The Core Formula Set for Helical Gear Calculations
All تروس حلزونية calculations derive from four fundamental relationships. Memorising these four and understanding their connections eliminates the need for tables in most practical design situations:
Four Core Helical Gear Formulas
1. Pitch diameter: d = Mn × z / cos β
2. Centre distance: C = Mn × (z₁ + z₂) / (2 × cos β)
3. Speed (gear) ratio: i = z₂ / z₁ = n₁ / n₂ = T₂ / T₁
4. Output torque: T₂ = 9550 × P × η / n₂ [N·m, kW, RPM]
Speed Ratio Calculation — Single Stage and Multi-Stage
Single-Stage Helical Gear Ratio
For a single تروس حلزونية pair (pinion driving gear):
i = z₂ / z₁ (always the driven gear teeth divided by the driver)
The output speed: n₂ = n₁ / i. The output torque (ignoring losses): T₂ = T₁ × i. Including mesh efficiency η (typically 0.985–0.995 for a ground تروس حلزونية stage):
T₂ = T₁ × i × η₁
Multi-Stage Helical Gear Train
For a k-stage تروس حلزونية gearbox, the total ratio is the product of all stage ratios, and the total efficiency is the product of all stage efficiencies:
i_total = i₁ × i₂ × i₃ × … × i_k
η_total = η₁ × η₂ × η₃ × … × η_k
T_output = 9550 × P × η_total / n_output [N·m]
How to Distribute Ratio Across Stages for Minimum Gearbox Size
For a 3-stage تروس حلزونية gearbox with total ratio i = 40, distributing the ratio as 8:1 × 5:1 × 1:1 is inefficient — the first stage gear wheel (8× the pinion diameter) dominates the housing size while the third stage adds nothing. The minimum-size distribution has approximately equal stage ratios: 3.4:1 × 3.4:1 × 3.4:1 ≈ 39.3 — close to the target 40. The practical rule: stage ratios should be within a factor of 1.5–2× of each other. Geometric progression (each stage ratio = i_total^(1/k)) is the mathematical optimum for minimum housing volume.
Centre Distance Calculation — The Critical Housing Dimension
The centre distance C is the distance between the input shaft centreline and the output shaft centreline of a تروس حلزونية stage. It is set by the housing and cannot be changed without remachining the housing. The centre distance formula:
C = Mn × (z₁ + z₂) / (2 × cos β)
Note the cos β term: a تروس حلزونية pair has a larger centre distance than a spur gear pair with the same Mn and tooth counts. For Mn = 5, z₁ = 20, z₂ = 60, at β = 25°:
C = 5 × (20 + 60) / (2 × cos 25°) = 5 × 80 / (2 × 0.906) = 400 / 1.812 = 220.8 mm
The equivalent spur gear pair would have C = 5 × 80 / 2 = 200 mm — 10.4% smaller. This is a frequent calculation error when a designer tries to retrofit تروس حلزونية into an existing spur gear housing: the same Mn and tooth counts will not fit the existing centre distance if β > 0°.
Module Selection from Required Torque — Starting a New Design
Simplified Module Sizing Formula
For a new تروس حلزونية design where the module is not yet determined, a simplified starting-point formula based on ISO 6336 tooth root bending strength gives the required module from the transmitted torque, shaft speeds, and material grade:
Mn_min ≈ 1.5 × ∛(T₁ × KA / (z₁ × b/d × σ_F_lim / SF))^(1/3) [mm, N·m]
This simplified form gives a first estimate only; the complete ISO 6336 calculation covers dynamic load factor K_v, face load distribution K_Hβ, and both root bending and contact fatigue. Korea Ever-Power’s engineering team performs the full calculation for all custom orders. For standard applications, the following empirical module ranges by input power cover the majority of cases for 20CrMnTi carburized تروس حلزونية at β = 20°, SF = 1.3:
| Input Power (kW) at n₁ = 1500 RPM | Starting Module Estimate (Mn) | Typical Face Width b | Typical Pinion Tooth Count z₁ |
|---|---|---|---|
| 5–15 kW | M2–M3 | 20–40 mm | 18–25 |
| 15–50 kW | M3–M5 | 30–60 mm | 20–28 |
| 50–150 kW | M4–M6 | 50–90 mm | 20–30 |
| 150–500 kW | M5–M8 | 80–140 mm | 22–32 |
| 500–2000 kW | M8–M14 | 120–240 mm | 22–35 |
| 2000–10,000 kW | M12–M22 | 200–450 mm | 20–30 |
Complete Three-Stage Gearbox Worked Example
Design a 3-Stage Inline Helical Gear Gearbox: 75 kW, 1500→38 RPM
Step 1: Total ratio
i_total = n_input / n_output = 1500 / 38 = 39.5
Step 2: Stage ratio distribution (geometric)
i_per_stage = 39.5^(1/3) = 3.41 per stage
Use: Stage 1 = 3.55, Stage 2 = 3.35, Stage 3 = 3.32 → total = 3.55 × 3.35 × 3.32 = 39.5 ✓
Step 3: Shaft speeds
n₁ = 1500 RPM n₂ = 1500/3.55 = 422 RPM n₃ = 422/3.35 = 126 RPM n₄ = 126/3.32 = 38 RPM
Step 4: Output torque (η_total = 0.97^3 ≈ 0.915 for 3 stages)
T₄ = 9550 × 75 × 0.915 / 38 = 17,278 N·m ≈ 17.3 kN·m
Step 5: Module selection (Stage 1 at 1500 RPM, P=75 kW)
Pinion torque T₁ = 9550 × 75 / 1500 = 477 N·m → from table: Mn = M4–M6 → select M5
Stage 3 at 126 RPM, P = 75 × 0.97² ≈ 70.6 kW → Pinion T₃ = 9550 × 70.6 / 126 = 5,348 N·m → Mn = M8–M10 → select M8
