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
or bi11d1i1g /Jftr/l,i1,g heet jo,iu -doz•etni/ - ••
.raJJ/t hrid._rre
~p,i,
<'llr//,/11,-[,
top ,,., flat 1977) ,,-,1 and E"..,,._,, of a ste.,1 (!:•ans The Parts ~ QUita, 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.
The back plate is also usually about 2 mm thick and has three structural struts of spruce for support. The resulting stiffness determines the amount the back plate couples to the sound waves in the air cavity of the guitar. Sometimes the back is made very stiff and polished smooth to reflect these sound waves. Others prefer a less stiff plate which will absorb and reradiate sound. Such guitars usually have less sustain, the energy being transferred quickly from the strings to a light body. Flamenco guitars are such light guitars and typically have poor sustain but a sharp attack. 15
2.1.3 The Braces
The typical classical guitar of Figure 1 exhibits the normal type of internal bracing of the lower bout of the top plate. The "fan bracing", using spruce braces has been used by most luthiers since
Torres. Only recently are top quality guitars using very different systems being built. Modern materials such as carbon fibre and epoxy resins are also being used.
The bridge is the only brace on the outside of the guitar body and is typically made of Brazillian Rosewood. It is a critical element of the guitar as it not only provides structural support, but is also the member through which energy os transferred from the strings to the sound board. Modern makers have experimented widely with different bridge geometry (Evans and Evans, 1977, Marty, 1985), confirming that it is very important to the guitar's sound, however most still use bridges of traditional shape.
2.1.4 Other Elements
The purfling and linings of the guitar exist mostly for structural reasons but both can have a great effect on the coupling of the top and bottom plates to the sides of the guitar.
The type of strings used on the guitar can have a great influence on the guitar sound. Different types of strings suit different guitars 16 however it will be assumed in the remainder of this work that for evaluation of guitars, the strings are largely irrelevant, though it would be a gross error to ignore them completely.
The final, and perhaps most important consideration is the player.
A good player can make even poor instruments sound fine. Later chapters will justify this reseach in terms of player's and listener's criteria.
2.3 Sound Production
The mechanism of sound production by the guitar has been the subject of previous research (Richardson, 1982, Cristensen and Vistisen,
1980, Christensen, 1983, Caldersmith, 1978, Caldersmith, 1981,
Jansson, 1971, Firth, 1977). Sound is produced by transfer of energy from a vibrating string to the top plate. Sound is produced until sufficient energy is lost in the form of acoustic radiation or due to the internal damping of the structure. Most of the sound is radiated from the top plate and from the sound hole. The sound is controlled by the normal modes of the guitar structure. The contention of this thesis is that the quality of the sound can be predicted to an extent from the knowledge of the normal modes of the top plate alone. In an assembled guitar the top plate vibrations
couple strongly with the sound waves in the body of the guitar.
Hence a model which does not include the air cavity can only be
approximate. - /"' ! - .... / / I f\ "' [\ , I I __ '" I ~ ~ I I .,__..__, I I '' I\ I\ I ,\ '' - J, A ...... f),, I\ , I, I,., .... i\ _,.,r-- \ I\ \ ,, r, l.J' t-."-l'I " I 1,., ,_ r-- \ ,- ~ .... \ l'i. v.~L,,,, ,.._r,.. I" I' '" ~ I'\ I\ /J "'I .... I\ V,W 1/1., h r---1\ '\[\ " \ I\ I 'J I\ V '\ " ~ J I '# :J/ I I I/ \.J'\ l\l\ ' " ~ I\ I/ ii I\ ""1,,r--, ~\ I\ \ 1 ~ I\ I ~ ! I fl "'' I I • l ,I.\ r--,._ L. ·1 j,' ') 1/ i ,1, I I/ ~1:1 \\ "'"'~~ ·--:;:f"'Y'l; ,;;, 'J/ \I\ \ I/', " ,r-- ~ \ /1/ .... ~ loo.I,,< ,. c., ""''- -?- ..... ""~i.- ... .. L,,,,
Monopole (200 Hz) Cross dipole (300 Hz)
,,.(1 I r-,. ~ ' /,..,, I I\ I ""I\ I 1, - I _._ \ - - ..,,v I I I I\ \ I i 1 I\ I\), I/ ' ~-r-- I\). .... 1, .. _I\ J ,,, "- [\ IJ I _1,.,, '-t-,. I'- ,, ~" 1, L,,,, \ vl) \ /Y \ '-N /'/ J ~ IJ' I'- I\ I i\ ' .... ,.Lo~ 't,I\ ...... I -,II f\ I. ~7'. j ,. ·--i ,, I 1, I IJ I?"'- 0,. J' ,-;.., " rt ir I I , ,.. ' I,., w"- Hi -'}.; I j N--J. 1\I~, L• i,. 11 1'1,, "~ ,,. ' :'J..,.'. i/'j/lA, \I' .... ~IJ t, " ~ ' f\ J '' ':' r, >/, t '·-':" t:,J--j.j' . I' ,_Vl,i ' "" - ' - .. - ... ,_
Long dipole (40Q Hz) Tripole (500 Hz)
Figure 2,3 First four top plate modes of a typical calssical guitar. Frequencies in brackets are indicative only. 18
In the model, only the solid elements of the guitar are included and approximations are made to determine the effect of the air cavity.
The first four normal modes of a typical guitar top plate are shown on Figure 2.3, as well as an average value for the frequencies at which they occur. These computer printouts show contours of equal displacement for the modes. These four modes occur on nearly every guitar so it is convenient to name them the monopole, the cross dipole, the long dipole and the tripole as indicated on Figure 2.3.
Other schemes for identifying modes have been used in the literature but they generally are confusing when higher order modes are discussed.
Figure 2.3 does not show that there is always a resonance due to air in the cavity of the guitar, sometimes erroneously named the
"Helmholtz" resonance. This is a misnomer since the sides of the body can vibrate and have the affect of lowering the frequency of resonance. The "Helmholtz" resonance would occur only if the guitar was rigid and there was no coupling between components. This mode will be termed the "cavity resonance" for the rest of this report.
The coupling tends to lower the cavity resonance and raise the first top plate resonance. Since the computer model predicts the frequency of the top plate modes in the absence of air in the cavity, it is important to understand the effect of the air cavity
(Christensen and Vistensen, 1980). 19
Let: f = uncoupled first top plate resonance (monopole) f f11 = uncoupled "Helmholtz" resonance
f+= coupled first top plate resonance
f_= coupled air cavity resonance.
Then
7. ~ f~ + f (2.1)
Hence if the computer model predicts a value for f., then one can ( use average values of existing guitars for f~and f_ to predict f+ , the actual value desired.
The higher modes also couple to the air if there is a net mass flow through the sound hole. The effect is therefore less than for the monopole and is ignored in this study. This study concentrates mainly on the frequencies and shapes of the first four or five modes of the top plate, assuming that a good guitar can be made by controlling these modes. The higher modes are more difficult to predict and vary widely between guitar, however their existence is critical as they cover the higher frequency notes of the guitar.
Some argue (e.g. Richardson, 1982) that it may be important to place the higher modes exactly where they will enhance the high frequency function of the guitar to greatest effect. 20
The bridge and strings are important in sound production
(Richardson, 1982). The height of the bridge governs the coupling between the longitudinal string motion and the body. The treble strings decay more rapidly beacause of their higher internal damping. When a player releases a string the body modes are excited by deformation and relax at a rate according to the damping of the guitar at the frequency of that mode.
2.4 Quality Studies
It is desirable that any results concerning the physical acoustics of a musical instrument can be related to the subjective quality of that instrument. An important study by Meyer (1983) is the first such work on this subject for guitars. In that work guitars were judged according to the results of many listening tests using taped guitar music. The rated guitars were then tested to determine their frequency response. The conclusion was that certain aspects of the frequency response of the guitars (i.e. the set of normal modes) were most important in producing the most desirable tone for a guitar. The nine most important aspects are listed in Table 2.1.
The modes can be described in terms of their frequency, gain and quality factor (Q). The gain of the mode is the peak level of the mode. The Q is a measure of the sharpness of the the mode on a frequency response curve and is determined by the damping of the structure at the frequency of the mode. The resonance rise of a 21 mode is the amount by which the peak level of the mode is lifted above the background of the mode.
TABLE 1.
The quality critia of guitars in descending order of importance
(Meyer, 1983)
1. Peak level of long dipole resonance (at about 400 Hz)
2. Resonance rise of long dipole resonance
3. Quality factor of long dipole resonance
4. Average level of third octaves 80-125 Hz
5. Average level of third octaves 250-400 Hz
6. Average level of third octaves 315-500 Hz
7. Average level of third octaves 80-1000 Hz
8. Peak level of monopole resonance 22
9. Average level of third octaves 800-1250 Hz.
It is clear the long dipole resonance is vital to a good guitar.
Guitar makers recognise the importance of the long dipole. In this study, the end result will be to produce a guitar with this resonance carefully controlled. Note that in Meyer's study, the frequency analysis of the guitars were obtained by exciting· the guitars at the centre of the bridge. Under these conditions the cross dipole is not excited since its nodal line runs through that point. Hence Meyer's evaluation ignores this mode. The contribution of the mode is small but not negligible (Caldersmith,
1985) especially off the axis normal to the guitar top plate.
Others (Richardson) have suggested other criteria for quality, including-
1. Loudness
2. Evenness of tone and ease of sound production
3. Range of sounds available
4. Sustain
5. Accuracy of intonation
6. Ease of playing
7. Appearance and durability. 23
3. THEORY OF FINITE ELEMENT ANALYSIS
3.1 Introduction
Finite element analysis is an established and accepted method in many aspects of engineering. A full treatment of the theory of finite element analysis is beyond the scope of this thesis but the general procedures and equations solved are described. More complete descriptions are given in the references (Cook 1981,
NASTRAN Manuals) which are the sources of the following outline.
The method involves describing a structure in terms of small simple pieces which themselves are easy to describe, for example regular solids and plates and beams. The interaction of the composite elements with each other yields information on the behaviour of the whole structure. The accuracy of the model depends upon the fineness of the elements and the solution method.
The finite element programme used in this analysis is MSC/NASTRAN which is the MacNeal Schwendler Corporation's version of the
National Aeronautics and Space Administrations Structural Analysis
Programme. This programme is used commercialy for structural analysis and is available on the University of New South Wales computers under a Special Universities Agreement which permits its use for teaching and research. The programme is extremely powerful allowing complex static and dynamic solutions including flui
structure interaction. 24
3.2 Notation
This is a list of symbols used in this chapter.
{} A vector.
A matrix.
Time differentiation.
d.o.f. Degree of freedom.
[B] Structure damping matrix.
{D} Nodal d.o.f. of a structure.
{d} Nodal d.o.f. of an element.
[K] Structure (global) stiffness matrix.
[k] Element stiffness matrix.
[M] Structure (global) mass matrix.
[m] Element mass matrix.
[N] Matrix of shape functions.
{R} Total load on structures.
{r} Forces applied by element to nodes.
{P(t)} Time dependent applied force vector.
{Q} Vector of forces resulting from constraints.
{u} Vector of grid point displacements. 25
3.2 Basic Relations
An element is a small fraction ·of a complicated structure. Elements generally have simple geometry and known properties. Adjacent elements are connected at nodes and neither separate or overlap when the strucure is stressed. The element arrangement used in the analysis of the guitar is shown in Chapter 4
A degree of freedom (d.o.f.) is defined as the displacements or rotations of a node. For an element with n d.o.f. we an write
k d + k d + .... + k d = r
11 1 12 2 ln n 1
k d + k d + .... + k d = r
21 1 22 2 2n n 2 (3.1)
k d + k d + ..•. + k d = r
nl 1 n2 2 nn n n
where d; is the ith d.o.f. and r is the corresponding force or moment applied to the element. The k-•AJ are the stiffness coefficients. In matrix form, equation 3.1 becomes 26
[k] {d} = {r} (3.2)
where [k] is the element stiffness matrix, {d} is the element nodal displacement vector and {r} is the vector of element nodal forces.
The matrices and vectors for each node are combined to form the global matrix and vectors. For example, if all stiffness matrices are expanded to operate on identical displacement vectors and all have as many rows and columns as [K] , elements can be directly added to the global matrix. The complete model is described by
[K] {D} = {R} (3.3).
After input of stiffness and force data for each node, equation 3.3 is solved for {D} (using Guassian elimination for example). Since
{d} for each element is contained in {D}, deformations of each element are known.
The generation of the element matrices (3.2) and the assembly procedures to give the global model (3.3) are based on classical properties in continuum mechanics such as the Principle of Minimum
Potential Energy usd by Cook (Cook, 1981). Reference should be made to his text if more details are required. 27
3.3 Dynamic Analysis
The main problem solved in this experiment is the extraction of eigenvalues or resonant frequencies of the guitar. The data input is similar to that of static analysis but the solution is more complicated, involving the set of second order differential equations:
[M] {u''} + [B] {u'} + [K] {u} = {P(t)} + {Q} (3.4)
where {u'} = {du/dt} and {u''} = {du'/dt} and the other symbols are as listed in section 3.2. Equation 3.4 is a statement in matrix form of the equilibrium of forces and moments at all grid points.
The off diagonal terms in the matrices represent coupling between the d.o.f.
To determine the eigenvalues, it is assumed that [B] and {P(t)} are null, and that all parts of the system are vibrating sinusoidally with the same radial frequency, w, and the same phase, i.e., that
(3.5)
where{~} is a vector of real numbers and cos wt is a scalar multiplier. Substitution of equation 3.5 into the simplified form of equation 4 gives 28
[K - w~M] {~} cos wt= 0 (3.6)
or, since equation 3.6 must be valid at all values of time,
(3.7).
This equation has non zero solutions if the matrix in the first term is singular. This happens only at frequencies called eigenfrequencies. Each eigenfrequency and eigenvector ( {Oi} for the ith mode) which satisfy equation 3.7 define a free vibration mode of the structure. In the absence of damping and non linear effects, the structure can vibrate indefinately at a modal frequency and with mode shape defined by {Oi}.
3.4 Methods of Eigenvalue Extraction
NASTRAN provides three methods for extracting the solutions to equation 3.7 (Gockel, 1983). The Givens Method and the Modified
Givens Method are 11 transformation11 methods in which the matrix in
3.7 is first transformed, while preserving its eigenvalues, into a special form from which eigenvalues may be easily extracted. The third method is the Inverse Power Method with Shifts which is a
11 tracking11 method in which the roots are extracted one at a time using iterative procedures applied to the original dynamic matrix. 29
The selection of an appropriate solution method is dependant on the complexity of the problem and the information required. The Givens and Modified Givens Methods are fastest for small problems and all the eigenvalues are extracted. For large problems dynamic reduction
(section 3.5) may be required or the Inverse Power method may be appropriate. This method will not extract all roots but is more accurate then the other methods.
3.5 Dynamic Reduction
A dynamic analysis can be divided into three phases: assembly of the dynamic equations, solution of those equations and recovery of response quantities such as force and stresses. As the problem size increases the cost of the first and third phases increases linearly with the number of degrees of freedom, whereas the cost of the second phase increases as the square or cube of that number, hence the second phase is nearly always dominant for problems with more than about 50 degrees of freedom (Gockel, 1983).
Dynamic reduction is a solution step between the first and second phases which reduces the cost of the second phase by condensing the physical degrees of freedom into a smaller analysis set. The procedure introduces approximations to the analysis but with careful
selection of the degrees of freedom to be retained the errors are minimised. The Guyan Reduction method (Schaeffer, 1982, Guyan,1965) 30 has been used in this project. Guidelines for selection of the degrees of freedom are (Levy, 1971):
1. Retain degrees of freedom with large inertia.
2. Retain degrees of freedom with large accelerations.
3. Retain approximately 3.5 times as many well chosed degrees of freedom as the number of accurate modes desired in the solution.
4. Retain displacement degrees od freedom for bending problems.
3.6 The NASTRAN Programme
After deciding which elements, materials properties and solution process to be used in an analysis, the data must be entered into the computer in a form which can be understood by the programme. For
NASTRAN this is achieved by using a data deck of "card images", one line in the deck for each card. The data deck is partitioned into three sections: the Executive Control deck; the Case Control deck and the Bulk Data deck (Schaeffer, 1982). The full data deck for the prototype guitar designed in this project is listed in Appendix
B. 31
3.6.1 The Executive Control Deck
The Executive Control deck provides the user with an interface to the host computer. Dynamic or static analysis is specified here as well as data blocks to be saved for use in later analysis. The user can specify the maximum time allowable for the solution to be computed, thereby ensuring that computing time is not wasted on a data deck which may be be ill defined. The Executive Control deck is terminated by the CEND card.
3.6.2 The Case Control Deck
The Case Control deck performs the following functions to give user control over the programmes input/output facilities:
1. Specifies the set of Bulk data to be included in the analysis set at execution time. This allows the user to specify more then one set of loads or constraints (for example) in the data deck, but to use only a subset of those constraints or loads.
2. Select solution techniques as appropriate.
3. Control the calculation and display of derived quantities, grid point displacements and plots for example. 32
4. Control subcases. Subcases allow changing the output directives for each loading condition.
3.6.3 The Bulk Data Deck
After the Case Control deck is placed the BEGIN BULK card indicating the beginning of the Bulk Data deck. The deck defines the physical problems or the system of equations to be solved. This is achieved by specification of such things as: the grid point locations in space; the element connectivities; the element properties; the material properties; the constraints; the loads; the eigenvalue extraction technique to be used etc.
The first words on the cards are generally descroptive of their function. For example, the GRIDG card defines a grid point location. The CQUAD4 card defines a quadrilateral shell element. A description of the data card is given in the MSC/NASTRAN Primer
{Schaeffer, 1982). The data deck is terminated with the ENDDATA bulk data card. coordinate system
top plate
back and sides
Figure 4.1 Arrangement of shel.l elements 34
4. MODELLING STRATEGY
4.1. Introduction
The selection element types to be used in a finite element model, the specification of those elements to the programme and the algorithms used to solve the equations is critical in obtaining accurate answers in minimum computing time. The programme used,
NASTRAN, is a very large, complicated and powerful programme.
Because of the complications, a significant proportion of the project involved learning the techniques of good modelling.
In the finite element model, most of the elements are modelled as two dimensional shell elements. The arrangement of these elements is shown on Figure 4.1. The ·shell element allows the stiffness to be different in two directions, critical for modelling wood. The side and bottom were modelled quite coarsely as shell elements.
Sometimes these elements were not included in the model and the behaviour of the top alone was analysed.
All the struts in the guitar are modelled as beam elements. NASTRAN only allows isotropic beams to be modelled, hence for wooden beams, only one stiffness value was allowed. The stiffness along the grain was used as it is obviously the most important. Errors must be
introduced for modes in which the beams were twisted. The increased overall stiffness would raise the natural frequencies of the modes. 35
Normal engineering theory of bending was used to determine the moments of inertia of the beams.
A regular grid pattern was chosen for the modelling of the top plate. Other studies (Schwab, 1975, 1983 and Richardson and Roberts
1985) have used element patterns which follow the lines of the bracing underneath the top plate. NASTRAN allows the user to specify an offset of the beam elements from the points to which they are connected. Since more than four different guitars were to be modelled this facility was used to model each guitar using the same grid pattern, greatly reducing the time necessary to prepare data.
To test succesive guitars, therefore, it was only necessary to input a new data set for the beams, and to change the values of element thickness and materials etc.
The linings around the join between the top and sides of the guitar were modelled as a series of beams. This was necessary as the modelling of that boundary is critical to accuracy. Slight changes in the thickness of the top plate at that boundary resulted in large changes in the results. 1. 70 Hz
2. 240 Hz
3. 453 Hz
4. 580 Hz
Figure +.:i. Modes of vibration of simply supported rectangular plate. Modelled using quarter symmetry.
'I 'I 37
4.2. Testing of Square Plates
A rectangular plate was analysed using finite element analysis for several reasons. Firstly to check that the programme was working correctly on a data set supplied in the NASTRAN Primer (Schaeffer,
1979). The vibration modes calculated by the programme were compared to those calculated from theoretical formulations (Hearmon,
1952). Also, by observing the computer time used for plates of different numbers of elements, one can approximate of the number of elements allowable in the guitar model which could be analysed using the allocated resources.
4.2.1. Theory
The exact solution for a plate with simply supported edges is given by Hearmon (1952) as
~ ~ k=wb (M/D)
where
D=Eh~ /12(1-v)~ 38
M=mass per unit area
w=angular frequency
b=length of side of plate
and k is a constant dependant upon the ratio of the length of the sides and the boundary conditions.
4.2.2. Results
The results for theoretical and computed modes of a rectangular plate are given in Table 4.1. Results were calculated using a plate with 4,16,264 and 400 elements. Plots of the modes are shown in
Figure 4.2 for the 4 and 16 element plate. 39
Table 4.1 Frequencies of vibration of rectangular plate
a=762mrn b=508mrn
E=2*10 Pa t=5.lmrn density=7833 kg/m.
No. Frequency of vibration [hertz]
Elements Mode Number Computing
1 2 3 4 5 Time [s]
2*2 66.4 244.8 442.2 623.7
4*4 68.5 235.5 448.2 580.5 582.1 :61
16*16 69.3 239.7 453.4 580.7 619.7 :149
20*20
Schaeffer 69.4 239.9 453.6 581.0 621.1 :1262(Vax)
:>1024
: (Cyber)
16*16
Guyan Red 69.4 240.6 457 598 633 :323 40
Theory 69.48
The results for the first mode of vibration show that good results are achieved with only a four element modelled and symmetry conditions used. Sine only one quarter of the plate was modelled this effectively increased by a factor of four the number of elements used in the complete model.
Table 4.1 also shows the results of a run using Guyan reduction.
For this run there were 16 points included in the set of allowed degrees of freedom. Note that although the first mode is predicted accurately, the results for the higher modes are usually predicted to be higher than those calculated without reduction, however the computation time is less than half for the same number of elements.
