考虑有限深度土体运动的Winkler地基梁自由振动分析
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摘要
利用Winkler地基梁理论,考虑有限深度土体运动的影响,建立了弹性地基梁的线性运动方程。采用分离变量法,求得弹性地基梁的模态构型、固有频率和有阻尼自由振动。通过数值计算和参数分析,揭示了有限深度Winkler地基上固支-自由梁的线性动力学特性,分析了土体质量、地基深度和阻尼等对系统固有频率和线性自由振动响应的影响。研究结果表明:若将有限深度土体运动引入到弹性地基梁的动力学模型,系统的固有频率将显著降低;土体质量和地基深度均抑制阻尼对弹性地基梁动力响应影响的发挥,在一定程度上减慢其动能耗散的速度。
Based on the Winkler foundation model, considering the effect of finite depth soil motions, the linear vibration equation of a beam on elastic foundation was derived. Using the separation of variables, the mode configuration, natural frequency and damped free vibration of the beam on elastic foundation were obtained. By means of numerical calculation and parameter analysis, the linear dynamic characteristics of the clamped-free beam on Winkler foundation of finite depth were revealed. Meanwhile, the effects of the soil mass, foundation depth and damping on the natural frequency and dynamic response of the system were analyzed. The numerical results show that the natural frequency of the beam on elastic foundation decreases significantly, when the motion of finite depth foundation is considered integratedly in the dynamic model of the system. Moreover, the mass and depth of foundation inhibit the effect of damping on the dynamic response of the beam on elastic foundation, and slow down the kinetic energy dissipation rate to some extent.
Based on the Winkler foundation model, considering the effect of finite depth soil motions, the linear vibration equation of a beam on elastic foundation was derived. Using the separation of variables, the mode configuration, natural frequency and damped free vibration of the beam on elastic foundation were obtained. By means of numerical calculation and parameter analysis, the linear dynamic characteristics of the clamped-free beam on Winkler foundation of finite depth were revealed. Meanwhile, the effects of the soil mass, foundation depth and damping on the natural frequency and dynamic response of the system were analyzed. The numerical results show that the natural frequency of the beam on elastic foundation decreases significantly, when the motion of finite depth foundation is considered integratedly in the dynamic model of the system. Moreover, the mass and depth of foundation inhibit the effect of damping on the dynamic response of the beam on elastic foundation, and slow down the kinetic energy dissipation rate to some extent.
引文
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[9]任青,黄茂松.分层地基中柔性高承台群桩基础的竖向振动特性[J].土木工程学报,2009,42(4):107-113.REN Qing,HUANG Maosong.Analysis of axial vibration of floating pile groups with flexible caps in layered soils[J].China Civil Engineering Journal,2009,42(4):107-113.
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[14]马建军,刘齐建,王连华,等.Winkler地基上有限长梁非线性自由振动[J].工程力学,2012,29(8):58-62.MA Jianjun,LIU Qijian,WANG Lianhua,et al.Non-linear free vibration of finite-length beams on the Winkler foundation[J].Engineering Mechanics,2012,29(8):58-62.
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[16]JAISWAL O R,IYENGAR R N.Dynamic response of a beam on elastic foundation of finite depth under a moving force[J].Acta Mechanica,1993,96(1/2/3/4):67-83.
[17]马建军,刘齐建,赵跃宇.端部横向荷载作用下改进Vlasov地基上梁的稳定性[J].岩土工程学报,2008,30(6):850-854.MA Jianjun,LIU Qijian,ZHAO Yueyu.Stability of beams on modified Vlasov foundation subjected to lateral loads acting on the ends[J].Chinese Journal of Geotechnical Engineering,2008,30(6):850-854.
[18]CLOUGH R,PENZIEN J.结构动力学[M].王光远,等,译.北京:高等教育出版社,2006.
[19]FRISWELL M I,ADHIKARI S,LEI Y.Vibration analysis of beams with non-local foundations using the finite element method[J].International Journal of Numerical Methods in Engineering,2007,71(11):1365-1386.
[20]ZHAOHUA F,COOK R D.Beam element on two-parameter elastic foundations[J].Journal of Engineering Mechanics,1983,109(6):1390-1402.
[2]龙驭球.弹性地基梁的计算[M].北京:人民教育出版社,1981.
[3]WANG Y H,THAM L G,CHEUNG Y K.Beams and plates on elastic foundations:a review[J].Progress in Structural Engineering and Materials,2005,7(4):174-182.
