscoelastic problems[J]. Engineering Mechanics, 2020, 37 (2): 23-33. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-GCLX202002005.htm
[21]	钟红, 贺帅, 李红军. 基于比例边界有限元的温度断裂研究[J]. 水电与抽水蓄能, 2017, 3 (3): 66-70. https://www.cnki.com.cn/Article/CJFDTOTAL-DBGC201703014.htm ZHONG Hong, HE Shuai, LI Hongjun. Thermal fracture analysis based on the polygon scaled boundary finite element method[J]. Hydropwer and Pumped Storage, 2017, 3 (3): 66-70. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-DBGC201703014.htm
[22]	SUN X, ZHANG P C, QIAO H, et al. High-order free vibration analysis of elastic plates with multiple cutouts[J]. Archive of Applied Mechanics, 2021, 91: 1837-1858. doi: 10.1007/s00419-020-01857-2
[23]	ZHANG P C, QI C Z, FANG H Y, et al. Bending and free vibration analysis of laminated piezoelectric composite plates[J]. Structural Engineering and Mechanics, 2020, 75 (6): 747-769.
[24]	ZHANG P C, QI C Z, FANG H Y, et al. Semi-analytical analysis of static and dynamic responses for laminated magneto-electro-elastic plates[J]. Composite Structures, 2019, 222: 110933. doi: 10.1016/j.compstruct.2019.110933
[25]	XIANG T S, NATARAJAN S, MAN H, et al. Free vibration and mechanical buckling of plates with in-plane material inhomogeneity: a three dimensional consistent approach[J]. Composite Structures, 2014, 118: 634-642. doi: 10.1016/j.compstruct.2014.07.043
[26]	LI J H, SHI Z Y, NING S W. A two-dimensional consistent approach for static and dynamic analyses of uniform beams[J]. Engineering Analysis With Boundary Elements, 2017, 82: 1-16. doi: 10.1016/j.enganabound.2017.05.009
[27]	LI J H, SHI Z Y, LIU L. A scaled boundary finite element method for static and dynamic analyses of cylindrical shells[J]. Engineering Analysis With Boundary Elements, 2019, 98: 217-231. doi: 10.1016/j.enganabound.2018.10.024
[28]	KANT T, SWAMINATHAN K. Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory[J]. Composite Structures, 2002, 56 (4): 329-344. doi: 10.1016/S0263-8223(02)00017-X
[29]	KHDEIR A A, REDDY J N. Free vibration of cross-ply laminated beams with arbitrary boundary conditions[J]. International Journal of Engineering Science, 1994, 32 (12): 1971-1980. doi: 10.1016/0020-7225(94)90093-0
[30]	谢瑾荣, 周翠英, 程晔, 等. 一种求解多阶梯悬臂梁自由振动问题的新方法与实验验证[J]. 工程抗震与加固改造, 2012, 34 (4): 52-55. https://www.cnki.com.cn/Article/CJFDTOTAL-GCKZ201204010.htm XIE Jinrong, ZHOU Cuiying, CHENG Ye, et al. A new method for solving free vibration of cantilever beam with multiple steps and its experimental validation[J]. Earthquake Resistant Engineering and Retrofitting, 2012, 34 (4): 52-55. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-GCKZ201204010.htm