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: 231
- HTML全文浏览量: 100
- PDF下载量: 33
- 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

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图 1 四层复合梁示意图

Figure 1. The 4 layered composite beam

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图 2 四阶谱单元

Figure 2. The 4th order spectral element

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图 3 梁层模型图

Figure 3. The model for the beam layer

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图 4 (0°/90°/0°/90°)四层复合梁

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

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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

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图 6 梁各阶谱单元

Figure 6. The different orders of spectral elements

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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

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图 8 三层夹层梁

Figure 8. The 3-layered sandwich beam

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图 9 三层简支夹层梁振动频率

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

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图 10 (0°/90°/0°)三层复合梁

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

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图 11 阶梯型悬臂梁(单位: cm)

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

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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

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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

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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

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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

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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

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