A study on nonlinear vibration of slightly curved hydraulic pipes under dual frequency excitation
FAN Xin1,2,SHU Song1,ZHANG Junning3,XIAO Lu3,MAO Xiaoye3,DING Hu3
1.The 5720th Factory of PLA, Wuhu 241007, China;
2.Department of Precision Machinery and Precision Instruments, University of Science and Technology of China, Hefei 230026, China;
3.Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China
摘要首次研究了双频激励下两端固支微弯液压管道的非线性振动，分析了拍振现象发生时管路非线性受迫振动特性。利用广义哈密顿原理建立微曲管道控制方程，使用Galerkin法将非线性偏微分积分方程离散为非线性耦合常微分方程组，采用龙格库塔法数值求解，并用微分求积单元法(Differential Quadrature Element Method, DQEM)进行数值验证，得到了微曲管道受迫振动响应。在此基础上，讨论了两个激励频率差值对管道中点受迫振动响应的影响，并分析了系统横向振动在一阶固有频率附近的分岔现象，考察了液压管道的混沌特性。分析结果表明，两个激励频率相差较小时会出现混沌现象，随着频率差值的减小，管道的混沌区域先增大后减小。这些结论为多频激励下液压管道系统复杂振动的研究提供了理论依据和研究方法。
Abstract：The nonlinear combined vibration of a slightly curved hydraulic pipe with two fixed supports under dual frequency excitation is studied. The nonlinear forced vibration characteristics are analyzed when the beat vibration phenomenon occurs. Generalized Hamiltonian principle is used to establish the governing equation of the slightly curved pipe. The nonlinear partial differential integral equation is discretized into a set of nonlinearly coupled ordinary differential equations (ODEs) via Galerkin method (GM). The numerical solutions of the forced vibration are obtained by the Runge Kutta method based on the discrete ordinary differential equations. The differential quadrature element method (DQEM) verifies the accuracy of it. On this basis, the influence of the frequency gap between two excitations on the forced vibration of the pipe’s midpoint is discussed. The bifurcation and chaos of the transverse vibration near the first order natural frequency is analyzed. The analysis results show that when the frequency gap is small, chaos will appear. With the decrease of frequency gap, the chaotic frequency band increases and then decreases. These conclusions provide theoretical basis and research methods for the study of complex vibration of hydraulic pipe systems under multi-frequency excitation.
范鑫1,2，舒送1，张俊宁3，肖璐3，毛晓晔3，丁虎3. 双频激励下微弯液压管道的非线性振动研究[J]. 振动与冲击, 2023, 42(20): 181-187.
FAN Xin1,2,SHU Song1,ZHANG Junning3,XIAO Lu3,MAO Xiaoye3,DING Hu3. A study on nonlinear vibration of slightly curved hydraulic pipes under dual frequency excitation. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(20): 181-187.
[1] GAO P, YU T, ZHANG Y, et al. Vibration analysis and control technologies of hydraulic pipeline system in aircraft: A review[J]. Chinese Journal of Aeronautics, 2021, 34(4): 83-114.
[2] ZHOU S, YU T J, YANG X D, et al. Global dynamics of pipes conveying pulsating fluid in the supercritical regime[J]. International Journal of Applied Mechanics, 2017, 9(02): 1750029.
[3] ZHANG Y F, YAO M H, ZHANG W, et al. Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance[J]. Aerospace Science and Technology, 2017, 68: 441-453.
[4] HU Y J, ZHU W. Vibration analysis of a fluid-conveying curved pipe with an arbitrary undeformed configuration[J]. Applied Mathematical Modelling, 2018, 64: 624-642.
[5] SINIR B G. Bifurcation and chaos of slightly curved pipes[J]. Mathematical and Computational Applications, 2010, 15(3): 490-502.
[6] 胡育佳,李海港.不同流体加速度描述对输液曲管稳定性的影响[J].振动与冲击,2015,34(15):168-174+187.
HU Yu-jia, LI Hai-gang. Effects of different fluid acceleration on stability of curved pipes conveying fluid[J]. Vibration and Shock, 2015,34(15):168-174+187.
[7] 李宝辉, 高行山, 刘永寿. 轴向可伸输液曲管平面内振动的波动方法研究[J]. 振动与冲击, 2013, 32(08): 128-133+142.
LI Bao-hui, GAO Hang-shan, LIU Yon-tao. Wave propagation method for in-plane vibration of an axially extensible curved pipe conveying fluid[J]. Vibration and Shock, 2013, 32(08): 128-133+142.
[8] 谢孝文. 不同边界条件下输流曲管的振动特性[J]. 通信电源技术, 2015, 32(04): 53- 56.
XIE Xiao-wen. Vibration characteristics of fluid-conveying curved pipe under different boundary conditions[J]. Telecom Power Technology, 2015, 32(04): 53- 56.
[9] OYELADE A O, EHIGIE J O, OYEDIRAN A A. Nonlinear forced vibrations of a slightly curved nanotube conveying fluid based on the nonlocal strain gradient elasticity theory[J]. Microfluidics and Nanofluidics, 2021, 25(11): 1-16.
