ARTICLE IN PRESS

Mechanical Systemsand

Signal Processing

0888-3270/$ - se

doi:10.1016/j.ym

�Tel.: +765

E-mail addr

Mechanical Systems and Signal Processing 21 (2007) 98–107

www.elsevier.com/locate/jnlabr/ymssp

Structural modification of stringed instruments

Mark French�

Department of Mechanical Engineering Technology, Purdue University, 138 Knoy Hall, 401 N. Grant St, West Lafayette, IN 47907, USA

Received 22 September 2005; received in revised form 23 December 2005; accepted 4 January 2006

Available online 18 April 2006

Abstract

Two degree of freedom structural models have been used for some time to model the low-frequency behaviour of

stringed instruments. Developing a closed form method of predicting the effects of structural modification can extend the

utility of these models. Analytical sensitivities can readily be developed for this purpose. This approach is described in

detail and demonstrated through example problems. In particular, the effect of structural modification on the coupled air-

structural modes is captured.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Guitar design; Musical acoustics; Luthiery; Structural modification

1. Background

It is encouraging to note that structural modelling of stringed instruments has generally kept pace with theavailable technology. Finite element models appeared as early as 1975 [1] and their sophistication hasimproved along with the capability of the hardware and software [2]. Recent work has progressed to the pointof coupling complete structural models of the body with numerical representations of the enclosed air [3,4].

It is clear that very sophisticated instrument models exist and are being improved continually.Understanding of the instrument characteristics that affect subjective sound quality has also progressed[5–7]. However, one side effect of the improvement in modelling is that the models themselves have becomeunwieldy. The increase in modelling precision comes at the price of discrete models that can have manythousands of degrees of freedom.

Simple discrete models that capture the acoustic-structural coupling for the first two or three modes [8] arequite useful when trying to understand the mechanics of stringed instruments. When the task at hand isdetermining the effect of structural modifications on the lower modes of an instrument, these models allowrapid evaluation of different options. This article describes a method of predicting the effect of designmodifications based on eigenvalue derivatives.

The method presented here is a member of a larger class of structural modification problems that is well-represented in the literature [9,10]. It is beyond the scope of this short article to summarise them completely,

e front matter r 2006 Elsevier Ltd. All rights reserved.

ssp.2006.01.002

494 7521.

ess: rmfrench@purdue.edu.

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107 99

but they can be collected roughly into three groups: parameter identification schemes, matrix model updatingand parametric modification.

Parameter identification schemes seek to identify unknown constants in mathematical models directly fromexperimental data with no initial estimate generally required [11–13]. Probably the most familiar example isexperimental modal analysis [14]. These schemes may also fall into the category of inverse problems [15]. Inperforming a dynamic structural response tests, one is assumed to be recording the solution to a system ofcoupled differential equations. The mathematical task is then to identify in some optimal sense the system ofequations that would yield the measured result.

Matrix model updating modifies elements in the matrix equations of motion of a dynamic system like avibrating structure. The system matrices generally result from finite element models and, thus, have a level ofgeometric realism not generally found in the first group. The object is to match the calculated eigenvalues andeigenvectors to the results of dynamic structural response measurements on a test structure. These techniquesare mathematical in nature and can often be specified with some formal rigor [16,17]. However, it is thepossible that the resulting solution is not physically realistic.

Parametric modification changes physical parameters of a structural model to either tune themodel to match experimental data or to predict the effect of a structural modification [18]. This stands incontract to the other two methods in which mathematical parameters are modified with no initialconsideration to the physical implications. The parameter modification approach tends towards the empirical,but has the advantage that modifications are made to physical parameters in the model rather than onmathematical coefficients; the resulting solution is naturally easier to understand in terms of structuralmodifications.

