Gear Mesh Stiffness — The Torsional Spring at the Tooth Contact
When force is applied to a спирална предавка tooth, both teeth deflect, both teeth deflect elastically under the Hertzian contact stress. The combined deflection of the tooth pair — the input tooth bending, the output tooth bending, and the Hertz contact deformation — creates an effective stiffness c’ [N/(µm per mm face width)] that characterises the gear mesh as a linear spring per unit face width. This value c’ is the tooth pair stiffness, and it is the fundamental mechanical property that determines the спирална предавка pair’s torsional characteristics:
Mean tooth pair stiffness (ISO 6336-1 notation):
c’_gamma ≈ c_th × C_M × C_R × C_B × cos β
where:
c_th = theoretical single pair stiffness ≈ 14–20 N/(µm·mm) for standard steel gear pair
C_M = correction for gear body mass (0.8 for solid gear body)
C_R = correction for rim thickness (1.0 for standard rim ≥ 2.5 × h_tooth)
C_B = correction for basic rack (1.0 for standard 20° pressure angle)
cos β = helix correction
Typical result for carburized steel gears, β = 20°:
c’_gamma ≈ 17 × 0.8 × 1.0 × 1.0 × cos 20° ≈ 12.8 N/(µm·mm)
This stiffness value varies periodically as the спирална предавка rotates — when two tooth pairs are simultaneously in contact, the total mesh stiffness is approximately 2 × c’_gamma × b; when only one pair is in contact, it falls to c’_gamma × b. This periodic variation in stiffness — at the tooth mesh frequency (z × RPM / 60) — is the primary source of transmission error and the dominant excitation of gear noise and vibration.
Total Mesh Stiffness and Torsional Spring Constant
The total mesh stiffness of a спирална предавка pair, referred to the input shaft, combines the per-unit-width stiffness c’_gamma with the face width and the mean number of tooth pairs in contact (related to the total contact ratio ε_γ):
C_mesh ≈ c’_gamma × b × ε_γ [N·m/µm, total mesh stiffness]
Torsional spring constant (referred to input shaft):
C_torsion = C_mesh × (d₁/2)² [N·m/rad]
Example: M5, z₁=24, z₂=72, β=20°, b=80mm, 20CrMnTi carburized, ε_γ=2.3
c’_gamma = 12.8 N/(µm·mm)
C_mesh = 12.8 × 80 × 2.3 = 2,355 N/µm = 2,355,000,000 N/m
d₁ = 127.8 mm
C_torsion = 2,355,000,000 × (0.0639)² = 9,615,000 N·m/rad ≈ 9.6 × 10⁶ N·m/rad
This is a very high torsional stiffness compared with mechanical shafts (a 50 mm diameter steel shaft of 200 mm length has a torsional stiffness of approximately 180,000 N·m/rad — 50× lower than the gear mesh). The спирална предавка mesh is the stiffest element in most drive trains — the limiting torsional compliance comes from the shafts, couplings, and rotor windup, not the gear teeth themselves.
Torsional Natural Frequency — The Two-Mass Model
The спирална предавка mesh in a servo drive system acts as a torsional spring C_torsion connecting the motor inertia J₁ and load inertia J₂. The torsional natural frequency f_n of this two-mass system must be above the servo control bandwidth — ideally 3–5× above it — to avoid resonance excitation from the servo torque commands
The simplest model for a servo-driven спирална предавка system is the two-mass torsional model: motor inertia J₁ connected to load inertia J₂ through the gear mesh torsional spring C_torsion. The undamped natural frequency of this system:
ω_n = √(C_torsion × (1/J₁ + 1/J₂)) [rad/s]
f_n = ω_n / (2π) [Hz]
Example with the gear from above (C_torsion = 9.6 × 10⁶ N·m/rad):
Motor inertia J₁ = 0.020 kg·m²
Load inertia J₂ = 0.060 kg·m² (reflected to input shaft via i²)
ω_n = √(9.6 × 10⁶ × (1/0.020 + 1/0.060)) = √(9.6 × 10⁶ × 66.7) = √(640 × 10⁶) = 25,298 rad/s
f_n = 25,298 / (2π) = 4,028 Hz
This is safely above any servo bandwidth (<1,000 Hz for most precision drives) — the gear mesh
torsional resonance is not a limiting factor for this compact gear pair.
