Answer
For a crest-to-crest wave spring with shim ends, the theoretical spring rate $k$ is derived from the linear elastic deflection of a curved beam. The formula is expressed as $k = \frac{E b t^3 N_w^4}{48 I D_m^3 N}$, where $E$ is the Modulus of Elasticity, $b$ is the radial width of the material, $t$ is the material thickness, $N_w$ is the number of waves per turn, $I$ is the moment of inertia, $D_m$ is the mean diameter, and $N$ is the number of active turns. It is critical to note that the spring rate is proportional to the fourth power of the number of waves ($N_w^4$). This means that even a minor increase in the wave count significantly increases the stiffness of the spring, allowing engineers to fine-tune loads within very tight axial spaces. In high-precision aerospace applications using 17-7PH stainless steel, this relationship is used to achieve high force-to-deflection ratios where traditional coil springs would be physically too large.
