This paper investigates the use of shear thickening fluid (STF) dampers for reducing vibrations in beam-based structures. Designing and analyzing these systems is challenging because STF's viscosity changes as the fluid's vibration frequency changes. To understand the impact of key design parameters, the behavior of STF dampers in various configurations of supports is analyzed. The governing equations are derived, considering the STF dampers as nonlinear damping forces, and solved/validated using Galerkin/Hencky's discretization methods. An energy reduction factor is introduced to compare the performance of different configurations in the frequency domain to obtain the optimal design. The results show that although the performance of STF dampers is highly dependent on the excitation frequency (with high efficiency in reducing vibrations near the resonance frequencies), the STF dampers fail to dissipate vibrations in some frequency regions. Surprising ranges of dynamic behavior, from Quasi-periodic to chaotic, are detected and analyzed. The findings also reveal that by analyzing only the shape of force-velocity curves of the STF damper, one can accurately predict the system's performance, without the need for a full system analysis.