非线性轴承—转子系统动力学行为及稳定性分析
|
摘要
大型旋转机械是国家基础设施和基础工业中关键的设备之一,随着旋转机械向着高速、高效、高可靠性发展,对旋转机械的稳定性提出了更高的要求,同时轴承-转子动力系统的研究也越来越受到重视。本文针对轴承-转子系统的非线性动力学行为进行分析。
论文的具体工作包括以下几方面：
1、首先,通过油膜力数据库方法对比了现有的非线性油膜力解析模型,讨论了现有非线性油膜力解析模型的适用范围和近似程度。接着对具有Reynolds边界条件的滑动轴承流体润滑的Reynolds原理进行了修正,在此基础上,采用8节点等参有限元方法,按照油膜的物理特性,形成修正的Reynolds方程的变分形式及其扰动方程,在不增加计算量的情况下,同时求得非线性油膜力和及其Jacobi矩阵。
2、分析了Newmark法在计算时间步长内的计算扰动,并且对其进行补偿,形成了一种有效的求解非线性动力系统行为的方法——结合扰动补偿的Newmark法。该方法由于考虑了计算扰动效应,使得计算结果在各个离散时刻既满足运动约束条件,又基本满足动力平衡方程。
3、基于Wilson-θ法,并将其改进,同时引入预估-校正机理,形成了一种有效的求解非线性动力行为的方法。该方法在迭代求解前对迭代的初值进行预估,使得迭代的初值尽可能接近真值,之后以预估值作为初值,运用Newton-Raphson方法对其进行校正,这样提高了计算效率,节约了计算时间。
4、最后,运用提出的方法并结合Floquet稳定性分岔理论和Poincare映射,针对动压轴承支承的转子系统,分析了其周期解的稳定性和分岔规律。通过数值仿真,揭示了轴承-转子系统的周期解、倍周期解、准周期解、跳跃和混沌等复杂丰富的非线性动力学行为。
论文得到的结果可为轴承-转子系统运动稳定性分析及产品的动力学设计提供理论参考,具有积极的指导意义。
Large rotating machinery is one of the equipments in national basic facilities and industries. A lot of high requirements for stability and dynamic characteristics of rotating machinery have been made along with the development of speed, efficiency and reliability, so research on the bearing-rotor system is increasingly emphasized. Nonlinear dynamic behaviors of bearing-rotor system are analyzed in the paper.
The main contents are as follow:
1. The applicable range and approximate extent of the current analytical model of nonlinear oil film force are discussed by comparing data base method with the current analytical method of nonlinear oil film force. The variational approch is revised in fluid lubrication of journal bearing with Reynolds boundary. According to the physical characteristics of oil film, a revised variational form of Reynolds equation and its disturbed equation are formulated by isoparametric finite element method with 8 nodes. So the nonlinear oil film force and its Jacobian matrix are obtained simultaneously without increase of computational costs.
2. The computational disturbances of Newmark method are analyzed at each time step. The computational disturbances of Newmark method are compensated, and then an effective numerical method, i.e. Newmark method with disturbance compensation, is presented to analyze dynamic behaviors of nonlinear dynamics system. Because the disturbance compensation is made in the presented method, the results obtained by the method can satisfy the movement constraints and dynamic balance equation simultaneously.
3. Based on Wilson-θmethod, an effective method is presented to analyze the dynamic behaviors of nonlinear dynamic system by improving Wilson-θmethod and combining with predictor-corrector mechanism. For solving the nonlinear equations by the method, initial values of iteration are predicted to approximate the values of the nonlinear equations. Furthermore obtained values are corrected by Newton-Raphson method. The presented method can improve calculation efficiency and save calculation time.
4 The nonlinear dynamic responses, stability of periodic solution and bifurcation of the bearing-rotor system are analyzed by the presented methods combining with Floquet theory and Poincare map. The numerical results reveal periodic, period-doubling, quasi-periodic, jump, chaos of rich and complex nonlinear behaviors of the bearing-rotor system.
The presented methods and numerical results can direct the design of practical bearing-rotor system, and provide references for practical application of bearing-rotor system.
