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移动车辆荷载过桥耦合振动精细积分算法

2016-04-20桂水荣万水陈水生

建筑科学与工程学报 2016年2期

桂水荣 万水 陈水生

摘要:根据模态综合叠加技术的优势,提出基于精细积分算法(PIM)的车桥耦合振动模型新算法。考虑积分步长内荷载协调分解,通过插值函数将移动车辆荷载等效到单元节点,利用科茨积分格式求解Duhamel非齐次项荷载。以移动常量力作用于简支梁桥为例,将解析解和多种迭代格式数值解进行对比,校验精细积分法结合科茨积分格式求解车桥耦合振动模型算法的准确性。以移动弹簧质量车模型作用于简支梁桥为例,分析积分步长、计算时间对RungKutta法、Newmarkβ法及PIM法计算结果的影响。结果表明:基于模态综合叠加法并结合精细积分格式求解车桥耦合振动问题不受积分步长限制,具有快速收敛的优势。

关键词:车桥耦合振动;移动弹簧质量;数值迭代格式;精细积分算法;模态综合叠加法

中图分类号:U443文献标志码:A

Abstract: According to the superiority of the modal superposition method, a new numerical algorithm based on precise integration method (PIM)was proposed to solve the problem of vehiclebridge coupling vibration. The load decomposition coordination in an integration step was considered, and moving vehicle load was equivalent to element point through interpolating function, then Cotes Integral format was introduced to solve Duhamel nonhomogeneous load. Taking a moving constant force on simply supported beam as an example, the veracity of Cotes Integral format was verified through comparing the analytical solution with several numerical integral results. Taking a moving spring mass vehicle model on simply supported beam as an example, the effects of integral time step and computing time on computing results using RungKutta method, Newmarkβ method and PIM were analyzed. The results show that the PIM lies in unlimited by integral step length, and has superiority of quick convergence in solving the problem of vehiclebridge coupling vibration.

Key words: vehiclebridge coupling vibration; moving spring mass; numerical iterative scheme; precise integration method; modal superposition method

0引言

移动车辆荷载与桥梁相互作用的数值模拟能高效准确地计算二者动力响应,可以用来研究车辆行驶过程中的行车舒适性及桥梁振动特性,对桥梁及车辆设计提供理论依据。通常考虑车桥耦合振动的数值模拟有2种方法:一种方法是直接建立桥梁全自由度的车桥耦合振动方程进行同步求解,这种方法称为全自由度耦合振动法[1];另一种方法则是利用结构模态正交特性,使用振型叠加技术,分别建立车辆与桥梁振动方程,使桥梁各阶模态广义坐标与车辆自由度位移协调,进行耦合求解,这种方法称为模态综合法[2]。求解车桥耦合振动问题的数值积分格式常用的有Newmarkβ法[1]、RungeKutta法[3]、翟婉明的显式积分法[4],乔宏等[5]基于Duhamel法积分求解,张楠等[6]运用全积分法求解,施颖等[7]运用ANSYS二次开发进行求解,张亚辉等[89]首先将精细积分法运用于求解车桥耦合振动方程。常规的逐步积分法计算移动车辆荷载与桥梁相互作用,在每一个积分步长内,荷载的大小及作用点位不变,导致从一个积分点到另一个积分点的“突变”,因而积分步长将影响数值计算精度。钟万勰[1011]提出的结构动力方程精细时程积分方法考虑荷载在积分步长内的连续变化。结合精细积分方法,研究移动荷载过桥问题,各学者研究了积分步长内荷载变化关系[710]、非齐次项荷载特解的求解问题[12]及将二者同时改进的算法[1314]。杜宪亭等[1516]运用精细积分法进行优化来求解车桥耦合振动问题。上述文献采用精细积分法求解移动荷载过桥问题,均采用全自由度耦合振动法[79,1214]或解析方法来进行求解,这些方法在计算公路桥梁车桥耦合振动问题时,因车辆在桥梁上行驶的横向、纵向位置的改变,桥梁有限元模型自由度将成倍增大,导致车桥耦合振动问题求解难以精确完成。

根据模态综合叠加法优势,结合精细积分格式,本文提出基于精细积分格式的车桥耦合振动模型的求解算法。考虑积分步长内外荷载按线性变化,运用插值函数将移动车辆荷载等效到单元节点,并利用指数矩阵及科茨积分格式求解非齐次项荷载。以移动弹簧质量车桥耦合模型为例,对比分析积分步长及计算时间对不同结果的影响,提出基于模态综合叠加法并结合精细积分格式求解车桥耦合振动问题的优势。

1移动弹簧质量车桥耦合振动模型

将车辆模型简化为由2个质量体系组成的移动弹簧质量车模型。假定车辆与桥梁始终保持接触,车辆模型质量由车轮质量m1及车体的簧载质量m2组成,车轮和悬架系统的弹簧刚度及阻尼分别等效为刚度k1、阻尼c1的弹簧阻尼系统。假设简支梁桥静止时为平衡位置,车辆以速度v行驶,梁的动挠度为y(x,t),簧载质量m2的动位移为z(t),车轮始终与梁体保持接触不脱离,移动弹簧质量车模型如图1所示。

5结语

将简支梁离散成欧拉梁单元,并结合模态综合叠加法建立了移动弹簧质量车桥耦合系统振动方程,考虑荷载在积分步内的线性变化关系,引入精细积分迭代格式,提出了基于模态综合叠加法的移动弹簧质量车桥耦合振动模型的精细求解算法。研究结果表明,车桥耦合振动系统采用模态综合法结合精细积分迭代格式,具有较好的通用性和准确性,能不受积分步长的限制,快速收敛,后期可推广用于求解长大跨桥梁及多车耦合振动响应。

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