Solution of Time-Dependent Problems Using Exact Modal Analysis with Illustrative Application to One-Dimensional Truss Structures

We proposed about two decades ago an advanced mode superposition technique for the solution of transient problems of potential and elasticity. It extends Pian’s hybrid finite element formulation and Przemieniecki’s displacement-based, frequency-dependent developments for the free vibration analysis of truss and beam elements. Our formulation has led to a hybrid finite element method (actually initially conceived in the frame of a variationally-based boundary element method) for the general analysis of transient problems. It was ultimately shown that the traditional structural dynamics taught in the textbooks is just a first-order truncation of a frequency power series for which there is an underlying complex-symmetric (if viscous damping is included), non-linear eigenvalue problem to be solved. In fact, whenever an effective stiffness matrix can be represented as an analytical function of frequencies we can formulate the exact - not just an improved or generalized - modal analysis of a given structural problem. This exact modal analysis may also be carried out if the problem’s generalized stiffness and mass matrices can only be expressed numerically, albeit exactly within machine precision, for a given frequency number (which only in passing resembles a Laplace-transform analysis), although this may become computationally intensive. The analytical developments apply directly also to some families of two- and three-dimensional finite elements. We restrict our numerical applications to the simple truss problem including viscous damping as just a proof of concept. With this novel formulation, an engineering structure - given its geometric and discretizing simplifications - can ultimately have its time response represented exactly, a feat that cannot be matched by any other technique.