High-frequency time-domain behaviour of a finite-element model of the eardrum

W.R.J. Funnell
McGill University, Montréal, Canada

24th Midwinter Res. Mtg., Assoc. Res. Otolaryngol., St. Petersburg Beach (2001)
Abstract

Complex vibration patterns and large phase lags are observed on the eardrum at high frequencies, both experimentally and in models (Decraemer et al., 1989, Hearing Res. 38: 1). The vibration patterns in response to pure tones seldom form nodal lines (i.e., lines of zero amplitude) but rather exhibit phases which change gradually across the eardrum, as demonstrated experimentally by Decraemer et al. (ARO Midwinter Mtg., 1997) and in a model by Funnell et al., (ARO Midwinter Mtg., 1997). These patterns can be described as combinations of standing waves and travelling waves and have been related to a delay-line model of the eardrum (Puria & Allen, 1998, J. Acoust. Soc. Am. 104: 3463). The presence of travelling waves has been described in an eardrum model which emphasizes pre-stress in the eardrum and nonuniform anisotropic material properties (Fay et al., ARO Midwinter Mtg., 1999). In the present work, the high-frequency time-domain behaviour is presented for a finite-element model similar to our previous ones, without pre-stress and with simplified material properties. In addition to step functions of uniform pressure, point loads are used to elucidate the nature of the travelling waves. The effects of changing the shape of the eardrum are investigated.

Supported by Canadian Institutes of Health Research.

1. Introduction

Complex vibration patterns and large phase lags are observed on the eardrum at high frequencies (e.g., Decraemer et al., 1989). The vibration patterns in response to pure tones seldom form nodal lines (i.e., lines of zero amplitude) but rather exhibit phases which change gradually across the eardrum, as demonstrated experimentally by Decraemer et al. (1997) and in models by Funnell et al. (1997) and Fay et al. (1999).

In the present work, the nature of these vibrations is explored in the time domain for a finite-element model of the cat eardrum. In addition to uniform pressures, concentrated point loads are used to elucidate the nature of the travelling waves. The effects of anisotropy and of changing the shape of the eardrum are investigated.

2. Finite-element model

The finite-element model used here is essentially the same as one we have used previously for the cat (Funnell & Decraemer, 1996; Funnell et al., 1997) except that the finite-element mesh of triangles has been modified to permit the definition of a radially oriented anisotropy of the material properties.

The shapes of the pars tensa and pars flaccida in this model were derived from a moiré measurement of an actual cat eardrum (Funnell & Decraemer, 1996). The pars tensa and pars flaccida are both modelled as uniform, homogeneous curved shells without pre-stress. For the pars tensa, the thickness (h) is taken to be 40 μm. The Young’s modulus (material stiffness, E) of the pars tensa is taken to be 2×108 dyn cm-2 in the isotropic case, with a Poisson’s ratio (ν) of 0.3; in the anisotropic case the Young’s modulus is again taken as 2×108 dyn cm-2 in the radial direction but is 100 times smaller in the orthogonal direction, with Poisson’s ratio and the shear modulus being taken as zero for simplicity (corresponding to no coupling between the radial and circular fibre layers). The pars flaccida is thicker but much less stiff than the pars tensa; it is modelled here as having a thickness of 80 μm, a Young’s modulus of 107 dyn cm-2 and a Poisson’s ratio of 0.3. Both pars tensa and pars flaccida are given a density (ρ) of 1 g cm-3.

A fixed ossicular axis of rotation is assumed, running from the anterior mallear process to the posterior incudal process. The combined ossicular and cochlear load is represented at the axis of rotation by a frequency-independent rotational stiffness of 14 kdyn cm and a moment of inertia of 0.2 mg cm2. It would be more accurate to model the elastic suspension of the ossicles, as we have done in some of our previous models (Funnell, 1996; Funnell et al., 1999, 2000; Abou-Khalil et al., 2000, 2001). Since the focus here is on the vibration of the eardrum, however, the simpler model is used for convenience.

The damping in the system is represented by a mass-proportional damping coefficient =1500 s-1 (Funnell et al., 1987). The effective damping ratio decreases with frequency and the damping has little effect on fast transients.

The input is either a pressure applied uniformly across the eardrum, or a concentrated force in the z direction (i.e., perpendicular to the tympanic ring) applied to a single point on the eardrum. In both cases the load is applied as a step function at time zero, and impulse responses are computed by differentiation. Only the z components of displacements are presented.

3. Results

A sequence of cases is considered, starting from a flat model with isotropic material properties and a load consisting of a concentrated force at a single node, and ending with a model having a realistic 3-D eardrum shape, anisotropic material properties, and a uniform pressure as the load.

