Modeling, numerical analysis, and simulations of nonlinear dynamic beams

This dissertation deals with modeling, analysis, and computational aspects of the dynamics and vibrations of extended versions of nonlinear Gao beams (D. Gao, 1996). The main interest lies in gaining insights into the dynamics as well as the construction of reliable and accurate computational algorithms for the vibrations and frequency spectra. These, in turn, may provide tools for better design of buildings, structures, machinery, and engineering components.;The extended beam models take into account frictionless contact with reactive or rigid supports, and the development of material damage in the form of a crack. In addition, a beam/rod system that includes thermal effects, which arises in MEMS (Micro-Electro-Mechanical Systems) actuators or grippers, is modeled and studied.;The Gao beam model allows for buckling, a topic of considerable interest to the civil engineer, as beams are the structural elements that support buildings, bridges, and most civil engineering structures. Their vibrations have considerable applied importance. The study of the model when a crack is growing is one of the main contributions of this dissertation. Preliminary results indicate that it may be possible to identify the position and the size of a crack from the vibrational spectrum of the beam.;The four main topics are: vibrations of the beam about a buckled state; dynamic contact of the beam with a support; dynamic beam with growth of a crack; and, the dynamics of a beam/rod system. The first topic is numerically simulated, depicting the vibrations of the beam about a buckled state.;For each one of the remaining topics a model, in the form of a variational equation or inequality is provided. The existence of the solution is proven. This involves the use of many tools from the theory of Variational Inequalities. Next, the variational formulation is used to construct Finite Elements algorithms for the models. These resulting computer codes provide the numerical simulations depicted here.;This work is a contribution to the emerging field of Mathematical Theory of Contact Mechanics, and, more generally, to nonlinear analysis of variational problems.