WARNING: This calculation tool is for semi-professional hobby purposes. Please check the relevant standards and calculations. For engineering studies, use more serious calculation software.

 

Planetary Gear System Calculation Formulas

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1. Basic Gear Dimensions

Module (m): The fundamental parameter for gear sizing

m=d/z​

Where:

• d: Pitch diameter (mm)

• z: Number of teeth

Pitch Diameter (d): d=m×z

Outside Diameter (da):

da=m×z+2×a

Addendum, typically a=m×ha∗

Root Diameter (df):

df​=m×z−2×f

f: Dedendum, typically f=m×hf∗

Tooth Height (h):

h=a+f

Base Circle Diameter (db):

db​=m×z×cos(α)

α: Pressure angle (in radians)

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2. Planetary Gear System Tooth Count

Ring Gear Teeth (z_r) :

zr​=zs​+2×zp​

• zs​: Sun gear teeth

• zp​: Planet gear teeth

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3. Transmission Ratios

Ring Gear Fixed:

i=1+ zr​​ / zs​

Carrier Fixed:

i=1/(1− zr/​​zs​​)

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4. Dynamic Factor (Kv)

Pitch Line Velocity (v):

v=π×d×n​/60

 

• d: Pitch diameter (mm)

• n: Rotational speed (rpm)

• Result: m/s

Kv Value:

• v<1 m/s → Kv=1.0

• 1≤v<3 m/s → Kv=1.1

• 3≤v<8 m/s → Kv=1.2

• 8≤v<12 m/s → Kv=1.3

• v≥12 m/s → Kv=1.4

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5. Size Factor (Ks)

• m<2 → Ks=1.0

• 2≤m<4 → Ks=1.1

• 4≤m<6 → Ks=1.2

• m≥6 → Ks=1.3

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6. AGMA Bending Stress

Torque (T):

T= P×60/2πn

• P: Power (Watt)

• n: Rotational speed (rpm)

Tangential Force (Ft):

Ft=2×T×1000​ / (m×z)

• Result: Newton (N)

Bending Stress (σ):

σ= [Ft×Ko×Kv×Ks×(Km×KB)×(Kf×KR×KT)​] / [b×m×J]

​Where:

• b: Face width (mm)

• J: Geometry factor (AGMA)

• Ko: Overload factor

• Kv: Dynamic factor

• Ks: Size factor

• Km: Load distribution factor

• KB: Rim thickness factor

• Kf: Surface factor

• KR: Reliability factor

• KT: Temperature factor

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7. AGMA Contact Stress

Contact Stress (σc):

σc​=Z × [[Ft×Ko×Kv×(Km×KR×KT)​​] / [d1​×b×I]]^0,5

Where:

• Z: Elastic coefficient

• d1​: Pitch diameter of the pinion

• I: Geometry factor for pitting resistance

Elastic Coefficient (Z):

For same material:

Z=[1/π(2×((1−v2​) / E))]^0,5

• E: Elastic modulus (MPa)

• v: Poisson’s ratio

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8. Safety Factor

SF= Material Yield Strength​ / Calculated Bending Stress

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9. Gear Tooth Profile Factors

• Addendum Factor (ha\*): Typically 1.0

• Dedendum Factor (hf\*): Typically 1.25

• Clearance Factor (c\*): Typically 0.25

• Profile Shift Factor (x): Range: -0.5 to +0.5

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10. Other Factors

• Load Distribution Factor (Km): Depends on face width.

• Overload Factor (Ko): Depends on application type.

• Rim Thickness Factor (KB): Depends on rim thickness/tooth height ratio.

• Surface Factor (Kf): Depends on surface finish.

• Stress Concentration Factor (Kt): Depends on fillet radius/module ratio.

• Reliability Factor (KR): Depends on required reliability.

• Temperature Factor (KT): Depends on operating temperature.

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References & Standards

• AGMA 2001-D04: Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth

• AGMA 2101-D04: Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth (Metric Edition)

• AGMA 6123-B06: Design Manual for Enclosed Epicyclic Gear Drives

• ISO 6336: Calculation of load capacity of spur and helical gears

• Shigley, J.E., Mischke, C.R., Budynas, R.G.: Mechanical Engineering Design, McGraw-Hill

• Dudley, D.W.: Handbook of Practical Gear Design, McGraw-Hill

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Note: All factors and coefficients should be selected according to the referenced standards and application-specific requirements. Ensure consistent units throughout all calculations.