Answer
The spring rate $k$ for a Crest-to-Crest wave spring is derived from the standard beam equation adapted for circular geometry. The primary formula is $k = \frac{E \cdot b \cdot t^3 \cdot N^4}{I_D^3 \cdot ID \cdot n}$ where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of waves per turn, and $n$ is the number of turns. However, as the spring approaches its solid height, the rate becomes non-linear due to the 'bottoming out' effect where the wave peaks begin to flatten against each other, effectively reducing the active length of the beam. To account for this, engineers apply a correction factor $K$ based on the ratio of $f/h$ (deflection to wave height). For high-precision applications in aerospace, a non-linear finite element analysis (FEA) is typically performed to map the rate change once deflection exceeds 80% of the available travel.