[4]赵明华,张玲,马缤辉,等.考虑水平摩阻力的Winkler地基有限长梁非线性受力分析[J].土木工程学报,2009,42(7):106-112.ZHAO Minghua,ZHANG Ling,MA Binhui,et al.Nonlinear analysis of finite-length beams on Winkler foundation with horizontal resistance[J].China Civil Engineering Journal,2009,42(7):106-112.
[5]蒲育,滕兆春.Winkler-Pasternak弹性地基FGM梁自由振动二维弹性解[J].振动与冲击,2015,34(20):74-79.PU Yu,TENG Zhaochun.Two-dimensional elasticity solutions for free vibration of FGM beams resting on WinklerPasternak elastic foundations[J].Journal of Vibration and Shock,2015,34(20):74-79.
[6]TIMOSHENKO S.Vibration problems in engineering[M].New York:Wolfenden Press,2008.
[7]吴志明,黄茂松,吕丽芳.桩-桩水平振动动力相互作用研究[J].岩土力学,2007,28(9):1848-1855.WU Zhiming,HUANG Maosong,LLifang.Research on pile-pile dynamic interaction of lateral vibration[J].Rock and Soil Mechanics,2007,28(9):1848-1855.
[8]杨志安,赵雪娟,席晓燕.非线性弹性地基上矩形薄板的非线性振动与奇异性分析[J].振动与冲击,2006,25(5):69-73.YANG Zhian,ZHAO Xuejuan,XI Xiaoyan.Nonlinear vibration and singularities analysis of a thin rectangular plate on nonlinear elastic foundation[J].Journal of Vibration and Shock,2006,25(5):69-73.
[9]任青,黄茂松.分层地基中柔性高承台群桩基础的竖向振动特性[J].土木工程学报,2009,42(4):107-113.REN Qing,HUANG Maosong.Analysis of axial vibration of floating pile groups with flexible caps in layered soils[J].China Civil Engineering Journal,2009,42(4):107-113.
[10]李顺群,郑刚.复杂条件下Winkler地基梁的解析解[J].岩土工程学报,2008,30(6):873-879.LI Shunqun,ZHENG Gang.Analytic solution of beams on Winkler foundation under complex conditions[J].Chinese Journal of Geotechnical Engineering,2008,30(6):873-879.
[11]LAI Y C,TING B T,LEE W S,et al.Dynamic response of beams on elastic foundation[J].Journal of Structural Engineering,1992,118(3):853-858.
[12]THAMBIRATNAM D,ZHUGE Y.Free vibration analysis of beams on elastic foundation[J].Computers&Structures,1996,60(6):971-980.
[13]楼梦麟,沈霞.弹性地基梁振动特性的近似分析方法[J].应用力学学报,2004,21(3):149-153.LOU Menglin,SHEN Xia.An approach for analyzing dynamic characteristic of reinforced concrete beam on elastic foundation[J].Chinese Journal of Applied Mechanics,2004,21(3):149-153.
[14]马建军,刘齐建,王连华,等.Winkler地基上有限长梁非线性自由振动[J].工程力学,2012,29(8):58-62.MA Jianjun,LIU Qijian,WANG Lianhua,et al.Non-linear free vibration of finite-length beams on the Winkler foundation[J].Engineering Mechanics,2012,29(8):58-62.
[15]IYENGAR R N,PRANESH M R.Dynamic response of a beam on a foundation of finite depth[J].Indian Geotechnical Journal,1985,15(2):53-63.
[16]JAISWAL O R,IYENGAR R N.Dynamic response of a beam on elastic foundation of finite depth under a moving force[J].Acta Mechanica,1993,96(1/2/3/4):67-83.
[17]马建军,刘齐建,赵跃宇.端部横向荷载作用下改进Vlasov地基上梁的稳定性[J].岩土工程学报,2008,30(6):850-854.MA Jianjun,LIU Qijian,ZHAO Yueyu.Stability of beams on modified Vlasov foundation subjected to lateral loads acting on the ends[J].Chinese Journal of Geotechnical Engineering,2008,30(6):850-854.
[18]CLOUGH R,PENZIEN J.结构动力学[M].王光远,等,译.北京:高等教育出版社,2006.
[19]FRISWELL M I,ADHIKARI S,LEI Y.Vibration analysis of beams with non-local foundations using the finite element method[J].International Journal of Numerical Methods in Engineering,2007,71(11):1365-1386.
[20]ZHAOHUA F,COOK R D.Beam element on two-parameter elastic foundations[J].Journal of Engineering Mechanics,1983,109(6):1390-1402.