分布式运动约束下悬臂输液管的参数共振研究
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摘要
输液管道结构在航空、航天、机械、海洋、水利和核电等工程领域都有广泛应用,其稳定性、振动与安全评估备受关注.针对具有分布式运动约束悬臂输液管的非线性动力学模型,分别采用立方非线性弹簧和修正三线性弹簧来模拟运动约束的作用力,研究了管道在脉动内流激励下的参数共振行为.首先,从输液管系统的非线性控制方程出发,利用Galerkin方法进行离散化;然后,由Floquet理论得出线性系统在失稳前两个不同平均流速下脉动幅值和脉动频率变化时的共振参数区域;最后,考虑系统的几何非线性项和分布式非线性运动约束力的影响,求解了管道的非线性动力学响应,讨论了非线性项及运动约束力对管道参数共振行为的影响.研究结果表明,系统非线性共振响应的参数区域与线性系统的共振参数区域是一致的,分布式运动约束力对发生参数共振时管道的位移响应有显著影响;立方非线性弹簧和修正三线性弹簧模型所预测的分岔路径存有较大差异,但都可诱发管道在一定的参数激励下出现混沌运动.
Pipes conveying fluid have been widely used in the fields of aerospace, mechanics, marine, hydraulic and nuclear engineering. The stability analysis, dynamic response and safety assessment of fluid-conveying pipes subjected to nonlinear constraints are particularly important for both engineering applications and scientific researches. Although the dynamical behaviors of fluid-conveying pipes subjected to single-point loose constraints have been discussed for four decades, the literature on the dynamics of fluid-conveying pipes subjected to distributed motion constraints is very limited.To obtain a better understanding of the dynamics of a cantilevered pipe conveying fluid subjected to distributed motion constraints, both cubic and modified trilinear spring models are employed in this study to describe the restraining force between the pipe and the motion constraints. This study is also concerned with the parametric resonance when the pipe is excited by an internal pulsating fluid. Firstly, the modified nonlinear equation of the pipe system was discretized via Galerkin's approach and solved using a fourth-order Runge-Kutta method. Via the Floquet theory, the nonlinear equation of motion was simplified to a linear one to calculate the parametric resonance regions versus the pulsating amplitude and frequency. Two representative values of mean flow velocity were employed to calculate the parametric resonance regions.Both the two values of mean flow velocity are assumed to be lower than the critical velocity for the cantilevered pipe system. Then, considering the geometric nonlinearity, the nonlinear dynamic responses focusing on the effect of external nonlinear restraining forces generated by the distributed motion constraints are discussed in detail. Results show that the stability regions of the nonlinear system agree well with that predicted by analyzing the linearized system. It is found that the distributed motion constraints would mainly affect the displacement amplitudes. Various oscillation types may arise when the pulsating frequency of the flow velocity is varied. Several bifurcation diagrams show that, however, a significant difference can be observed between the routes to chaos for the two constraint models, i.e., the pipe with a trilinear spring model can exhibit chaotic oscillations more easily than that with a cubic spring model.
Pipes conveying fluid have been widely used in the fields of aerospace, mechanics, marine, hydraulic and nuclear engineering. The stability analysis, dynamic response and safety assessment of fluid-conveying pipes subjected to nonlinear constraints are particularly important for both engineering applications and scientific researches. Although the dynamical behaviors of fluid-conveying pipes subjected to single-point loose constraints have been discussed for four decades, the literature on the dynamics of fluid-conveying pipes subjected to distributed motion constraints is very limited.To obtain a better understanding of the dynamics of a cantilevered pipe conveying fluid subjected to distributed motion constraints, both cubic and modified trilinear spring models are employed in this study to describe the restraining force between the pipe and the motion constraints. This study is also concerned with the parametric resonance when the pipe is excited by an internal pulsating fluid. Firstly, the modified nonlinear equation of the pipe system was discretized via Galerkin's approach and solved using a fourth-order Runge-Kutta method. Via the Floquet theory, the nonlinear equation of motion was simplified to a linear one to calculate the parametric resonance regions versus the pulsating amplitude and frequency. Two representative values of mean flow velocity were employed to calculate the parametric resonance regions.Both the two values of mean flow velocity are assumed to be lower than the critical velocity for the cantilevered pipe system. Then, considering the geometric nonlinearity, the nonlinear dynamic responses focusing on the effect of external nonlinear restraining forces generated by the distributed motion constraints are discussed in detail. Results show that the stability regions of the nonlinear system agree well with that predicted by analyzing the linearized system. It is found that the distributed motion constraints would mainly affect the displacement amplitudes. Various oscillation types may arise when the pulsating frequency of the flow velocity is varied. Several bifurcation diagrams show that, however, a significant difference can be observed between the routes to chaos for the two constraint models, i.e., the pipe with a trilinear spring model can exhibit chaotic oscillations more easily than that with a cubic spring model.
