There seems to be a fair amount of confusion as to how to engineer an ax weapon. The mechanics of an ax weapon can be very simple or very complicated. It really depends on how much energy you want from it and how much weight you can spare on your bot. In this tutorial, we'll examine the basic physics and provide some hints on designing and building an effective ax weapon.

Energy. What is it?

What is energy? Well in simple terms, you can equate it with the ability to do 'work'. For example, when you strike a nail with a hammer, you've done work against the nail to push it into the wood. The work cost you some energy. In this case, the energy came from you raising the hammer, then swinging it down at a high velocity. The energy was stored in the hammer in the form of velocity, or more properly 'kinetic energy'. When the hammer hit the nail, the hammer stops suddenly. The kinetic energy doesn't just go away. Instead, it gets transferred into the nail in the form of work, the work being spent by the nail forcing its way through the wood. This illustrates an important concept of physics that 'energy must be conserved'. This means that the total energy placed into a system (for example, the kinetic energy of our hammer) has to equal the total energy taken out. It's ok to take it out a little at time if we choose or we can use it up in one big bang. (For example, this might happen when the hammer skips off the nail due to a bad swing. Some of the energy gets put into the nail, the rest goes into putting a pucker mark on your nice woodwork or smashing your thumb.)

So let's take a closer look at 'work'. It turns out that work is measured in the same units as energy. This should be no surprise given the conservation of energy rule. In the english system of measurement, energy/work is often measured in lb-ft (pound feet). Using the metric system, it's measured as Nm (Newton-meters). A lb-ft is the amount of energy required to push an object with one pound of force over a distance of one foot. Likewise, a Nm is the energy/work required to push an object with a force of one Newton over a distance of one meter. So you may have guessed at this point the we can find the energy of a system by multiplying the force exerted against on object by the distance traveled. This is in fact the case. There are other ways to find the energy in a system (such as the energy in a moving object), but they all can be derived from the definition given here.

Finally, there is one other type of energy to know about and it's called 'potential energy'. This is energy that is stored, waiting to be released. For example, a stretched rubber band has potential energy. So does a compressed spring or more importantly to us, a bottle of compressed gas. Our job in designing an ax weapon is to find ways to store potential energy and use it to give our ax as much kinetic energy as possible when we need it. The more energy our ax gains, the more damage it can cause.

Energy is your friend and your opponent's worst nightmare

The ax weapon is second only to the spinner style bot in terms of energy it can deliver against an opponent. However, the ax weapon isn't subject to gyroscopic effects which can be very significant and can make driving a spinner bot a challenge. The ax weapon is also friendlier to the bot's internal workings since most of the impact shock is absorbed by the ax head itself rather than in the chassis as is the case with the spinner. (This doesn't mean an ax is free of side effects. Oh contrair. It has some nasty ones which will be discussed in part 2.)

So why do we care about the energy delivered? To answer this, think about our hammer example. If we swing our hammer with little baby taps, we won't drive the nail very effectively. The energy is too low to do anything with. (The hammer has too little kinetic energy in each swing.) If we take a manly swing at the nail, we could drive it with one whack. (Lots of kinetic energy this time.) Even though the total energy exerted tapping the nail with a hundred little taps might equal the energy spent in one big whack, the results are very different. The difference comes from the ability of the nail to absorb small amounts of energy. To make the nail do something, we have to exceed the level that it can absorb. Now think about what we are trying to do with our ax. We want to drive our opponent through the floor of the arena. But he likely has armor which can absorb energy without damage. We need to exceed our opponent's ability to absorb energy. When we do this, something will give and damage will be inflicted. Therefore, the more energy we can deliver, the better our chances of reducing our opponent's bot to metallic mush.

The physics of the ax weapon

If you're having trouble grasping the ideas of kinetic and potential energy and concept of work and energy being the same thing, then you may want to find someone to help explain these ideas more thoroughly before going further. The physics is going to get tougher from this point on.

