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Coupled Modes and Time-Domain Simulations of a Twelve-String Guitar with a Movable Bridge

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Coupled Modes and Time-Domain Simulations of a Twelve-String Guitar with a Movable Bridge Coupled Modes and Time-Domain Simulations of a Twelve-String Guitar with a Movable Bridge Miguel Marques ?, Jose´ Antunes ?, Vincent Debut ? ? Applied Dynamics Laboratory, Campus Tecnologico´ e Nuclear, Instituto Superior Tecnico´ Universidade Tecnica´ de Lisboa, Estrada Nacional 10, 2695-066 Bobadela, Portugal and Mechanical Engineering Department, Instituto Superior Tecnico´ Universidade Tecnica´ de Lisboa, Av Rovisco Pais 1, 1049-001 Lisboa. Portugal [email protected], [email protected], [email protected] ABSTRACT model the dynamics of an archtop twelve strings acous- tic guitar. For such purpose, we built a conceptual model Coupling between the different vibrating sub-systems of a for a set of twelve (six pairs of) strings coupling, at some musical instrument is an important feature in music acous- point of the strings (where the bridge would be located), tics. It is the reason why instruments of similar families to a body, through the bridge, which will be modelled as a have such different and characteristic sounds. In this work, spring with a very large stiffness. In this paper, the body we propose a model for a twelve strings (six pairs) guitar, will be simplified as a thin wood plate. Other coupling such that the strings are coupled with the instrument body techniques have already been proposed in the literature of through the moving bridge, which is the relevant compo- musical acoustics. Specifically for guitars, different mod- nent for energy transmission from the strings to the guitar els, which do not take into account the vibrations of the body and back. In this preliminary study, the guitar body is dead side of the string, have already been proposed [1–4], modelled as a simple plate, strings being assumed to dis- but to our knowledge, the sympathetic vibrations of a gui- play planar-only vertical motions. However, the coupled tar with a movable bridge, displaying a dead side of the equations thus obtained can be readily extended to cope string after the bridge, has never been addressed nor mod- with real guitar body modes and orbital string motions. Af- elled. A model of coupling between one string and the ter obtaining the coupled modes of the instrument, we il- violin body, which has the same features of our model, has lustrate the instrument time-domain coupled dynamics, by been proposed by some of the present authors, in order to considering the characteristic modal frequencies typical of address the problem of the wolf note and the nonlinear be- a Portuguese guitar. In particular we show how, when only haviour of the string/bow interaction [5]. one string alone is plucked, energy is transmitted to all other strings, causing sympathetic vibrations, which con- The Portuguese guitar (Cithara lusitanica) is a pear-shaped tribute to give this guitar its own characteristic sound iden- instrument with twelve metal strings (six courses), descen- tity. dant from the renaissance european cittern. This instru- ment is widely used in Portuguese traditional music, mainly 1. INTRODUCTION in Fado, and more recently also started to play a consider- able role among urban Portuguese musicians. Unlike most The acoustic guitar has been the subject of many studies, common guitars, this guitar has a bent soundboard (arched both on its theoretical description and on the experimen- top) with a bridge somewhat similar, although smaller in tal side. Yet, there still are limitations when it comes to size, to the bridge of a violin, a neck typically with 22 fixed the description of real guitars with movable bridges, i.e. metal frets and it is tuned by a fan-shaped tuning mecha- when the bridge role is not simplified as a simple enforced nism, consisting in twelve screws, acting as pegs, mounted pinned boundary condition. Overcoming this limitation, with small gliding pins where the strings are attached to could surely help improving the guitar engineering, e.g. it adjust its tension. It has the typical tuning of the Euro- could allow us to optimise the bridge position in order to pean cittern tradition, and has kept an old plucking tech- achieve the best radiative acoustic response, without loss nique, described in sixteenth century music books. The of the archtop guitar sound characteristics. first courses are composed by plain steel strings and tuned Motivated by the wish to extend our understanding of in unison, and the remaining are combinations of a plain the acoustics of the Portuguese guitar, we set ourselves to steel string and an overspun copper on steel string tuned one octave bellow. There are two different models of the Copyright: c 2013 