This webpage provides supplementary material for the Dissertation “Simulation of Distributed Parameter Systems by Transfer Function Models” by Maximilian Schäfer, Erlangen, 2019.
Both, videos and sound examples have been created for the publications [Schäfer16, Schäfer18, Schäfer19a, Schäfer19b].
They are presented here to complement Part II of the dissertation, where the developed methods are applied to different fields of application.
(3+1)D Diffusion Process – Chapter 6.1
This section provides supplementary material for the modelling of a (3+1)D cylindrical diffusion process in Sec. 6.1.
The results are illustrated by videos of the spatio-temporal dynamics of the particle concentration pxy(r,ϕ,t) in a circular cross section of the cylinder at z = 0 (see Fig. 6.3). The results correspond to the presented results in Sec. 6.1.7 for the circular disk.
All simulations have been performed with the discrete-time state space description in Sec. 6.1.6, i.e., Eqs. (6.40) and (6.43).
An exact description of the considered scenario and all simulation parameters can be found in Sec. 6.1.7.
- The influence of the magnetic force is neglected, i.e., u = 0.
- The video shows the pure diffusion process.
- Due to the reflective boundaries (see Eq. (6.7)), no particles can leave the disk.
- The video can be also compared to Fig. 3.2 in Sec. 3.3
- The video shows the diffusion process including the influence of the magnetic force, i.e., u = 2.
- The scenario is comparable to Fig. 6.5 and 6.6 in Sec. 6.1.7.
- The injected particles diffuse and are attracted by the magnetic force in negative y-direction.
- The scenario is similar to the preceding one.
- The video shows the process for a higher magnetic force, i.e.,
u = 5.
(1+1)D Guitar String – Section 7.1
This section provides supplementary material for the modelling of a guitar string in Section 7.1. The string is attached to a bridge model at x = 0.
The results are illustrated by videos of the simulated guitar string deflection y(x,t) and by sound examples for two different kinds of strings.
The underlying simulations have been performed with the discrete-time state space description as presented in Sec. 4.8.3, i.e., Eqs. (4.84), (4.85).
An exact description of the considered scenario and all simulation parameters can be found in Sec. 7.1.
Videos
Both videos show the first mili-seconds of the spatio-temporal dynamics of the deflection y(x,t) of a guitar string.
- The influence of the bridge is neglected, i.e., Yb(s) = 0.
- This scenario models a guitar string with simply supported edges.
- It is comparable to the black curves in Fig. 7.5 – 7.7 in Section 7.1.
- The video explains the existence of dispersion in the string: Higher frequency oscillations travel faster than the lower ones.
- The video explains the influence of the frequency-dependent damping term d3: Higher frequency oscillations attenuate faster than the lower ones.
- The influence of the bridge is reduced to a frequency independent model, i.e., Yb(s) = Yb (red curve).
- The blue curve serves as a ground truth with Yb = 0.
- The scenario is comparable to Fig. 7.5.
- The video explains the damping influence of the bridge at x = 0: All frequency components are damped equally (see left side of the video screen).
Sound Examples
Several sound examples have been created for a nylon guitar B-string and a steel guitar high-E-string.
The parameters for the nylon strings are taken from [Schäfer16] and for the steel strings from [Schäfer18].
Nylon guitar B-string with zero admittance (top), a mid ranged admittance (mid) and a large admittance (bottom).
Steel guitar high E-string with zero admittance (top), a mid ranged admittance (mid) and a large admittance (bottom).
(2+1)D Membrane – Section 7.2
The modelling of a vibrating membrane has been discussed in Sec. 7.2. The membrane model has been extended by a general feedback structure to modify its timbre.
Several sound examples for different feedback matrices and time-variant modifications have been produced for [Schäfer19b]. They can be found on the webpage for this paper.
References
[Schäfer2016] M. Schäfer, P. Frenstátský, and R. Rabenstein, “A Physical String Model with Adjustable Boundary Conditions,” in 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, Sep. 2016, pp. 159 – 166. [Online]. Available: http://dafx16.vutbr.cz/
[Schäfer2018] M. Schäfer, and R. Rabenstein, “Physical Modelling of Guitar Strings with Realistic Boundary Conditions,” in Jahrestagung für Akustik (DAGA), München, Germany, Mar. 2018, pp. 713–716.
[Schäfer2019a] M. Schäfer, W. Wicke, R. Rabenstein, and R. Schober, “Analytical Models for Particle Diffusion and Flow in a Horizontal Cylinder with a Vertical Force,” in Proc. IEEE International Conference on Communications (ICC 2019), Shanghai, China, May 2019, pp. 1–7.
[Schäfer2019b] M. Schäfer, S. J. Schlecht, and R. Rabenstein, “Feedback Structures for a Transfer Function Model of a Circular Vibrating Membrane,” in Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), New Paltz, NY, Oct. 2019, pp. 1–5.