Answer
The spring rate $k$ for a Crest-to-Crest wave spring is derived from the beam deflection formula adapted for a circular geometry with $N$ waves and $Z$ turns. The standard formula is $k = \frac{E \cdot b \cdot t^3 \cdot N^4}{1.68 \cdot D_m^3 \cdot Z}$, where $E$ is the Young's modulus, $b$ is the radial wall, $t$ is the material thickness, and $D_m$ is the mean diameter. In high-precision applications, parasitic loads arise from the friction between the waves and the contact surfaces of the housing or shaft. As the spring compresses, the mean diameter $D_m$ slightly increases, which can lead to a non-linear stiffening effect near the end of the stroke. Designers must ensure that the operating height $H$ does not result in the spring reaching its solid height $H_s$, as the stress level $S = \frac{3 \cdot \pi \cdot P \cdot D_m}{4 \cdot b \cdot t^2 \cdot N^2}$ will spike exponentially, leading to plastic deformation or fatigue failure.
