一种数值雅可比隐式积分方法在旋翼动力学中的评估与应用

doi:10.3901/JME.2025.01.162

摘要/Abstract

摘要： 近年来，几何精确梁理论被广泛应用于直升机旋翼桨叶结构动力学大变形建模。基于该理论建立的桨叶运动与变形之间的关系非线性较强，广义力项复杂而庞大。因此，推导解析形式的桨叶动力学方程雅可比矩阵比较困难，给隐式积分方法带来了巨大挑战。在求解旋翼桨叶非线性动力学方程中，提出一种采用数值雅可比矩阵的隐式直接积分方法，其中雅可比矩阵和刚度矩阵由结合了外推方法的中心差分法给出。以解决无铰式和铰接式旋翼桨叶的动力学问题为例，评估了外推差分法和隐式积分方法的计算精度。通过改变典型桨叶的网格划分和雅可比矩阵的更新策略，在缩比气动载荷作用下，进一步研究了该方法的数值稳定性和积分效率。结果表明，这种数值雅可比积分法在预测旋翼桨叶的固有频率和瞬态响应等方面表现良好。

关键词: 旋翼动力学, 数值积分, 雅可比矩阵, 外推法, 拟牛顿法

Abstract: In recent years, geometrically exact beam theory is widely used in rotor structural dynamics to model composite blade undergoing large deflections. The relationship between movements and elastic deformations based on this theory shows strong nonlinearity and the generalized forces term is complex and huge. Therefore, it is difficult to derive the Jacobian matrix and determinant of the blade dynamics equation in analytical form, which brings great challenges to the implicit integration method. A trapezoidal integration utilizing numerical Jacobian matrix is proposed to solve the dynamic equations of helicopter rotor blade, in which the Jacobian matrix and stiffness matrix are numerically computed by central difference method with Romberg extrapolation scheme. Accuracy of the methods is evaluated on dynamic problems from one cantilevered beam and the other fully articulated rotor blade. Numerical stability and efficiency are also investigated by changing update policies of Jacobian matrix and mesh configurations of typical rotor blade. Numerical results show that the integration method with numerical Jacobian performs well on predicting natural frequencies and transient responses in rotor dynamics.

Key words: rotor dynamics, numerical integration, Jacobian matrix, extrapolation method, quasi-Newton method

中图分类号:

V214

吴杰, 虞志浩. 一种数值雅可比隐式积分方法在旋翼动力学中的评估与应用[J]. 机械工程学报, 2025, 61(1): 162-171.

WU Jie, YU Zhihao. Comprehensive Application of Implicit Trapezoidal Rule with Numerical Jacobian in Rotor Dynamics[J]. Journal of Mechanical Engineering, 2025, 61(1): 162-171.

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