Vibration response analytical solutions of cantilever beam with tip mass and spring constraints
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摘要:
根据端部带质量和弹簧约束悬臂梁的特征值条件,提出了一种特征变换方法,获得了带约束悬臂梁广义质量和振动响应的解析解。通过分析根部弯矩、端部位移、速度和加速度放大系数的变化特征可知,端部弹簧的刚度对静态和一阶载荷响应有明显的影响,减载设计时可以放宽对端部质量的限制,载荷响应分析阶次介于速度和加速度的分析阶次之间。提出的特征变换方法可应用于求解其他载荷分布、边界条件和端部约束悬臂梁的振动响应解析解。
Abstract:According to the eigenvalue condition of cantilever beam with tip mass and spring, this paper proposes a characteristic transformation method, and obtains the analytical solutions of generalized mass and vibration response of cantilever beam with constraints. By analyzing the variational regularities of amplification factors of root bending moment, tip displacement, tip velocity, and tip acceleration for this cantilever beam, the results indicate that the stiffness of tip spring has notable effect on static and first-order load responses, the restriction of tip mass can be relaxed in the load reduction design, and the analysis order of load response is between the analysis orders of velocity and acceleration. The proposed characteristic transformation method can be used to obtain the vibration response analytical solutions of cantilever beam with other loading distribution, boundary conditions and tip constraints.
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Key words:
- cantilever beam /
- tip mass /
- tip spring /
- characteristic transformation method /
- vibration response
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表 1 端部带不同质量和弹簧约束悬臂梁的特征值λn
Table 1. Eigenvalues λn of cantilever beam with different tip mass and spring constraints
特征值阶次 无约束 硬弹簧 中等弹簧中等质量 软弹簧 小质量 大质量 小质量 大质量 一阶 1.875 22 2.623 23 1.622 63 1.552 26 1.710 28 1.048 66 二阶 4.694 17 4.149 42 3.950 55 3.983 54 4.114 54 3.949 92 三阶 7.854 61 7.192 99 7.082 58 7.102 83 7.190 48 7.082 54 四阶 10.995 6 10.298 8 10.219 9 10.234 1 10.298 5 10.219 9 表 2 端部带不同质量和弹簧约束悬臂梁无量纲广义质量的解析解和数值解
Table 2. Analytical and numerical solutions of dimensionless generalized mass for cantilever beam with different tip mass and spring constraints
特征值阶次 无约束 硬弹簧 中等弹簧中等质量 软弹簧 小质量 大质量 小质量 大质量 一阶 0.250 0 1.314 8/1.314 6 3.375 8/3.375 3 1.244 3/1.244 2 0.568 0/0.568 1 0.681 9/0.681 8 二阶 0.250 0 0.271 6 0.251 6 0.253 6 0.259 9 0.251 4 三阶 0.250 0 0.254 1 0.250 5 0.251 2 0.253 7 0.250 5 四阶 0.250 0 0.252 0 0.250 2 0.250 6 0.252 0 0.250 2 注:表格中“/”前数据为解析解,“/”后数据为数值解,无“/”栏中的解析解与数值解相同。 -
