具有任意弹性边界单跨梁结构的振动特性分析
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摘要
为了获得任意边界条件下单跨梁结构的振动特性,采用改进傅里叶级数方法建立振动模型。由于传统傅里叶级数在边界处存在导数不连续和收敛性较差的问题,使用改进傅里叶级数可以有效改善其解的收敛性与准确性。在梁的两端边界处引入横向约束弹簧与旋转约束弹簧以通过改变其刚度值来模拟任意边界条件。在求解过程中首先建立梁的能量表达式,再通过瑞利-里兹法对其进行求解,最后使用数值计算软件对其进行数值仿真验证,并与已有文献的结果进行比较。该文所提方法合理,误差可控制在极小范围内,当边界条件改变时不需要对理论推导与数学计算过程重新推导,从而实现了单跨梁结构振动特性最为一般情况的预报。
Considering the bad convergence and discontinuities of displacement derivatives of traditional Fourier series at boundary points,we constructed an improved Fourier series to enhance the convergence and correctness of the series solution as a single-span beam free vibration analysis model with arbitrary boundary conditions was built. Linear and rotational springs were settled at both ends of the beam and the stiffness constants of springs were set to simulate arbitrary boundary conditions. In the solving process,we described the beam system based on energy principle and used the Rayleigh-Ritz method. Finally,we implemented numerical simulations to validate the method and compared the results with those in the existing literature. The results show that this method is reasonable and takes good accuracy. Therefore,when the boundary conditions change,there is no need to reformulate the theoretical model or modify the simulation in forecasting the vibration characteristics of single-span beam structure.
Considering the bad convergence and discontinuities of displacement derivatives of traditional Fourier series at boundary points,we constructed an improved Fourier series to enhance the convergence and correctness of the series solution as a single-span beam free vibration analysis model with arbitrary boundary conditions was built. Linear and rotational springs were settled at both ends of the beam and the stiffness constants of springs were set to simulate arbitrary boundary conditions. In the solving process,we described the beam system based on energy principle and used the Rayleigh-Ritz method. Finally,we implemented numerical simulations to validate the method and compared the results with those in the existing literature. The results show that this method is reasonable and takes good accuracy. Therefore,when the boundary conditions change,there is no need to reformulate the theoretical model or modify the simulation in forecasting the vibration characteristics of single-span beam structure.
引文
[1]李麦侠.梁振动问题的MOL数值方法研究[D].西安:西安建筑科技大学,2014.
[2]梁利娜,柯燎亮,汪越胜.原子力显微镜微悬臂梁的振动分析[C]//2015中国力学大会论文集.上海:中国力学学会,2015.
[3]王其申,吴磊,王大钧.多跨梁离散系统的频谱和模态的定性性质[J].力学学报,2009,41(6):947-952.
[4]程远胜.多跨梁在移动车载作用下的动响应分析[J].华中科技大学学报(自然科学版),2002,30(1):81-83.
[5]KIM H K,KIM M S.Vibration of beams with generally restrained boundary conditions using fourier series[J].Journal of Sound&Vibration,2001,245(5):771-784.
[6]LI W L.Free vibrations of beams with general boundary conditions[J].Journal of Sound&Vibration,2000,237(4):709-725.
[7]吴琛,周瑞忠.无单元法求解欧拉梁及梁系的自由振动问题[J].计算力学学报,2009,26(6):856-861.
[8]周渤,石先杰.连续多跨梁结构振动特性分析[J].机械设计与制造,2017(8):43-46.
[9]石先杰,李春丽,蒋华兵,等.弹性边界条件下圆板横向自由振动特性分析[J].振动、测试与诊断,2016,36(5):984-989.
[10]史冬岩,石先杰,李文龙.任意边界条件下环扇形板面内振动特性分析[J].振动工程报,2014,27(1):1-8.
[11]许得水,杜敬涛,李文达,等.任意边界条件弹性杆结构扭转振动特性分析[J].振动与冲击,2017,36(1):161-166.
[12]杜敬涛,许得水,吕朋,等.任意边界条件非局部弹性杆纵振特性分析[J].振动工程学报,2016,29(5):787-794.
[13]赵玫,陈光冶,周海亭,等.机械振动与噪声学[M].北京:科学出版社,2004:159-161.
[2]梁利娜,柯燎亮,汪越胜.原子力显微镜微悬臂梁的振动分析[C]//2015中国力学大会论文集.上海:中国力学学会,2015.
[3]王其申,吴磊,王大钧.多跨梁离散系统的频谱和模态的定性性质[J].力学学报,2009,41(6):947-952.
[4]程远胜.多跨梁在移动车载作用下的动响应分析[J].华中科技大学学报(自然科学版),2002,30(1):81-83.
[5]KIM H K,KIM M S.Vibration of beams with generally restrained boundary conditions using fourier series[J].Journal of Sound&Vibration,2001,245(5):771-784.
[6]LI W L.Free vibrations of beams with general boundary conditions[J].Journal of Sound&Vibration,2000,237(4):709-725.
[7]吴琛,周瑞忠.无单元法求解欧拉梁及梁系的自由振动问题[J].计算力学学报,2009,26(6):856-861.
[8]周渤,石先杰.连续多跨梁结构振动特性分析[J].机械设计与制造,2017(8):43-46.
[9]石先杰,李春丽,蒋华兵,等.弹性边界条件下圆板横向自由振动特性分析[J].振动、测试与诊断,2016,36(5):984-989.
[10]史冬岩,石先杰,李文龙.任意边界条件下环扇形板面内振动特性分析[J].振动工程报,2014,27(1):1-8.
[11]许得水,杜敬涛,李文达,等.任意边界条件弹性杆结构扭转振动特性分析[J].振动与冲击,2017,36(1):161-166.
[12]杜敬涛,许得水,吕朋,等.任意边界条件非局部弹性杆纵振特性分析[J].振动工程学报,2016,29(5):787-794.
[13]赵玫,陈光冶,周海亭,等.机械振动与噪声学[M].北京:科学出版社,2004:159-161.