Timoshenko连续梁模态分析的传递矩阵法

TRANSFER MATRIX METHOD FOR THE MODE ANALYSIS OF TIMOSHENKO CONTINUOUS BEAM

  • 摘要: 虽然Timoshenko梁理论被命名为两广义位移梁理论,但是其位移和转角间存在相互关系,并不独立。为建立Timoshenko连续梁振动的模态函数的理论解析式,确定位移、转角、弯矩和剪力的模态间关系,利用Timoshenko梁自由振动的初参数解建立Timoshenko连续梁自由振动的传递矩阵方法、状态向量经过刚性节点的传递矩阵形式,进而推导频率方程和模态函数解析式。结合二等跨Timoshenko连续梁频率方程解的特征,得到了模态函数的简化形式,以及位移、弯矩和剪力与转角的模态峰值比的理论解析式。同时,利用有限元和传递矩阵法对二等跨Timoshenko连续梁进行模态分析,得到前4阶模态。此外,还对三等跨Timoshenko连续梁模态、二等跨连续梁位移和转角模态的Timoshenko梁理论及与Euler–Bernoulli梁理论间的差别进行了分析。结果表明,Timoshenko梁的模态函数理论解与有限元计算的特征向量一致;竖向位移与转角间存在相关关系;模态峰值比的理论解析与数值结果一致。得到用传递矩阵法可建立准确的Timoshenko连续梁振动模态函数、将位移模态与转角模态的幅值同步归一化的传统方法不合理等结论。

     

    Abstract: Although the Timoshenko beam theory is termed as the two generalized displacement beam theory, there is an interrelationship between the displacement and rotation, which are dependent. To derive the analytical functions of mode shape in Timoshenko continuous beam, and determine the relations among the displacement, rotation, moment and shear force modes, the initial parameter solutions to the free vibration of Timoshenko beam are employed to establish the transfer matrix method and the transfer matrix formulation of state vector passing through the rigid node. Consequently, the frequency equation and the analytical functions of shape mode are constructed. Based on the properties of the solutions to the frequency equation of a two–span Timoshenko continuous beam, the simplified functions of shape mode is demonstrated, and the peak value ratios of vertical displacement, bending moment and shear force modes to the rotation mode are derived. At the same time, the mode shapes of a two–span Timoshenko continuous beam are calculated by the finite element method and the transfer matrix method. The vibration figures of the first four mode shapes are achieved. Furthermore, the mode shapes of a three–span Timoshenko continuous beam is analyzed, the mode differences of a two–span continuous beam are captured by the Timoshenko beam theory and the Euler–Bernoulli beam theory in the present paper. Results demonstrate that the shape modes of the analytical solutions presented by the transfer matrix method agree with the eigenvectors of the finite element method. Shape modes of vertical displacement and rotation are interdependent. The analytical solutions of peak values of shape modes are consistent with the numerical values. Some conclusions are drawn that the transfer matrix method can be used to derive the exact mode shapes of Timoshenko continuous beam; the traditional method is not rational to simultaneously normalize the shape modes of the vertical displacement and rotation.

     

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