...From a modelling viewpoint, what are the things to avoid when using long drive shafts? Obviously the amount of power being transmitted is a factor, how about physical size?

Well, I tried every trick in the book to avoid this, tried to change the subject, threw in historical anecdotes, etc., but I see some of you are absolutely bound and determined to find out how to actually design one of these long shaft systems. I don't know if there's any Freudian basis to that interest, I'll leave that to others to decide, although that could help explain the apparent passion inspired by this topic. ;-)

However, since you asked, I'll try to at least discuss the QUALITATIVE basics. I won't go into the quantitative issues very much (if at all) for reasons that should soon be apparent:

Don...

From : Don Stackhouse

Let's start with a little review of Mechanical Vibrations 101, for those who either missed that course, or were fortunate enough to successfully forget it.

Almost any physical system that contains both mass and stiffness can vibrate. There will be at least one, and quite possibly a number of frequencies where it vibrates the best. These are called "natural" frequencies. A single weight hanging on a single spring has one natural frequency. A pendulum with a single pivot point has one natural frequency. A torsion spring with a single flywheel on the end has a single torsional natural frequency. A torsion spring system with multiple weights at different locations along the spring will have multiple natural frequencies. A cantilever beam has an infinite number of natural frequencies.

The behavior of a vibrating system varies quite dramatically depending on where the vibrating frequency is relative to the system's natural frequency. Let's assume we have a "single degree of freedom" system (which, BTW, therefore has only a single natural frequency). A weight bobbing up and down as it hangs from a spring would be an example, such as an old 7-cell 500mah battery pack hanging from about 10 chained-together #64 rubber bands.

WARNING: This experiment has been performed by specially trained Rubber Band Professionals, be very careful if you try this in your own home!!

Let's also assume it's being fed a "forced vibration" by moving the other end of the rubber band chain up and down at some frequency, with an amplitude (the distance it is being moved) of one inch. Now, let's vary the frequency (how fast the end is moving up and down, measured in "cycles per second", or "Hertz") while keeping the amplitude constant, and watch what happens to the amplitude of the bobbing weight.

At very low frequencies, the motion of the weight is almost perfectly in step with the motion of the other end of the spring. Your hand holding the end of the rubber band moves up and down 1", and the weight moves up and down 1", in step with your hand. Now move your hand very fast (much faster than the system's natural frequency), but still 1" amplitude. The weight can't keep up, and so it moves much less than 1".

Now let's find the system's "natural frequency", somewhere between those other two. It's equal to the square root of K/M, where "K" is the stiffness of the spring (also called "spring rate", typically measured in pounds per inch of deflection), and "M" is the mass of the weight. You'll know when you've found it because two things will happen: 1. The "phase" between your hand and the weight will be exactly 180 degrees. This means that the weight will be moving down when your hand is moving up, and vice versa. 2. The amplitude of the weight will go CRAZY!!! (Careful, don't hit yourself in the hand with the battery!)

When systems are excited at their natural frequency, their amplitude goes up, usually WAY up. The case of an opera singer shattering a wine glass with his/her voice is a classic example. The singer holds the glass by the stem and adjusts the note they are singing till they find the natural frequency of the glass. The amplitude of the vibration of the glass in response to the forcing vibration from the singer gets so large that it overstresses the glass. POW!!

This phenomenon is what limits the "frequency response" of stereo speakers, for all you audiophiles out there. The speaker magnet and the cone that supports it is a spring-mass system. If you try to play a sound through a speaker with a frequency lower than the speaker's natural frequency, the speaker's magnet will oscillate back and forth proportionally to the amplitude of the electrical signal, faithfully reproducing the sound. As you approach the speaker's natural frequency, the magnet's amplitude increases, giving a much louder sound than called for by the electrical signal. This is that familiar "Boom" you get from bass speakers when they are driven outside their limits. If you hit the natural frequency exactly, it's not hard to tear the speaker cone with a relatively low-power input signal.

The only thing that limits the amplitude of the system at its natural frequency is something called "damping". The shock absorbers in your car's suspension are one example. Damping elements create a force opposite to the direction of the vibration, trying to stop it. In the case of your car's shocks, they keep the wheels from bouncing uncontrollably on the road. In our little experiment, the hysteresis of the rubber bands, and the air resistance acting on the moving battery pack both supplied the damping.