The computation times shown, except for the 20 * 20 result, are for the Cyber computer at the University of New South Wales. 41
4.3. Selection of Solution Method
4.3.1. Inverse Power Method
This method was used to compute resonant frequencies of complete guitars. It uses more computer resources than the other method and only finds a few modes, but it is the most accurate.
4.3.2. Givens Method and Guyan Reduction
This method was used for most of the computing as it requires less computer time to extract all the modes of vibration. Comparison with results of the Inverse Power Method showed that the accuracy was not overtly compromised by using this method.
The procedures outlined in chapter 3 for choosing the degrees of freedom to be included in the reduced set were followed.
Approximately 40 points could be included in the set (due to computing time limits) and these were chosen from the points distributed over the top plate and at the edges of the top plate.
The points were symmetric with respect to the centreline of the guitar.
If the whole guitar, including the bottom and sides, was analysed as
if it were in free space then the programme wasted time by
calculating the rigid body modes for a free body, which should be 0 42
Hz. Since only a limited number of modes can be calculated accurately this was a waste of resource. Also, some of these modes did not turn out to be at O Hz, indicating that the Guyan reduction was not acting normally.
A constraint in six degrees of freedom was placed at every point around the bottom of the guitar. This allowed for no rigid body modes and that only the top plate modes would be extracted. The extra constraints did raise the frequencies of the top plate modes, but the errors were small at low frequencies.
To furthur reduce the number of degrees of freedom and the computing time required, those degrees of freedom which would have small displacements or rotations were purged from the data set. The points at the sides of the top plate were allowed to remain free in six degree of freedom. The other top plate points were allowed only vertical displacement and rotation about the x and y axes. The x axis was defined to be parallel with the top plate and along the centre of the guitar and theh y axis was parallel to the top plate and normal to the x axis. These axes are shown Figure 4.1.
4.4. Static Analysis
Static analysis was used to check the deformation of the top plate under the influence of strings. In this way, the structural
integrity of the new design could be checked by comparing the 43 deformation with that of existing guitars which do not fall apart under influence of that force.
4.5. Element Properties
Finite element analysis allows the analyst to choose any value for material properties of an element. Knowledge of the material properties is essential for accurate modelling, and the aquisition of those properties can sometimes be difficult, especially for dynamic analysis, where damping and frequency responses can be relevant.
4.5.1. Shell Elements
The top, bottom and sides of the guitar have been modelled as two dimensional "shell elements". NASTRAN allows selection of different material properties for bending behaviour, membrane behaviour, transverse shear behaviour and membrane-bending coupling. In this analysis, properties for only the first two of these have been entered, and in general the same properties for bending and membrane behaviour have been used.
NASTRAN allows for the shell elements to be orthotropic. Different
stiffness values have therefore been entered for wood stiffness along and across the grain. Values used for wood are generally 44 those given by Haines (1979,1980) and are listed in Table 1 below.
Note that the material properties of wood, particularly density and stiffness, vary from sample to sample. Consequently it would be essentially impossible to build two identical guitars, and modelling can never be exact. Construction details also contribute to this.
The properties used in this analysis are density, stiffness, and Poisson's ratio. Poisson's ratio is assumed to be 0.3 when an empirical value is not available. 45
Table 4.2. Mechanical properties of guitar woods
Species Density Youngs Modulus [MPa]
3 Along Across
[kg/m Grain Grain
European Spruce 460 15000 760
Western Red Cedar 390 9100 720
Indian Rosewood 730 13000 2600
Brasillian Rosewood 830 16000 2800 1 } []
x- axis along centre of beam :> y !"ocal coordinate system z a) Beam of rectangular cross section
rd carbohribre faces l.
balsa core- groin parallel to direction of beam
b) Cross sectiooof sandwich beam
j!='igure 4.3 Beam Geometries 47
4.5.2. Beam Elements
To model the braces or struts of the guitar, the standard NASTRAN beam elements were used. The beams are connected to the grid points which define the geometry of the model. Each brace was modelled as several beam elements end to end. A new element was formed each time the brace crossed the point of connectivity of two adjacent shell elements in the top plate. The ends of the beams were connected to the grid points which most nearly coincided with the end of the beam element. The NASTRAN manual states that four elements are usually enough to accurately model a beam and this criterion was generally achieved except for very short braces.
Because the regular array of grid points did not allow coincidence exactly, a NASTRAN feature was utilised which allows the user to specify the distance by which the ends of the beam are offset from the grid point. By using this feature, it was possible to model all the guitars by using the same data set for all of the guitar except the braces. Different guitars were analysed by simply exchanging the brace data subset. This was considered the most efficient way of comparing the different guitars and was valid because the most significant difference betweem the guitars was the bracing pattern.
Beam geometry is shown on Figure 4.3 for both rectangular cross section beams and sandwich beams. Other beam geometries used were 49
Iyy= (4.2)
12
where:
Izz is the second moment of area about the local beam z axis;
Iyy is the second moment of area about the local beam y axis; w is the width of the beam; his the height of the beam;
For the sandwich beams, the properties are as follows if the core is ignored:
EA= Ef*Af + Ec*Ac (4.3)
where:
Eis the Young's modulus of the combined elements;
Ef is the Young's modulus of the faces (125000 MPa for carbon
fibre);
Af is the area of the faces;
Ee is the Young's modulus of the core (120 MPa for balsa). 48 triangular and "C channel" beams for the bridges of the guitars.
The beam properties were adjusted on the PBEAM cards in the NASTRAN bulk data deck.
4.5.3. Beam Properties
Apart from material properties such as Young's modulus and density, the geometry of the beams must be described for the programme to interpret its effect on the structure.
The cross sectional area of the beam is entered as well as the
"moments of inertia" or "second moments of area". These are described in full in the references (Timoshenko, Singer) and the results are only given here for the relevent geometries. With reference to Figure 4.3, the second moment of area for a beam of rectangular cross-section are:
Izz= (4.1)
12 50
Izz= *2 (4.4)
12
'3 2. wt d
Iyy=(----)*2 + wt(---) )*2 + (4.5)
12 4
where:
t is the thickness of the faces and dis the distance between the centres of the faces.
4.5.4 Modelling Limitations
The programme only permits the use of beam elements with isotropic stiffness. The stiffness along the grain, and therefore along the beams is most important, however cross grain stiffness can be important for modes where a beam may be twisted.
The shear modulus, G, of an isotropic beam is given by 51
E
G = ------(4.6)
2(1 + v)
where Eis the Young's modulus along the grain and vis Poisson's ratio. The calculated value from the equation gives G=5770 MPa, using typical values of 15000 MPa for E and 0.3 for v. The value has been measured to be 850 MPa (Haines). To compensate for this discrepancy we followed a method by Roberts (Roberts 1986) of introducing a correction factor to the beam geometry.
The torsional constant of an isotropic beam is defined as
MT
J = ------(4.7) ,G
J= 1/16*hw~[16/3-3.36*w/h(1-w/12h+)] (4.7a)
for w where MT is the torque and i the angle of twist per unit length (Timoshenko, 1956). 52 The torsional rigidity, C, of the beam is given by C=JG. Therefore to change J to suit the orthotropic beams, the calculated J is multiplied by 850/5770, being the ration of the measured rigidity to the calculated rigidity. This method of attempting to increase the accuracy of the model was found to be limited in practice as it was found that changing the value of the torsional constant had little effect on the modes of vibration. Results for different values of torsional constant are given in later chapters. 4.6 Damping The resonant modes of a structure are described in terms of frequency, gain and Q factor. The Q factor is a measure of damping in the structure at that frequency. Although it was not done for this project, it is possible, using finite element analysis, to calculate a complete frequency response of a structure, if the damping is known. This chapter describes the importance of damping to the project and reports on the experimental determination of the damping of some of the struts used in guitar making. 4.6.1 Modal Damping The importance of the damping of the modes of guitars is illustrated by the study of Meyer (1983). This study has been mentioned earlier 53 as being the source of the premises upon which the guitar designed in this project was based. Meyer found that the resonance around 400 Hz should have a high Q factor. Most of the good guitars analysed by modal analysis (Chapter 6) showed this feature, particularly the Kohno SO and the Gilet classical. Some, however, had Q factors as low as or lower than the Aria plywood guitar for that mode. It is also considered that the first top plate mode at about 200 Hz should have fairly low Q, otherwise the notes around that mode will be uneven in tone and loudness. The design of the new guitar has taken these factors into account qualitatively only due to the difficulties in aquiring accurate damping data. The damping of the entire structure must be known for analysis. This can be approximated with knowledge of the damping of the composite materials, yet is complicated by many factors. The damping of a type of wood can vary for different samples. Energy dissipation at the interface of components, by slip and other effects, must be considered and these depend upon so mccy factors of construction that prediction of damping is extremely difficult. According to Lazan (1968) the damping in a flat plate can be due to the damping in the plate material itself in many cases. (beam clomped to table t"---.!•~======~======i=''-~ metal disc table top ~ displacement measuring probe ~ proximitor to power probe B p 0 oscilloscope Figure 4.4 Schematic of damping measurement experiment 55 4.6.2 Tests of Materials An experiment was performed to determine the relative damping in struts made from spruce and from carbon fibre/balsa sandwich. 4.6.3 Instrumentation and Measurement The instrumentation used is illustrated in Figure 4.4. The struts were clamped to a table top with a length poking over the edge. The experiment would measure the damping in the struts when vibrating as a cantilever beam. A small metal disc was attached to the end of the beam. An eddy current proximity probe, Bently Nevada 7000, was used to measure the displacement of that metal disc. The probe in combination with a Bently Nevada proximitor outputs a voltage directly proportional to the distance to the metal, provided that the metal stays within a certain range. The beam was excited by an impuse and the resulting movement of the metal disc was observed on an oscilloscope, Tektronix type 5103N. Expected page number is not in the original print copy. 10 ms/div a) Length= 190 mm Frequency= 77 Hz Logarithmic decrement= 0.06 f • • ,,- I I , . . t--t-t-- 1 I b) Length= 110 mm Frequency= 316 Hz Logarithmic decrement= 0.09 Figure 4.5 Oscillation of Spruce Beam 58 4.6.4 Results Figure 4.5 shows the results of the vibration for a spruce beam 4 mm wide and 5 mm high. Figure 4.6 shows the results for a carbon fibre/balsa sandwich beam 4 mm wide and 4 mm high, the carbon fibre faces being 0.5 mm thick. The damping is determined measuring the amplitude of succesive peaks. The natural logarithm of the ratio of those amplitudes is the logarithmic decrement of the material (Harris and Crede, 1976). The logarithmic decrement for the spruce beam was 0.06 at 77 Hz and 0.09 at 316 Hz. This is higher than the value for quarter sawn wood of 0.021 along the grain and 0,064 across the grain, both measeured at 1000 Hz (Haines, 1979). The logarithmic decrement for the sandwich beam was 0.05 at 354 Hz and 0.03 at 200 Hz. Atypical value for the logarithmic decrement of carbon fibre/epoxy composite is 0.01 (Rao et al, 1985). Hence the addition of the balsa to the composite increases the damping significantly. Although the damping is difficult to determine with accuracy by inspection of the photograph of the oscilloscope trace, approximate values can be determined which allow qualitative comparisons of the beams. The damping in the sandwich beam was slightly less than the damping in the spruce beam. 59 5. RESULTS OF FINITE ELEMENT ANALYSIS OF EXISTING GUITARS 5.1 Introduction Results are given here for the calculated modes of several classical guitars. The results were obtained using the finite element method described in previous chapters. Comments concerning the accuracy of the results are given in Chapter 7 which compares the results of finite element analysis with the results of experimental methods of determining modes of existing guitars. 5.2 Results The tables below give the calculated frequencies of the modes of the guitars. Those given as calculated using the Givens method refer to a guitar which is clamped at the bottom of the sides of the guitar body. Each clamped node is fixed in six degrees of freedom. Those listed as being calculated using the inverse power method refer to the whole guitar (minus neck and strings) in free space. This method only calculates modes in a specified frequency domain and not many modes are listed due to the expense of the calculation. For each guitar there is a table of mode frequencies and a figure showing the shapes of the modes. The figures show countours of equal displacement of the guitar soundboard at the resonant frequency listed. Although for some guitars the modes were 60 calculated under several conditions of constraint and construction, only one set of mode shapes is shown for each guitar since they are essentially the same for all such minor changes. The frequencies on the figures refer to the calculation with the bottom of the guitar clamped. For each of the existing guitar designs analysed in this section, it was assumed that the top plate was the most important acoustic element to be analysed. Hence although the construction of the back and sides of the real guitars is significantly different, for simplicity of modelling each of the models used the same elements for the back and sides of the guitar. 7 X 13 7 X 13 Fon braces - European Spruce 4 x 5 mm Top - Western Red Cedar 2.2 mm thick approx. Figure 5.1 Guitar by Ramirez 1 1 ~ ~ I.. I.. I\ I\ ,,, ,,, I\ I\ / / I/ I/ ! ! _\ _\ \ \ ~ ~ J J I I j j '/ '/ J\ J\ _l _l I I \ \ \ \ I I ·1: ·1: 11,r:,, 11,r:,, I I --. --. Hz Hz \ \ ,·r- I\ I\ I\ I\ \j,)' \j,)' ,:. ,:. ~ ~ I I ,~I'\ ,~I'\ \ \ \I \I •· •· l' l' i i ...... "".., "".., - .,-11/, .,-11/, I\ I\ ,...,...,... ,...,...,... -- Hz Hz r-.K r-.K _ _ .. .. ) ) ~ ~ _;,, _;,, \\ \\ lj lj ' ' \ \ -- ~ .,<~ .,<~ 1373 1373 _i"' _i"' ~l' ~l' ,, ,, ' ' ..,.I\ ..,.I\ "'l!u "'l!u !' !' ~ ~ ., ., ,. ,. _/ _/ ...... • • 683 683 , , I\ I\ ll'.l\ ll'.l\ ' ' -- u u J J ...... I I I I ~ ~ _l _l t\ t\ >,, >,, """' """' ~\l\(A.\l ~\l\(A.\l 10. 10. ...... \ \ ...... ~ ~ -- ,.,, ,.,, 5. 5. ~ ~ I\ I\ 1 1 - \ \ I' I' l,I l,I - I I /r /r \ \ [\ [\ \ \ ' ' J J ' ' \ \ ~I\ ~I\ '. '. ,,,,,, ,,,,,, I I ...... ...... ' ' \ \ I I I I \ \ " " I I L'_L L'_L " " N N I I \ \ /~) /~) L L \ \ ' ' '\ '\ I I I I I I I\, I\, 1..