2. Introduction

The three elements in the structure of the instrument that couple to form the first two or three modesare the top and back of the body and the enclosed volume of air. Typically, the first two modes aredistinguished largely by the phase relationship between the top and the back. In the first mode, thetop and back move in opposite directions so that the volume of the body changes. In the second modethey move in the same direction so that there is no nominal change in body volume. The third top mode isroughly anti-symmetric about the centreline of the instrument and is more strongly affected thebracing pattern [19]. In the course of developing new instrument designs, it is often necessary to modifyprototypes to improve the sound. It would be very helpful at this stage to have an approximatestructural model that could predict accurately the effect of different structural changes. In that case,modifications could be directed so as to shift the first few natural frequencies of the instrument. The effect ofstructural modifications can be described by calculating gradients of the eigenvalues of the equations ofmotion of the discrete models described by Christensen and Vistisen [8]. This paper shows how these modelscan be tuned to match experimental results and how analytical gradients can be used to predict the effect ofstructural modifications.

The methods presented here uses eigenvalue derivatives to predict the effect of basic structural changes onthe first few natural frequencies of an instrument. While it is possible to compute the changes in the equationsof motion required to yield a desired set of eigenvalues, the result is not necessarily easy to interpret orimplement physically. This approach operates directly on the physical parameters because they are moreimmediately useful to instrument makers.

3. Discrete structural models

Simple discrete models for the low-frequency behaviour of stringed instruments have appeared in theliterature for some time. The two DOF model couples a flexible top surface with a Helmholz resonator asshown in Fig. 1. The walls are assumed to be rigid everywhere except for the flexible section.

ARTICLE IN PRESS

Volume, V

Ap, mp, xpAh, mh, xh

F

kp

Fig. 1. Two degree of freedom discrete model of a guitar body.

M. French / Mechanical Systems and Signal Processing 21 (2007) 98–107100

3.1. Equations of motion

The equations of motion for the two DOF model are:

mp 0

0 mh

" #€xp

€xh

( )þ

Rp 0

0 Rh

" #_xp

_xh

( )þ

kp þ mA2p mAhAp

mAhAp mA2h

" #xp

xh

( )¼

F

0

� �, (1)

where mp is the mass of the top plate, mh is the mass of the moving air, kp is the stiffness of the top plate, Ap isthe effective plate area, Ah is the sound hole area and m is a proportionality constant between changes involume, V, and changes in pressure

m ¼ c2r=V . (2)

Here, c is the speed of sound and r is the density of air. The 2-DOF model can be extended simply by addinga flexible back.

3.2. Calculating sensitivities for structural modification

In order to tune the math model or to predict the effect of structural modifications using the tuned mathmodel, it is most helpful to understand the effect of variations of model parameters on the natural frequencies.Fortunately, it is not difficult to derive analytically the variation of natural frequencies with respect to somedesign variable [20,21]. Consider the familiar eigenvalue equation

½K �fxg ¼ l½M�fxg, (3)

where [K] is the stiffness matrix, [M] is the mass matrix. Note that expressions in square brackets are matricesand those in braces are vectors. Designate the design variable as a and take the partial derivative of theeigenvalue equation

qK

qa

� �fxg þ ½K�

qx

qa

� �¼

qlqa½M�fxg þ l

qM

qa

� �fxg þ l½M�

qx

qa

� �. (4)

Pre-multiply by {x}T. If [K] and [M] are symmetric, the left and right eigenvectors are identical and

qlqafxgT ½M�fxg þ lfxgT

qM

qa

� �fxg � fxgT

qK

qa

� �fxg ¼ 0. (5)

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107 101

Finally, it is simple to solve explicitly for the eigenvalue derivative

qlqa¼fxgT ½qK=qa�fxg � lfxgT ½qM=qa�fxg

fxgT ½M�fxg. (6)

If there are no variable parameters that appear in both the mass and stiffness matrices, stiffness and massmodification can be treated as separate problems. For stiffness modification, q[M]/qa ¼ 0. Conversely, formass modification, q[K]/qa ¼ 0.

3.3. Experimental data

Before the effect of structural modification can be predicted, the math model must be tuned usingmeasured dynamic response data. For this step, a small accelerometer was fixed to the top of theinstrument just behind the bridge. Input force was provided by a small modal hammer tapping nextto the accelerometer. The instrument was supported on several soft foam blocks placed under thehead stock and at the perimeter of the body. These provided an effectively free boundary condition for theinstrument. Input and response data were collected using a small PC-based data acquisition system asshown in Fig. 2. Note that the accelerometer is placed slightly off-centre on the soundboard. This is becausetops typically have a node line for the third mode near the centreline. This top is braced using a slightlyasymmetric X pattern.

