Gyroscopes — Page 139, Lesson 168

Let’s pick this up right where the gyro’s two basic properties left off, because now we’re going to watch what happens when you actually try to push one of these spinning rotors around. This is the phenomenon called precession, and it’s the single most counter-intuitive thing in gyroscopic theory, so I want you to park your everyday intuition at the door. Look at Figure 11.5. We have a gyro spinning clockwise, viewed from the front. Now imagine an upward force — a torque — acting on the spin axis. Think of that as the force going into the rotor at the 12 o’clock position. Your first instinct, and mine, is that the gyro should tip backwards about the horizontal axis, right at the point where the torque is applied. But that is exactly what does not happen. Instead, that torque is precessed through 90° in the direction of rotation. The rotor moves inwards to the page about the vertical axis, as though the force had actually been applied at the 3 o’clock position to a stationary rotor. So the rule you must engrave: a torque applied to a spinning gyro does not produce motion at the point of application. It produces motion 90° around, in the direction of rotation. That’s precession in a nutshell. Now let’s put that into a real gimballed gyro, as in Figure 11.6. The gyro is spinning about its spin axis, which we label XX. On the inner gimbal, in line with the XX axis, we attach a small mass M. That mass pulls the inner gimbal down, producing a torque about the YY axis. Effectively, this is a force applied to the 6 o’clock position of the rotor. Here’s the sequence of events, and I want you to follow it carefully. Initially, the gyro axis tilts through a small angle, which we call φ (phi). At that instant, the spin axis is no longer pointing at the original fixed point in space. But — and this is crucial — after that small initial tilt, no further movement takes place about the YY axis. The gyro does not keep tipping over. It stops. Why? Because that torque about YY is precessed through 90° in the direction of the gyro’s rotation. The rotary motion takes place at the 3 o’clock position, and the spin axis starts to turn at a constant velocity about the ZZ axis. So the applied torque about YY produces steady rotation about ZZ, not continued tipping about YY. Now, two conditions to lock in. If the torque at M is withdrawn, precession ceases immediately. But if the torque application at M continues, and remains in the same relative position on the gimbal ring, then the gyro spin axis will continue to rotate at that constant velocity. Constant torque, constant rate of precession. Remove the torque, precession stops. Now let’s connect precession to the other property, rigidity, because they are opposite faces of the same coin. The rigidity of a gyro depends on two properties. First, Moment of Inertia — that’s a combination of the mass and the effective radius at which that mass operates. Second, rotor rpm. Let me unpack moment of inertia, because it’s a measure of how big and how heavy the gyro is. A gyro with a greater radius will have a larger moment of inertia than a smaller one with the same mass. And a gyro with a greater mass will have a larger moment of inertia than one of the same radius but less mass. So it scales with both size and weight. Now, here’s the clever bit: to minimize weight — mass — while still getting a large moment of inertia, the mass is often concentrated at the rim of the gyro. Think of a bicycle wheel with spokes. All the heavy material is pushed out to the edge, so you get maximum inertia for minimum mass. Second property: rotor rpm. The faster the rotor spins, the greater the gyro’s rigidity. Straightforward. So rigidity is increased if you increase the mass, or increase the effective radius at which the mass operates, or increase the rotor rpm. Any of those three. Now the relationship that ties it all together. The rate of precession is directly proportional to the applied torque, but inversely proportional to the moment of inertia and the rotor rpm. Let me say that again slowly, because it’s the heart of the matter. More torque means faster precession. But more inertia, or more rpm, means slower precession. And that is exactly why precession and rigidity are opposite characteristics. If a gyro has a lot of rigidity — high inertia, high rpm — it will not precess very much. It resists being pushed around. Conversely, if it precesses a lot, it cannot be very rigid. You can’t have both. That trade-off is what you’ll carry into every gyroscopic instrument we look at next.