论文的具体工作包括以下几方面：
1、首先,通过油膜力数据库方法对比了现有的非线性油膜力解析模型,讨论了现有非线性油膜力解析模型的适用范围和近似程度。接着对具有Reynolds边界条件的滑动轴承流体润滑的Reynolds原理进行了修正,在此基础上,采用8节点等参有限元方法,按照油膜的物理特性,形成修正的Reynolds方程的变分形式及其扰动方程,在不增加计算量的情况下,同时求得非线性油膜力和及其Jacobi矩阵。
2、分析了Newmark法在计算时间步长内的计算扰动,并且对其进行补偿,形成了一种有效的求解非线性动力系统行为的方法——结合扰动补偿的Newmark法。该方法由于考虑了计算扰动效应,使得计算结果在各个离散时刻既满足运动约束条件,又基本满足动力平衡方程。
3、基于Wilson-θ法,并将其改进,同时引入预估-校正机理,形成了一种有效的求解非线性动力行为的方法。该方法在迭代求解前对迭代的初值进行预估,使得迭代的初值尽可能接近真值,之后以预估值作为初值,运用Newton-Raphson方法对其进行校正,这样提高了计算效率,节约了计算时间。
4、最后,运用提出的方法并结合Floquet稳定性分岔理论和Poincare映射,针对动压轴承支承的转子系统,分析了其周期解的稳定性和分岔规律。通过数值仿真,揭示了轴承-转子系统的周期解、倍周期解、准周期解、跳跃和混沌等复杂丰富的非线性动力学行为。
论文得到的结果可为轴承-转子系统运动稳定性分析及产品的动力学设计提供理论参考,具有积极的指导意义。
Large rotating machinery is one of the equipments in national basic facilities and industries. A lot of high requirements for stability and dynamic characteristics of rotating machinery have been made along with the development of speed, efficiency and reliability, so research on the bearing-rotor system is increasingly emphasized. Nonlinear dynamic behaviors of bearing-rotor system are analyzed in the paper.
The main contents are as follow:
1. The applicable range and approximate extent of the current analytical model of nonlinear oil film force are discussed by comparing data base method with the current analytical method of nonlinear oil film force. The variational approch is revised in fluid lubrication of journal bearing with Reynolds boundary. According to the physical characteristics of oil film, a revised variational form of Reynolds equation and its disturbed equation are formulated by isoparametric finite element method with 8 nodes. So the nonlinear oil film force and its Jacobian matrix are obtained simultaneously without increase of computational costs.
2. The computational disturbances of Newmark method are analyzed at each time step. The computational disturbances of Newmark method are compensated, and then an effective numerical method, i.e. Newmark method with disturbance compensation, is presented to analyze dynamic behaviors of nonlinear dynamics system. Because the disturbance compensation is made in the presented method, the results obtained by the method can satisfy the movement constraints and dynamic balance equation simultaneously.
3. Based on Wilson-θmethod, an effective method is presented to analyze the dynamic behaviors of nonlinear dynamic system by improving Wilson-θmethod and combining with predictor-corrector mechanism. For solving the nonlinear equations by the method, initial values of iteration are predicted to approximate the values of the nonlinear equations. Furthermore obtained values are corrected by Newton-Raphson method. The presented method can improve calculation efficiency and save calculation time.
4 The nonlinear dynamic responses, stability of periodic solution and bifurcation of the bearing-rotor system are analyzed by the presented methods combining with Floquet theory and Poincare map. The numerical results reveal periodic, period-doubling, quasi-periodic, jump, chaos of rich and complex nonlinear behaviors of the bearing-rotor system.
The presented methods and numerical results can direct the design of practical bearing-rotor system, and provide references for practical application of bearing-rotor system.
引文
[1]J.I.Nikolajsen, et al. Investijation of Squeeze-Film Isolators for the Vibration Control of a Flexible Rotor[J]. ASME Journal of Mechanical Science,1979,21(4),247-252.
[2]X.H.Li, D.L.Taylor. Non-synchronous Motion of Squeeze-Film Damper Systems[J]. ASME Journal of Tribolojy,1987,109,169-176.
[3]F.F. Ehrich. High Order Sub-harmonic Response of High Speed Rotor in Bearing Clearance[J]. ASME Journal of Vibration, Acoustics, Stress and Reliability in Design,1988,110,9-16.
[4]J. Kicinski, et al. The Nonlinear Analysis of the Effect of Support Construction Properties on the Dynamics Properties of Multi-Support Rotor Systems[J]. Journal of Sound and Vibration, 1997,206(4),523-539.
