The focus of this work is to advance the theoretical and modeling techniques for the fields of hybrid simulation and multi-slider friction pendulum systems (MSFPs). Hybrid Simulation is a simulation technique involving the integration of a physical system and a computational system with the use of actuators and sensors. This method has a strong foundation in the experimental mechanics community where it has been used for many years. The hybrid simulation experiments are performed with the assumption of an accurate result as long as the main causes of error are reduced. However, the theoretical background on hybrid testing needs to be developed in order validate these findings using this technique. To achieve this objective, a model for hybrid simulation is developed and applied to three test cases: an Euler-Bernoulli beam, a nonlinear damped, driven pendulum, and a boom crane structure. Due to the complex dynamics that these three test cases exhibit, L2 norms, Lyapunov exponents, and Lyapunov dimensions, as well as correlation exponents were utilized to analyze the error in hybrid simulation tests. From these three test cases it was found that hybrid simulations are highly dependent on the natural frequencies of the dynamical system as well as how and where the hybrid split is located. Thus, proper care must be taken when conducting a hybrid experiment in order to guarantee reliable results.

Multi-stage friction pendulum systems (MSFPs), such as the triple friction pendulum (TFP), are currently being developed as seismic isolators. However, all current analytical models are inadequate in modeling many facets of these devices. Either the model can only handle uni-directional ground motions while incorporating the kinetics of the TFP system, or the model ignores the kinetics and can handle bi-directional motion. And in all cases, the model is linearized to simplify the equations. The second part of this dissertation presents an all-in-one model that incorporates the full nonlinear kinetics of the TFP system, while allowing for bi-directional ground motion. In this way, the model presented here is the most complete single model currently available. It was found that the non-linear model can more accurately predict the experimental results for large displacements due to the nonlinear kinematics used to describe the system. The model is also able to successfully predict the experimental results for bi-directional ground motions.