- / / ~ ~ I\ I\ I\J\ I\J\ I\\\ I\\\ '.I '.I Ill Ill ' ' \ \ \ \ ...... I I L L \ \ I I 1 1 ', ', di di ...,v ...,v I\ I\ I\ I\ 1\. 1\. ' ' ii ii I\ I\ I/ I/ .J .J ,_ ,_ ,.., ,.., 1,.1 1,.1 Hz Hz ...... ,, ,, - ), ), \\I~ \\I~ .,__ .,__ Hz Hz r-...1\. r-...1\. ,• ,• \ \ / / ' ' t t ' ', ', . . ' ' "~ "~ r-" r-" "'"" "'"" " " \ \ ...... -- r,1--.,... r,1--.,... I\ I\ 610 610 i..·.\ i..·.\ .J. .J. L.o L.o I I r\ r\ I\ I\ ...... ~ ~ I I l ,, ,, ' ' ! ! "- 1134 1134 4. 4. ~ ~ l-0 l-0 - -- ~ ~ -- ,. ,. \ \ ~ ~ i---.. i---.. 11- - ,-1-' ,-1-' 9. 9. \\ \\ - ' ' ~ ~ _ _ -s: -s: 1,/ 1,/ - f.. f.. l' l' f f \ \ ...... - ii" ii" \ \ ~1-1\ ~1-1\ yr,_ yr,_ ...... /,,,. /,,,. ,, ,, \ \ ,.. ,.. - 'I 'I \ \ ' ' I' I' /_,; /_,; - ,,---~ ,,---~ \ \ I I J J •c., •c., I I \ \ I I \ \ I I I I I~ I~ \ \ / / r. r...... I I /J /J -, -, ' ' I I , , I\ I\ ,, ,, I/ I/ /') /') \ \ I I \1\ \1\ I I / / I/ I/ I\ I\ 1\1\ 1\1\ //1 //1 V V I I \ \ \ \ ~J ~J \ \ I I ~ ~ Hz Hz IJ/ IJ/ t\ t\ 'JI/ 'JI/ I I I I I ~ ~ I I '.'.'. '.'.'. .., .., I I Hz Hz I'!\.. I'!\.. - "rs:~ "rs:~ t:: t:: " " ,-~ ,-~ I/ I/ ,,; ,,; r1.l r1.l Jr--H-,li Jr--H-,li ( ( 71 71 ~ ~ ;., ;., \ \ I-' I-' I I I I I I I I I I\ I\ 1009 1009 4J[JffiT 4J[JffiT 459 459 I I I,. I,. I\ I\ Ji Ji 8. 8. . . \i\ \i\ -- 3. 3. :a. :a. ·-i ·-i H-J H-J ITTl ITTl ~ ~ ...... I/ I/ r-1++--N r-1++--N ...... ' ' 1 1 \ \ .,,,.,. .,,,.,. -->::::: -->::::: i,,- I/ I/ I I I I - ' ' ' ' I I _.Pi,- A A .-1. .-1. \ \ I I 1 1 J/ J/ /l /l /1 /1 fflrJiffil fflrJiffil I I ITTTTlrHTTlTn ITTTTlrHTTlTn '' '' I I '\ '\ I\ I\ /./ /./ /i\i/1 /i\i/1 ~~~-s ~~~-s \ \ Smffl~ Smffl~ I I I I l l \ \ . . / / , , \ \ ' ' I\ I\ '/• '/• ' ' \ \ I I \ \ \ \ I I I I / / \ \ I I I I i-,,""'' i-,,""'' V V I\ I\ \ \ [\ [\ -:"'.I~.: -:"'.I~.: \_i':/;; \_i':/;; - ~ ~ 10:-: 10:-: IJ'-1', IJ'-1', Ramirez Ramirez I I i.-./: i.-./: , , Hz Hz .. .. _J _J Hz Hz .) .) ,, ,, \\" \\" ~-nl\I\ ~-nl\I\ ~~·. ~~·. ...... r-...t\. r-...t\. I\' I\' 'J7h~., 'J7h~., ...... I\~ I\~ . I I . . . I . •7"7 •7"7 ,' ,' - \/ \/ -7 -7 - ~ ~ for for - I I ~ ~ 339 339 ~'.\ ~'.\ J J ;... ;... -'I,. -'I,. 913 913 ----- \ \ \l": \l": \ \ I\\.,.. I\\.,.. l1n--;. l1n--;. I I ...... I,-:- I\ I\ I I 2. 2. 7. 7. -- -- ~ ~ I\ I\ \ \ I I plate plate ...... \). \). - ji,'J: ji,'J: I\. I\. i\ i\ 11',LI 11',LI -~ri::; -~ri::; . . ::,, ::,, ...... ' ' "- -- '---7:''' '---7:''' I I ~-,,=; ~-,,=; ,.. ,.. - I\ I\ ~- ~-K. ~-K. .,,.-~ .,,.-~ \ \ \ \ _.,_ _.,_ ,,,.,,. ,,,.,,. top top -~-- / / I I \ \ ' ' \ \ I I \ \ '- \ \ r-..1'- ...... I I J J \ \ I_,... I_,... I I I I ' ' , , I I I I ;\ ;\ I\ I\ /./1 /./1 I I of of \ \ \ \ I I I I Modes Modes \ \ I\ I\ 1 1 l l / / ~ ~ J\ J\ f' f' VI/ VI/ \ \ I I I\ I\ ,i ,i 1.'1' 1.'1' : : I I I I \ \ \ \ I/ I/ I I \J\I\ \J\I\ ...... \ \ 11' 11' \' \' I I , , °:hi· °:hi· ,1,1 ,1,1 ·J ·J 5,2 5,2 ,; ,; Hz Hz " " I I ' ' i i ...... Hz Hz ,\\ ,\\ ...... I\ I\ r-., r-., I"-~ I"-~ '-i '-i I\ I\ '-- r--' r--' II II lo' lo' i-.i,.,.., i-.i,.,.., -.,,, -.,,, I'~ I'~ ~--· ~--· ...... " " ,, ,, I' I' 1_: 1_: ,n•.,; ,n•.,; \ \ .. .. ,•, ,•, r"i r"i f- ~ ~ 791 791 ~ ~ 219 219 ' ' ~ ~ ii. ii. I,. I,. <~ <~ 11,, 11,, r4-:, r4-:, - ...... I\ I\ ' ' ,w-: ,w-: Figure Figure I'. I'. I"' I"' 6. 6. -- 1. 1. \ \ I ' ' :\; :\; -- I I l,, l,, ...... i i ? ? ~ ~ -·1·_i\ -·1·_i\ ). ). ~- r,:;: r,:;: ~ ~ V V I I I\ I\ ,. ,. ,, ,, ' ' / / -r-- ,j ,j ~-- .,,. .,,. \ \ ,.., ,.., /' /' - ,.. ,.. ,~ ,~ II' II' ~-1,. ~-1,. .. .. D~~ D~~ I I ,I ,I "'\ "'\ 1,1 1,1 I I :\ :\ \1\ \1\ ~ ~ ', ', /.,,. /.,,. \ \ I I \ \ /. /. ' ' ~ ~ ...... 1, 1, '~ '~ /. /. I I ' ' '\ '\ ,, ,, I I I I , , ...... I\ I\ ,, ,, \ \ " " I I \ \ It It ' ' l'I l'I 63 5.2.1 Ramirez Table 5.1 gives the modes calculated for the guitar shown on Figure 5.1, reputed to be a copy of a guitar made by one of the famous Spanish Ramirez family. The guitar is typical of the design used by modern luthiers. The figures in this chapter showing the guitar designs depict the underside of the guitar top plate and describe the bracing pattern, which is normally hidden. A stiking feature of the Ramirez guitar is the "treble bar" shown on the figure as a large brace oriented diagonally across the body and adding stiffness to the lower bout on the treble side of the guitar, that is, the side to which the higher pitched strings are attached. Much of the development work of concerning the finite element model was done on this guitar so there are many results. The initial results obtained matched the expected set of results more closely than any of the others given, however it was later discovered that the density of the wood, as specified on a program data card, was in error. Also, inconsistent notation on the drawing of the guitar, on which the data set for calculation was based, meant that all the early results were wrong. The drawing showed the fan braces being 4 mm high and 5 mm wide, but this should be 4 mm wide and 5 mm high. The results for both are shown for reasons of comparison. 64 TABLE 5.1 Calculated modes for guitar - Ramirez ---Fan braces the wrong way up--- Conditions and method of extraction* Mode l 2.2 mm thick at edges ll.4 mm thick at edges NumberlBridgellBridge2lLess :cedar top :spruce lOffsetsl lTop linv lG+G ------: 1 230: 22s: 191 l 21s: 211: 2 360: 361: 318: 334l 343: 3 465: 461: 438: 459l 462l 4 623l 623l 553: 604l 6oa: 5 692: 12s: 615: 676: 612: 6 803: a13: 121: 1ao: 799l 7 943: 947l 856: a9a: 916: 8 1001 l 1003: ass: 9as: 995l 9 1097: 1109: 1004: 1106: 111s: 10 131a: 1399: 1226: 1347 l 1339: ------: 65 TABLE 5.1 continued ---Fan braces corrected--- Mode :cedar Top :spruce :3 Ply Number: :spruce :Top :rP :G+G ------: 1 219: 201: 220: 231: 2 339: 310: 347: 398: 3 459: 335l 462: 475: 4 610: 397: 615: 617: 5 683: 397: 678: 781: 6 791: 809: 851: 7 913: 930: 1119: 8 1009: 1018: 1204: 9 1134: 1143: 1442: 10 1373 l 1364: 1671: * G+G- the method used was Givens method with Guyan Reduction - bottom clamped IP - the method used was the inverse power method - guitar in free space fan strut cross section j_ 5 V T -i15 j(-- Top plate European Spruce or Western Red Cedar Figure 5.3 Guitar by Fleta ' ' I I .. .. , , d\ d\ V V "~ "~ ~, ~, I\, I\, ~ ~ v:r;, v:r;, I I I I ' ' I I \ \ I I "l "l I\ I\ y y tt-1/IJ" tt-1/IJ" :Xlll :Xlll ,, ,, L'>d\ L'>d\ I\ I\ ~ ~ ~ ~ 'r/ 'r/ I\ I\ - "'"' "'"' ~,, ~,, ...... 'f(. 'f(. jff, jff, I I r r ...... -~ -~ - ', ', i-- - Hz Hz /!Y.. /!Y.. ' ' .r .r ...... I I ~ ~ I I ~ ~ \ \ - I) I) I\ I\ ,,. ,,. ~ ~ .I. .I. I I ' ' \ \ 716 716 :\ :\ \ \ I\ I\ I I - -r-. -r-. / / \ \ I I ~ ~ -- -- • • ), ), I I ), ), 5. 5. V\ V\ / / ~ ~ I/ I/ I/ I/ ~ ~ 1, 1, ...... I I I" I" - Ji Ji j j - J J I I )( )( ,,, ,,, ~ ~ / / !( !( -- {' {' /,,, /,,, r r \ \ ~ ~ i\ i\ I"' I"' / / !'I:"- I I \ \ \ \ ...... J J ...... J J \ \ I I \ \ ...... I I ,~ ,~ , , 1• 1• ' ' I,..,, I,..,, \ \ , , -..- I I \ \ 'I 'I I' I' I I \r-.~ \r-.~ I I .~ .~ I I I I I I l\ l\ , , / / I\ I\ ., ., I I 1/,/ 1/,/ '/ '/ I I ~ ~ / / I I V// V// ~ ~ \ \ I I I\ I\ I I I I \ \ ~I/ ~I/ ,~ ,~ ,.~ ,.~ 1/1,' 1/1,' ...... I I ~v ~v "- r,.. r,.. ~ ~ I I ', ', - I's!,;.._ I's!,;.._ \ \ .-;.., .-;.., .. .. I I Hz Hz r--,1"\ r--,1"\ r~ r~ ~ ~ ~ ~ r-. r-. I I ""i'-. ""i'-. i;r-r-, i;r-r-, -. -. ,-, ,-, ~, ~, ..I. ..I. - \..,. \..,. - \ \ -7 -7 l,..J l,..J 583 583 -- ,- \ \ r-i- --._,,_ --._,,_ If'\ If'\ -- ' ' i.. i.. ~ ~ -- 4. 4. -~ -~ i.. i.. _,... _,... " " ~ ~ I I ...... ,.J ,.J .L .L - /I /I r r ,. ,. - IP/ IP/ II.~ II.~ \ \ '- - lr lr I I ,,- ,, ,, ,. ,. ,,, ,,, /,,, /,,, \ \ ) ) '\ '\ I I K. K. / / ,\ ,\ \ \ I I \ \ i\ i\ "- ...... I I ~ ~ I I 11 11 }/, }/, \ \ I, I, ~ ~ l"I l"I \ \ 1 1 1 '\ '\ 11 11 11Vi, 11Vi, ,, ,, v' v' I I I I l l \ \ / / \11, \11, /' /' I I I I 'r 'r ' ' /\ /\ ' ' I I ,I\ ,I\ /I/ /I/ V V t t \ \ I I '\ '\ V V \ \ I I i,,-"/ i,,-"/ ..., ..., "" "" :\f\ :\f\ I\ I\ ~ ~ ~ ~ \ \ I I 1 1 ( ( I\ I\ ) ) I'd\ I'd\ ~\ ~\ ' ' 1 I I /1,1 /1,1 \ \ )1,-'1 )1,-'1 I I - f\ f\ "'- ',\ ',\ I I ,.,.v ,.,.v \ \ /1 ,,, ,,, \ \ -- I\ I\ Hz Hz JA/ JA/ ,r<::~ ,r<::~ r--. r--. ( ( _,./ _,./ // // .. .. v- \ \ ..I. ..I. I/ I/ _ _ I\ I\ ,,. ,,. \ \ I\ I\ I\ I\ - I I ::. ::. \ \ \ \ '\ '\ 445 445 - ...... -- /\ /\ ). ). - II II -- I I I I ...... '""' '""' ~~ ~~ I I - 3. 3. \ \ ...... Ill Ill I'-.\ I'-.\ ~ ~ ' ' ,, ,, / / ...... '- 1 1 i--1:l., i--1:l., 1/ 1/ j j 1\ 1\ J J ,,.. ,,.. - " " - I/ I/ I I '" '" ~J ~J I/' I/' f..- /,,, /,,, \ \ \ \ \ \ I I A' A' 'I,,. 'I,,. 11\ 11\ \ \ '-- J J i,.- z::: z::: I I \ \ I I lr\ lr\ -~ -~ ' ' '\I', '\I', )/ )/ I\ I\ \.' \.' I I I I \.._ \.._ ,,,,, ,,,,, 1r-.- ~ ~ JI/ JI/ I I il, il, I) I) \ \ I I I\ I\ ,I ,I I/ I/ I\ I\ / / ·, ·, ~ ~ I I I I I I \ \ .\":\ .\":\ ,\ ,\ ) ) I I I I /I/ /I/ :,.,.,y- I I "I\ "I\ ,,,v ,,,v ~ ~ I\ I\ \ \ I I \ \ I\ I\ ~ ~ ~/ ~/ J J I/ I/ ' ' ,. ,. '\/ '\/ ,-l"\" ,-l"\" I I \1,4 \1,4 _, _, -- r--,1"\ r--,1"\ - l\v- Hz Hz I'- ...... \ \ ,,..,....., ,,..,....., \ \ - - "'""' "'""' -r-- I I \ \ ~ ~ - ..I. ..I. 11 11 - 323 323 - ,,. ,,. ./ ./ I\ I\ \ \ ,, ,, ~ ~ ...... \ \ \ \ '\ '\ ~ ~ r--- - -- 2. 2. Ii. Ii. ,,.. ,,.. -- ), ), I I ' ' - ir-, ir-, \ \ - I\ I\ / / ...... ., ., - -1)( -1)( I"\ I"\ ...... I I - /,,, /,,, V V 7 7 /,,, /,,, I\ I\ Fleta Fleta 1/ 1/ ' ' L'. L'. \ \ I I 11\ 11\ 1--1 1--1 \ \ '" '" I I I I ., ., ':,i. ':,i. I I '1 '1 t,.I& t,.I& lj lj '\ '\ I I JV/2 JV/2 I\ I\ \J \J / / ;,,r, ;,,r, •.1/ •.1/ / / by by I I \ \ / / I \ \ I\ I\ [// [// ~ ~ ,, ,, r r I I I I \ \ r, r, :'/ :'/ I/ I/ \ \ I I I'- [\ [\ I\ I\ ~\:\ ~\:\ ,\ ,\ ,,,.I\ ,,,.I\ r1 r1 I I \ \ Guitar Guitar ,/ ,/ ~~ ~~ I I I I ✓,; ✓,; ,\ ,\ :\ :\ I I \ \ i-', i-', ::-,· ::-,· ;x ;x .:,,r,,..[\ .:,,r,,..[\ - I\ I\ ', ', rl rl r, r, \I\ \I\ ,['\ ,['\ r r ' ' i--,.i'I\ i--,.i'I\ -- .,, .,, ' ' ·~ ·~ i::'r-... i::'r-... i'-. i'-. ~ ~ - )< )< "'I, "'I, /'~ /'~ ...... 5.4 5.4 ,' ,' Hz Hz ~f- ,~- '-r- ,I, ,I, r--, r--, ~-- j j I I Hz Hz \ \ I \ \ ~ ~ \ \ ...... - \ \ r--..._ r--..._ [\1\ [\1\ .,0..,, .,0..,, ~ ~ -- I'<- I I >, >, r- ' ' 852 852 ...... '\ '\ ,.I'. ,.I'. ~ ~ ..-, ..-, 210 210 - -- 7 7 -~ -~ --~ --~ D D ,-p<; ,-p<; ...... Figure Figure 6. 6. ., ., 7' 7' - - - ,.,__ ,.,__ ,,, ,,, - l. l. I I - '~ '~ ...... _.._ _.._ / / I)(/ I)(/ I I I_.. I_.. ,. ,. I I \ 'I 'I \ \ JI JI ~ ~ J J \ \ 'I, 'I, /. /. ,-- \ \ ",;:::.- r-.. r-.. }( }( 1 1 /,,,, /,,,, I I I I ~ ~ ,'i:1'r--.. ,'i:1'r--.. \ \ \~PI \~PI I I v v # # \ \ ~ ~ I I ,r--:::; ,r--:::; /' /' ' ' /l1 /l1 \\' \\' It It 14 14 68 5.2.2 Fleta Table 5.2 gives the calculated results for the guitar shown in Figure 5.3, a design by the Spanish maker, Ignatio Fleta. The results are for a guitar with a western red cedar top plate. The Fleta guitar is also of a typical Spanish design and is one of the most highly regarded of all guitars. The shapes of the natural modes are shown on Figure 5.4. TABLE 5.2 Calculated modes for guitar - Fleta : Conditions and method of extraction Mode :rnv Number:Power G+G: ------: 1 210: 2 323: 3 445: 4 583: 5 716: 6 852: upper bout very stiff Extent of carbon fibr mat on sides of top Typical beam cross section ---- lbalsa core carbon fibre faces 0.38 mm thick Beam width- 2.8 mm Top - Western Red Cedar - 1.5 mm thick Figure 5.5 Guitar by Gilet Expected page number is not in the original print copy. ,, I' ~ /./,....-V"' ,.. ,1 \ L L"I" ·-''·" i I. ii VL"v - I'' J I I ,-- J II 1- -- --7 ·~!::I:: ' ~ 2: "'~ .. P )...... - .J.- -- '-- i.. i \ h :;: I ) ) !~~~~ I/ I\V I\ ' ~ f /1'- } X I) I' l,'Q I 1\V ~ I\ /r,. I, I\V 1111 ) ll " I ~ '\ ; 1\ X X )· X .1 j I ~,l I ) \ .) ;~ ::,,v ~ I ! ',\ /1\.1/ I\ V X J ) \ ': ~~ fl / \ I\ XJ ) X J \ I/ ) ;) /\ I . ' \II II l\ll' V . ~ !/ X I/ \V" ) .\ .... - .:.-- 1. 143 Hz 2. 278 Hz 3. 416 Hz 4. 435 Hz 5. 673 Hz ,, - ', / " '\ ;1 \ I \ I -- I I -- \ .... 'i -- I \_ -- I --~ .J. \. ~r- / . lr r " I/~ I\ :II X \ ~ \ I( J' ) /,~I tl "' ~ '( " /\ \ ' I"'.) \ r/ '\ /I\ 1,n \ --1~ \J A I\ i I ~ X ~ ~ I I I)( 1/.:..£" , ~ , I I \ I~ l/ I-~/ ) l)r.. )l I ,. I \ i I ~ ~ ~ ~ rii I '/ )( \ X lV ; r/ \ ~I/) 1 ;x ~ I)( l(J/ ·, "1 ~IXl'i ~ I/ X I/:\"" \ l.J \ 6. 689 Hz 7. 835 Hz 8. 889 Hz 9. 1036 Hz 10. 1050 Hz Figure 5.6 Modes for guitar by Gilet 72 5.2.3 Gilet Gerard Gilet is a Sydney based luthier. Figure 5.5 shows the bracing pattern of one of his recent classical guitars. Table 5.3 gives the calculated mode frequencies. Those measured by modal analysis and holographic interferometry are compared later. The mode shapes are shown on Figure 5.6. The guitar is far from traditional, the bracing pattern being a lattice of carbon fibre/balsa/carbon fibre sandwich beams. The top plate itself is very thin and needs a sheet of carbon fibre mat to reinforce it at the edges of the lower bout. Because the top is light it would break if there was significant force on opposing sides of the guitar. To prevent this there is a "tone rim" installed at the top of the sides. This is effectively a large beam running around the whole guitar, just underneath the top plate, which inhibits movement of the edge nodes in the plane parallel to the top plate. The tone rim was modelled by increasing the cross sectional area of the beams used to model the linings. 73 TABLE 5.3 Calculated modes of guitar - Gilet Conditions and method of extraction* Mode :cedar :with Number:Top :Tone Rim :G+G :G+G ------: 1 143: 147: 2 218: 284: 3 416: 408: 4 435: 486: 5 673: 667: 6 689: 677: 7 835: 825: 8 889: 868: 9 1036: 1015: 10 1050: 1020: "Star II braces - spruce or carbon fibre composite Top - Western Red Cedar 2.5 mm thick Figure 5.7 Guitar by Marty l l f f I\ I\ ' ' " " IJ IJ 1/ 1/ I\ I\ ' ' + + l\ l\ I I ',.1/ ',.1/ 17 17 I I . . I I I I I I II\ II\ 1\ 1\ N N 1,il' 1,il' I\ I\ ' ' II II II II ,.- ,- )( )( ,,..I\' ,,..I\' t--V t--V I/ I/ Ui Ui ,.-I.I ,.-I.I ~ ~ Hz Hz i... i... r-. r-. ...... r r I I ~IA'" ~IA'" I" I" ,, ,, - --r. --r. ...... I I Hz Hz ...... I I l/ l/ ~ ~ . . ,, ,, ...... I I ~-- ~~ ~~ , , _ _ ,i--- V,-i>, V,-i>, ~ ~ ~ ~ , , I/ I/ I\ I\ '\ '\ ...... 1,,_ 1,,_ I I ,,.,. ,,.,. 574 574 ...... II. II. ~ ~ /J /J '-' '-' .. .. 937 937 \/\ \/\ I I /,'"' /,'"' 5. 5. i/1'1." i/1'1." \ \ I/ I/ -- ·,:. ·,:. Ill Ill ). ). ,, ,, ~ ~ -" -" 1, 1, -- ). ). I I 10 10 ~ ~ \ \ _,... _,... \' \' ""r-- ,I} ,I} -, -, - ~ ~ r'I r'I -1/ -1/ 'f.__. 'f.__. , , "' "' "i..,~ "i..,~ l't'\~ l't'\~ ,\ ,\ I'; I'; or or - \JIJ \JIJ - ( ( ,,,,.. ,,,,.. r, r, r-..'- _.,. _.,. I/, I/, I\ I\ ...... /,,,. /,,,. ' ' I I ' ' '""t:: '""t:: 1 1 I I \ \ ...... / / I., I., ' ' I I \ \ I I '\ '\ 1\ 1\ ~ ~ .I .I \ \ \"-~~ \"-~~ \I\' \I\' \ \ I I I, I, I I ,~ ,~ I I I\ I\ 1 1 ' ' I\ I\ II II ,, ,, I\ I\ v/ v/ I/IV I/IV ' ' r r 'I 'I Hz Hz I\ I\ IX IX Hz Hz r-r,. r-r,. I',. I',. :::.~ :::.~ 1/V 1/V ']'• ']'• ...... .. .. --."'r-.. --."'r-.. \ \ II"'. II"'. ...... "' "' 923 923 437 437 J. J. - 7" 7" ;/J ;/J ~ ~ 9. 9. \ \ 4. 4. -- >,, >,, ,. ,. I I -~ -~ ,,. ,,. - ...... ' ' ,,.. ,,.. ~r,; ~r,; ...... / / ; ; i..- I\ I\ ,..._' ,..._' ' ' \' \' /1 /1 • • ...... \' \' I I ~ ~ I/ I/ I I '\..., '\..., .i .i / / \ \ j j ! ! I I / / / / I\ I\ " " / / I I I I '11! '11! I\ I\ I\ I\ II' II' , , / / 11v 11v I I I I ' ' .. .. ' ' I I J J ,, ,, '\ '\ \ \ (L (L I I ,\V ,\V I I . . 'I 'I \ \ I', I', \ \ .,, .,, ...... J J 'J 'J /l /l Ill\!" Ill\!" • • . . Hz Hz - i--, i--, .~ .~ _,. _,. .,,J .,,J ~ ~ 'lfl 'lfl --. --. Hz Hz ...... I/ I/ l/ l/ ~ ~ .. .. ':;J ':;J -.,.-,;"' -.,.-,;"' .;::: .;::: 7' 7' \ \ - I,,< I,,< I\ I\ I'\ I'\ I/ I/ ,,~ ,,~ '~'.~~:-.:✓/ '~'.~~:-.:✓/ ,.~ ,.~ r..,.- I I " I I _, _, I\ I\ ...... 405 405 rLo rLo - I/\ I/\ = = 793 793 l l 111, 111, \ll \ll l\, l\, - 3. 3. - }. }. '\ '\ ~ ~ l/ l/ I\ I\ 8. 8. ...... 1/IJ 1/IJ t:::lx t:::lx ...... ...... 1,11-.. 1,11-.. - V V I\ I\ I', I', ~ ~ '\\ '\\ ,r ,r ...... i-. i-. I\: I\: I\ I\ '"- ...... ':-. ':-. n n f(.r, f(.r, - .. .. ...... ~ ~ - ,..._ ,..._ '"-ls '"-ls :-. :-. IJ IJ - ...... '" '" \ \ /-' /-' •- -- I I J J /';;; /';;; ' ' t,; t,; , , \._ \._ ' ' I'-, I'-, J J I I - ~ ~ ...... \ \ ...... ,_ ,_ I I /, /, I I • • '-. '-. - , , ' ' //"\ //"\ / / \ \ jl"\. jl"\. \ \ /I /I I I I I '• '• " " '/ '/ V V / / J ,\ ,\ 1 1 ,, ,, I I I I I I I I ,~ ,~ '' '' I I ~ ~ ,,.,,,, ,,.,,,, fj fj I I I\ I\ -.'I -.'I i'\ i'\ ...... ' ' ( ( .,.,,,. .,.,,,. ( ( , , ' ' .I .I ::: ::: , , l l \\I \\I j j - .-+ .-+ -- ...... ,... ,... - l.J l.J ...... t-( t-( ...... I I ~ ~ ...... ' ' )-- A A ,, ,, } } Hz Hz ,.. ,.. I I ,,-1\. ,,-1\. I• I• ~ ~ ~ ~ Hz Hz ' ' \ \ ~ ~ II\ II\ ' ' \, \, .,j .,j / / 1t 1t ...... .,j .,j ...,r- I I •"''""" •"''""" -- - Marty Marty 343 343 - l\J l\J /~ /~ ,,,.