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A modal analysis of the violin
Finite Elements in Analysis and Design 5 (1989) 269-279
269
Elsevier Science Publishers B.V., Amsterdam - - Printed in the Netherlands
A MODAL ANALYSIS OF THE VIOLIN G.A. KNOTT Department of Physics, Naval Postgraduate School, Monterey, CA 93943, U.S.A.
Y.S. SHIN Department of Mechanical Engineering Naval Postgraduate School, Monterey, CA 93943, U.S.A.
M. CHARGIN NASA/Ames Research Center, Mountain View, CA 94035, U.S.A. Received February 1989 Revised March 1989 Abstract. The MSC//NASTRANfinite element computer program was used to study the modal characteristics of a violin with the Stradivari shape. The violin geometry was modeled using an arcs of circles scheme with PATRAN, a finite element graphics pre/postprocessor program. Belly, back, sound post, bassbar, neck, bridge, tall-piece, strings, rib linings, end and corner blocks are the components of the model. Mode shapes and frequencies were calculated for free top and back plates, the violin box, and the complete violin system, including the strings. The results obtained from the finite element technique were compared to experimental data collected from real violins by other investigators.
Introduction
Use of the finite element method in the analysis of stringed instruments The application of the finite element method to the field of stringed instrument acoustics began in 1974 with Schwab's finite element description of the modal shapes of a guitar top plate [10]. Richardson and Roberts [6] continued the work on guitar plates in 1983. In 1984 Richardson and Roberts [7] and Rubin and Farrar [8] applied the finite element method to the study of violin plate vibrations. The purpose of this work was to employ finite element analysis to a violin, and to make a catalog of violin mode shapes available to violin researchers. A more complete description of the model (and the mode contour plots and deformed mesh plots) than what is given in this paper is available from Knott. Mode shapes for the free violin plates and the violin box model, as well as a comparison of the results to those obtained by other researchers, Marshall in particular, is also available. The plates
Before attempting to describe the modal characteristics of the finite element violin as a complete system the constituent parts must be studied as entities; then the knowledge about each part can aid in explaining the physical characteristics of the system as a whole. First, the 0168-874)(/89/$3.50 © 1989, Elsevier Science Publishers B.V.
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G.A. Knott et ell / Modal analysis of the violin
~j ttOI,E
TOP PLATE
Fig. l. Top and back plate, plan view.
top and back plates of the violin (Fig. 1) were modeled with the finite element graphics pre/postprocessor language PATgaY. The natural frequencies for each plate were then obtained using MSC/NASTgaN, a finite element analysis program. (All analyses were performed on a CRAY-XMP computer.) Then the dimensions and material properties of the plates were varied to discover how such variations affect the mode shapes and their frequencies. Most of the effects of the modifications were intuitive. For example, increasing the thickness of a plate shifts the natural frequencies up. Or, for another example, increasing the across-grain stiffness raises the frequency of those modes which have predominant bending action across the grain. The violin box The second stage of the study puts the belly (the top plate) and back together, joined with the ribs, a luthier's term for the side walls of the violin. Five other components of the violin were included in this second stage: (1) the soundpost, a dowel of wood attached vertically between the belly and back; (2) the bassbar, a strip of wood attached to the underside of the belly; (3) the two endblocks, blocks of wood attached to the inside ends of the violin box for the purpose of structural rigidity; (4) the four corner blocks, again solid blocks of wood attached to the "points", or corners of the violin for the same purpose as the end blocks; and (5) the rib linings, wood firring strips which strengthen the bond between the ribs and the belly and back (Fig. 2). With the " b o x " modeled, the second-stage analysis could begin. The first nine natural frequencies and mode shapes of the box model were calculated using the MSC/NASTRAN software. The mode shapes and corresponding frequencies were then compared to those obtained from an impact hammer modal test conducted by Marshall [5] of a real violin (neck. bridge and strings attached). An experimental modal test of a violin box without attachments would have been desirable but it was found that Marshall's results confirmed the results of the finite element run in most respects, The mode shapes experimentally obtained were similar to those generated by the finite element run for those modes involving the violin box alone. Of
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CORNERBLOCK Fig. 2. Cutawayview of the violin. course the neck, bridge and string modes were absent from the finite element model results since these components were not yet included in the model. In addition the violin finite element model was in vacu so none of the air modes or their influence on the wood modes was observed. The modal frequencies were also verified to be in the same general order and close to those obtained by Marshall. The whole violin
The final stage of the study began with the modeling of the remaining components of the violin.With neck, bridge, strings and tail piece models added to the data base a complete violin system was realized (Fig. 3). With the entire violin modeled, the final step in the study was to run a normal mode analysis and once again compare the results to those obtained by Marshall's modal test on a real violin. This final analysis proved to be the most difficult portion of the study. All of the previous runs had used canned solution sequences included in the MSC/NASTRAN software; linear static and dynamic analyses of relatively simple problems can usually be solved using these preprogrammed sequences of matrix manipulations. The addition of the strings and the requirement that they be under tension complicated the system to the extent that the rigid format solution sequences were no longer applicable to the structure. Specifically, off-axis deformations of the strings and large relative displacements between the string elements and those in the rest of the violin model introduced geometric non-linearities which must be included in the finite element model. The strings are under an enormous amount of static tension and the vertical component of this tension is directed through the bridge and onto the top plate creating large strains in the plate. In other words, the violin is a highly pre-stressed system. The geometric non-linearities in the violin system under tension were incorporated into the model by using the MSC//NASTRANDMAP language to alter the rigid format solution sequences. The details of the alterations are discussed in the analysis portion of this paper.
J
Fig. 3. Whole violin.
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Once the solution sequence was altered the normal modes of the entire system were computed to 1,200 Hz. Thirty-nine mode shapes and frequencies were obtained and compared to those found by Marshall on a real violin. This time, as was expected, the agreement between physical experiment and the finite element simulation was even better than in the case of the box alone.