[1] 铁摩辛柯 S, 杨 D H, 小韦孚 W.工程中的振动问题[M].胡人礼, 译.北京: 人民铁道出版社, 1978.TIMOSHENKO S, YOUNG D H, JR WEAVER W.Vibration problems in engineering[M].HU R L, translated.Beijing: People Railway Publishing House, 1978(in Chinese). [2] 铁摩辛柯S, 沃诺斯基S.板壳理论[M].《板壳理论》翻译组, 译.北京: 科学出版社, 1977.TIMOSHENKO S, WOINOWSKY-KRIEGER S.Theory of plates and shells[M].Translation Team of Theory of Plates and Shells, translated.Beijing: Science Press, 1977(in Chinese). [3] 王俊奎, 张志民.钣壳的弯曲与稳定[M].北京:国防工业出版社, 1980.WANG J K, ZHANG Z M.Bending and vibration of plates and shells[M].Beijing:National Defense Industry Press, 1980(in Chinese). [4] 屈维德, 唐恒龄.机械振动手册[M].北京:机械工业出版社, 2000.QU W D, TANG H L.Mechanical vibration manual[M].Beijing:China Machine Press, 2000(in Chinese). [5] 尹传家, 黄怀德.机械振动学[M].北京:科学出版社, 1979.YIN C J, HUANG H D.Mechanical vibration[M].Beijing:Science Press, 1979(in Chinese). [6] 刘树林, 王金东, 李凤明, 等.冲击与振动手册[M].5版.北京:中国石化出版社, 2007.LIU S L, WANG J D, LI F M, et al.Shock and vibration manual[M].5th ed.Beijing:China Petrochemical Press, 2007(in Chinese). [7] WANG H J, MENG Q F, FENG W W.Discussion of the improved methods for analyzing a cantilever beam carrying a tip-mass under base excitation[J].Shock and Vibration, 2014, 2014:981053. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=Doaj000004232726 [8] 陆海桃, 仝艳文.集中质量对悬臂梁振动模态影响的分析研究[J].装备制造技术, 2014(10):122-124. doi: 10.3969/j.issn.1672-545X.2014.10.043LU H T, TONG Y W.Analysis of the effect of concentrated mass on the cantilever beam socle modal[J].Equipment Manufacturing Technology, 2014(10):122-124(in Chinese). doi: 10.3969/j.issn.1672-545X.2014.10.043 [9] 陈娟娟, 刘杰.悬臂梁质量摆杆结构一阶模态减振控制分析[J].三峡大学学报(自然科学版), 2014, 36(6):67-72. http://d.old.wanfangdata.com.cn/Periodical/whsldldxxb-yc201406016CHEN J J, LIU J.First order model's vibration reduction control analysis of cantilever beam with a mass swinging rod structure[J].Journal of China Three Gorges University(Natural Sciences), 2014, 36(6):67-72(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/whsldldxxb-yc201406016 [10] 杨一柳, 佘东生, 魏泽飞.一种双桥臂微悬臂梁谐振频率的影响因素研究[J].渤海大学学报(自然科学版), 2015, 36(2):183-187. doi: 10.3969/j.issn.1673-0569.2015.02.017YANG Y L, SHE D S, WEI Z F.Study on the influencing of resonance frequency for microcantilever with double beams[J].Journal of Bohai University(Natural Science Edition), 2015, 36(2):183-187(in Chinese). doi: 10.3969/j.issn.1673-0569.2015.02.017 [11] 赵存生, 李海峰, 朱石坚.悬臂梁动力吸振器的理论分析与试验[J].噪声与振动控制, 2015, 35(4):175-178. http://d.old.wanfangdata.com.cn/Periodical/zsyzdkz201504039ZHAO C S, LI H F, ZHU S J.Theoretical analysis and experimental study of cantilever beam type dynamic vibration absorbers[J].Noise and Vibration Control, 2015, 35(4):175-178(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/zsyzdkz201504039 [12] SWAMINADHAM M, MICHAEL A.A note on frequencies of a beam with a heavy tip mass[J].Journal of Sound and Vibration, 1979, 66(1):144-147. doi: 10.1016/0022-460X(79)90609-6 [13] 王栋.附带有考虑集中质量的转动惯性的梁固有振动分析[J].振动与冲击, 2010, 29(11):221-225. doi: 10.3969/j.issn.1000-3835.2010.11.048WANG D.Vibration analysis of a beam carrying lumped masses with both translational and rotary inertias[J].Journal of Vibration and Shock, 2010, 29(11):221-225(in Chinese). doi: 10.3969/j.issn.1000-3835.2010.11.048 [14] LAJIMI S A M, HEPPLER G R.Comments on natural frequencies of a uniform cantilever with a tip mass slender in the axial direction[J].Journal of Sound and Vibration, 2012, 331(12):2964-2968. doi: 10.1016/j.jsv.2012.01.038 [15] 杨帅, 王太勇.