- -- - t--,-.J t--,-.J r\...:: r\...:: -- V V ' ' - >,, >,, >,, >,, \, \, 778 778 I I 2. 2. J J \ \ ~ ~ by by ' ' I I - I I ' ' ...... 1 1 I\ I\ -~ -~ I\ I\ ~ ~ 7. 7. '\ '\ - ""'loo ""'loo ,I\ ,I\ - ~ ~ ':::,. ':::,. ;1' ;1' l'I. l'I. ~ ~ /I /I /,,.. /,,.. ~ ~ ['a"'~ ['a"'~ I I ' ' \V \V . . I I ' ' / / \ \ I I (J (J I I I I I I '" '" '\ '\ 1I 1I \. \. guitar guitar I. I. \:~\L'-.., \:~\L'-.., ' ' \ \ I, I, of of ·, ·, II II ~ ~ ., ., LI' LI' ~ ~ ,~ ,~ 7 7 l l f f I I \ \ ' ' .. .. 'f\ 'f\ i::: i::: i;;:: i;;:: . . Modes Modes , , r-.. r-.. \ \ • • ., ., ' ' " " I'',, I'',, I"\ I"\ ...... ::l\ ::l\ ':::~ ':::~ "'r-.. "'r-.. I I ,; ,; 'I. 'I. l\ l\ ' ' \ \ 5.8 5.8 Hz Hz Hz Hz .,j .,j ' ' ,_ ,_ • • '' '' J J -- -- I, I, \. \. \ \ 152 152 719 719 ' ' I I I -- ~ ~ \' \' \ \ 1. 1. 6. 6. J J Figure Figure - ' ' I I I\ I\ ,:., ,:., .,, .,, ~ ~ ./" ./" ,,._ ,,._ ...... /,,. /,,. v v \ \ J J ~~ ~~ ' ' ' ' ...... ...... J J I I ~ ~ I I \ \ \ \ I I I I J J l\ l\ / / ,, ,, // // ~l'\ ~l'\ " " 76 5.2.4 Marty A guitar by Sydney luthier Simon Marty is shown on Figure 5.7 and the modes of vibration of that guitar are shown on Figure 5.8. The guitar was analysed using beams made from spruce or carbon fibre/spruce sandwich beams. Only the last column in the table has results for a guitar with the sandwich beams. The guitar is unusual in that the braces on the lower bout form a star pattern instead of the more normal fan pattern. The sides and bottom of the guitar are also stronger than on Spanish guitars. Table 5.4 gives the calculated modes for this guitar. 77 TABLE 5.4 Calculated modes of guitar - Marty Conditions and method of extraction* Mode :sides 2.2 mm :sides 1.5 mm Number:G+G :no :cf/bals: :linings:G+G :rnv :G+G ------: 1 176: 150: 153: 144: 156: 2 405: 368: 343: 325: 346: 3 467: 413: 405: 405: 4 472: 442: 437: 439: 5 609: 589: 574: 585: 6 110: 748: 719: 738: 7 835: 012: 110: 783: 8 851: 021: 793: 835: 9 997: 958: 923: 915: 10 1029: 976: 937: 994: 78 5.3 Discussion The discussion in this chapter is concerned with the accuracy of the results as well as the effect of changes to the model, that is the changes to the modal frequencies caused by constructional changes. To aid this discussion, the frequencies of vibration of the open strings of the normally tuned guitar are listed in Table S.S. Also note that adjacent semitones of the musical scale vary in frequency by about 6 percent. Table 5.5 Frequencies of open strings of normally tuned guitar String :Note :Frequency: I I I [hertz] I ------1 E 329.6 2 B 246.9 3 G 195.9 4 D 146.8 5 A 110.0 6 E 82.4 79 5.3.1 Ramirez In general the first two modes of the Ramirez guitar were predicted by the programme to be at expected frequencies, that is at about 200 Hz and 300 Hz. However the higher modes, particularly number 4 on Figure 5.2 were calculated to be at higher frequencies than expected. The third mode on Figure 5.2 does not generally show up for other tests of guitar modes so it is expected that this mode would not radiate strongly. It is important to remember that all the modes, and particularly the lower ones, are coupled strongly to the air in the guitar cavity, which tends to raise the frequency of the modes. The small changes to the guitar construction changed the resonant frequencies only slightly. For example, two bridges were modelled and the results of the change are in the first two columns of Table 5.1. The two bridges were nearly the same size, however the moments of inertia for the second bridge were calculated more accurately. The first bridge was made from beams of rectangular cross-section whereas the second bridge was made from shapes more closely resembling actual bridges. The modes were affected somewhat but never more than a semitone. Hence, even though makers stress the importance of the bridge, small changes may be too subtle to be 80 recognised by knowing the frequencies of the modes alone. Other factors such as gain may be important. The columns headed "less offsets" show the effect of reducing the amount which the braces are offset from the soundboard, as if they were embedded in it. The resulting loss in stiffness reduced the frequency of the modes. These results are of limited practical· value since they describe an impossible situation. The last columns in the top part of Table 5.1 are the results after the edge elements of the model were thinned to 1.4 mm. Most guitar makers do thin the top plates at the edges resulting in greater rotational freedom at the boundaries of the top plate. All the lower modes were lowered in frequency due to this extra freedom. All the results discussed so far were using fan braces wider than they were high. In fact they should be higher than they are wide and the results for such a case are a in the lower section of Table 5.1. This meant that the second moments of area of the beam elements were increased and the consequent raise in frequency of the modes is evident from the table. The differences in modes for top plates made from western red cedar and European spruce were very small. The 3 ply top in the final column of the table was formed from three layers of spruce one third their normal thickness. The central laminate was oriented across 81 the guitar, thereby increasing the stiffness across the guitar and reducing the stiffness along the guitar. This had a significant effect on the cross dipole mode (the second on Figure 5.2) due to the cross guitar stiffness increase. The other lower frequency modes (less than 1000 Hz) were not significantly affected whereas the higher frequency modes were raised in frequency by about 20 percent. This could result in the guitar having too few modes in the appropriate frequency range. Note that modes calculated using the inverse power technique are lower than those calculated using Givens method with Guyan reduction. This is because overall stiffness was reduced when the constraint was taken away from the bottom of the guitar and it was allowed to "vibrate" in free space. 5.3.2 Fleta The construction of the Fleta is similar to that of the Ramirez, hence the mode shapes are also similar. The frequencies of the lower modes are higher than expected. 3.3 Gilet The modes calculated for the Gilet guitar are closer to the expected frequencies than those calculated for the Spanish guitars. Although 82 the first mode was calculated to occur at 147 Hz, this should rise to about 190 Hz when coupled to the air cavity. The installation of the tone rim, inhibiting the movement of the sides of the guitar, caused an increase in the frequencies of some modes and a decrease in others. The change was always small. Note that the mode shapes are different to those of the spanish guitar. The guitar is constructed in such a way that the lower bout section of the top plate vibrates in the manner of a circular plate, hence more symmetric modes are evident. 3.4 Marty The second mode of the Marty guitar was calculated higher than expected, but the other modes were at reasonable frequencies. The soundboard, like the Gilet, is somewhat like a circular plate and the mode shapes are therefore quite symmetric, though different to the Gilet. Note that there was little difference between the guitar braced with spruce and the guitar braced with carbon fibre/balsa sandwich struts. Although the sandwich beams are about 50 percent stiffer than the spruce beams, they have lower second moments of area as discussed in Chapter 4. The bracing pattern obviously controls the shape of the modes to a large extent, but it may be that the 83 frequency ranges of the modes is controlled more by the thickness and stiffness of the top plate and its supports. The monopole, cross dipole and long dipole modes are the only modes common to all the guitars. The shapes of the higher frequency modes depends greatly on the bracing type. All the bracing patterns produce good quality guitars, so to produce a guitar of good quality, one is not confined to producing one of particular mode shapes. It is more important to have a proper frequency distribution of the modes. 5.3 Effect on Mode Frequencies of Torsional Constant Adjustments Roberts (1986) reports that the adjustment has increased the accuracy of his finite element models. In order to determine the possibility of using the adjustment in this analysis, the modes were calculate for the Ramirez guitar, using the standard value, the corrected value and no value at all for the torsional constant of the beam elements used to model the braces of the guitar. Table 5.6 gives the mode frequencies calculated using Givens method with Guyan reduction. In all cases the bottom of the guitar was clamped. 84 TABLE 5.6 Mode frequencies [Hertz] for Ramirez guitar for different values of beam torsional constants. Torsional Constant Mode No. No Value Standard Corrected ------1 226 228 227 2 359 364 360 3 461 461 461 4 618 623 620 5 730 740 732 6 798 821 804 7 945 990 954 8 1021 1063 1031 Table 5.6 shows that the different values used for the torsional constant had little effect on the frequencies of the top plate modes. The higher frequency modes are noticeably different, the modes on the guitar with the highest value for torsional constant being higher. The difference for the lower frequency modes is very small, and since these are the modes in which we are most interested, the 85 torsional constant was not used in any of the other guitar models. 5.4 Computer Resources Finite element analysis can requires large amounts of computing time for all but trivial problems. 5.4.1 Static Analysis The static analysis of the four by four plate, having 48 degrees of freedom in the problem required 29 seconds of Central Processor Unit (CPU) time. A similar analysis on the guitar with 621 elements and 983 displacement degrees of freedom required 276 seconds. 5.4.2 Dynamic Analysis Table 5.7 lists the CPU time required for the eigenvalue extraction of the guitars. For the Modified Givens methael the number of displacement degrees of freedom was 983 and there were 40 degrees of freedom in the reduced mass matrix after Guyan reduction. For the inverse power method there were 1214 displacement degrees of freedom. 86 TABLE 5.7 CPU times for eigenvalue extraction Guitar Number Method* Number Number CPU of of of Time Elements modes Iterations [s] Extracted Required Ramirez 539 MG+G 386 539 Inv 1 14 1153 539 Inv 2 19 517 Marty 543 MG+G 554 Gilet 576 MG+G 611 Fleta 561 MG+G 478 561 Inv 1 5 981 561 Inv 1 7 1025 *MG+G - Modifified Givens Method with Guyan Reduction Inv - Invers Power Method with shifts 87 Since the Givens method extracts all the eigenvalues of the structure, it would appear more prudent to use it then th~Inverse Power method. The InversePower method is, however, more accurate even though the analyst may miss some modes. The amount of computer time required is fairly large and almost certainly beyond the resources of the average instrument maker. 88 6. EXPERIMENTAL DETERMINATION OF GUITAR MODES 6.1 Introduction Several methods have been used by researchers to analyse the normal modes of guitars, including holographic interferometry and measurement of sound from a guitar whose top plate is being driven by an external force. It is important in finite element models to compare results with experimental data in order to validate and refine the model. An experiment was performed to measure the modes of several of the guitars analysed by finite element analysis, and to compare them with other guitars of varying quality and type. The fast fourier transform technique was used to analyse the sound output of the instruments. ~ Guitar V Exciter Digital Oscilloscope ' Function Disc Storage Generator Microcomputer Line Printer Trigger Pulse Figure 6.1 Block diagram of experimental arrangement. 90 6.2 Fast Fourier Transform Measurement 6.2.1 Instrumentation and Method The experiment was performed in conjunction with guitar makers Marty and Gilet (Marty, 1985). A block diagram of the instrumentation set up is shown in Figure 6.1. The guitars were stood on their side on a table and the strings damped with a cloth and the neck supported by foam rubber. An impulse force was applied to the bridge using an Advance vibrator type 6 which consists of a voice coil placed in a magnet. A brass rod of 3 mm diameter and mass 5.4 g was attached to the voice coil. A single pulse from an I. E. C. F53A function generator was supplied to the vibrator. Two impact points on the bridge were tested, the centre of the bridge and the end of the bridge, noted as positions 1 and 2 respectively. The strength of the impulse has to be such that the detection equipment is triggered by the instrument's response but that no damage is done to the guitar. For the exciting apparatus used here a square pulse of 7 ms duration and with an amplitude of 30 volts proved satisfactory. To measure the length of contact between the rod and the guitar, a strip of aluminium foil was attached to the guitar bridge and the collision completed an electric circuit. A typical impact duration time was 0.5 ms. 91 To analyse the sound produced by the impact, a microphone signal was input to a digital oscilloscope. The microphone was an electret type having a frequency response flat to within 1 dB in the range from 50 Hz to 10 kHz. The microphone was placed 500 mm from the guitar, directly in front of the bridge. The microphone signal triggered a Nicolet 3091 digital oscilloscope. The sampling interval was 0.1 ms whichgives a maximum frequency of 5 kHz. For a total sampling time of 0.4 seconds a frequency resolution of 2.5 Hz is obtained. A Sirius microcomputer applied a fast fourier transform to the digitised signal from the oscilloscope to produce a frequency analysis of the signal. Because an impulse force was used, modes of all frequencies were excited. No force transducer measured the force input to the guitars, so the resultant frequency analysis graphs are scaled such that the maximum sound pressure level is at the top of the chart. Thus the absolute sound pressure level is not given, however the frequency analysis charts were repeatable between impacts so the experimental method was satisfactory. A more rigourous analysis of the characteristics of the guitars would have required a more controlled and complex experiment, for example using several microphone positions instead of one and measuring in an anechoic room instead of a reverberant laboratory. However, the experiment did yield valuable information on many guitars in a very short time, and since the aim of the experiment 92 was to assess the modes of the guitars and not to completely analyse their acoustics, the more complex experiment was not considered necessary. Frequency peak : 0.24170 Khz Spectral analysis of ramirez.1 ( 4.13737 ms) 0.137 I I ' 8.109 I'\ 3 E 0.882 \I ~ µl 0.055 3 0 l 11. I I I 0.027 ( ' I \, J I I! l I .I I I\ I J . .,. \ ,,.,.,I ' t Ih ...______.- I I l.. __ •!"· \,.~.. ..(' I ;' I / \,\ 1\ 0.000 ' J I -....__. ,': 0.00 0.20 0.40 0.tiO 0.80 1.00 FREOUI:HCY (kHz) Figure 6.2 Frequency response of Ramirez guitar excited at bridge centreline. 94 6.2.2 Results For brevity, the result charts are shown for only one guitar in Figure 6.2, which showa the frequency response of a Ramirez guitar excited at the bridge centreline. The remainder of the charts are reproduced in Appendix A. The frequency, gain and Q factor of the major modes below 1000 Hz of each guitar are listed in the tables below. The gain of each mode is given in decibels with respect to the mode of highest gain. It is possible to extract a great deal more information from the charts thus the tables.are only a way of summarizing the most significant characteristics. The mode shapes could not be determined, however it is possible to be reasonable certain of the position of the major modes, viz the air resonance, the top plate monopole, cross dipole and long dipole. The columns in the tables are headed 11 111 or 11 2 11 indicating that the guitar was excited at the centre or the side of the bridge respectively. The major difference between the two positions is that the excitation at the side of the bridge imparts energy to the cross dipole mode, whereas excitation at the centre of the bridge is usually at a point of small displacement for this mode. Not all the columns are complete as the complete analysis was not performed on every guitar. 