⚠ If the gear pair were much larger (J₁=0.5, J₂=5.0, C_torsion=9.6×10⁶):
ω_n = √(9.6×10⁶ × (1/0.5 + 1/5.0)) = √(9.6×10⁶ × 2.2) = 4,596 rad/s → f_n = 731 Hz
With servo bandwidth 300 Hz, the ratio f_n/BW = 2.4× — dangerously close to resonance.
How DIN Accuracy Class Affects Torsional Stiffness Variation
The torsional mean stiffness of a спирална предавка is not strongly affected by DIN accuracy class — the tooth contact area and deflection are determined by the module and material, not the surface finish. However, the variation of mesh stiffness around the mean (at tooth mesh frequency) is strongly accuracy-class dependent:
| Клас на точност по DIN | Profile Deviation ff (µm) for M5 | Stiffness Variation at Mesh Frequency | Transmission Error (TE) Amplitude | Resonance Excitation |
|---|---|---|---|---|
| DIN Class 4–5 | 3–5 µm | ±3–5% of mean stiffness | 2–5 µm | Low — servo bandwidth limited by machine dynamics, not gear resonance |
| DIN Class 6–7 | 8–16 µm | ±8–15% of mean stiffness | 8–18 µm | Moderate — gear resonance contributes to positioning error at mesh frequency |
| DIN Class 8–9 | 18–36 µm | ±20–35% of mean stiffness | 20–40 µm | High — gear resonance typically visible in position error spectrum; limits servo gain |
Backlash and the Dead Zone — Non-Linear Torsional Behaviour
Unlike shaft compliance, a спирална предавка pair with backlash has a non-linear torsional spring characteristic: within the backlash zone (±j/2 at the pitch circle, where j is the total backlash), the mesh transmits no torque — the teeth are not in contact and the torsional stiffness is effectively zero. Outside the backlash zone, the mesh restores to the full torsional stiffness C_torsion. This “dead zone” non-linearity has two important servo system consequences:
- Position error during direction reversal: When the servo drive reverses direction, the motor shaft rotates through the backlash angle before re-engaging the load. This reversal error (equal to the backlash at the output) is directly visible as a step in the position feedback signal and as an overshoot on the output shaft. For a DIN 3967 class ef gear at M5 (backlash j_t ≈ 0.12 mm at pitch circle, d₁ = 127.8 mm), the output shaft reversal angle = arcsin(0.12/127.8) ≈ 0.054° — translating to a linear error of 0.054 × 500/57.3 = 0.47 mm at a 500 mm arm.
- Limit cycle instability at high servo gain: If the servo loop gain is increased to improve response, the closed-loop system may enter a limit cycle — an oscillation where the servo repeatedly drives the output across the backlash zone, with the stiffness switching between zero and C_torsion at each reversal — the practical upper boundary on servo gain for a спирална предавка drive with finite backlash, and is the primary reason why anti-backlash gear pairs (Art47) are used in high-performance servo axes.
Tip Relief and Its Effect on Stiffness Variation
Tip relief applied to a спирална предавка directly reduces mesh stiffness variation by smoothing the stiffness transition at tooth entry and exit. Without tip relief, the stiffness jumps from its single-pair value to its double-pair value at the moment of tooth engagement — creating a sharp stiffness step that excites the torsional natural frequency. With optimised parabolic tip relief, the stiffness transition is spread over the roll angle of the approach zone — reducing the stiffness step amplitude and therefore the resonance excitation amplitude. For servo спирална предавка applications, Korea Ever-Power applies parabolic tip relief as standard (see Art46 for the tip relief design methodology), because the stiffness variation reduction from tip relief is as important for servo dynamic performance as the noise reduction for NVH applications.
