Answer
The classical linear spring rate ($k$) of a multi-turn Crest-to-Crest wave spring is given by the modified Timoshenko wave spring equation:
$$k = \frac{E \cdot b \cdot t^3 \cdot N_w^4}{P_m^3 \cdot N} \times K_g$$
Where:
- $E$ is the Young's Modulus of Elasticity ($N/mm^2$)
- $b$ is the radial wall thickness ($mm$)
- $t$ is the material thickness ($mm$)
- $N_w$ is the number of active waves per turn
- $P_m$ is the mean spring diameter ($mm$), computed as $(D_{out} + D_{in}) / 2$
- $N$ is the number of active turns
- $K_g$ is a correction factor based on the expansion of diameter during deflection
Limitations of Linear Equation:
1. Friction and Hysteresis: Contact between wave crests during axial deflection generates friction, resulting in hysteresis and an increase in effective spring rate during loading vs unloading.
2. Deflection Limits: The formula is strictly linear only up to approximately 80% of its total available deflection. Beyond 80%, the waves begin to bottom out or form line-contact, exponentially increasing the spring stiffness.
3. Shim Ends: If shim ends (flat ends) are specified to distribute load evenly, their contribution must be accounted for as they increase structural rigidity and decrease total effective active turns.
