Vibration attenuation devices are used to reduce the vibrations of various mechanical systems and structures. In this work, an analytical method is proposed to provide the means to investigate the influence of system parameters on the dynamic response of a system. The method of multiple scales is used to calculate an approximate broadband solution for a two degree-of-freedom system consisting of a linear primary structure and a nonlinear tuned mass damper. The model is decoupled, approximate analytical solutions are calculated, and then they are combined to produce the desired frequency-response information. The approach is initially applied to a linear two degree-of-freedom system in order to verify its performance. The approach is then applied to the nonlinear system in order to study how varying the values of parameters associated with the nonlinear absorber affect its ability to attenuate the response of the primary structure. Finally, the analytical solution is compared to a numerical solution in order to determine how well it approximates the nonlinear system frequency-response.