95 6.2.2.1 Classical guitar by Kohno Table 6.1 list the resonances of an high quality guitar by the Japanese maker, Kohno. The mode of highest gain was the air resonance at 99 Hz. Several modes around 200 Hz show the effect of the coupling of the top plate, the bottom plate and the air in the cavity. The strong mode at 405 Hz is almost certainly the long dipole mode. This is consistent with the hypothesis that a strong, high Q mode at about 400 Hz is common to nearly all good guitars. 96 TABLE 6.1 Resonances of guitar:Kohno 50 Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 ------1 99 21 21 0 2 183 -8 3 198 21 28 -3 4 226 16 -5 5 245 -12 6 287 17 7 405 43 -6 8 608 43 51 -8 9 777 -24 10 796 -24 97 6.2.2.2 Classical guitar by Gilet Table 6.2 lists the modes of vibration of a classical guitar by Gilet. The guitar is one with the lattice brace pattern using carbon fibre and balsa composites. This was one of the guitars for which finite element analysis of the modes was also done. A comparison of the experimentally determined and calculated modes is presented in Chapter 7. 98 TABLE 6.2 Resonances of guitar:Gilet classical Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 ------ 1 107 18 0 2 206 17 0 3 262 -8 4 270 -8 5 374 -9 6 437 55 -8 7 461 -8 8 485 -8 9 509 64 -5 10 875 73 -5 99 6.2.2.3 Classical guitar by Ramirez Table 6.3 lists modes of the guitar by Ramirez. The frequency response of this guitar was shown in Figure 6.2. 100 TABLE 6.3 Resonances of guitar:Ramirez Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 ------1 103 27 27 0 2 226 19 -3 3 238 25 25 0 4 276 -9 5 315 13 26 -10 6 438 38 28 0 7 584 43 -3 8 761 66 66 -14 9 838 -18 ------101 6.2.2.4 Aria classical guitar Table 6.4 lists the modes of a guitar from the Aria company. This is a guitar with a plywood top and is ot considered to be a guitar of high quality. The modes above about 300 Hz are in general about 5 dB lower in gain with respect to the loudest mode when compared with the higher quality guitars. 102 TABLE 6.4 Resonances of guitar:Aria Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 1 99 21 21 0 2 187 -4 3 198 7 7 -4 4 216 -9 5 348 -15 6 400 -15 7 424 30 30 -9 8 528 37 -4 9 584 -12 10 735 -18 11 792 -17 12 898 -18 ------103 6.2.2.5 Classical guitar by Marty Table 6.5 lists the modes of a guitar by Marty. The guitar is one with the Marty Star bracing system described in Chapter 5. 104 TABLE 6.5 Resonances of guitar:Marty 8 Resonance Frequency Q Gain (dB} Number [Hz] 1 2 1 2 ------1 89 12 19 0 2 169 -6 3 174 7 0 4 183 39 0 5 202 28 -6 6 216 30 -12 7 259 -10 8 273 -10 9 367 -12 10 405 34 21 -2 11 481 -12 12 664 47 -2 13 768 -12 14 792 -12 15 839 -12 ------105 6.2.2.6 Gilet steel string guitar Table 6.6 lists the modes of a steel string guitar by Gilet. It is a small bodied steel string, approximately the same size as a classicsl guitar. Since a steel string guitar has to withstand greater string tension then classical guitars, the construction is necessarily heavier. - 106 TABLE 6.6 Resonances of guitar:Gilet steel string Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 ------1 106 22 22 -3 0 2 183 19 25 0 -3 3 235 25 -11 -14 4 334 35 -18 -5 5 377 -20 -15 6 405 -12 -12 7 452 48 48 -7 -7 8 565 120 -24 -7 9 726 77 -14 -11 10 782 55 -11 -18 11 820 87 -18 -6 ------107 6.2.2.7 Maton steel string guitar Table 6.7 lists the modes for a Maton CW80 steel string guitar. This is a larger guitar then the others measured. The Maton steel string has fewer prominent modes then the Gilet steel string. The guitar sounds more "hollow" because of this, however it could also be judged as more "mellow" for the same reason. Although the Maton guitar is generally considered to be a moderately high quality instrument, the quality must take the acoustic requirement into account. 108 TABLE 6.7 Resonances of guitar:Maton CW80(steel string) Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 1 97 21 -2 2 186 15 0 3 440 -18 4 608 -12 5 697 -13 6 829 -12 109 6.2.2.8 Yairi flamenco guitar Table 6.8 lists the modes for a Yairi flamenco guitar. Flamenco guitars are generally lighter in construction then classical guitars. Their sound is one of sharp attack and short sustain. The low Q of the mode at 420 Hz implies that this guitar would be judged as poor if assessed as a classical guitar. 110 TABLE 6.8 Resonances of guitar:Yairi flamenco Resonance Frequency Q Gain (dB) Number [Hz] 1 2 1 2 ------ 1 94 13 19 0 0 2 193 -4 -8 3 217 13 -1 -6 4 235 25 -6 -6 5 264 -8 -4 6 283 -9 -12 7 420 22 22 -6 -8 8 552 58 58 -1 -8 9 698 -12 -10 10 726 -11 -11 ------111 6.3 Holographic Studies of the Guitar 6.3.1 Introduction Holgraphic studies of the modes of guitars have been made by several researchers (Jannsen, 1971, Richardson, 1982, Richardson and Roberts, 1983). Their results are generally intended to describe the physical characteristics of guitars. The experiments described here were designed to assess various innovations in guitar design by guitar makers Marty and Gilet whose guitars were analysed by finite element analysis by the author. The results of the finite element analysis were reported in the Chapter 5. The same guitars were analysed by fast fourier transform techniques as described in Section 6.2. Results of the holography are given here are extracted from the report by Marty (Marty, 1985). The results are given for three guitars, one by Marty, one by Gilet and one by the Ramirez. The Marty and the Gilet guitars were the same guitars which were analysed by the other two methods. The Ramirez is not the same guitar and the construction details may be different for the Ramirez analysed by finite element analysis and the Ramirez analysed in Section 6.2. Note especially that the Ramirez guitar used in this experiment was a left handed guitar. 112 The holograms shown can be interpreted in the same way as the contour maps of mode shapes as determined by finite element analysis. 6.3.2 Results Table 6.9 lists the resonant frequencies extracted by the holographic analysis. Figure 6.3 (a-h) show the modes for the Ramirez. Figure 6.4 (a-h) show the modes for the Marty. Figure 6.5 (a-i) show the modes for the Gilet. The figures are in a pocket on the back cover of the binding of this report. The Marty 6 guitar was braced with spruce and the Marty 8 guitar was braced with carbon fibre/balsa composites. The guitars were identical in all other respects. The mode shapes were similar for both guitars and are only shown for the Marty 6. 113 TABLE 6.9 Modes of Guitars as Measured by Holography Mode Frequency [Hz] and Mode Identification* Guitar:- I I Ramirez Marty 6 I Gilet I Marty 8 ------a 190 M 172 M 185 M 180 M b 285 CD 290 CD 270 CD 275 CD C 380 LD 390 LD 410 LD 395 LD d 525 T 565 CD 470 T 483 e 700 638 T 570 540 f 775 692 710 640 g 905 738 805 h 1090 905 i 1085 * M - Monopole CD - Cross Dipole LD - Long Dipole T - Tripole 114 The mode shapes and frequencies of the Ramirez guitar are as expected from the previous work cited (Jansson,1971). Those of the Gilet guitar are similar up to the tripole mode. Above that mode, the symmetric modes discussed in Chapter 5 are evident. The Marty guitar had two monopole modes close together. It had two cross dipole modes which were different in shape from the other guitars. Its higher frequency modes exhibited the characteristic of a circular plate, similar to the behaviour of the Gilet guitar. Further comments ar made in Chapter 7 which compares the results of the three methods of modal analysis. 115 7. COMPARISON OF FINITE ELEMENT ANALYSIS AND EXPERIMENTAL RESULTS 7.1 Introduction A comparison of the modal analysis results can indicate deficiencies in the finite element model. The model can then be changed so that it predicts modes more accurately. In this way future modelling of unbuilt structures can be done with more confidence. 7.2 The Results 7.2.1 Ramirez The Ramirez design was the first to be analysed. The initial results were so close to the expected mode frequencies for a guitar that the model was believed to be an accurate representation of the guitar. It was later found, however, that the value of the density of the top plate timber had been specified 50 percent too high on the material card in the Bulk Data deck. When this was corrected the modes were not as close to the expected frequencies as before and deficiencies in the model had to be sought. The thickness of the elements which connected the top plate to the sides of the guitar were found to be very important. Most guitar makers make the edges of the top plate thin to allow more rotational flexibility. When the edge elements were made thinner then the rest of the top plate the frequencies were closer to those expected. Table 7.1 116 lists the mode frequencies of the Ramirez analysed by finite element analysis and two other guitars by the same maker, one analysed using holography and one using fast sourier transforms. The following abreviations are used on the tables below: FEA - finiete element analysis M Giv - Modified Givens Method Guyan - Guyan reduction FFT - Fast Fourier Transform. The mode identifications are as for Chapter 6. For the finite element evaluation of the monopole, two numbers are given. The first, denoted M, is as calculated by the programme. The second, denoted M', is the corrected value as calculated by the procedure outlined in Chapter 4. The value for the Helmholtz resonance of the cavity is assumed to be 128 Hz (Christensen and Vistisen, 1980). The value of the coupled air resonance is as measured by the fast fourier transform experiment. 117 TABLE 7.1 Mode frequencies of Ramirez guitars. Guitar analysed by: FEA FEA Holog FFT M Giv IP Guyan 219 M 201 M 190 M 103 231 M' 215 M' 226 M 339 CD 310 CD 285 CD 238 459 380 LD 276 610 LD 525 T 315 CDj 683 T 700 438 LD'? 791 775 585 Tj 913 905 761 1009 1090 838 1134 1373 The FFT measurement was the only one to detect the cavity resonance radiated through the sound hole. Although the three methods were used on guitars by the same maker, they differed widely in mode 118 frequencies. The two for which the monopole mode can be identified by almost 30 Hz which which is almost three and a half semitones. The FEA computed monopole lies between the two. The other modes varied in frequency for the guitars, however it is reasonable to state that the FEA computations for the modes higher in frequency then the cross dipole are much higher than can be reallistically expected. 7.2.2 Marty. The frequencies of the Marty 6 guitar are listed in Table 7.2 119 TABLE 7.2 Mode frequencies of Marty 6 Guitar analysed by: FEA FEA Holog M Giv IP Guyan ------ 155 M 144 M 178 M' 170 M' 172 M 343 CD 325 CD 270 CD 405 410 LD 437 LD 470 TP 574 570 719 TP 710 778 805 793 905 923 1085 927 The lower modes are predicted with reasonable accuracy, with the exception of the cross dipole wich was calculated too high, possibly because of incorrect modelling of Marty's complex bridge design. The mode shapes are correct but above the tripole there is little 120 correlation between the two mode shape sets. Both techniques show that the top plate behaves as a circular plate at high frequencies. 7.2.3 Gilet The frequencies for the lattice braced Gilet classical guitar are given in Table 7.3. 121 TABLE 7.3 Mode frequencies of Gilet lattice braced classical guitar Guitar analysed by: FEA Holog FFT M Giv Guyan ------143 M 107 161 M' 185 M 183 M 278 CD 270 CD 198 416 LD 410 LD 226 435 470 T 245 673 T 570 287 CD - 689 710 405 LD 835 805 608 889 905 777 1036 1085 796 1050 ------ The dipole modes were predicted accurately for this guitar, however the monopole was calculated too low. The higher order modes for this guitar show that the top plate behaves as a clamped circular 122 plate, similar to the Marty. The correlation between FEA and the other methods was poor for the higher modes even though mode shapes are accurately predicted. 7.3 Discussion The finite element analysis of the guitars was most accurate for the lower frequency modes. The monoploe and the two dipole modes were predicted most accurately. The higher modes were generally calculated to be higher than they are in reality. The use of the inverse power method was more accurate than the Givens method with Guyan reduction. The shapes of the modes of the top plate were predicted accurately. The least accurate models were the those for the two Spanish guitars, the Fleta and the Ramirez, particularly at high frequencies. This was possibly due to the fact that the damping of guitar woods increases with frequency (Haines, 1980 and Chapter 4). A more accurate model would take the damping into account, however it increase the calculation time required enormously. Of the two experimental guitars by Gilet and Marty, the Gilet guitar was modelled most accurately. This is possibly due to the modelling of the sandwich beams which braced the structure. 123 The stiffness of timber can vary greatly from sample to sample. Environmental factors also affect the stiffness. Changes in relative humidity can vary the sound of a guitar dramatically. Gilet reports that the cross dipole mode of a guitar built in country NSW by luthier Greg Smallman changed by three semitones when brought to the higher humidity of Sydney. This change is in the order of 15 percent. Since the properties of wood are not static, no model of a wooden structure can be exact. Perhaps, therefore, the accurate prediction of mode frequencies of the guitar is not viable or even necessary. Since all the guitars analysed had very different mode frequencies and shapes, yet were all considered to be high quality guitars, there may be no set of modal frequencies which is optimum. If one could determine the sound of a guitar by knowing its modal properties, then accurate prediction of the modes would be a more valuable tool in guitar making. Expected page number 124 is not in the original print copy. Extent of carbon fibre Lattice beams as for Gilet (figure 5.5) Top - Western Red Cedar 1.5 mm thick - 0.7 mm thick at edges 0.38 mm carbon fibre ma trot edges of lower bout Figure 8.1 New guitar design 126 8. A New Guitar Design 8.1 Introduction A new guitar was designed based on the findings of Meyer (1983) as discussed in Chapter 2. The aim of the design make a guitar which was louder than normal and having a good quality sound. The new guitar would be light and have a greater radiating surface area than normal to achieve this. To ensure that quality would be maintained, the long dipole mode would be strongly radiative by making the lobes of the dipole greatly different in area. Carbon fibre/balsa sandwich beams were chose because of their lower damping and high stiffness. 8.2 The Modes of the New Design The new design is an extension of the lattice braced guitar by Gilet, and is shown on Figure 8.1. The modes calculated are shown on Figure 8.2 and the frequencies listed on Table 8.1. The enhancement to the long dipole was made by increasing the radiating area by deleting the middle structural brace and extending the lattice up to the top structural brace. The top plate is very thin and therefore less massive than a traditional guitar is therefore excited into vibration more easily. There are two long dipole modes near 400 Hz which should increase the output in that frequency range. 127 The mode frequencies in Table 8.1 are for the same guitar with two defferent tops. The first is Western Red Cedar and the second is 0.5 mm thick carbon fibre mat. Hz Hz 1230 685 10. 5. • T\ l't, Hz r-,, ~~_;:-:.-:-··--- 1205Hz ; I 447 9. I I 4. --...L A' I Hz Hz 349 961 3" e.. I I I I/ I I\ fTI. L\. v'I\ ~ ',] I I \ ~I{ I 'I ..,;· \ \ 1\\ / r\ " l'I.' IX ' l"1 J ~ ~ \ .,,. ~ .- , \ "' ·,11,- I :· ~ -, 'i. Hz 'lo )( \/ r-.1\,...... '.,.: =·· Hz "" I\ ... I/ l"i:x' l\f/ - ' ~,- ). ): design 1\; lX J ~ :(I x~ ~ ~ /},,,. ) ~ K ~ ~ l, .. I,{ !loo 304 926 ~ ) - ~ 1-.wn t? ~ - " \ ,1/ I\ ~/ - 2. - 7. .11N ~ )( DC l'I[ Ii\~ guitar I A ~ - ~ J( ~ IX ~ J 'If. ~ ~ )( ... 'X J ,11 I ,Ii ., I ~ ,,. / "'"121 7 I .J. ~ ... _.,.. - ~ / ~/ \ ~ ' / ,J)( ~ I \) 7 new J \ J I ~ A '. I I ,t t\ I \ r,.. !'.'I, 1.\ 1/ .. ,) \ \ I ~~ " I ~ 'Do: ' 9X /'1,"} \,}i of . \; I"- /, I. l1 ~ u I I / 1 f Moclles •I I\ 1\ I , I II', I ' • II ,, J ' ,)(l'\ t\_\ ) )( ' , ~ /~ -.:;,,Y,/ >, ~" IJ I\ v l,,l ·' r,. 8.2 ..... )( ~ii ) [\.'\[\ 1, IJ I\ - i:::" r- ~Kl\' I' L.