It should be noted that the mass of the accelerometer has the potential to alter the measured frequencies ofthe instrument. For this reason, I used the smallest accelerometer available to me—approximately 4 gincluding the mounting. Since the effective mass of the top was 200 g, the mass of the accelerometer can beignored. Certainly, the lightest possible sensor should be used; models weighing less than 2 g are now readilyavailable. If sensor mass is a potential problem, it can simply be added to the equivalent mass term in theequations of motion.

Fig. 3 shows the instrument with structural modifications to provide information for tuning thediscrete model. The sound hole has been covered and two steel nuts have been fixed to the soundboardto increase its mass without affecting stiffness. They are slightly off-centre so as to not get in the wayof the modal hammer. This off-centre placement may slightly shift the node line for the third andmode. However, only the first two natural frequencies of the instrument are needed here and neitherhas a node line near the bridge. The sound hole cover eliminates the first mode that is approximately aHelmholz mode.

Fig. 4 shows a typical frequency response function, this one from the unmodified instrument. The first fournatural frequencies are 99.4, 185, 338 and 426Hz.

Fig. 2. Test setup for determining structural frequencies.

ARTICLE IN PRESS

Fig. 3. Sound hole cover and top masses.

100

10-1

10-2

10-3

Nor

mal

ized

Mag

nitu

de

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

Fig. 4. FRF of baseline instrument.

M. French / Mechanical Systems and Signal Processing 21 (2007) 98–107102

3.4. Tuning the model

The model tuning process requires identifying correct values for the physical parameters. The modelparameters are representative of physical properties of the instrument, but cannot necessarily be measureddirectly from the instrument. Rather, they are usually inferred from natural frequency measurements.Measuring natural frequencies of the instrument in several different configurations results in several datapoints for tuning.

Differences in the configuration of the instrument must be able to be expressed in terms of modelparameters. Ideally, there should be one configuration for every unidentified parameter. Practically, a few setsof results are usually sufficient. Three configurations are easy to test for: nominal (no modifications), soundhole covered ðAh ¼ 0Þ and additional mass on top.

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107 103

Simple methods are available to identify the lower resonant frequencies of instruments without the need forexpensive equipment. By making simple changes to the structure in a known way, one can change the resonantfrequencies of the instrument so that multiple sets of experimental data are available to identify unknownparameters.

Whereas modal parameter estimation typically identifies all the unknown values at once, the process usedhere essentially decouples the problem so that unknown structural constants can be identified a few at a time.Since we are only concerned with natural frequencies, damping is ignored, thus reducing the number ofunknown values.

Table 1 shows the results of dynamic testing in several different configurations. Note that the frequencyresolution 0.625Hz. All frequencies are in Hz.

The response of these different configurations is easily predicted from the 2 DOF model. For example, if thesound hole is covered and damping is ignored, the equations of motion reduce to

mp €xp ¼ ðkp þ mA2pÞxp (7)

and the resulting resonant frequency is

o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikp þ mA2

p

mp

s. (8)

Using a least-squares error minimisation approach, the parameters for the two-DOF model were identifiedas shown in Table 2. This process used data from the unmodified instrument and from the instrument withadditional masses.

Any set of identified parameters can be assumed correct only if it can be used to predict measured valuesthat were not used as part of the identification process. Two such values are the rigid wall Helmholz frequencyand the fundamental frequency with the sound hole covered.

The rigid wall Helmholz resonance is defined as

f H ¼1

2p

ffiffiffiffiffiffiffiffiffimA2

h

mh

s(9)

and is identified by Christensen and Vistensen [2] as the first anti-resonance of the transfer function –115Hz asmeasured. The calculated value is 118.9Hz.

