In the previous two parts, we examined the math behind behind an ax being swung at great velocity and why there is an upper limit to the energy it can deliver into an opponent. Now we're going to look at a way to get around the limits imposed by the resulting centrifugal forces. We'll also look at why there is still a limit of how hard we can swing an ax without being bothered by centrifugal force.

A question of balance

To get around our centrifugal force problem, let's examine something called a counterweight. A counterweight is just a lump of something that has mass. It has many applications, but for our purposes we want to use it to balance our spinning ax. You have probably seen counterbalances at work on your car. Those little lead weights on your car's wheels are there to make the wheel spin smoothly. It becomes pretty obvious when one is missing when traveling down the highway. The car will have a very noticeable vibration. As we saw in part 2, an 8lb weight can produce a very large 'vibration' which can cause the bot to go airborne or slide wildly across the floor. To stop this from happening, we need to balance the ax about its center of rotation (in other words, balance it about the axle). To figure out how to do this, we need to take another look at what is happening. Remember from the first part that the ax is moving in a circle and is therefore subject to centrifugal force. The direction of the force will always point along a line that starts at the axle (the center of rotation) and passes through the center of gravity of the ax head. So if we place a weight on the opposite side of the axle so that it is in line with the thing we want to counterbalance, and if we select a weight such that its centrifugal force is the same as that of the ax, then the two centrifugal forces will cancel each other and our bot will stay firmly on the floor. With the centrifugal force out of the way, we can crank up the energy in our ax and promote it from 'nasty' to 'devastating'. So, now we know what needs to be done and why. We just need a way to figure out how much weight to use for our counterweight and how far it needs to be from the axle. As a quick review, I'll re-state the equation for centrifugal force:

We have already determined that the centrifugal force from the counterweight must match that of the ax. We can modify the equation to describe the requirement:

Finally, let's solve for m_cw so that we can set the length of the counterweight swing arm to whatever we like:

As you can see, our counterweight's weight is equal to the weight of the ax times the ratio of the two arm lengths. So for example, If the ax is mounted on a 3 ft arm, and the counterweight is on a 1 ft arm, then the counter weight needs to be 3/1 times the weight of the ax head. Using our 8lb ax as an example, this means we need a 24lb counterweight. Suddenly our 8lb ax is weighing in at 8lbs + 24lbs = 32lbs for the entire assembly! (It gets even worse if we add the weight of the swing arms needed to support all this.) It doesn't take a lot of imagination to see this sort of thing can eat your weight allowance in a hurry. This is the first problem when using a counterweight. They tend to be very heavy.

Problem #2. There are no free lunches

We have just seen the first reason why counterweights are not used all that often when building a bot. It does bad things to our weight budget. It gets even worse. Keep in mind that in order for the counterweight to do its job, it needs to move along with the ax. But we've already seen that it takes energy to put something into motion. Guess where the energy comes from? If you guessed from the pneumatic cylinder, then you win a cookie. The counterweight steals some of the energy supplied by the pneumatic cylinder away the from the ax. This means we have to need a bigger cylinder to make up the difference. Bigger means still more weight on our bot. So how much additional energy do we need? We already know how to find this. Just treat the counterweight assembly as if it were an ax and solve for its moment of inertia. Once you have that, use this equation:

Just so we can get a feel for how much energy we're losing to the counterweight, let's stay with our hypothetical weapon and see what we get. Given an ax that weighs 8lbs mounted on an arm that's 2.5 long we find that the ax has a moment of 1.55 ft^2-slugs. If we make the arm on the counter weight 1/3 the length of the ax's arm, we get a 24 pound counterweight mounted on a 2.5/3.0 or 0.833 ft arm. The counterweight moment is then 0.52 ft^2-slugs. This means that if the ax is to carry 500 lb-ft of energy, another 168 lb-ft of energy will need to be supplied to the counterweight for a total of 668 lb-ft of energy which has to come from the pneumatic cylinder. Put another way, we lose about 25% of the supplied energy to the counterweight. A way to help this situation is to reduce the length of the arm supporting the counterweight. Cutting the arm by half cuts its moment of inertia down by half. But it also doubles the weight needed for the counterweight. If you can tolerate the added weight, then keep the counterweight arm as short as possible.

Problem #3. The kangaroo bot.