-" I' Hz _.,.. )(r-,_I\ "l\ ,,, Hz ,,, ll J ~.,-· .x1. )j "' X ... ' ...... IX ... ~"" lil:i/ ., :x ) ~&, 695 .. - II 1, - [7 143 -~ ,r I\V \I/ I\ l'I - - , Figure ' 6. ) I !'I. ) l'I. Ill 1. ,--~ ~Ill I ) ~ 1, ~ ~ I' ~ J ~ IJ )I l( " I) I .. A lll1/ l " r ~ !/ I\ 1\ - ~ __, ./' I \i '11 I\ L_,; 'J ,., ~~ u ':" I 'J \ , ,~· \ I ' , 14 IS::1',r/. "' \t\/ ' XL, ' 1\ / / I') ~ I \. \'ll I / 129 TABLE 8.1 Calculated Modes of New Design Mode Frequency [Hz] Cedar Top Carbon Fibre Top ------ 1 M 143 143 2 CD 304 310 3 LO 349 345 4 447 442 5 685 673 6 T 695 693 7 926 923 8 961 1021 9 1205 1267 10 1230 1332 The frequencies for the modes were nearly the same for the two tops, as were the mode shapes. The top lobe of the long dipole is larger than on standard guitars. It was lower in frequency than desired but there was a new mode at 447 Hz which does not normally appear. This mode was a "long tripole" which should have good radiative 130 properties and therefore make up for the lower frequency of the long dipole. 8.3 The Prototype It was decided to build a guitar with this design. Rather than build a whole guitar for a radical design, a broken top plate was taken off an old guitar and replaced with the new top plate. This was considered valid as the top plate is the most important element. After completion the guitar was tested using the Fast Fourier Transform experiment described in Chapter 6. The resultant frequency response charts are shown on Figures 8.3 to 8.6, in the back cover of the binding. Table 8.2 Lists the most prominent modes measured, as well as the modes calculated by FEA. As before the guitar was excited by an impuse on the bridge, first at the centre and then at the first string position. 131 TABLE 8.2 Modes of New Guitar Calculated: Measured Freq [Hz] Excited at Freq Bridge First String [Hz] Gain Gain ------93 -12 -10 143 200 0 0 304 280 -15 -2 349 320 -16 -10 447 410 -15 -5 685 500 -16 -12 695 600 -15 -10 926 820 -10 -5 Note that the monopole mode frequency in the first column of Table 8.2 is for top plate uncoupled to the air. Using the value of 126 Hz for the helmholtz resonance and 93 Hz for the coupled air resonance (as measured}, the coupled monopole frequency is 166 Hz, almost 15 percent lower than that measured. The first four modes were predicted with an accuracy of 10 to 15 percent. Without doing 132 the holography experiment, the higher mode shapes can not be determined. The quality factor, Q, of the measured resonance at 410 Hz is 26. This is not abnormally high or low for a good guitar. The mode has high gain and is almost positioned at the same frequency as the G string of the guitar. When the guitar was excited at the centreline of the bridge, the monopole mode radiated about SdB louder with respect to the other modes than when excited at the first string point. A definite subjective assessment of the guitar has as yet not been made. The guitar is very loud, fulfilling one of the objectives of the design exercise. This is particularly notable since the guitar is a small body type. The sound of guitars generally changes during the first few weeks after construction completion, so all judgements are tentative. The guitar has been played by several people. One commented on the brilliance of the G string and the next player ( who was not present when the previous person was playing) commented on the dullness of the G string. This exemplifies the difficulties of making general statements about guitar quality, since different players may be after very different sounds. Most listeners and players, so far, though that the guitar is good, indicating that the design is not intrinsically bad. 133 The most adverse criticism has been that the bass sounds "lack definition". All players said something similar. This is possibly due to over thinning the edges of the top plate. Another possible cause is that the guitar back plate is very thin and the resonances of that plate are blurring the more precise sounds required. The overall balance of the guitar, that is the relative amplitude of the sound from each string, is good. 8.4 Static Analysis An analysis of the response of the guitar box due to the application of a static force ( the force normally supplied by the strings) was used to check that the new design was structurally sound. The deformation of the new guitar top was compared to the deformation of one of the other guitars (Ramirez). Since the old design is known to be structurally sound, if the deformations of the new guitar were similar to the deformations of the Ramirez, then one could conclude that the new guitar would not fall apart under the tension of the strings. 8.4.1 Analysis of a Square Plate To check that the static analysis of the guitar would be valid, a square plate was analysed and the results compared with theory. The plate had the following characteristics. The plate characteristics were chosen to be similar to the top plate of a guitar, hence giving 134 an estimate of the deflection to be expected for the guitar. a=400 mm length b=300 mm width E=16000 kPa t=2.2 mm v=0.3 The plate was simply supported at all edges(translations disallowed). If a load P is applied to the centre of the plate, the maximum deformation at that point is given by; ,. Wmax=qPa /D where :) "2. D=Et /12(1-v ) and q is a constant determined by the ratio a/b.(Timoshenko) For a load of 6N, the maximum deformation is then 0.48mm. The deformation calculated by NASTRAN using a four by four element plate was 0.51mm. The agreement between the programme and theory was therefore good. The discrepancy was mainly due to the determination 135 of the constant q and, and the use of only 16 elements in the plate model. 8.4.2 Static Analysis of Guitars To model the force of the strings, extra elements were added to the data deck to model the saddle of the guitar, the point to which the strings are connected. The saddle was a series of 5 pins between the bridge nodes and the point at which the top of the saddle would be. Equal force was applied to the 5 pins in the direction of the nut at the far end of the neck. The total force applied was 70N, that is 14N for each of the five new elements. Five elements were used instead of six, the number of strings on a guitar, because there happened to be five lattice grid points at were the saddle is situated. This was considered satisfactory for the accuracy required. The guitars were clamped at the bottom of the sides. Table 8.3 lists the displacement of the guitar top plate at several positions for each of the guitars. The Ramirez guitar is used as a control. The two sets of displacements for the Jenner are for a guitar with standard linings and for a guitar with heavy tone rims as discussed above. The displacements are given with respect to the position of those points when there is no external force on the guitar. The horizontal displacement (H) is in the direction of the string, that is along the length of the guitar. A positive number means that the points move toward the bottom end of the guitar. The 136 vertical displacement (V) is in the direction normal to the top plate. A negative number means that the point has moved toward the back plate of the guitar. 137 TABLE 8.3 Displacement [mm] of guitar tops due to string force. Position*l Ramirez Jenner Jenner with Standard linings tone rims H V H V H V 1 1.2E-2 -1.2E-3 8.SE-3 -1.lE-3 8.3E-3 -1.lE-3 2 1.2E-2 -8.8E-4 8.SE-3 -3.SE-4 8.3E-3 -3.3E-4 3 1.2E-2 7.lE-8 8.SE-3 9.7E-8 8.3E-3 3.GE-8 * Position- !. At centre of bridge 2. Just below sound hole 3. At top of upper bout (top edge of the guitar} Table 8.3 shows that the new design is much stronger then the traditional Ramirez design. The displacent at the point just below the sound hole is less than half of the displacement at the same point for the Ramirez. This point is usually the place where displacement is greatest and a structurally weak guitar will buckle most at this point. The tone rims made little difference to the strength of the guitar in this test. 138 After building the guitar,but before the strings were attached, the guitar appeared to be not as stiff as predicted around the sound hole area. This was determined by pressing down vertically on the top plate. Since stringing, however, the guitar top has not changed shape and appears to be structurally sound. 139 9. CONCLUSIONS 9.1 Use of FEA for Design Finite element analysis accurately predicted the mode shapes of several guitars, and to less accuracy predicted the mode frequencies of those guitars. Since the frequencies were generally overestimated it may be that the damping of the guitars is important in determining the resonant freqencies. Great accuracy is difficult because of the complex structure of guitars, and more particularly the differences in the way they may be constructed. The way luthiers adjust the top plate, for example, after construction can drastically alter the modes of the guitar. If the thickness of the top plate is not accurately known, particularly at the edges of the guitar, then good modelling is difficult. A new design was analysed using FEA and the resulting guitar was judged favourably, indicating that the design concept was good, and that the assumed precepts for good guitars was valid. 9.2 Traditional and Modern Guitars The analysis showed that the similarity in mode shapes between the traditional fan braced guitars and the modern designs is most evident for the lowest modes. At high frequency modes, the guitars modes for every guitar are very different. Good guitars can be 140 built using any of the designs, though the new designs which utilise the strength of modern materials such as carbon fibre are significantly louder. This is very important now that the guitar is expected to be heard in the large auditoria and concert halls which are the venues for recitals. 9.3 Future Research To improve the value of computer aided design of guitars, future research should include improving the accuracy of FEA models, and a more detailed understanding of the quality of a guitar with respect to its physical behaviour characteristics. A more detailed knowledge of the effect of the structural damping would be valuable, though difficult to include in the model. If damping could be determined, then the finite element programme can be used to calculate the frequency response of the guitars by simulating a forcing frequency at the bridge and outputting the displacement at the top plate at various frequencies. In this way, the gain as well of the frequencies of the modes could be established. The importance of boundary conditions, particularly at the join between the top and sides of the guitar, is well known. A more detailed or accurate representation of that boundary would improve the computer model. 141 The coupling between the top plate and the air in the cavity could be investigated by modelling the air as solid elements attached to the body. The resulting model would be more complicated but the understanding of the interaction between the fluid and the body would be enhanced. 142 10. REFERENCES Allen, H., Analysis and Design of Structural Sandwich Panels, Pergamon,1969. Blakeley, L., L., "Comparison of Modal Analysis, Energy Methods and Finite Element Analysis for Estimation of Natural Frequencies" pp 580-584 from Proceedings of 2nd International Modal Analysis Conference, 1984, Orlando Florida. Caldersmith, G., "Guitar as a reflex enclosure," J. Acoust. Soc. Amer. 63, 1566-1575, 1978. Caldersmith, G., "Plate fundamental coupling and its musical importance," Catgut Acoust. Soc. News Letter 36, 21-27, 1981. Caldersmith, G., "Monopoles, dipoles and tripoles", J. Australian Association of Musical Instrument Makers!, May, 7-18, 1985. Cook, R., Concepts and Applications of Finite EleMent Analysis, 2ed, John Wiley and Sons, 1981. Christensen, O., "The response of played guitars at middle frequencies", Acustica 53, 45-47, 1983. 143 Christensen, O. and Vistisen, B., "Simple model for low frequency guitar function," J. Acoust. Soc. Amer. 68, 758-766, 1980. Evans, T., and Evans, M., A., Guitars: Music, History and Players from the Renaissance to Rock, Paddington Press Ltd., 1979. Firth, I., M., "Physics of the guitar at the Helmholtz and first toplate resonances," J. Acoust. Soc. Amer. 61, 588-593, 1977. Gockel, M., Ed. Handbook for Dynamic Analysis, MSC/NASTRAN Version 63, MacNeal- Schwendler Corp., 1983. Guyan, R., J., "Reduction of stiffness and mass matrices", AIAA J., l, pp38o, 1965. Haines, D., W., "On musical instrument wood", Catgut Acoust. Soc. Newsletter 31, 23-32, 1979. Haines, D., W., "On musical instrument wood part II - surface finishes, plywood, light and water exposure", Catgut Acoust. Soc. Newsletter 33, 19-23, 1980. Harris, C. and Crede, C., Shock and Vibration Handbook, 2Ed, McGraw Hill,1976. 144 Hearmon, R., "The frequency of vibration of rectangular isotropic plates", J. App. Mech 19, 402-403, 1952. Jansson, E., V., "A study of acoustic and hologram interferometric measurements of the top plate vibrations of a guitar," Acustica 25, 95-100, 1971. Levy, R., "Guyan reduction solutions recycled for improved accuracy", NASTRAN User's Experiences, NASA TM X-2378, Sept. 1971, 201-220. Lazan, B., J., Damping of Materials and Members in Structural Mechanics, Pergamon, 1968. Meyer, J., "Quality aspects of the guitar tone", seminar, Function, construction and quality of the guitar, E. Jansson, ed., Royal Swedish Academy of Music No. 38, 1983. Meyer, J., "The function of the guitar body and its dependance upon constructional details", seminar, Function, construction and quality of the guitar, E. Jansson, ed., Royal Swedish Academy of Music No. 38, 1983. Marshall, K., D., "Modal analysis of a violin",J. Acoust. Soc. Amer 77, 695-709, 1985. 145 Marty, S., The assessment of innovations in the construction of the classical guitar using hologram interferometry, report for Australia Council Project, 1985. Plantema, F., Sandwich Construction, John Wiley & Sons, 1966. Rao, M., Raju, P., Crocker, M. and Zhu, G., "Experimental evaluation of damping of graphite fiber composites", Proceedings Noise-Con 85, Singh, R. Ed., 1985. Richardson, B. , E. , and Roberts, G. , W. , "The adjustment of mode frequencies in guitars: a study by means of holographic interferometry", SMAC 1983. Proc. Stokholm Music Acoustics Conference, July 28 - August 1, 1983, Val II,285-302. Richardson, B., E., "A physical investigation into some factors affecting the musical performance of the guitar", thesis for Ph.D., University of Wales, 1982. Richardson, B., E., and Roberts, G. ,W., "A finite element model of the guitar top plate", J. Acoust. Soc. Amer. 77 Suppl. l,Spring 1985. Roberts, G., W., "Finite Element Analysis of Guitar Top Plate", 12th ICA, Toronto, 1986 (in publication). 146 Schaeffer, H., G., MSC/NASTRAN PRIMER, Static and Normal Modes Analysis, Wallace Press, Inc., Milford, New Hampshire, 1982. Schwab, H., L., "Vibrational analysis of a guitar soundboard," Masters thesis, University of Missouri-Rolla, 1975. Schwab, H., L., and Chen, K., C., "Finite element analysis of a guitar soundboard-part II," Catgut Acoust. Soc. Newsletter 39, 13-15, 1983. Singer, F., L. and Pytel, A., Strength of Materials, Harper and Row, New York, 1980. Timoshenko, S., P., Strength of Materials, D. Van Nostrand, 1958. Young, D., "Vibration of rectangular plates by the ritz method", J. App. Mech, 17, 448-453, 1950. 147 11. BIBLIOGRAPHY Anderson, R., Irons, B. and Zienkiewicz, O. "Vibration and stability of plates using finite elements", Int. J. Solids and Structures, 1968(4), 1031-1055. Anton, H., Elementary Linear Algebra, Wiley, 1977, 2nd edition. Bernard, R., J., Price, S. M. and Seidel, M., R., "Acoustic modes of 3-D instrument cavities calculated with 1- and 2-D finite elements", J. Acoust. Soc. Amer. 77 Suppl. l,Spring 1985. Blakeley, L., L., "Comparison of Modal Analysis, Energy Methods and Finite Element Analysis for Estimation of Natural Frequencies" pp 580-584 from Proceedings of 2nd International Modal Analysis Conference, 1984, Orlando Florida. Boullosa, R., R., "The use of transient excitation for guitar frequency response testing," Catgut Acoust. Soc. Newsletter 36, 17-20, 1981. Caldersmith, G., "Guitar as a reflex enclosure," J. Acoust. Soc. Amer. 63, 1566-1575, 1978. Caldersmith, G., "Plate fundamental coupling and its musical importance," Catgut Acoust. Soc. News Letter 36, 21-27, 1981. 