Frequency Spectrum of Tine -> Pickup Signal

118 Hz fundamental, plucked

275 Hz fundamental, plucked

118 Hz fundamental, struck

275 Hz fundamental, struck

118 Hz fundamental, plucked (zoomed in)

(Click to enlarge!)

These spectra come from 4096 point FFTs of the signal generated by the tine vibrating in front of the pickup. The samples were taken during the steady state portion of the oscillation (after the attack/decay portion) and each plot is an average of 25 trials or more. The spectra are of two different tines, one lower with a fundamental frequency of 118 Hz (B-flat 2), the other higher with a fundamental at 275 Hz (D-flat 4). These frequencies appear as peaks on the graphs and can be verified aurally. There are two plots for each tine, one where I plucked the end of the tine with my finger, and one where I struck the tine with a steel bolt.

The first thing to note is the appearance of harmonic overtones (integer multiples of the fundamental). These are added by the pickup and do not reflect the motion of the tine which has a second natural frequency at around 6.27 times the fundamental [Whitney, 1999]. The 6.27 ratio is for a specific case in the Whitney paper and I haven't figured out how this might vary with cantilever beams of different size, mass, etc. The important thing is that the second overtone is certainly not two times the fundamental.

The spectrum of the longer tine looks similar to that of a guitar string oscillating perpendicularly across the axis of a guitar pickup (instead of along the axis) where even harmonics are much more present than the odd harmonics [Horton, 2008 - Fig. 16, plot labeled "Horizontal"]. Indeed, the motion of the tine is (mostly) in this perpendicular plane.

The spectrum from the shorter tine does not exhibit the same attenuated odd harmonics, and I wonder if the effect is caused by wider deflection of the tine from the pickup axis (deflection is greater with lower frequency tines).

The struck tine spectra have much more high frequency content than the plucked tines. The fundamental anti-node of the tine is at its free end, so plucking it at the end adds energy at the fundamental frequency. A strike is similar to an impulse (equal energy across all frequencies) so it makes sense that we see more energy at higher frequencies and not just at the fundamental. I'm not able to explain why clusters appear where they do in the higher frequency range for the struck tine plots. I also don't totally understand where the small peaks immediately adjacent to the harmonics come from, though Matt suggested I look into sum and difference tones as a possible explanation. This is on my list of things to do.