{"id":2393,"date":"2026-06-26T04:00:16","date_gmt":"2026-06-26T04:00:16","guid":{"rendered":"https:\/\/helicalcutgears.top\/?p=2393"},"modified":"2026-06-26T04:00:16","modified_gmt":"2026-06-26T04:00:16","slug":"helical-gear-torsional-stiffness-servo-resonance","status":"publish","type":"post","link":"https:\/\/helicalcutgears.top\/ja\/helical-gear-torsional-stiffness-servo-resonance\/","title":{"rendered":"Helical Gear Torsional Stiffness \u2014 How It Affects Servo Bandwidth"},"content":{"rendered":"
\n
\n
<\/div>\n
\n

Helical Gear Torsional Stiffness \u2014 Servo Bandwidth, Resonance and Backlash Dead Zone<\/h1>\n

A \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> pair is not a rigid torsional connection between its input and output shafts \u2014 it is a torsional spring whose stiffness is set by the tooth contact geometry and the number of tooth pairs simultaneously in contact. This spring stiffness creates a mechanical resonance between the motor inertia and the load inertia that, if not accounted for in the servo control design, limits the achievable bandwidth and can cause vibration that appears as servo instability. Understanding and calculating the torsional stiffness of a \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> pair is therefore an essential step in any precision servo drive design.<\/p>\n

Request Stiffness Data for Your Gear \u2192<\/a><\/p>\n<\/div>\n<\/div>\n

Gear Mesh Stiffness \u2014 The Torsional Spring at the Tooth Contact<\/h2>\n

When force is applied to a \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> tooth, both teeth deflect, both teeth deflect elastically under the Hertzian contact stress. The combined deflection of the tooth pair \u2014 the input tooth bending, the output tooth bending, and the Hertz contact deformation \u2014 creates an effective stiffness c’ [N\/(\u00b5m 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<\/em>, and it is the fundamental mechanical property that determines the \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> pair’s torsional characteristics:<\/p>\n

Mean tooth pair stiffness (ISO 6336-1 notation):
\nc’_gamma \u2248 c_th \u00d7 C_M \u00d7 C_R \u00d7 C_B \u00d7 cos \u03b2
\nwhere:
\nc_th = theoretical single pair stiffness \u2248 14\u201320 N\/(\u00b5m\u00b7mm) for standard steel gear pair
\nC_M = correction for gear body mass (0.8 for solid gear body)
\nC_R = correction for rim thickness (1.0 for standard rim \u2265 2.5 \u00d7 h_tooth)
\nC_B = correction for basic rack (1.0 for standard 20\u00b0 pressure angle)
\ncos \u03b2 = helix correction<\/p>\n

Typical result for carburized steel gears, \u03b2 = 20\u00b0:
\nc’_gamma \u2248 17 \u00d7 0.8 \u00d7 1.0 \u00d7 1.0 \u00d7 cos 20\u00b0 \u2248 12.8 N\/(\u00b5m\u00b7mm)<\/p>\n

This stiffness value varies periodically as the \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> rotates \u2014 when two tooth pairs are simultaneously in contact, the total mesh stiffness is approximately 2 \u00d7 c’_gamma \u00d7 b; when only one pair is in contact, it falls to c’_gamma \u00d7 b. This periodic variation in stiffness \u2014 at the tooth mesh frequency (z \u00d7 RPM \/ 60) \u2014 is the primary source of transmission error and the dominant excitation of gear noise and vibration.<\/p>\n

Total Mesh Stiffness and Torsional Spring Constant<\/h2>\n

The total mesh stiffness of a \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> 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 \u03b5_\u03b3):<\/p>\n

C_mesh \u2248 c’_gamma \u00d7 b \u00d7 \u03b5_\u03b3 [N\u00b7m\/\u00b5m, total mesh stiffness]
\nTorsional spring constant (referred to input shaft):
\nC_torsion = C_mesh \u00d7 (d\u2081\/2)\u00b2 [N\u00b7m\/rad]<\/p>\n

Example: M5, z\u2081=24, z\u2082=72, \u03b2=20\u00b0, b=80mm, 20CrMnTi carburized, \u03b5_\u03b3=2.3
\nc’_gamma = 12.8 N\/(\u00b5m\u00b7mm)
\nC_mesh = 12.8 \u00d7 80 \u00d7 2.3 = 2,355 N\/\u00b5m = 2,355,000,000 N\/m
\nd\u2081 = 127.8 mm
\nC_torsion = 2,355,000,000 \u00d7 (0.0639)\u00b2 = 9,615,000 N\u00b7m\/rad \u2248 9.6 \u00d7 10\u2076 N\u00b7m\/rad<\/p>\n

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\u00b7m\/rad \u2014 50\u00d7 lower than the gear mesh). The \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> mesh is the stiffest element in most drive trains \u2014 the limiting torsional compliance comes from the shafts, couplings, and rotor windup, not the gear teeth themselves.<\/p>\n