Sources for the input data and experimental verification The correct implementation of any numeric method requires that the input data be accurate, the methods be sound, and that the results be experimentally verifiable. Other researchers were the source for much of the input data and for all of the experimental data. Specifically, Eckwall [1] and Sacconi [9] provided the geometry for the violin model. Eckwall's arcs of circles scheme for describing a Stradavari violin saved hundreds of hours in data preparation. Haines [2] and the United States Department of Agriculture [11] were the source for most of the material properties. Marshall's recent study [5], a modal analysis of a real violin, was an extremely valuable source for experimental verification. Hutchen's work on plate tuning [3,4] provided the experimental basis for comparison of the finite element plate mode shapes with the mode shapes of real violin plates.
The finite element model of the violin Ekwall's work describes "II Cremonese 1715 ex Joachim" entirely in arcs of circles [1]. These arc lines are ideally suited as inputs to the PATRAN preprocessing graphics software. Once the outlines and archings of the violin box were described with the arcs, adjacent arcs were used to describe surfaces. The components of the box were then attached to the surfaces, resulting in a highly accurate numerical representation of the violin geometry. The finite elements were defined from the surface geometry and consist of triangular and quadrilateral shell elements for the belly, back, and tail piece, beam elements for the strings, and solid elements for the rest of the components. The material properties used for the analysis runs are listed in Table 1. The data listed can be considered as a good starting point since it represents "average" values obtained from the
Table 1 Material properties ~
E 1 (Young's modulus, Pa × 109) E2 E3 G1 (sheer modulus, PaX 109) G2 G3 v12 (Poisson's ratio) 1~13 v23 v32 v21 v3] p (density, k g / m 3)
Neck, back, tailpiece, ribs (maple)
Belly, riblinings, post, bridge, blocks (spruce)
17.000 1.564 0.731 1.275 1.173 0.187 0.318 0.392 0.703 0.329 0.030 0.019 800.000
13.000 0.89 0.649 1.015 0.715 0.416 0.375 0.436 0.468 0.248 0.034 0.022 460.000
1 - - A l o n g grain; 2--Across grain; 3--Tangential.
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) Fig. 4. Back plate thickness profile (ram).
wood handbook [11] and from a study by Haines [2]. The belly, soundpost, rib linings, bridge, bassbar, end and corner blocks are modeled as Sitka spruce and the back, neck, ribs and tailpiece are modeled as Norwegian maple. The thickness of the belly was defined in accordance with Sacconi [9] to be 3 mm except at the edges and area around the two ff holes (these holes can be seen in fig. 1), where it was 3.8 mm. In the violin model the top plate was modeled with a uniform thickness of 3.0 mm. The back was defined to the thickness of 4.5 mm at the center, descending to 3.8 mm on the first circles, to 3.5 mm on the second, to 2.5 mm in the upper lung (the tops and bottoms of violin plates are often called the "lungs") and to 2.6 mm in the lower one. Figure 4 shows a contour map of the back thicknesses. The neck dimensions were unavailable for the Stradivari violin, so the modeling of the neck was accomplished using neck dimensions from the author's violin. Hexahedral and wedge elements were used for the neck model. For the tail-piece quadrilateral and triangular shell elements were used. The geometry of the tail-piece was obtained from the author's violin. The bridge model was created using the classical bridge form found in most violins. The model realization is, of necessity, a simplification of the true geometry as can be seen in Fig. 5. However, the obvious design objectives are retained. The thickness of the bridge varies from 5.0 mm at the feet to 0.5 mm at the top. The strings were modeled as beam elements. Beam elements were chosen instead of rod elements to allow the inclusion of bending stiffness into the string model, which the MSC/NASTRAN rod element does not consider. The " G " , " D " , and " A " string's material properties were modeled as gut cores with steel windings. The windings were modeled as a nonstructural mass since their contribution to bending stiffness is assumed to be negligible. Solid steel was used for the " E " string. Table 2 lists the dimensions and material properties for
Fig. 5. Bridge detail.
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G.A.
Knott et al. / Modal analysis of the violin
Table 2 String dimensions and material properties String "'G.
.
.
.
D.
.
.
.
A.
.
.
.
F?"
Core mass (g) N.S.M. ( g / m )
0.110 3.314
0.065 1.352
0.034 0.557
0.13 0.00
Total mass (g) Core diameter ( m m ) E (Young's modulus, Pa × 109)
1.200
0.750
0.260
0.13
0.600
0.460
0.330
0.27
30.000
30.000
30.000
195.0
v (Poisson's ratio) 0 (density, k g / m 3)
0.4000 1200
0.400 1200
0.400 1200
(I.28 770(t
Number of elements
110
82
46
33
the strings. The number of beam elements for each string was determined from the desired frequency response of about 8,000 Hz.
The analysis As mentioned in the introduction the transition from the violin box model to the model of the violin with all of its components was difficult. The greatest difficulty lay in the fact that the finite element "strings" had to be stretched a large amount relative to other strains in the violin. Large errors in strain energy were experienced when the complete violin system under string loading was programmed for a static analysis. The rigid formats available in the NASTRAN software had to be modified, and the following is a step by step explanation of the process: (1) Assemble the stiffness matrix for the violin box with the neck included, but without the bridge/tail piece/strings system. A static analysis, SOL 61, was run with an alteration in line 355 of the preprogrammed solution sequence (Fig. 6). The ALTER statement simply provides the capability to extract the stiffness matrix for the follow-on runs (named KGGW).
SOL 61 ALTER 355 //NE/V,N,NOADD/SEID/0 $ PARAM NOADDI,NOADD $ COND MATGEN EQEXIN S/INTEXT/9/0/LUSETS $ INTEXT,KGG,/KGGE/1 $ MPYAD KGGE,INTEXT,/KG GN///// $ MPYAD MATMOD KGGW,EQEXINS,,,,,/MATP 1,/16/1 $ EXIT $ LABEL NOADDI $ CEND TITLE = VIOLIN SUBTITLE = BOX AND NECK ONLY LABEL = CALCULATE THE EQUIVALENT STIPI~NESSMATRIX ECHO = NONE SEALL = ALL SPC = 1 SUBCASE 1 SUPER = 1 SUBCASE 2 BEGIN BULK PARAM,PRGPST,NO SPC1,123456,481 SESET,1,565 SESET,1, E T C ......
