COMPUTER AIDED DESIGN of ACOUSTIC GUITARS George
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COMPUTER AIDED DESIGN OF ACOUSTIC GUITARS George Norman Jenner Project report submitted in partial fulfilment of the degree Master of Science (Acoustics), 1986. 2 TABLE OF CONTENTS Acknowledgements - Page 4 Declaration - Page 5 Summary - Page 6 1. Introduction - Page 7 2. Guitar Construction, Sound Production and Quality - Page 10 3. Finite Element Analysis Theory - Page 23 4. Modelling Strategy - Page 33 5. Results of FEA of Existing Guitars - Page 59 6. Experimental Determination of Guitar Modes - Page 88 7. Comparison of FEA and Experimental Determination of Modes - Page 115 8. A New Guitar Design - Page 124 3 9. Conclusions - Page 139 References - Page 142 Bibliography - Page 147 Appendix A - Results of experimental determination of modes - Page AO Appendix B - NASTRAN Data Deck for Guitar Model - Page BO 4 ACKNOWLEDGEMENTS The assistance from the following people is gratefully acknowledged: Associate Professor Anita Lawrence, Dr. Don Kelly and David Eden for advice and supervision; Gerard Gilet, Simon Marty and Mike Penberthy for help in constructing and analysing guitars and Alan Raymond for assistance in preparing the manuscript. 5 DECLARATION I hereby declare that this thesis is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgement is made in the text of the thesis. 6 SUMMARY A computer has been used to analyse the physical characteristics of classical guitars. To do this, finite element models were built for several existing guitars, and the models analysed using the MSC/NASTRAN finite element analysis programme. Both traditional guitars and modern guitars were analysed and compared. The results of the finite element analysis were compared with results obtained using Fast Fourier Transform techniques and hologram interferometry, so that the validity of the model could be assessed. The physical characteristics analysed were the frequencies and shapes of the resonances of the top plates of the guitars. This was done using the dynamic analysis capability of the programme. Based on the results of the analysis, and on work by previous workers concerning the relationship between guitar quality and its physical characteristics, a new guitar was designed and built. The aim was to build a new guitar which would be louder than traditional guitars without loss of quality. Static analysis was used to ensure the structural integrity of the new design. The guitar was built and the modes of vibration as calculated using finite element analysis were compared with the resonances measured using Fast Fourier Transforms. The guitar was evaluated subjectively and it was concluded that finite element analysis can be a valuable tool in musical instrument design. 7 1. INTRODUCTION A study of the physical characteristics and behaviour of a musical instrument can aid in the understanding of the production of sound by that instrument, and in many instances it can be used to improve the instrument. Although the quality of any instrument is highly subjective with regard to the listener and player, it is usually possible to identify some of the parameters that govern that quality. If those parameters can be controlled by the instrument maker, then the maker can be more confident that an instrument of high quality will be produced. This project is an assessment of the above statement with respect to acoustic guitars. The results of previous workers in the field are reported to identify both important aspects of physics in sound production and the psychoacoustics of guitar quality. The finite element analysis method is used to study the modes of vibration of existing guitars and then to predict the modes of a new design. Results are reported for modal analysis of guitars using Fast Fourier Transform techniques and holographic interferometry. These experiments were performed on: high quality traditional classical guitars; new classical guitars by experimentalist makers; low quality classical guitars and on steel string guitars. The results are compared with those of other workers and are used to evaluate the various criteria which have been shown to be relevant to guitar 8 quality. Those criteria can then be used with confidence to evaluate guitars based on their frequency reponse alone. Guitar quality has been shown to be dependant on the frequency response of the top plate of the guitar (Meyer 1983). If the guitar maker can predict the frequency response of a certain construction, then it is possible to predict the quality of the guitar. The prediction of the response of a new construction is, however, quite difficult, especially for one as complex as a guitar. The finite element analysis method is commonly used for evaluating new engineering constructions but has been used little in musical instrument analysis (Schwabb 1975 and 