一种针对多裂纹弹性基础Timoshenko梁振动分析的解析方法

doi:10.7527/S1000-6893.2024.30280

固体力学与飞行器总体设计

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一种针对多裂纹弹性基础Timoshenko梁振动分析的解析方法

伍科¹^,², 彭祺擘¹(), 武新峰¹, 刘朋博³, 寇亚军¹

^1.中国航天员科研与训练中心，北京 100094
^2.中国载人航天工程办公室，北京 100071
^3.航天跃盛（杭州）信息技术有限公司上海分公司，上海 201107

收稿日期:2024-02-02 修回日期:2024-03-28 接受日期:2024-05-30 出版日期:2024-11-25 发布日期:2024-06-14
通讯作者: 彭祺擘 E-mail:poochie003@163.com

Analytical method for vibration analysis of multi-cracked Timoshenko beam structures with elastic foundations

Ke WU¹^,², Qibo PENG¹(), Xinfeng WU¹, Pengbo LIU³, Yajun KOU¹

^1.China Astronauts Research and Training Center，Beijing 　100094，China
^2.China Manned Space Agency，Beijing 　100071，China
^3.HTYS Information Technology Co. ，Ltd. Shanghai Branch，Shanghai 　201107，China

Received:2024-02-02 Revised:2024-03-28 Accepted:2024-05-30 Online:2024-11-25 Published:2024-06-14
Contact: Qibo PENG E-mail:poochie003@163.com

摘要/Abstract

摘要：

针对具有弹性基础的任意多开裂纹Timoshenko阻尼梁结构，基于分布传递函数法（DTFM）的基本形式，提出了一种用于振动分析的新颖的解析方法。对于弹性基础，采用了考虑剪切变形和转动变形的二参数连续弹簧模型；对于梁结构上的局部开裂纹，引入了一种描述裂纹处广义位移不连续的局部附加柔度矩阵，然后建立了弹性基础多裂纹阻尼梁的控制方程；采用增强型分布传递函数形式，对裂纹梁组件和非裂纹组件分别建立了局部边界矩阵，并在全局坐标系中组装成全局边界方程；最后，以积分的形式得到了多裂纹阻尼梁的封闭形式的解析解；应用解析表达式，对裂纹梁结构的模态频率、模态振型及频域响应进行分析。在数值仿真算例中，将本文方法得到的动力学响应结果与参考文献和用有限元方法得到的结果进行了对比，验证了提出方法的正确性。与有限元方法相比，提出的方法对于中、高频动力学响应计算具有更高的精度和效率，可以有效地应用于弹性基础梁的裂纹检测问题中。

关键词: 分布传递函数法, 弹性基础, 多裂纹梁, 模态频率, 频域响应, 高频动力学

Abstract:

This study proposes a novel analytical method for vibration analysis of arbitrary multi-crack Timoshenko damping beam structures with elastic foundations based on the Distributed Transfer Function Method （DTFM）. A two-parameter continuous spring model considering shear and rotational deformation is adopted for the elastic foundation. For local cracks on the beam structure， a local additional compliance matrix induced by the crack is employed， and the governing equation of the multi-cracked Timoshenko damping beam with an elastic foundation established. Using the augmented distributed transfer function method， we establish the local boundary matrices of the non-cracked and cracked beam components， respectively， and then obtain the global boundary equation in the global coordinate system. Finally， a closed-form analytical solution is derived in integral form， and the natural frequencies， mode shapes， and frequency responses of the cracked beam structures can be derived. In numerical examples， the results from previous papers and the Finite Element Method （FEM） are employed to validate the correctness of the proposed method， and the effect of the cracks and the elastic foundations on dynamic behaviors investigated. With higher computational accuracy and efficiency than the FEM for mid-to-high frequency dynamic responses， the proposed method can be used for crack detection of elastic foundation beams.

Key words: Distributed Transfer Function Method (DTFM), elastic foundation, multi-cracked beam, natural frequency, frequency response, high-frequency

中图分类号:

V414.3

伍科, 彭祺擘, 武新峰, 刘朋博, 寇亚军. 一种针对多裂纹弹性基础Timoshenko梁振动分析的解析方法[J]. 航空学报, 2024, 45(22): 230280.

Ke WU, Qibo PENG, Xinfeng WU, Pengbo LIU, Yajun KOU. Analytical method for vibration analysis of multi-cracked Timoshenko beam structures with elastic foundations[J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(22): 230280.

