Answer
The spring rate $K$ for a multi-turn Crest-to-Crest wave spring is derived from the beam deflection formula adapted for a curved geometry. It is expressed as $K = \frac{E b t^3 N^4}{R^3 5.88 n}$, where $E$ is the modulus of elasticity, $b$ is the radial wall thickness, $t$ is the material thickness, $N$ is the number of waves per turn, $R$ is the mean radius, and $n$ is the number of turns. Non-linearity occurs primarily at the extremes of the deflection curve. At the start, 'bedding in' of the waves against the mating surfaces causes a lower initial rate. As the spring approaches 'solid height,' the waves begin to flatten and touch, causing a sharp increase in the rate. In high-precision applications, engineers must account for the $K_{factor}$ which adjusts for the change in the moment arm as the wave crests shift radially during compression.
