Ye Lu 2011-4-11 Slide 2 A bar with non-uniform cross-sectional area Clamped to the mouthpiece at one end Additional constraints provided by the mouthpiece profile and interaction with the lip Slide 3 Slide 4 Slide 5 Compute the Inverse Laplace transform to get impulse response of the analogue filter Sample the impulse response (quickly enough to avoid aliasing problem) Compute z-transform of resulting sequence Slide 6 Linear multistep methods used for the numerical solution of ordinary differential equations Slide 7 A. M. Schneider, J. T. Kaneshige, and F. D. Groutage. Higher Order s-to-z Mapping Functions and their Application in Digitizing Continuous-Time Filters. Proc. IEEE, 79(11):16611674, Nov. 1991. Slide 8 Slide 9 Used for a generic linear system and are based on a polynomial interpolation of the system input Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Typical resonance frequency lie in the high frequency region, non-critical in helping self- sustained oscillations the reed resonance has a role in adjusting pitch, loudness and tone color, and in helping transitions to high regimes of oscillation (S. C. Thompson. The Effect of the Reed Resonance on Woodwind Tone Production. J. Acoust. Soc. Am., 66(5):12991307, Nov. 1979.) Slide 17 Euler Method provide poor accuracy even with Fs=44100Hz Results for the AM methods are in good agreement with theoretical predictions the magnitude of AM2 amplifies the magnitude of the resonance the methods becomes unstable at Fs = 190000Hz Slide 18 Results for the WS methods are in excellent agreement with theoretical predictions, even at low sampling rates Numerical dissipation is introduced, the amplitude responses is smaller The phase responses are well preserved by both methods WS methods better approximate the reed frequency response than AM methods Slide 19 The quasi-static estimated value underestimates the true pt (D. H. Keefe. ) Slide 20 For all the digital reeds, pt converges to the dynamic estimate pressure1802 1-step methods exhibit robustness with respect to the sampling rate Slide 21 Slide 22 the clarion register can be produced without opening the register hole, if the reed resonance matches a low harmonic of the playing frequency and the damping is small enough an extremely low damping causes the reed regime to be produced Slide 23 Slide 24 Slide 25 One-dimensional: assume no torsional modes in the reed No attempt to model the air flow in the reed channel or to simulate the acoustical resonator Slide 26 Slide 27 http://www.tandfonline.com/doi/abs/10.1080/10236190802385298#preview Slide 28 Slide 29 The reed tip can not exceed a certain value The tip is not stopped suddenly but rather gradually Slide 30 Slide 31 Slide 32 Slide 33 The cause of the discontinuity in the one- dimensional case is not the omittance of the reeds torsional motion Slide 34 The reed-lay interaction exhibits a stronger non- linearity when (1) The players lip moves towards the free end of the reed (2) When a thinner reed is used Slide 35 The closer the lip is positioned towards the free end of the reed, the stronger the non-linear behavior of S becomes Slide 36 F. Avanzini. Computational Issues in Physically-based Sound Models. Ph.D. Thesis, Dept. of Computer Science and Electronics, University of Padova (Italy), 2001. M. van Walstijn and F. Avanzini. Modelling the mechanical response of the reed-mouthpiece-lip system of a clarinet. Part II. A lumped model approximation. Acta Acustica united with Acustica, 93(3):435-446, May 2007. F. Avanzini and M. van Walstijn. Modelling the Mechanical Response of the Reedmouthpiece- lip System of a Clarinet. Part I. A One-Dimensional Distributed Model. Acta Acustica united with Acustica, 90(3):537-547 (2004). A. M. Schneider, J. T. Kaneshige, and F. D. Groutage. Higher Order s-to- z Mapping Functions and their Application in Digitizing Continuous- Time Filters. Proc. IEEE, 79(11):16611674, Nov. 1991. C.Wan and A. M. Schneider. Further Improvements in Digitizing Continuous-Time Filters. IEEE Trans. Signal Process., 45(3):533542, March 1997. V. Chatziioannou and M. van Walstijn. Reed vibration modelling for woodwind instruments using a two-dimensional finite difference method approach. In International Symposium on Musical Acoustics, Barcelona, 2007. Slide 37 Thank You!