Table 1

Natural frequencies of instrument in different configurations (Hz)

Configuration F1 F2 F3 F4

1 Baseline 99.4 185 338 426

2 18.6 g top mass 98.1 179 336 424

3 37.3 g top mass 97.5 170 335 424

4 37.3 g top mass and covered sound hole — 156 325 345

5 Covered sound hole — 174 196 329

Table 2

Variable parameters identified from comparison with test

Variable parameter Value

Ap, top plate area 671.3 cm2

Kp, top plate stiffness 182,000N/m

Mp, top plate mass 198.9 g

th, effective air column height 11.77 cm

V, body volume 14.11L

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107104

For the case of the sound hole being covered, the fundamental frequency is defined as

f ¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKp þ mA2

p

mp

s. (10)

The value was measured as 170Hz and calculated as 170Hz. Finally, when two top masses were added tothe instrument with the sound hole covered, the measured resonance frequency was 156Hz and the calculatedfrequency was 156Hz. Thus, the 2-DOF model can be considered tuned. The first two measured frequencies ofthe unmodified instrument were 99.4 and 185Hz, respectively while the two calculated frequencies were 99.4and 182Hz. Thus, the 2-DOF model can be considered tuned.

Note that the topology of the function space described by the discrete models is probably quite complicatedwith many places at which the difference between the calculated and measured natural frequency is at a localminimum. The numbers presented here represent only one of those minima and, thus, one of many possible‘tuned’ models. It is assumed here that for local minima reasonably close to the global minimum, results arequalitatively similar.

4. Structural modification

With a tuned model available, it is possible to calculate the effect of design changes. The next task is tocalculate design sensitivities for the likely modifications to the structure.

4.1. Top mass

It is particularly simple to increase the mass of the top, though this may not be desirable because it tends todecrease the volume of the instrument. If the design variable is the top mass, mp, then only the system massmatrix has a partial derivative

qM

qmp

¼1 0

0 0

� �(11)

and the sensitivity of the nth eigenvalue is

qln

qmp

¼ �

ln x1n x2n

� � 1 0

0 0

� �x1n

x2n

( )

x1n x2n

� � mp 0

0 mh

" #x1n

x2n

( ) ¼ � lnx21n

mpx21n þmhx2

2n

. (12)

Note that the truncated braces indicate row vectors so that xb c ¼ fxgT . For the tuned model, theeigenvalues and eigenvectors are:

l1 ¼ 3:9006� 105; f1 ¼0:0363

�0:999

� �; l2 ¼ 1:309� 106; f2 ¼

0:160

0:987

� �(13)

Table 3

Physical constants

Constant Value

r, sound hole radius 5.08 cm

c, speed of sound 338m/s

r, air density 1.23 kg/m3

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107 105

and the sensitivities are ql1/qmp ¼ �3.586� 105 kg�1 s�2 and ql2/qmp ¼ �5.377� 106 kg�1 s�2. Note themagnitudes of these numbers are large because l ¼ o2 and because a 1 kg change in the mass of the top isextremely large (Table 3).

4.2. Body volume

Another modification is to increase the volume of the body. A change in body volume is practical onlyduring construction when it is easy to make the sides a little taller or shorter while cutting them out. This is astiffness change, so only qK/qV need to be calculated.

qK

qV¼

qmqV

A2p AhAp

AhAp A2h

" #¼ �

c2rV 2

A2p AhAp

AhAp A2h

" #, (14)

qln

qV¼fxgT ½qK=qV �fxg

fxgT ½M�fxg¼

�ðc2r=V2Þ x1n x2n

� � A2p AhAp

AhAp A2h

24

35 x1n

x2n

( )

x1n x2n

� � mp 0

0 mh

" #x1n

x2n

( )

¼�ðc2r=V ÞðA2

px21n þ 2AhApx1nx2n þ A2

hx22nÞ

mpx21n þmhx2

2n

ð15Þ

and the sensitivities are ql1/qV ¼ �1.572� 107m�3 s�2 and ql2/qV ¼ �3.902� 107m�3 s�2.

4.3. Sound hole area

One simple modification is to change the area of the sound hole. If the area of the sound hole is changed,both mass and stiffness matrices are modified, so

qK

qAh

¼ m0 Ap

Ap 2Ah

" #qM

qAh

¼0 0

0 rth

" #(16)

and Eq. (7) becomes

qln

qAh

¼

m x1n x2n

� � 0 Ap

Ap 2Ah

" #x1n

x2n

( )� ln x1n x2n

� � 0 0

0 rth

" #x1n

x2n

( )

x1n x2n

� � mp 0

0 mh

" #x1n

x2n

( )

¼2m Apx1nx2n þ Ahx2

2n

� � lnrthx2

2n

mpx21n þmhx2

2n

ð17Þ

and the resulting sensitivities are ql1/qAh ¼ 3.933� 107m�2 s�2 and ql2/qAh ¼ 2.946� 107m�2 s�2.

