Torque Transfer and Gear Ratios: How to Calculate the Work Rate of a Gear Drive

[LearningCurve]

I am trying to determine both a gear ratio AND rpm/ft.lbf of a driver gear.

My selected gear ratio is 9:1, I am using a 1HP motor (driver) @ 400RPM, making 13.13ft.lbf.
The driven side must rotate at 3600RPM.

I need to proof the following statement:
9:1 gr
400RPM*9=3600RPM (absolute)
1HP@400RPM=13.13ft.lbf (known)
13.13ft.lbf*9= {unknown}


Again, this is a search to see how the torque transfers through a gear drive (preferably, a simple two-gear setup, but will entertain all setups as long as the gear ratios are given specifically).

Object:
prove the {unknown} value, and specify gear ratio.


Thanks for any help :)

 

[LearningCurve]

Anyone?

 

[DickL]

If you can neglect friction and similar losses, then energy and power is conserved through the gear unit. That said, look at your 2nd equation, for power and torque. Can you now figure out the torque on the other side of the gear unit.

Re-think that 3rd equation.

 

[LearningCurve]

So it's 13.13ft.lbf* 9, or 13.13/9?

In other words, if the gr is set at 9:1, and RPM is* 9, then does the torque mirror that, or will the ft.lbf be reduced by 1/9?

 

[DickL]

Power is torque times rotational speed: P = k*T*rotational speed. The value of k is dependent on the units being used for power, torque and rotational speed. Using the mks system of units: power would be Watts (W), Torque in Newton-meters (Nm), and rotational speed in radian/sec (rad/s) and k would equal 1. I'll let you find out the value of k when using horsepower, ft-lbf, and rpm (look it up in a reference, do not use the data in your problem statement to back solve). Oh, and I'm also letting you figure out if you multiply of divide by the gear ratio.

I also recommend you re-check the values in your problem statement, I suspect something is amiss.

 

[LearningCurve]

I have done the math to my understanding, and hp & torque follow this example:

T=(hp x 5252)/rpm
so, a 1hp motor turning at 400rpm, puts out 13.13ft.lbf and at 9x gr would make 3600rpm.

Please tell me I missed something?

 

[DickL]

I've always used a conversion of 1 hp = 33,000 ft-lb/min = 550 ft-lb/sec = 746 Watts. There are a number of different horsepower definitions. You might check this reference http://en.wikipedia.org/wiki/Horsepower.

 

[LearningCurve]

I am certainly grateful for your additional input, and as heavy as my workload is irl, research is a luxury unfortunately for me at this time :(

We seem to be going the same place mathematically, but in the simplest and most direct terms, I m still at a loss to determine whether the torque divides or multiplies at the 9:1 gr.

Please be direct, as time is the essence on my end.

 

[DickL]

Okay, you have an input of 1 HP at 400 rpm which gives an input torque of 825 ft-lbs. Assuming a loss-less gear unit with a 9:1 step up ratio, that is an output of 1 HP at 3600 rpm and a torque of 9.17 ft-lbs.

A real gear unit will have losses. Efficiencies for typical units can be estimated in the 96% - 98%, which can be used to reduce the useful power and torque output.

 

[SteamKing]

The correct value of torque to produce 1 hp @ 400 rpm is 13.13 lb-ft.

HP = T * N / 5252 where T is in lb-ft and N is in RPM

Increasing speed from 400 RPM to 3600 RPM using a gear ratio of 9:1 means the output torque on the 3600 RPM shaft will be 13.13 / 9 = 1.46 lb-ft. Assuming no loss in the gears, power in = power out.

If T = 825 lb-ft and N = 400 RPM, P = 62.83 hp.

 

[DickL]

Conventionally rpm stands for revolutions per minute, not radians per minute. If your are using radians, then the usage should be rad/min. Using all caps RPM still means revolutions per minute.

HP = T * rpm / 33,000 or HP = T * rad/min / 5252

Outside of a few rather technical areas, rad/min (or rad/sec) is not a common unit for angular velocity. Revolutions per minute is the common unit. Your use of rpm to stand for rad/min has caused confusion. Even within those areas that use radians, rad/sec is more common than rad/min.

 

[SteamKing]

I don't know how common rpm is for rad/min, but I know that 5252 = 33000 / 2pi rad/rev

If you check this link: http://en.wikipedia.org/wiki/Horsepower

you will see that N is measured in revs. per minute or RPM

I stand by my calculation that 1 hp = 13.13 lb-ft of torque @ 400 RPM.

I don't know where the torque of 825 lb-ft came from, but it is not correct.

 

[DickL]

1 HP = 33,000 ft-lb/min = T * 400 rpm (revolutions /min)
T = 33,000 ft-lb-rev/min / 400 rpm = 82.5 ft-lb

If 400 rpm is to be taken as 400 rad/min, then
1 HP = 33,000 ft-lb-rad/(2pi-min) = 5252 ft-lb-rad/min = T * 400 rad/min
T = 5252 ft-lb-rad/min / 400 rad/min = 13.13 lb-ft

We often think of units such as revolutions, radians degrees and cycles as being dimensionless, but they do have specific meanings and need to be included in keeping units straight.

 

[SteamKing]

That's just it. The definition of 1 hp is the work of 33000 ft-lbs expended in 1 minute or 550 lb-ft in one second. Radians are unitless, since a radian represents the ratio of a length of arc equal to the radius of a circle to that radius. Revolutions, on the other hand, are considered to be units, although not the usual L, T, or M.

1 Rev = 2 pi rad

The work rate equivalent to 1 horsepower is 33000 ft-lb/min, not 33000 ft-lb-rev/min.

If the shaft speed is 400 rev/min, then the revs are converted to rads by multiplying 400 rev/min by 2 pi radians/rev = 800 pi rad/min.

Taking a torque of 13.13 ft-lbs and multiplying by the angular velocity 800 pi rad / min gives:
Power = 13.13 ft-lbs x 800 pi rad / min = 33000 ft-lbs / min = 1 hp

The radians are by definition unitless and are not carried over to the final work rate result.