Axles and Counter Weights (CW's)
 
You've built your nice new machine, cleared the firing range, checked the location of the dog, got the neighbor to record the event on video and pull the trigger.  The payload goes somewhere but your not sure where and for some reason your machine no longer looks like it did a moment ago.  A closer look reveals that the axle has broken for some reason and the CW has smashed down through the frame and now you have a pile of kindling for the winter coming up that you'll use to figure out the next design.  During which you might keep the following in mind...

A general rule-of-thumb about Counter Weights is that they will weigh 10 times as much at the stall point as they do at rest.  It's like holding some object (maybe a bowling ball?) on the end of a string as compared to dropping the bowling ball and trying to stop it when it reaches the end of the string.

So, you have to design an axle that is strong enough to support not only the beam, counter weight, hanger, sling, payload and pouch, but beef it up to do this while all these things are in motion.  This is something of an over simplification, but the main item is the counter weight.

There are machines out there that look like they're on steroids, but at least they don't break!  Over building a machine may be a time waster and costly, but on the other hand it's not as bad as trashing the machine on its first shot.

Keeping the axle short will dramatically improve its bending strength, so will increasing the diameter of the axle.  Both these answers though have their own problems. (Almost everything about any machine is a compromise, so don't get discouraged.) If you have a short axle, then the frames of the Trebuchet have to be closer at the top, which means the CW may not have clearance to pass through.  If the axle is too long, then it may bend at critical moments.  If the axle has a large diameter, then the size of the hole in the Beam that is passes through (if that is your type) must be larger, which in turn weakens the Beam.

Here's a trick you can try on a possible axle selection.  Support the axle at the same points as it would be when it's placed in your machine, but do it close to the ground, a pair of bricks perhaps would work.  Now load the middle of it with 10 times the weight of your CW.  If it doesn't collapse immediately, unload the weights and examine the axle.  Is it still straight?  If while under the load it bends noticably, then a bit of caution is advised.

By the way, are you going to have bearings on your axle?  If so, what kind and where will you place them?  In most cases, haveing the axle go through a simple hole is good enough.  Yes, there is friction there and that means a loss of energy, but it isn't much compared to the total energies involved.  On the other hand, bearings can be helpful during construction, like this;  Self-aligning bearings allow for a bit a 'slop' in the alignment of the axle with the rotational axis of the bearing housing.

Another possible choice is to place the bearings on the Throwing arm instead of at the ends of the axle.  This means adding mass to the Throwing arm, but now the axle itself doesn't have to turn, which means less mass to move.  Placement, or even the use, of bearings may be dictated by how much room you have to work in.

Counter weights can be made off almost anything, diving weights, barbell weights, clay, pennies, bags of cement, bricks, sheets of steel, etc.  The machine in my yard uses 3 railroad wheels, each weighing 375 pounds each.  Traditionally, CW's were often large wooden boxes filled with rocks and dirt, some weighing over 16 tons!  Enough to toss an enemy Knight, his armor AND his horse back where he came from!  Interesting...

Again, there are as many CW designs as designers.  Some points to keep in mind while designing yours:  Will the CW fit through the framework of your machine during firing?  Does it hit the frame or throwing arm in the cocked position? (If it's against the throwing arm you now have a proped CW!)  Will it clear the ground and/or the frame base during the low point of its swing?  Is the hanger capable of holding that mass? (Remember the 10 times rule-of-thumb for the axle?) Will it spin during fireing?  Can you figure where the center of mass is? (An important number to know while designing the machine.)  Can you make the hanger length adjustable? (Not required, but handy.)  Can you adjust the mass of the CW easily? (Again handy.)  Also keep in mind that the CW is going to be changing angle during firing, will yours dump the CW material when it does?

Water is often asked about.  Sure, it'll work fine.  Keep in mind though that there are many other materials that are denser and so won't take as much room.  Also, can you get enough water at the location where your firing from?  Are you allowed to dump that much water there or are you going to haul it around with you?

Here is one CW design of my own.  The material is primarily plate steel and the whole assembly, including chain (hanger) and locating tube is 1500 pounds.  This is not for everybody, but perhaps it can inspire an idea for your own machine.  The individual plates have a pair of slots near the edges for hand holds.  (I've already replaced this idea with straps, but don't have a graphic of it.)  There is a slot that runs from one edge of each plate to a hole in the center.  The slot is big enough to allow the chain to pass through but not the locating tube.  The bottom of the locating tube has a plate that is larger than the outside diameter of the tube, to hold the plates on.  The chain passes through this plate and is fastened in some manner, perhaps with a larger removable link, a pin, welded, etc. The locating tube protects the chain a bit as well as limiting the amount of movement between plates, although this isn't a biggy in this case.  Further efforts could keep the plates from spinning relative to each other, a nicety that I didn't bother with.

