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Chapter 1

CHAPTER 1

INTRODUCTION

Statement of the Problem

To begin a discussion regarding the structural dynamics problem a guitar presents requires an introduction to the primary mechanisms at work. In standard practice, the player first rests the arched rib between the lower bout and upper bout on the leg and holds the neck with the left or right hand. These are the boundary conditions imposed on the guitar structure during normal playing, simply modeled as a fixed constraint normal to the side at the leg and fixed in 2 directions at the neck, but free to slide in the hand in the third direction. In both cases friction can be assumed neglected. The strings of the guitar are attached to the tuning machines at the head end of the guitar and the bridge at the body end. The strings are tuned to a given fundamental frequency based on their mass properties and stiffness properties which include material density, length, and diameter of the string, and string tension, and string boundary conditions. Using a finger or pick, the player will pluck 1, or strum many, of the guitar strings. This is the forcing function into the structural dynamic system of the guitar. The string input can be modeled as some form of impulse function depending on the technique implemented by the player. Literature is available that outlines modeling suggestions for the wide range of techniques used by the player. The impulse function by the player on the strings causes excitation and resonance in the structure and the surrounding medium, air, dictated by the modal characteristics of the complex coupled dynamic system of the guitar. The fluid-structure interaction of the resonating top plate, back plate, and air cavity causes rarefaction and compression of the surrounding air, thus creating sound waves, due to the harmonic nature of the excitation, that eventually travel to the ears of the listener.

The guitar is generally divided into 2 families--the classical and the steel-string. The classical guitar uses nylon strings and the steel-string makes use of, as the name suggests, steel strings. The structural characteristic of the latter is different than the former, specifically in the bracing of the top plate, due to the need to support the increased tension from the steel strings. Focus in the following discussion will be placed on a specific guitar of the steel-string family, the Martin D-28 folk guitar.

Objectives

Given such a complex problem, the luthier--guitar builder, could hope to achieve a complete understanding of the mechanisms and interactions involved in the guitar after completing a Ph.D. in both physics and structural dynamics, along with a lifetime of research and application. For this reason, it is important for the researchers to hand down bits and pieces of the problem in simpler forms so that the luthier can apply the techniques without a complete understanding of the theoretical background.

As stated in the previous section, the discussion will focus on the Martin D-28 folk guitar. The process plan and objectives include the following. The first part will be the creation of a finite element model, using I-DEAS 10 software, of the Martin D-28 folk guitar from blueprints obtained from an online source [5]. Once the model is complete, including all components of the guitar but not including modeling the surrounding air, an attempt will then be made to correlate the fundamental mode (0, 0) of the top and back plate using test data and boundary conditions presented by Rossing [9]. Once satisfactory correlation is obtained - satisfactory being defined for this discussion as the fundamental frequency of mode (0, 0) of the top and back plate within 1-2 Hz of presented test data [9], further work will be performed on the finite element model.

First, a sensitivity analysis will be performed to quantify the influence that each feature of the top plate has on the frequency and modal effective weight of mode (0, 0). Each feature will be added in succession, simulating the building process that the luthier will use, and a normal modes analysis will be run. The natural frequency and modal effective weight of mode (0, 0) will be extracted. Next, the same procedure will be followed, except this time varying the material parameters and thickness of the top plate by ± 15%. Again, the frequency and modal effective weight of mode (0, 0) will be extracted using the normal modes analysis. Finally, the top plate will modeled at 3 stages: flat, arched to an initial radius, and then arched an increased radius. Again a normal modes analysis will be run at each modeling step. Once complete, the whole process will be repeated for the bottom plate. The goal is to determine the relative influence different parameters and features have on the frequency and modal effective weight of mode (0, 0). The first portion of the results section will include the natural frequencies and mode shapes of the top and back plate obtained from the FEM compared to the actual testing performed by Rossing [9] to show level of correlation. For the sensitivity analysis, results will include tables and graphs that show the frequency and modal effective weight changes due to feature inclusion, material parameter change, thickness adjustment, and arching of the top and back plate.

In this paper, an attempt is made to perform a sensitivity analysis of the features and parameters of the top and back plate of a Martin D-28 Steel String Guitar using the finite element method, and report on the influence each feature and parameter has on the natural frequency and modal effective weight of mode (0, 0).