IAN LANG ELECTRONICS

# Fulcrum

Everything has a combined centre of gravity if it exists in three dimensions. You, me, the cats, a salt pot, you name it it has a CCOG. Below is a drawing of a piece of metal standing up on its beam end and some vandal has come along and superimposed some faint red lines over it:

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# Z

In fact what those red lines represent is not an incredibly geometric grafitti artist but the co-ordinates of the piece of metal's combined centre of gravity along the three dimensions ( X Y Z) in which it exists. Since it's rectangular and of uniform density and thickness throughout, they'll actually converge on the centre. If we welded another piece on top to make a T shape the CCOG would have to reflect the new mass and would in fact move upwards on the Y axis taking the Z with it, but the X axis would remain the same.

You could flick this over with your finger if you felt so inclined; and in fact if you banged the table or bench it's sitting on hard enough it would topple over. Why should this be? Let's look at it directly side on.

In the diagram on the right we are looking at the side of the piece of metal and considering the Y axis as above which is marked by the red line. You can see the red line sits well inside the base in the first (leftmost) representation and so the piece of metal is stable and stands up. A slight movement in the next representation puts the red line at the extreme end of the base. If the force causing this movement stops, the metal will topple back on to its base and in theory should stand up (though in practice the impetus will rock over the other side and probably topple it the other way). In the third representation the red line has gone outside of the base. The piece of metal will topple over to the left.

The moral of this story is that anything tall standing on a thin base is not stable. More so if it moves and turns, because the turning produces a sideways force that can easily move the structure base outside of its centre of gravity line.  Consider this double-decker bus:

Whilst it does indeed pander to my vanity (look at the advertisement......) it does have a serious point to make in this article. A bus such as this is 27 ft 6 in (8.38 m) long, 8 ft (2.44m) wide and

14 ft 4 1⁄2 in (4.38 m) high. That's quite tall, but the width is more than half the height. A bus like this should be able to lean up to 28 degrees from the perpendicular and still return upright Moral: make your base as wide as you can. Note where the wheels on the bus are too. They are as far out as you can get whilst still being in the body of the bus. So, here's the general gist: carry the load for your bots inside the wheelbase as far as you can, and a low, wide bot is better than a narrow, tall one. If you must make a tall robot, try to put as much weight at the bottom as you can and keep it all inside the wheelbase.

If the bot needs to move quickly in a straight line, consider slowing the speed down when it turns to reduce the lateral forces trying to topple it. Range Rovers can have this problem. If you are belting along and you make a violent turn, you will topple that car on to its roof. You have to slow down and turn carefully. It's because Range Rovers are very tall but have not that much more width than other cars and it's easy if you're moving fast to tilt the car outside of the COG along the Y axis. If you turn a fork-lift with the forks in the air on your fork-lift driving test it is an instant fail. When the forks are up the CCOG is over the roof of the cab and it doesn't take much to topple it. With the forks down the CCOG is somewhere under the driver's seat. It takes much more to topple it that way.

So, let's consider the wheel base and some physics behind it. Starting with the priniple of a lever of the first order. Eh? Levers? Bear with me. All will become clear. Here's the principle:

# Effort

We know what happens here, as you push down on one side of the lever the other side goes up provided you push hard enough, and the load goes up with it. If you let the lever go then the load goes crashing back down again as it's full of potential energy being under gravity and accelerating at 9.8 metres per second per second until it hits the floor. The maths of this are moment = force x distance or

M = Fd and there are two moments in a lever, one at either side of the fulcrum. So, if the load weighed 10 kg and the centre was 1metre from the fulcrum you'd have 1 x 10 kg/m at that side. If you pushed down 2m from the fulcrum, you'd only need to have an effort of 5 kg/m on that side (as 2 x 5 = 10 ) and if it was 4m long you'd only need to put an effort of 2.5 kg/m in. If you pushed down 1m from the fulcrum you'd need to put 10kg in to balance the load. Simple yes? But what's it got to do with wheelbases?

Consider a wheel and axle. Half the wheel is one side of the axle and half on the other. The axle is attached to a chassis through some sort of bearing. The axle is therefore acting as a fulcrum. If you put too much weight on the side of the wheel that is outside the wheelbase, what will happen is that the vehicle will tip around the wheel in the direction of the weight. For example:

What we have on the left here is a counterbalance fork lift truck. It's used for getting pallets of goods on to and off from the sides or backs of lorries and that big number 25 painted on the side there means it's rated for lifting 2.5 metric tonnes. But- and this is an important but- ONLY AT LOAD CENTRE. This is an important point that idiot drivers often ignore. What it means is that if you've got your forks under the pallet so that the centre of the pallet is closer to the safety guard than the tip of the forks, and that pallet weighs 2.5 tonnes, the truck will lift it safely. If the centre of the pallet is closer to the tip of the forks, the weight on the forks will be greater than the weight behind the centre of the front wheels and the truck will tip forwards. Your workmates will invariably see this.

You will be the butt of all jokes for the rest of the week. So if you do it, do it on Friday teatime.

It works because everything behind the centre of the front wheel is weighted to balance a load of 2.5 tons plus the weight of the lifting assembly on the front, which is in front of the front wheels. Notice that safety cage above where the driver sits and how every part of it is behind the centre of the front wheels. That cage is made of big bits of steel and it is heavy. The body is made of even bigger bits of steel. The motor is huge and sits low down in the middle, between the front and back wheels and central to the axis running from side to side. The battery is made of tons of lead and sits at the back. All of this means that if a fork-lift runs over your foot (whether or not it's carrying a load at the time) it won't matter if you are wearing your most expensive Caterpillar boots at the time because at least five tons is going to introduce itself to your pink little toesy-wosies and you won't be playing football for a while. Notice how far apart the wheels are set too. This means that everything that isn't supposed to be lifting a weight off a lorry can be put to the back of the front wheels to help balance the weight of the goods outside the wheel base.

All of this is important to the design considerations of your bots. If you don't want your bot to tip over, make sure there are no heavy weights outside the wheelbase, and anything heavy is as low down as you can get it and central to the front-to-back axis of your bot. If you must put something heavy outside the wheelbase, put something equally heavy inside the wheelbase to counterbalance (aha!) the outer weight according to the equation M = Fd as above considering the axle of the wheel as the fulcrum.

# Ian Lang, August 2013

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