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For an aircraft engine, what is the difference between a gearbox and a transmission?

I was of the understanding that a transmission is a type of gearbox. For example, in a helicopter, a transmission is used in to convert horizontal rotation of the turbo-shaft to a vertical shaft rotation. Is an aircraft engine gearbox and aircraft engine transmission the same thing?

Are there other types of gearboxes used in aircraft engines besides the transmission design?

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In aircraft, transmission usually refers to the whole system, that transmits power between two points, say between the MGB and the tail rotor. Gearbox is usually a component of the transmission system (along with shafts, for example) which is used for changing rotational speed.

In a sense, a full transmission system is found only in helicopters, and not in fixed wing aircraft. The closest thing a fixed wing aircraft has to a transmission is the variable pitch propeller, which is analagous to the car transmission.

Variable pitch propeller

By nydano - Cut away view of Hamilton Standard Propeller, CC BY-SA 2.0, $3

On the other hand, a number of aircraft have gearboxes, especially the turboprops (and some piston engines), for reducing the engine output speed before turning the propeller. The following image shows the gearbox in a GE turboshaft engine.

Gearbox

Image from mrlarry.org

Turbofan engines do have a gearbox, which is used to run the accessories. The compressor is usually run through a shaft from the turbine, with various turbine stages running various compressor stages, while some turbofans do use gears between turbine and compressor, called geared turbofans.

Aircraft transmissions (whether helicopter or fixed wing) is usually one speed from one point to another, with speed reduction (or direction change) offered by the gearboxes.

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A transmission is something usually used on a car. They have multiple "gears" that allow the engine to rotate at the same speed while the output rotates at progressively higher speeds. Airplanes don't use transmissions (that I know of!).

A gearbox is simply a 1-gear transmission. I suppose technically it could be called a transmission. But it usually is reduction gearing that reduces the engine RPM to propeller RPM to increase the amount of torque produced by the prop. (ex: engine RPM of the turbine is 35,000 - while the prop spins at only 1700 RPM!)

Gearboxes are most commonly used on turboprops, but some piston engines, like Rotax, use reduction gearboxes.

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