Zenith Division

Technical Bulletin

Page 1 of 2

Parker Hannifin Corp.•••• Zenith Division •••• Sanford, NC 27330 •••• 1-877-959-4327 •••• Fax: 919-774-5952

Date: January 9, 2001No: ZTB-006To: ATCs, ETMs, RMCc: AEs

Subject: Squaring of Gear Ratio for inertia MatchingEffective: Long time ago

Gearhead Sizing

When sizing a gearhead for inertia matching the squaring of the ratio is often used without explanation.Here are the simple equations and the algebra to explain why if anyone ever asks.

Variable definitionsT1 = motor torque

J1 = motor inertia

α1 = motor angular acceleration

T2 = load torque

J2 = load inertia

α2 = load angular acceleration

To determine why the gear ratio is squared we must first start with the following fundamental equations.

1. T2= J2* α2

2. T1 / R1 = T2 / R2

3. T1 = J1 * α1

4. T2= J2* α2

5. T1 / R1 = T2 / R2

6. R1 * α1= R2 * α2

7. Gear ratio = R2 / R1

With simple algebra we manipulate equations 6 & 4 to get8. α1 = (R2 / R1) * α2

9. α2 = T2 / J2

Then with substitution of equation 8 for α1 into 3,10. T1 = J1 * (R2 / R1) * α2

Next we substitution of equation 9 for α2 into 10,11. T1 = J1 * (R2 / R1) * (T2 / J2)

Zenith Division

Technical Bulletin

Page 2 of 2

Parker Hannifin Corp.•••• Zenith Division •••• Sanford, NC 27330 •••• 1-877-959-4327 •••• Fax: 919-774-5952

Bring T2 to the left side of the equation…12. T1 / T2 = (R2 / R1) * (J1 / J2)

And cross-multiply on equation 5…13. T1 / T2 = R1 / R2

Equate equation 12 & 13…T1 / T2 = R1 / R2 = (R2 / R1) * (J1 / J2)

Finally bring the (R2 / R1) and J2 to the left. J2 * 1 / (R2 / R1) ² = J1

∴ J2 * 1 / (gear ratio) ² = J1