图/表 16

图 1

图 2

图 3

图 4

表1

无裂纹弹性基础简支梁的一阶归一化自然频率

$K ¯ w$	$K ¯ p / π$	无量纲自然频率
		$λ = 1 / 120$			$λ = 1 / 15$		$λ = 1 / 5$
		本文	文献［31］	文献［32］	本文	文献［31］	本文	文献［31］
0	0	3.141 40	3.141 43	3.141 5	3.129 91	3.130 24	3.045 33	3.047 99
	0.5	3.476 59	3.476 59	3.476 7	3.467 11	3.466 71	3.398 64	3.394 58
	1.0	3.735 87	3.735 87	3.736 0	3.727 40	3.726 56	3.667 05	3.658 02
	2.5	4.296 89	4.296 86	4.297 0	4.289 72	4.288 09	4.239 47	4.218 34
10²	0	3.748 23	3.748 23	3.748 3	3.739 80	3.738 94	3.679 77	3.670 50
	0.5	3.960 68	3.960 67	3.960 8	3.952 86	3.951 68	3.897 52	3.883 97
	1.0	4.143 58	4.143 56	4.143 7	4.136 15	4.134 71	4.083 87	4.066 36
	2.5	4.582 28	4.582 26	4.582 4	4.575 46	4.573 47	4.527 93	4.499 13
10⁴	0	10.024 1	10.024 0	10.024	10.015 0	9.995 82	7.376 58	7.340 81
	0.5	10.036 1	10.036 1	10.036	10.027 0	10.007 7	7.376 58	7.340 88
	1.0	10.048 2	10.048 1	10.048	10.039 0	10.019 6	7.376 58	7.340 95
	2.5	10.084 0	10.083 9	10.084	10.074 8	10.055 1	7.376 58	7.341 16
10⁶	0	31.623 0	31.621 7	31.623	12.776 6	12.772 2	7.356 58	7.350 81
	0.5	31.623 4	31.622 1	31.623	12.776 6	12.772 2	7.356 58	7.350 81
	1.0	31.623 8	31.622 4	31.624	12.776 6	12.772 2	7.356 58	7.350 81
	2.5	31.625 0	31.623 6	31.625	12.776 6	12.772 2	7.356 58	7.350 81

表1

表2

无裂纹弹性基础梁的前三阶及第100阶归一化自然频率（K¯w=1/30, α1=0, h/L=0.2, v=0.3, κ=5/6）

边界	$α 2$	$K ¯ p$	方法	无量纲自然频率
边界	$α 2$	$K ¯ p$	方法	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 100$
H-H	0	0	本文	9.275	32.166	61.458	1 443.213
			LSM（n=20）	9.274	32.15	61.29
			LSM（n=100）	9.274	32.17	61.45
			TMM	9.274	32.17	61.46
	1/2	0	本文	9.531 9	32.239	61.495	1 443.214
			LSM（n=20）	9.531	32.22	61.33
			LSM（n=100）	9.532	32.24	61.49
			TMM	9.532	32.24	61.49
	1/2	$π 2 / 300$	本文	11.681	34.730	64.458	1 469.177
			LSM（n=20）	11.67	34.41	64.40
			LSM（n=100）	11.68	34.67	64.64
			TMM	11.68	34.72	64.45
	1	0	本文	9.784 8	32.311	61.532	1 443.216
			LSM（n=20）	9.784	32.29	61.38
			LSM（n=100）	9.784	32.31	61.53
			TMM	9.785	32.31	61.53
	1	$π 2 / 300$	本文	13.804	37.524	67.734	1 483.261
			LSM（n=20）	13.79	37.44	67.42
			LSM（n=100）	13.80	37.52	67.72
			TMM	13.80	37.52	67.73
C-H	2/3	$π 2 / 300$	本文	15.196	40.336	68.965	1 469.177
			LSM（n=20）	15.18	40.09	68.40
			LSM（n=100）	15.20	40.34	68.96
			TMM	15.20	40.34	68.97
C-C	2/3	$π 2 / 300$	本文	19.225	42.246	70.305	1 483.623
			LSM（n=20）	19.20	42.29	70.00
			LSM（n=100）	19.22	42.24	70.31
			TMM	19.22	42.24	70.31