If the damping is high enough, it's possible to run safely at the system's natural frequency. However, this is generally the exception rather than the rule, unless the amplitude of the signal that's exciting the system is VERY low. In full scale propellers I know of cases of this involving Kevlar blades (Kevlar has a very high structural damping coefficient, it makes the world's worst church bell). OTOH, graphite or aluminum blades ring very nicely, and if you excite them at their natural frequency (other than just momentarily as you accelerate through a critical rpm), they WILL break.

So how does this relate to rotating shafts? A rotating shaft with a rotating mass (such as a propeller on one end and the armature of an electric motor on the other) is a vibrating system. In this case, the "K" is the torsional stiffness of the shaft (which goes down as the shaft gets longer, and up as the shaft gets bigger in diameter), and the "M" is the mass moments of inertia of the prop and armature. Since this system has two masses, each of which can choose their own way to move in a vibration response, then it has two degrees of freedom and therefore two natural frequencies. If you run the system at an rpm corresponding to either of those, the system could very easily decide to ruin your whole day! Don't worry, we're not done yet. The shaft can vibrate in bending as well as in torsion, and this introduces more natural frequencies. The shaft and masses are mounted on an airframe, and that airframe structure with attached masses plus its own weight has additional natural frequencies. Run the system at an rpm corresponding to one of those and it will also decide to ruin your whole day. In addition, the propeller blades are cantilever beams, which have their own set of natural frequencies. If any of these guys manage to agree with each other on the same natural frequency, things can get very nasty indeed!

So how do we avoid this? Remember how a longer spring has a lower "K"? For a direct-drive setup, the length of the shaft is very short, so its stiffness is relatively high. This increases the value of SQRT(K/M), the natural frequency. If the natural frequency is sufficiently high, then our rpm always stays below it, and we don't get into any trouble. However, as the shaft gets longer, its mass goes up and its stiffness in both bending and torsion goes down. The numerator and denominator of the ratio K/M both get worse, and the natural frequencies go down. If they're still comfortably above our operating rpm range we're still safe, but obviously we're starting to eat into our safety margins. Push this issue far enough and eventually it will bite you!

So how far is pushing it too far? For some good examples of successful long-shaft drives, look at model helicopters. Main rotor shafts have very high inertia rotors on the end, but tend to be stiff, reasonably short (although they tend to have significant overhangs between the upper bearing and the main rotor assembly, which is bad), but fairly low rpms, which is good. Tail rotor shafts are a little trickier. They are VERY long, and although the inertia of a tail rotor is lower (probably similar to some of our props), the rpm is much higher (also probably similar to some of our props).

I have seen cases where tail rotor drives tended to be troublesome, particularly on the GMP "Cricket". This used a long 1/16" music wire drive shaft for the tail rotor, with a slight angle flexed into it at one end (which gave it a once-per-rev bending stress), driving a prop about the same size and rpm as a geared Speed 400. The natural frequency of this system was relatively low, and the failure rate of tail rotor drive shafts was relatively high. This made it an excellent trainer, because this disgusting thing was such a "maintenance hog" that I came to despise it with a passion. As a result, I really didn't care if I splattered it, which made me willing to try new things with it that I might have been timid to try with a helicopter I really cared about. It did wonders for my flying skills, at least during those brief periods it managed to stay airworthy.

Other helicopters tend to use either a flat belt drive (which can have a lot of damping available via slippage of the belt on the pulleys), or much more substantial driveshafts with much better supports. Speaking of that, the positioning of those supports is critical. They don't have much influence on torsional vibration, but they dramatically effect the shaft's bending frequencies. Remember how the "K" value goes up as the spring gets shorter? A support somewhere in the middle of a long shaft breaks the shaft up into shorter segments as far as bending is concerned, each of which has lower effective mass and higher bending stiffness, which tends to increase the segment's natural frequency.

However, be careful to position the supports so that they do NOT divide the shaft into even multiples (halves, thirds, fourths, etc.). If you have a support exactly in the middle, dividing the shaft into halves, then the halves both have the same natural frequencies in bending, which means they can work together to get into trouble. (Two kids working together can get into FAR MORE trouble than one kid alone!) If the segments are all different length's, they will all have different frequencies and be less likely to excite each other.

