Gyroscopes — Page 139, Lesson 168

Let’s pick up right where we left off — we’ve got the gyro spinning, and now we’re going to look at the second of the two fundamental properties: precession. This is the behaviour that makes gyros feel almost counter-intuitive, so I want to walk you through it carefully. First, the definition you need to hold onto: precession is the reaction of a gyro to an applied torque, and that reaction occurs 90° away from the point of application, in the direction of rotation. That’s the whole secret. When you push on a spinning gyro, it doesn’t move the way a stationary object would. It moves as though the force had been applied a quarter of a turn around the rotor, in the direction the rotor is spinning. Let me set up the picture. In Figure 11.5, the gyro is rotating clockwise. Now assume an upward force — a torque — acts on the spin axis. Think of that as the force going into the rotor at the 12 o’clock position. Your instinct might be that the gyro would rotate backwards about the horizontal axis, from the point where the torque is applied. But that’s not what happens. Instead, the torque is precessed through 90° in the direction of rotation, and the rotor moves inwards to the page about the vertical axis — as though the force had been applied at the 3 o’clock position to a stationary rotor. So the applied force at 12 o’clock produces motion as if it had been applied at 3 o’clock. That’s the 90° shift, and it’s always in the direction of spin. Now let’s put this into a gimballed gyro, as in Figure 11.6. Here the gyro is spinning about its spin axis, which we label XX. A small mass, labelled M, is applied on the inner gimbal, in line with the XX axis. That mass acts to pull the inner gimbal down, and that produces a torque about the YY axis. Effectively, this is a force being applied to the 6 o’clock position of the rotor. Now watch what happens, step by step. Initially, the gyro axis tilts through a small angle, which we call φ — that’s the Greek letter phi. At that moment, the spin axis is no longer pointing to the original fixed point in space. But here’s the key: after that small initial tilt, no further movement takes place about the YY axis. The downward pull doesn’t keep tipping the gyro over. Instead, the torque 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 rotation about ZZ — again, 90° away, in the direction of spin. Two more important behaviours. If the torque at M is withdrawn, the precession ceases — the turning about ZZ stops. 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 a constant velocity. So precession is a steady, continuous turning, not a one-off kick, as long as the torque is maintained. Now I want to connect precession to the other property we already covered — rigidity — because they’re two sides of the same coin. The rigidity of a gyro depends on two properties. The first is moment of inertia, which is a combination of the mass and the effective radius at which that mass operates. The second is rotor rpm — how fast the rotor spins. Let me define moment of inertia properly, because it’s a term you’ll see again and again. Moment of inertia is 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 a practical trick: in order to minimise weight — that is, mass — but still get a greater moment of inertia, the mass is often concentrated at the rim of the gyro. A bicycle wheel, which uses spokes, is a clear example — the heavy part is at the rim, not spread through the middle. The second factor is rotor rpm. The faster the rotor spins, the greater the gyro’s rigidity. So, to summarise: the rigidity of a gyroscope is increased if the mass is increased, if the effective radius at which the mass operates is increased, or if the rotor rpm is increased. Any one of those three will stiffen the gyro. Now for the relationship between precession and rigidity — and this is the elegant part. The rate of precession is directly proportional to the applied torque, but inversely proportional to the moment of inertia and the rotor rpm rate. Let me unpack that. Directly proportional to torque means: push harder, and the gyro precesses faster. Inversely proportional to moment of inertia and rpm means: the more rigid the gyro is — the heavier, the bigger, the faster it spins — the slower it precesses for a given torque. Effectively, this means precession and rigidity are opposite characteristics. If a gyro has a lot of rigidity, it will not precess very much. If it precesses a lot, it cannot be very rigid. That’s the trade-off you’ll carry through every gyroscopic instrument we look at next — the attitude indicator and the directional gyro both live on this balance between rigidity and precession.