Answer
The spring rate $k$ for a crest-to-crest wave spring is fundamentally derived from the deflection of a curved beam. For a spring with $N$ turns and $n$ waves per turn, the load $P$ is expressed as $P = _x000c_rac{E imes b imes t^3 imes n^4 imes f}{C imes D_m^3 imes N}$, where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the thickness, $f$ is the deflection, and $D_m$ is the mean diameter. The constant $C$ varies based on end conditions. Shim ends provide a flat surface for load distribution, which effectively adds two non-functional half-waves to the stack. While this increases the solid height $H_s = (N imes t) + (2 imes t_{shim})$, it significantly improves the linearity of the spring rate at the beginning and end of the stroke by preventing the 'point-loading' of wave peaks against the mating surfaces.