[10] OYELADE A O, IKHILE O G, OYEDIRAN A A. Stability and dynamics of a slightly curved viscoelastic pipe resting on linear and nonlinear viscoelastic foundation[J]. Iranian Journal of Science and Technology, Transactions of Civil Engineering, 2022, 46(2): 1329-1343.
[11] LI Y D, SUN Z G. Analysis of planar motion for curved pipe conveying fluid with different types of initial configuration[J]. Journal of Vibration Engineering & Technologies, 2021: 1-16.
[12] ZHOU K, NI Q, CHEN W, et al. New insight into the stability and dynamics of fluid-conveying supported pipes with small geometric imperfections[J]. Applied Mathematics and Mechanics-English Edition, 2021, 42(5): 703-720.
[13] ZHOU K, NI Q, CHEN W, et al. Static equilibrium configuration and nonlinear dynamics of slightly curved cantilevered pipe conveying fluid[J]. Journal of Sound and Vibration, 2021, 490: 115711.
[14] 袁嘉瑞, 丁虎, 陈立群. 微曲输流管道振动固有频率分析与仿真[J]. 应用数学和力学, 2022, 43(07): 719-726.
YUAN Jia-rui, DING Hu, CHEN Li-qun. Analysis and simulation of natural frequencies of slightly curved pipes[J]. Applied Mathematics and Mechanics, 2022, 43(07): 719-726.
[15] ZHAI Y J, MA Z S, DING Q, et al. Nonlinear transverse vibrations of a slightly curved beam with hinged–hinged boundaries subject to axial loads[J]. Archive of Applied Mechanics, 2022: 1-14.
[16] 曹建华,黄颖青,仝国军,赵年顺.非线性输流曲管面内振动的样条小波有限元方法研究[J].振动与冲击,2022,41(18):66-74+103.
CAO Jian-hua, HUANG Ying-qing, TONG Guo-jun, ZHAO Nian-shun. Wavelet-based finite element method for the nonlinear in-plane vibration of curved pipes conveying fluid[J]. Vibration and Shock, 2022,41(18):66-74+103.
[17] CZERWINSKI A, LUCZKO J. Non-planar vibrations of slightly curved pipes conveying fluid in simple and combination parametric resonances[J]. Journal of Sound and Vibration, 2018, 413: 270-290.
[18] YE S Q, DING H, WEI S, et al. Non-trivial equilibriums and natural frequencies of a slightly curved pipe conveying supercritical fluid[J]. Ocean Engineering, 2021, 227: 108899.
[19] YE S Q, DING H, WEI S, et al. Nonlinear forced vibrations of a slightly curved pipe conveying supercritical fluid[J]. Journal of Vibration and Control, 2022, 0(0): 1-12.
[20] ZHANG J N , YANG S P, LI S H, LU Y J, ET AL. Influence of vehicle-road coupled vibration on tire adhesion based on nonlinear foundation[J]. Applied Mathematics and Mechanics-English Edition, 2021, 42(5), 607-624.
[21] WU P, GUO J, WU H, WEI J. Influence of DC-link voltage pulsation of transmission systems on mechanical structure vibration and fatigue in high-speed trains[J]. Engineering Failure Analysis, 2021, 130, 105772.
[22] ZHAO F T, JING X D, YANG M S. Experimental study of rotor blades vibration and noise in multistage high-pressure compressor and their relevance[J]. Chinese Journal of Aeronautics, 2020, 33(3), 870-878.
[23] TAN X, DING H, CHEN L Q. Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model[J]. Journal of Sound and Vibration, 2019, 455, 241-255.
[24] 徐鉴, 杨前彪. 输液管模型及其非线性动力学近期研究进展. 力学进展, 2004, 02, 182-194
XU Jian, YANG Qian-biao. Recent development on models and nonlinear dynamics of pipes conveying fluid [J]. Advances in Mechanics, 2004, 02, 182-194.
[25] YANG T Z , LIU T, TANG Y, HOU S, LV X F. Enhanced targeted energy transfer for adaptive vibration suppression of pipes conveying fluid[J]. Nonlinear Dynamics, 2019, 97(2), 1937-1944.
[26] 汪博, 高培鑫, 马辉,等. 航空发动机管路系统动力学特性研究综述[J]. 航空学报, 2021, 40: 1-25.
WANG Bo, GAO Pei-xin, MA Hui, et al. Dynamic characteristics of aero-engine pipeline system: Review[J]. Acta Aeronautica et Astronautica Sinica, 2021, 40: 1-25.
[27] Mao X Y, Shu S, Fan X, et al. An approximate method for pipes conveying fluid with strong boundaries[J]. Journal of Sound and Vibration, 2021, 505: 116157.
[28] 毛晓晔,邵志华,舒送等.中间支撑刚度对双跨梁屈曲稳定性的影响[J].振动与冲击,2022,41(11):1-9+17.
MAO Xiao-ye, SHAO Zhi-hua, SHU Song, et al. Effect of intermediate support stiffness on buckling stability of a double-span beam[J]. Vibration and Shock, 2022,41(11):1-9+17.
[29] Rosenstein M T, Collins J J, De Luca C J. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D: Nonlinear Phenomena, 1993, 65(1-2): 117-134.