1983, Richardson 1985, Roberts 1986). Finite element analysis is a well known engineering procedure whereby an object is divided into a large number of small elements. The elements are of simple geometry, and approximations can readily be made to describe their physical behaviour. The whole object is then assembled from a knowledge of the constituent elements and the way in which they are connected to each other. The basis of the element matrix geneneration and assembly procedures lies in the classical principles of minimum potential energy or virtual work, giving the procedure a sound theoretical foundation. The limitation on accuracy is defined by the computing resources available for the analysis with higher accuracy being achieved by dividing the object into larger numbers of smaller elements. 9 The validity of the procedure is shown by analysis of existing guitars. A new guitar design is then devised and analysed, based on precepts of guitar quality determined by other workers. A guitar is designed and built, thereby confirming that the finite element analysis predicted the behaviour of the guitar correctly and that the assumptions about aspects of good guitars are validated. The overall utility of the procedure is discussed, with particular reference to analysis time involved and the cost of the large amounts of computer time which is needed for the analysis. Ji11gcrl•oanl 11 2. GUITAR CONSTRUCTION, SOUND PRODUCTION AND QUALITY 2.1 Introduction The classical guitar has remained virtually unchanged for over a century. Although many small changes have been introduced by luthiers, their guitars are essentially the same construction as developed by the Spanish luthier Antonio de Torres Jurado (1817-1892). The elements of a modern classical guitar are shown in Figure 2.1 (Evans and Evans, 1977). Modern six and twelve string steel string guitars are generally heavier in construction to ensure structural integrity with the increased string tension. The bracing patterns of steel string guitars are different to those of the classical guitar. A common type of construction is illustrated on Figure 2.2 (Evans and Evans, 1977). The remainder of this chapter is concerned with the classical guitar. Sound is radiated from the guitar due to the vibration of the top plate as energy is imparted to it from the strings. The top plate interacts with the rest of the guitar, including the air in the body cavity, and radiation from the whole guitar and from the sound hole determines the pressure field around the guitar. The description of the sound field has been the subject of previous research (Richardson, 1982, Cristensen and Vistisen, 1980, Christensen, 1983, Caldersmith, 1978, Caldersmith, 1981, Jansson, 1971, Firth, 1977). ~,,,,, ./titger/,onr,/ or_fi-e16onr,t /Jftr/l,i1,g or bi11d1i1g heet - •• -doz•etni/ jo,iu hrid._rre .raJJ/t ~p,i, <'llr//,/11,-[, The Parts of a ste.,1 ,,-,1,,., flat top ~ QUita, (!:•ans and E"..,,._,, 1977) 13 The quality of a guitar's sound is judged subjectively. By surveying the properties of guitars, both acoustically and subjectively, the quality of a guitar can be predicted by knowing its physical characteristics (Meyer, 1983). 2.1.1 The Top Plate The top plate or soundboard is the most important radiating element of the guitar. It is usually made from European Spruce or Western Red Cedar, though others timbers are occasionally used and some experiments are even being performed on Australian timbers. The soundboard should be made from high quality quarter sawn wood, though the need for superior quality is argued. Some makers of high renown do not use wood of the best quality but can make high quality guitars by careful control of internal strutting. Richardson (1985) used a finite element model of a guitar to determine the appropriate thickness of a top plate which had been sawn incorrectly and hence had different physical properties to properly sawn wood. Top plates are generally between 1.5 and 2.2 mm thick. Most are thinner at the edges of the guitar which tends to decrease the rotational stiffness of the top plate about the supporting edges. It will be shown that this has the effect of lowering the frequencies of the natural modes. 14 2.1.2 Back and Sides. The back plate and the sides of guitars are nearly always made from Indian or Brazillian Rosewood. The physical properties of both are similar, and there is very little agreement over what is the best construction of the back and sides. The sides are generally about 2 mm thick and because the cross grain stiffness is finite, the compression and bending of the sides must affect the vibrations of both the top and bottom plates. Some makers prefer to make the sides thicker to decrease the effect of this coupling, maintaining that the top plate is the most important radiator and no energy should be wasted by exciting other parts of the guitar.