[5]Renato Brancatiand Michele Russo.On the Stability of an Unbalanced Rigid Rotor on Lubricated Journal Bearing[J]. Journal of Nonlinear Dynamics,1996,10:175-185.
[6]GAdileta and A.R.Guido. Chaotic Motions of a Rigid Rotor in Short Journal Bearings[J]. Journal of Nonlinear Dynamics,1996,10:251-269.
[7]孟泉,陈予恕.非线性转子-轴承系统油膜失稳新机理研究[J].非线性动力学学报,1995,2(3)：189-197.
[8]张宇,陈予恕,毕勤.转子-轴承-基础非线性动力学研究[J].振动工程学报,1998,11(1)：24-30.
[9]陈予恕,丁千.非线性转子动力学的稳定性和分叉[J].非线性动力学学报,1996，3(1)：14-21.
[10]赵玉成,袁树清,肖忠会.轴承转子系统分叉参数的分维数识别法[J].应用数学和力学,2000,21(2)：126-130.
[1l]焦映厚,陈照波,夏松波等.转子轴承系统非线性动力学行为的研究[J].中国机械工程,2000,11(6)：697-699.
[12]Cheng Zhaobo, Jiao Yinjhou. Non-synchronous Response Bifurcation Analysis for a Rigid-Bearing System Using Oil Film Force Database[J]. Journal of Harbin Institute of Technology,2000,7(2): 87-90.
[13]和兴锁,赵渊.转子-非线性支承系统振动响应的优化计算[J].计算力学学报,1997,14(1)：85-90.
[14]J.Y.Zhao, E.J.Hahn. Subharmonic、 Quasi-Periodic and Chaotic Motions of a Rigid Rotor Supported by an Eccentric Squeeze Film Damper[J]. Journal of Mechanical Engineering Science, Part C 1993,207:383-392.
[15]J.YZhao, IW.Linnett. Subharmonic and Quasi-Periodic Motions of an Eccentric Squeeze Film Damper-Mounted Rigid Rotor[J]. ASME Journal of Vibration and A coustics,1994,116:357-363.
[16]郭银朝,孟光.挤压油膜阻尼器支承的柔性转子的双稳态响应特性[J].振动工程学报,1997,10(1)：66-70.
[17]郭银朝,孟光,戈立春.惯性力对挤压油膜阻尼器支承转子的响应的影响[J].西北工业大学学 报,1997,15(2)：295-299.
[18]张家忠,徐庆余,郑铁生.阻尼器滑动轴承转子系统的非协调运动及突跳特性分析[J].航空动力学报,1998,13(2)：165-168.
[19]赵永辉,王莉,邹经湘.非线性转子SFD支承系统的分叉[J].航空动力学报,1999,14(4)：353-356.
[20]陈照波,焦映厚.结构对称的SFD柔性转子系统双稳态现象发生规律的研究[J].航空动力学报,1999,14(4)：443-448.
[21]顾致平,孟光,支希哲等.非线性多转子支承系统的偏置同步响应[J].振动工程学报,1998,11(3)：322-327.
[22]刘恒,虞烈,谢友柏.非线性不平衡轴承转子系统全局特性及其稳定性准则的研究[J].机械工程学报,1999,35(2)：61-65.
[23]J.Kicinski, R.Drozdowski, P.Materny. The Nonlinear Analysis of the Effect of Support Construction Properties on the Dynamic Properties of Multi-support Rotor Systems[J]. Journal of Sound and Vibration,1997,206(4):523-539.
[24]丁千,陈予恕.非线性转子-密封系统的亚谐共振失稳机理研究[J].振动工程学报,1997,10(4)：404-412.
[25]何立东,夏松波,闻雪友.转子轴承密封系统中的若干非线性动力学问题[J].哈尔滨工业大学报,1998,30(增刊)：14-17.
[26]崔升.非线性转子-轴承系统分析与支承参数及载荷识别研究[D].哈尔滨工业大学博士论文,1996：81-121.
[27]张家忠,许庆余,郑铁生.具有局部非线性动力系统周期解及稳定性方法[J].力学学报,1998,30(5)：572-579.
[28]郑铁生.高维局部非线性转子-轴承动力系统的稳定性和分叉[J].航空学报,1998,19(3)：284-292.