Fig. 6. Superelement statistics solution for
reduced box and neck stiffness matrix,
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275
SOL 64 ALTER 4 KGGW = SAVE $ FILE ALTER 66 ,,MATI~OOL,GEQEXINS ,GSILS JKGOW,,/V,N,LUSET/S ,N,NO KOGW $ MTRXIN GPECT,KDICTI,KELM 1,GBGPDT,GSILS,GCSTMS/KGG2,/ $ E/VIA KGG2,KGGW/KGGl $ ADD ALTER 188,188 HGPECT,KDICT,KELM,GBGPDT,GSILS,GCSTMS/KJJX,/- t $ EMA KGGW,KJJX/KJJ $ ADD ALTER 23t,231 KGGWJ-1UGV,/I~X $ MPYAD ADD FGX,FG/IZl" $ ADD HPG,FT/PG 1//(- l.,0.) $ ALTER 321 DBFL:ri"CI-I /XKGG l ,XKJJ,,,/MODEL/0/1 $ ADD XKJJ,XKGG I/YKJJ//( - 1.,0. ) $ FEQEX:INS/n~rrEXT~tLUSET $ MATGI~I INTEXT,YKIJ,/KGGE/1 $ MPYAD KG GE JNTEXT,/KGGZ/////6 $ MPYAD MATMODKGGZ,FEQEXINS,,,JMATP1,/16/1 $ ALTER 285 VECPI.JgT FUG V,H) GPDT,FEQEX]NS ,FCSTMS,CASEXX,/UG V'B]0/0/5 $ SDR2 CASEXX,,,,FEQEXINS ....... UGVB,,/,,OUGVX.,,/APP/0 $ O ~ OUGVX//0/tt $ (END TrILE = VIOLIN SUBTITLE = STRINGS AND BRIDGE LABEL = NON-LINEAR ANALYSIS - DMIG INPUT FOR BODY ECHO = NONE SPC= i TEMP(LOAD) = l SET 66 = 2740,2878,2971,3024 ELFORCE = 66 SUBCASE 1
SUBCASE 15 DISP = ALL ELFORCE = ALL BEGIN BULK PARAM, MAXJ~TIO, 1.+20 $ SPC FOR STRING ENDS SPC 1,1,456,2187,THRU,2266 SPC1,1,456.2268,THRU,2277
Fig. 7. Static analysis with geometric nonlinearity solution sequence for the strings and bridge.
(2) Perform static condensation of the above stiffness matrix "eliminating" all degrees of freedom except at those 14 locations where: the strings attach to the neck (4 nodes), where the bridge feet attach to the top plate (8 nodes), and where the two gut cords from the tail piece attach to the body (2 nodes). MSC/NASTRAr~'S substructuring capability was utilized for the condensation. The SESETcards in the data deck (see Fig. 6) define the substructure, in this case all of the nodes in the violin box/neck that are to be omitted from the KGGW matrix. In other words, the stiffness of the entire box/neck structure is defined for each unit stiffness of each of one of the 14 points, effectively "eliminating" all degrees of freedom except at those 14 points. (3) Run an iterative, non-linear static solution sequence for the bridge/tail piece/strings system (Fig. 7) using the 84 × 84 KGGW matrix (14 nodes times 6 degrees of freedom per node) matrix obtained in step 2 as the boundary conditions. The stiffness matrix KGGW is essentially the stiffness of the neck and body that the string ends and bridge feet would experience. Problems of large relative strains are avoided since the box/neck system is present only as a boundary condition (strains are not calculated for the box/neck). The first iteration uses the
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276
SOL 63 CEND TITLE = VIOLIN SUBT1TLE = BOX, NECK, BRIDGE AND STRINGS LABEL = RESTART W1TH DIF-F'~RENTIAL ECHO = NONE SEALL = ALL SPC = 1 DYNRED = 1 METHOD = 1 BEGIN BULK PARAM, MAXRATIO. 1.+20 SPC 1,1,0,9900 I,THRU.99150 SPOINT,99001,THRU,99150 AS ET 1,0,99001 ,THRU,99150 QSET1.0,99001 ,THRU,99150 DYNRED, 1,1200. EIGR.I ,GIV,0., 1200.,,,,1 .-8.MAX READ DMIG
F i g . 8. N o r m a l
modes
analysis of the en-
tire violin system (restart with differential stiffness).
rigid format for a static analysis in which the strings are subjected to differing amounts of thermal stress. In essence, the strings were "cooled down" to shrink them and bring them to the required tensions. The desired tensions were those that would place the natural frequencies of the strings as close as possible to the tones found on a real violin, that is, G, D, A, and E. The following subcases (or iterations) of the run successively add the geometric non-linear terms and compute the stiffness matrix for each added term until an equilibrium of forces exists.
F=P+Q, where F is the vector of element forces, P is the vector of applied thermal loads, and Q is the vector of forces of constraint. The string/bridge system stiffness matrix is not modified for the stress of the thermal loads on the strings and is saved for the next step. (4) The final run puts the violin system back together (Fig. 8). The string/bridge system, now in equilibrium, is added to the b o x / n e c k system. Then the rigid format solution sequence computes the normal modes of the entire violin model.
Comparison to experiment The finite element method is a numerical technique that cannot stand alone; this is especially so when applied to a complex structure such as the violin. Ideally, as each subsystem is simulated using finite elements, a parallel verification of the model is accomplished using experimental methods. For the violin, an exhaustive verification would involve: (1) Top and back plate. (2) Bassbar. (3) Top plate with bassbar. (4) Violin box without sound post. (5) Violin box with sound post. (6) Neck. (7) Violin box with neck. (8) Bridge. (9) Tail piece. (10) Whole violin. In the ideal case the material properties and precise geometry of each component used for experimental verification would be known and would constitute the data base for the finite element modeling process.
