Computational mechanics of the classical guitar

192 Pages·2005·5.562 MB·English

by  Rolf Bader

Computational Mechanicsofthe ClassicalGuitar
Rolf Bader
Computational Mechanics
of the Classical Guitar
With65Figures
ABC
Dr.RolfBader
UniversityofHamburg
MusikwissenschaftlichesInstitut
NeueRabenstr.13
20354Hamburg,Germany
e-mail:[email protected]
LibraryofCongressControlNumber:2005925685
ISBN-10 3-540-25136-7SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25136-1SpringerBerlinHeidelbergNewYork
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To my parents for their love and support in good and in bad
times.
To Prof. Albrecht Schneider for his critical and broad ranged
teachings about most I know about music, his support and his
endless and patient flow of ideas, energy and wisdom.
Preface
Acknowledgements
Thanks to Dr. Maria Haase, Gunter Faust and the people at ISD at the
University of Stuttgart for their teachings, their support and encouragement.
ThankalsotoElizabethHicksformostofthetranslationfromGermanto
English.
Contents
1 Introduction............................................... 1
1.1 General Remarks ........................................ 1
1.2 Brief Summary.......................................... 5
2 Musical Transient-Modeling Software MTMS.............. 7
2.1 Part of the MTMS Program for Plates ..................... 9
2.2 MTMS Software for Guitar Parts.......................... 10
2.3 Software WAVELET..................................... 17
2.4 The MAS Program as a Signal-Processing Tool for Music..... 18
3 Continuous and Discrete Mechanics ....................... 23
3.1 Overview............................................... 23
3.1.1 Strong Formulation................................ 23
3.1.2 Weak Formulation................................. 24
3.1.3 Action Principle................................... 24
3.2 Overview of Continuum Mechanics ........................ 27
3.2.1 Definition of the Variables.......................... 28
3.2.2 Mechanical Laws .................................. 29
3.3 Discrete Mechanics ...................................... 31
4 Studies of the Guitar To Date ............................. 33
4.1 Strings................................................. 34
4.1.1 Transverse Movement.............................. 34
4.1.2 Longitudinal Movement ............................ 35
4.2 Strings and Corpus Impedance ............................ 37
4.3 Guitar Body Modes...................................... 38
4.4 Nonlinearities and Transient Analyses...................... 43
4.5 Discrete Instrument Models............................... 49
X Contents
5 The Discrete Guitar Model ................................ 51
5.1 Top and Back Plate ..................................... 52
5.1.1 Coupling within Top and Back Plate................. 52
5.1.2 Boundary Conditions of Top and Back Plate.......... 53
5.1.3 Mass Distribution of the Top and Back Plate ......... 53
5.2 Ribs ................................................... 55
5.3 The Neck of the Guitar .................................. 55
5.4 The Strings of the Guitar................................. 56
5.5 Air in the Guitar ........................................ 56
5.6 Sound Radiation of the Guitar Body....................... 56
5.7 The String Movement and Its Discrete Form ................ 58
5.8 Longitudinal Waves and Their Discrete Form ............... 59
5.9 Bending Waves and Their Discrete Form ................... 62
5.10 Boundary Conditions .................................... 65
5.10.1 Boundary Conditions with Longitudinal Boundaries ... 66
5.10.2 Boundary Condition of Bending Movements .......... 69
6 Bending, Damping and Coupling .......................... 73
6.1 Coupling of Longitudinal and Bending Waves ............... 74
6.1.1 LT – First Type Coupling .......................... 74
6.1.2 LT – Second Type Coupling ........................ 83
6.2 Time-Stepping Method................................... 93
6.3 Energy Conservation..................................... 97
6.4 Spacial-Temporal Differences in Discrete
and Continues Mechanics.................................102
6.5 Results for the Plate.....................................109
6.6 Damping ...............................................117
6.6.1 Velocity Term as Damping Term
with the Harmonic Oscillator .......................117
6.6.2 Force Term as Damping Term with Bending Waves ....119
6.7 Coupling ...............................................128
7 Results ....................................................131
7.1 Parameters of the Guitar Model...........................132
7.2 Sounds of the Individual Parts of the Guitar ................134
7.3 Couplings ..............................................135
7.4 Spectra of the Parts of the Guitar .........................135
7.5 Development of the Initial Transient .......................143
7.6 Aural Evaluation of the Calculated Sounds
of the Parts of the Guitar ................................156
7.7 Coupling of the String to the Top Plate ....................159
7.8 Pre-Scratch.............................................161
Contents XI
8 Summary and Outlook.....................................167
References.....................................................173
Audio Files and Videos on the Web............................181
Glossary of Variables
Coordinates
Ω domain
Γ boundary
R boundary condition
Γ
x x-coordinate
y y-coordinate
z z-coordinate
u displacement vector {r, s, t}
r or u displacement in x-direction
x
s or u displacement in y-direction
y
t or u displacement in z-direction
z
n unit outward normal vector
Matrices
D n × m matrix of spatial vectors of the central points of the
top plate elements
B n × m matrix of spatial vectors of the central points of the
back plate elements
Z n × m matrix of spatial vectors of the central points of the
rib elements
H n × m matrix of spatial vectors of the central points of the
neck elements
L n × m × o matrix of spatial vectors of the central points
of the enclosed air space elements
S n × s matrix of string elements of a string s
s
XIV Glossary of Variables
D n × m matrix of displacements of the central points
u
of the top plate elements
B n × m matrix of displacements of the central points
u
of the back plate elements
Z n × m matrix of displacements of the central points
u
of the rib elements
H n × m matrix of displacements of the central points
u
of the neck elements
L n × m × o matrix of displacements of the central points of
u
the enclosed air space elements elementmittelpunkte
S n × s matrix of string elements of a string s
s
D n × m matrix of displacements of the central points of the
v
top plate elements
B n × m matrix of velocities of the central
v
. . points of the back plate elements .
. .
S n × s matrix of velocities of the string elements of a string s
v
D n × m matrix of accelerations of the central points of
a
the top plate elements
B n × m matrix of accelerations of the central points
a
of the back plate elements
. .
. .
S n × s matrix of accelerations of string elements of a string s
a
D intermediate memory of the spatial vectors of the central
a
points of the top plate elements
B intermediate memory of the spatial vectors of the central
a
points of the back plate elements
. .
. .
S intermediate memory of the spatial vectors of string
a
elements of a string s
G rib geometry vectors
u
Physical Parameter
D mass of each element (in this case in the top plate)
m
D density of the wood (in this case in the top plate)
(cid:1)
D Youngs modulus in the x direction (in this case in the top plate)
Ex
D Youngs modulus in the y direction (in this case in the top plate)
Ey
D height of each element (in this case in the top plate)
h