Mathematical and Physical Models for Coupled Oscillators
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Ashley Pacheco
College:
The Dorothy and George Hennings College of Science, Mathematics, and Technology
Major:
Computational Science & Engineering
Faculty Research Advisor(s):
Edward Farnum
Abstract:
Sympathetic resonance on a stringed instrument is the tendency for a mechanical vibration, possibly from one string, to excite other strings, leading to additional sound production. This can be modeled by a partial differential equation, which we have solved both analytically and numerically. We have a novel stringed instrument which demonstrates this sympathetic resonance, subject to a variety of mechanical driving forces. This will be of interest both musically as well as a teaching tool. The short term goal is to use a synthetically generated driving force to excite the strings to resonance, without directly touching them, as in plucking or bowing. We will study this induced resonance to optimize its efficiency and sustain, in an effort to design better prototypes.
To accomplish this, a physical and mathematical model will be required. The physical instrument resembles a zither with a sound hole large enough to be placed on top of a speaker, which provides a controllable driving force. We record the plucked instrument directly, identify its dominant frequencies, and then use the gathered information to construct a driving signal. This signal will be played through the speaker, and we measure the induced excitation on the physical model. Preliminary tests suggest that the resonance is feasible. We hypothesize that the amplitude of the response will depend on the duration of the driving force. However, we don’t know if all frequencies will be equally excitable, nor whether upper harmonics are easily induced. Once we better understand the response from a controlled driving signal, we consider the much more complicated acoustic input signals from a trumpet.