We're not done with our headaches just yet. There's one more to deal with. Our counterweight will make us face a new physical property called momentum. We've already seen this concept. It's a direct result of one of Newton's laws we saw earlier. In particular, the one that says a body in motion tends to stay in motion. It turns out there is a physical law which says momentum be conserved (just like energy). Whatever momentum you put into something has to come out somewhere if we expect to change its velocity. If you were to try to catch a bowling ball dropped of a tall building, you'd get feel for the momentum carried by the ball in a big hurry. You'd find it would take a lot of strength to catch the ball and keep it from hitting the pavement. This brings us to our next point: It takes a force to change a body's momentum. The faster we want to change the momentum, the more force it requires. This is what is causing the damage to our opponent when the ax hits him. The ax is losing its momentum very quickly, which we now know requires a lot of force. The force is coming from our opponents armor trying to resist our best efforts to punch through it. If the armor can supply the needed force without damage, the ax just bounces off. If it can't supply the needed force to stop the ax, the ax will continue to bury itself into our opponent until the momentum is used up. (Think of a nail being hit with a hammer into a block of wood.)

So we see that the sudden change in momentum in the ax is what produces the damage when it hits something. Our ax was moving at a high speed and then is suddenly brought to rest upon impact. But what about the counterweight? It's heavy and it's moving along pretty fast too. It doesn't hit anything, so the momentum is still around. This brings us back to the conservation of momentum. The momentum has to go somewhere unless we want our counterweight to just fly off into space. (The safety monitors would likely object to that.) To see what happens, we need to formally define how we measure momentum:

I should point out the velocity is the tangential velocity. You can use the equation from part 1 to find this. You should already know the angular velocity (w) at this point. And you know should know the mass of the counterweight as well. (If you're really quick, you may have noticed I used pounds for the units of mass even though a pound is a unit of force. Don't worry, for our purposes, it will work just fine.) This makes finding the counterweight's momentum easy. Once we have the momentum of the counterweight, what do we do with it? We apply the conservation of momentum law which will tell us how fast the bot will leave the ground when the ax hits something.

At this point, I'm going to make a major simplification. I'm going to assume the counterweight is positioned directly over the center of gravity of your bot. I'm also going to ignore the fact it's on a lever arm and pretend it's connected directly to the axle supporting the ax assembly. The reason I'm doing this is because the actual layout and materials used in your bot can effect the results quite a bit. Don't worry though. These assumptions create a worse-case scenario. We now have all the pieces. Simply solve for the momentum of the counterweight and divide the result by the total weight of the bot. The result is the speed the bot will leap into the air.

We can now estimate how high the bot will jump. We do this using the conservation of energy principle and equating the potential energy with the kinetic energy. (Just take my word for it.)

So, if we have a 210lb bot using the weapon we've proposed, how far will it jump off the ground? Crunching the numbers using a calculator we get about 0.1 feet (roughly an inch). What if we want to crank up the energy to 1000 lb-ft in our ax. How much then? Doing some more crunching we get a solution of about 0.18 feet or a little over 2 inches. This probably isn't going to hurt anything, but it does require that the bot's insides be securely tied down.

In the rush to apply our momentum law, we skipped talking about why it predicts that the bot will jump. I've already mentioned that momentum must be conserved, and that the counterweight is carrying momentum. When the ax head strikes something, it stops. But the counterweight is still going. Since the arm is no longer allowed to swing freely (the ax head is now stuck against something), the counterweight now behaves as if it were rigidly attached to the frame of the bot. The remainder of the bot is now forced to share the momentum once exclusively held by the counterweight. Conservation of momentum says the net momentum can't change (unless we provide an outside force). Since the counterweight is now effectively part of the bot's frame, we have to work the momentum equation backwards using the total weight of the bot and the original momentum to get the new velocity of the bot/counterweight combination. This meets our momentum conservation requirement.

So you are probably thinking that a 2 inch hop is nothing to worry about. But now for some bad news. Let's look a little deeper as to how the momentum was shared. Using the results from our 1000 lb-ft hammer example, we found that the counterweight was traveling about 30 ft/sec which gives it a momentum of 720 lb-ft/sec. When the momentum gets split between the counterweight and the rest of the bot, the new velocity is 3.4 ft/sec. Now solve for the momentum contained in the counterweight itself: 3.4 ft/sec * 24lbs = 82.3 lb-ft/sec. The counterweight gave up nearly 640 lb-ft/sec of momentum to the rest of the bot. Remember the rule that it takes force to change momentum? Remember that the faster we want to change momentum, the more force it takes? Well slamming the counterweight to a halt produces a big change in momentum in a big hurry. The forces necessary to do this are being supplied by the axle and the swing arm that the counterweight is mounted to and can be very large. Depending on the rigidity of the arm and its mounts, the force will likely be many 1000's of pounds in the form of a shock load. Make sure your axle and its mounts can take this sort of abuse.

If the idea of conservation of momentum is a little murky, don't worry about. The equations used to predict how high the bot will jump are only rough estimates anyway. They assume everything is perfectly rigid which won't be the case. I just brought the subject up so that if you see your bot hop up, you'll at least have a partial explanation for it. But most importantly, it warns us about the loads our swing arm and its axle are going to have to face so that we can make a design that will last for more than a couple of swings.

Conclusions