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277
In the present study steps (1), (3), (5), and (11) were the only ones which were compared to experimental results. The experimental data was obtained from Marshall [5] and Hutchins (3,4]. There exists no geometry or material property data for Marshall's violin SUS #295 (SUS #295 is the model number of the violin used by Marshall for his modal test) or Hutchen's violin plates so the examination of the results in the present study must be done with the realization that natural frequencies and mode shapes were not expected to match exactly. A basic assumption to be tested in this study is that there exists a fundamental commonality of mode shapes among all but the most poorly constructed violins. This assumption has been shown to hold in the case of free violin plates, at least among the lower modes of vibration [5]. That is, even plates with differing geometries and material properties show a clear similarity in the ordering and shape of their vibration patterns. That all violins sounds like a violin (some sound better than others!), is the intuitive and rather obvious basis for the mode shape similarity assumption. As mentioned in the introduction a detailed comparison to experiments (as well as catalog of mode shapes for the free plates, the violin box, and the whole violin system) is available from Knott. In this paper a short explanation of the comparison is provided. Top plate, back plate, and violin box
The free top and back plate mode shapes of the finite element analysis were found to compare well with experiments done by Hutchins using Chladni pattern and holographic interferometry techniques. The violin box model results were, as mentioned earlier, found to compare reasonably well with Marshall's data on a real violin. Contour plots and deformed mesh views of the free plate and violin box model are available from Knott along with tables comparing the finite element results and experiment. The entire viofin
The complete violin model compared well with the real violin tested by Marshall in mode shapes, frequencies and relative order of the modes. The source of discrepancies is due to a number of factors, the most obvious being that the degree to which the material properties and geometries of the finite element violin and the real violin match is unknown. Any firm conclusions about the origins of the discrepancy between the finite element model and the real violin would require the actual dimensions and material properties of violin SUS # 295. The following are other causes of disagreement between mode shapes and frequencies between the finite element model and SUS #295: - The air modes of the real violin can be expected to couple to varying degrees with the structural modes of the violin. The model does not contain "air" elements. The air modes are absent from the finite element model and any effect they may have had on the structural modes is a matter for future research. In the finite element model the strings were not tuned to the exact frequencies used on a real violin. For the G string the model was off by + 16 Hz; the D string, + 18 Hz; the A string, + 46 Hz; and the E string, - 7 Hz. The differences in tuning could have caused some of the discrepancies in mode shapes and ordering of the modes. Fixing the tuning problem would merely require an adjustment of the thermal loads applied to the string model. A real violin is glued together in most places. In the finite element model there is no attempt to model the glue joints. - Forces from the plates and strings hold the soundpost and bridge to the real violin body. -
-
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G.A. Knott et a{ / Modal analysis oJ the violin
Fig. 9. Mode 24.
Fig. 10. Mode 26.
In the finite element model most of these forces exist (not the stress mounting of the soundpost in the violin box), but the components are "attached" to the plates and the bridge. In other words, there is no provision for slipping in the case of the strings or decoupling from the plates in the case of the soundpost. Figures 9-11 show hidden line deformed meshes corresponding to modes 24, 26, and 32. A complete catalog of all of the the mode shapes using hidden line deformed meshes (and contour plot views as well) is available from Knott.
Fig. l 1. Mode 32.
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279
Conclusion and future work The work described in this paper resulted in a catalogue of a finite element violin's mode shapes. The purpose of the study was to provide this preliminary catalogue and to support the idea that the mode shapes of a violin, complete with the string system, can be described using the finite element method. It should be remembered that the method's real utility lies not in mere reproduction and description of physical reality but in sensitivity analyses and design optimization. However, before research can progress to violin design optimization still more will be required in the effort to accurately describe the violin. Further work might first concentrate on gaining a more complete data base of real violins and components, with known geometries and material properties. A coincident finite element analysis can then be expected to agree with experimental modal tests of the real violins and components. Different violins could be tested and modeled and the mode shapes compared, perhaps leading to an understanding of what differentiates a " g o o d " sounding violin from a " p o o r " one. The incorporation of air elements into the model is a long-term but necessary goal for further research. The extent of the air mode's coupling effects with the structural modes could be easily determined through the study of simple finite element models. Radiation and string excitation studies using finite element models are also promising areas for further work. After developing the finite element model of the violin the authors found the model to be as full of mystery as a real violin! A methodical, step-by-step analysis should clear up many of the grey areas. The finite element method is an extremely powerful descriptor of physical reality and we believe that much of the violin "puzzle" will be solved as research continues.
References [1] EKWALL,A., "A study of the Stradivari-Sacconi violin arching for getting a computerizable arching of the same type", Catgut Acoust. Soc. Newslett. 35, pp. 33-36, 1981. [2] HAINES,D.W., "On musical instrument wood", Catgut Acoust. Soc. Newslett. 31, pp. 23-32, 1979. [3] HtrrcmNS, C.M., "The acoustics of violin plates", Sci. Am. 245, pp. 171-186, October 1981. C.M. [4] HtrrcniNS, C.M., K.A. STETSONand P.A. TAYLOR,"Clarification of free plate tap tones by hologram interferometry", Catgut Acoust. Soc. Newslett. 16, pp. 15-23, 1971. [5] MARSHALL,K.D., "Modal analysis of a violin", J. Acoust. Soc. Am. 77, pp. 695-709, 1985. [6] RICHARDSON,B.E. and G.W. ROBERTS,"The adjustment of mode frequencies in guitars: a study by means of holographic interferometry and finite element analysis", Proc. SMAC 1983 Conference, 1983. [7] RICHARDSON,B.E. and G.W. ROBERTS,"Predictions of the behavior of violin plates by the finite element method", Proc. Inst. Acoust., 1985. [8] RUBIN,C. and D.F. FAm~R, "Computer analysis of the vibrations of a violin back", Department of Mechanical and Materials Engineering, Vanderbuilt University, Nashville, Tennessee, 1984. [9] SACCONI,S.F., Die Geheimnisse Stradivaris, Verlag Das Musikinstrument, Frankfurt/M, F.R.G., 1976. [10] SCrlWAB,H.L., "Finite element analysis of a guitar soundboard", Catgut Acoust. Soc. Newslett. 24 & 25, 1974. [11] Wood Products Handbook, U.S. Forest Products Lab, Madison, Wisconsin, p. 79, 1955.