Torsional Natural Frequency \u2014 The Two-Mass Model<\/h2>\n

\"precision<\/p>\n

\u306e \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> mesh in a servo drive system acts as a torsional spring C_torsion connecting the motor inertia J\u2081 and load inertia J\u2082. The torsional natural frequency f_n of this two-mass system must be above the servo control bandwidth \u2014 ideally 3\u20135\u00d7 above it \u2014 to avoid resonance excitation from the servo torque commands<\/p>\n

The simplest model for a servo-driven \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> system is the two-mass torsional model: motor inertia J\u2081 connected to load inertia J\u2082 through the gear mesh torsional spring C_torsion. The undamped natural frequency of this system:<\/p>\n

\u03c9_n = \u221a(C_torsion \u00d7 (1\/J\u2081 + 1\/J\u2082)) [rad\/s]
\nf_n = \u03c9_n \/ (2\u03c0) [Hz]<\/p>\n

Example with the gear from above (C_torsion = 9.6 \u00d7 10\u2076 N\u00b7m\/rad):
\nMotor inertia J\u2081 = 0.020 kg\u00b7m\u00b2
\nLoad inertia J\u2082 = 0.060 kg\u00b7m\u00b2 (reflected to input shaft via i\u00b2)<\/p>\n

\u03c9_n = \u221a(9.6 \u00d7 10\u2076 \u00d7 (1\/0.020 + 1\/0.060)) = \u221a(9.6 \u00d7 10\u2076 \u00d7 66.7) = \u221a(640 \u00d7 10\u2076) = 25,298 rad\/s
\nf_n = 25,298 \/ (2\u03c0) = 4,028 Hz<\/p>\n

This is safely above any servo bandwidth (<1,000 Hz for most precision drives) \u2014 the gear mesh
\ntorsional resonance is not a limiting factor for this compact gear pair.<\/p>\n

\u26a0 If the gear pair were much larger (J\u2081=0.5, J\u2082=5.0, C_torsion=9.6\u00d710\u2076):
\n\u03c9_n = \u221a(9.6\u00d710\u2076 \u00d7 (1\/0.5 + 1\/5.0)) = \u221a(9.6\u00d710\u2076 \u00d7 2.2) = 4,596 rad\/s \u2192 f_n = 731 Hz
\nWith servo bandwidth 300 Hz, the ratio f_n\/BW = 2.4\u00d7 \u2014 dangerously close to resonance.<\/p>\n

Minimum separation rule for servo gear drives:<\/strong> The torsional natural frequency f_n must be at least 3\u00d7 the servo control bandwidth to provide adequate phase margin. Below this ratio, the servo loop gain must be reduced to maintain stability \u2014 directly limiting the achievable servo bandwidth and therefore the positioning speed and accuracy of the machine. If the calculation shows f_n\/BW < 3, the drive designer must either: increase C_torsion (larger module, wider face width, higher accuracy class to reduce stiffness variation), reduce inertia (hollow gear body, lighter load coupling), or widen the gear mesh (increase \u03b5_\u03b3 by increasing \u03b2 or b).<\/div>\n

How DIN Accuracy Class Affects Torsional Stiffness Variation<\/h2>\n

The torsional mean<\/em> stiffness of a \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> is not strongly affected by DIN accuracy class \u2014 the tooth contact area and deflection are determined by the module and material, not the surface finish. However, the variation<\/em> of mesh stiffness around the mean (at tooth mesh frequency) is strongly accuracy-class dependent:<\/p>\n

\n\n\n\n\n\n\n\n
DIN\u7cbe\u5ea6\u7b49\u7d1a<\/th>\nProfile Deviation ff (\u00b5m) for M5<\/th>\nStiffness Variation at Mesh Frequency<\/th>\nTransmission Error (TE) Amplitude<\/th>\nResonance Excitation<\/th>\n<\/tr>\n<\/thead>\n
DIN Class 4\u20135<\/td>\n3\u20135 \u00b5m<\/td>\n\u00b13\u20135% of mean stiffness<\/td>\n2\u20135 \u00b5m<\/td>\nLow \u2014 servo bandwidth limited by machine dynamics, not gear resonance<\/td>\n<\/tr>\n
DIN Class 6\u20137<\/td>\n8\u201316 \u00b5m<\/td>\n\u00b18\u201315% of mean stiffness<\/td>\n8\u201318 \u00b5m<\/td>\nModerate \u2014 gear resonance contributes to positioning error at mesh frequency<\/td>\n<\/tr>\n
DIN Class 8\u20139<\/td>\n18\u201336 \u00b5m<\/td>\n\u00b120\u201335% of mean stiffness<\/td>\n20\u201340 \u00b5m<\/td>\nHigh \u2014 gear resonance typically visible in position error spectrum; limits servo gain<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Backlash and the Dead Zone \u2014 Non-Linear Torsional Behaviour<\/h2>\n

Unlike shaft compliance, a \u30d8\u30ea\u30ab\u30eb\u30ae\u30a2<\/strong> pair with backlash has a non-linear torsional spring characteristic: within the backlash zone (\u00b1j\/2 at the pitch circle, where j is the total backlash), the mesh transmits no torque \u2014 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:<\/p>\n