What the translation accomplishes
A gearbox connects a drive shaft (motor) to an output shaft (application), thereby changing the speed and torque in proportion to the gear ratio i. With a reduction ratio (>1), the speed at the output decreases, while the torque increases in the same proportion—minus the losses described by the efficiency η.
n₂ = n₁ / i
n₁ = drive speed (min⁻¹) · n₂ = output speed (min⁻¹) · i = gear ratio
Torque increases in direct proportion to the decrease in speed—though not entirely, because some of the power is lost as frictional heat in the transmission. This proportion is represented by the efficiency η:
M₂ = M₁ · i · η
M₁ = Drive torque (Nm) · M₂ = output torque (Nm) · η = Efficiency (0…1)
From Torque and Speed to Power
Torque and speed together determine mechanical power. For the units commonly used in drive technology (P in kW, M in Nm, n in min⁻¹), the following applies:
P = M · n / 9550
P = Power (kW) · M = Torque (Nm) · n = Speed (min⁻¹)
Because an ideal, lossless gearbox merely converts power rather than increasing it, the output power actually decreases by exactly the factor η compared to the input power—as can be seen in the worked example below.
Multi-stage gear transmissions
If a single stage is insufficient for the desired gear ratio or would be uneconomical, multiple gear stages are connected in series. The total gear ratio is then calculated as the product of the individual gear ratios—not as their sum:
i_ges = i₁ · i₂ · … · i_n
i_ges = Total reduction ratio · i₁…i_n = Reduction ratio per stage
Two stages, each with i = 5, therefore result in a total ratio of i_ges = 5 · 5 = 25—the same total ratio that a single, heavily reduced-ratio stage would achieve, but with significantly lower efficiency (more on this in the next unit).
Mnemonic
Worked example
Given
A motor rotates at n₁ = 1,000 min⁻¹ with a torque of M₁ = 8 Nm. The downstream gearbox has a gear ratio of i = 25 and an efficiency of η = 0.85.
Calculation
- Output speed: n₂ = n₁ / i = 1,000 / 25 = 40 min⁻¹
- Output torque: M₂ = M₁ · i · η = 8 · 25 · 0.85 = 170 Nm
- Drive power: P₁ = M₁ · n₁ / 9550 = 8 · 1,000 / 9550 ≈ 0.84 kW
- Output power: P₂ = M₂ · n₂ / 9550 = 170 · 40 / 9550 ≈ 0.71 kW
Verification: P₂ = P₁ · η = 0.84 kW · 0.85 ≈ 0.71 kW – matches as expected.
Knowledge check
Answer all three questions, then click "Check". From 2 of 3 correct answers, the unit counts as completed. You can retry at any time.
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