Can a brace be used to control the frequencies of a plate?

Abstract

Although many improvements in the manufacturing of guitars have been made recently, one aspect that has often been overlooked is that of the acoustical consistency of the final manufactured product. The aim of this paper is to create a better understanding of the effect of a brace on the frequencies of vibration of the brace-soundboard system. This paper seeks to shed light on why a luthier ‘tunes’ braces when a guitar soundboard is hand-manufactured. A simple analytical model of a rectangular brace and soundboard is derived from first principles using Kirchhoff plate theory in order to develop insight into the effect of the soundboard’s stiffness and brace thickness on the frequencies of the combined system. Natural frequencies and modeshapes of the combined system are calculated via the assumed shape method. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies of the combined system can be adjusted to meet a desired value. However, simultaneously adjusting several natural frequencies cannot be done with a rectangular brace. Therefore modifications to the shape of the brace are explored.

Mathematical Modelling and Acoustical Analysis of Classical Guitars and Their Soundboards

“…Dumond and Baddour [25,26] studied the effects of scalloped braces on mode shapes. A simple analytical model based on Kirchhoff plate theory was used to study the vibration of a rectangular board with and without braces.…”

Structured Approaches to General Inverse Eigenvalue Problems

“…The assumed shape method is an energy method, which uses global plate elements with the kinetic and strain energy plate equations in order to determine the system's equations of motion, from which the mass and stiffness matrices are extracted [18]. For the details of the development of the large mass and stiffness matrices, the reader is referred to the study of Dumond and Baddour [19]. The system is assumed simply supported, conservative and the material properties are assumed orthotropic.…”

“…These equations are broken into three sections as shown in Figure 1 in order to take into account presence of the brace. This procedure is described in detail in the study of [19]. It is assumed that the E y is known and used as input information into the stiffness matrix.…”

A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass

“…This implies that for situations where the lower natural frequencies are of concern, lower degrees of discretization suffice for the engineering design purposes which imply lower order matrices, in addition to lower order polynomials and thus less computation (see Tables 4 and 5). For example, in the design of musical instruments, there is common consensus that the lower modes of vibration are of greater importance [22][23][24].…”

“…The variables 1 , 2 , 1 , and 2 are the unknown variables of the inverse problem. Substituting (24) into (23), a system of four equations and four unknowns is obtained whose solution is presented in Table 6 for the case of a simply-supported beam. In these simulations, = 8 was used and both fsolve and DirectSearch were used to solve the inverse problem.…”