移动荷载作用下曲线轨道振动响应解析解研究
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摘要
将曲线轨道模拟为周期性离散支撑的曲线Timoshenko梁,首次推导了移动荷载作用下曲线轨道平面外振动的响应解析解。首先以Duhamel积分为基础,应用动力互等定理,得到沿曲线移动荷载作用下,半无限弹性空间体上任意点动力响应的一般表达式;然后在该式的基础上,针对轨道结构的周期性支撑特点,将荷载沿钢轨的移动问题转化为拾振点以轨枕间距为周期反方向跳跃式移动与荷载只在一个轨枕间距内移动的组合问题。以此,将一个频域积分问题转化为拾振点频域周期解的叠加问题,从而得到了曲线轨道在移动荷载作用下动力响应的解析解。曲线梁的传递函数用传递矩阵法求解。通过对曲线简支梁在移动荷载下振动响应的求解,验证了计算模型的正确性;对比曲线轨道和直线轨道在移动荷载下的振动响应结果,发现:相同荷载条件下,曲线轨道的振动响应大于直线轨道的响应,轨道振动响应频谱与荷载速度密切相关,并且曲线轨道的响应频谱更为丰富。
To study the vibration of the curved track structure,a periodical solution on the out-of-plane vibration response of the curved track structure,modeled as periodically supported curved Timoshenko beam,subjected to moving loads is determined here.Firstly,the general dynamic response induced by moving loads along curved path on an elastic semi-infinite space is obtained on the basis of Duhamel Integral and Dynamic Reciprocity Theorem.Then,in the case of periodic curved track structure,the general dynamic response equation in the frequency domain is simplified in a form of summation within the track sleeper spacing instead of integral.In this way,the track response induced by moving loads is obtained by the calculation program.The transfer function of the curved track is solved using the transfer matrix approach.To verify the analytical model,the vibration of simply supported curved beam under moving loads is obtained and compared with the existing results.Besides,the vibrations for the curved and straight track structure under single and series of moving loads are all obtained and compared,indicating that: under the same moving loads,the vibration of the curved track is bigger than that of straight track,the spectrum is closely related to the loads speed,and in addition the spectrum of the curved track is more abundant.
To study the vibration of the curved track structure,a periodical solution on the out-of-plane vibration response of the curved track structure,modeled as periodically supported curved Timoshenko beam,subjected to moving loads is determined here.Firstly,the general dynamic response induced by moving loads along curved path on an elastic semi-infinite space is obtained on the basis of Duhamel Integral and Dynamic Reciprocity Theorem.Then,in the case of periodic curved track structure,the general dynamic response equation in the frequency domain is simplified in a form of summation within the track sleeper spacing instead of integral.In this way,the track response induced by moving loads is obtained by the calculation program.The transfer function of the curved track is solved using the transfer matrix approach.To verify the analytical model,the vibration of simply supported curved beam under moving loads is obtained and compared with the existing results.Besides,the vibrations for the curved and straight track structure under single and series of moving loads are all obtained and compared,indicating that: under the same moving loads,the vibration of the curved track is bigger than that of straight track,the spectrum is closely related to the loads speed,and in addition the spectrum of the curved track is more abundant.
引文
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[2]Wu J S,Chiang L K.Out-of-plane responses of a circularcurved Timoshenko beam due to a moving load[J].International Journal of Solids and Structures,2003,40(26):7425-7448
[3]魏双科.曲线梁桥的固有振动特性及地震反应分析[D].南京:南京工业大学,2006(Wei Shuangke.Natural dynamic characteristics and seismic responseanalysis of curved girder bridges[J].Nanjing:NanjingUniversity of Technology,2006(in Chinese))
[4]Rao S S.Effects of transverse shear and rotatory inertia onthe coupled twist-bending vibrations of circular rings[J].Journal of Sound and Vibration,1971,16(4):551-566
[5]Montalvo E Silva J M,Urgueira A P V.Out-of-planedynamic response of curved beams—an analytical model[J].International Journal of Solids and Structures,1988,24(3):271-284
[6]Kawakami M,Sakiyama T,Matsuda H,et al.In-planeand out-of-plane free vibrations of curved beams withvariable sections[J].Journal of Sound and Vibration,1995,187(3):381-401
[7]Yang Y B,Wu C M,Yau J D.Dynamic response of ahorizontally curved beam subjected to vertical andhorizontal moving loads[J].Journal of Sound andVibration,2001,242(3):519-537
[8]孙建鹏,李青宁.曲线箱梁桥地震反应的频域精细传递矩阵法[J].地震工程与工程振动,2009,29(4):139-146(Sun Jianpeng,Li Qingning.Precise transfer matrixmethod for solving earthquake response of curved boxbridges[J].Earthquake Engineering and EngineeringVibration,2009,29(4):139-146(in Chinese))
[9]刘维宁,张昀青.轨道结构在移动荷载作用下的周期解析解[J].工程力学,2004,21(5):100-102(LiuWeining,Zhang Yunqing.A periodic analytical solution ofrailway track structure under moving loads[J].EngineeringMechanics,2004,21(5):100-102(in Chinese))
[10]贾颖绚.基于解析的车轨耦合模型及地铁对环境的振动影响研究[D].北京:北京交通大学,2009(JiaYingxuan.Study on analytical model of coupled vehical&track and effect to environment by metro train-inducedvibrations[D].Beijing:Beijing Jiaotong University,2009(in Chinese))
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