[29]张家忠,许庆余,郑铁生.一种分析具有局部非线性多自由度动力系统动力特性的方法[J].计算力学学报,1999,16(1)：74-78.
[30]Sundararajan. P. Dynamics of forced nonlinear system using shooting/arc-lenghth continuation method-application to rotor system[J]. ASME Journal of Vibration and Acoustics,1997,119(1):9-20.
[31]刘恒.转子轴承系统稳定性分岔问题的研究[D].西安：西安交通大学机械工程学院,1998.
[32]Zhang Wen, Zhang Huisheng, Xu Xiaofeng. Study of general nonlinear formular of oil-film force acting on a journal with unsteady motion. Asia-Pacific Vibration Conference'97[C], Kyongju, Korea: 1997,824-829.
[33]吕延军,虞烈,刘恒.非线性转子-轴承系统的动力学特性及稳定性[J].机械强度,2004，26(3)：242-246.
[34]彭超英,朱均,陈瑞琪.弹性支撑滑动轴承系统的稳定性理论及应用[J].摩擦学学报,1999,
19,(1):66-71.
[35]Yan H T, Chen C K, Chen C L. Chaos and bifurcation analysis of a flexible rotor supported by short journal bearings with non-linear suspension[J]. Proceedings of the IMechE, Part C, Journal of Mechanical Engineering Science,2000,214(7):931-947.
[36]Brown R D, Drummond Q Addison P S. Chaotic response of a short journal of bearing[J]. Proceedings of the IMECH E, Part C, Journal of Engineering Tribology,2000,214(4):387-400.
[37]Chen C K, Yan H T. Bifurcation in a flexible rotor supported by short journal bearings with nonlinear suspension[J]. Journal of Vibration and Control,2001,7(5):653-673.
[38]Choy F K. Nonlinear effects in a plain journal bearing:Part 1-Analytical study[J]. ASME Journal of Tribology,1991,113(3):555-526.
[39]Choy F K. Nonlinear effects in a plain journal bearing:Part 1-Results[J]. ASME Journal of Tribology, 1991,113(3):563-570.
[40]Yu Q L. Prediction of cavitation in journal bearings using a boundary element method[J]. ASME Journal of Tribology,1995,117(3):411-421.
[41]Rao H V V L N, Biswas B, et al. Analytical approach to evaluate dynamic coefficients and transient analysis of a hydrodynamic journal bearing[J]. Tribology Transactions,2001,43(1):109-115.
[42]刘大全,肖忠会,张文,郑铁生.滑动轴承非线性油膜力的一维直接解法[J].机械工程学报,2005,41(2)：51-55.
[43]Lu YanJun, Dai Rong, Liu Heng, Yu Lie, et al. Stability and coupling dynamic behavior of nonlinear journal active electromagnetic bearing rotor system[J]. Front. Mech. Eng. China,2008,3(2): 193-199.
[44]虞烈,刘恒.轴承-转子系统动力学[M].西安：西安交通大学出版社,2001.
[45]Y J Lu, Y F Zhang, R Dai, et al. Non-linear analysis of a flexible rotor system with multi-span bearing supports[J]. Proceedings of the IMechE, Part J, Journal of Engineering Tribology,2008,222: 87-95.
[46]吕延军,李旗,张志禹.转子-机匣系统的碰摩振动响应[J].润滑与密封,2005，167：76-78.
[47]Lu YanJun, Liu Heng, Yu Lie, et al. Analysis of stability and nonlinear response of rotor system with elliptical sliding bearing supports[J]. Front. Mech. Eng. China,2007,2(1):37-45.
[48]黄庆丰,王全凤,胡云昌Wilson-θ法直接积分的运动约束和计算扰动[J].计算力学学报,2005,22(4)：477-481.
[49]晏启祥,刘浩吾,何川.一种改进的Wilson-0法及其算法稳定性[J].土木工程学报,2003,36(10)：76-79.
[50]晏启祥,刘浩吾,饶俊勇.用改进的Wilson-θ法分析流固耦合表面波的影响[J].四川大学学报,2002,34(4)：19-25.
[2]X.H.Li, D.L.Taylor. Non-synchronous Motion of Squeeze-Film Damper Systems[J]. ASME Journal of Tribolojy,1987,109,169-176.
[3]F.F. Ehrich. High Order Sub-harmonic Response of High Speed Rotor in Bearing Clearance[J]. ASME Journal of Vibration, Acoustics, Stress and Reliability in Design,1988,110,9-16.
