变截面连续梁动力特性的差分解法
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摘要
针对变截面连续梁在公路、铁路桥梁、建筑、机械结构中广泛使用的情况,为解决变截面连续梁的变系数控制微分方程无法得到解析解这一困难,提出了运用有限差分方法解此问题的方法。通过推导连续梁动力特性的振型方程,建立其差分格式,利用对称性、反对称性、边界条件,将差分方程组简化,求解振型系数行列式为0这一方程,得到自振频率,相应地求出振型,并应用到实际跨线桥的计算。结果表明,差分方解法解决连续梁的振动问题非常有效,应用步骤简洁、固定。按照此思路编制计算机专用程序能对任意多跨变截面梁进行动力分析,可适用于土木、机械、舰船等结构。
In highway and railway engineering,continuous bridges are usually made up with variable section along the span in order to reduce its designed positive bending moment in the mid-span,and also to provide convenience of arranging the prestressed bars on the top of the cross section. In the paper,the vibrating mode equations were deduced at first,and then,the difference schemes of the equations were established. By utilizing the symmetry,antisymmetry,and boundary conditions,the difference equations were simplified. Based on the fact that it is impossible for all the deflections being zero in each node,taking the coefficient determinant to be zero,an equation with the square of natural frequencies as unknown was generated. By solving the equation,the natural frequencies were obtained in the end.
In highway and railway engineering,continuous bridges are usually made up with variable section along the span in order to reduce its designed positive bending moment in the mid-span,and also to provide convenience of arranging the prestressed bars on the top of the cross section. In the paper,the vibrating mode equations were deduced at first,and then,the difference schemes of the equations were established. By utilizing the symmetry,antisymmetry,and boundary conditions,the difference equations were simplified. Based on the fact that it is impossible for all the deflections being zero in each node,taking the coefficient determinant to be zero,an equation with the square of natural frequencies as unknown was generated. By solving the equation,the natural frequencies were obtained in the end.
引文
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[2]王赞芝,胡如成,张鹏,等.简支梁随机振动问题的简化解法[J].河北农业大学学报,2009,32(3):121-124.
[3]王赞芝,胡如成,吴辉琴.成层地基自由场地面运动的随机模型[J].世界地震工程,2009(2):17-20.
[4]李富文.钢桥[M].北京:中国铁道出版社,1992.
[5]王晓臣.蒲军平.变截面梁有限元分析[J].浙江工业大学学报,2008,36(3):311-315.
[6]赵明洁.范庆林.运用局部有限差分法确定变截面梁最大挠度位置[J].内蒙古大学学报:自然科学版;2007,38(5):514-516.
[7]张元海.李乔.变截面梁的应力计算及其分布规律研究[J].工程力学,2007,24(3):78-82.
[8]徐芝纶.弹性力学[M].第4版.北京:高等教育出版社,2008.
[9]王赞芝,胡如成,张鹏,等.多跨连续梁弯曲自由振动的波动解法[J].四川建筑科学研究,2009,32(5):33-36.
[10]王赞芝,江林雁,辛立凤,等.与粘弹性地基相互作用的梁的自由振动分析[J].工业建筑,2009,39(10):65-70.
[11]赵玉青,霍洪媛,邢振贤,等.差分法在混凝土导温系数反演分析中的应用[J].人民黄河,2008,30(7):93-94.
[12]贺铁山,陈文革.二阶非线性差分方程多重周期解的存在性[J].数学杂志,2009,30(3):300-306.
[13]皮红梅,蒋先艺,刘财,等.波动方程数值模拟的三种方法及对比[J].地球物理学进展,2009,31(2):391-397.
[14]崔成贤,汪宏远,温绍泉,等.利用Liapanov直接方法判定差分方程稳定性[J].佳木斯大学学报:自然科学版,2009,27(2):288-300.
[15]陈娟,张鲁明.Klein-Gordon-Zakharov方程的一类初边值问题的数值解[J].数学物理学报:A辑,2009,29(2):494-504.
[16]赵明洁,范庆林,杜浩.运用局部有限差分法确定变截面梁最大挠度的位置[J].内蒙古大学学报:自然科学版,2007,38(5):514-516.
[17]燕柳斌,陈艳香,苏国韶.岩体弹性模量反分析的进化差分方法.[J].广西大学学报:自然科学版,2008,33(4):346-348.
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