Dynamic analysis of complex flexible structures is readily accomplished using discrete parameter models usually formulated by the finite element method. Whereas the finite element method lends itself to modeling very complex structures, it has the major disadvantage that it often requires a very large number of degrees of freedom to obtain accurate estimates of the vibration modes, where the number can reach into the tens of thousands. Although many available computer programs have been developed specifically to manipulate and solve large complex models, the eigenvalue solution procedure still requires considerable computation time to obtain desired results. In order to reduce the computational effort in the solution of the associated eigenvalue problem, component mode synthesis method has been developed.
The mode synthesis method is a technique of matrix order reduction for structural dynamic analysis and aims to determine the lower vibration modes of the complex structures. It is based on the Rayleigh-Ritz procedure and treats the structure as an assemblage of connected substructures, each of which is analyzed separately. The dynamic behavior of each substructure is represented by a reduced set of substructure modes that accommodates the structure eigenvalues and modes of interest. Depending on the method, the substructure mode set consists of the free, fixed, or loaded interface modes of the lower vibration modes of the substructure. Computationally, this represents a much more desirable method of analysis than analyzing the whole structural system, as long as the reduced-order structure can adequately respond in the dynamic loading environments. Although the mode synthesis methods have been successfully employed for the structure eigenvalue analysis, they have been rarely applied for the structure forced responses analysis, particularly, for the nonlinear structures.
A new mode synthesis method using substructure hybrid interface modes and Lagrange multiplier technique is presented for the dynamic analysis of linear and locally nonlinear structural systems. Compared with the conventional mode synthesis methods, the suggested method does not construct the governing equations of motion of the reduced-order model of the whole system. Only modal parameters of each substructure and constraints of geometric compatibility are required. For the substructure model, both the distributed and discrete parameter models, and any combination of these modes can be employed. The generalized coordinates defined at the substructure boundary result in an eigenvalue problem of much smaller order than those of the conventional methods, which results in a large reduction in the numerical computations.
A computationally efficient response analysis method based upon the recurrence discrete-time state equations is also presented for the forced response analysis in the time domain. As the modal parameters of the whole system are not required, the method can be efficiently utilized especially for the structure subjected to parameter changes with respect to time such as a missile launcher. From the view-point of the computational efficiency and solution accuracy, the superiority of the suggested method to the conventional methods is proved from several numerical examples.