摘要高模态密度结构的宽频振动分析问题是声振分析领域内关注的重点问题之一，可实现宽频振动预测的数值分析方法是该领域内重要的研究内容，有效的宽频振动数值分析方法应在低频至高频域可同时提供精准的数值解。然而，由于明显的耗散误差和计算成本过高导致传统有限元方法（Traditional Finite Element Methods, TFEMs）在对高模态密度结构进行宽频振动分析时，难以在高频域提供精准的数值解，致使无法实现有效的宽频振动分析。而小波有限元分析方法（Wavelet Finite Element Methods, WFEMs）在进行结构分析时具有潜在的求解效率优势，并且可大幅度降低耗散误差带来的影响。为此，本文首先构造了基于小波有限元理论进行宽频振动分析时的自耦合算法，并据此介绍了小波有限元方法对高模态密度结构进行宽频振动分析的架构，形成了宽频小波有限元分析方法（Wide Wavelet Finite Element Method, WWFEM）。随后，采用数值分析研究方法，基于WWFEM对具有解析解的高模态密度薄板结构进行了宽频振动分析。最后，采用实验分析研究方法，预测了高模态密度结构在宽频域内的振动响应。在此基础上，对比分析了小波有限元方法在进行高频振动分析时的收敛性和宽频振动分析的有效性等。可为依据小波有限元分析方法解决圆柱壳、曲壳等高模态密度结构宽频振动分析问题提供理论参考。
Abstract：The dynamic analysis of high-modal density structures in a wide-frequency domain is a significant problem in the field of acoustic and dynamic analysis. As well known, the numerical methods should provide the accurate numerical solutions in both low and high frequency domains when proceeding the dynamic analysis in a wide-frequency domain. However, due to the obvious dispersion errors and high computational cost, Traditional Finite Element Methods (TFEMs) are difficult to provide accurate numerical solutions in the high-frequency domain when performing the dynamic analysis of high-modal-density structures. Fortunately, the Wavelet Finite Element Methods (WFEMs) have potential solution to provide the accurate numerical solutions with low computational cost when proceeding the structural analysis. Meanwhile, the WFEMs can greatly reduce the impact of dispersion errors when proceeding the dynamic analysis with refined mesh. Firstly, this paper will mainly introduce the formula to construct the self-coupling algorithm based on the WFEMs and discusses the detailed procedure of the wavelet finite element methods when performing the dynamic analysis of the high-modal density structures in a wide-frequency domain. In view of the above, a thin plate, which has analytical solutions, with high modal density is adopted for analyzing the numerical stability and validity of wavelet finite element in a wide-frequency domain when proceeding the dynamic analysis. Therefore, the convergence and validity of wavelet finite element method in high-frequency domain and wide-frequency domain is compared with the traditional finite element methods based on the numerical and experimental studies. It provides a theoretical reference for solving the problem of wide-frequency dynamic analysis of high-modal density structures, such as cylindrical and shells based on wavelet finite element methods.
耿佳1,2,李明1,2,张兴武1,2,杨来浩1,2,陈雪峰1,2. 高模态密度结构宽频振动分析的小波有限元方法实现[J]. 振动与冲击, 2023, 42(1): 54-65.
GENG Jia1,2, LI Ming1,2, ZHANG Xingwu1,2, YANG Laihao1,2, CHEN Xuefeng1,2. Implementation of WFEM for broadband vibration analysis of high modal density structures. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(1): 54-65.
[1] 张 月, 孙 伟. 考虑应变依赖性的硬涂层圆柱壳振动特性有限元分析[J]. 振动与冲击, 2018, 37 (12):17-22.
Zhang Yue, Sun Wei. A finite element analysis of the vibration characteristics of a hard-coating cylindrical shell considering the strain dependence of hard coating [J]. Journal of Vibration and Shock, 2018, 37 (12): 17-22.
[2] 王 迪, 朱 翔, 李天匀，等. 基于能量有限元法的功能梯度梁振动分析[J]. 振动与冲击, 2018, 37 (3): 119-124.
Wang Di, Zhu Xiang, Li Tianyun, et al. Vibration analysis of a FGM beam based on energy finite element method [J], Journal of Vibration and Shock, 2018, 37 (3): 119-124.
[3] 刘文玺, 周其斗, 谭 路,等. 尾后轴承刚度对潜艇结构声辐射特性的影响[J]. 船舶力学, 2018, (2): 235-247.
LIU Wenxi, ZHOU Qidou, TAN Lu, et al. Effect of rear bearing stiffness on vibro-acoustic radiation of submarine[J] Journal of Ship Mechanics, 2018, (2): 235-247.
[4] 楼京俊, 张阳阳, 俞翔. 桨-轴-艇纵向耦合振动机理研究[J]. 应用力学学报, 2018, (1)：54-59.
LOU Jingjun, ZHANG Yangyang, YU Xiang. Study on longitudinal coupling vibration mechanism of propeller-shaft-hull system [J] Chinese Journa of Applied Mechanics, 2018, (1): 54-59.
[5] 乐瀚洋. 潜艇结构模型的全频段振动声辐射分析[D]. 大连：大连理工大学, 2015.
LE Hanyang. Analysis of vibration and Acoustics Radiation for a submarine in Full-frequency Range [D], Dalian: Dalian University of Technology, 2015.