竖直方向弹性约束悬臂梁的固有频率分析[J].天津大学学报, 2011, 44(1):18-22. doi: 10.3969/j.issn.0493-2137.2011.01.004YANG S, WANG T Y.Nature frequency analysis of a cantilever beam with elastic restraint in vertical direction[J].Journal of Tianjin University, 2011, 44(1):18-22(in Chinese). doi: 10.3969/j.issn.0493-2137.2011.01.004 [16] 蔡国平, 洪嘉振.考虑附加质量的中心刚体-柔性悬臂梁系统的动力特性研究[J].机械工程学报, 2005, 41(2):33-40. doi: 10.3321/j.issn:0577-6686.2005.02.007CAI G P, HONG J Z.Dynamics study of hub-beam system with tip mass[J].Chinses Journal of Mechanical Engineering, 2005, 41(2):33-40(in Chinese). doi: 10.3321/j.issn:0577-6686.2005.02.007 [17] 闫安志, 滕军, 鲁志雄, 等.动质量对悬臂梁振动抑制的数值分析和实验研究[J].机械科学与技术, 2007, 26(1):122-126. doi: 10.3321/j.issn:1003-8728.2007.01.030YAN A Z, TENG J, LU Z X, et al.Experiment study and numerical analysis of a moving mass's vibration suppression of a cantilever beam[J].Mechanical Science and Technology, 2007, 26(1):122-126(in Chinese). doi: 10.3321/j.issn:1003-8728.2007.01.030 [18] 夏季, 朱目成, 马德毅.带集中质量和弹性支承梁的横向固有振动分析[J].力学与实践, 2000, 22(5):27-30. doi: 10.3969/j.issn.1000-0879.2000.05.008XIA J, ZHU M C, MA D Y.Analysis of lateral natural vibration of beams with lumped masses and elastic supports[J].Mechanics in Engineering, 2000, 22(5):27-30(in Chinese). doi: 10.3969/j.issn.1000-0879.2000.05.008 [19] 闫安志, 陶天增, 张振华.端部质量对有无损伤悬臂梁的模态影响分析[J].郑州大学学报(工学版), 2016, 37(5):39-42. http://d.old.wanfangdata.com.cn/Periodical/zzgydxxb201605008YAN A Z, TAO T Z, ZHANG Z H.The mode analysis of the cantilever beam with concentrated mass on its free end in the case of damage and no damage[J].Journal of Zhengzhou University (Engineering Science), 2016, 37(5):39-42(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/zzgydxxb201605008 [20] 郭金泉, 陈垂福, 杨晓翔, 等.辅助质量块-单裂纹悬臂梁耦合系统固有频率的理论研究及其应用[J].固体力学学报, 2016, 37(3):264-272. http://www.cqvip.com/QK/95077X/20163/71847688504849544851484854.htmlGUO J Q, CHEN C F, YANG X X, et al.Theoretical and applied research on natural frequencies of cracked cantilever beams with an auxiliary mass[J].Chinese Journal of Solid Mechanics, 2016, 37(3):264-272(in Chinese). http://www.cqvip.com/QK/95077X/20163/71847688504849544851484854.html [21] YOUNG D, FELGAR R P.Tables of characteristic functions representing normal modes of vibration of a beam: 4913[R].Austin: The University of Texas at Austin, 1949. [22] 周民强, 孙山译, 王跃东, 等.数学手册[M].北京:工人出版社, 1987.ZHOU M Q, SUN S Y, WANG Y D, et al.Mathematical manual[M].Beijing:China Worker Publishing House, 1987(in Chinese). [23] 叶其孝, 沈永欢.实用数学手册[M].2版.北京:科学出版社, 2008.YE Q X, SHEN Y H.Practical mathematical manual[M].2nd ed.Beijing:Science Press, 2008(in Chinese).