4.4. Sound hole sleeve

Note that changing mh (without a corresponding change in Ah) would require adding a sleeve to the soundhole in the manner of a tuned port on a stereo speaker. In this way, the air column set in motion duringHelmholz resonance would be taller and thus more massive. Assuming the sleeve is light in comparison withthe top and does not significantly change the volume of the body, the sleeve results only in a change in the

ARTICLE IN PRESSM. French / Mechanical Systems and Signal Processing 21 (2007) 98–107106

mass matrix. The sensitivity is then

qln

qt¼�lnfxg

T ½qM=qt�fxg

fxgT ½M�fxg¼

�ln x1n x2n

� � 0 0

0 rAh

" #x1n

x2n

( )

x1n x2n

� � mp 0

0 mh

" #x1n

x2n

( ) ¼ �lnrAhx2

2n

mpx21n þmhx2

2n

(18)

and the resulting sensitivities are ql1/qt ¼ �2.69� 106m�1 s�2 and ql2/qt ¼ �1.984� 106m�1 s�2.

4.5. Top stiffness

Decreasing the stiffness of the top is a common modification, in part, because it is easy to do. A luthier caneasily reach through the sound hole with a small plane and remove material from the top braces in an effort toimprove the sound quality. The sensitivity due to a stiffness change is

qln

qkp

¼fxgT ½qK=qkp�fxg

fxgT ½M�fxg¼

x1n x2n

� � 1 0

0 0

� �x1n

x2n

( )

x1n x2n

� � mp 0

0 mh

" #x1n

x2n

( ) ¼ x21n

mpx21n þmhx2

2n

(19)

and the resulting sensitivities are ql1/qkp ¼ 0.918 kg�1 and ql2/qkp ¼ 4.11 kg�1.

5. Example of structural modification

Baritone guitars are becoming popular and builders often modify designs of existing instruments to makethem into baritones. A baritone guitar is typically tuned a musical fifth (four semitones) lower than a standardguitar. Consider then that the problem is to find the most effective way to lower the resonant frequency of theinstrument by four semi-tones.

A semi-tone represents a frequency ratio of 21/12E1.0595. Thus, the factor corresponding to four semi-tonesis 2�1/3E.794—roughly a 20% decrease in the fundamental frequency of the instrument. An instrument withthe first two resonant frequencies at 100 and 200Hz, respectively, should presumably be modified so that thefirst two resonant frequencies are 80 and 159Hz.

There is no assurance that both natural frequencies will be lowered by the same proportion. Thus,predictions are based on reducing the first frequency by the desired amount. The first eigenvalue of theunmodified design is 3.901� 105 and the desired eigenvalue is 2.458� 105. The first natural frequency of theunmodified instrument of 99.4Hz is to be reduced to 78.9Hz. The second frequency is then to be reduced from185 to 147Hz.

A prediction of the effect of a change in some design variable, a, is expressed as a first order Taylor seriesapproximation

lnew ¼ lold þ@l@a

Da. (20)

The most effective modification is a change in the sound hole area. The necessary reduction in sound holeradius is 0.69 cm. Table 4 summarises the different possible modifications. These results suggest that reducingthe stiffness of the top would be the best approach because it reduces both of frequencies by approximately thesame proportion. It is, however, a difficult modification to reverse. Building a guitar with a reduced sound holearea has the advantage that the sound hole can always be enlarged if necessary.

Note that the predicted change in the design variables are not necessarily close to the actual changerequired. The first-order approximation is not always accurate for large changes in the design variable. Onesolution is to calculate the second derivative of the eigenvalues and use a second-order Taylor seriesapproximation. Another approach is to move from the initial position in design space to the final position(initial design to final design) in two or more successive steps.

