Keywords: ear, elastic wave, sound transfer function, acoustic modeling, ANSYS, hearing, finite element method.

The results presented in this paper demonstrates the possibilities of mathematical modeling for the studies of hearing mechanics. The finite element system ANSYS is used as for finite element model preparation as for the determination of the transmission characteristics. From the point of view of sound transmission the following are taken into account: the acoustic subsystem of the external auditory canal (EAC), the elastic structure of the tympanic membrane (TM), the middle ear cavity acoustic subsystem, the stiff ear ossicle chain with moveable joints, the coupling of individual ossicles to the walls of middle ear cavity via ligaments and muscles and the coupling of stapes footplate to the enter of cochlea. For the determination of the geometry of ear cavities the transformation of computer tomography data into a finite element model has been used [1].

The mode frequencies and the shapes of modes of EAC acoustic subsystem were firstly determined for the rigid eardrum and the total reflections on the walls of EAC, five modal frequencies in audible range were determined. The pressure distribution in the acoustic subsystem of the EAC and the pressure dependence upon the distance from EAC axis (shape of mode) for the first modal frequency is shown in the Figure .

This frequency corresponds to the resonance in the EAC cavity. As it follows from the acoustic pressure map, this frequency is the most important with regard to sound transmission because the pressure level distribution on the eardrum is for this frequency homogenous. The results of modal analysis of eardrum indicated that TM modal frequencies start in lower frequency range (=655 Hz).

As a next step, the harmonic analysis has been performed to determine the transfer function of the stapes footplate movement on harmonic pressure oscillations in the entrance of EAC. As it followed from animation, for the first TM modal frequency the movement of malleus was 'piston like' for frequencies up to at least 2000Hz. For higher frequencies the motion of malleus head was 'elliptical' and stapes displacements diminished, which shows that the higher modes of tympanic membrane has only small effects on the motion of the stapes. The change of the direction of the chain rotation axis with frequency is mentioned also by Wada [2]. The middle-ear cavity pressure practically did not vary spatially up to at least the frequency 3kHz, the angle shift between the pressures in the external ear canal and in the middle ear cavity was observed.

From the point of view of hearing the stapes velocity is probably one of the most important for acoustic quantity. The calculated dependence of the stapes velocity on the frequency was similar to the results of the stapes velocity measurements made on cadaver ears [3]. It is interesting that the curves of the inverse value of stapes velocity for different frequencies were in good agreement with well-known curves of the same loudness. The calculations performed are a good example that the mathematical modeling by finite element method could have a significant role in the understanding of the mechanisms for hearing.

Acknowledgment

The investigation was supported by the Ministry of Education and Youth (Project MSM No 262100001).

1

P. Kršek, P. Krupa, "Human Tissue Geometrical Modeling", Proceedings of 12th IASTED Int. Conf. "Applied Simulation and Modelling", 357-360 Marbella, Spain, 2003.

2

H. Wada, "Middle ear mechanics from the standpoint of dynamics", Proceedings of 7th Int. Congress on Sound and Vibration, 2275-2282, Garmisch-Partenkirchen, 2000.

3

S.E. Voss, J.R. Rosowski, S.N. Merchant, W.T. Peake, "Acoustic responses of the human middle ear", Hearing research 150,43-69,2000. doi:10.1016/S0378-5955(00)00177-5

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