表2

图 5

表3

部分支撑的双裂纹弹性基础梁自然频率

边界	裂纹尺寸			$K ¯ w 1$	$K ¯ p 1$	无量纲自然频率
	裂纹尺寸			LSM方法（n=100）				本文方法
	$δ 1 h$	$δ 2 h$	$δ 3 h$	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 100$
C-F	0	0	0	0	0	3.410	18.36	43.71	72.56	3.409 7	18.363	43.709	72.567	1 441.97
				1/30	0	4.635	18.61	43.81	72.62	4.635 4	18.614	43.811	72.626	1 441.97
				1/30	$π 2 / 300$	7.563	24.58	50.07	79.59	7.563 3	24.583	50.073	79.609	1 467.70
	0.2	0.1	0.2	0	0	3.212	17.27	39.68	69.82	3.213 2	17.252	39.678	69.937	1 428.00
				0	0	3.212	17.27	39.68	69.82	3.211 2	17.173	39.544	69.341	1 412.10
				1/30	0	4.491	17.53	39.79	69.89	4.491 7	17.518	39.789	69.998	1 428.01
				1/30	0	4.491	17.53	39.79	69.89	4.490 3	17.438	39.654	69.402	1 412.10
				1/30	$π 2 / 300$	7.517	23.87	46.59	76.93	7.518 4	23.864	46.594	77.042	1 458.04
				1/30	$π 2 / 300$	7.517	23.87	46.59	76.93	7.504 3	23.613	46.404	75.837	1 446.27
H-H	0	0	0	0	0	9.274	32.17	61.45	93.24	9.274 0	32.166	61.458	93.259	1 443.21
				1/30	0	9.785	32.31	61.53	93.28	9.784 8	32.311	61.532	93.308	1 443.21
				1/30	$π 2 / 300$	13.80	37.52	67.72	100.6	13.804	37.524	67.734	100.62	1 483.26
	0.2	0.1	0.2	0	0	8.345	29.03	60.03	91.26	8.340 4	29.047	60.107	91.139	1 442.66
				0	0	8.345	29.03	60.03	91.26	8.401 2	28.940	59.671	90.416	1 419.19
				1/30	0	8.909	29.18	60.11	91.31	8.904 3	29.205	60.183	91.189	1 442.66
				1/30	0	8.909	29.18	60.11	91.31	8.961 3	29.099	59.747	90.466	1 419.19
				1/30	$π 2 / 300$	13.20	34.85	66.41	98.69	13.199	34.876	54.413	66.489	1 469.17
				1/30	$π 2 / 300$	13.20	34.85	66.41	98.69	13.224	34.719	65.902	97.630	1 450.33
C-H	0	0	0	0	0	13.44	36.88	65.19	95.75	13.436	36.877	65.195	95.769	1 456.50
				1/30	0	13.79	37.01	65.26	95.79	13.794	37.005	65.266	95.817	1 456.50
				1/30	$π 2 / 300$	17.14	41.74	71.18	102.9	17.140	41.742	71.186	102.95	1 483.26
	0.2	0.1	0.2	0	0	12.67	34.07	63.64	93.84	12.662	34.074	63.711	93.728	1 442.94
				0	0	12.67	34.07	63.64	93.84	12.678	33.927	63.333	93.020	1 419.19
				1/30	0	13.04	34.21	63.71	93.89	13.041	34.211	63.783	93.777	1 442.94
				1/30	0	13.04	34.21	63.71	93.89	13.056	34.063	63.405	93.069	1 419.19
				1/30	$π 2 / 300$	16.59	39.27	69.77	101.1	16.591	39.284	69.842	100.95	1 469.17
				1/30	$π 2 / 300$	16.59	39.27	69.77	101.1	16.581	39.076	69.317	100.02	1 456.21
C-C	0	0	0	0	0	17.99	41.19	68.64	98.06	17.994	41.189	68.646	98.083	1 469.17
				1/30	0	18.27	41.31	68.71	98.11	18.265	41.305	68.714	98.130	1 469.17
				1/30	$π 2 / 300$	20.95	45.71	74.40	105.1	20.953	45.708	74.410	105.11	1 485.45
	0.2	0.1	0.2	0	0	17.60	38.36	67.04	96.26	17.591	38.347	67.111	96.149	1 443.16
				0	0	17.60	38.36	67.04	96.26	17.570	38.158	66.765	95.442	1 419.89
				1/30	0	17.88	38.49	67.10	96.31	17.868	38.470	67.180	96.196	1 443.17
				1/30	0	17.88	38.49	67.10	96.31	17.847	38.279	66.834	95.489	1 419.89
				1/30	$π 2 / 300$	20.62	43.23	72.95	103.3	20.616	43.229	73.023	103.17	1 469.17
				1/30	$π 2 / 300$	20.62	43.23	72.95	103.3	20.569	42.954	72.544	102.25	1 465.25

