耦合输流管系统的非线性动力学行为研究
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摘要
本文研究了耦合输流管系统的振动稳定性和非线性动力学行为,主要考查三种典型的耦合输流管系统：(1)输流管道-地基耦合系统；(2)输流管道-输流管道耦合系统；(3)输流管道-梁式结构耦合系统。重点考查动力耦合效应作用下耦合输流管系统各子结构在多种参数组合下的复杂分岔路径和混沌运动现象,并对分析该动力学模型的半解析方法(DTM)进行了分析与探讨。通过数值计算得到了一些极具价值的结果,发现了一些以往未曾发现的现象。本文所的研究工作具有一下几个方面的特色：
(1)研究了弹性地基上悬臂输流管的振动稳定性及非线性动力学行为。建立了非线性Pasternak弹性地基上输流管道的非线性运动方程,重点考查了悬臂输流管的内共振、模态转换及内流作用下的非线性动力学行为。结果表明,悬臂输流管-地基系统在内流速的作用下一、二阶模态之间可能会发生3：1内共振,1：1内共振(模态转换)；地基剪切刚度对系统的振动稳定性有很大影响,其改变会导致系统出现模态转换,从而导致失稳形式和发生失稳的模态发生变化；增大地基剪切刚度会增大系统Hopf分岔临界流速,并且对系统的混沌运动有较明显的抑制作用。同时,在连接刚度和剪切刚度的作用下,系统可能会出现双退化分岔。
(2)研究了基础激励作用下非线性Pasternak地基上悬臂输流管的受迫振动。结果表明,系统在基础激励作用下具有非常复杂的动力学行为,包括多种形式的周期、概周期和混沌运动；地基剪切刚度对系统的概周期运动和混沌运动有抑制作用,当地基剪切刚度足够大时,该输流管系统将始终处于周期运动状态；非线性地基刚度对系统的动力学行为也有较大影响。
(3)研究了脉动流作用下简支输流管-非线性地基系统的动力学行为。详细分析了Galerkin模态截断对系统动力学行为的影响,发现脉动频率越高,则需要越高阶的模态截断才能得到较为准确的结果。基于以上结论,对脉动流作用下简支输流管-非线性地基系统的动力学方程进行求解。结果表明,地基剪切刚度对系统的动力学行为的影响十分明显,而地基的线性支承刚度和非线性支承刚度的影响则较为有限。增加地基剪切刚度,不仅可以增强系统的稳定性,还可以抑制系统的混沌运动,同时会导致系统动力学行为发生根本性的变化。
(4)研究了由具有相同平均内流速度的两根简支输流管组成的耦合输流管系统的动力学行为。使用分布线性弹簧来模拟两管之间的连接,建立了具有内流脉动的耦合输流管系统的运动方程；通过数值计算考查了不同激励频率组合和连接刚度对系统动力学行为的影响。数值分析表明,由于连接刚度的影响,耦合输流管系统两管的振动状态之间会出现同步现象,连接刚度越大,同步现象越明显。系统在不同激励频率组合下也会出现十分丰富的动力学行为,在某些激励频率组合下,系统的动力学行为与初始条件密切相关。
(5)进一步研究了不同平均流速组合下耦合简支输流管系统的非线性动力学行为。主要考虑两种不同的流速组合：(1)两管分别具有超临界平均流速和亚临界平均流速脉动流；(2)其中一根管道具有超临界平均流速,另一根则为简支梁(平均流速为零)。结果表明,在连接刚度的作用下,两种组合下耦合系统均表现出同步现象。不同的流速组合会导致系统出现截然不同的动力学行为；对于流速组合(1),系统的动力学响应会由具有亚临界流速的管道所支配；对于流速组合(2),系统的动力学响应会由具有两根管道共同支配。
(6)将DTM推广到输流直管的振动稳定分析当中,具体考察了不同支承条件下输流管道的振动频率及临界流速,用以检验DTM法的有效性。结果显示,与DQM及文献[1,110]中的结果相比,DTM具有极高的精度,并且待解微分方程阶数的增加并不会使DTM的计算过程变得复杂。
综上所述,本文对耦合输流管系统的稳定性机理和非线性动力学行为进行了深入的研究,揭示了不同参数组合下耦合输流管系统的分岔路径和混沌运动的形成机制,这对工程输流管线的合理设计具有重要的指导意义。
The vibration stability and nonlinear dynamics of coupled pipe system are studied in the present paper. Three typical coupled pipe systems are considered in the current work: (1) fluid-conveying pipe-nonlinear elastic foundation system; (2) coupled two-pipe conveying fluid system; (3) coupled fluid-conveying pipe-beam system. The complex bifurcation path and chaotic motion of substructures of coupled pipe system are mainly investigated under different combination of system parameters. The semi-analytical methods employed to analyze the dynamical model of coupled pipe system, such as DTM, are also discussed. Based on numerical calculations, some interesting phenomena were observed, and some valuable results were put forward.