[6] 王 仲. 湍流激励下潜器蒙皮减阻降噪性能研究[D]. 哈尔滨：哈尔滨工程大学, 2017.
WANG Zhong, Research on the Drag & Noise Reduction Performance of Submersible Skin under Turbulent Excitation [D], Harbin: Harbin Engineering University, 2017.
[7] Brian Mace, Desmet W, Pluymers B. Mid-Frequency Methods in Sound and Vibration—Part A [J]. Journal of Sound and Vibration, 2013, 332: 1895-1896.
[8] Mace B, Desmet W, Pluymers B. Mid-Frequency Methods in Sound and Vibration—Part B [J]. Journal of Sound and Vibration, 2013, 332 (9): 2131.
[9] 纪 琳. 中频振动分析方法[M], 北京: 机械工业出版社, 2013.
JI Lin，Mid-frequency Vibration Analysis Method[M], China Machine Press, Beijing, 2013.
[10] 刘忠文, 李天匀, 骆东平. 某潜艇综合声纳声腔自噪声的统计能量方法分析[J]. 舰船科学技术, 2000, (3): 61-64.
LIU Zhongwen, LI Tianyun, LUO Dongping. Statistical Energy Analysis of the Self-noise of a Submarine Integrated Sonar Cavity [J], Ship Science and Technology, 2000, (3): 61-64.
[11] 所 俊, 张文光. 水面舰艇宽频振动噪声结构影响因子量化分析[J]. 海军航空工程学院学报, 2016, 31 (2): 179-184.
SUO Jun, ZHANG Wenguang. Reasearch on structural influence factors of broadband vibration and noise of a typical surface ship [J], Journal of Naval Aeronautical and Astronautical University, 2016, 31 (2): 179-184.
[12] 于 汉, 李 清, 杨德庆. 水面舰船粘性流场和流噪声的数值计算[J]. 中国舰船研究, 2017, (6): 22-29.
YU Han, LI Qing, YANG Deqing. Numerical simulation of viscous flow and hydrodynamic noise in surface ship [J], Chinese Journal of Ship Research, 2017, (6): 22-29.
[13] 刘卫东. 潜艇隔振仪器中的动力吸振装置设计与仿真[J]. 舰船科学技术, 2017, (10): 119-121.
LIU Weidong. Design and simulation of dynamic vibration absorber in submarine vibration isolation instrument [J], Ship Science and Technology, 2017, (10): 119-121.
[14] 孙卫红, 晏 欣. 潜艇振动与噪声控制技术的最新研究进展[J]. 噪声与振动控制, 2012, 32 (5): 6-10.
SUN Weihong, YAN Xin. Some Progress of the research of vibration and noise control technology for submarine [J]. Noise and Vibration Control, 2012, 32 (5): 6-10.
[15] 姜荣俊, 何 琳. 有源振动噪声控制技术在潜艇中的应用研究[J]. 噪声与振动控制, 2005, 25 (2): 1-6.
JIANG Rongjun, HE Lin. Application research of active noise and vibration control technology in submarines [J]. Noise and Vibration Control, 2005, 25 (2): 1-6.
[16] 江文成. 潜艇流噪声与流固耦合作用下流激噪声的数值模拟[D]. 上海：上海交通大学, 2013.
JIANG Wencheng. Numerical simulation on flow noise and flow-excited noise of submarine under fluid structure interaction [D], Shanghai: Shanghai Jiao Tong University, 2013.
[17] Ko J , Kurdila A J , Pilant M S. Triangular wavelet based finite elements via multivalued scaling equations [J]. Computer Methods in Applied Mechanics & Engineering, 1997, 146 (1-2): 1-17.
[18] Diaz L A , Martin M T , Vampa V. Daubechies wavelet beam and plate finite elements [J]. Finite Elements in Analysis and Design, 2009, 45 (3): 200-209.
[19] Xiang J, Chen X, He Z, et al. A new wavelet-based thin plate element using B-spline wavelet on the interval [J]. Computational Mechanics, 2007, 41 (2): 243-255.
[20] Mi T L, Sun B B, Yang H Z. Adaptive mesh based on second generation wavelet simulation wave propagation for 2D acoustic wave equation [J]. Chinese Journal of Geophysics, 2009, 52 (11): 2862-2869.
[21] Wang Y, Chen X, He Y, et al. New decoupled wavelet bases for multiresolution structural analysis [J]. Structural Engineering and Mechanics, 2010, 35 (2): 175-190.
[22] Xiang J W, Jiang Z S, Xu J Y. A Wavelet-Based Finite Element Method for Modal Analysis of Beams [J]. Advanced Materials Research, 2010, 97-101:2728-2731.
[23] Xiang J, Chen X, He Z, et al. A new wavelet-based thin plate element using B-spline wavelet on the interval [J]. Computational Mechanics, 2007, 41 (2): 243-255.
[24] Hughes, T J R. The finite element method: linear static and dynamic finite element analysis [M]. Chicago: Courier Corporation, 2012.
[25] 姚德源. 统计能量分析原理及其应用[M]. 北京：北京理工大学出版社, 1995.
YAO Deyuan. Principle and application of statistical energy analysis [M]. Beijing: Beijing Institute of Technology Press, 1995.
