人行荷载下梁式结构振动分析的DQ-IQ混合法
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摘要
为了评估人行荷载作用下梁式结构的振动舒适度,利用微分求积-积分求积,即DQ-IQ混合法求解移动荷载作用下梁的振动响应。人行荷载作用下梁式结构的振动控制方程是含Dirac函数的偏微分方程,首先利用IQ法离散与时间相关的Dirac函数,再利用DQ法把控制方程转化为二阶常系数微分方程,最后利用Newmark算法求解微分方程。以某钢结构连廊为例,利用DQ法计算结构自振频率并与解析解进行对比,结果验证了节点选取和边界条件施加的合理性,再利用DQ-IQ混合法和振型叠加法分别计算了不同行走步频下连廊的响应,计算结果表明,DQ-IQ混合法具有较高的可靠性和精确性。DQ-IQ混合法也可以推广到诸如车辆荷载作用下路面或桥梁的动力响应等其他移动荷载下结构的振动分析。
This paper evaluates the vibration serviceability of a girder beam structure under a moving load.Its vibration control equation is a partial differential equation containing Dirac function.Dirac function is discretized by IQ method for time-dependent and the control equation is transformed into a differential equation with constant coefficients of the second order by DQ method.The solution to the differential equation is obtained by using Newmark algorithm.Differential quadrature-integral quadrature hybrid method is used for solving the vibration response of a beam under the action of pedestrians.Take a steel corridor as an example.The natural frequencies of the structure are calculated by DQ method and compared with the analytical solution to illustrate the accuracy of node selection and boundary conditions.The corridor responses at different walking frequencies under corridor response are calculated by using the hybrid DQ-IQ method and modal superposition method,which verify the reliability and accuracy of the DQ-IQ hybrid method.The DQ-IQ hybrid method also can be applied to vibration analysis of a structure under other moving loads such as the dynamic responses of pavements or bridges under vehicle loads.
This paper evaluates the vibration serviceability of a girder beam structure under a moving load.Its vibration control equation is a partial differential equation containing Dirac function.Dirac function is discretized by IQ method for time-dependent and the control equation is transformed into a differential equation with constant coefficients of the second order by DQ method.The solution to the differential equation is obtained by using Newmark algorithm.Differential quadrature-integral quadrature hybrid method is used for solving the vibration response of a beam under the action of pedestrians.Take a steel corridor as an example.The natural frequencies of the structure are calculated by DQ method and compared with the analytical solution to illustrate the accuracy of node selection and boundary conditions.The corridor responses at different walking frequencies under corridor response are calculated by using the hybrid DQ-IQ method and modal superposition method,which verify the reliability and accuracy of the DQ-IQ hybrid method.The DQ-IQ hybrid method also can be applied to vibration analysis of a structure under other moving loads such as the dynamic responses of pavements or bridges under vehicle loads.
引文
[1]许立言,陶慕轩,樊健生,等.大跨度钢-混凝土组合人行天桥舒适度分析[J].建筑结构学报,2016,37(5):138-145.(XU Li-yan,TAO Mu-xuan,FAN Jiansheng,et al.Vibration comfort analysis on large-span steel-concrete composite pedestrian[J].Journal Building of Structure,2016,37(5):138-145.(in Chinese))
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[9]Bellman R,Kashef B G,Casti J.Differential quadrature:A technique for the rapid solution of nonlinear partial differential equations[J].Journal of Computational Physics,1972,10(1):40-52.
[10]Bert C W,Malik M.Differential quadrature method in computational mechanics:A review[J].Applied Mechanics Reviews,1996,49(1):1-28.
[11]张琼,杜永峰,朱前坤.基于微分求积法的EulerBernoulli梁的大变形力学行为研究[J].工程力学,2014,31(s1):1-4,10.(ZHANG Qiong,DU Yongfeng,ZHU Qian-kun.Study on large deformation mechanical behavior of Euler-Bernoulli beam using DQM[J].Engineering Mechanics,2014,31(s1):1-4,10.
[12]聂国隽,朱佳瑜.纤维曲线铺放的复合材料层合板的自由振动分析[J].力学季刊,2016,37(2):274-283.(NIE Guo-jun,ZHU Jia-yu.Free vibration analysis of composite laminates with curvilinear fibers[J].Chinese Quarterly of Mechanics,2016,37(2):274-283.(in Chinese))
[13]Fung T C.Solving initial value problems by differential quadrature method,Part 1:First-order equations[J].International Journal for Numerical Methods in Engineering,2001,50(6):1411-1427.
[14]李莹,程昌钧,胡育佳.移动周期载荷作用下孔隙热弹性地基的动力学响应[J].固体力学学报,2012,33(2):125-135.(LI Ying,CHENG Chang-jun,HUYu-jia.Dynamic response of thermoelastic foundation with voids under a moving periodic load[J].Chinese Journal of Solid Mechanics,2012,33(2):125-135.(in Chinese))
[15]廖旭,李鸿晶,孙广俊.基于DQ原理的结构弹塑性地震反应分析[J].工程力学,2013,30(7):161-166.(LIAO Xu,LI Hong-jing,SUN Guang-jun.Structure elasto-plastic seismic response analysis by differential quadrature method[J].Engineering Mechanics,2013,30(7):161-166.(in Chinese))
[16]Eftekhari S A,Jafari A A.Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates[J].Applied Mathematics and Mechanics,2012,33(1):81-98.