(1) The vibration stability and nonlinear dynamical behavior of cantilevered fluid-conveying pipe-nonlinear elastic foundation system are studied. The equation of motion of a cantilevered pipe conveying fluid rested on nonlinear elastic foundation is obtained based on the nonlinear Pasternak foundation model. The internal resonance, mode exchange and nonlinear dynamics induced by internal flow are mainly investigated. Results show that the internal flow may induce 3:1 internal resonance,1:1 internal resonance; the shear stiffness has significant influence on the stability of the system; mode exchange may occur leading to the change of mode and type of instability with the variety of shear stiffness; the critical flow velocity corresponding to Hopf bifurcation increases and chaos may be controlled with the increasing of shear stiffness.
(2) The forced vibration of cantilevered fluid-conveying pipe rested on nonlinear elastic foundation system under foundation excitation is further studied. The effect of excitation frequency and shear stiffness on the dynamical behavior of system is mainly investigated. The results demonstrate that the pipe system has complex response with the variation of foundation excitation, and period motions, quasi-periodic motions, chaos are observed; with the increase of foundation shear stiffness, the regions of quasi-periodic motions, chaos decrease. The system undergoes period motion with the variation of foundation excitation.
(3) The dynamical behavior of simply supported pipe conveying pulsating fluid rested on nonlinear elastic foundation is studied. The influence of Galerkin mode truncation on the dynamics of the system is discussed. The results show that, the Galerkin mode truncation number may chose to be bigger when the forcing frequency is higher. Then, the Equation of motion of the pipe-foundation system is solved based on Galerkin method. Results show that, the shear stiffness has more significant influence on the dynamics of the system than linear stiffness and nonlinear stiffness. With the increasing of shear stiffness, the stability of the system can be reinforced and the chaotic motion may be controlled.
(4) The nonlinear dynamical behavior of two coupled pipes conveying pulsating fluid is studied. The mean flow velocities of the two pipes are the same to each other. The pipe system is composed of two colligated pipes. The connection between two pipes taken into account is considered as a linear spring. Base on this consideration, the nonlinear equations of motion of the coupled two pipe system are derived, and solved numerically. The effect of connection stiffness and different combination of forcing frequencies on the dynamics of the system is explored. Results show that the connection stiffness has significant effect on the dynamic behavior of the coupled pipe system. The pipe system exhibits extremely rich dynamical behaviors when the connection stiffness is taken into account. The most interesting results obtained are that the motion types of the two pipes might be synchronous. The dynamical behavior may be very rich in different combination of the forcing frequencies, and in some cases, the stable response of the system is related to the initial conditions.
(5) The nonlinear dynamical behavior of two coupled pipes conveying pulsating fluid with different mean flow velocities is further studied. Two combinations of the mean flow velocities considered are:the mean flow velocity of one pipe is in sub-critical region, and of another is in super-critical region; the mean flow velocity of one pipe is in super-critical region, and of another is zero (considered as simply supported beam). Results show that, for two cases, the synchronous phenomenon can be observed; for different combinations, the coupled pipe system exhibits entire different dynamical behaviors; the dynamical response of the system is dominated by the pipe with the sub-critical mean flow velocity for case (1); the dynamical response of the system is dominated by both pipes for case (2).
(6) The differential transformation method (DTM) is extended to analyze the vibration and stability of straight pipe conveying fluid. The vibration frequencies and critical velocities of different boundary conditions are obtained by using DTM and compared with that obtained by DQM and reported in Refs.[1,110]. Results show that, the DTM is an effective and precision method for solving the vibration problem of pipes conveying fluid. The calculation procedure of DTM does not get more complicated when the order of differential equation increases.
From these contents listed above, the vibration stability and nonlinear dynamical behavior of coupled pipe system conveying are deeply investigated in the present paper. Especially, the bifurcations and chaotic mechanism of the coupled pipe system under different combination of system parameters are explored. The conclusions obtained in this study are significant for the design of pipes conveying fluid.
