Research on Reduced-order Nonlinear Modal Equations for Arbitrary Continuous Structures

本文研究了各种连续结构的降阶非线性模态方程的推导过程。利用虚功原理推导了考虑几何非线性的非线性有限元列式。通过将运动方程从物理坐标转换到模态坐标，得到了降维模型。将牛顿-拉夫森方法应用于一个近似的非线性有限元方程，确定了伪振型矢量。利用有限元分析得到的非经典振型向量，对非线性有限元方程进行了模态分析。将本方法应用于由弹簧支承的梁和曲梁。仅用几个振型矢量就建立了它们的非线性振型方程，并对基本离面振型和内共振进行了数值计算，结果与其他文献的结果吻合较好。对于曲线梁，考虑其变形相对于中性轴的不对称性来确定振型矢量。这种方法的计算时间比其他任何方法都要短，并且保持了足够的精度。文中还提出了利用商用有限元软件的数值结果确定振型方程的非线性项系数的方法。通过对固支梁第一阶和第三阶振型的数值计算，验证了该方法的有效性。

Procedures to derive reduced-order nonlinear modal equations for various continuous structures have been studied in this research. Nonlinear finite element formulation was derived by the principle of virtual work taking into account geometrical nonlinearity. Reduced-order model was derived by transforming the equations of motion from the physical coordinates to the modal coordinates. Pseudo mode vectors were determined by applying the Newton-Raphson method to an approximated nonlinear finite element equation. Modal analysis is applied to the nonlinear finite element equation by using non-classical mode vectors obtained by the finite element analysis. Present method was applied to a beam supported by a spring and a curved beam. Nonlinear modal equations of them were derived by using only a few mode vectors, and numerical results for the fundamental out-of-plane mode and an internal resonance showed good agreement with those presented in other papers. In the case of the curved beam, asymmetry of its deformation with respect to the neutral axis was taken into consideration to determine the mode vectors. Computational time of this method is very shorter than that of any other methods, and the present method maintains enough accuracy. In this study, a procedure was also proposed to determine the coefficient of nonlinear term of the modal equation by using numerical results of commercial finite element software. Validity of the procedure was verified by the numerical results on the first and third modes of a clamped beam.