normal modes multi-DOF vibration

Join two equal masses with springs. The classic arrangement here uses three identical springs. Pull only m₁ and let go. m₁ swings, then soon m₂ takes over the motion, m₁ goes quiet, and the motion returns to m₁. This pattern, energy traveling back and forth, is called beating. A single mass in the earlier lessons was one clean beat, but this coupled motion looks complicated and does not repeat in a simple way. The middle spring is the messenger that carries energy between the two. The apparent disorder hides a simple structure, and the next two steps bring that structure out. The messenger splits the energy between them.

Here is the hidden order. Place the two masses at the same size and in the same direction, then let go. There is no beating at all. They move together, in step, at one single clean frequency. This is the first normal mode, the in-phase mode. Because the two masses move together, the spring between them neither stretches nor shortens. It is simply carried along. Only the outer springs work, so the frequency is the lower one, ω₁ = √(km). It is the same as one mass on one spring. A resting middle spring is the mark of this mode. A quiet middle spring marks this mode.

Now do the opposite. Place them at the same size but in opposite directions, then let go. Again there is one clean frequency and no beating. The two masses always move in opposite directions, like a mirror. This is the second normal mode, the out-of-phase mode. This time the middle spring is pressed and stretched hardest on every cycle, and it adds its own stiffness to the restoring force. More stiffness means a higher frequency. ω₂ = √(3km) is faster than mode 1. A system of two masses has exactly two modes, and that is the rule. N masses mean N modes, each with its own frequency. The middle spring is what raises the frequency.

Now the decisive point. Any motion of the two masses, including the messy beating of the first step, is only the sum of these two modes. Take some of mode 1 and some of mode 2. The top row is mode 1 alone, the middle row is mode 2 alone, and the bottom row is their sum, the actual motion. Move the mix. When the motion is purely mode 1 or purely mode 2, the bottom row stays clean. Between those, the two frequencies overlap and the beating of block 1 comes back. A complicated motion was never complicated. It was always two simple modes added. Split any motion into modes, and each mode is a simple oscillator with its own natural frequency. Adding the modes brings the beating back into view, which is the whole point of the picture.

This understanding is a design tool, not only an explanation. If a tall building has a dangerous resonance, an engineer adds a second mass on purpose. A tuned mass damper (TMD) hangs from its own spring at the top, and its mode is set on the building's resonance frequency. The system now has two modes, not one. At the old resonance the added mass swings hard, out of phase, and soaks up the energy, so the building stays nearly still. Toggle it. Without the TMD the building shakes hard. With it, the small mass moves instead and the building hardly moves. Taipei 101 hangs a steel ball of 660 tons near the top to do exactly this. Bridges, chimneys, and car engines use the same trick. Add one degree of freedom and cancel a troublesome vibration. A small mass takes the shake in place of the large structure.