(1)研究了弹性地基上悬臂输流管的振动稳定性及非线性动力学行为。建立了非线性Pasternak弹性地基上输流管道的非线性运动方程,重点考查了悬臂输流管的内共振、模态转换及内流作用下的非线性动力学行为。结果表明,悬臂输流管-地基系统在内流速的作用下一、二阶模态之间可能会发生3：1内共振,1：1内共振(模态转换)；地基剪切刚度对系统的振动稳定性有很大影响,其改变会导致系统出现模态转换,从而导致失稳形式和发生失稳的模态发生变化；增大地基剪切刚度会增大系统Hopf分岔临界流速,并且对系统的混沌运动有较明显的抑制作用。同时,在连接刚度和剪切刚度的作用下,系统可能会出现双退化分岔。
(2)研究了基础激励作用下非线性Pasternak地基上悬臂输流管的受迫振动。结果表明,系统在基础激励作用下具有非常复杂的动力学行为,包括多种形式的周期、概周期和混沌运动；地基剪切刚度对系统的概周期运动和混沌运动有抑制作用,当地基剪切刚度足够大时,该输流管系统将始终处于周期运动状态；非线性地基刚度对系统的动力学行为也有较大影响。
(3)研究了脉动流作用下简支输流管-非线性地基系统的动力学行为。详细分析了Galerkin模态截断对系统动力学行为的影响,发现脉动频率越高,则需要越高阶的模态截断才能得到较为准确的结果。基于以上结论,对脉动流作用下简支输流管-非线性地基系统的动力学方程进行求解。结果表明,地基剪切刚度对系统的动力学行为的影响十分明显,而地基的线性支承刚度和非线性支承刚度的影响则较为有限。增加地基剪切刚度,不仅可以增强系统的稳定性,还可以抑制系统的混沌运动,同时会导致系统动力学行为发生根本性的变化。
(4)研究了由具有相同平均内流速度的两根简支输流管组成的耦合输流管系统的动力学行为。使用分布线性弹簧来模拟两管之间的连接,建立了具有内流脉动的耦合输流管系统的运动方程；通过数值计算考查了不同激励频率组合和连接刚度对系统动力学行为的影响。数值分析表明,由于连接刚度的影响,耦合输流管系统两管的振动状态之间会出现同步现象,连接刚度越大,同步现象越明显。系统在不同激励频率组合下也会出现十分丰富的动力学行为,在某些激励频率组合下,系统的动力学行为与初始条件密切相关。
(5)进一步研究了不同平均流速组合下耦合简支输流管系统的非线性动力学行为。主要考虑两种不同的流速组合：(1)两管分别具有超临界平均流速和亚临界平均流速脉动流；(2)其中一根管道具有超临界平均流速,另一根则为简支梁(平均流速为零)。结果表明,在连接刚度的作用下,两种组合下耦合系统均表现出同步现象。不同的流速组合会导致系统出现截然不同的动力学行为；对于流速组合(1),系统的动力学响应会由具有亚临界流速的管道所支配；对于流速组合(2),系统的动力学响应会由具有两根管道共同支配。
(6)将DTM推广到输流直管的振动稳定分析当中,具体考察了不同支承条件下输流管道的振动频率及临界流速,用以检验DTM法的有效性。结果显示,与DQM及文献[1,110]中的结果相比,DTM具有极高的精度,并且待解微分方程阶数的增加并不会使DTM的计算过程变得复杂。
综上所述,本文对耦合输流管系统的稳定性机理和非线性动力学行为进行了深入的研究,揭示了不同参数组合下耦合输流管系统的分岔路径和混沌运动的形成机制,这对工程输流管线的合理设计具有重要的指导意义。
The vibration stability and nonlinear dynamics of coupled pipe system are studied in the present paper. Three typical coupled pipe systems are considered in the current work: (1) fluid-conveying pipe-nonlinear elastic foundation system; (2) coupled two-pipe conveying fluid system; (3) coupled fluid-conveying pipe-beam system. The complex bifurcation path and chaotic motion of substructures of coupled pipe system are mainly investigated under different combination of system parameters. The semi-analytical methods employed to analyze the dynamical model of coupled pipe system, such as DTM, are also discussed. Based on numerical calculations, some interesting phenomena were observed, and some valuable results were put forward.
(1) The vibration stability and nonlinear dynamical behavior of cantilevered fluid-conveying pipe-nonlinear elastic foundation system are studied. The equation of motion of a cantilevered pipe conveying fluid rested on nonlinear elastic foundation is obtained based on the nonlinear Pasternak foundation model. The internal resonance, mode exchange and nonlinear dynamics induced by internal flow are mainly investigated. Results show that the internal flow may induce 3:1 internal resonance,1:1 internal resonance; the shear stiffness has significant influence on the stability of the system; mode exchange may occur leading to the change of mode and type of instability with the variety of shear stiffness; the critical flow velocity corresponding to Hopf bifurcation increases and chaos may be controlled with the increasing of shear stiffness.