The next thing we need to know about is a concept called the (rotational or angular) "moment of inertia". This is a measure of a spinning body's resistance to changes in its spinning speed, or more properly, its "angular velocity". A body with a high moment of inertia resists changes in its angular velocity more than body with a low moment of inertia. So how does this effect our ax? Well our ax is a spinning body. It has mass. Therefore, it has a moment of inertia. (Think about it. Our ax is spinning in a circle whose center is the axle used to hold the ax handle.) So why do we care about the moment of inertia? Well we can get along without knowing it, but it can be very useful. For one thing, if we know the moment of inertia, we can predict the speed the ax will travel. Once we know the speed, we can predict the amount of energy being delivered into our opponent. Besides, it's actually easy to measure.

Before we can measure the moment of inertia, we need to understand what "mass" is. Mass is a collection of matter. In this case, our ax head. In the english system of measurement, we often measure mass using the "pound". For example, you might hear someone say that his ax weighs 8 pounds. Unfortunately, a 'pound' is a unit of force, not a unit of mass. The proper unit is called a "slug". (I have no idea who came up with that name.) Our bathroom scale is measuring the Earth's gravitational effect exerted against the mass expressed as pounds of force. We use pounds so much, we tend to equate pounds of force with particular quantity of mass. This is a bad thing since an ax on the moon weighs 1/6 of what it does here on Earth even though it's the same amount of mass. The important point here is that weight depends on the local gravity. Mass doesn't. If this seems a little cloudy, don't worry about it. Just know that to convert pounds to slugs, simply divide the weight by 32.2 (which is the gravitational accelleration here on Earth) or more formally:

Those of you on the metric system don't have to worry about this. Your scale already does this conversion. (But it only works when the scale is here on Earth.) A kilogram is a proper measure of mass.

Now we can define the equation used to determine the moment of inertia of a swinging ax. The symbol normally used for the moment of inertia is usually 'I'. So here's the equation (metric units are in parenthesis):

As you can see, the moment of inertia is easy once we get things into the proper units. But I should point out that this equation only works when the ax head is shaped more or less like a cylinder. If you're swinging a broad ax, then all bets are off. This equation also won't work for finding the moment of inertia of the ax handle which needs to be included to get reasonably accurate results from our predictions. The moment of inertia for the ax handle can be found using this equation:

To get the total moment of inertia, simply add the results of each equation together. As you can see, there's not a lot of effort required to find the moment of inertia for something as simple as an ax head on a stick. Now let's do a practical example. Suppose we want to find the moment of inertia of an 8 lb. ax head attached to an arm (handle) that weighs 2 lbs.. The arm is 2.5 feet long. This would be typical of an ax used in the heavyweight or super-heavyweight class of bot.

Step 1: Convert pounds to slugs.
mass of ax head = 8 lbs * 0.031 = 0.248 slugs,
mass of handle = 2 lbs * 0.031 = 0.062 slugs.

(Remember that if you're using metric units, you don't need to do this step.)

Step 2: Find the moment of inertia of the ax head.
I_ax = 2.5 * 2.5 * 0.248 = 1.55 ft^2-slugs

Step 3: Find the moment of the inertia of the arm.
I_arm = 0.333 * 0.062 * 2.5 * 2.5 = 0.125 ft^2-slugs

Step 4: Find the total moment of inertia
I_total = I_ax + I_arm = 1.55 + 0.125 = 1.675 ft^2-slugs

That's all there is to it. We now know the moment of inertia of our ax along with its handle. Before we can use it, we now need to know how much energy we can put into our ax. This brings us to an introduction to the wonderful world of pneumatics.

Pneumatics and the magic of air pressure

This is another area that causes a lot of people a good deal of confusion. But it's actually rather straight forward. All you need to know is the amount of air pressure you intend to apply, the diameter of the pneumatic cylinder, and the distance the piston travels. Once you have this information, you can easily find the energy output of the cylinder.

As before, we need to define a consistent set of units to work with in order for our equations to produce meaningful results. For air pressure, the common unit is 'PSI' which stands for pounds per square inch or lbs/in^2. Metric units would be Newtons per square meter or N/m^2. We can understand this better using an example. Let's say that we have an air pressure of 150 PSI (a typical air pressure used in a bot). This means that 150 of force will be exerted for every square inch of surface area on the face of the piston in our pneumatic cylinder. So if the piston has 10 square inches of surface area, the force will be 10 * 150 or 1500 pounds. As you can see, we want as much surface area as we can get (within reason) since surface area directly effects how much force our cylinder can produce. So how do we find the the surface area? Some times it will be given in the specs of the cylinder. More commonly, the cylinder's diameter will be listed on the specs. Surface area can be found using the equation:

Note: You may see air pressure listed in 'bars' instead of PSI. This is no big deal. To convert 'bars' to 'PSI', just multiply by 14.7. One bar is 14.7 PSI. Where did the bar come from? It came from meteorology. One bar is the air pressure at sea level which happens to be 14.7 PSI. You may see the barometric pressure listed in millibars on a weather report. There are 1000 millibars in a bar. So a barometric pressure of 980 millibars is 0.98 bars or 0.98 * 14.7 = 14.4 PSI.

