基于高斯小波函数和线性表达法的开口板自由振动特性研究

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振动与冲击 ›› 2024, Vol. 43 ›› Issue (6) : 29-37.

论文

郭文杰，柴天建，颜建伟，洪显

作者信息 +

Natural vibration characteristics of an open plate based on the Gaussian wavelet function and linear expression

GUO Wenjie，CHAI Tianjian，YAN Jianwei，HONG Xian

Author information +

文章历史 +

摘要

研究开口板自振特性的求解方法，将开口部位视为厚度为零的板，引入局域化特性的高斯小波函数作为位移形函数来捕捉厚度突变的情况，提高解的精确度。本文提出线性表达方法解耦位移形函数与边界条件，基本思路是通过高斯消元法找到约束条件矩阵中线性无关的列向量，将位移形函数中的未知系数转变为线性无关系数列向量的线性表达，从而将有约束问题转变为无约束问题。对四边简支和四边固定的开口板进行分析，结合有限元方法计算结果，讨论解的收敛性和准确性。研究了不同开口尺寸、开口形状对自振频率的影响，得到开口尺寸、开口形状与自振频率关系曲线，对影响的原因进行了解释。最后计算了不同边界约束条件下多开口板的自振频率。

Abstract

Research the solution method for the natural vibration characteristics of open plates, treating the open area as a plate with zero thickness, and introducing localized Gaussian wavelet functions as displacement shape functions to capture sudden changes in thickness, thereby improving the accuracy of the solution. In this paper, a linear expression method is proposed to decouple the displacement shape function and boundary conditions. The basic idea is to find the Linear independence column vector in the constraint matrix through the Gauss elimination method, and transform the unknown coefficient in the displacement shape function into the linear expression of the linearly independent sequence vector, thus transforming the constrained problem into the unconstrained problem. Analyze the simply supported and fixed open plates on four sides, and discuss the convergence and accuracy of the solution based on the finite element method calculation results. We have studied the influence of different opening sizes and shapes on natural frequency, obtained the relationship curve between opening size and shape and natural frequency, and explained the reasons for the influence. Finally, the natural frequencies of multi opening plates under different boundary constraints are calculated.

导出引用

郭文杰，柴天建，颜建伟，洪显. 基于高斯小波函数和线性表达法的开口板自由振动特性研究[J]. 振动与冲击, 2024, 43(6): 29-37

GUO Wenjie，CHAI Tianjian，YAN Jianwei，HONG Xian. Natural vibration characteristics of an open plate based on the Gaussian wavelet function and linear expression[J]. Journal of Vibration and Shock, 2024, 43(6): 29-37

