点阵夹芯圆柱壳自由振动分析的波有限元方法

韩少燕; 高汝鑫

doi:10.21656/1000-0887.440130

点阵夹芯圆柱壳自由振动分析的波有限元方法

doi: 10.21656/1000-0887.440130

韩少燕^1,,
高汝鑫^{2, 3, ,}

1.
西安交通大学城市学院机械工程学院，西安 710018
2.
北京理工大学先进结构技术研究院，北京 100081
3.
大连理工大学工业装备结构分析国家重点实验室，辽宁大连 116024

基金项目:

国家自然科学基金青年科学基金 12102045

详细信息

作者简介:
韩少燕(1987—)，女，讲师(E-mail: hanshaoyan0809@outlook.com)

通讯作者:
高汝鑫(1991—)，男，副教授，博士生导师(通讯作者. E-mail: gaoruxin@bit.edu.cn)

中图分类号: O302
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- 被引次数: 0
出版历程
- 收稿日期: 2023-04-30
- 修回日期: 2023-10-10
- 刊出日期: 2024-01-01

A Wave Finite Element Method for Free Vibration Analysis of Lattice Core Sandwich Cylindrical Shells

HAN Shaoyan^1
,,
GAO Ruxin^{2, 3
, ,}

1.
School of Mechanical Engineering, Xi'an Jiaotong University City College, Xi'an 710018, P.R.China
2.
Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing 100081, P.R.China
3.
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

摘要

摘要: 针对点阵夹芯圆柱壳的自由振动分析发展了波有限元法. 首先基于自由波的传播规律，建立了点阵夹芯圆柱壳胞元的二维波有限元控制方程，相比于全尺寸有限元模型，显著缩减了控制方程的自由度规模；其次，基于Neumann级数推导了约束动刚度矩阵求逆的显式表达式，不仅可以提高计算效率，而且使得固有频率从控制方程中分离出来，从而将点阵夹芯圆柱壳的固有频率求解转化为单个胞元自由度规模的二次特征值问题；最后，根据结构振动模态与自由波的关系，给出了圆柱壳轴向和周向的波传播参数的表达式，进而求得点阵夹芯圆柱壳的固有频率和模态. 数值算例考虑了多种边界条件下的点阵夹芯圆柱壳自由振动问题，验证了该方法的正确性和高效性.
- 点阵夹芯 /
- 圆柱壳 /
- 自由振动 /
- 波有限元 /
- 波传播
Abstract: A wave finite element method was developed for the free vibration analysis of lattice core sandwich cylindrical shells. Firstly, based on the propagation law of free waves, governing equations for a core element of the lattice core sandwich cylindrical shell was established. Compared with the full-scale finite element model, degrees of freedom of the governing equations for a core element are significantly reduced. Secondly, an explicit expression for the inverse of the constrained dynamic stiffness matrix was derived based on the Neumann series, which not only improves computation efficiency but also separates the natural frequency from the governing equations, thereby transforming the natural frequency solution of the lattice core sandwich cylindrical shell into a quadratic eigenvalue problem of a core element. Finally, according to the relationship between the structural vibration mode and the free wave, the expressions of the axial and circumferential wave propagation parameters of the cylindrical shell were given, and the natural frequencies and modes of the lattice core sandwich cylindrical shell were obtained. Numerical examples of the free vibration analysis on a lattice core sandwich cylindrical shell under different boundary conditions verify the validity and efficiency of the proposed method.
- lattice core /
- cylindrical shell /
- free vibration /
- wave finite element /
- wave propagation

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图 1 点阵夹芯圆柱壳胞元的有限元模型

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. The finite element model for a core element in a lattice core sandwich cylindrical shell

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图 2 点阵夹芯圆柱壳及其胞元示意图

Figure 2. The lattice core sandwich cylindrical shell and a pyramidal lattice truss core element

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图 3 本文方法和有限元方法计算的模态对比(m=4, n=3)

Figure 3. Natural mode shapes (m=4, n=3) obtained with the 2 methods

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表 1 两端简支边界条件下(SS)点阵夹芯圆柱壳的自由振动频率对比