Miguel Marques et al. This is an open-access article distributed Portuguese guitar: the Lisbon guitar and the Coimbra gui- under the terms of the Creative Commons Attribution 3.0 Unported License, which tar, named after two Portuguese cities where the two most permits unrestricted use, distribution, and reproduction in any medium, provided important Fado styles emerged. They differ in some de- the original author and source are credited. tails, such as the body measurements, the string length (the where j = 1; :::; J is related to the number of nodal lines along the xx direction, l = 1; :::; L is the number of nodal lines along the yy direction, mjl are the modal masses, !jl are the modal angular frequencies, ζjl are the modal damping coefficients, and Fjl are the generalized forces. To simplify the formulation, these equations can be or- ganised in order to have a single modal index g, with a total of G = J × L modes, by ordering the frequencies in an ascendent form. The modal masses of the body are LxLy mg = ρ 4 , with ρ being the wood surface density, and Lx;Ly the plate dimensions. Each string will have its own set of modal dynamical equations, given as s s 2 s s mnq¨n(t) + 2mn!nζnq_n(t) + mn!nqn(t) = Fn(t); (3) Figure 1. The master musician Carlos Paredes, playing the where n = 1; :::; N are the modes of each string s. The Coimbra Portuguese Guitar. Photo credits: Egidio Santos. s Ls modal masses of each string are mn = µ 2 , with µ being the linear density of the string, and Ls the size of the string Coimbra guitar has a larger arm), and the Lisbon guitar is (throughout this work, we will assume that all strings have tuned (each string) a whole tone above the Coimbra guitar. the same size, which is the case for the Portuguese guitar). Strings used in these two guitars are therefore different in Furthermore, the string and the body will have modeshapes size and linear mass. More about the Portuguese guitar can respectively given by be read in [6, 7] and about experimental vibrational mea- nπx surements of this instrument in [8]. In figure1, a Coimbra s (x ) = sin s ; n s L model of the Portuguese guitar is shown as an example. s jπx lπy b (x ; y ) = sin b sin b ; (4) jl b b L L 2. GUITAR MODEL x y and eigenfrequencies (assuming no inharmonicity) respec- Throughout this section, we will give a full description of tively given by the computational model proposed in this paper. For sim- plicity, in this conceptual model, we will replace the guitar s s !n = n!0; body by a thin wood rectangular soundboard plate. Al- s " # E jπ 2 lπ 2 though it would be easy to implement in the model, we !b = h y + ; (5) jl b 2 will neglect any energy transfer occurring through the nut 12ρ 1 − vp Lx Ly or the fingerboard, as there are experimental studies reveal- where !s is the s string fundamental frequency, h is the ing that this effect is not much relevant, and we will also 0 b plate height, E is the Young bulk modulus, and v is the assume the energy transfer occurring at the tailpiece to be y p the Poisson coefficient. Throughout the paper, we will re- negligible, due to its location on the Portuguese guitar ge- fer to !b as !b, and to b (x ; y ) as b(r), assuming a ometry. Therefore, our fully coupled system will consist jl g jl b b g proper indexation mapping g ! (j; l), according to what in a rectangular plate with fixed ends, and twelve strings, has been previously stated. Given a general force F (x; t), attached to the plate trough a set of springs, displayed in acting on a surface S of a vibrating system with mode- six pairs. For simplicity, we will only consider string vi- shapes (x), the corresponding forces on the modal space brations in the direction normal to the soundboard plate, m will be although it would be relatively straightforward to imple- Z ment the other direction as well. We will decompose the Fm(t) = F (x; t) m(x)dS: (6) S strings and the plate displacements respectively as s b s Let us define Fc (xc; t) and Fc (rc; t) as being the force ex- 1 erted on the bridge, as seen, respectively, from each string X s s s Ys(xs; t) = n(xs)qn(t); and the from body at each coupling point rc (we are as- n suming that all strings will meet the bridge at the same dis- 1 tance from the nut, xc). We will consider that the bridge X Z (x ; y ; t) = b (x ; y )q (t); (1) his thin enough such that it can be modelled as a spring b b b j;l b b j;l with very large stiffness. Therefore, these forces are j;l s s s Fc (xc; t) = Kc [δ(xs − xc)Ys(xc; t) + δ(r − rc)Z(rc; t)] with the script s = 1; :::; S, for a total of S strings; where h s s i +Cc δ(xs − xc)Y_s(xc; t) − δ(r − r )Z_ (r ; t) Y is the displacement for each string, Z is the displace- c c s b 8 9 Ns ment of the body, stand for the modeshapes and q stand < X = = δ(x − x ) s (x )[K qs (t) + C q_s (t)] for the modal amplitudes.
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