A vibration analysis of a structure with joints is performed. The simulation is conducted with finite element software capable of performing a numeric modal analysis with hysteretic damping assumption. The joints are modeled with thin layer elements, representing dissipation and stiffness of the joints. The matrices describing the system consist of the mass, as well as real and complex-valued stiffness matrices. If the eigenvalues of this system are found in one step, due to the mode crossing occurring for the closely spaced modes, it is difficult and time consuming to assign calculated modal damping factors to the corresponding undamped eigenvalues. In order to avoid this problem, an eigenvalue following method is used. The outcome of the solution is the graphical presentation of continuous eigenvalue paths, showing the change in the eigenvalues from the undamped to the fully damped case. For every undamped eigenvalue exists its equivalent eigenfrequency and damping factor that can be used for further numerical analysis.In scope of this article a Predictor-Corrector and a Rayleigh-Quotient Iteration algorithms are applied to the problem. The algorithms are tested specifically on the type of matrices resulting from the weakly damped hysteretic formulation arising from the simulation of metallic structures with joints.
A local error estimate of the method of multi-scale asymptotic expansions for elliptic problems with rapidly oscillatory coefficients