So how do we design-calculate-measure the system's natural frequencies? Unless you have access to some fairly exotic design tools and/or test instrumentation, it's probably going to be very difficult. Your only recourse is likely to be old fashioned cut-and-try methods. Run up the system on the ground, watching carefully for unusually large buzzing or other vibrations, suggesting a resonance at a natural frequency of something. Bending vibrations should be fairly easy to spot, but torsionals are probaby going to be harder to see, at least until something breaks! If you find something (either the easy way or the hard way!), then you will have to determine the best way to either change the stiffness and/or the mass in the system to move that frequency to a safe area, away from your operating range. In some cases it may be best to move it down below the operating range (add mass or reduce stiffness), but usually it's safest (but also most difficult) to try to move it up (less mass or more stiffness), above your max rpm. The tricky part will be determining the system behavior at flight rpm's. You probably don't have instrumentation to measure vibrations in flight. If you try to get the rpm on the ground pushed up to in-flight values by installing a smaller prop, you've just changed the mass of the system and therefore all of the frequencies. Probably your best bet is to install a prop of the same diameter and weight (and hopefully about the same mass moment of inertia), but less pitch.

If you do find something, be careful how you fix it. Stiffening something by adding material can also add mass, for little or no net effect on natural frequency. In the case of a torsional or bending problem in a shaft, tube construction can increase both the bending and torsional stiffness without adding weight. This is why tubes are often used for long driveshafts. Be careful, though, with composite tubes. The lengthwise fibers contribute the bending stiffness, but very little torsional stiffness. A kite spar tube is likely to have no bending problems, but could have massive torsional problems if you don't add some 45 degree wraps to provide sufficient torsional stiffness.

In any case, hope for the best but expect the worst, and keep your eyes and ears open!

Are there things I can get away with on a Sp400 plane, that would be a problem with a bigger design? How about Sp600?

A Speed 400 has relatively low armature inertia, as do the props for this size motor. OTOH, rpms are generally higher, which could cause trouble, but in general it shouldn't be too difficult to come up with a system that works. As you get larger, it's likely to get increasingly difficult to stay out of trouble. Because the amount of energy in the system will increase in larger systems, the results are also likely to get more spectacular (and expensive) when something does go wrong. This also means that scaling up a successful Speed 400 system to a larger size is not a reliable approach.

I'm not trying to scare anyone off here, it's very possible to develop a system that works. It's just that it's also very possible to develop one that doesn't! Just be careful, and don't go into it with unrealistic expectations that it's going to work perfectly the first time. And by all means, please stay well clear of the plane of the prop disk while testing!

I was planning on a rod of some sort (probably a carbon tube), attached directly to the motor shaft at one end and running through a ball race at the other end. Is this okay? What should I do to minimise vibration?

Make sure your tube has adequate bending stiffness (lengthwise fibers), AND adequate torsional stiffness (45 degree carbon wraps). Make sure all components are balanced, and long components like the shaft are dynamically balanced (i.e.: the balancing weight is located at the same place along the shaft as the unbalance it's correcting). DO NOT add weight at one end to correct an unbalance at the other end.

Mid-span bearings can dramatically help the bending vibration issues, but then you have to deal with the problem of keeping more than two bearings in line with each other. This one can be a killer, especially when you add deflections of the supporting structure to the mix. An alternative is to break the shaft into sections, with universal joints at each intermediate support bearing, but that introduces all sorts of other problems.

Does it matter if I use a folding prop?

Folders will probably have more mass and rotating inertia, will probably have less structural tolerance for torsional stresses (that's spelled t-h-r-o-w-n b-l-a-d-e-s), and complicate the whole vibrational spectrum because of the extra degrees of freedom added by the blade hinges. That doesn't mean they won't work, it just means there's additional ways to get into trouble.

Is a gearbox a factor with our models? Again I'm only thinking of Sp400 or Sp600 applications, probably something like a Northrop N9M.

It can be a factor. It changes the vibrational characteristics by altering the apparent mass of the attached component. For example, if you have a Speed 400 geared to a shaft by a 2:1 gearbox, from the perspective of the shaft, the motor's armature will appear to have twice the mass moment of inertia. This lowers the natural frequency to about 71% of the direct-drive value. In addition, the gearbox introduces additional exciting vibrations, the tooth passing frequencies and the once-per-rev frequencies of each of the gears. OTOH, it's possible for a gearbox, especially a belt drive, to add some damping to the system. The lowered rpm of the shaft also helps; in general the lower the rpm of the shaft, the better.

Gearboxes are just one more factor, adding their own set of pros and cons. The more elements you have in your system, the more potential you have to get either in or out of trouble. How you use that potential is up to you.

Don Stackhouse
DJ Aerotech