269
Elsevier Science Publishers B.V., Amsterdam - - Printed in the Netherlands
A MODAL ANALYSIS OF THE VIOLIN G.A. KNOTT Department of Physics, Naval Postgraduate School, Monterey, CA 93943, U.S.A.
Y.S. SHIN Department of Mechanical Engineering Naval Postgraduate School, Monterey, CA 93943, U.S.A.
M. CHARGIN NASA/Ames Research Center, Mountain View, CA 94035, U.S.A. Received February 1989 Revised March 1989 Abstract. The MSC//NASTRANfinite element computer program was used to study the modal characteristics of a violin with the Stradivari shape. The violin geometry was modeled using an arcs of circles scheme with PATRAN, a finite element graphics pre/postprocessor program. Belly, back, sound post, bassbar, neck, bridge, tall-piece, strings, rib linings, end and corner blocks are the components of the model. Mode shapes and frequencies were calculated for free top and back plates, the violin box, and the complete violin system, including the strings. The results obtained from the finite element technique were compared to experimental data collected from real violins by other investigators.
Introduction
Use of the finite element method in the analysis of stringed instruments The application of the finite element method to the field of stringed instrument acoustics began in 1974 with Schwab's finite element description of the modal shapes of a guitar top plate [10]. Richardson and Roberts [6] continued the work on guitar plates in 1983. In 1984 Richardson and Roberts [7] and Rubin and Farrar [8] applied the finite element method to the study of violin plate vibrations. The purpose of this work was to employ finite element analysis to a violin, and to make a catalog of violin mode shapes available to violin researchers. A more complete description of the model (and the mode contour plots and deformed mesh plots) than what is given in this paper is available from Knott. Mode shapes for the free violin plates and the violin box model, as well as a comparison of the results to those obtained by other researchers, Marshall in particular, is also available. The plates
Before attempting to describe the modal characteristics of the finite element violin as a complete system the constituent parts must be studied as entities; then the knowledge about each part can aid in explaining the physical characteristics of the system as a whole. First, the 0168-874)(/89/$3.50 © 1989, Elsevier Science Publishers B.V.
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G.A. Knott et ell / Modal analysis of the violin
~j ttOI,E
TOP PLATE
Fig. l. Top and back plate, plan view.
top and back plates of the violin (Fig. 1) were modeled with the finite element graphics pre/postprocessor language PATgaY. The natural frequencies for each plate were then obtained using MSC/NASTgaN, a finite element analysis program. (All analyses were performed on a CRAY-XMP computer.) Then the dimensions and material properties of the plates were varied to discover how such variations affect the mode shapes and their frequencies. Most of the effects of the modifications were intuitive. For example, increasing the thickness of a plate shifts the natural frequencies up. Or, for another example, increasing the across-grain stiffness raises the frequency of those modes which have predominant bending action across the grain. The violin box The second stage of the study puts the belly (the top plate) and back together, joined with the ribs, a luthier's term for the side walls of the violin. Five other components of the violin were included in this second stage: (1) the soundpost, a dowel of wood attached vertically between the belly and back; (2) the bassbar, a strip of wood attached to the underside of the belly; (3) the two endblocks, blocks of wood attached to the inside ends of the violin box for the purpose of structural rigidity; (4) the four corner blocks, again solid blocks of wood attached to the "points", or corners of the violin for the same purpose as the end blocks; and (5) the rib linings, wood firring strips which strengthen the bond between the ribs and the belly and back (Fig. 2). With the " b o x " modeled, the second-stage analysis could begin. The first nine natural frequencies and mode shapes of the box model were calculated using the MSC/NASTRAN software. The mode shapes and corresponding frequencies were then compared to those obtained from an impact hammer modal test conducted by Marshall [5] of a real violin (neck. bridge and strings attached). An experimental modal test of a violin box without attachments would have been desirable but it was found that Marshall's results confirmed the results of the finite element run in most respects, The mode shapes experimentally obtained were similar to those generated by the finite element run for those modes involving the violin box alone. Of
271
G.A. Knott et al. / Modal analysis of the violin
CORNERBLOCK Fig. 2. Cutawayview of the violin. course the neck, bridge and string modes were absent from the finite element model results since these components were not yet included in the model. In addition the violin finite element model was in vacu so none of the air modes or their influence on the wood modes was observed. The modal frequencies were also verified to be in the same general order and close to those obtained by Marshall. The whole violin
The final stage of the study began with the modeling of the remaining components of the violin.With neck, bridge, strings and tail piece models added to the data base a complete violin system was realized (Fig. 3). With the entire violin modeled, the final step in the study was to run a normal mode analysis and once again compare the results to those obtained by Marshall's modal test on a real violin. This final analysis proved to be the most difficult portion of the study. All of the previous runs had used canned solution sequences included in the MSC/NASTRAN software; linear static and dynamic analyses of relatively simple problems can usually be solved using these preprogrammed sequences of matrix manipulations. The addition of the strings and the requirement that they be under tension complicated the system to the extent that the rigid format solution sequences were no longer applicable to the structure. Specifically, off-axis deformations of the strings and large relative displacements between the string elements and those in the rest of the violin model introduced geometric non-linearities which must be included in the finite element model. The strings are under an enormous amount of static tension and the vertical component of this tension is directed through the bridge and onto the top plate creating large strains in the plate. In other words, the violin is a highly pre-stressed system. The geometric non-linearities in the violin system under tension were incorporated into the model by using the MSC//NASTRANDMAP language to alter the rigid format solution sequences. The details of the alterations are discussed in the analysis portion of this paper.
J
Fig. 3. Whole violin.
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Once the solution sequence was altered the normal modes of the entire system were computed to 1,200 Hz. Thirty-nine mode shapes and frequencies were obtained and compared to those found by Marshall on a real violin. This time, as was expected, the agreement between physical experiment and the finite element simulation was even better than in the case of the box alone.
Sources for the input data and experimental verification The correct implementation of any numeric method requires that the input data be accurate, the methods be sound, and that the results be experimentally verifiable. Other researchers were the source for much of the input data and for all of the experimental data. Specifically, Eckwall [1] and Sacconi [9] provided the geometry for the violin model. Eckwall's arcs of circles scheme for describing a Stradavari violin saved hundreds of hours in data preparation. Haines [2] and the United States Department of Agriculture [11] were the source for most of the material properties. Marshall's recent study [5], a modal analysis of a real violin, was an extremely valuable source for experimental verification. Hutchen's work on plate tuning [3,4] provided the experimental basis for comparison of the finite element plate mode shapes with the mode shapes of real violin plates.
