Within this dissertation work, numerical, analytical, and experimental studies are conducted with macro-scale and micro-scale elastic structures in the presence of nonlinear force interactions. The specific physical systems explored within this work are an atomic force microscope (AFM) micro-cantilever and a macro-scale cantilever experiencing similar tip interaction forces as the AFM cantilever operated in tapping mode. The tip sample forces in an AFM operation are highly nonlinear, with long-range attractive forces and short-range repulsive forces. In the macro-scale case, magnetic attractive forces and repulsive forces, which arise due to impacts with a compliant surface are used to generate similar nonlinear tip interaction forces. For elastic structures subjected to off-resonance base excitations, bifurcations close to grazing events are studied in detail, and the observed nonlinear phenomena are found to be common across the considered length scales. The dynamics of the considered systems are studied with a reduced-order computational model based on Galerkin projection with a single mode approximation. Along with studies on the bifurcation behavior, the effects of added Gaussian white noise on the system dynamics are also examined. Non-smooth system dynamics is studied by constructing local maps near the discontinuity. Period-doubling events are examined by using Poincaré maps and discontinuity mapping analysis. An important component of this dissertation research is the investigations into the effects of noise on the dynamics of these structures. Experimental and numerical efforts are used to examine the stochastic dynamics of the cantilever structures when a random component is added to the harmonic input. The noise effects are studied when the excitation frequency is close to a system resonance as well as when it is off-resonance. An analytical-numerical method with moment evolution equations is used to study the effects of noise. The effects of noise on contact and adhesion phenomena are explored. Through this dissertation work, the importance of considering noise-influenced dynamics in micro-scale applications such as AFM operations is illustrated. In addition, this work helps shed light on universality of nonlinear phenomenon across different length scales.