Ok, so we now know how the find the surface area of the piston in the pneumatic cylinder. That surface area times air pressure is the force exerted against the piston. How do we find the energy output? Simple. All we do is multiply the force times the distance traveled by the piston when the ax moves. Remember from our discussion earlier that work is expressed as a force that moves something over distance. In this case, the something that moves is the piston inside the cylinder. So the work (energy) delivered by the cylinder is then force * distance traveled. More formally it's:

I can't tell you how to find the distance traveled because this is determined by the linkage system you use to couple the cylinder to the ax and it can vary quite a bit. But a typical distance might be 6 inches or 0.5 feet. Let's say the cylinder bore is 3 inches and the piston really does travel 0.5 feet. How much energy can we expect?

Step 1: Find the area of the piston
A = 0.785 * 3 * 3 = 7.065 square inches

Step 2: Determine the air pressure
p = 150 psi (we specify this. 150 psi is a typical limit for a cylinder)

Step 3: Find the force on the piston
F = A * p = 7.065 * 150 = 1059.8 pounds

Step 4: Multiply by the distance traveled E = F * d = 1059.8 * 0.5 = 529.9 lb-ft

That's it. We now know the energy produced by our cylinder. If we realize that some of this energy is lost to friction in the cylinder, we can arbitrarily round this result to 500 lb-ft. So how much energy is this in terms we can understand? Well, as a comparison, it's roughly the amount of energy a full-grown man of average build can produce swinging a sledge hammer. (An athletic individual could easily exceed this, so your mileage can vary considerably.)

Finding the speed of the ax

If you used the equations for the moment of inertia and energy delivered from the pneumatic cylinder correctly, then you are now ready to get an estimate of how fast your ax is going to swing. We are going to do this by recognizing that energy is energy. It may take different forms, but it is still energy. We are also going to take advantage of the fact that energy must be conserved. So if we take 500 lb-ft of energy from our cylinder and dump it into our ax in the form of kinetic energy, we can predict how fast the ax is traveling. We just need an equation that describes how much kinetic energy there is in a moving object and set it equal to the energy delivered by the cylinder. Rotational (angular) kinetic energy can be expressed by the equation:

In case you didn't know, one complete turn is 6.28 radians or in other words, 360 degrees per second is 6.28 radians per second. Now look at the equation. You see that big 'E' in there? It's the same 'E' that we found earlier when we found how much energy our cylinder can give us. If we plug in 'E' from the previous equation and work our kinetic energy equation backwards, we can solve for the angular velocity:

We're almost done. Just one more equation and we'll know how fast our ax is traveling. All we need to do is solve for tangential velocity of an object traveling in a circle. (all right, I used a new term without defining it. The 'tangential velocity' is the speed an object would travel if you straightened out the circular path into a straight line. Think about driving your car in a big circle. The speedometer would indicate its tangential velocity.) This equation is a toughy:

Ok. So it wasn't tough. If you plowed through all this and managed to keep your sanity, you now know how to find the theoretical speed of your ax. I say theoretical because some of the energy produced by the cylinder never makes it to the ax. Part of it is lost to friction inside the cylinder. Some more is lost because the piston has mass and it's moving along with all the linkages that connect the piston to the ax. (Whenever you put something in motion, you give it kinetic energy. That energy has to come from somewhere.) This gives us motivation to keep all the moving parts (except the ax head) as light as possible. Finally, it also assumes that we can deliver the full air pressure to the cylinder which is unlikely. This is because there is a small pressure loss when we pass air through tubing and valves. To get a more realistic value, you can apply a fudge factor and multiply the resulting tangential velocity by 0.8. This should get you pretty close to reality.

Summary