参考文献

[1] 张俊, 李天匀, 朱翔. 多开口矩形板自由振动特性分析[J]. 振动与冲击, 2020,39(14):142-147. Zhang Jun, Li Tian-yun, Zhu Xiang. Free vibration characteristics analysis on a rectangular plate with multiple cutouts[J]. Journal of Vibration and Shock, 2020,39(14):142-147. [2] 高世兵. 钢弹簧浮置板减振轨道在城市地铁中的应用[J]. 铁道工程学报, 2008,25(3):88⁃91. Gao Shi-bing. Application of floating slab damping roadbed with steel spring in metro[J]. Journal of Railway Engineering Society,2008(3): 88⁃91. [3] Yang Guoshu. The effect of cutout on natural frequencies of thin plate [J]. Journal of Beijing Institute of Technology(English Edition), 1996(1):7-12. [4] CHANG C N, CHIANG F K. Vibration analysis of a thick plate with an interior cutout by a finite element method[J]. Journal of Sound&Vibration,1988,125(3):477-486. [5] 张媛, 傅成, 孟春玲. 开孔对薄板振动频率影响的有限元分析[J]. 机械设计与制造, 2005(7): 127-128. ZHANG Yuan, FU Cheng, MENG Chun ling. Finite element analysis on the effect of cut-out to the natural frequencies of thin plate［J］.Machinery Design and Manufacture,2005(7):127-128. [6] 邱昌林，陈志刚，邓轶，等.开孔平板水下振动及声辐射特性［J］.中国舰船研究，2013，8（6）：75⁃80. QIU Chang⁃lin，CHEN Zhi⁃gang，DENG Yi，et al. The characteristics of vibration and sound radiation of underwater perforated plates［J］. Chinese Journal of Ship Research，2013，8（6）：75⁃80. [7] 华洪良，廖振强，张相炎. 轴向移动悬臂梁高效动力学建模及频率响应分析. 力学学报, 2017, 49(6): 1390-1398 Hua Hongliang，Liao Zhenqiang，Zhang Xiangyan. An efficient dynamic modeling method of an axially moving cantilever beam and frequency response analysis. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(6): 1390-1398. [8] Qu Y , Wu S , Chen Y , et al. Vibration analysis of ring-stiffened conical–cylindrical–spherical shells based on a modified variational approach[J]. International Journal of Mechanical Sciences, 2013, 69:72-84. [9] TAKAHASHI, Shin. Vibration of Rectangular Plates with Circular Holes[J]. Transactions of the Japan Society of Mechanical Engineers, 2008, 24(139):183-187. [10] G. MUNDKUR, R. B. BHAT, S. NERIYA. Vibration Of Plates With Cut-Outs Using Boundary Characteristic Orthogonal Polynomial Functions In The Rayleigh-ritz Method[J]. Journal of Sound & Vibration, 1994, 176(1): 136-144. [11] D.R. AVALOS, P.A.A. LAURA. Transverse vibrations of simply supported rectangular plates with two rectangular cutouts[J]. Journal of Sound and Vibration, 2003, 267(4): 967-977. [12] H.A. LARRONDO, D.R. AVALOS, P.A.A. LAURA, et al. Vibrations of simply supported rectangular plates with varying thickness and same aspect ratio cutouts[J]. Journal of Sound and Vibration, 2001, 244(4): 738-745. [13] 王栋,禹志刚.四边形板开孔动力学优化设计研究[J].机械强度,2012,34(03):459-464. WANG Dong YU ZhiGang. optimization of cutout inside quadrilateral plate for maximization of eigenfrequency[J]. Journal of Mechanical Strength,2012,34(03):459-464. [14] 王栋, 钟习建, 禹志刚. 开孔对矩形板动力性能影响分析[J]. 噪声与振动控制, 2009, 29(6): 53-57. WANG Dong ZHONG Xi-ji YU Zhi-gang. effects of a cutout inside a rectangular plate on the plate dynamics. [J] Noise and Vibration Control, 2009, 29(6): 53-57. [15] GUOYONG JIN, XIANGLONG MA, SHUANGXIA SHI, et al. A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions[J]. Applied Acoustics, 2014, 85: 82-96. [16] GUOYONG JIN, TIANGUI YE, XUEREN WANG, et al. A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions[J]. Composites Part B: Engineering, 2016, 89: 230-252. [17] JIN G, YE T, WANG X, et al. A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions[J]. Composites Part B Engineering, 2016, 89(Mar.): 230-252. [18] 王旻昊, 李凯, 邱永康, 等. 基于傅里叶级数法的开孔板振动固有特性分析[J]. 中国舰船研究, 2017, 12(4): 102-109. Wang Minhao, Li Kai, Qiu Yongkang, et al. Natural characteristics analysis of perforated plate vibration based on Fourier series method [J] Chinese Ship Research, 2017, 12 (4): 102-109. [19] 鲍四元, 邓子辰. 薄板弯曲自由振动问题的高精度近似解析解及改进研究[J]. 应用数学和力学, 2016, 37(11): 1169-1180. Bao Siyuan, Deng Zichen. A high-precision approximate analytical solution and improvement study for the free vibration problem of thin plate bending [J] Applied mathematics and Mechanics, 2016, 37 (11): 1169-1180. [20] 邱永康, 李天匀, 朱翔, 等. 任意边界条件下中心开口矩形板自由振动特性分析[J]. 振动与冲击, 2017, 36(20): 112-117. Qiu Yong⁃kang，Li Tian⁃yun，Zhu Xiang，et al. The free vibration characteristics analysis of rectangular plate with central opening using in arbitrary boundary conditions［J］. Journal of Vibration and Shock，2017，36（20）：112⁃117. [21] SHUANGXIA SHI, BIN XIAO, GUOYONG JIN, et al. Modeling and Simulation of Transverse Free Vibration Analysis of a Rectangular Plate with Cutouts Using Energy Principles[J]. Shock & Vibration, 2018, 2018(PT.6):1-16. [22] JIE DENG, YUXIN XU, ORIOL GUASCH, et al. Nullspace technique for imposing constraints in the Rayleigh–Ritz method[J]. Journal of Sound and Vibration, 2022, 527: 116812. [23] 陆斌, 陈跃华, 冯志敏, 等. 基于Chebyshev-变分法的复杂开口形状矩形薄板弯曲振动特性分析[J]. 振动与冲击, 2020, 39(2): 178-187. Lu Bin，Chen Yuehua，Feng Zhimin，et al. Bending vibration characteristics analysis of a rectangular plate with complexopening using Chebyshev⁃variational method［J］. Journal of Vibration and Shock，2020，39（2）：178⁃187. [24] WENJIE GUO, ZHOU YANG, QINGSONG FENG, et al. A new method for band gap analysis of periodic structures using virtual spring model and energy functional variational principle[J]. Mechanical Systems and Signal Processing, 2022, 168: 108634. [25] 杨舟, 冯青松, 邓杰, 等. 采用高斯展开法分析组合式开口矩形Mindlin板的弯曲自振特性[J]. 振动工程学报, 2022, 35(5): 1130-1137. YANG Zhou1，FENG Qing-song1，DENG Jie. et al. Flexural vibration analysis of rectangular Mindlin plate with combined openings using Gaussian expansion method. [J]. Journal of Vibration Engineering, 2022, 35(5): 1130-1137. [26] JIE DENG, LING ZHENG, PENGYUN ZENG, et al. Passive constrained viscoelastic layers to improve the efficiency of truncated acoustic black holes in beams[J]. Mechanical Systems and Signal Processing, 2019, 118: 461-476. [27] JIE DENG, LING ZHENG, ORIOL GUASCH, et al. Gaussian expansion for the vibration analysis of plates with multiple acoustic black holes indentations[J]. Mechanical Systems and Signal Processing, 2019, 131: 317-334. [28] JIE DENG, LING ZHENG, ORIOL GUASCH, et al. Gaussian expansion for the vibration analysis of plates with multiple acoustic black holes indentations[J]. Mechanical Systems and Signal Processing, 2019, 131: 317-334. [29] JIE DENG, ORIOL GUASCH, LAURENT MAXIT, et al. An artificial spring component mode synthesis method for built-up structures[J]. International Journal of Mechanical Sciences, 2023, 243: 108052. [30] 徐芝纶著. 弹性力学第5版(下册)[M]. 高等教育出版社, 2016：19-22 Xu Zhilun Elastic Mechanics 5th Edition [M] Higher Education Press, 2016：80-100 [31] J.W. YAN, J.H. ZHU, C. LI, et al. Decoupling the effects of material thickness and size scale on the transverse free vibration of BNNTs based on beam models[J]. Mechanical Systems and Signal Processing, 2022, 166: 108440.