Korea Ever-Power — Torsional Stiffness Data with Precision Gear Orders
The tooth contact detail of a спирална предавка pair — the elastic deflection at this contact zone, described by the tooth pair stiffness c’_gamma, is the fundamental parameter for torsional stiffness and resonance frequency calculation. Korea Ever-Power provides the calculated c’_gamma and C_torsion values with every precision servo gear order
Korea Ever-Power provides the calculated tooth pair stiffness c’_gamma, total mesh stiffness C_mesh, and torsional spring constant C_torsion (referred to the input shaft) for every precision servo and robot спирално нарязано зъбно колело order — giving the servo engineer the exact спирална предавка values needed for the two-mass torsional model and bandwidth calculation. As a direct производител на спирални зъбни колела, Korea Ever-Power also provides the stiffness variation amplitude (ΔC/C) based on the measured DIN class and tip relief specification — enabling the resonance excitation level to be quantified before installation. Browse the продуктова гама от спирални зъбни колела for precision servo and robot applications.
Често задавани въпроси
Increasing helix angle β reduces the спирална предавка torsional stiffness slightly (through the cos β factor in c’_gamma) but significantly increases the contact ratio ε_γ (more tooth pairs in simultaneous contact). The combined effect: mean C_torsion decreases slightly with β, but the stiffness variation at mesh frequency decreases much more — because higher ε_γ means the transition between single-pair and double-pair contact is smoother. For servo спирална предавка applications, increasing β from 15° to 25° reduces the transmission error excitation amplitude by approximately 20–35% — a meaningful improvement in servo dynamic performance at the cost of slightly more axial thrust on the shaft bearing.
Yes — the torsional natural frequency f_n = (1/2π) × √(C_torsion × (1/J₁ + 1/J₂)) can be increased by reducing J₁ or J₂. In practice, J₂ (load inertia referred to input shaft) = J_actual_load / i². Increasing the gear ratio i reduces J₂ by i² — the most effective route to raising f_n when the спирална предавка cannot be changed. Alternatively, replacing a solid gear wheel with a hollow-rim version (same tooth form, but internal hub material removed) reduces the спирална предавка body inertia contribution to J₂ by 20–50%, raising f_n by √(1/(0.5–0.8)) = 11–41%.
Yes, slightly — but less than intuitively expected. Спирална предавка mesh stiffness in the Hertzian contact zone is not strictly linear: at higher contact force, the contact ellipse grows and the effective spring stiffness increases slightly (the stiffness is proportional to the contact area). For спираловидни зъбни колела, the stiffness variation with load is approximately 5–15% over the typical operating load range (20–120% of rated) — small enough that the linear spring model (constant C_torsion) is adequate for most servo control designs. For very precise torsional natural frequency calculations (e.g. noise-source identification in gearbox vibration analysis), the load-dependent спирална предавка stiffness variation should be modelled — Korea Ever-Power provides the nonlinear c'(F) characteristic on request for precision vibration applications.
In an industrial спирална предавка gearbox, the torsional natural frequency (typically 1,000–10,000 Hz for compact gears) is well above the operating frequency range of interest (0–200 Hz for most industrial machinery). The gear mesh resonance is excited by the stiffness variation at mesh frequency, but the response decays rapidly in a well-damped industrial gearbox housing and does not create operationally significant vibration. In a servo drive, the torsional natural frequency (which can fall to 100–800 Hz for large gears with high inertia) is within the servo control bandwidth (50–500 Hz for modern servo amplifiers). When the servo’s commanded position error contains frequency components near f_n, the resonance amplifies the gear mesh compliance fluctuation into a visible position tracking error and potential instability. This is why the torsional natural frequency check is standard practice for servo спирална предавка drive design, but rarely performed for industrial gearbox applications.
Torsional Stiffness Data for Your Servo Helical Gear
Provide your module, tooth count, face width, and inertia values. Korea Ever-Power calculates c’_gamma, C_mesh, C_torsion, torsional natural frequency f_n, and bandwidth separation ratio — as standard documentation with every precision servo gear order.
c’_gamma · C_torsion · f_n calculation · ΔC/C variation · Tip relief stiffness smoothing · Backlash dead zone quantification
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