Sunday, August 9, 2015

Stability - or - What Happened to Homer's Rocket (Part 2)

Click here for Part 1

When I was a kid, I found this strange book in a library - I forget the title. It was written in the 1960's, and was a kind of activity book with various crafts you could do to pretend you were living in the stone age. One of these crafts was to make a spear from a broom handle. You used colored tape to decorate it, and there you go, kid. You're a caveman!

Problem is, I could never through the darned thing straight. I thought paleolithic man must have had some secret spear-throwing technique this book didn't cover. It wasn't until I started building rockets that I thought back to that book and realized what the problem was.

. . .

There are three types of stability you may encounter in rocketry: positive stability, neutral stability, and negative stability.

Positive Stability

In the first post, I mentioned the concepts of center of gravity (CG) and center of pressure (CP). These are both central points, of sorts, on a rocket. The CG is the balance point of the rocket (or of any object). It is an imaginary point at the center of all mass on the rocket. The rocket (or any object flying or falling through space) will rotate around this point, always, 100% of the time. The CP is the center of all aerodynamic pressure on a rocket, and is where it is due to the surface area of the rocket.

Although gravity acts on all points of a rocket equally (the motor hook is pulled toward the Earth with as much force as the nose cone), we say that gravity acts through the center of gravity. If you balance an object on your finger, at its center of gravity, it won't fall. Although gravity is pulling on the ends of the object just as much as it's pulling on the center of gravity or mass, you can hold it up just by balancing it at that point.

Similarly, although the air acts on all points of a rocket, we say that it acts through the center of pressure.

I also briefly mentioned the proper relationship between the CG and CP - that the center of gravity must be ahead of the center of pressure. If the rocket is built this way, it will fly straight up, in the direction intended. It may oscillate - wobble back and forth - a bit as it does so, the fins correcting its trajectory as it flies upward. As the fins dampen the oscillation and the rocket accelerates, it will oscillate less and less. A rocket built this way is what is known as positively stable, or, simply, stable.

In this slo-mo supercut, you can best see the slight oscillation in the Der Red Max launches. It's not very prominent, but you can see it in the smoke trail.

Negative Stability

If a rocket is built with its center of pressure ahead of its center of gravity, it will not fly straight. It will fly erratically, flipping and flopping around the sky, and probably crashing to the ground in the process. This can be dangerous, with a larger, heavier rocket. In this case, the rocket is called negatively stable, or just unstable.

In this great video from KQED Public Television, at about 3:40, you can see what some unstable rocket flights really look like.

Neutral Stability

A neutrally stable rocket is one in which the CG and CP are at roughly the same point. A neutrally stable rocket can have a very strange flight. It won't necessarily flop around in flight, but neither will it necessarily fly straight. In fact, it will go in any direction it is pointing. That might sound like it will fly straight up as desired, but in fact, any small gust of wind, or anything which disturbs its straight flight, will change its direction. A neutrally stable rocket cannot correct its trajectory, as a stable rocket will.

An illustration of the three types of stability: positive, neutral, and negative. From Centuri Technical
Information Report 30: Stability of a Model Rocket In Flight
, by Jim Barrowman, 1970. We'll discuss
Barrowman and his contribution to the understanding of rocket stability in the next post in this series.
The oscillation depicted in the stable rocket illustration is exaggerated.
But neutral stability is interesting, because with it, we can do a little experiment to show us how we make a rocket stable. The key ingredient: fins. The fins on a rocket have two purposes.

An Experiment in Stability

Take a foot long dowel rod and balance it on your finger. The point where it balances is its center of gravity - it's the center of all its mass. The mass on one side of your finger is equal to the mass on the other side of your finger.

If you flip the dowel rod in the air, it will rotate around that balance point exactly. If you try to force it to rotate around some point closer to one end, it simply won't work.

Now try throwing the dowel like a spear straight across the room, and see if you can get the front end to stay at the front. Perhaps you can do it, but most times, it will simply point wherever. It probably won't spin fast, but it will end up turning sideways, maybe backwards. The dowel rod is neutrally stable. Can you throw it straight? Maybe, but not reliably. Because the point at which the surface area is equal is also the point at which the mass is equal on all sides, the center of gravity and center of pressure are in the same spot.

This is why I could never throw my broomstick spear straight! The center of pressure and center of gravity were essentially the same, so the spear (like a rocket) was neutrally stable!

Now take three bits of masking tape, a couple of inches long. Tape one to an end of the dowel rod, making a T shape. Then fasten the other bits of tape on the same way, so that the three pieces of tape come in contact with both the dowel rod and the sticky parts of each other. You've now made fins. They're a bit janky, but they'll do the trick.

If you re-balance the dowel rod, you may notice that the center of gravity has moved rearward just a little bit, due to the mass of the tape being added to one end. But you've also moved the center of pressure rearward, and by much more than the center of gravity. That is the first purpose of the fins - to move the center of pressure rearward.

If you now throw the dowel across the room, it will go straight, with the pointy end forward and the fins aft. The dowel is now stable!

If I'd added a bit of weight - like a clay spearhead - to the tip of my pretend spear, moving the center of gravity forward, or if I'd added some kind of fletching to the rear, moving the center of pressure aftward, or a little of both, I could have thrown the broomstick spear with no trouble.

