# Homework question about gears.

Asked by Mariah (17358 ) April 3rd, 2012

This question has stumped my TA!

The setup is this: I have a line or “train” of four interlocking gears (sort of like this, except there’s 4) of different sizes. The radius of each gear is known, the number of teeth on each gear is known, and the angular velocity of the “input” gear is known. The efficiency of each gear interaction is 93%. I am to find the torque available on the output gear.

We have learned some formulas for finding torques in gear trains, but they are all expressed as ratios, e.g. (torque in)/(torque out) = (number of teeth on input gear)/(number of teeth on output gear). So without knowing the torque on any of the other gears, I don’t know how to find the torque on the output gear. I’ve researched other relationships between torque and gear properties and haven’t found anything useful for the information I have.

Any jelly that can help gets my eternal gratitude.

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I think you’re missing some information. In order to fin the output torque you’ll need some information about the forces on the input. The angular velocity on the input gear is not enough.

Given the angular velocity on the input you could figure out the velocity of the output – but that is not the question. And you’re (torque in)/(torque out) equation with a little addition for the efficiency at each interface could give you the output, but we still need the input torque.

Do you have the actual wording of the problem? Post that maybe there’s some unobvious reference to input force.

dabbler (11575 )

@dabbler My thoughts exactly. I wouldn’t be altogether surprised if this professor had assigned an impossible problem. I don’t think I missed anything, but here’s the exact wording (parenthesized comments are mine):

“The following parameters apply to the gear train shown below:”
(there is then a picture similar to the one I linked above, but with four gears labelled A, B, C, D)
“Na = 25; Nb = 10; Nc = 30; Nd = 15” (N is the number of teeth on the gear)
“Ra = 5; Rb = 2; Rc = 6; Rd = 3” (radii)
“Assume the efficiency for all gear interactions is 93%. The input gear has an angular velocity of +4.75 rad/s. For the gear train, determine the torque available at the output gear.”

Thanks for taking the time to help.

Mariah (17358 )

Can I glean any information from the fact that the the angular velocity is constant (it’s not accelerating)?

Mariah (17358 )

Nah, that’s a badly written question. The author either wanted to ask for something else like the output velocity, or they left something out.

If the output velocity was the intended question then it’s a trick question because the efficiency doesn’t matter. And the radii, or the number of teeth, are superflous since presumably the teeth are the same size and the ratios of teeth will be the same as the ratios of circumference.

And output torque… where? On the central shaft of the last gear? Force available from a tooth on the last gear (then you’ll need the radius of the last gear)? We can’t tell the torque anyway but this question seems sloppy.

dabbler (11575 )

The whole course is sloppy. Won’t be studying under this professor again. Thanks.

One question of yours I think I can answer though, I think the output torque is on the central shaft. We’re interested in knowing with how much torque is available for turning a wheel if we attach a motor to the input gear, for instance.

Mariah (17358 )

Hey, ..You could express the output torque as a fraction of input torque, whatever that is.
T(output)= T(input) * 10/25 * 30/10 * 15/30 * (.93)^3 <= that’s .93*.93*.93

dabbler (11575 )

If I use the letters A-D to mean the radii of the four gears, then you could set it up in stages:

torque of gear A / torque of gear D = (A/B * 0.93) * (B/C * 0.93) * (C/D * 0.93)
= A/B * B/C * C/D * 0.93^3
= A/D * 0.93^3

A neat result of this is that the radii of the intermittent gears don’t matter at all.

PhiNotPi (9827 )

Yeah that’s been my thought process… it is easy to find information about the ratios of the torques from gear to gear, but then of course I need to know at least one other torque, and none are given. I will talk to the professor.

Mariah (17358 )

Saw the professor today. He read over the problem, stared at it for a moment, then said “oops.” He said the input torque should have been provided. Gah.

Mariah (17358 )

or