引文
1 Pa?doussis MP.Fluid-Structure Interactions:Slender Structures and Axial Flow.Volume 1.London:Academic Press,1998
2 Pa?doussis MP.Fluid-Structure Interactions:Slender Structures and Axial Flow.Volume 2.London:Academic Press,2004
3 Pa?doussis MP,Semler C.Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end.International Journal of Non-Linear Mechanics,1998,33(1):15-32
4金基铎,邹光胜,张宇飞.悬臂输流管道的运动分岔现象和混沌运动.力学学报,2002,34(6):863-873(Jin Jiduo,Zou Guangsheng,Zhang Yufei.Bifurcations and chaotic motions of a cantilevered pipe conveying fluid.Chinese Journal of Theoretical and Applied Mechanics,2002,34(6):863-873(in Chinese))
5 Nikoli′c M,Rajkovi′c M.Bifurcations in nonlinear models of fluidconveying pipes supported at both ends.Journal of Fluids and Structures,2006,22(2):173-195
6 Namchchivaya NS.Non-linear dynamics of supported pipe conveying pulsating fluid-I.Subharmonic resonance.International Journal of Non-Linear Mechanics,1989,24(3):185-196
7 Namchchivaya NS,Tien W.Non-linear dynamics of supported pipe conveying pulsating fluid-II.Combination resonance.International Journal of Non-Linear Mechanics,1989,24(3):197-208
8 Pa?doussis MP,Li GX,Moon FC.Chaotic oscillations of the autonomous system of a constrained pipe conveying fluid.Journal of Sound and Vibration,1989,135(1):1-19
9 Mostafa NH.Effect of a viscoelastic foundation on the dynamic stability of a fluid conveying pipe.International Journal of Applied Science and Engineering,2014,12(1):59-74
10 Pa?doussis MP,Semler C.Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid:A full nonlinear analysis.Nonlinear Dynamics,1993,4(6):655-670
11 Jin JD.Stability and chaotic motions of a restrained pipe conveying fluid.Journal of Sound and Vibration,1997,208(3):427-439
12 Xu J,Huang YY.Bifurcations of a cantilevered pipe conveying steady fluid with a terminal nozzle.Acta Mechanica Sinica,2000,16(3):264-272
13 Hassan MA,Weaver DS,Dokainish MA.A simulation of the turbulence response of heat exchanger tubes in lattice-bar supports.Journal of Fluids and Structures,2002,16(8):1145-1176
14 Hassan MA,Weaver DS,Dokainish MA.A new tube/support impact model for heat exchanger tubes.Journal of Fluids and Structures,2005,21(5-7):561-577
15 Wang Y,Wang L,Ni Q,et al.Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints.Nonlinear Dynamics,2018(https://doi.org/10.1007/s11071-018-4206-1)
16金基铎,杨晓东,尹峰.两端铰支输流管道在脉动内流作用下的稳定性和参数共振.航空学报,2003,24(4):317-322(Jin Jiduo,Yang Xiaodong,Yin Feng.Stability and parametric resonances of a pinned-pinned pipe conveying pulsating fluid.Acta Aeronautica et Astronautica Sinica,2003,24(4):317-322(in Chinese))
17 Jin JD,Song ZY.Parametric resonances of supported pipes conveying pulsating fluid.Journal of Fluids and Structures,2005,20(6):763-783
18金基铎,梁峰,杨晓东等.两端固定输流管道参数共振的实验研究.振动与冲击,2007,26(11):169-173(Jin Jiduo,Liang Feng,Yang Xiaodong,et al.Experiments on parametric resonance of clamped-clamped pipes conveying fluid.Journal of Vibration and Shock,2007,26(11):169-173(in Chinese))
19 Panda LN,Kar RC.Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances.Nonlinear Dynamics,2007,49(1-2):9-30
20 Wang L.A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid.International Journal of Non-Linear Mechanics,2009,44(1):115-121
21张紫龙,唐敏,倪樵.非线性弹性地基上悬臂输流管的受迫振动.振动与冲击,2013,32(10):17-21(Zhang Zilong,Tang Min,Ni Qiao.Forced vibration of a cantilever fluid-conveying pipe on nonlinear elastic foundation.Journal of Vibration and Shock,2013,32(10):17-21(in Chinese))
22李云东,杨翊仁,文华斌.非线性弹性地基上悬臂管道的参数振动.振动与冲击,2016,35(24):14-18(Li Yundong,Yang Yiren,Wen Huabin.Parametric vibration of a cantilevered pipe conveying pulsating fluid on a nonlinear elastic foundation.Journal of Vibration and Shock,2016,35(24):14-18(in Chinese))
