Overview

1
Overview

• Introduction Solving 1-D systems
• Derivation of PDEs for MD systems
• Solving MD systems on the example of a vibrating
  string
• Laplace transformation
• Sturm-Liouville transformation
• Transfer function model
• Inverse transformations
• Realization with recursive systems
• Different string excitations
• Conclusions

2
Introduction Solving 1-D systems (I)
Example 1-D system only dependent on time t
Electrical network with lumped parameters
Ordinary differential equation (ODE)
R
L
u
(t)
(t)
C
u
c

• ODE cannot be implemented directly in the
  computer
• Direct discretization ? computational
  ineffective
• Laplace transformation and discretization ?
  comp. effective

3
Introduction Solving 1-D systems (II)
Outline
Laplace-Transformation
Differentiation theorem
Laplace transf. of ODE
Transfer function model (TFM)
4
Derivation of PDEs for MD systems (I)

• Physical based MD systems are often time and
  space dependent
• they can be described by partial differential
  equations (PDEs)
• derivation of the PDEs by basic laws of physics
• here example of a longitudinal vibrating string

• String with
• A cross section area
• E Youngs modulus
• ? density
• F force on string element

String element of length dx
5
Derivation of PDEs for MD systems (II)
Strain on string element
Stress on string element
Hookes law
Force on string element
Fundamental law of dynamics
PDE after force elimination
with
? PDE describing longitudinal string vibrations
6
PDE of a transversal vibrating string
Now PDE for a transversal vibrating string with
dispersion and losses
? density A cross section area E Youngs
modulus I moment of inertia tension on the
string damping
PDE
Initial conditions
Boundary conditions
? will be solved in the temporal and spatial
frequency domain
7
Solving the PDE - Laplace transformation
Outline
Laplace transformation

• removes temporal derivatives
• includes initial conditions as additive terms

ODE
Boundary cond.
8
Solving the PDE - Sturm-Liouville (SL)
transformation
SL transformation
Differentiation theorem
Eigenvalue problem
?
SL transformation on ODE

• removes spatial derivatives
• includes boundary conditions as additive terms

9
Solving the PDE - Transfer function model
TFM
Inverse Laplace transformation of the TFM
with
10
Inverse transformation with respect to space
with

• inverse spatial transformation only a sum over
  discrete frequencies µ

? a discretization with respect to time must be
done for computer implementation
11
Discretization with respect to time

• discretization of time tkt
• impulse invariant transformation (e.g.
  z-transformation)
• ? discrete transfer function model

z discrete frequency variable

• inverse Z transformation by applying the
  shifting theorem
• ? leads to a recursive system
• ? summation for inverse spatial transformation
• N recursive systems in parallel (not infinity to
  avoid aliasing)

12
Realization with recursive systems

• structure contains only multipliers and delays
• can easily be implemented in real-time on DSPs

13
Excitation models (I) Plucked
E-bass
guitar to xylophone
spinet
14
Excitation models (II) Struck
(0)
v
y
(0)
h
h
Rec. System
(Hammer)
h
(
)
y
,k
x
a
-
NL
a
(0)
a
(N)
...
...
(0)
b
(N)
b
Rec. Systems
S
....
d
(string)
)
(
y
,k
x
a
Lowest piano note
15
Excitation models (III) Bowed
Finale
16
Conclusions

• Solution of 1-D systems with transfer function
  models
• Description of MD systems by PDEs
• Derivation of a MD system of a vibrating string
  by basic physical laws
• PDE of a transversal vibrating string was solved
  by
• Laplace transformation
• Sturm-Liouville transformation
• Transfer function model
• Inverse transformations
• Discretization
• Examples of excitation functions (plucked,
  struck, bowed)
• ? System can be implemented on a DSP for real
  time sound synthesis
• It is realized on a SHARC DSP (60MHz) with two
  voices
• and 100 harmonics per voice.