Originally Posted by emsvitil
Fundamental frequency, first natural frequency, lowest vibratory mode. Different users may notate it otherwise, but here's the picture.
Aircraft propeller mass moment of inertia is almost always much higher than powerplant mass moment of inertia, so imagine a simple two element system as a large flywheel connected to a smaller flywheel with a shaft, hanging free in space....two intertias connected by a stiffness.
Using your giant magical hands, hold the big wheel stationary and turn the small wheel a few degrees, in a manner which twists the shaft, now a torsional spring. Release both wheels at the same time. The wheels will begin to oscillate in opposition. The angular displacement of the large inertia will be less than the angular displacement of the small inertia, but the period (rate, cycles per second, hertz) will be identical. The period at which they oscillate is the natural frequency (F1) of the simple two element system, a function of the inertias and the connecting spring rate. Change an inertia or the spring rate to change the frequency.
A real system can (and does) have multiple inertias and connecting stiffnesses. The number of available natural frequencies (F1, F2, Fx, etc) and vibratory modes is always total inertias less one. I've attached some examples below. Note the "mode shapes", a name which seems to stem from the 2D drawings depicting them in old texts (example below). At F2 and higher, the various inertias take up different rotation directions in relation to other inertias.
Note that natural frequencies are passive. They are the frequency at which the mode oscillates if excited,
which is the key concept in application. When excited at a forcing frequency other than the system natural frequency, nothing much happens. When excited at the same frequency, the system resonates; the amplitude of the oscillation grows, and without damping, can theoretically reach infinity. It's theoretical because (1) there is always some damping, and (2) no connecting stiffness is infinitely strong...