ARTICLE IN PRESS

Table 4

Design modifications

Variable Predicted Da f1 using predicted Da (Hz) Actual Da for f 1 ¼ 78:9Hz f2 using actual Da (Hz)

Mass 4.02 g 78.7 398 g 132

Body volume 9.13L 83.9 13.2L 165

Sound hole area 36.7 cm2 57.3 �18.8 cm2 176

Sound hole sleeve 6.37 cm 82.4 8.21 cm 176

Top stiffness �157,200N/m 46.2 �94900N/m 157

M. French / Mechanical Systems and Signal Processing 21 (2007) 98–107 107

6. Summary

The techniques now used to manufacture acoustic guitars are good enough that product variation is low. Inthis environment, the effect of small design changes can be differentiated from product variation andinstruments can be tailored for specific applications. A simple discrete model can be tuned so that is describesthe first few natural frequencies of the instrument and then used to predict the effect of design changes.Deriving analytical gradients of the eigenvalues of the discrete model allows the gross effects of design changesto be predicted quickly.

Acknowledgements

I would like to thank Dave Hosler and Bob Taylor of Taylor Guitars in El Cajon, CA for theirencouragement and for providing the instruments shown in Figs. 2 and 3.

References

[1] H.L. Schwab, Finite element analysis of a guitar soundboard, Catgut Acoustical Society Journal 24 (1975) 13–15.

[2] A. Chaigne, Numerical simulations of stringed instruments—today’s situation and trends for the future, Catgut Acoustical Society

Journal 4 (5) (2002) 12–20.

[3] A. Isaksson, H.O. Saldner, N.-E. Molin, Influence of enclosed air on vibration modes of a shell structure, Journal of Sound and

Vibration 187 (3) (1995) 446–451.

[4] M.J. Elejabarrieta, A. Ezcurra, C. Santamaria, Coupled modes of the resonance box of the guitar, Journal of Acoustical Society of

America 111 (5) (2002) 2283–2292.

[5] F. Bustamante-Orduna, Experiments on the relationship between acoustical properties and the subjective quality of classical guitars,

Catgut Acoustical Society Journal 2 (1) (1992) 20–23.

[6] R.R. Boullosa, F. Orduna-Bustamante, A. Perez Lopez, Tuning characteristics, radiation efficiency and subjective quality of a set of

classical guitars, Applied Acoustics 56 (1999) 183–197.

[7] B. Richardson, The acoustical development of the guitar, Catgut Acoustical Society Journal 2 (5) (1994) 1–10.

[8] O. Christensen, Bo B. Vistisen, Simple model for low-frequency guitar function, Journal of the Acoustical Society of America 68 (3)

(1980) 758–766.

[9] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, Journal of Sound and Vibration 167 (1993)

347–375.

[10] M. Imregun, W.J. Visser, A review of model updating techniques, Shock and Vibration Digest 23 (1991) 9–20.

[11] W.P. Rodden, A method for deriving structural influence coefficients from ground vibration tests, AIAA Journal 5 (1967) 991–1000.

[12] M. Baruch, Optimization procedure to correct stiffness and flexibility matrices using vibration tests, AIAA Journal 16 (1978)

1208–1210.

[13] J.J. Hollkamp, S.M. Batill, Automated parameter identification and order reduction for discrete time series models, AIAA Journal 29

(1991) 96–102.

[14] D.J. Ewins, Modal Testing Theory and Practice, Wiley, New York, 1985.

[15] R. Aster, C.H. Thurber, B. Borchers, Parameter Estimation and Inverse Problems, Elsevier, Amsterdam, 2005.

[16] P.W. Moller, O. Friberg, Updating large finite element models in structural dynamics, AIAA Journal 36 (1998) 1861–1868.

[17] C. Minas, D.J. Inman, Matching finite element models to modal data, Journal of Vibration and Acoustics 112 (1990) 84–92.

[18] M. French, F.E. Eastep, Aeroelastic model design using parametric identification, Journal of Aircraft 33 (1996) 198–202.

[19] N. Fletcher, T. Rossing, The Physics of Musical Instruments, Springer, Berlin, 1998.

[20] R.B. Nelson, Simplified calculation of eigenvector derivatives, AIAA Journal 14 (1976) 1201–1205.

[21] R.L. Fox, M.P. Kapoor, Rates of change of eigenvalues and eigenvectors, AIAA Journal 6 (1968) 2426–2429.