148 Firth, I., M., "Physics of the guitar at the Helmholtz and first toplate resonances," J. Acoust. Soc. Amer. 61, 588-593, 1977. Fletcher, N. H. and Legge, K. A., "Non linear generation pf missing modes on a vibrating string", J. Acoust. Soc. Amer. 76, 5-12, 1984. Jansson, E., v., "Acoustics for the guitar player," seminar, Function, construction and quality of the guitar, E. Jansson, ed., Royal Swedish Academy of Music No. 38, 1983. Jansson, E., V., "Acoustics for the guitar maker", seminar, Function, construction and quality of the guitar, E. Jansson, ed., Royal Swedish Academy of Music No. 38, 1983. Marshall, K., D., "Modal analysis of a violin",J. Acoust. Soc. Amer 77, 695-709, 1985. Mohanan, V., "Resonance frequency and half-width of assymetrical singlets in high-Q systems", App. Acoust 18, 159-170, 1985. Pollard, H., F., and Jansson, E., V., "A tristimulus method for the specification of musical timbre", Acustica (51), 1982,162-171. Pollard, H., F., and Jansson, E., V., "Analysis and assessment of musical starting transients", Acustica 51, 249-262, 1982. 149 Richardson, B., E., "The influence of strutting on the top plate modes of a guitar," Catgut Acoust. Sec Newsletter 40, 13-17, 1983. Richardson, B., E., "Investigations of mode couplong in the guitar", Proceedings Inst. Ac. 81-88, 1984. Richardson, B. , E. , and Walker, G. , P. , "Mode Coupling in the Guitar", 12th ICA, Toronto, 1986 (in publication}. Roberts, G., W., "Predictions of the modal behaviour of violin plates by the finite element method", Pree Inst. Ac ,89-96, 1985. Weinreich, G., "Sound hole sum rule and the dipole moment of the violin", J. Acoust. Soc. Amer. 77, 1985, 710-718. APPENDIX A AO FREQUENCY RESPONSE OF GUITARS AS MEASURED BY FAST FOURIER TRANSFORMS. Al Frequenc!J peak 0. 10010 Khz Spectral analysis of kohno50.1 ( 9.99024 ms) I I I Q) > Q) _J ...Q) ::> 1/) 1/) ...Q) 0.. "'O C :') 0 V) 0.000 ' 0.00 0.20 0.40 0.60 0.80 1.00 FREQUENCY (kHz) A2 Frequency peak 0.10010 Khz Spectral analysis of kohno50.2 ( 9.99024 ms) I I Cl) > Cl) _J Cl) w ::, II) II) Cl) a..w "O 1, C ::, 0 V) l liJ )\ ,1\l I. "' \ I I /I i ,,.,.1 \ /i 1 1t . /V---. 0, 000 '--~,,_•..:,,··--=~-"--.·:....111.1...---~-"-=----~.__,·· \..,,. __,-.i.!l...,:-:i... 1 _'··;::..,,=~::.;..-·'·'\..:..·.::::...-·' _ •.!...:::''-~,-__,~---' 0.00 0.20 - 0.40 0.60 0.80 1.00 FREQUENCY (kHz) A3 Frequency peak 0.10010 Khz Spectral analysis of kohno50.1 ( 9.99024 ms) 0.00 ,-----,---"T-----.------...... ------, -12.00 I'\ cc "tl -24.00 V Q) > j -36.00 Q) i.. ::, II) II) ~ -48.00 c.. "'O C 6 -60.80 V> n nn ll ?l'A ll .m A ll'A n or.. 1 nn V,VV V,i..V V,'fV V,tlV V,UV 1,VV FREQUENCY (kHz) A5 Frequenc!:! peak 0.20264 Khz Spectral analysis of gilet.1 · 1 1.93494 ms) (I) > (I) ...J (I) ...:, "' "'...(I) 0.. ""O C :, 0 V) 0.20 0.40 0.60 0.80 1.00 FRf:OUDiCY (kHz) A6 Frequency peak : 0.20264 Khz Spec!ral analysis of gilet.1 ( 4.93494 ms) -0.00 ,------,,------.,------r------, I~ l'\ . I . I I i\ 11 n' , /Y' A ~ JI,. f\\ j t \...,.1,t "' -...,.• ..,,.l..._/\ j'--,iv,J ,\ ' ~ / \ -22.74 I \ I \.,,. " )-"[ ~~ l\~l\.} \ (',-...., ~ I ·- r· r" , \. / I'\ \_-..., I ' '',..-.1 \ ,4 \ I ...M \ l 1, V "!j -45.47 \I Q) > -68.21 Q) -' Q) i.. ::, 1/) 1/) -90. 94 Q) a..i.. -0 I C: I I :> -113.&8 0 0.20 0.40 0.60 0.88 1.00 V) 0.00 FR£QU£HCY (kHz) A7 Frequency peak : 0.20264 Khz Spectral analysis of gilet.1 ( 4.93494 ms) -0.00 ~------~ -22.74 /'\ -~ ~ -45. 47 V Q) > j -68.21 ...Q) ::> 1/) ...~ -90. 94 0.. '"'C 5C -113. 68 ...______. _____. _____.______,______. V) 0.00 1.00 2.00 3.00 4.00 5.00 FREOUEHCY (kHz) A8 Frequency peak : 0.24170 Khz Spectral analysis of ramirez.1 ( 4 .13737 ms) I I a; l > 4) I _J . I 4) '• ,i ... IR -~ :> I II) ,'~ i II) I ...4) !I ) . a. "O If C :> I \ ) 0 V) I ~J \k. \ ,r ~...... , 0.08 0.20 0.40 0.Ete 0.80 1.00 FRI:OU£NCY (kHz) A4 Spectral analysis of kohno50.1 -21.38 I"', ~ ~ -42.7? V Q) ~ -64.15 ...J ...Q) ::> ::l -85.54 ...Q) a.. ""O § -106. 92 '-----'------.l'---_J,'----L'------1 ~ o.eo 1.eo 2.00 3.00 4.00 5.00 FREOU£1iCY (kHz) A9 Frequenc!:J peak 0.10498 Khz Spectral analysis of ramirez.2 ( 9.52558 ms) h 'Ji/ Q.) > Q.) ..J ...Q.) ::> Ill Ill ...Q.) a.. -0 C ::> ~. 0 II V) I\ ' .l '-- u.00 0.20 0.40 0.60 0.80 1.00 FREOUEiiCY (kHz) AlO Frequency peak : 0.24170 Khz Spectral analysis of ramirez.1 ( 4.13737 ms) -0.00 l""T"""'1,-,-----,------.------r-----,------, Q.) > Q.) ;:;:• -., --1 A.IL Q.) :,... 1/) ...~ li8.% a.. ""O C :, 0 36.20 V') 0.00 1.00 2.00 3.00 4.00 5.00 fREOUEHCY 0.Hz) All Frequency peak : 0.24170 Khz Spectral analysis of ramirez.1 ( 4.13737 ms) -0.00 --..----ir--T"----,----r----.-----.------, \ I Nit-. I /\ ,' I / \. j i\ I\ I / \ J\ j .,./ \ -17.24 ) \._... ,..... \,} -....~, \ i '•••,., Jt' l..!1<-,...Y\ ' !... I"\ { '-,\ (\ l c:Q ·~ { ;1U 'v ~- r \~-M '\ tr·J "'C -34.48 l,.i 'iII t'\i ,., 1 ~ I a:; ~i.72 u ' > Cl) J Cl) 5 ~18. 96 II) II) ...Cl) a.. -o 86.20 I I I I § 0.00 0.20 0.48 O.t0 0.80 1.00 0 V) FREOUI:h'CY (kHz) Al2 Frequency peak 0.10010 Khz Spectral analysis of aria.I ( 9.99024 ms) I I I Q) -> Q) _J Q) 1-o ::, II) II) Q) a.1-o ~ -0 C ::, 0 V) I l ~ 0. 000 ...__..... ,e__:: ,--_J"_/ ..1.. 1 _\~.:.../'.-==-.. _A:..:.' ~_JL-...,f,...::.\~... ..:.-_,):....} ...::l:.:::..· l'--....1::'~=::c:::::,,,..c:.r-::.::,-'- .....i.....c... rr-.c::....-~..-.1 0.00 0.20 0.40 0.60 0.80 1.00 FREQUENCY (iHz) r ' · " Frequency peak 0.10010 Khz ~pectra I Iana ~sis or aria.2 ( 9.99024 ms) QJ -> QJ _j QJ ...::, Ill Ill ...QJ 0.. "'C C ::, 0 V) 0.20 0.40 0.60 0.80 1.00 FREOIJENCY (kHz ) AJ4 Frequency peak 0.10010 Khz Spectral analysis of aria.1 ( 9.99024 ms) -0.00 -18. 10 I"\ CQ "d -36. 19 V Q) -> Q) ..J -54.29 ...Q) :::> V) V) ...Q) -72.39 0.. -0 C: :::> 0 I I I V> -90.48 0.00 0.20 0.40 0.60 0.80 1.00 FR£0UrHCY (kHz) A15 Frequency peak 0.17578 Khz Spectral analysis of '5.68889 ms) Q) > Q) _J ...Q) :::> 1/) 1/) ...Q) (l_ "'O C :::> 0 V) 0.000 0.00 0.20 0.40 0.60 0.80 1.00 FREOUEiiC''I' (k.Hz) A16 Frequenc!:J peak 0.27344 Khz Speciral analysis of mariy8.2 1 3.65714 ms) I J l Q) -> Q) _J I II I ....Q) ::> II) II) / 11 \ Q) ~ .... I I H I I I a.. I ( ,, ff I I it #' -0 C ::> 1· ' t•. 0 l 11\/1 · ir 11~ f·1 ,/\ V> j \ ' ·1 I ~. I 1 \ I, I l ~. ~ A1 \ 0. 000 r-,-.-1'-' •"'IJ ' I ~ \ ! ,._,,.J I \J \I 'M \ i.! \.,,", ,,._~ i ' ...., 0.00 0.28 0.40 0.60 0.80 1.00 FREQUENCY (kHz) A17 m~rt· i:O 1 Frequency peak · 0.17578 Khz Spectral analysis of a ~u, J. ( 5.68889 ms) 0.00 -12.00 I'\ CQ ',:i -24.00 \,I (I) > (I) -36.00 -' (I) ...::, .r, ...(I) 0. '"C § -ti0.00 I I I 0 ,'l l:ICI 1:1 ?li 1:1 ,Al), 1:1 lr. li Of.I I (ll:I V) V,VV V,L.V V,'tV V,CIV V,UV 1 • U'U FREQUENCY (kHz) A18 Frequency peak : 0.17578 Khz Spectral analysis of : marly8.1 ( 5.68889 ms) ,-.. CQ "ti V II) -> II) _J ...II) :> Ill Ill II)... -81.25 a.. "O C :> -101.57~---'-1 ----'1 0 '-----.....•--__.• ___ •__ __, V) 0.00 1.00 2.00 3.00 4.00 5.00 FR£0UEHCY (kHz) A 19 D. 1 m:::c:;:; VJ, ~ U, J.U•J-JJ J\ lk gilet2.1 { L, ---H-l44 ·t ..., ., ) ._i 1 ,-#U _.,. J. I llt ..J- I' ! ; L j \1 I II,·, ! i! n Q) !! > il Q) l! _J {! ...Q) Ii ::, il 1/J ;1 7 1/J Ii Q) I·, ' ... i j 0.. I; t' i '"O i C: ! ::, 0 V) fREQUF.NCV ( kHz ) ~\\J A20 n 1Ll7J? 1n u,.1u11L f\hz n:l~-4-?? ~ ' ' c; 1,1... • I.. ,---r.,... -~------~------. QJ -> QJ _j i ...QJ ,I :::> H t V, "II V, }! ;' QJ fl j ... Jl ii 0. H ,Ii ll !i "U il ! C _ij i! ; I :::> Ii _! ! i'; l1 0 Ii i l \i ii .) ii V) n i1 1 i !, .1l F 1•, '1 ! ,\ 11 ,iJ h l\ 1 i I ) ;_ (... -. Ii 11 ·,_~ _; ·, .l; • I_}. (iOO .,;,1...·, ~-=--· _,IL~--=-;.:.-/_,.,J'•-:.,!_· ··.:.-..,...... :.·,,;.;.... _ _;··,,c../·-...l_'·-=·-._"-/ _··_--_-..,·-=--···_·:..• --'-....:,--••c.:··-c..! _'•. .::;1·._;,-··-:.:.·,··TC~;:..:"-' _.,..·-.o.'..,~-----•I 0.00 0.20 0.40 0.60 0.80 1.00 fRtoUENCY (kHz) c.:\~ A2l r,:1--1-? 1 J;=,.r,.,.-. , .• ,,. pr.-.1. o. 1 or.:cr.: v1. - , :t1 l l. C 1.L., J. l l C:'-{Ur, 1lt..8 l,aJ\ U, J.UJJ-..1 1\llk 0v,vv f,fl -tZ,00 L •,j Q) -> Q) ...J ...Q) ::::> Ill Ill ...Q) a.. ""C C ::::> 0 -60.00 V) 0.00 0.20 0..10 0.(-;0 0.80 1.00 FR£0UEHCV (kHz) A22 i;t W7.1? VI.~ Frequency peak U, .LUI J.L.. l\Jli, Spectral analysis of gilet2.Z ( ' /'\ Cl) -> Cl) ..J ....Cl) I :> f Vl i Vl -AJ!, (\(\ L ....Cl) i Q. -0 C ::, 0 -60.00 V') ,"< All .... /..f) 0.00 0.20 V,"fV v.ov i.00 fREQUE'nCY ( kHz ) A23 U 10.c;c:r: VL~ , 1 I'"' 1 U , U,.t-.J-1 I\JI l, g ,J l "-'·,..t:," (pl.,. I ill "t -Z~.00 '-; Q) -> Q) ...J ....Q) ::> 1/) 1/) ....Q) a.. -0 5 -bU.00 0 fl Ar, V, v. vv LOO 2.00 3.00 4.00 5.00 fR£0ifi:.1iCY ( kHz ) A24 f, Freqnenr:u peak 0.18B66 Khz Spectral analysis ot maton.1 ( 5.53513 ms) f.; IP7 \.-·• .!V, I I ji ii .'I ii H II II Ii !i I (I) > (I) j _j ii ii ....(I) ii - ( ::, !, ' Ill ii i Ill II I ....(I) I! l a.. J! l i'; i' 1 -a Ii i' I C -,, i ::, \ __ _ !\. ,/ .r-, I 0 ; ·-- ' ...... V) ,' I .-/'-..• ===' 4-·-- ·-- -0--s J 0.20 0.40 0.60 v.uvc. or. 1.00 A25 frequenGH peak r.Jpectra , I analysis of iiiafon .1 ( 5. 53513 r:is) f; Hfl V • i•"\:" /'\ ..... --24.ov ~ > Cl) -36.00 _J Cl) ...:::, 1/) 1/) -48.00 ...Cl) a.. "O C: :::, 0 (\ '){I. V) 'V,L'V 0.40 0.60 0.80 1.00 fR£QUI:riCY (kHz) A26 . ~rrequency peaK1 0.18666 Khz Spectral analysis of ma ton .1 0.00 -24.00 (I) > (I) _j iI ...(I) :> 1'A\JJ;I V) ,f f'l!fir,._i V) -48.00 . 1 '1 ilil (I)... . . ll 0.. !I -0 H C ! :> -Ml.00 0 A {V'; V) 0100 1.00 3.00 'i,'VV 5.00 FREGUEHCV (kHz) A27 0.09756 Khz Spectral analysis of yairi .1 I I i I I I I (I) !l -> (I) !l,! ..J II II ...(I) !l ::> j II) II) I ...(I) I Cl.. -0 c:: ::> 0 V) 0.000 0.00 0.20 0.40 0.60 0.80 1.00 FREQUENCY (kHz) A28 f.l ClQ7C..t: Vl ~ . . '] Frequency peak U,UJIUU J'>JIL- ya l r-I. L. Q) -> Q) _J Q) '"':> V) V) I Q) r: a..'"' I, ji, "'O C ii l/l_/11 :> ii t ~ 0 ii i '.I V') fi I ii ,1 q n. • I. h' I'"·. tiUi '11' ·,.. __ • I ., I . /\_ !\_J~ A. t\ .. ~ -· 0.000 ·--.r' . r. 'J{\ Cl AC. {\ t.r:. i Ci{\ V,LV V,'tV v.ov 0.80 1,VV FREOUEHC\' 0:.Hz) A29 Frequenc!;! peak 0.09766 Khz Spectral analysis of yairi.1 ti 'J4WMii. m<' ·, (\ 1u, La ~,uu ~ .1 0.00 ,, 1 I~ _.}... , H il _, i/ 1·~ , ! !, I~ 1 ~ =-~ \ i ~ /\ I \ ,1 I ,J ~j~1i;, ..\ l \, .r i -lZ,00 ! i, r- {\ i \ { \,,-, . ,•, t l-.t nt , ji,.._tl ·J \ ,◄ \ 1, r~ .i '\i' ~,,I.,_.,/ ' ... I ·,, .A, t!\,I \ ,· •, ', 'i \1 \ I '\ / 11 I I, I ,, I ·, .. , • ,\/ \ • -j. ••,' I ' ·, ·11 I ; '· I ~ b .,,..,, ' "' ',i 0l . I l ! ' j . ,,) ~ r,J _lf\_r, 1!1lH !' I 1 , '· .1 i,/ \ ,I ~,1,, I 1( \ -L4,\J!J ttl I} V ~// \ ( lj V' ! ' \i !!!i i1l1 . ; H,, V !IIL"I' .,l'i.. ,., ' I I ll i{i,! \If ii Q.) II -> 'II :i~•.I II Q.) -.:m.00 h ....J i! i ...Q.) I :> I I II) II) -48.00 i ...Q.) i a.. I ""'O ! C 1 :> ------~-----' 0 -60.00 ~ V') 0.00 0.20 0.40 0.80 1.00 fR[QU:DiCY (l.Hz) A30 Frequency peak 0.09766 Khz Spectral analysis of !Ja i ri .1 , rn J 41Ai;tti ,.. ... ·) \.J.U,1..11.1.UUU IIIUI AM V•VV -12,00 .,. ., ~ -24.60 ai > Q) J Q) -3b.OO i.. ::> Ill Ill Q) i.. a. --48.00 --0 C ::> 0 V) -60.00 0.00 l.00 2.00 3.00 4.00 5.00 FREQUENCY (kHz) A31 . . ,, Frequency peak 0.09766 Khz Spectral analysis of ~jalrl ,(. AMVr"\.4'V I ! I -12,00 l 'I. 'ii ,., .J .l\..r-, Ii -L4,t:,;JU If. if 1\-,/ V !, > ll::, V ;p _j V I-, :::> V) V) V I-, 48.00 0.. -0 C: :::> 0 V) 0.20 0.40 0.60 0.80 l.00 fREOUEHCY (kHz) BO APPENDIX B NASTRAN DATA FOR NEW GUITAR DESIGN 81 I Jr':3 J~~~E~,C~200CGO,T5020. US~R,J1P4~~5,G~C~G~. ATTJCH.L~KFIL~/UN=~ASTRAN GET,~!STRAN/U~=NASTPAN N,STP.A~(,~Ilf.,CA~~S) ~~PLACF(FILE=GfStUT) ~crPY,Ffl~,SUTPUT. G~T,~ASPLCT/LN=NASTRAN LI 8R.f.i~V, ?l,.,TLI ?.. ~~SPLCT. CALCOMP GI~rCSE,TAPc2=PL/J~=JE~~ER. 1t:c;=:, NAST~AN P~~FCDT=2 IO GUllAR ~4fN PL~1E ~ 1'1t. ] TIJ,lc 2-0C,'.) C Ea,,n Tille= ~~a 4 l~-?-q6 CECAq TCP- ALL CF .76~~ T ► ICK TCTAL SLgTJTL~= e~TTCM CCNST?A!NEC L!REL= MGI~+GUYAN. GUI1AR C □ ~PLETE :,,:.:THCD=2:9 !:CJ-'O=SC~T MCCi:S=lO 5ET 15= 6,lOCC5,10505,llCD~,50,lCC006,lCC05C CISPLACEME~T(SC1Tl)=l5 ~FJ~ST PLCTS A~E ~~ TOP PLATE (S~T 1) S:=>C=lOl GUTPUT(PLrT> SFT 2=2 THRU 60203 1\X::<; 7.,X,Y VIEW O,C,O CCNTCUR ZCIS? EVE~lO FINO SET 2 PLOT MCCAl DEFCR~ATIO~ CCNTCUR SET 2 RcGIN J.!ULK EIGR,2G,~GJV.,,,10,,,+El +El.~ASS ISUPPORT CARC AT TCP CF GUlTJR !SUPOQ.T.6,12?45'.J \GUYAN ~E£tCTICN OVEP All ~SG~ES~ CN TOP PLATE !~~LY ~E~1lCAL DISPLACE~ENT ALLCkE~ if-[~ST A~E TCP CF TCP PLATE AS~il,3,40202,50202 $LI]\-;~~- :3CUT ASET1.~,10003,10006,102C3,10207,1C403,10407 AS~Tl,3,1C603,10607,108C3,10S07 A5fTl,~,llOC3,110J7,112C!,ll206 ~PC[NTS ON SIDf. CF TCP ANO ~GTTC~ 1At:;~Tl,12345f,1C0034,1COC7C,1CCC22,100C82 ASFT1,1234ry6,34,7G,22,~2 ~Grci:;::r.,,,,,, 12 ~LARGEST GRIC ON Lr.WF.R BGUT FGPID,ll,C,20C.,-120.,C. r-~P.ID,12,C,470.,-120.,C. Fr.~IC,13,,47C.,12J.,O. CGRIC,14,,20C.,l20.,C. r,~J~G,1,G,126,9,-11,-12,-13,,+r,1 +Gl,12,-14 [GEN,9UA~4,3~U1~3,1 !.GqJr~ EIT~EP ~IDE 0~ l~qGES1 PLATF. fGRIC,74,0,2~C.,-l67.,C. ~GRfC,?l,C,440.,-156.,C. ~G~1c,22.c,4~J.,-133.,c. FGPJC.?3,G,2~: •• -144.,G. B2 c~r~c,2,c.12~,1,-21,-2?,-l1,,+c2 +;z,:,-24 C~FN,QL~~4,2CC,~D,2,,JC. ~~Rlr,J4,0,2SC.,144.,J. fGO!C,31,C,440.,13B.,O. fG~IC,32,C,440.,15~.,0. ~GPID,33,G,Z~C.,167.,J. ~~IDG,3,0,l26,l,-Jl,-32,-33,,+G3 +.:;i,5,-3 1t CGf~,CUA~4,5C0,50,3,,;c. ~GRIDS ~IT~ER SICE CF ~CLE r:GR!0,41,0,51.,-120.,J. c~PIC,42,c,110.,-120.,c. ES 0 In,43,C,170.,-4C.,O. EGPIC,44,C,5C.,-4J.,C. G~IDG,4,0,1?6,4,-~l,-42,-43,,+G4 .. ,:;4, 4, -44 CGFN,OU!"4,10J,43,4, EGRI0,51,0,SC.,40.,0. FGPIC,52,:,17 □ .,4~.,0. rc~I □ ,~3,J,1,~.,120.,0. =~rI~,~~,0,5C.,l?O.,C. G~I~G,5,C,l26,4,-5l,-57,-53,,+G5 +r;5,4,-'34 C~EN,JUlr4,6C0,43,~ iG?IO I~CLUSI~G P~PT CF FI~GE~ECARC f~RIC,6l,C,2C.,-3J.,J. f0DID,62,C,J2J.,-30.,0. ~JRID,63,0,12C.,3o.,o. FG~r0,64,C,20.,30.,J. f:~IDG,t,0,126,3,-61,-62,-63,,+(6 +G6,2,-64 CGfN,OUA04,8C0,4S,6 IMOCIFY EDrE PCINTS CF GRICS iGRIDMOO,l,00,00,,,, $G~ID~CD,1,0C,19,,,, $GRIOMCC,1,12,CO,,,, !GPlCMCn,t,12,C9,,,, ~Gf!CMOD,2,CC,00,,,, !~0 IDMn0,3,CC,Oi,,,, l~RIO~CJ,4,00,0J,,,, 5GRICHCD,5,04,00,,,, iGD{CMQ0,2,C5,00,,,, $GPIOMOD,3,05,0l,,,, t~L~~E~TS RElWEEN ~RICS CJUAD4,205,5G,20C01,1CCCB,10CC7,2Cl01,90. C0UA~4,2Ct,5C,2~101,1CCC7,10CC6,2C201,<;0. C~UAG4,2~7,5C,7C201,100C6,10C05,2C3Jl,g □• CJ l: AO'• , 2 0 ,S , 5 C , 20 3 :J 1 , l O C Ci:; , 10 CO 4 , 2 G4 0 1 , <; 0 • C-,ll1Af14,2·19,5C,2C 1t0l, 10CC4, IOCC3,2C501,<;0. c.; U ,'\ r.. 4 , 5 8 5 , SC , 11? 0. :l , 3 0 C CC , 3 0 1 0 0 , l 1 2 0 7 , <; J • C J ll A C.,.. , 5 D l: , 5 C , 1 12 Q 7 , ? ,) 1 G O , 3 0 2 C C , 1 I ? 0 6 , 9 0 • C~UAn4,5 □ 7,5C,11?06,3G2CC,30300,112 □ 5,~0. C;~U~C4,5(1P.,SC,112J5,3G30C,304CC,11204,<;0. COUA04,5~q,5c,ll?04,1G4C0,3C50C,11203,90. C~UA04,12J.,43.40004,l0CC0,101C0,4Cl04 Ci~UA84, l ~l .43.40104, lCICO, 102cc ,4C2J4 C1UAC4,123,43,4J204,102CC,l03CC,4C304 ,: ;) U .4 C 4 , l 2 4 , 4 ~ , 4 0 3 0 4 , 1 Q 3 C C , I O 4 C C , 4 C 40 4 C'HJA'J4,o22',43., c;uOJ4, lO~CO, 1090S,5Cl04 C )I!~ C4, ':, 21, 4 3 , 5010 4 • l O c; GC, 11 CO C , :i C 20 't ~lUA~4,~?2,4?.50204,11CCC,lllCC,SC304 C1UA~4,623,43,~0304,lllCC,112CC,5C~04 IGRI~S °CINTS AROUND 0 ERl~ITER CF LA~GE GRICS $D(INTS STAPl AT 2 ARCU~C scu~c ~[LE THE~CE ~NTICLCCK OR(U~ SRIC,2,,136.,-~o.,o. ~r[C,4,,125.,-30.,~. r,!-' I~ ,6,, D • ., C., 0. 83 S~Ic.g,.3.,-30.~0. :,'~ IC,lC,,?0.,-40.,0. G~IC,14,,11.,-60 •• 0. f,~TD,16,.lH.,-eJ.,O. ~r1c,1a,,31.,-100 •• J. r, •~ I D , 2 C , , 6 Q • , - 13 B • • 0 • r~1c.22,.11c.,-141.,o. ~QI~,24,.140.,-13S.,0. r;ii r a.~ c,, 17 J., -12:.,., o. G~I □ ,28,.210.,-132.,0. G~I0,3C,,26C.,-l4i •• O. G~!~,32,,320,,-1~1.,C. G~!C,34,.350.,-l~J.,C. G~I~,36,,3qo.,-1P6,,c. fRJC,3~,,410.,-l?j.,C. f:RID,4C,.4r.~.,-13S.,C. ~JID,42,,477.,-100.,C. CRID,44,,4q2.,-ao.,o. G~IC,46,,4~7.,-61.,J. ~RTD,48,.4~~.,-40.,J. G?[C,5G,,490.,-zo •• ~. r;~yn,52,.~91.,n.,o. G~IC,54,,490.,2C.~C. ~~IJ,56,,439.,40.,0. f:~JC,?e,,487.,60.,0. ~~IO,oC,,4?.2.,~0.,0. r~TC,62,,477.,100.,Q. G~ID,64,,459.,13B.,~. GRID,66,,410.,176.,0. Gt1D,6P,.380.,186.,0. ~~ID,70,,350.,1~5.,D. C~J0,72,,320.,181.,0. GKIC,74,,26C,,1~~.,o. GRIC,76,,23C.,132.,0. GP-IJ,7~,,170.,124.,0. f,~lD,~C,,140,,135.,0. GKI~,a2,,110.,141.,o. ~RID,84,,80.,136.,C. GRf0,~6,,30.,100.,0. GRI □ ,~e.,1e.,so •• ~. ~RIC,9C,,ll •• ~o.,o. G~I0,94,,3 •• 30.,0. f~TD,Q6,,20 •• 40.,0. GR!H,1c3,,1s6.,~o.,3.G~tP-,9f. ,l?5.,3Q.,O '********************************************************** ~EL~~~NTS ARCUND GPICS C~ TOP PLATE CTrtA3,2,43,1J50 □ ,106DG,2,go. CTRIA3,4,43,1J400,1050C,2,gQ. CT~Il3,b,43,40404,1C40C,2 C0U~14,~,43,4J402,4C403,4,60003 CTRlA3,10,45,60U03,4,6ClC3 CJUA~4,1?,43,404Cl,4C4C~,6C003,6C002 C1Ul\f'l4,l4,43,40~01,6COC2,6COC1,40400,CJC. CJUA04,l6,44,10,4J400,6CCJ1,6CCCC C~UI04,1B,43,60100,6,3,ECOCO CQUA04,2G,44,l0,60COC,8,14,;C. f.