Each plate is light enough to be handled by one person (in decent health, which leaves me out).  The plate is placed by passing the slot past the chain and then dropping it down on the locating tube.  The overall mass of the CW can be changed easily by one plate increments.  The CW can be taken apart for transportation, in several vehicles if needed.  The plates can be made thinner for easier handling, though you'd need more of them to get the same mass but the overall stack height remains the same.

 

To continue the axle selection diatribe....

For many, finding the right axle is a real problem.  Several factors contribute to the problem of selecting the right axle, knowledge of beam bending, material properties and the load on the axle.  None of these are simple facets by themselves, yet a proper axle selection needs to include all three, making the selection that much tougher.

The simple round steel bar is very common, frequently found in the size needed and not terribly expensive, (excluding some of the more exotic materials.) The equations for round bars are easy enough to solve, at least relative to other shapes. Round stock can be easily fitted with bearings, weather they are ball, roller, self-aligning, just a simple bushing or simply a hole drilled through the frame or beam.

However, to simply say that the round bar is made of steel says next to nothing about how well it will resist bending under a load. The variety of available steels ‘off the shelf’ is so large that it actually contributes to the problem of selection.

Other considerations aside, the thing to look for in a steel selection is the Yield Strength.  The topic of Yield Strength leads into a whole host of inter-related topics, which I don’t have room for here.  Suffice to say that when making your selection, find out what the Yield Strength is, the higher the better (roughly speaking).  The common range for Yield Strength for steel ranges from under 45,000 psi to over 200,000 psi. If that information is not readily available, find the AISI number (i.e. AISI 1080) and other manufacturing processes, (i.e. hot rolled, as rolled, quenched, tempered, etc.)  Then you’ll need to find a table showing this code and properties and look up the Yield Strength therein.  The Machinery’s Handbook is a good source for this information.

Other information needed to solve the axle selection problem is how much of a load is the axle going to have to support? It is NOT simply the weight of the components added together. For a Hinged Counter Weight Trebuchet, the force on the axle is a result of some complicated motions in combination with the masses involved.  Maximum loads are assumed to occur during a ‘dry fire’, no projectile at all.  Since this does happen in the real world, it is a good assumption to work from.  Further, the throwing arm is considered to be without mass and the entire structure is thought of as being perfectly rigid. There is no air, so there is no drag coefficient due to friction. This allows the CW to fall unhindered, accelerating freely and gaining thereby a maximum amount of velocity.  Now as the CW stalls, all that kinetic energy is applied suddenly to the axle.  Up until that point, the CW was falling freely but now stops instantly.  This is the model that is used for determining an axle, it provides the conditions that will produce the greatest strain on the axle.  Picking an axle that will survive this, will survive any lesser forces, thus making it safer.

Physics tells us that this instant stop takes an infinite amount of force.  Obviously we are not going to get anything near that, buy why?  The deceleration of the CW actually takes place over a distance (which is how we avoid that infinite force), that distance being determined by how rigid the support structure is.  The frame is going to ‘squish’ a bit, the hanger is going to stretch some and even the CW itself will deform to some extent.  All of this adds up to the fact that the CW does not stop instantly, but over a fairly short distance.  It is this distance, which, combined with the masses involved and the time to cover that distance, gives us the actual load on the axle.  Trying to measure this distance though can prove to be more than a little difficult, things are in motion and the observable time element is so short as to be almost impossible to measure.

Fortunately the empirical method (trial and error) over the years has shown that the rule-of-thumb of CW * 10, works well in axle selection and includes a generous safety factor.

Out of interest, let’s take some numbers and play a little.
Given: A CW with a velocity of 25 ft/s and a mass of 1000 pounds
Determine: What is the force needed to stop the CW in a given distance?

The equation for this is a = (v^2) / (2 * x)  where v is the velocity and x is the distance.
If the distance is 1 foot
a = (25^2) / (2 * 1) = 62.5 ft/s^2

Now take 62.5 and divide by 32 to get G’s and we have 1.95.   Take 1.95 and multiply the CW mass (1000 pounds) by that number and you’ll have the force on the axle.  In this case it nearly doubles the weight to 1950 pounds.

For other values of x (in feet still) we get:

0.5 feet = 3.9 g’s = 3900 pounds
0.1 feet = 19.5 g’s = 19,500 pounds!

So if the CW stops in 1/10 of a foot (1.2 inches) we need an axle that will support nearly 20 times the original weight of the CW!  Now that *10 rule doesn’t look so odd.

There is much more to understand if one chooses to select an axle by rigorous examination of all the factors.  Differential calculus is required as well a thorough understanding of material properties and beams under a load.  Hopefully though, this brief look at axle selection will shed a little light on the topic.  It is not intended to allow one to select an axle, but to impart some idea of the difficulties involved in determining an exact answer.