Table 1. Comparison of natural frequencies of the lattice core sandwich shell with simply supported boundary conditions at both ends

m	n=2			n=3			n=4			n=5
m	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%
1	63.85	63.84	0.015	72.41	72.36	0.069	123.59	123.55	0.032	192.13	192.09	0.021
2	188.95	188.97	-0.011	127.99	127.91	0.063	145.66	145.51	0.103	204.58	204.41	0.083
3	321.37	321.43	-0.019	217.41	217.32	0.041	194.79	194.54	0.129	232.03	231.71	0.138
4	431.87	431.95	-0.019	311.86	311.77	0.029	261.98	261.67	0.118	275.34	274.87	0.171
5	516.00	516.07	-0.014	398.52	398.42	0.025	335.31	334.96	0.104	330.19	329.62	0.173

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表 2 两端固支边界条件下(CC)点阵夹芯圆柱壳的自由振动频率对比

Table 2. Comparison of natural frequencies of the lattice core sandwich shell with clamped boundary conditions at both ends

m	n=2			n=3			n=4			n=5
m	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%
1	122.00	101.70	19.961	93.63	87.42	7.104	131.34	127.91	2.682	196.78	193.72	1.580
2	256.90	212.26	21.031	170.71	151.09	12.986	167.13	158.03	5.758	216.22	210.19	2.869
3	380.00	331.64	14.582	265.10	235.26	12.684	226.97	210.42	7.865	251.90	241.61	4.259
4	477.01	436.91	9.178	356.53	323.48	10.217	298.44	276.14	8.076	301.67	286.81	5.181
5	549.70	520.28	5.655	437.62	406.78	7.581	371.90	347.02	7.170	360.24	341.75	5.410

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表 3 两端自由边界条件下(FF)点阵夹芯圆柱壳的自由振动频率对比

Table 3. Comparison of natural frequencies of the lattice core sandwich shell with free boundary conditions at both ends

m	n=2			n=3			n=4			n=5
m	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%
-1/2(λ_x=0)	22.80	22.71	0.396	63.59	63.33	0.411	119.66	119.10	0.470	189.09	188.16	0.494
1	28.18	24.04	17.221	64.68	65.04	-0.554	120.42	120.84	-0.348	189.79	189.79	0.000
2	122.00	129.83	-6.031	93.63	97.27	-3.742	131.34	134.23	-2.153	196.78	199.29	-1.259
3	256.90	269.57	-4.700	170.71	177.66	-3.912	167.13	172.35	-3.029	216.22	220.94	-2.136
4	380.00	396.43	-4.144	265.10	275.78	-3.873	226.97	234.54	-3.228	251.90	258.51	-2.557
5	477.01	492.12	-3.070	356.53	369.24	-3.442	298.44	308.07	-3.126	301.67	309.92	-2.662

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表 4 一端固支一端简支边界条件下(CS)点阵夹芯圆柱壳的自由振动频率对比

Table 4. Comparison of natural frequencies of the lattice core sandwich shell with one end clamped and the other end simplify supported

m	n=2			n=3			n=4			n=5
m	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%
1	91.16	82.72	10.203	81.16	78.88	2.890	126.75	125.35	1.117	194.12	192.80	0.685
2	223.16	201.60	10.694	148.62	139.91	6.225	155.54	151.58	2.612	209.89	207.13	1.332
3	351.49	326.67	7.598	241.27	226.64	6.455	210.42	202.59	3.865	241.48	236.58	2.071
4	406.80	434.46	-6.367	288.68	317.77	-9.154	244.21	269.05	-9.232	263.21	280.86	-6.284
5	497.23	518.16	-4.039	377.88	402.64	-6.149	316.87	341.08	-7.098	315.70	335.73	-5.966

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表 5 一端固支一端自由边界条件下(CF)点阵夹芯圆柱壳的自由振动频率对比

Table 5. Comparison of natural frequencies of the lattice core sandwich shell with one end clamped and the other end free

m	n=2			n=3			n=4			n=5
m	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%	present F/Hz	FEM F/Hz	error δ/%
1	28.18	31.89	-11.634	64.68	64.82	-0.216	120.42	119.95	0.392	189.79	189.01	0.413
2	122.00	110.66	10.248	93.63	91.28	2.574	131.34	130.96	0.290	196.78	196.49	0.148
3	256.90	239.50	7.265	170.71	163.47	4.429	167.13	164.90	1.352	216.22	215.57	0.302
4	380.00	363.26	4.608	265.10	254.43	4.194	226.97	221.95	2.262	251.90	249.97	0.772
5	477.01	465.45	2.484	356.53	345.96	3.055	298.44	291.65	2.328	301.67	298.22	1.157

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