23 Ni Q,Tang M,Wang Y,et al.In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid.Nonlinear Dynamics,2013,75(3):603-619
24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.应用数学和力学,2016,37(4):345-351(Mao Xiaoye,Ding Hu,Chen Liqun.Forced vibration responses of supercritical fluidconveying pipes in 3:1 internal resonance.Applied Mathematics and Mechanics,2016,37(4):345-351(in Chinese))
25 Mao XY,Ding H,Chen LQ.Steady-state response of a fluidconveying pipe with 3:1 internal resonance in supercritical regime.Nonlinear Dynamics,2016,82(2):795-809
26陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报,2005,37(1):57-63(Chen Shuhui,Huang Jianliang.On internal resonance of nonlinear vibration of axially moving beams.Chinese Journal of Theoretical and Applied Mechanics,2005,37(1):57-63(in Chinese))
27张艳雷.超临界输液管横向振动的非线性动力学分析.[博士论文].上海:上海大学,2012(Zhang Yanlei.Nonlinear dynamics of transverse vibrations of pipe conveying fluid in the subcritical regime.[PhD Thesis].Shanghai:Shanghai University,2012(in Chinese))
28 Wang Y,Ni Q,Wang L,et al.Nonlinear impacting oscillations of pipe conveying pulsating fluid subjected to distributed motion constraints.Journal of Mechanics of Materials and Structures,2017,12(5):563-578.
29 Ni Q,Wang Y,Tang M,et al.Nonlinear impacting oscillations of a fluid-conveying pipe subjected to distributed motion constraints.Nonlinear Dynamics,2015,81(1-2):893-906
30 Wang L,Liu ZY,Abdelkefi A,et al.Nonlinear dynamics of cantilevered pipes conveying fluid:Towards a further understanding of the effect of loose constraints.International Journal of Non-Linear Mechanics,2017,95:19-29
31 Semler C,Pa?doussis MP.Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe.Journal of Fluids and Structures,1996,10(7):787-825
2 Pa?doussis MP.Fluid-Structure Interactions:Slender Structures and Axial Flow.Volume 2.London:Academic Press,2004
3 Pa?doussis MP,Semler C.Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end.International Journal of Non-Linear Mechanics,1998,33(1):15-32
4金基铎,邹光胜,张宇飞.悬臂输流管道的运动分岔现象和混沌运动.力学学报,2002,34(6):863-873(Jin Jiduo,Zou Guangsheng,Zhang Yufei.Bifurcations and chaotic motions of a cantilevered pipe conveying fluid.Chinese Journal of Theoretical and Applied Mechanics,2002,34(6):863-873(in Chinese))
5 Nikoli′c M,Rajkovi′c M.Bifurcations in nonlinear models of fluidconveying pipes supported at both ends.Journal of Fluids and Structures,2006,22(2):173-195
6 Namchchivaya NS.Non-linear dynamics of supported pipe conveying pulsating fluid-I.Subharmonic resonance.International Journal of Non-Linear Mechanics,1989,24(3):185-196
7 Namchchivaya NS,Tien W.Non-linear dynamics of supported pipe conveying pulsating fluid-II.Combination resonance.International Journal of Non-Linear Mechanics,1989,24(3):197-208
8 Pa?doussis MP,Li GX,Moon FC.Chaotic oscillations of the autonomous system of a constrained pipe conveying fluid.Journal of Sound and Vibration,1989,135(1):1-19
9 Mostafa NH.Effect of a viscoelastic foundation on the dynamic stability of a fluid conveying pipe.International Journal of Applied Science and Engineering,2014,12(1):59-74
10 Pa?doussis MP,Semler C.Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid:A full nonlinear analysis.Nonlinear Dynamics,1993,4(6):655-670
11 Jin JD.Stability and chaotic motions of a restrained pipe conveying fluid.Journal of Sound and Vibration,1997,208(3):427-439
12 Xu J,Huang YY.Bifurcations of a cantilevered pipe conveying steady fluid with a terminal nozzle.Acta Mechanica Sinica,2000,16(3):264-272