The finite element model of the violin Ekwall's work describes "II Cremonese 1715 ex Joachim" entirely in arcs of circles [1]. These arc lines are ideally suited as inputs to the PATRAN preprocessing graphics software. Once the outlines and archings of the violin box were described with the arcs, adjacent arcs were used to describe surfaces. The components of the box were then attached to the surfaces, resulting in a highly accurate numerical representation of the violin geometry. The finite elements were defined from the surface geometry and consist of triangular and quadrilateral shell elements for the belly, back, and tail piece, beam elements for the strings, and solid elements for the rest of the components. The material properties used for the analysis runs are listed in Table 1. The data listed can be considered as a good starting point since it represents "average" values obtained from the
Table 1 Material properties ~
E 1 (Young's modulus, Pa × 109) E2 E3 G1 (sheer modulus, PaX 109) G2 G3 v12 (Poisson's ratio) 1~13 v23 v32 v21 v3] p (density, k g / m 3)
Neck, back, tailpiece, ribs (maple)
Belly, riblinings, post, bridge, blocks (spruce)
17.000 1.564 0.731 1.275 1.173 0.187 0.318 0.392 0.703 0.329 0.030 0.019 800.000
13.000 0.89 0.649 1.015 0.715 0.416 0.375 0.436 0.468 0.248 0.034 0.022 460.000
1 - - A l o n g grain; 2--Across grain; 3--Tangential.
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G.A. Knott et al. / Modal analysis of the violin
) Fig. 4. Back plate thickness profile (ram).
wood handbook [11] and from a study by Haines [2]. The belly, soundpost, rib linings, bridge, bassbar, end and corner blocks are modeled as Sitka spruce and the back, neck, ribs and tailpiece are modeled as Norwegian maple. The thickness of the belly was defined in accordance with Sacconi [9] to be 3 mm except at the edges and area around the two ff holes (these holes can be seen in fig. 1), where it was 3.8 mm. In the violin model the top plate was modeled with a uniform thickness of 3.0 mm. The back was defined to the thickness of 4.5 mm at the center, descending to 3.8 mm on the first circles, to 3.5 mm on the second, to 2.5 mm in the upper lung (the tops and bottoms of violin plates are often called the "lungs") and to 2.6 mm in the lower one. Figure 4 shows a contour map of the back thicknesses. The neck dimensions were unavailable for the Stradivari violin, so the modeling of the neck was accomplished using neck dimensions from the author's violin. Hexahedral and wedge elements were used for the neck model. For the tail-piece quadrilateral and triangular shell elements were used. The geometry of the tail-piece was obtained from the author's violin. The bridge model was created using the classical bridge form found in most violins. The model realization is, of necessity, a simplification of the true geometry as can be seen in Fig. 5. However, the obvious design objectives are retained. The thickness of the bridge varies from 5.0 mm at the feet to 0.5 mm at the top. The strings were modeled as beam elements. Beam elements were chosen instead of rod elements to allow the inclusion of bending stiffness into the string model, which the MSC/NASTRAN rod element does not consider. The " G " , " D " , and " A " string's material properties were modeled as gut cores with steel windings. The windings were modeled as a nonstructural mass since their contribution to bending stiffness is assumed to be negligible. Solid steel was used for the " E " string. Table 2 lists the dimensions and material properties for
Fig. 5. Bridge detail.
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G.A.
Knott et al. / Modal analysis of the violin
Table 2 String dimensions and material properties String "'G.
.
.
.
D.
.
.
.
A.
.
.
.
F?"
Core mass (g) N.S.M. ( g / m )
0.110 3.314
0.065 1.352
0.034 0.557
0.13 0.00
Total mass (g) Core diameter ( m m ) E (Young's modulus, Pa × 109)
1.200
0.750
0.260
0.13
0.600
0.460
0.330
0.27
30.000
30.000
30.000
195.0
v (Poisson's ratio) 0 (density, k g / m 3)
0.4000 1200
0.400 1200
0.400 1200
(I.28 770(t
Number of elements
110
82
46
33
the strings. The number of beam elements for each string was determined from the desired frequency response of about 8,000 Hz.
The analysis As mentioned in the introduction the transition from the violin box model to the model of the violin with all of its components was difficult. The greatest difficulty lay in the fact that the finite element "strings" had to be stretched a large amount relative to other strains in the violin. Large errors in strain energy were experienced when the complete violin system under string loading was programmed for a static analysis. The rigid formats available in the NASTRAN software had to be modified, and the following is a step by step explanation of the process: (1) Assemble the stiffness matrix for the violin box with the neck included, but without the bridge/tail piece/strings system. A static analysis, SOL 61, was run with an alteration in line 355 of the preprogrammed solution sequence (Fig. 6). The ALTER statement simply provides the capability to extract the stiffness matrix for the follow-on runs (named KGGW).
SOL 61 ALTER 355 //NE/V,N,NOADD/SEID/0 $ PARAM NOADDI,NOADD $ COND MATGEN EQEXIN S/INTEXT/9/0/LUSETS $ INTEXT,KGG,/KGGE/1 $ MPYAD KGGE,INTEXT,/KG GN///// $ MPYAD MATMOD KGGW,EQEXINS,,,,,/MATP 1,/16/1 $ EXIT $ LABEL NOADDI $ CEND TITLE = VIOLIN SUBTITLE = BOX AND NECK ONLY LABEL = CALCULATE THE EQUIVALENT STIPI~NESSMATRIX ECHO = NONE SEALL = ALL SPC = 1 SUBCASE 1 SUPER = 1 SUBCASE 2 BEGIN BULK PARAM,PRGPST,NO SPC1,123456,481 SESET,1,565 SESET,1, E T C ......
