Passive vibration suppression devices known as shimmy dampers are vital in maintaining stability and safety of certain landing gears. Yet, systematic design, performance analysis, and optimization of such devices are rarely discussed in the literature. This paper presents structural design optimization of a novel shimmy damper for nose landing gears. This new design is a multifunctional mechanism integrating the shimmy damper into the torque link system of the nose landing gear. It features symmetric load distribution, and it can be tailored to existing nose landing gears. Here, the damper design concept is developed in the structural sense and optimized for the nose landing gear of a Piper Cheyenne aircraft. Dynamic loads from a representative shimmy scenario are employed in the analysis and design procedures. Utilizing equivalent static load method, topology optimization with transient loads is performed to obtain optimal material distribution satisfying the objective function and constraints. Flexible multibody dynamics analysis based on a high-fidelity finite element model is utilized in the analysis of design candidates and for validating the final design. To ensure adequate strength under the dynamic torque loads and to offer sufficient damping needed for stabilizing the nose landing gear, a three-piece torque link mechanism emerged through multiple design iterations guided by the topology optimization. Using numerical simulations, the final design is shown to satisfy the strength requirement while providing sufficient damper stroke. The results from the present study signal a vast potential in improving shimmy mitigation strategies by eliminating the need for costly redesigns of landing gears susceptible to shimmy.