This paper presents the design and synthesis of four-bar mechanisms with integrated springs for applications requiring nonlinear stiffness. The proposed method systematically identifies a total of 42 distinct mechanism configurations, each composed of a planar four-bar linkage with strategically placed springs to achieve specific torque-angle profiles. The method combines free-body diagram analysis with particle swarm optimization to determine the optimal geometric and spring parameters that minimize the error between the desired and actual torque outputs over a specified angular range. The significance of this method lies in the practical realization of nonlinear stiffness characteristics using simple and compact structures. Numerical simulations across various target stiffness profiles demonstrate the accuracy and adaptability of the synthesized mechanisms with minimal torque deviations. Physical prototypes and experimental validation further confirm the effectiveness of the method, highlighting its potential in applications such as compliant actuators, adaptive robotic manipulators, and vibration isolators.
