The challenge of nose landing gear shimmy vibrations is largely addressed through adding passive shimmy suppression devices to the gear, known as shimmy dampers. Despite the wide treatment of the shimmy modeling in the literature using various approaches, little is known about shimmy dampers' performance and their influence on the dynamics. This article presents a multi degree of freedom nose landing gear model including a two-piece strut, with a generic shimmy damper added in between of the strut parts as it is the case in the actual system. Using linear stability analysis and nonlinear time domain simulations, we identify the required equivalent stiffness and damping to be supplied by the shimmy damper in order to ensure stability of the dynamic system. Furthermore, a parametric study of three shimmy dampers is presented in which effect of design parameters on the equivalent stiffness and damping of shimmy dampers are highlighted and compared. Integrating the learnings from both parts of the study, strategies for arriving at an optimized shimmy damper design are offered.