Answer
The theoretical spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived from the formula $P/f = (E b t^3 n^4) / (D_m^3 N Z)$, where $E$ is the Modulus of Elasticity, $b$ is the radial width, $t$ is the material thickness, $n$ is the number of waves per turn, $D_m$ is the mean diameter, and $N$ is the number of turns. The inclusion of shim ends introduces a correction factor to the active number of turns. Non-linearity typically occurs as the spring approaches its solid height, usually around $80\%$ of the total available travel. This is caused by the 'bottoming out' effect where the wave peaks begin to flatten against the contact surfaces, effectively reducing the active length of the beam and increasing the spring rate exponentially. In high-precision applications, designers must account for the expansion of the radial wall $b$ during compression using $D_{outer-max} = D_{outer} + (0.015 \cdot f \cdot n^2 / D_m)$ to prevent binding in the housing.