[17]Eftekhari S A.A differential quadrature procedure with regularization of the Dirac-delta function for numerical solution of moving load problem[J].Latin American Journal of Solids and Structures,2015,12(7):1241-1265.
[18]Eftekhari S A.A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads[J].Applied Mathematical Modelling,2015,39(20):6223-6242.
[19]Kerr S C,Bishop N W M.Human induced loading on flexible staircases[J].Engineering Structures,2001,23(1):37-45.
[20]丁丽娟,程杞元.数值计算方法(第二版)[M].北京:北京理工大学出版社,2005.(DING Li-juan,CHENGQi-yuan.Numerical Method(Second Edition)[M].Beijing:Beijing University of Technology Press,2005.
[2]朱前坤,陈凯,张举涛,等.某梁式钢楼梯在人致激励下的振动舒适度分析[J].建筑结构,2015,45(19):53-57.(ZHU Qian-kun,CHEN Kai,ZHANG Ju-tao,et al.Vibration serviceability analysis of a beam-typed steel stair under human induced excitation[J].Building Structure,2015,45(19):53-57.(in Chinese))
[3]Tubino F,Piccardo G.Tuned Mass Damper optimization for the mitigation of human-induced vibrations of pedestrian bridges[J].Meccanica,2015,50(3):809-824.
[4]朱前坤,李宏男,杜永峰,等.不同行走步速下人行桥振动舒适度定量化评估[J].工程力学,2016,33(10):97-104.(ZHU Qian-kun,LI Hong-nan,DU Yongfeng,et al.Quantitative evaluation of vibration serviceability of pedestrian bridge under different walking speed[J].Engineering Mechanics,2016,33(10):97-104.
[5]■ivanovic'S,Pavic'A.Quantification of dynamic excitation potential of pedestrian population crossing footbridges[J].Shock and Vibration,2011,18(4):563-577.
[6]Venuti F,Racic V,Corbetta A.Modelling framework for dynamic interaction between multiple pedestrians and vertical vibrations of footbridges[J].Journal of Sound and Vibration,2016,379:245-263.
[7]ivanovic'S,Pavic'A,Reynolds P.Probability-based prediction of multi-mode vibration response to walking excitation[J].Engineering Structures,2007,29(6):942-954.
[8]李红利,陈政清.考虑荷载随机性的人行桥人致振动计算方法研究[J].湖南大学学报(自然科学版),2013,40(10):22-31.(LI Hong-li,CHEN Zhengqing.A calculation method for footbridge vibration under stochastic pedestrian loading[J].Journal of Hunan University(Natural Science),2013,40(10):22-31.(in Chinese))
[9]Bellman R,Kashef B G,Casti J.Differential quadrature:A technique for the rapid solution of nonlinear partial differential equations[J].Journal of Computational Physics,1972,10(1):40-52.
[10]Bert C W,Malik M.Differential quadrature method in computational mechanics:A review[J].Applied Mechanics Reviews,1996,49(1):1-28.
[11]张琼,杜永峰,朱前坤.基于微分求积法的EulerBernoulli梁的大变形力学行为研究[J].工程力学,2014,31(s1):1-4,10.(ZHANG Qiong,DU Yongfeng,ZHU Qian-kun.Study on large deformation mechanical behavior of Euler-Bernoulli beam using DQM[J].Engineering Mechanics,2014,31(s1):1-4,10.
[12]聂国隽,朱佳瑜.纤维曲线铺放的复合材料层合板的自由振动分析[J].力学季刊,2016,37(2):274-283.(NIE Guo-jun,ZHU Jia-yu.Free vibration analysis of composite laminates with curvilinear fibers[J].Chinese Quarterly of Mechanics,2016,37(2):274-283.(in Chinese))
[13]Fung T C.Solving initial value problems by differential quadrature method,Part 1:First-order equations[J].International Journal for Numerical Methods in Engineering,2001,50(6):1411-1427.
[14]李莹,程昌钧,胡育佳.移动周期载荷作用下孔隙热弹性地基的动力学响应[J].固体力学学报,2012,33(2):125-135.(LI Ying,CHENG Chang-jun,HUYu-jia.Dynamic response of thermoelastic foundation with voids under a moving periodic load[J].Chinese Journal of Solid Mechanics,2012,33(2):125-135.(in Chinese))
[15]廖旭,李鸿晶,孙广俊.基于DQ原理的结构弹塑性地震反应分析[J].工程力学,2013,30(7):161-166.(LIAO Xu,LI Hong-jing,SUN Guang-jun.Structure elasto-plastic seismic response analysis by differential quadrature method[J].Engineering Mechanics,2013,30(7):161-166.(in Chinese))
[16]Eftekhari S A,Jafari A A.Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates[J].Applied Mathematics and Mechanics,2012,33(1):81-98.
[17]Eftekhari S A.A differential quadrature procedure with regularization of the Dirac-delta function for numerical solution of moving load problem[J].Latin American Journal of Solids and Structures,2015,12(7):1241-1265.
[18]Eftekhari S A.A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads[J].Applied Mathematical Modelling,2015,39(20):6223-6242.
[19]Kerr S C,Bishop N W M.Human induced loading on flexible staircases[J].Engineering Structures,2001,23(1):37-45.
[20]丁丽娟,程杞元.数值计算方法(第二版)[M].北京:北京理工大学出版社,2005.(DING Li-juan,CHENGQi-yuan.Numerical Method(Second Edition)[M].Beijing:Beijing University of Technology Press,2005.
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