~UAD4,2~,44,14,40~00,4C40C,10 CQUAD4,24,44,l6,402DC,40300,14 CQUAD4,26,44.1B,40100,4C20C,16 CTPIA3,2~,44,4000J,4ClCC,le,go. CTPIA3,3~,44,21,40001,4COO~,go. CQUAn4,32,44,22,400J2,4:C01,?C,~O. c:uAC4,34.44.24,400Q3,4CCJ2,22,~C. fJUAC4.36,44,26,40C04,4CC0~,24,gC, CTR[13,38,44,1Q000,4C004,26 CT~I~3.40,,l,10001•1COCC,23 f. ,J t, A fJ 4 • 4 ? , :; l • 3 0 , 1 G C v 2 • l CC 0 l , .-~ 8 • g D • 84 r)UAr4,44,~l,2050L,1CCC3,lCCf2,3r,9J, cTrIA3,~6.~1,20~co,zo;c1,1c,so. (T~IA3,4],~l,3?,?0~00,2C::JC,90. C~UA~~.~:,31,34,2J11G,2C4JC,32,gQ, C.~ I J '\ r, 4 , 5 :> , 5 l , J 6 , ? D 2 G C, ~ C.! 0 C , 3 ·• , 1 C: • C '; U .II C,; , •;; 4 , -; 1 , 3 t: , 2 J 1 J C , 1 C '2 0 C, 3 6 , ':i C • rr~It3,5~.~1,2010J,1~,2CCCC,1J. rX~IA3,5~,~l,4~~~1CJl~~~CUQ ~ ~ C. ,d,".i.D ~~!t~icb6jJ~~26C.,-143.,-lC~. ~~l~,1CJC31,,2~C.,-167.,-1CG. G~TC,1CJJ32,,J2J.,-l~L.,-l~O. r. ;: I r, • l C ,JC 3 4, , 3 i:; C •, -1 :::> ·5 • • -1 CO • G~fD,100C36,,3QO.,-l~6.,-lCG. ~~I □ ~lCJC3~,,41J.,-17~.,-lC~. ~~IJ,1C~13Q,,440.,-156.,-1CJ. ~~,~,1CJC4~,,%~7.,-133.,-1co. G,r0,1coo4I,,\7J.,-120.,-1c0. r~r1,1con4~,,411.,-1oc.,-1co. ~~1 □ ,1cJc4~,,~P2.,-s~.,-1ac. ~~ID,1C0046,,4~7.,-6C.,-10C. ~~1~,1cao4~,,4q9.,-~c.,-1oc. ~~1a,1c0030,,4qJ.,-2c.,-1oc. G~1 □ ,1coos2,,~q1.,c.,-1cr. ~~,0,1can~4,,4sc.,?o.,-1c~. ~1I1,1COJ56,,~~~.,4J.,-1ao. ~?TD,1C9053,,437.,6C.,-ICO. G~IJ,1C00A0,,4£2.t?J.,-1CO. G~I~,1COJ6?,,477.,1JC.,-1oc. ~~I~,1C0063,,47G.,12G.,-1oc. G~IC,1CJ064,,459.,133 •• -1oc. G~!D,1COC65,,440.,l56~,-1oc. G~lC,lCOJ66,,4I0.,17~.,-1JC. ~1!D,lCOCbR,,3eO.,l16 •• -1oc. ~R!J,1c0a10,,3~0.,1~s.,-1oc. Gqrc,1c1n12,,320.,1~1.,-1ac. G~TJ,1C0013,a2QC.,167.,-1JC. r~rJ,1cao14,,26c.,140.,-1oc. G1IQ,1COC76,,23C.,132.,-1oc. r,~JC,1C0G77,,20C.,1ZC.,-1oc. G~t~,1C007~,.17G.,124.,-1cc. ~RID,1C0080,,l40.,135.,-1oc. G~ID,1C00~2,,110.,141.,-1oc. GRIC,100~94,,80.,136.,-100. G~JJ,1C00~5,,50.,12C.,-1CG. G~ID,lGJ03~,,30.,lOO.,-lOJ. GRID,1CQCj3,,13.,jo.,-1cc. GRI0,1C0090,,ll.,6C.,-1cc. r~1n,1cnJ~4,,3.,30.,-1cc. '**~*******************************************************' ~ELE~~~TS ARCUNO SIDE ~NC □ ~l~~TATIQN GlV~~ ~~U~~ER IS ~~JAC~~T T~P PLATF ~L~~~~ PLUS ICCCCC ~~T~QT t..T l!PPl::R. l'.}iJUT fHEN .6NTICLGCK LOCKINC CC\\I\ C]UA04,lOJOl@,47,100006,lGCOC8,3,E C ".; IJ An 4, l CO C2 C, 4 7, 10 0 0 0 3 , 1 t'J C :J 14, 14 • ~ C0UAJ4,10JC24,47,100Cl4,100016,l6,l4 C~UA~4,1C0026,47,100016,10C01~,13,16 CQU~04,l00028,47,lOJClA,lJCOl~,40001,18 c~uAry4,lCJC3~,47,100Cl~,1JC02C,2C,4000C C::) IJ Af) -'t , 1 J ;J O3 2 , 4 7 , 10 CC 2 C , 1 JC O 2 2 , 2 2 , 2 •J CQUA □ 4.10JC34,47,10CC22,1JC024,24,22 CQU1D4,10JG36,47,100024,10CJ25,26,24 CQIJ ~ 0 4, 1ODO3°:,47, lOC 02 ~, 1ocO27, 10 COG, U: COUAn4,10004C,47,lOOC27,10C023,2?,10COC C~UAD4,lCJ042,47,lOOC2~.10cc3c,3a,2~ CJU~C4,l00046,47,lOCClC,lOC031,2050J,3C C~UA~4,10004P,47,100C31,1JC032,32,20~00 CJUAn4,lOOC5C,47,lOOC32,lJC034,34,32 CJUAr4,100052,47,lOOC34,lOC036,36,34 C')IJ Ar,:+, 1JCH~54, 4 7, 10 J 03 6, l JC Q 3 ;J, 3 3 , 36 CJIJ AD •t, l C: i) :J 5 6, 4 7, 1 CJ C3 ~ , 1 JC :J 3 q, 2 0 C ~O, 3 8 C ::1 U AC'.:,., 1 CJ C5 ?- , '• 7, l lJ GC 3 c; , 1 JC O4 C , 4 Q , 2 0 CO C ~1~AD4,lOJ06C,47,10004C,lJCJ41,1CC □ ~,4C r,,;uA~4,lOOC62,47,lC0041,10CC42,42,lOC09 C~Ula4,lC0064,47,lOC042,1JC044,44,42 r ,·I! Ar. 4 • 1 CO C 6 ii' • 4 7 • l O v Q 4 4 , 1 JC J 4 S , 4 6 • 44 86 C ~, IJ ~ ': ;. ~ IC JC 6:: ~ 4 7 ~ I~'.) C-t f:. ~ 1:; Co 4:) , ti 3 , 4 ~ f ~; U Ar') 4 , 10 :; :) 7 S , 4 7 , l J .JO 4 <; , 1 iJ C -J c:; ~ , 5 C , 4 '3 f '". I_; 4 r. 4 , 1 0 u J 7 ? , 4 7 , l OO C c; r, , l ') CO r: 2 , 5 ? t i:: ;J r. 1 'JA~" ,t • 1 c o71 :, , " 1 , 1 n 0 c ') -;i , 1 ) c 1J r:; 4 • 5 4 , s l r,uab4,10077G,47,1CJC54,l~JC5t,~t,54 C ; !Jt1 G 't , 10 J 7 6? , t.. 7, 10 J ·'J 'i 6 , 1 JC') r, 3 , 5 q , 'i:, C) '. J ta " ·+ , 1 0 CJ 7 6 A , 4 l , l D ;) C 5 f' , 1 j C Ot O , c G , t; ,; r :•J 1 r:: 4. to J 7 6 4 , 41, 1 c Jc'., c , 1 Jc o6 2, c ~, s J CJUA~4,lCJ762,~7,lJOCS2,lJCG63,l12Cq,62 CJUA04,l0076C,47,lJOCb3,1JQJ64,64,ll2C9 C~UA~4,18J 7 52,47,l0:03~,lJ0065,3CCOl,64 f~UA~4,1~07]6,47,10JC~~,1GC06J,66,3JC01 ciu~~4,1JJ754,47,l0~C6t,10C06J,f3,6~ CJ I J AD 't , 1 0 ,) 7 5 ?. , '• 7 • l ') J C 6 :~ , 1 u C O7 J , 1 0 , 6 ~ CJG~~4,lCJ75C,47,100C7C,1GC072,72,70 C0UAD•,lOJ74~,47,lOOC72,1CC073,38501.,72 C~UA~4,100746,47,lCOC73,lGC:74,74,305Jl CQUl04,11~742,47,100074,lGC076,76,74 f1Ul~4,l0J74~,47,1J0~7t,1JC □ 77,ll20G,76 (~U4n4,l~G73q,47,10~C77,lOCa?e.7~,ll2GC f(.)IJ~D4,lGJ736,47,lOG0!5,lJCC3C,jC,7:, C~UAC4,lCJ?~4,47,lOJ03C,lJCJP2,82,8J c; l' .H) '· , 1 GO:"~ 2 , t.. 7 , 1 CJ O3 ~ , 1JCO94 , 8 4 , 8 2 r, ~ U .I\~ 't, 1 /J J 7 3 G , tt 7 , 1 0 0 0 t: 4 , 1 iJ CO ~ 5 , .: C 4 Q O , :_: 4 CQUlC4,lCJ72~,47,1000J5,lOC036,~t,5040C ClUA~4,lOG726,47,10JOR6,lOCJ8B,29,S6 C1UA~~,10J724,47,100C8~,1JC09J,gc,~s r~uAn4,1oa120,41~1accgc,1acaq4,;4,go C~UAJ4,lQJ71~?47,100Cg4,lJCJCt,6,94 $GOTTJ~ r>L.~TE '********************~*************************************: ~***************************************•******************= S3CTTO~ ELEME~TS-USE □ NLV PERIMITER G~IC PCI~TS-START CR(~ I CTRIA3,10J300,47,100054,10J8~0,1C0052,~0. CGUA~4,10QjJ2,47,lOOCS4,10005C,1CC043,lGCC~6,9C. C0Uft04,l00304,47,lOC056,lOC043,l □ C046,lOC058,9C. CQUAD4,l10~0t,47,lOOC5e,1oco46,lCC044,lJCOEC,9C. CQUAD~,lOO~oe,47,lOOC6C,10C044,lOOC42,lOCJ(:2,gc. CQUA04,lOJ~lC,47,lOOC62,lJCC42,lOC041,10COE3,9C. CO t• Ar.: 't , l OJ 8 1 ? , 4 7 , 100 0 5 3 , 10 CC 4 1 • l OC O4 C , 1C CCH: 4 , q C • C;: 1 l! Ar: .:i- , 1 0 CH:: l 4 , 4 7 , l O ,J C,:, 4 , 1 0 n O4 C , 1 0 0 0 3 'h 10 CC t 5 , 9 C • r r, U 4 0 4 , 1 0 0 i:' 1 (: , 4 7 , 1 0 v IJ :) 5 , 1 0 C O ~ :; , 1 0 C O3 A , I O CO l: 6 , 9 C • C ~111 AC4, 10 '.J :! 13,47, 10 0 C 6 6, 1 CC O 3 :3, l C ~ 0 3 6, 10 CO f !3, 9 C. CJUA04,lJ022J,47,l~JC6B,1C~036•l0CC34,lOCD7~,qc. CJU~r't,lOCP22,47,100~7C,1UC0?4,lOGQ32,1JC072,9C. cou,~4,lQJ~24,~7.1ocr72,lCC032.IOCC31,JJCJ73,9C. CJUAC4,lOJ~2f,47,lCOCr~,10C031,lOCC30,lCCC14,9C. C 1lJ AC It, 10 J Q 2 8, 4 7, l G;; C 7 4 , 10 CO 1 C, 10 CO? P,, 10 CO 16, ➔ C • C0UAD4,lQ0?32,47,la0076,lOCJ22,10C027,l0C~77,1C. C '} IJ AIH, 1 Q 0 'l 3 2, 4 7 ,, 1J~C77 , 10 C :> 2 7 , 1GCO26, 10 CJ 7 3 , 9 C. CJUAD~,,10 □ 334,47,1CJC75,IOCC26,1G:o24,1cc~~c.gc. C~UAC4,lOOB3t,47,lC~C8C,10C024,lGC022,1CC~E2,9C. rJUA"4,lOD83?,47,lOCC32,lJCC22,10CC20,lCC0~4,9C. c0u~o~,100~4n,41,1ooc~4,1cco?c,10co1q,10co~~.9c. c~uAr4,lOJ~42,47,1GJCj~,1cco11,1cco19,10co36,9C. CuVA14,lCOq44,47,lOJC66,10CJ13,1CCOl6,lJC088,1C. CJI.I.~ I)~ , l OO 3 '1 f, 4 7, 10 0 C3 P , 1 CC O16, 10 CO 14, 1 CCC c; 0, 1 C • C. J !_! <\ ~ 4 , 1 'J J ~ 4 'l , 4 7 , 1 0 G C J C , 1 0 C O 14 , l OO O O r , 1 0 CO c; 1-1 , 9 C • CT~IA3,100~5C,47,10J094,1CO~G3,1CC006,~J. ~,f~~s ALG~G grrrG~ PLaT~ r e o:: ,A:, , g ') 0 0 , :5 1 , l C r,:) 2 2 , l CCC 3 2 , 1 • , 1. , 0 • , , + C 1:1 1 +(rl,,,J.,C.,7.6,G.,C.,1.6 C~EA~,~J11,Sl,1CC02Dtl:Cr76,1.,1.,0.,,+Ce?. +cq2,,,J.,0.,7.6,C.,G.,l.6 f \3 E Mi , c; 0 e L , ~; 1 , l OO iJ 3 o , 1 COO-'>~ , l • , 1 • , 0 • , , + C Jl 3 + r. l:l ~ ••• ·., •• n •• 1 • 6 • o • , r •• 7. ·:> 87 '***~~~~~•ii~~io~~***~ioioooo***************************** ~**~****************************************************** 88 89 ~PROPE 0 TY (A~~ F~r LINT~( ~~A~S ~3J::.'.1: 1 ,72,7J,t,().,53.,313. ~MtT P0 JPEPT[~S ~F eqA?ILLIAN RCSE~GSJ w.iTl,71,1~00G.,,.3,33J.-12~~~-~-~**·-·•·**~~-~~~,e--**~~~~-~-~~*~*~ i~XTQA c~ ~E~~s t~nu1 ~CGiS CBEAM,J20j,lCOOl,lGOC3~3c,1.,1.,o.,,+C~l205 +cr120P,,,-1c.,-i •• -1.1,4.,10 •• -.6 ceEj~,1107,lCOCl,2040l,20SCJ,l.,l.,O.,,+C311C7 +ca1101,,,-5.,-J.,-1.1,1a.,1c.,-.b C2~A~,1004,1CQ01,20281,2C3CJ,l.,l.,O.,,+CBICC4 +C~l0J4,,,-1C.,-6,,-l.l,-b.,-10,,-.6 CBEA4,2lC5,lOJOl,20201,2GZCC,1.,l •• O.,,+CB2lCS +C921os,,,-1c.,-6.,-1.1,10.,-s.,-.6 ce~A~,2ZC~,1C001,201 □ 1,2CCCG,l.,l.,0.,,+C322C3 +CB2?Cc,,,J.,o.,-1.1,-1c.,10.,-.6 ce~Ar,23CG,lCOCl,1ClC~,lCCC~,1.,l.,G.,,+CB23C9 +cs23cq,,,-2.,-a.,-1.1,-1z.,c.,-.6 CBEA~,24C~,1C001,1030~,102C9,1.,l.,O.,,+C824C~ +CR24C8,,,11.,-9.,-1.l,G.,-4.,-.6 C8EA~,1003,lCOGl,10208,103C9,l.,l.,0.,,+CBlOC3 +CA.1003,,,s.,2.,-1.1,4.,6.,-.b ce~~~,25C~,1COC1,10501,104C9,1.,l.,O.,,+CB25C3 +CR25C2.,,14.,-4.,-l.L,5.,-2.,-.6 CBEA~,1106,l0001,1050q,l06Cq,l,,l.,O.,,+CB11C6 +C~ll06,,,l4.,-4.,-1.1,4.,-u.,-.6 CBf~M,26C7,lCQCl,107Cj,lC5C9,l.,l.,C.,,+CB26C7 +:B26C7,,,14.,6.,-l.1,9,,a.,-.~ CF.!;-:-.~~-', 1?. 0 7, 1 CJ O l • l O 7 0 '\ , l G 8 0 9 , 1 • , 1. , 0 • , , +CB 12 C 7 +cr1201.,.14.,6.,-1.1,6.,4.,-.6 ceEAV,27C4,lCJCl.11CGJ,lC9C9,1.,1., □ .,,+CB27C4 +C°-270~.,,6.,4.,-l,l,+.,G.,-.6 c~~a~,13Gd,lCOOl,l~qcJ,llOCJ,l.,l.,O.,,+~dl3CB +C~l30S,,,IG.,1J.,-1.1,-3.,~.,-.6 ca~A~,l40~,1CJJl,lllCJ,112C9,l.,l.,Q.,,+C~l4C3 +Cql4J8,,,-4.,7.,-l.l,-12.,3.,-.6 C8EAM,15G3,lCJOl,JOlC~,1CJC1,1.,l,,O.,,+C~15C8 +CR15~a.,,o.,-2.~-1.1,-1c.,-?.,-.6 C8f4M,16C7,1C8Cl,30300,3C2~1,l.,1.,~.,,+C816C7 +CP1607,,,14.,10,,-l.l,1C.,1C.,-.6 C8E~~,27C3,10001,302C~,3C3Cl~I •• I.,O.,,+CJ27C3 +C827C3,,,-10.,10.,-l.l,-u.,9.,-.6 CeEA~,2bC6,lGOOl,304CJ,3C5Cl,l.,1., □ .,,+cg26C6 +CPZ6C6,,,-2.,c.,-1.1,1c.,-1c.,-.6 CeFA~,2]C7,lOJOl,11203,74,l.,1.,0.,,+Ce2cc7 +CB25C1,,,-1C,,J.,-l.i,5.,-14 •• -.6 eriN ~PACES ••• ~M~*4MM SP~JCE ( e f j ii , l J O C , 1 r: :; G , 2 G ? 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C8~A~,1Sl0,1232,l04JC,4C304,l.,l.,0.,,+ce151c ~c~1~1c,.,4 •• o.,-J.,1J.,~.,-?.~ · c~~A~,l~ll,122l,4U304,4C204,l.,l.,O.,,+Ce1~11 +C~l511,,,1G.,0.,-2.~,-12,,G.,-2. C~~AW,1512,1ClG,4C2J4,4ClU3,l.,l.,O.,,+C21512 •~BlS12,,,-12.,J.,-l.6,-6.,0.,-!.l CEEA~,l513,1CJ0,4Jl03,4C002,l.,l.,O.,,+CE151~ ,.~CR151~,,.-6.,C.,-1.1,c.,o.,-1.1 ie~A~,l~C0,1234,10701,lOSJl,1.,l.,O.,,+CeltOC +Srl600,,,-4.,q,,-3.,l0.,0.,-3.5 ce~A~,16Cl,1244,to~a1,1c902,1.,1.,o.,,+ce1tc1 +CBlfOI,,,l0.,0.,-3.5,2.,0.,-3.5 CS~A~,16C2,1243,10902,11C03,l.,1.,0.,,+ce1602 +C91602,,,2.,0.,-3.~,-5.,0.,-J. C~fA~,loJ3,1C32,llCG3,lllC4,l.,1.,0 •• ,+cP.1to3 +CP1603,,,-5.,C.,-,.6,-12.,0.,-2.L ~c~1~~l~~!!l9:!o;!!~!r!f~~!0!:!l:60.,,+Celt04 cec~~,16C5,1Cl0,11204,3C3JO,l,,l.,O.,,+Ce1tC~ +C9ltJ5,,,12.,0.,-l.6,15.,lO.,-l.l •c:_: ~ C I', CJ 1•i ce~A~,1603,l2ll,10701,106J □ ,1.,l.,D.,,+Ce1ECe +CP.l60j,,,-4.,S.,-2.5,C.,-4.,-l.5 C8Elr,l6Gg,1ca1,40403,4c~02,1.,1.,o.,,+Ce1EaG +cs1~Js,,.~.,o.,-1.1,12.,c.,-1.o ce~~-~,lb10,1Cl0,4G302,4C202,l.,1.,0.,,+CeltlC +S8161C,,,12.,C.,-l.6,-1C.. ■ ,O.,-l.l ";, ce~A~,17C0,1221,110Ql,11102,l.,l.,C.,,+Cel70C +C~1700,,,-5.,-l.,-2.5,-2.,0.,-2. C8E~~,17~1,1Cl0,11102,ll203,l,,1.,0.,,+C~l1Cl +C817~1,,,-3.,G.,-1.6,4.,-1J.,-l.l ,;;,y:: ·""1 A C:J~ ..; cecAP,1702,1221,llOCl,lCcJc,1.,l.,O.,,+CPl702 •~Rl 7 C2,,,-~.,-2.,-z.s,2.,c.,-2. ~eEAM,17C3,12ll,!0°JC,SCGJ4,l.,1.,0.,,+C~l703 •~91 7 03,,,?.,J.,-2.,a.,c.,-2. ~ C8EAM,1~no,1c1 □ ,112Jl,lllJC,l.,l.,O.,,+Cel30C +S812JC,,,-S.,4,,-l,6,-~.,c.,-l.1 ~EF4~,1JC1,L?l2,ll1JC,SC204,l.,l.,0.,,+CR1E01 812 •~~1~01,.,-1.,c.,-~ •• -:.,0 •• -2.5 c~~i~,1~a2,1c21,~u2u4,5Cl03,l.,1.,o.,,+ce1~02 •~?i~J2,,,-z.,~ •• -2.1,2.,j.,-L.t r e =- r:.. i , 1 • !- 1T !-! ::: R ';I ~ Y ! r: ,, '-' S cerA~,2,)CJ,lOC1,10003,!ClC2,l.,1., □ .,,+CE2CCC &C?20CC,,,-1c.,-6.,-1.1,-1c •• ~ •• -1.~ ~eFAM,?JCl,!?12,10102,!C2~1,1.,1.,0.,,+C82CCl +c12co1,,,-1c.,c.,-2.,-~.,G.,-,.5-~ r.: e::: .'.j M, 2 :JC 2 , l 2 2 l , l 'J 2 8 1, l C'3 J ;J , l • , l • , 0 • , , + C e 2 C C 2 +cn2Jc2,,,-~.,c.,-?.5,3.,J.,-1. c~ca~,2JC3,1Cl0,1~~1C,4C404,l.,l.,n. ,,+C~2CC3 +co.200~.,,1.,o.,-1.~,1c.,J.,-i.1 ·,t CB~A~,21CC,lCJ~,202Jl,lC:~s,1.,1.,n. ,,+ce21cc +:8?1CC,,,-1C.,-~.,-I.l,-14.,C.,-l.l c~rAH,2iCl,lCOl,100G~,1Cl04,l.,l.,0.,,+:P21Cl +cqz101,,,-l4.,o •• -1.1,-1a.,r.,-1.6 rerA~,?lC2.10i2,101J4,lC~J3,1.,l.,O.,,+CP21C2 •~B21C2,,.-l~.,C.,-!.6,-4.,C.,-2.l cecA~,21CJ,1221,lJ203,lC302,1.,l.,O.,,+CP21C3 +CB21C3,,,-4.,0.,-?.~,c.,~.,-3. c~~~1/,21C4~1?~2,1]3C2,1C~Jl,1.,1.,0.,,+CE21C4 +.. 1..,? ,.: 1 C {. , , , ll • , .J • , - ··' • , - 't • , ., • , - ?. • c; CBFAM,21C6,l221,l040l,1C~JC,l.,l.,O.,,+Cr21cE +C92106,,,-4.,9.,-2.5,C.,1C.,-2. r~ca~,21C7,lC10,~0003,~ClC2,l ■ ,1 ■ ,0.,,+Ce21C7 +C~21C7,,,0.,0.,-1.o,4.,C.,-l.l ~ ieEA~,22cc.1co~,201J1,1coo1,1.,1.,o.,,+ce22cc +:: !J Z 2 CC , , , 4. , 6., -1 • !. ,-r;. , 0 • , - l • 1 ceEA~,22c1,1co1,1Jca1,1c106,l.,1.,o.,,+ce22c1 +CP22C1,,,-9.,0.,-1.l,-~.,C.,-l.6 ceFa~.2?C2,lC12,10106,lC205,1.,l.,O. ,,+ce22c2 +C~2~C2,,,-3.,C.,-l.6,-4.,C.,-2.1 (8FAV,22C~,1CZ3,1020S,1C3C4,l.,1.,0.,,+Ce22c3 +CS22C3,,,-4.,0.,-2.1,2.,2.,-2.t C2~A~,2:C4,l034,l03Q4,lC403,1 •• 1.,G., •• ce22c4 +C~22C4,,,2.,0.,-3.l,3.,C.,-3.6 C8~AM,2~C~,104~,l04J3,1C~02,l.,l.,O.,,+Ce22c5 +CR22C~, •• j.,G.,-3.1,12.,G.,-3.6 C~EA~,2~C6,l25~,10502,1CtJ2,l.,1.,0.,,+C~22Ct +C~2!C6,,,12 •• a.,-4.C,-lr.,J.,-4.C (~~1~,22C7.1234,lu6 □ 2,1C1Jl,l.,l.,O.,,+Ce22c1 +c~22c1,.,-1c.,~.,-4.~,-4.,-2.,-?.5 cerA~.22c;,1244,10701,l~~OG,1.,1.,J •• ,+Ce22CS +CB22CQ,,,-4.,-z.,-3.5,C.,C.,-3.~ ceFA~,221C,1243,1030C,5Cl04,l.,1.,1.,,+Ce221c +c~221c.,,c.,~.,-J.5,3.,c.,-?.~ C~F6~,22ll,1232,5Jl04,5C213,1.,1.,l.,,+CP2211 +CB2211,,,3.,0.,-3.l,1.,C.,-2.~ C3CA~,2~12,1C20,~1203 ■ 3C~J2,1.,1.,1.,,+CP2212 +(~2212,,,q.,J •• -2.1,12.,J.~-1.t t rerA~,2~Cl,1CJl,1110P,lC207,l.,l.,O.,,+C~2~Cl •CA23Cl,,,-3.,-5.,-l.l,-4.,0.,-l.6 cecAv,~1c2,1c1~,1 □ 2J7,1~3J6,1.,1.,o.,,+cez~c2 +c~~~c~ ••• -~.,n •• -1.6,n.,G.,-2.1 813 ceEj~,~1C),1823,1~ 2 06,lC4J~,1.,l.,O.,,+CP2?C3 +CR2?C?,,,n •• ~.,-2.1,1.,c.,-2.~ ~A~AM,2~C4,LC34,JJ4J~,JC~G4,1.,l.,O.,,+C?Z3C4 •~P2JQ4,,,,.,J.,-2.½,12.,C.,-3.l ca~AM,23C~.1c~~,105o4,1C6J4,l.,l.,o •• ,+:e2~cs +CR~3C'.',,,12.,0.,-3.l,-ll.,0.,-3.t c~~,~.~JC6,lC5~,10~04,1C703,1.,l.,O.,,+Cez3c6 +~P230~,,,-11.,J.,-J.5,-~.,0.,-Ja6 ce~A~,23C7,12i4,107G3,1C2J2,1.,i.,O.,,+C~23C7 +~f23C7,,,-6.,r.,-4.,0.,J.,-3.~ ce~~~,?3C2,1~44,l □ SJ2,1C9~1,l.,l.,n.,,+ce23ce +:q2~0~,,,o.o,.o,-~.~,-4.,q.,-3.5 ~ ce~1~,23lC,1243,lOQJJ,11CJC,1.,l.,1.,,+C~2?1C +CP2JiC,,,-s •• ~ •• -J.~,1c.,c.,-1. ca~A~,2311,1232,llQJ~,lllJC,l.,1.,1.,,+C~2~11 +C?2311,,,10.,c •• -~.,12.,0.,-2.s rBrAV,2312,lC2C,111J~.~C4J4,1.,1.,1.,,+CP2?12 •~32312,,,12.,G.,-2.l,-D.,C.,-l.l ~ cs~aY,24Ct,1cJ □ ,lu3U~,,c,a7,1.,l.,O.,,+CP24CC +Cr24CC,,,11.,-g.,-1.1,c.,C.,-1.1 cac6V,74Cl,lC11,l0407,lC~0~,1.,1.,0.,,+CP24Cl +CB2\0l,,,6.,J.,-l.1,l?.,J.,-l.~ C~[AV,24C~,lC13,10106,lC606,!.,l.,O.,,+CE24C2 +CB24C2,,,12.,~.,-l.6,-12.,0.,-2.6 cera~,24C3,1C34,lJ606,lC7J~,l.,l.,O.,,+C~24C3 +~924C3,,,-12.,0.,-~.6,-6.,G.,-3.1 ce~.~.2~C4,1C;4,10?0~,1CPU4,l.,l.,O.,,+ce24c4 +CR24C4,,,-6.,0.,-?.1,0.,J.,-3.l f~~A 1~,24C5,lC44,lOPQ4,lCQ0~,1.,l.,O.,,+C?24C5 •~RZ405,,,0.,0.,-J.l,6.,0.,-3.l ce~AM,2~C6,1C43,l01J3,11002,1.,l.,O.,,+CB24C6 +C324C6,,,6.,0.,-3.1,12.,J.,-2.c ce~AM,24C7,1232,llC02,11102,l.,1.,0.,,+Ce24c7 +CR2407,,,12.,0.,-3.,-12.,-2.,-2.5 ! CP.FA~,24C9,1C10,11C02,11201,l.,l.,1.,,+Ce24c~ +CP24cg,,,-12.,-2.,a.,-5.,-z.,c. « ta~A~,25C0,1C00,10~0P,1C608,l.,l.,~ •••• cez~cc +C825C~,,,14.,-4.,-l.l,-14.,C,,-I.l ce~AM,~.c5Cl,lC011J06C8llC7071l•tl•tO.,,+Ce2sc1 +C3Z~Ct,,,-14.,~.,-l. ,-P..,u.,-1.6 C?fA~,25C2,1Cl2,1070 7 ,IOrJE,l.,l.,O.,,+Ce25c2 +C32~02,,,-j.,C.,-l.6,G.,J.,-2.l ce~A~,25C3,1C21,103J6,1C1J~,1 •• 1.,o.,,+C825C3 +cqz50~,.,o.,o.,-2.1,s.,a.,-~.t CBEA~,~5C4,1032,1J9J5,11C04,!.,1.,J.,,+C82~C4 +C~25C4,,,~.,0.,-2.6,ll.,0.,-2.l (2El~,2~C5,1C~l,!1004,lll04,l.,l.,O.,,+Ce2,c5 +CP~~o~,,,11.,0.,-2.1,-11.,J.,-!.6 c~~a~,2~C~,1010,lll04,ll203,1.,l.,O.,,+C925C6 +CE25C6,,,-11.,1.,-1.~,-1 □ .,4.,-1.1 CGFAM,26CO,lCGC,lJ70~,lC30S,1.,l.,O.,,+Ce26CC +cn2~oc,,,14.,5.,-1.1,-2.,o.,-1.1 ce~A~,26c1,1001,11jo~,1cgo1,1.,1.,o.,,+ce2tc1 +C~260l,,,-2.,0.,-l.l,4.,J.,-l.6 ~ecA~.~6C2,1012,l0907,11006,l.,l.,O.,,+C~26C2 +(~2~C2,,,4.,J.,-l.~,1C.,J.,-2.l r~rA~,26C3,lC21,11006,lll06,1.,l.,O.,,+C~26C3 +CP26C3,,,1G.,C.,-2.l,-ll.,J.,-l.6 r~FAM,26C4,1C10,lll □ 6,ll20~,l.,l.,O.,,+Ce2fC4 +CA26C~,,.-11.,G.,-1.~,-4.,0.,-1.1 C8~Ar,25C5,1C00,11?05,3C400,1.,1.,0.,,+Ce2EC~ !C82fC5,,,-4.,o.,-1.1,-c.,c.,-1.1 ~ 814 r~~l~.21cc,1~cc,1100~,111JR,1.,1.,1.,,+c~z1cc + C: ". ;:_ "7 0 C • , , 6 • , ) • , - 1 • l , - l :' • , t, • , - l • l C~~!~t~7Ci,lCJf,111Jµ,112J7,l.,l.,O.,,+CP~7C1 +C82 7 ~1,,,-l?.,C.,-1.i,-A.,0.,-l.l r~r~~.21:2,1~co,!1~O1,Jc2Jc,1.,1.,1.,,+ce21c2 +CP2?C?,,,-s.,: •• -1.1,-1c •• 1G.,-1.1 ,t, C !J: ~ ~.- , -;; -. C O , 1 C :J l , 1 .J n J Cl , 1 G 1 ') C , l • • 1 • , 1 • , , + C P. 2 2 CC +r~21cc,,,12.,c.,-1.1,-~ •• c.,-1.s CB[A4,2jCl,1Cll,lD10C,4G2J4,l.,l.,1.,,+C?29Cl •~a2sc1,,,-s •• ~.,-1.h,~ •• 0.,-1.6 C~~A~,2SC2,1Cll,4D2J4,~C?J~,l.,l.,1. ,,+C?28C2 +cq2~c?,,,o.,a •• -1.~,J.,~.,-1.6 ~ecA~,2~C3,1ClJ,4~3S3,4C4J2,l.,l.,l.,,+:~23C3 +:u29a3,,,6.,J.,-1.0,12.,a.,-1.1.. fEFA~,21ca,1c10,40004,4c10~,1.,1.,1.,,+ce2scc +(f2QCO,,,-~.,-J.,0.,-6.,~.,0. C E!:: ll ,,, , '•J :: l , 1 C l,C , .!. G 1 :) '3 , 4 C 2 J 2 , 1 • ., l • , 1 • , , + C e 2 <; C 1 +CP?901.,., •• ,-~. •• c.,o.,~ •• c.,c • •~cpc~~y CA~~S F~? 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