最佳平方逼近下的轴向功能梯度梁自由振动分析

Free-vibration analysis of axial functionally graded beams based on best square approximation

为解决轴向功能梯度梁的自由振动问题,满足实际工程环节,本文采用最佳平方逼近方法求解梁固有频率,并拟合出梁的挠度和转角曲线。推导基于正交Legendre多项式理论的最佳平方逼近方法,在Gauss点处将Timoshenko梁的振动控制方程离散,结合投影矩阵方法处理边界条件,得到离散后Timoshenko梁的广义特征方程。运用QR矩阵分解法求解该方程,可得Timoshenko梁各阶固有频率以及对应的挠度和转角曲线。对本方法进行收敛性分析,并与已有的文献结果进行比较,验证方法的正确性。对比挠度和转角曲线的拟合结果:相比于线性或二次函数曲线拟合,正交Legendre多项式拟合具有较高的拟合精度,但端点处边界条件只能近似处理,无法真正地满足边界条件。

To solve the free-vibration problem of axial functionally graded beams, we use the best square approximation (BSA) to determine the natural frequencies, deflections, and rotation angle curves. Based on the basic theory of orthogonal Legendre polynomials, we determine the BSA and discretize the governing equation of the Timoshenko beam using Gaussian sampling. By considering the boundary conditions of the projection matrices, we obtain the generalized characteristic equations of the Timoshenko beam after discretization, after which we apply QR matrix decomposition to solve the equations and enable identification of the natural frequencies, corresponding deflections, and rotation angle curves of the beam. We confirm the validity of the BSA by convergence analysis and comparison with the results reported in the literatures. Based on the curve-fitting results, we conclude that compared to linear or quadratic curve fitting, orthogonal Legendre polynomial fitting has higher precision, but the boundary conditions at the endpoints can only be approximated, so the boundary condition cannot be established with certainty.