How do you calculate the spring rate ($k$) for a Crest-to-Crest multi-turn wave spring and what role does the number of waves ($N$) play in its linearity?

The spring rate for a multi-turn wave spring is inversely proportional to the number of turns ($Z$) and sensitive to the number of waves per turn ($N$). The fundamental formula is $k = _x000c_rac{E imes b imes t^3 imes N^4}{ID_m^3 imes Z} imes f$, where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the material thickness, $D_m$ is the mean diameter, and $f$ is a correction factor for nonlinearity. As $N$ increases, the spring rate increases by the fourth power, making it a critical design parameter. Linearity is typically maintained between 20% and 80% of the total available deflection. Beyond this, 'bottoming out' occurs as the waves flatten, leading to an exponential increase in rate due to the reduction of effective moment arms.