(2) The forced vibration of cantilevered fluid-conveying pipe rested on nonlinear elastic foundation system under foundation excitation is further studied. The effect of excitation frequency and shear stiffness on the dynamical behavior of system is mainly investigated. The results demonstrate that the pipe system has complex response with the variation of foundation excitation, and period motions, quasi-periodic motions, chaos are observed; with the increase of foundation shear stiffness, the regions of quasi-periodic motions, chaos decrease. The system undergoes period motion with the variation of foundation excitation.
(3) The dynamical behavior of simply supported pipe conveying pulsating fluid rested on nonlinear elastic foundation is studied. The influence of Galerkin mode truncation on the dynamics of the system is discussed. The results show that, the Galerkin mode truncation number may chose to be bigger when the forcing frequency is higher. Then, the Equation of motion of the pipe-foundation system is solved based on Galerkin method. Results show that, the shear stiffness has more significant influence on the dynamics of the system than linear stiffness and nonlinear stiffness. With the increasing of shear stiffness, the stability of the system can be reinforced and the chaotic motion may be controlled.
(4) The nonlinear dynamical behavior of two coupled pipes conveying pulsating fluid is studied. The mean flow velocities of the two pipes are the same to each other. The pipe system is composed of two colligated pipes. The connection between two pipes taken into account is considered as a linear spring. Base on this consideration, the nonlinear equations of motion of the coupled two pipe system are derived, and solved numerically. The effect of connection stiffness and different combination of forcing frequencies on the dynamics of the system is explored. Results show that the connection stiffness has significant effect on the dynamic behavior of the coupled pipe system. The pipe system exhibits extremely rich dynamical behaviors when the connection stiffness is taken into account. The most interesting results obtained are that the motion types of the two pipes might be synchronous. The dynamical behavior may be very rich in different combination of the forcing frequencies, and in some cases, the stable response of the system is related to the initial conditions.
(5) The nonlinear dynamical behavior of two coupled pipes conveying pulsating fluid with different mean flow velocities is further studied. Two combinations of the mean flow velocities considered are:the mean flow velocity of one pipe is in sub-critical region, and of another is in super-critical region; the mean flow velocity of one pipe is in super-critical region, and of another is zero (considered as simply supported beam). Results show that, for two cases, the synchronous phenomenon can be observed; for different combinations, the coupled pipe system exhibits entire different dynamical behaviors; the dynamical response of the system is dominated by the pipe with the sub-critical mean flow velocity for case (1); the dynamical response of the system is dominated by both pipes for case (2).
(6) The differential transformation method (DTM) is extended to analyze the vibration and stability of straight pipe conveying fluid. The vibration frequencies and critical velocities of different boundary conditions are obtained by using DTM and compared with that obtained by DQM and reported in Refs.[1,110]. Results show that, the DTM is an effective and precision method for solving the vibration problem of pipes conveying fluid. The calculation procedure of DTM does not get more complicated when the order of differential equation increases.
From these contents listed above, the vibration stability and nonlinear dynamical behavior of coupled pipe system conveying are deeply investigated in the present paper. Especially, the bifurcations and chaotic mechanism of the coupled pipe system under different combination of system parameters are explored. The conclusions obtained in this study are significant for the design of pipes conveying fluid.
引文
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[71]Chatjigeorgiou I. K., Mavrakos S. A. Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications. Applied Ocean Research,2002,24:341-354
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[86]Pramila A. On the gyroscopic terms appearing when the vibration of fluid conveying pipe is analyzed using the FEM. Journal of Sound Vibration,1986,105: 515-516
[87]Zhang Y. L., Gorman D. G., Reese J. M. A finite element method for modelling the vibration of initially tensioned thin-walled orthotropic cylindrical tubes conveying fluid. Journal of Sound Vibration,2001,245:93-112
[88]Pramila A., Laukkanen J., Liukkonen S. Dynamics and stability of short fluid-conveying Timoshenko element pipes. Journal of Sound Vibration,1991,144: 421-425
[89]Ni Q., Zhang Z. L., Wang L. Application of the differential transformation method to vibration analysis of pipes conveying fluid. Applied Mathematics and Computation,2011,217:7028-7038
[90]Dutta S. C., Roy R. A critical review on idealization and modeling for interaction among soil-foundation-structure system. Computers and Structures,2002, 80(20-21):1579-159
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[92]Lottati I., Kornecki A. The effect of an elastic foundation and of dissipative forces on the stability of fluid-conveying pipes. Journal of Sound and Vibration,1986, 109:327-338
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