What is the Mechanical Advantage of this lever? (for pulling tree stumps)

[Christopher M Hack]

I am interested in building a lever system for pulling up small tree stumps. I have seen this demonstrated on youtube videos. For example...

However, I am more interested in calculating the various forces and mechanical advantage for such a system so I can figure out the best lever length, chain strength, and pulling angle. I have yet to find an example calculation for such a system and I'm not even certain which class the lever is. I would assume either class 1 or class 2. Anyway, here is a crude drawing of the system. Any help with identifying class and formula for calculating MA would be greatly appreciated. I would like to know what happens as the lever moves toward the force of pull as it first straightens up then collapsed toward the ground again which also alters the angles during the progress of the pull.

 

[stockzahn]

Momentum 1: $M_g=F_g l sin\theta$
Momentum 2: $M_p=F_p l cos\theta$

Lever: $\frac{cos\theta}{sin\theta} = cot\theta >\frac{F_g}{F_p}$ ... to pull out the stump

 

[sophiecentaur]

formula for calculating MA would be greatly appreciated.

Until you know the details of the construction, you cannot predict what will be the 'Mechanical Advantage' of any machine. MA is the actual Ratio of load force to effort force. Other forces such as friction can reduce the MA and I can see a beam there of which the mass may not be negligible. That beam can be referred to as Dead Weight which may or may to be relevant. What you can calculate from the geometry of the machine is the Velocity Ratio (known also by other names), which is the ratio of distance moved by the effort divided by the the distance moved by the load. This involves either some calculus or just a choice of a 'small' movement or sometimes just the lengths of levers or diameters of wheels. You can see that one angle increases as the other decreases so simple lever lengths won't do the job.
It isn't necessary to actually name the class of lever to work out. the VR The fulcrum is on the ground and that's the best place to take Moments about. There are two angles involved (Note @stockzahn ) because the rope doesn't emerge in a horizontal direction or may not be symmetrical. If we call the angle of the beam to the vertical as θ_stump, there will be an anticlockwise moment of length_beamWeight_stump Sin(θ_stump)
If we call the angle between the rope and the beam as θ_rope the clockwise Moment will be length_beam Tension Sin(θ_rope) then we can equate those two and solve the equation for the unknown Tension.
It may not be convenient but the maximum moment from the rope is when θ_rope is 90° which requires it to be higher than horizontal. A second beam can help you here, with a 'lifting triangle', rather than just a simple jib. Look at the design of many cranes and derricks to see what I mean.
The maximum force on the tree will be obtained with the beam just off vertical. But you would not get much movement at that angle so you need to tilt the beam a bit (enough to allow the tree root to be lifted out of the hole.
The weight of the beam also adds an anticlockwise moment (1/2 length_beam weight_beam Sin(θ_stump. Using that could give you an idea of the actual MA. Two beams in a triangle could actually balance each other and remove that moment.

 

[A.T.]

Momentum 1: $M_g=F_g l sin\theta$
Momentum 2: $M_p=F_p l cos\theta$

Lever: $\frac{cos\theta}{sin\theta} = cot\theta >\frac{F_g}{F_p}$ ... to pull out the stump

Note that this assumes the strings are at 90°.

 

[stockzahn]

Note that this assumes the strings are at 90°.

True, but it vividly shows the idea. Start simple and evolve...

 

[sophiecentaur]

True, but it vividly shows the idea. Start simple and evolve...

But isn't a practical scenario, is it? How can you have a horizontal rope if the beam is more than chest height? I have had experience of this sort of thing when raising and lowering a 10m sailboat mast. The rope would probably need to be attached to a ground picket if any significant lifting force is needed. This is why I was suggesting the use of two beams and not just one. It's a win win design - pulling at near 90 degrees for the derrick and almost zero degrees for the lifting section.
PS Momentum is Mass times velocity. The Moment of a Force is Force times perpendicular distance.

 

[dirtybagg]

Three years ago several small elm tree stumps rendered my square body 4x4 F250 un-worthy as a puller or anchor. The truck's bald tires were not optimal, but momentum is momentum, and you can purchase a whole lot of rigging for the cost of F-250 tires.

I sold the truck and have pulled 30-ish stumps/roots using other nearby trees as anchors (trees are really strong anchors for horizontal pull near the ground (use a basket hitched strap above tapered zone to keep anchor tree healthy). Remember, the primary purpose of the A-frame/column is to exchange horizontal pull for vertical pull, M.A. is an incidental benefit.

My first column was an 8ft tall scissor style A-frame from pressure treated 4x4 lumber hinged on a 3/8in or 1/2in? through bolt, and it was awkward to rig (watch your teeth on swinging chain). My original was trimmed down to 4-ish ft tall for easier rigging. After adding pulley M.A. to the rope winch end, I'm twisting 4x4's - the scissor design is compact and convenient, but compounds the twisting moment.

I have more stumps to pull, and I did some math to entertain adding more M.A. and building a stronger A-frame. I sketched a free body diagram with vectors summing to zero (static situation), to estimate the forces supported by my choker chain, a-frame, rigging, etc. Like others on this forum, I also calculated that mechanical advantage increases as the a-frame/column approaches the vertical position. 45deg = 1.4:1, 80deg = 5.7:1 (vertical stump pull force : horizontal rope pull force). The column length does not change Mechanical Advantage, but a taller column facilitates more ground clearance & horizontal reach, to achieve a more vertical position with more M.A.

Tuning the a-frame's vertical angle is a simple way to optimize for stroke length or pull force. Column strength decreases with length, and knowing my my A-frame' strength would be complicated. I do have hard specs on my choker chain and rigging. With 4800lbf theoretical pull from my rope winch, and 80deg column angle, I'm potentially exceeding the 26,400 lb minimum breaking strength (MBS) of my 3/8in G70 choker chain. For smaller roots, I use a 5/16 G70 with 18,000lb MBS.

Perhaps my A-frame is strong enough...is it safer to break a timber A-frame or chain/strap/shackle?...it's probably a better policy to not put rigging at risk of breaking. If my winch is rated like an automotive jack, it's probably 30% to 50% over-rated for average conditions in practical application. Also, I'm losing 10 to 20% to friction through each pulley. I'm guessing my 4800 lbf theoretical horizontal pull is closer to 2400lbf, and my A-frame imparts a maximum of 4:1 under peak loading, and my chain sees ~9600lbf - that's not bad for a kit which fits in a backpack/on a shoulder, and requires no battery/fuel tank. I'd love to play with a dynamometer.

Happy Geeking!

 

[sophiecentaur]

Happy Geeking!

A good fun / highly practical set of exercises but "Geeking" would require details of all the gear involved. For circumstances where things are 'marginal' then some calculations could take you from 'not working' to 'working like a dream' and that would be where accurate sketches / diagrams ( even) would come in handy.
I like the idea of 'taking a run at it' on bald tyres. There is a brilliant YouTube video I which the stump turns into a missile when a long rope has stretched a lot and its energy turns up as Kinetic Energy of the stump. Nylon rope can be evil in some circumstances and other fibres can be a lot safer. But that's something an experienced worker would already know.

 

[jrmichler]

Some things to keep mind when pulling stumps:

If a rope or chain fails in tension, it can go flying at high speed.
If the A-frame in Post #1 fails in compression, the pieces are pushed down into the ground.
A slow steady pull from a winch needs less force than a jerk using truck momentum.