This book lays the foundation of knowledge that will allow a better understanding of nonlinear phenomena that occur in structural dynamics. This work is intended for graduate engineering students who want to expand their knowledge on the dynamic behavior of structures, specifically in the nonlinear field, by presenting the basis of dynamic balance in non-linear behavior structures due to the material and kinematics mechanical effects. Particularly, this publication shows the solution of the equation of dynamic equilibrium for structure with nonlinear time-independent materials (plasticity, damage and frequencies evolution), as well as those time dependent non-linear behavior materials (viscoelasticity and viscoplasticity). The convergence conditions for the non-linear dynamic structure solution  are studied and the theoretical concepts and its programming algorithms are presented.