Fig. 6. Superelement statistics solution for
reduced box and neck stiffness matrix,
G.A. Knott et al. / Modal analysis of the violin
275
SOL 64 ALTER 4 KGGW = SAVE $ FILE ALTER 66 ,,MATI~OOL,GEQEXINS ,GSILS JKGOW,,/V,N,LUSET/S ,N,NO KOGW $ MTRXIN GPECT,KDICTI,KELM 1,GBGPDT,GSILS,GCSTMS/KGG2,/ $ E/VIA KGG2,KGGW/KGGl $ ADD ALTER 188,188 HGPECT,KDICT,KELM,GBGPDT,GSILS,GCSTMS/KJJX,/- t $ EMA KGGW,KJJX/KJJ $ ADD ALTER 23t,231 KGGWJ-1UGV,/I~X $ MPYAD ADD FGX,FG/IZl" $ ADD HPG,FT/PG 1//(- l.,0.) $ ALTER 321 DBFL:ri"CI-I /XKGG l ,XKJJ,,,/MODEL/0/1 $ ADD XKJJ,XKGG I/YKJJ//( - 1.,0. ) $ FEQEX:INS/n~rrEXT~tLUSET $ MATGI~I INTEXT,YKIJ,/KGGE/1 $ MPYAD KG GE JNTEXT,/KGGZ/////6 $ MPYAD MATMODKGGZ,FEQEXINS,,,JMATP1,/16/1 $ ALTER 285 VECPI.JgT FUG V,H) GPDT,FEQEX]NS ,FCSTMS,CASEXX,/UG V'B]0/0/5 $ SDR2 CASEXX,,,,FEQEXINS ....... UGVB,,/,,OUGVX.,,/APP/0 $ O ~ OUGVX//0/tt $ (END TrILE = VIOLIN SUBTITLE = STRINGS AND BRIDGE LABEL = NON-LINEAR ANALYSIS - DMIG INPUT FOR BODY ECHO = NONE SPC= i TEMP(LOAD) = l SET 66 = 2740,2878,2971,3024 ELFORCE = 66 SUBCASE 1
SUBCASE 15 DISP = ALL ELFORCE = ALL BEGIN BULK PARAM, MAXJ~TIO, 1.+20 $ SPC FOR STRING ENDS SPC 1,1,456,2187,THRU,2266 SPC1,1,456.2268,THRU,2277
Fig. 7. Static analysis with geometric nonlinearity solution sequence for the strings and bridge.
(2) Perform static condensation of the above stiffness matrix "eliminating" all degrees of freedom except at those 14 locations where: the strings attach to the neck (4 nodes), where the bridge feet attach to the top plate (8 nodes), and where the two gut cords from the tail piece attach to the body (2 nodes). MSC/NASTRAr~'S substructuring capability was utilized for the condensation. The SESETcards in the data deck (see Fig. 6) define the substructure, in this case all of the nodes in the violin box/neck that are to be omitted from the KGGW matrix. In other words, the stiffness of the entire box/neck structure is defined for each unit stiffness of each of one of the 14 points, effectively "eliminating" all degrees of freedom except at those 14 points. (3) Run an iterative, non-linear static solution sequence for the bridge/tail piece/strings system (Fig. 7) using the 84 × 84 KGGW matrix (14 nodes times 6 degrees of freedom per node) matrix obtained in step 2 as the boundary conditions. The stiffness matrix KGGW is essentially the stiffness of the neck and body that the string ends and bridge feet would experience. Problems of large relative strains are avoided since the box/neck system is present only as a boundary condition (strains are not calculated for the box/neck). The first iteration uses the
G.A. Knott et al. / Modal analysis of the violin
276
SOL 63 CEND TITLE = VIOLIN SUBT1TLE = BOX, NECK, BRIDGE AND STRINGS LABEL = RESTART W1TH DIF-F'~RENTIAL ECHO = NONE SEALL = ALL SPC = 1 DYNRED = 1 METHOD = 1 BEGIN BULK PARAM, MAXRATIO. 1.+20 SPC 1,1,0,9900 I,THRU.99150 SPOINT,99001,THRU,99150 AS ET 1,0,99001 ,THRU,99150 QSET1.0,99001 ,THRU,99150 DYNRED, 1,1200. EIGR.I ,GIV,0., 1200.,,,,1 .-8.MAX READ DMIG
F i g . 8. N o r m a l
modes
analysis of the en-
tire violin system (restart with differential stiffness).
rigid format for a static analysis in which the strings are subjected to differing amounts of thermal stress. In essence, the strings were "cooled down" to shrink them and bring them to the required tensions. The desired tensions were those that would place the natural frequencies of the strings as close as possible to the tones found on a real violin, that is, G, D, A, and E. The following subcases (or iterations) of the run successively add the geometric non-linear terms and compute the stiffness matrix for each added term until an equilibrium of forces exists.
F=P+Q, where F is the vector of element forces, P is the vector of applied thermal loads, and Q is the vector of forces of constraint. The string/bridge system stiffness matrix is not modified for the stress of the thermal loads on the strings and is saved for the next step. (4) The final run puts the violin system back together (Fig. 8). The string/bridge system, now in equilibrium, is added to the b o x / n e c k system. Then the rigid format solution sequence computes the normal modes of the entire violin model.
Comparison to experiment The finite element method is a numerical technique that cannot stand alone; this is especially so when applied to a complex structure such as the violin. Ideally, as each subsystem is simulated using finite elements, a parallel verification of the model is accomplished using experimental methods. For the violin, an exhaustive verification would involve: (1) Top and back plate. (2) Bassbar. (3) Top plate with bassbar. (4) Violin box without sound post. (5) Violin box with sound post. (6) Neck. (7) Violin box with neck. (8) Bridge. (9) Tail piece. (10) Whole violin. In the ideal case the material properties and precise geometry of each component used for experimental verification would be known and would constitute the data base for the finite element modeling process.
