This paper reports on the development of the component mode synthesis (CMS) method using hybrid coordinates for a localized nonlinear dynamic system. As is well known, the CMS method is effective in reducing the degrees-of-freedom (DOF) of the system. In contrast to most existing CMS methods, which are usually developed for linear systems, a new CMS method using hybrid coordinates for nonlinear dynamic analysis has been developed in this paper. Generally, the system is divided into two parts, namely, a linear component and a nonlinear component. The equations of the linear component can be transformed into the modal coordinates using its linear vibration modes. To improve the accuracy, the equivalent higher-order matrix of the system is developed to capture the effects of the neglected higher-order modes. Quite different from early works, the flexibility attachment matrix can be obtained without using the inverse of the stiffness matrix by using an equivalent higher-order matrix, thus making it easier to deal with those components that have rigid-body freedom when formulating the residual flexibility attachment matrix. By introducing the residual flexibility attachment matrix and the retained lower-order modes, the dynamic governing equations of the linear component can be converted into the modal space and expressed by a few modal coordinates. To adopt the entire set of nonlinear terms into the final equations, the equations of the nonlinear component are kept in their original form. Compatibility conditions at the interface are used to combine the nonlinear component and the linear component to form the synthesis equations, which are expressed in hybrid coordinates. Finally, the computational efficiency and accuracy of the presented method is demonstrated using numerical examples.
Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity
