Mechanical advantage question.

July 26, 2021, 03:37 PM

OK a rainy day question that may result in me adapting something into a project.

How much mechanical advantage would one gain by using a simple crank handle on an axle to pull an object with a rope.

Say a one inch axle turned by a 10 inch crank.
Using a rope that winds up on an axle to lift an object.
(Multiple layers of wraps of the rope on the axle can be ignored)

Rope is 1/2 inch so the center to center diameter of a single wrap around a 1 inch axle would be 1.5 inches. So the rope take up is almost 4 inches per turn. A 10 inch long crank would would travel about 62” to make one complete turn.

So it would seem that the simple travel ratio would be about 15 to 1.

But does that equate to a 15 to 1 mechanical advantage for lifting force? (minus friction and inefficiencies etc)

Could a 10 pound force on the crank produce 150 pounds (roughly) of force on the rope? again... (minus friction and inefficiencies etc)

Here is a photo of a simple crank on an axle for an example of what I'm referring to.

July 26, 2021, 03:54 PM

sigmonkey

It is simple torque multiplication.

You are generating nearly 1 pound feet of torque with that lever, measuring from the centerline of the rotating shaft at the work to the centerline of the handle shaft of the crank.

The increasing the amount of rope on subsequent wraps will reduce the multiplication requiring more effort as you go, but yes, you are pretty close to your numbers on effort to do the work.

That is why a small drive gear turning a larger output gear with longer lever can produce more work, but at the cost of your arm moving a greater distance. (so the advantage is work over time) that produces greater ability to accomplish heavy work with less effort.

July 26, 2021, 03:56 PM

YellowJacket

Ideal mechanical advantage of a wheel and crank is R/r.

So if you have a 1" axle and a 10" crank handle, you're at 10/1, or 10. As you wind more rope onto the axle, your mechanical advantage is decreasing because the smaller radius is increasing. A couple of winds in and you're at 10/2, then 10/3, etc.

July 26, 2021, 04:01 PM

cne32507

quote:
Originally posted by YellowJacket:
Ideal mechanical advantage of a wheel and crank is R/r.

So if you have a 1" axle and a 10" crank handle, you're at 10/1, or 10. As you wind more rope onto the axle, your mechanical advantage is decreasing because the smaller radius is increasing. A couple of winds in and you're at 10/2, then 10/3, etc.

That's what I remember from Engineering Mechanics 1.

July 26, 2021, 04:31 PM

cparktd

10 to 1. Handle to axle. Seems right enough, simple lever 10 to 1 advantage.

But the traveled distance of the handle vs axle/rope is 15 to one.

July 26, 2021, 04:40 PM

honestlou

Overton's has trailer winches starting at $30:

https://www.overtons.com/overt...-winches-74898G.html

And they come ready to mount, including the strap.

July 26, 2021, 04:42 PM

229DAK

You place a pulley on the end you are pulling, loop around and back, and increase your mechanical advantage by a factor of 2.

July 26, 2021, 08:18 PM

cparktd

quote:
Originally posted by honestlou:
Overton's has trailer winches starting at $30:

Yea, got one of those, except it's the upgraded two speed version.

July 26, 2021, 08:18 PM

cparktd

quote:
Originally posted by 229DAK:
You place a pulley on the end you are pulling, loop around and back, and increase your mechanical advantage by a factor of 2.

Yep, been doing that, and making an extra loop in the rope for effectively the same thing on various things for decades... comes in pretty handy sometimes.

July 26, 2021, 09:07 PM

cparktd

Thanks guys...

Just to be clear, I'm wasn't looking for ideas for a solution to a problem. I'm just into another one of my "what if" wild goose chases that I might pursue.

July 26, 2021, 09:19 PM

maladat

All the "10 to 1" answers are wrong - the reason being that the 10" crank has a 10" RADIUS of travel and the 1" axle has a 1" DIAMETER of travel.

The crank handle is 10" from the axis of rotation. The surface of the axle is 1/2" from the axis of rotation.

At the surface of the axle, there is a 20-to-1 mechanical advantage.

This will be somewhat reduced due to the thickness of the rope, but probably not exactly what using the centerline of the 1/2" diameter rope would indicate.

Ignoring deformation of the rope, the centerline on the rope is an additional 1/4" from the axis of rotation - giving a force at 3/4" from the axis of rotation. This would yield a mechanical advantage of 10 / 3/4 = 13 1/3 to 1.

However, the rope will flatten somewhat against the axle and depending on the elasticity of the rope, there may be more tension in the rope closer to the axle or further from the axle than the centerline, so even the centerline of the flattened rope isn't necessarily the right value to use.

July 26, 2021, 09:27 PM

cparktd

Very interesting! I missed that... Thanks for that maladat.

July 26, 2021, 10:11 PM

YellowJacket

quote:
Originally posted by maladat:
All the "10 to 1" answers are wrong - the reason being that the 10" crank has a 10" RADIUS of travel and the 1" axle has a 1" DIAMETER of travel.

The crank handle is 10" from the axis of rotation. The surface of the axle is 1/2" from the axis of rotation.

At the surface of the axle, there is a 20-to-1 mechanical advantage.

This will be somewhat reduced due to the thickness of the rope, but probably not exactly what using the centerline of the 1/2" diameter rope would indicate.

Ignoring deformation of the rope, the centerline on the rope is an additional 1/4" from the axis of rotation - giving a force at 3/4" from the axis of rotation. This would yield a mechanical advantage of 10 / 3/4 = 13 1/3 to 1.

However, the rope will flatten somewhat against the axle and depending on the elasticity of the rope, there may be more tension in the rope closer to the axle or further from the axle than the centerline, so even the centerline of the flattened rope isn't necessarily the right value to use.

Good call on the radius vs diameter.

In an easier math/physics class they would probably give you an ideal situation and tell you have some sort of levelwind mechanism that lays the rope beside itself on an extruded axle, thus leaving the ratio constant throughout the winding. Kind of like assuming a frictionless surface when doing an acceleration problem.