13 Hassan MA,Weaver DS,Dokainish MA.A simulation of the turbulence response of heat exchanger tubes in lattice-bar supports.Journal of Fluids and Structures,2002,16(8):1145-1176
14 Hassan MA,Weaver DS,Dokainish MA.A new tube/support impact model for heat exchanger tubes.Journal of Fluids and Structures,2005,21(5-7):561-577
15 Wang Y,Wang L,Ni Q,et al.Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints.Nonlinear Dynamics,2018(https://doi.org/10.1007/s11071-018-4206-1)
16金基铎,杨晓东,尹峰.两端铰支输流管道在脉动内流作用下的稳定性和参数共振.航空学报,2003,24(4):317-322(Jin Jiduo,Yang Xiaodong,Yin Feng.Stability and parametric resonances of a pinned-pinned pipe conveying pulsating fluid.Acta Aeronautica et Astronautica Sinica,2003,24(4):317-322(in Chinese))
17 Jin JD,Song ZY.Parametric resonances of supported pipes conveying pulsating fluid.Journal of Fluids and Structures,2005,20(6):763-783
18金基铎,梁峰,杨晓东等.两端固定输流管道参数共振的实验研究.振动与冲击,2007,26(11):169-173(Jin Jiduo,Liang Feng,Yang Xiaodong,et al.Experiments on parametric resonance of clamped-clamped pipes conveying fluid.Journal of Vibration and Shock,2007,26(11):169-173(in Chinese))
19 Panda LN,Kar RC.Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances.Nonlinear Dynamics,2007,49(1-2):9-30
20 Wang L.A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid.International Journal of Non-Linear Mechanics,2009,44(1):115-121
21张紫龙,唐敏,倪樵.非线性弹性地基上悬臂输流管的受迫振动.振动与冲击,2013,32(10):17-21(Zhang Zilong,Tang Min,Ni Qiao.Forced vibration of a cantilever fluid-conveying pipe on nonlinear elastic foundation.Journal of Vibration and Shock,2013,32(10):17-21(in Chinese))
22李云东,杨翊仁,文华斌.非线性弹性地基上悬臂管道的参数振动.振动与冲击,2016,35(24):14-18(Li Yundong,Yang Yiren,Wen Huabin.Parametric vibration of a cantilevered pipe conveying pulsating fluid on a nonlinear elastic foundation.Journal of Vibration and Shock,2016,35(24):14-18(in Chinese))
23 Ni Q,Tang M,Wang Y,et al.In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid.Nonlinear Dynamics,2013,75(3):603-619
24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.应用数学和力学,2016,37(4):345-351(Mao Xiaoye,Ding Hu,Chen Liqun.Forced vibration responses of supercritical fluidconveying pipes in 3:1 internal resonance.Applied Mathematics and Mechanics,2016,37(4):345-351(in Chinese))
25 Mao XY,Ding H,Chen LQ.Steady-state response of a fluidconveying pipe with 3:1 internal resonance in supercritical regime.Nonlinear Dynamics,2016,82(2):795-809
26陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报,2005,37(1):57-63(Chen Shuhui,Huang Jianliang.On internal resonance of nonlinear vibration of axially moving beams.Chinese Journal of Theoretical and Applied Mechanics,2005,37(1):57-63(in Chinese))
27张艳雷.超临界输液管横向振动的非线性动力学分析.[博士论文].上海:上海大学,2012(Zhang Yanlei.Nonlinear dynamics of transverse vibrations of pipe conveying fluid in the subcritical regime.[PhD Thesis].Shanghai:Shanghai University,2012(in Chinese))
28 Wang Y,Ni Q,Wang L,et al.Nonlinear impacting oscillations of pipe conveying pulsating fluid subjected to distributed motion constraints.Journal of Mechanics of Materials and Structures,2017,12(5):563-578.
29 Ni Q,Wang Y,Tang M,et al.Nonlinear impacting oscillations of a fluid-conveying pipe subjected to distributed motion constraints.Nonlinear Dynamics,2015,81(1-2):893-906
30 Wang L,Liu ZY,Abdelkefi A,et al.Nonlinear dynamics of cantilevered pipes conveying fluid:Towards a further understanding of the effect of loose constraints.International Journal of Non-Linear Mechanics,2017,95:19-29
31 Semler C,Pa?doussis MP.Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe.Journal of Fluids and Structures,1996,10(7):787-825