[4]J. Kicinski, et al. The Nonlinear Analysis of the Effect of Support Construction Properties on the Dynamics Properties of Multi-Support Rotor Systems[J]. Journal of Sound and Vibration, 1997,206(4),523-539.
[5]Renato Brancatiand Michele Russo.On the Stability of an Unbalanced Rigid Rotor on Lubricated Journal Bearing[J]. Journal of Nonlinear Dynamics,1996,10:175-185.
[6]GAdileta and A.R.Guido. Chaotic Motions of a Rigid Rotor in Short Journal Bearings[J]. Journal of Nonlinear Dynamics,1996,10:251-269.
[7]孟泉,陈予恕.非线性转子-轴承系统油膜失稳新机理研究[J].非线性动力学学报,1995,2(3)：189-197.
[8]张宇,陈予恕,毕勤.转子-轴承-基础非线性动力学研究[J].振动工程学报,1998,11(1)：24-30.
[9]陈予恕,丁千.非线性转子动力学的稳定性和分叉[J].非线性动力学学报,1996，3(1)：14-21.
[10]赵玉成,袁树清,肖忠会.轴承转子系统分叉参数的分维数识别法[J].应用数学和力学,2000,21(2)：126-130.
[1l]焦映厚,陈照波,夏松波等.转子轴承系统非线性动力学行为的研究[J].中国机械工程,2000,11(6)：697-699.
[12]Cheng Zhaobo, Jiao Yinjhou. Non-synchronous Response Bifurcation Analysis for a Rigid-Bearing System Using Oil Film Force Database[J]. Journal of Harbin Institute of Technology,2000,7(2): 87-90.
[13]和兴锁,赵渊.转子-非线性支承系统振动响应的优化计算[J].计算力学学报,1997,14(1)：85-90.
[14]J.Y.Zhao, E.J.Hahn. Subharmonic、 Quasi-Periodic and Chaotic Motions of a Rigid Rotor Supported by an Eccentric Squeeze Film Damper[J]. Journal of Mechanical Engineering Science, Part C 1993,207:383-392.
[15]J.YZhao, IW.Linnett. Subharmonic and Quasi-Periodic Motions of an Eccentric Squeeze Film Damper-Mounted Rigid Rotor[J]. ASME Journal of Vibration and A coustics,1994,116:357-363.
[16]郭银朝,孟光.挤压油膜阻尼器支承的柔性转子的双稳态响应特性[J].振动工程学报,1997,10(1)：66-70.
[17]郭银朝,孟光,戈立春.惯性力对挤压油膜阻尼器支承转子的响应的影响[J].西北工业大学学 报,1997,15(2)：295-299.
[18]张家忠,徐庆余,郑铁生.阻尼器滑动轴承转子系统的非协调运动及突跳特性分析[J].航空动力学报,1998,13(2)：165-168.
[19]赵永辉,王莉,邹经湘.非线性转子SFD支承系统的分叉[J].航空动力学报,1999,14(4)：353-356.
[20]陈照波,焦映厚.结构对称的SFD柔性转子系统双稳态现象发生规律的研究[J].航空动力学报,1999,14(4)：443-448.
[21]顾致平,孟光,支希哲等.非线性多转子支承系统的偏置同步响应[J].振动工程学报,1998,11(3)：322-327.
[22]刘恒,虞烈,谢友柏.非线性不平衡轴承转子系统全局特性及其稳定性准则的研究[J].机械工程学报,1999,35(2)：61-65.
[23]J.Kicinski, R.Drozdowski, P.Materny. The Nonlinear Analysis of the Effect of Support Construction Properties on the Dynamic Properties of Multi-support Rotor Systems[J]. Journal of Sound and Vibration,1997,206(4):523-539.
[24]丁千,陈予恕.非线性转子-密封系统的亚谐共振失稳机理研究[J].振动工程学报,1997,10(4)：404-412.
[25]何立东,夏松波,闻雪友.转子轴承密封系统中的若干非线性动力学问题[J].哈尔滨工业大学报,1998,30(增刊)：14-17.
[26]崔升.非线性转子-轴承系统分析与支承参数及载荷识别研究[D].哈尔滨工业大学博士论文,1996：81-121.