表3

图 6

图 7

图 8

表4

无裂纹无弹性基础二维框架结构模态频率

边界（i-j-k-l）	模态自然频率/Hz
	FEM方法						本文方法（5组件）
	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 100$		$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 100$
	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	260单元	1 721单元	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 100$
C-H-H-C	63.887	113.156	130.954	148.367	23 867.01	23 723.38	63.886	113.151	130.951	148.360	23 629.031
C-C-C-C	64.108	125.551	150.633	179.262	23 877.11	23 761.48	64.107	125.549	150.625	179.257	23 642.645
H-H-H-H	44.102	86.858	116.941	142.524	23 570.97	23 403.27	44.101	86.857	116.935	142.519	23 298.577
H-C-C-H	44.211	87.704	139.165	154.692	23 879.99	23 757.23	44.210	87.702	139.161	154.685	23 634.636

表4

图 9

图 10

表5

多段弹性基础多裂纹框架结构模态频率

边界	$K ¯ w 1$	$K ¯ p 1$	$K ¯ w 2$	$K ¯ p 2$	裂纹尺寸				自然模态频率/Hz
边界	$K ¯ w 1$	$K ¯ p 1$	$K ¯ w 2$	$K ¯ p 2$	$δ 1 h$	$δ 2 h$	$δ 3 h$	$δ 4 h$	$Ω 1$	$Ω 2$	$Ω 3$	$Ω 4$	$Ω 100$
C-H-H-C	0	0	0	0	0.1	0.1	0.1	0.1	63.867	113.084	130.930	148.272	23 628.63
	1/30	0	1/30	0	0	0	0	0	120.543	148.866	215.241	462.730	39 557.27
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0	0	0	0	120.631	148.901	215.258	462.869	40 546.91
	0	0	0	0	0.2	0.2	0.2	0.2	63.812	112.890	130.869	148.020	23 627.67
	1/30	0	1/30	0	0.2	0.3	0.3	0.2	120.024	148.377	215.058	460.048	39 405.16
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0.3	0.4	0.4	0.3	119.560	147.865	214.874	457.645	39 300.34
C-C-C-C	0	0	0	0	0.1	0.1	0.1	0.1	64.089	125.542	150.553	179.248	23 639.87
	1/30	0	1/30	0	0	0	0	0	153.416	216.941	259.464	526.709	40 373.36
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0	0	0	0	153.518	216.990	259.464	526.872	40 381.05
	0	0	0	0	0.2	0.2	0.2	0.2	64.036	125.521	150.343	179.223	23 637.89
	1/30	0	1/30	0	0.2	0.3	0.3	0.2	152.837	216.050	259.059	523.455	40 217.22
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0.3	0.4	0.4	0.3	152.361	215.156	258.586	520.579	40 087.45
H-H-H-H	0	0	0	0	0.1	0.1	0.1	0.1	44.088	86.853	116.858	142.474	23 285.47
	1/30	0	1/30	0	0	0	0	0	120.543	148.866	215.241	462.730	39 557.27
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0	0	0	0	120.631	148.901	215.258	462.869	39 561.00
	0	0	0	0	0.2	0.2	0.2	0.2	44.047	86.841	116.634	142.343	23 247.32
	1/30	0	1/30	0	0.2	0.3	0.3	0.2	120.024	148.377	215.058	460.048	39 405.16
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0.3	0.4	0.4	0.3	119.560	147.865	214.874	457.645	39 300.34
H-C-C-H	0	0	0	0	0.1	0.1	0.1	0.1	44.197	87.701	139.129	154.634	23 627.03
	1/30	0	1/30	0	0	0	0	0	153.416	216.941	259.464	526.709	40 373.36
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0	0	0	0	153.518	216.990	259.464	526.872	40 381.05
	0	0	0	0	0.2	0.2	0.2	0.2	44.157	87.699	139.037	154.490	23 604.17
	1/30	0	1/30	0	0.2	0.3	0.3	0.2	152.724	215.660	258.732	523.331	40 150.16
	1/30	$π 2 / 300$	1/30	$π 2 / 300$	0.3	0.4	0.4	0.3	152.361	215.156	258.586	520.579	40 087.45

表5

图 11

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一种针对多裂纹弹性基础Timoshenko梁振动分析的解析方法

Analytical method for vibration analysis of multi-cracked Timoshenko beam structures with elastic foundations

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