G.A. Knott et al. / Modal analysis of the violin
277
In the present study steps (1), (3), (5), and (11) were the only ones which were compared to experimental results. The experimental data was obtained from Marshall [5] and Hutchins (3,4]. There exists no geometry or material property data for Marshall's violin SUS #295 (SUS #295 is the model number of the violin used by Marshall for his modal test) or Hutchen's violin plates so the examination of the results in the present study must be done with the realization that natural frequencies and mode shapes were not expected to match exactly. A basic assumption to be tested in this study is that there exists a fundamental commonality of mode shapes among all but the most poorly constructed violins. This assumption has been shown to hold in the case of free violin plates, at least among the lower modes of vibration [5]. That is, even plates with differing geometries and material properties show a clear similarity in the ordering and shape of their vibration patterns. That all violins sounds like a violin (some sound better than others!), is the intuitive and rather obvious basis for the mode shape similarity assumption. As mentioned in the introduction a detailed comparison to experiments (as well as catalog of mode shapes for the free plates, the violin box, and the whole violin system) is available from Knott. In this paper a short explanation of the comparison is provided. Top plate, back plate, and violin box
The free top and back plate mode shapes of the finite element analysis were found to compare well with experiments done by Hutchins using Chladni pattern and holographic interferometry techniques. The violin box model results were, as mentioned earlier, found to compare reasonably well with Marshall's data on a real violin. Contour plots and deformed mesh views of the free plate and violin box model are available from Knott along with tables comparing the finite element results and experiment. The entire viofin
The complete violin model compared well with the real violin tested by Marshall in mode shapes, frequencies and relative order of the modes. The source of discrepancies is due to a number of factors, the most obvious being that the degree to which the material properties and geometries of the finite element violin and the real violin match is unknown. Any firm conclusions about the origins of the discrepancy between the finite element model and the real violin would require the actual dimensions and material properties of violin SUS # 295. The following are other causes of disagreement between mode shapes and frequencies between the finite element model and SUS #295: - The air modes of the real violin can be expected to couple to varying degrees with the structural modes of the violin. The model does not contain "air" elements. The air modes are absent from the finite element model and any effect they may have had on the structural modes is a matter for future research. In the finite element model the strings were not tuned to the exact frequencies used on a real violin. For the G string the model was off by + 16 Hz; the D string, + 18 Hz; the A string, + 46 Hz; and the E string, - 7 Hz. The differences in tuning could have caused some of the discrepancies in mode shapes and ordering of the modes. Fixing the tuning problem would merely require an adjustment of the thermal loads applied to the string model. A real violin is glued together in most places. In the finite element model there is no attempt to model the glue joints. - Forces from the plates and strings hold the soundpost and bridge to the real violin body. -
-
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G.A. Knott et a{ / Modal analysis oJ the violin
Fig. 9. Mode 24.
Fig. 10. Mode 26.
In the finite element model most of these forces exist (not the stress mounting of the soundpost in the violin box), but the components are "attached" to the plates and the bridge. In other words, there is no provision for slipping in the case of the strings or decoupling from the plates in the case of the soundpost. Figures 9-11 show hidden line deformed meshes corresponding to modes 24, 26, and 32. A complete catalog of all of the the mode shapes using hidden line deformed meshes (and contour plot views as well) is available from Knott.
Fig. l 1. Mode 32.
G.A. Knott et al. / Modal analysis of the violin
279
Conclusion and future work The work described in this paper resulted in a catalogue of a finite element violin's mode shapes. The purpose of the study was to provide this preliminary catalogue and to support the idea that the mode shapes of a violin, complete with the string system, can be described using the finite element method. It should be remembered that the method's real utility lies not in mere reproduction and description of physical reality but in sensitivity analyses and design optimization. However, before research can progress to violin design optimization still more will be required in the effort to accurately describe the violin. Further work might first concentrate on gaining a more complete data base of real violins and components, with known geometries and material properties. A coincident finite element analysis can then be expected to agree with experimental modal tests of the real violins and components. Different violins could be tested and modeled and the mode shapes compared, perhaps leading to an understanding of what differentiates a " g o o d " sounding violin from a " p o o r " one. The incorporation of air elements into the model is a long-term but necessary goal for further research. The extent of the air mode's coupling effects with the structural modes could be easily determined through the study of simple finite element models. Radiation and string excitation studies using finite element models are also promising areas for further work. After developing the finite element model of the violin the authors found the model to be as full of mystery as a real violin! A methodical, step-by-step analysis should clear up many of the grey areas. The finite element method is an extremely powerful descriptor of physical reality and we believe that much of the violin "puzzle" will be solved as research continues.
References [1] EKWALL,A., "A study of the Stradivari-Sacconi violin arching for getting a computerizable arching of the same type", Catgut Acoust. Soc. Newslett. 35, pp. 33-36, 1981. [2] HAINES,D.W., "On musical instrument wood", Catgut Acoust. Soc. Newslett. 31, pp. 23-32, 1979. [3] HtrrcmNS, C.M., "The acoustics of violin plates", Sci. Am. 245, pp. 171-186, October 1981. C.M. [4] HtrrcniNS, C.M., K.A. STETSONand P.A. TAYLOR,"Clarification of free plate tap tones by hologram interferometry", Catgut Acoust. Soc. Newslett. 16, pp. 15-23, 1971. [5] MARSHALL,K.D., "Modal analysis of a violin", J. Acoust. Soc. Am. 77, pp. 695-709, 1985. [6] RICHARDSON,B.E. and G.W. ROBERTS,"The adjustment of mode frequencies in guitars: a study by means of holographic interferometry and finite element analysis", Proc. SMAC 1983 Conference, 1983. [7] RICHARDSON,B.E. and G.W. ROBERTS,"Predictions of the behavior of violin plates by the finite element method", Proc. Inst. Acoust., 1985. [8] RUBIN,C. and D.F. FAm~R, "Computer analysis of the vibrations of a violin back", Department of Mechanical and Materials Engineering, Vanderbuilt University, Nashville, Tennessee, 1984. [9] SACCONI,S.F., Die Geheimnisse Stradivaris, Verlag Das Musikinstrument, Frankfurt/M, F.R.G., 1976. [10] SCrlWAB,H.L., "Finite element analysis of a guitar soundboard", Catgut Acoust. Soc. Newslett. 24 & 25, 1974. [11] Wood Products Handbook, U.S. Forest Products Lab, Madison, Wisconsin, p. 79, 1955.