Gear Mesh Stiffness⚠ unverified
Mechanical / Gears · Compute gear mesh stiffness from load and tooth deflection
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Wt | Wt | N | 1.0 | Transmitted tangential load |
| delta | δ | — | 1.0 | Tooth deflection under load, in length units |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | k | — | Mesh stiffness, in force per unit length. Returns 0.0 when delta is non-positive |
The science & history
Understanding the Parameters
-
Tangential load $W_t$ — the load pressing the engaged teeth together. For a linear mesh model, stiffness is the constant ratio $W_t/\delta$; real tooth stiffness is mildly nonlinear (Hertzian flattening) and load-dependent.
-
Tooth deflection $\delta$ — the total compliance of the tooth pair at the contact: bending of each tooth as a cantilever, shear, Hertzian contact flattening at the surface, and gear-body (foundation) flexibility. All four add in series to give $\delta$, so mesh stiffness is softer than bending alone would suggest.
-
Why the fluctuation matters — over one mesh cycle the contact ratio means the load is shared by one or two tooth pairs alternately, so $k$ swings between a lower (single-pair) and higher (double-pair) value. This periodic stiffness variation is the primary source of transmission error and gear whine; helical gears smooth it out (more overlap → more constant $k$), which is why they are quieter.
Derivation (Approaching a Proof)
Mesh stiffness is the elastic-element definition — force over deflection, $k = F/\delta$ — applied to the tooth pair. The subtlety is what makes up $\delta$. A loaded tooth pair deflects by the sum of several compliances in series (each carries the same load $W_t$, and their deflections add):
$$\delta = \delta_{\text{bend}} + \delta_{\text{shear}} + \delta_{\text{Hertz}} + \delta_{\text{body}},$$
so the mesh stiffness is the series combination
$$\frac{1}{k} = \frac{1}{k_{\text{bend}}} + \frac{1}{k_{\text{shear}}} + \frac{1}{k_{\text{Hertz}}} + \frac{1}{k_{\text{body}}}.$$
- Bending treats the tooth as a cantilever (Gear Bending Stress Lewis / Beam Bending Stress);
- Hertzian contact adds surface flattening (Gear Contact Stress);
- the body term accounts for the flexible rim/hub the tooth grows from.
Given a computed or measured total deflection $\delta$ at load $W_t$, the calculator returns the secant stiffness $k = W_t/\delta$. Feeding the resulting time-varying $k(t)$ (as teeth engage and disengage) into a torsional model of the gear pair predicts the dynamic response — the basis of gear-whine and transmission-error analysis.
Dimensional check. $[k] = \dfrac{\text{N}}{\text{m}} = \text{N/m}$. ✓
History and Development
Mesh stiffness and its periodic variation are central to gear dynamics, developed from the mid-20th century (Tuplin, and later the ISO 6336 dynamic factor and modern finite-element/analytical mesh-stiffness models). Time-varying mesh stiffness is the standard explanation for gear whine and a key input to noise-vibration-harshness (NVH) design of transmissions. It also underlies the empirical velocity/dynamic factors in the Lewis and AGMA stress equations.
Related Concepts: Gear Bending Stress Lewis, Gear Contact Stress, Beam Bending Stress, Bearing Stiffness Radial, Gear Scoring Index
Notes: Secant stiffness $W_t/\delta$; total $\delta$ combines bending, shear, Hertzian, and body compliance in series. $\delta$ in m, $k$ in N/m (see note). Its periodic variation drives gear dynamics/whine.