变截面曲梁面外自由振动的一种新型级数分析法

A New Series Method for Out-of-plane Free Vibration Analysis of Curved Beams with Variable Cross-sections

给出曲梁面外自由振动问题的一种数值分析模型,该模型基于改进傅里叶级数法假设位移,并使用瑞利-里兹方法进行求解。首先,应用改进傅里叶级数法表示曲梁面外弯扭振动时的3个位移函数,利用改进傅里叶级数法解决位移函数在边界导数不连续的问题。其次,对曲梁采用人工虚拟弹簧模拟边界条件,通过改变横向位移约束弹簧、扭转约束弹簧和旋转约束弹簧的刚度值来实现任意弹性约束边界,得到曲梁结构振动时的刚度矩阵和质量矩阵。最后,通过求解矩阵特征值问题得到平面曲梁面外自由振动的频率值。此方法克服了以往只能求解某些特定的常规边界条件下振动问题的缺陷。数值算例验证了该方法良好的收敛性和精确性。还采用该方法分析不同弹性约束和截面变化系数对曲梁面外自由振动频率的影响。算例结果表明:对于两端固定圆弧形曲梁,无量纲自振频率随着圆心角变大而变小,但随着长细比增大而增大。

A numerical model for analysis of out-of-plane free vibration of curved beams is presented.In this model,the displacement is expressed by the improved Fourier series,and solved by the Rayleigh-Ritz method.First of all,the improved Fourier series method is used to express the three displacement functions of the curved beam in out-of-plane bending and torsional vibration,which can solve the problem of discontinuity of boundary derivatives of displacement functions.Secondly,artificial virtual springs are used to simulate the boundary conditions of the curved beam,and the constraint conditions on arbitrary boundaries are realized by changing the stiffness values of the additional transverse displacement constraint spring,the torsion constraint spring and the rotation constraint spring.The stiffness matrix and the mass matrix are obtained when the curved beam structure is in vibration.Finally,the frequency values of the out-of-plane free vibration of the plane curved beam are obtained by solving the matrix eigenvalue problem.This method overcomes the defect of some methods that vibration problems can only be solved under certain specific boundary conditions.Numerical examples show that the convergence and accuracy of the method are good.This method also analyzes the influence of different elastic constraints and section variation coefficient on out-of-plane free vibration frequency of the curved beam.It can be found that the dimensionless frequencies decrease with the increase of the central angle of circular curved beam with two ends clamped,and increase with increasing of the slenderness ratio.