Now, this is not because the fins are merely guiding the rocket. If that were the case, the fins could be anywhere. If you try to throw the dowel with the fins first, you cannot do it. The dowel will flip around in flight and fly fins last. If the part with the fins were your "nose cone," you'd have an unstable dowel rocket, because the CP would be ahead of the CG. This is why we do not put the fins at the front of the rocket. If we did that, we'd have a terribly unstable rocket, which would try to fly backwards.

Of course, a rocket cannot fly backwards, because it has the continual thrust of the motor coming out the back, trying to push the rocket forward. The result is that the rocket flips around all over the place, constantly trying to "correct" itself and fly with the CG ahead of the CP, but the thrust keeps pushing it in different directions.

If the fins were merely guiding the rocket, it might seem like having fins all the way up the side of the rocket would make it go straighter. Let's take a look at why that's not such a good idea.

Here's a simple design of a 4-finned rocket.

Click the pictures to enlarge.

You can see that the CG is ahead of the CP, and that the stability is positive - in this case, 1.35 caliber. We'll get more into caliber next time. For now, you can see that this is a stable rocket, and will perhaps fly over 800 feet on a C6-5 motor.

Now, what happens if we add some fins to the front end as well - in an attempt to make the rocket fly even straighter.

The rocket is now negatively stable. It's stability is -0.915 caliber. The simulation also tells us that the rocket may hit 98 feet. But it won't be pretty.

Why is this?

Let's illustrate the CG/CP relationship. Here we see a cardboard cutout of a simple rocket design - Sounder IB.

As I mentioned in Part 1, the center of gravity is indicated by the blue and white circle (really tiny in this image), and the center of pressure is the red circle with the red dot in the middle. Note that the CG is ahead of the CP.

As the rocket flies straight, assuming there's no wind coming from the side, the rocket experiences the wind coming at it straight on, flying past the body and fins at an angle of 0 degrees. This angle is known as angle of attack, and we'll talk more about it later.

For now, though, just imagine the pencil as indicating the airflow past the rocket. The thumbtack is on the center of gravity, because a free-floating object in space can only rotate around its center of gravity.

Now imagine that something disturbs the rocket in flight. It could be a gust of wind, an odd bump of plastic on the nose cone - anything. The rocket will turn slightly, so that the apparent wind is coming at it at an increased angle (of attack). The rocket is still flying upwards - how does it correct itself?

Well, as mentioned before, the air pressure acts on all parts of the rocket. But the center of pressure is the point the air pressure acts through. It is as though the air pushes right on that spot.

Because the air pressure is essentially pushing on the CP, it will cause the rocket to rotate around the CG, making it fly straight forward again. When a rocket is flying at an angle of attack above 0 degrees, the airflow over the fins creates high pressure on one side and low pressure on the other side. This creates an aerodynamic force called lift, which we'll talk about in more depth later, and straightens the rocket out. The rocket corrects its trajectory. This is the second purpose of the fins - to correct a rocket's trajectory in flight.

It may overcorrect, and flip the other direction. In that case, the air pressure once again causes the rocket to rotate around its CG, in the other direction. This is the oscillation you may notice in rocket flight.

Eventually, the fins dampen out the oscillation and it becomes less and less, until the rocket flies more or less straight up. That's a stable rocket.

But what happens if the CG is behind the CP? Using the same cutout (because it took me a lot of effort to cut it cleanly and reinforce it), let's imagine that scenario.

Maybe this is because we have a really heavy motor, or really tiny fins, or we thought it would be a good idea to put a second set of fins near the front of the rocket to provide "more guidance."

Here, I've moved the imaginary CG behind the CP.

As the rocket flies, the air pressure, acting through the CP, rotates the rocket around it's CG - this time, flipping the rocket around so that it's trying to fly backwards!

Of course, with the thrust coming out the motor, the rocket continually tries to fly forward, so the apparent wind continually changes direction, and the rocket flips and flops around in the air until it crashes.

A neutrally-stable rocket has its CG and CP at roughly the same spot. The fins cannot correct its trajectory, so it's free to fly wherever it happens to be pointing.

"Whatever... I do what I want."

Just remember: G comes before P - in the alphabet, and in your rockets.

How do you know where the CP is? And does it matter how far behind the CG you put the CP? And what if you have an unstable rocket design - what are some things you can do to fix it without throwing the whole thing out?

Well, we'll talk about that in the next couple posts in this series.

But why do we need to know this, if kits are designed to be stable in the first place?

Because I want to show you how easy it is to design your own rockets. But in order to do that, you need to understand the basics of stability. And of all the different aspects of rocketry, stability might be my favorite subject.

Click here for Part 3.

. . .

When I put the first part of this series on Twitter, Homer Hickam himself posted it on his Facebook page, and even commented on the blog post itself. You can imagine how thrilled I was at that!

He mentioned a couple of things which he thought might be other explanations for the lack of stability in that early flight - including non-vertical launch, poorly mixed propellant, or a poorly-machined throat nozzle. I might talk a little about those in a future post in this series - especially the non-vertical launch possibility (something he corrected in later launches, if you read the book, which I highly recommend).

Of course, this series isn't meant to be an analysis of his rockets per se - that would be nearly impossible to do for a rocket launch which took place nearly 60 years ago and was not filmed!

I mention Mr. Hickam's book (and the movie), because once you gain some understanding of the basic principles of rocketry, you can make an educated guess as to what happened when you get a weird flight. And I think that makes reading the book more fun - you can understand and appreciate some of the technical aspects of what he's talking about.

. . .

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