Assignment #1 – EECE 300 Molecules to Mechanisms

Paper: On the Synthesis of Guitar Plucks [1]

The experiment consists in synthesise the pluck response from a guitar through several different

approaches, trying to match in as much detail as possible the measured behaviour of real guitars. The

study is based on the fundamental proprieties of the guitar strings, which are tension, mass per unit

length, bending stiffness and damping. These proprieties are used to model equations of angular

frequency, Q-factor, input admittance and any other relevant proprieties that are useful to synthesise

the vibrations.

The first class of synthesis method is the Modal Synthesis, that works by computing the coupled

string/body modes together with appropriate frequencies and damping factors, then using modal

superposition to construct the transient response. For this it was chosen a set of generalized

coordinates to describe the motion of the strings and body of the guitar. These coordinates consists of

the mode amplitudes of the body in the absence of strings, a set of Fourier series coefficients for the

string displacement, and one or two constraint modes, to allow the string to move at the end attached

to the body. Being able to model the proprieties of the mass, stiffness and dissipation into matrices, and

applying the first coordinates, it is possible to come to an equation for the free motion of the system.

Solving that equation and finding the eigenvalues and eigenvectors will give the complex vectors

associated with the natural string frequencies. This method gave very good results, but can be extremely

slow if all the degrees of freedom of the body manufacturing are considered.

Another method consist in making the synthesis in the frequency domain by applying an inverse

fast fourier transform on the strings and body to create the required time-varying transient response.

This method has a great conceptual simplicity, but in practice it is extremely slow.

A third method of time domain synthesis was also tested that work basically o use a finite

difference approach to integrate directly the differential equation governing string motion or using a

digital waveguide to make the calculations. This method unfortunately did not show accurate and

efficient synthesis method for guitar plucks, and it was not considered to go further with the

experiment.

Of the three methods, the modal and frequency domain synthesis gave predictions in accurate

agreement with one another, and the frequency domain was considered the best option because it

proved itself faster than the modal synthesis.

This study showed an application of vibration theory to synthesise plucks on guitar strings. It

was a macroscale application of the same laws that is shown in the study of MEMS resonators of Dr.

Phani[2]. As the study of Dr Phani shown, as we add degrees of freedom in the sensitive resonator, the

degree of sensitivity also increase. The same happen with the modal synthesis method used by Dr.

Woodhouse. Greater the accuracy of the synthesis is, slower the experiment become. The MEMS

resonators also have a similar tradeoff. It is not possible to have good sensivity and a large measurable

range.

It is interesting to see that the same eigenvalue/eigenvector modal calculations is used in both

studies, proving that the vibration theory is universal and functional in both macro and nano scale. If the

studies in the same area of vibration synthesis of Dr. Woodhouse and vibration sensitivity of Dr Phanis

go further, both MEMS and instrument manufacturing can help each other to improve.

References:

[1] – Woodhouse, Jim. "On the synthesis of guitar plucks." Acta Acustica united with Acustica

90.5 (2004): 928-944.

[2] – Phani, A. S. . “Some Applications of Vibration Theory at Micro and Nanoscales” – Lecture.

March 10, 2015.