Deriving formula for MA of differential windlass

[TheKShaugh]

Hi,

I am trying to find an equation that would give the mechanical advantage of this system:

I am fairly new to this kind of analysis, but my understanding is that to determine the MA I need to consider that:

$$W_{in} = W_{out}$$

$$F_{in} d_{in} = F_{out} d_{out}$$

$$F_{out} / F_{in} = d_{in} / d_{out} = MA$$

So all I need to do to determine the mechanical advantage is to find the ratio of the distances of the applied and effort forces.

d_in:

The crank and large drum act as a wheel and axle, so the distance the large drum travels per full revolution of the crank shaft is:

$$\frac{r_{large \, drum}}{r_{crank}}$$

d_out

The movable pulley is going to rise a distance that equals one half the difference in circumferences between the large and small drum:

$$\pi(r_{large \, drum} - r_{small \, drum})$$

Which give a rough mechanical advantage (ignoring the levers created by the crank) of:

$$2 \times \frac{r_{large \, drum}}{\pi r_{crank}(r_{large \, drum} - r_{small \, drum})}$$

I have some concerns though, the main one being that this is different from the only equation I could find online, which is:

$$\frac{2 \, r_{crank}}{r_{large \, drum} - r_{small \, drum}}$$

Is my analysis correct or have I missed something? Also, if anyone could tell me why the pulley moves only one half the difference between the two radii and not the full difference between the two I would really appreciate it.

Thanks!

 

[TheKShaugh]

I've thought about it some more and I think I've got it. This complex machine can be broken up into three simple machines: the movable pulley, the wheel and axle, and the rope wrapped around the two drums.

The movable pulley has a MA of 2.

The wheel and axle has an MA of $$\frac{d_1}{d_2} = \frac{r_{crankshaft}}{r_{large \, drum}}$$

The rope wrapped around the two drums has an MA of $$\frac{d_1}{d_2} = \frac{r_{large \, drum}}{r_{large \, drum} - r_{small \, drum}}$$

All these mechanical advantage are multiplied to give the MA listed at the end of my post.

It that it?

 

[Dr.D]

Looks like you got two different results; which are you going with?

I suggest that you look at this by the principle of virtual work (if you are familiar with that approach).

 

[TheKShaugh]

Looks like you got two different results; which are you going with?

I suggest that you look at this by the principle of virtual work (if you are familiar with that approach).

Well, I found that equation in a mechanical engineering reference book, and it makes sense, so I'm inclined in that direction. Also, I'm not sure if my d_in equation makes sense, it's a unitless ratio of distances and the MA if the force is applied to the drum, and not an actual distance, isn't it? It seems to be simpler to just break it down into simple machines and take the product. I'll look into virtual work, thanks for that.

 

[pmacias]

Hi there,

Your analysis seems to be correct. The mechanical advantage (MA) of a differential windlass can be calculated using the equation you provided:

MA = d_in / d_out = (r_large drum / r_crank) / (pi(r_large drum - r_small drum))

The reason why the pulley only moves one half the difference between the two radii is because of the way the system is designed. The movable pulley is attached to the rope, which is wrapped around both the large and small drums. As the large drum rotates, the rope will move along with it, causing the pulley to rise. However, since the rope is also wrapped around the small drum, the movement of the pulley will be half the distance between the two drums. This is due to the principle of conservation of energy, where the work done by the effort force (F_in) must equal the work done by the load (F_out). In this case, the effort force is applied over a longer distance (d_in) compared to the load moving a shorter distance (d_out), resulting in a mechanical advantage.

As for the difference between the equation you found online and the one you derived, it could be due to different assumptions or simplifications made in the calculations. It's important to keep in mind that the equation you derived is a rough approximation and may not take into account all the factors and variables in a real-world scenario. Further analysis and experimentation may be needed to obtain a more accurate equation.

I hope this helps clarify your understanding of the mechanical advantage of a differential windlass. Keep up the good work with your analysis and don't hesitate to reach out if you have any further questions. Best of luck with your research!