Dynamic response and vibration isolation effect of generalized fractional-order van der Pol-Duffing oscillator
TANG Jianhua1, LI Xianghong1,3, WANG Min1, SHEN Yongjun2,3, LI Zhuangzhuang2
1.Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2.School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
3.State Key Lab of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Abstract:By means of the average method, this paper studied the dynamic behavior and force transmissibility of the van der Pol-Duffing oscillator with fractional derivative term. First, the first-order analytical solution of the van der Pol-Duffing oscillator with fractional derivative term was calculated. Further, the expressions of the amplitude-frequency curve and the phase-frequency curve for the steady solution were obtained. The correctness of the analytical solution was verified by comparing with the numerical solution. Finally, the influence of different parameters on the amplitude-frequency curve and the force transmissibility was analyzed. The results show that, the analytical solution is in good agreement with numerical solution. In the dimensionless case, some parameters can restrain the resonance peaks of the amplitude-frequency curve and the force transmissibility within the resonance region, such as the fractional order coefficients, nonlinear parameters, fractional order, and damping ratio. In the low-frequency vibration isolation area, the nonlinear parameters and amplitude have a significant effect on the vibration isolation effect, the smaller the nonlinear parameters and amplitude, the better the vibration isolation effect may be. In the high-frequency vibration isolation area, the increase of some parameters helps to improve the vibration isolation effect, such as the non-linear parameters, amplitude and damping ratio.
唐建花1,李向红1,3,王敏1,申永军2,3,李壮壮2. 广义分数阶van der Pol-Duffing振子的动力学响应与隔振效果研究[J]. 振动与冲击, 2022, 41(1): 10-18.
TANG Jianhua1, LI Xianghong1,3, WANG Min1, SHEN Yongjun2,3, LI Zhuangzhuang2. Dynamic response and vibration isolation effect of generalized fractional-order van der Pol-Duffing oscillator. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(1): 10-18.
[1] SHEN Y, YANG S P, XING H J and GAO G S. Primary resonance of Duffing oscillator with fractional-order derivative[J]. CNSNS, 2012, 17(7): 3092-3100.
[2] 姜源, 申永军, 温少芳, 杨绍普. 分数阶杜芬振子的超谐与亚谐联合共振[J]. 力学学报, 2017, 49(5): 1008-1019.
JIANG Yuan, SHEN Yongjun, WEN Shaofang, YANG Shaopu. Super-harmonic and sub-harmonic simultaneous resonances of fractional-order Duffing oscillator[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1008-1019.
[3] 冯志华, 胡海岩. 内共振条件下直线运动梁的动力稳定性[J]. 力学学报, 2002(3): 389-400.
FENG Zhihua, HU Haiyan. Dynamic stability of a slender beam with internal resonance under a large linear motion[J]. Chinese Journal of Theoretical and Applied Mechanics, 2002, 34(3): 389-400.
[4] 孙春艳, 徐伟. 含分数阶导数项的随机Duffing振子的稳态响应分析[J]. 振动工程学报, 2015, 28(3): 374-380.
SUN Chunyan, Xu Wei. Stationary response analysis for a stochastic Duffing oscillator comprising fractional derivative element[J]. Journal of Vibration Engineering, 2015, 28(3): 374-380.
[5] 陈树辉. 应用于具有二次,三次非线性系统的增量谐波平衡法[J]. 非线性动力学学报, 1993, 1(1): 73-79.
CHEN Shuhui. Application of the Incremental Harmonic Balance Method to Quadratic and Cubic Non-Linearities systems[J]. Journal of Nonlinear Dynamics in Engineering. 1993, 1(1): 73-79.
[6] LI X H, HOU J Y, CHEN J F. An analytical method for Mathieu oscillator based on method of variation of parameter[J]. Communications in Nonlinear ence and Numerical Simulation, 2016, 37: 326-353.
[7] LI X H, TANG J H, WANG Y L, et al. Approximate analytical solution in slow-fast system based on modified multi-scale method[J]. Applied Mathematics and Mechanics, 2020, 41(4): 605–622.
[8] NAYFEH A H. Quenching of primary resonance by a superharmonic resonance[J]. Journal of Sound & Vibration, 1984, 92(3): 363-377.
[9] GLEBOV S G, KISELEV O M. Applicability of the WKB method in the perturbation problem for the equation of principal resonance[J]. Russian Journal of Mathematical Physics, 2002, 9(1): 60-83.
[10] SUN Z K, YANG X L, XU W. Resonance dynamics evoked via noise recycling procedure[J]. Physical Review E Statistical Physics Plasmas Fluids & Related Interdisciplinary Topics, 2012, 85(6): 061125.
[11] 毛晓晔, 丁虎, 陈立群. 3∶1内共振下超临界输液管受迫振动响应[J]. 应用数学和力学, 2016, 37(4): 345-351.
MAO Xiaoye, DING Hu, CHEN Liqun. Forced vibration response of of supercritical fluid⁃conveying pipes in 3: 1: internal resonance. Applied Mathematics and Mechanics, 2016, 37(4): 345-351.
[12] XIA T, SUN J Q, DING H, et al. Primary and super-harmonic resonances of Timoshenko pipes conveying high-speed fluid[J]. Ocean Engineering, 2020, 203: 107258.
[13] MAO X Y, SUN J Q, DING H, et al. An approximate method for one-dimensional structures with strong nonlinear and nonhomogenous boundary conditions[J]. Journal of Sound and Vibration, 2020, 469: 115128.
