Answer
The theoretical spring rate $k$ for a Multi-Turn Crest-to-Crest Wave Spring is derived from the formula $k = _x000c_rac{E b t^3 N^4}{I D_m^3 Z}$ 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 $Z$ is the number of turns. For shim-end springs, the rate is often adjusted because the shim ends do not contribute to deflection but add to the solid height. The linearity of the spring rate is highly sensitive to the $N$ value; as the spring compresses towards its solid height, the contact points between waves shift radially, leading to a non-linear increase in the spring rate (rate-up). For precision aerospace applications, we aim for an operating range between 20% and 80% of the available deflection to maintain a linear response where $P = k imes f$ holds true within $\pm 10\%$.
