平面杆系结构自由振动的一种解析解法
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摘要
介绍了一种平面杆系结构自由振动的解析解法。即将计算无限自由度平面杆系结构的自振频率和主振型的广义特征值问题转换为典型的常微分方程边值问题,构造了一系列平凡ODE,建立了相应的常微分方程组,并利用常微分方程求解器COLSYS予以求解。该方法将一根杆件视为一个单元,直接求解其运动微分方程,是一种数值解析法,与有限元法相比,无需通过增加单元数提高计算精度,可精确求解平面杆系结构的任意阶自振频率和主振型。并利用该方法求解了一般约束、弹性支座以及变截面条件下的平面杆系结构无阻尼弯曲自由振动的任意阶自振频率和主振型,与精确解和现有软件相比,其计算结果表明,该方法的求解精度和效率较高。
The analytic method for the free vibration of plane frame structure is presented. It is achieved by transforming the general eigenvalue problem of natural frequency and vibration mode of continuously distributed property system into typical boundary value problem of ordinary differential equation (ODE). A series of ODE, corresponding to the general eigenvalue problem, is formed, and then is solved by an general ordinary differential equation solver-COLSYS. Each component of a frame structure is regarded as an element in the analytic method. This makes it more efficient than that of finite element method which needs to increase elements to get the value of high-class natural frequency and vibration mode accurately. Some applications of the method show that it can solve the free bending vibration of plane-frame structure with various types of displacement constraint. The result indicates that this method is precise and efficient.
The analytic method for the free vibration of plane frame structure is presented. It is achieved by transforming the general eigenvalue problem of natural frequency and vibration mode of continuously distributed property system into typical boundary value problem of ordinary differential equation (ODE). A series of ODE, corresponding to the general eigenvalue problem, is formed, and then is solved by an general ordinary differential equation solver-COLSYS. Each component of a frame structure is regarded as an element in the analytic method. This makes it more efficient than that of finite element method which needs to increase elements to get the value of high-class natural frequency and vibration mode accurately. Some applications of the method show that it can solve the free bending vibration of plane-frame structure with various types of displacement constraint. The result indicates that this method is precise and efficient.
引文
[1] 包世华.高层建筑结构解析、半解析常微分方程求解器方法系列[A].现代土木工程的新发展[C].南京:东南大学出版社,1998.
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[4] 袁驷.介绍一个常微分方程边值问题求解通用程序———COLSYS[J].计算结构力学及其应用,1990,7(2):104-105.
[5] 包世华.结构力学Ⅱ[M].北京:高等教育出版社,2001.
[6] 袁驷,张亿果.常微分方程特征值问题的求解器解法[J].地震工程与工程振动,1993,13(2):94-102.
[7] 龚耀清,包世华,龙驭球.半无限大弹性地基上变截面筒中筒高层建筑结构的自由振动[J].工程力学,1999,16(3):7-14.
[2] ASCHERU,CHRISTIANSENJ,RUSSELLRD.CollocationSoftwareforBoundary-valueODEs[J].ACMTransMathSoftware,1981,7(2):209-222.
[3] ASCHERU,CHRISTIANSENJ,RUSSELLRD.Algorithm569,COLSYS:CollocationSoftwareforBoundary-valueODEs[D2][J].ACMTransMathSoftware,1981,7(2):223-229.
[4] 袁驷.介绍一个常微分方程边值问题求解通用程序———COLSYS[J].计算结构力学及其应用,1990,7(2):104-105.
[5] 包世华.结构力学Ⅱ[M].北京:高等教育出版社,2001.
[6] 袁驷,张亿果.常微分方程特征值问题的求解器解法[J].地震工程与工程振动,1993,13(2):94-102.
[7] 龚耀清,包世华,龙驭球.半无限大弹性地基上变截面筒中筒高层建筑结构的自由振动[J].工程力学,1999,16(3):7-14.
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