[14]胡宇达, 戎艳天. 磁场中轴向变速运动载流梁的参强联合共振[J]. 中国机械工程, 2016, 27(23): 3197-3207.
HU Yuda, RONG Yantian. Combined parametric and forced resonance of an axially accelerating and current carrying beam under magnetic field[J], 2016, 27(23): 3197-3207.
[15] 胡宇达, 王彤. 磁场中导电旋转圆板的磁弹性非线性共振[J]. 振动与冲击, 2016, 35(12): 177-181.
Hu Yuda, Wang Tong. Nonlinear resonance of conductive rotating circular plate in the magnetic field. Journal of Vibration and shock, 2016, 35(12): 177-181.
[16] SHEN Y J, WEN S F, LI X H. Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method[J]. Nonlinear Dynamics, 2016, 85(3): 1457-1467.
[17] WEN S F, SHEN Y J, YANG S P, et al. Dynamical response of Mathieu-Duffing oscillator with fractional-order delayed feedback[J]. Chaos, Solitons, Fractals, 2017, 94: 54-62.
[18] 王军, 申永军, 杨绍普, 温少芳, 王美琪. 一类分数阶分段光滑系统的非线性振动特性[J]. 振动与冲击, 2019, 38(22): 216-223.
WANG Jun, SHEN Yongjun, YANG Shaopu, et al. Nonlinear vibration performance of a piecewise smooth system with fractional-order derivative[J]. Journal of Vibration and Shock, 2019, 38 (22): 216-223.
[19] CHEN L, WANG W, LI Z, et al. Stationary response of Duffing oscillator with hardening stiffness and fractional derivative[J]. International Journal of Non-Linear Mechanics, 2013, 48: 44-50.
[20] LEUNG A Y T, GUO Z. Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(4): 2169-2183.
[21] MALTI R, MOREAU X, KHEMANE F, et al. Stability and resonance conditions of elementary fractional transfer functions[J]. Automatica, 2011, 47(11): 2462-2467.
[22] 李园园, 陈国平, 王轲. 直升机旋翼/机身动力反共振隔振器的优化设计[J]. 振动与冲击, 2016, 35(15): 115-121.
LI Yuanyuan, CHEN Guoping, WANG Ke. Optimization design for dynamic anti-resonance isolators of helicopters’ rotor / fuselage[J]. Journal of Vibration and Shock, 2016, 35(15): 115-121.
[23] 李壮壮, 申永军. 单自由度系统强迫激励下惯容对Kelvin模型和Maxwell模型的影响[J]. 石家庄铁道大学学报: 自然科学版, 2019, 32(1): 28-34.
LI Zhuangzhuang, SHEN Yongjun. Influence of inerter on Kelvin and Maxwell models under forced excitation for single-degree-of-freedom system [J]. Journal of Shijiazhuang Tiedao University: Natural Science, 2019, 32 (1): 28-34.
[24] 李壮壮, 申永军, 杨绍普等. 基于惯容-弹簧-阻尼的结构减振研究[J]. 振动工程学报, 2018, 31(6): 157-163.
LI Zhuangzhuang, SHEN Yongjun, YANG Shaopu, et al. Study on vibration reduction based on ISD (I-Inerter, S-Spring, D-Damping) structure [J]. Journal of Vibration Engineering, 2018, 31 (6): 157-163.
[25] 罗昌杰, 刘荣强, 邓宗全等. 泡沫铝填充薄壁金属管塑性变形缓冲器吸能特性的试验研究[J]. 振动与冲击, 2009, 28(10): 26-30.
LOU Changjie, LIU Rongqiang, DENG Zongquan, et al. Experimental study on the energy absorption characteristics of plastic deformation buffer of aluminum foam filled thin-walled metal pipes [J]. Journal of Vibration and Shock, 2009, 28 (10): 26-30.
[26] 周超, 吴庆鸣, 张强等. 粘弹性阻尼隔振体的非线性振动分析[J]. 工程设计学报, 2009, 16(3): 207-209.
ZHOU Chao, WU Qingming, ZHANG Qiang, et al. Nonlinear vibration analysis of viscoelastic isolator [J]. Journal of Engineering Design, 2009, 16(3): 207-209.
[27] MAKRIS N, CONSTANTINOU M C. Fractional‐derivative Maxwell model for viscous dampers[J]. Journal of Structural Engineering, 1991, 117(9): 2708-2724.
[28] ALBERTO C, TRINDADE M A, RUBENS S. Frequency-dependent viscoelastic models for passive vibration isolation systems[J]. Shock & Vibration, 2002, 9: 253-264.
[29] KUMAR P, KUMAR A, ERLICHER S. A modified hybrid van der Pol-Duffing-Rayleigh oscillator for modelling the lateral walking force on a rigid floor[J]. Physica D: Nonlinear Phenomena, 2017, 358: 1-14.
[30] FRANCESCUTTO A, CONTENTO G. Bifurcations in ship rolling: experimental results and parameter identification technique[J]. Ocean Engineering, 1999, 26(11): 1095-1123.
[31] PETRAS Ivo. Fractional-order nonlinear systems: modeling, analysis and simulation. Higher Education Press, 2011: 18-19.
[32] 张小龙, 东亚斌. Duffing型隔振的力传递率及跳跃现象的理论分析[J]. 振动与冲击, 2012, 31(16): 43-47.
ZHANG Xiaolong, DONG Yabin. Theoretical analysis on force transmissibility and jump phenomenon of Duffing spring type vibration isolator [J]. Journal of Vibration and Shock, 2012, 31(16): 43-47.