In power/energy systems, free-piston linear machines are referred to as a mechanism where the constrained crank motion is eliminated and replaced with free reciprocating piston motion.Depending on the application, the piston motion can be converted into other types of energy and includes compressed air/fluid, electricity, and high temperature/pressure gas. A research group at West Virginia University developed a free-piston linear engine alternator (LEA) in 1998 and have achieved significant accomplishment in the performance enhancement of the LEAs to date. The present LEA design incorporates flexure springs as energy restoration components and as bearing supports. The advantages of using flexure springs are threefold and include: (1) it increases the LEA's stiffness and resonant frequency, and hence the power density; (2) it eliminates the need for rotary or linear bearings and lubrication system; and (3) it reduces the overall frictional contact area in the translator assembly which improves the durability.The current research focuses on the design and optimization of the flexure springs as the system's resonant dominating component for a 1 kW free-piston LEA. First, the flexure springs were characterized according to the LEA's target outputs and dimensional limitations. The finite element method (FEM) was used to analyze the stress/strain, different modes of deformation, and fatigue life of a range of flexure spring designs under dynamic loadings. Primary geometric design variables included the number of arms, inside and outside diameter, thickness, and arm's length.To find the near-optimum designs, a machine learning algorithm incorporating the FEM results was used in order to find the sensitivity of the target outputs to the geometrical parameters. From the results, design charts were extracted as a guideline to flexure spring selection for a range of operations. Then, methods were introduced, investigated, and analyzed to improve the overall energy conversion performance and service life of the flexure springs and the overall LEA system.