[27]张家忠,许庆余,郑铁生.具有局部非线性动力系统周期解及稳定性方法[J].力学学报,1998,30(5)：572-579.
[28]郑铁生.高维局部非线性转子-轴承动力系统的稳定性和分叉[J].航空学报,1998,19(3)：284-292.
[29]张家忠,许庆余,郑铁生.一种分析具有局部非线性多自由度动力系统动力特性的方法[J].计算力学学报,1999,16(1)：74-78.
[30]Sundararajan. P. Dynamics of forced nonlinear system using shooting/arc-lenghth continuation method-application to rotor system[J]. ASME Journal of Vibration and Acoustics,1997,119(1):9-20.
[31]刘恒.转子轴承系统稳定性分岔问题的研究[D].西安：西安交通大学机械工程学院,1998.
[32]Zhang Wen, Zhang Huisheng, Xu Xiaofeng. Study of general nonlinear formular of oil-film force acting on a journal with unsteady motion. Asia-Pacific Vibration Conference'97[C], Kyongju, Korea: 1997,824-829.
[33]吕延军,虞烈,刘恒.非线性转子-轴承系统的动力学特性及稳定性[J].机械强度,2004，26(3)：242-246.
[34]彭超英,朱均,陈瑞琪.弹性支撑滑动轴承系统的稳定性理论及应用[J].摩擦学学报,1999,
19,(1):66-71.
[35]Yan H T, Chen C K, Chen C L. Chaos and bifurcation analysis of a flexible rotor supported by short journal bearings with non-linear suspension[J]. Proceedings of the IMechE, Part C, Journal of Mechanical Engineering Science,2000,214(7):931-947.
[36]Brown R D, Drummond Q Addison P S. Chaotic response of a short journal of bearing[J]. Proceedings of the IMECH E, Part C, Journal of Engineering Tribology,2000,214(4):387-400.
[37]Chen C K, Yan H T. Bifurcation in a flexible rotor supported by short journal bearings with nonlinear suspension[J]. Journal of Vibration and Control,2001,7(5):653-673.
[38]Choy F K. Nonlinear effects in a plain journal bearing:Part 1-Analytical study[J]. ASME Journal of Tribology,1991,113(3):555-526.
[39]Choy F K. Nonlinear effects in a plain journal bearing:Part 1-Results[J]. ASME Journal of Tribology, 1991,113(3):563-570.
[40]Yu Q L. Prediction of cavitation in journal bearings using a boundary element method[J]. ASME Journal of Tribology,1995,117(3):411-421.
[41]Rao H V V L N, Biswas B, et al. Analytical approach to evaluate dynamic coefficients and transient analysis of a hydrodynamic journal bearing[J]. Tribology Transactions,2001,43(1):109-115.
[42]刘大全,肖忠会,张文,郑铁生.滑动轴承非线性油膜力的一维直接解法[J].机械工程学报,2005,41(2)：51-55.
[43]Lu YanJun, Dai Rong, Liu Heng, Yu Lie, et al. Stability and coupling dynamic behavior of nonlinear journal active electromagnetic bearing rotor system[J]. Front. Mech. Eng. China,2008,3(2): 193-199.
[44]虞烈,刘恒.轴承-转子系统动力学[M].西安：西安交通大学出版社,2001.
[45]Y J Lu, Y F Zhang, R Dai, et al. Non-linear analysis of a flexible rotor system with multi-span bearing supports[J]. Proceedings of the IMechE, Part J, Journal of Engineering Tribology,2008,222: 87-95.
[46]吕延军,李旗,张志禹.转子-机匣系统的碰摩振动响应[J].润滑与密封,2005，167：76-78.
[47]Lu YanJun, Liu Heng, Yu Lie, et al. Analysis of stability and nonlinear response of rotor system with elliptical sliding bearing supports[J]. Front. Mech. Eng. China,2007,2(1):37-45.
[48]黄庆丰,王全凤,胡云昌Wilson-θ法直接积分的运动约束和计算扰动[J].计算力学学报,2005,22(4)：477-481.
[49]晏启祥,刘浩吾,何川.一种改进的Wilson-0法及其算法稳定性[J].土木工程学报,2003,36(10)：76-79.
[50]晏启祥,刘浩吾,饶俊勇.用改进的Wilson-θ法分析流固耦合表面波的影响[J].四川大学学报,2002,34(4)：19-25.