Step 6: Tooth counts for Stage 1 (i = 3.55, Mn = 5, β = 20°)
z₁ = 24, z₂ = 24 × 3.55 = 85.2 → round to z₂ = 85, actual i = 85/24 = 3.542
Revised chain: 1500 → 423.1 → 126.3 → 38.1 RPM (error <0.3%)
Step 7: Centre distance Stage 1
C₁ = 5 × (24 + 85) / (2 × cos 20°) = 5 × 109 / 1.879 = 290.1 mm
Back-Calculating Tooth Counts from a Fixed Centre Distance
When replacing gears in an existing housing with a fixed centre distance C, the challenge is finding tooth counts z₁ and z₂ that give the required ratio i and fit the fixed C, given the selected Mn and β. The procedure:
From C = Mn × (z₁ + z₂) / (2 cos β): z₁ + z₂ = 2C × cos β / Mn
From i = z₂ / z₁: z₂ = i × z₁
Combining: z₁ × (1 + i) = 2C cos β / Mn → z₁ = 2C cos β / (Mn × (1 + i))
The result must be rounded to an integer. If the rounded values do not give exactly the required ratio, the actual ratio i_actual = z₂_rounded / z₁_rounded differs slightly from i_target. For most replacement applications, ±1–2% ratio tolerance is acceptable. If tighter ratio accuracy is required, profile shift can adjust the operating centre distance to accommodate the rounded tooth counts precisely.
Korea Ever-Power — Gear Train Calculations as Part of Standard Engineering Service
Korea Ever-Power provides complete gear train calculation support as part of every تروس القطع الحلزونية order — confirming tooth counts, centre distance compatibility, gear ratio accuracy, and output torque before production begins
Korea Ever-Power’s engineering team performs complete gear train calculations for every custom تروس القطع الحلزونية order — confirming tooth counts, centre distance compatibility, gear ratio accuracy, module selection from torque and speed, and face width adequacy. As a direct مصنع التروس الحلزونية with over 10,000 تروس حلزونية orders of experience, Korea Ever-Power’s team identifies calculation errors and specification conflicts before production begins — preventing the most common and expensive mistake in gear drive procurement: receiving gears that technically meet the drawing but do not fit the housing or achieve the specified output speed.
الأسئلة الشائعة
The centre distance formula C = Mn × (z₁ + z₂) / (2 × cos β) contains cos β in the denominator. Increasing β decreases cos β, which increases C. For example, at Mn = 5, z₁ + z₂ = 80: C at β = 0° (spur) = 200 mm; C at β = 20° = 212.8 mm; C at β = 25° = 220.8 mm. This means you cannot simply increase the helix angle of a تروس حلزونية without checking whether the housing centre distance must also change — or whether profile shift (addendum modification) must be applied to bring the operating centre distance back to the original housing dimension.
The minimum tooth count for a standard addendum تروس حلزونية without undercutting is approximately: z_min ≈ 17 / cos³β (for α_n = 20°). At β = 0° (spur), z_min ≈ 17. At β = 20°, z_min ≈ 17 / cos³(20°) = 17 / 0.830 ≈ 20.5 → round to 21. At β = 30°, z_min ≈ 17 / 0.650 ≈ 26. This means helical gears with higher helix angles have a higher minimum tooth count than spur gears — important for compact high-ratio designs where the pinion tooth count might otherwise fall below the undercutting limit.
i_total = n_input / n_output. For the worked example above: i = 1500 / 38 = 39.47. To find the individual stage ratios from the tooth counts, measure the tooth counts of all gears in the gearbox (count directly for small gears; use gear analyser for large gears where direct counting is difficult). Stage ratio = z_driven / z_driver for each stage. The product of all stage ratios should equal i_total within ±0.5% (the rounding of tooth counts introduces a small discrepancy).
Yes — this is a standard service. Provide: centre distance C (measured from the housing), required gear ratio i (from input/output shaft speed measurement or nameplate), helix angle β (measured from the existing gear by gear analyser), and normal module Mn (calculated from tooth count and OD measurement). Korea Ever-Power calculates the exact tooth count combination that fits C and gives i within the acceptable rounding tolerance, specifies any required profile shift, and produces the replacement تروس حلزونية pair as a matched set.
Need Gear Train Calculation Support?
Provide your input speed, output speed, power, housing centre distance, and helix angle. Korea Ever-Power calculates tooth counts, verifies ratio accuracy, checks centre distance compatibility, and produces the specification within 24 working hours.
Ratio calculation · Centre distance check · Tooth count optimisation · Profile shift · Worked example · MOQ 1 piece
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