3.1 Flat model. As the simplest case, the model is flattened by setting all z coördinates to zero. The material properties are isotropic, and a force is applied at a single point near the tympanic ring in the inferior-posterior quadrant.

Fig. 1. Time courses of impulse responses at five points on flat model, from 0 to 100 μs. Units on vertical scale are arbitrary. Fig. 1 shows the time courses of the displacements at a few points. At the point where the force is applied, the maximal displacement is reached at about 8 μs. The displacement maximum moves to neighbouring points and gradually dies out. Note that the waveform changes from one location to the next. This is because flexural wave propagation is dispersive, that is, different frequency components travel at different speeds. For flat isotropic plates, the speed is (e.g., Gorman, 1991). For the model parameters used here, this gives speeds of about 100 and 1000 cm s-1 at 100 Hz and 10 kHz, respectively. In Fig. 1, the first maximum moves at about 3400 cm s-1 from the first node to the second, and at about 1400 cm s-1 from the second to the third, consistent with the speeds expected for high-frequency components of the waves.

Fig. 2 shows the evolution of the impulse-response displacement patterns over the entire eardrum, from time zero to 1 ms. The displacement starts by spreading radially from the point where the stimulus is applied, and has largely died out by 1 ms.

Fig. 2. Evolution of displacement pattern on flat model, 0 to 1 ms, globally normalized displacements.

To permit closer examination of the patterns, Fig. 3 shows the first 400 μs of the same response, with the displacements at each time step normalized independently, so the details of the later responses are more clearly visible.

Fig. 3. Evolution of displacement pattern on flat model, 0 to 400 μs, individually normalized displacements.
It can be seen that by 25 μs the circular waves in the posterior region have already encountered the manubrium, which in turn has evoked wavefronts parallel to its length. Anteriorly these secondary waves advance to the anterior edge of the eardrum. Posteriorly they are reflected back from the manubrium and start to interfere with the original circular waves; the interference pattern is well established by about 150 μs. By about 200 μs the circular waves in the inferior region have rounded the umbo and start to advance superiorly; by 350 μs they have reached the superior limit of the eardrum and obliterated the original waves parallel to the manubrium.

3.2 3-D model. Fig. 4 shows the evolution of the displacement pattern for a model in which the shape of the eardrum is as measured using moiré topography, in the same format as Fig. 3. The patterns are similar to those for the flat model in Fig. 3: circular waves again radiate from the point of stimulation and are reflected from the manubrium. The most noticeable difference is that in the anterior region no waves are seen parallel to the manubrium. It is to be expected that the behaviour should be somewhat different because of the 3-D curvature (cf. the discussion of hoop stress by Fletcher, 1982).

Fig. 4. Evolution of displacement pattern on 3-D model, 0 to 400 μs, individually normalized displacements.
3.3 Anisotropy. The ultrastructure of the pars tensa suggests that it is anisotropic with greater stiffness in the radial direction. Fig. 5. Static responses to point loads: (a) isotropic; (b) 100:1 anisotropy. Fig. 5 shows the static displacement patterns for a concentrated load applied in the posterior superior quadrant of the pars tensa for (a) isotropic stiffness, and (b) anisotropic stiffness with a 100:1 ratio of radial to circular stiffness. The essentially circular pattern in (a) is significantly narrowed in (b), although not as much as might be expected from a 100:1 stiffness ratio. To confirm that the width of the oval pattern is not caused by the coarseness of the finite-element mesh, the same stimulus was simulated with a mesh having twice as many elements both radially and circularly; the resulting pattern was practically the same. Fig. 6 shows the evolution of the displacement pattern for this anisotropic model, in the same format as Fig. 3. The pattern is very different from the isotropic case. The wavefronts radiating from the point of stimulation are no longer circular, and they give rise to almost straight waves moving in a direction perpendicular to the radial fibres. The largest displacements are on a radial line through the point of stimulation, and the maximum can be seen to move toward the manubrium, ‘bounce’ off and move back toward the tympanic ring. (It must be recalled that the displacements are again normalized separately for each time value in Fig. 6. The displacement maximum moving back and forth is actually decaying quite rapidly as it moves.)

Fig. 6. Evolution of displacement pattern on anisotropic 3-D model, 0 to 400 μs, individually normalized displacements.

3.4 Pressure stimulus. Fig. 7 shows the evolution of the displacement pattern in response to an impulse of pressure on the eardrum. In this case the format of Fig. 2 is used, with a longer time scale and global normalization of the displacements. There is no evidence of the distinct wavefronts seen with point stimuli in the other figures. The response starts out being quite uniform over the surface of the drum, but as it develops it focusses on a small area in the posterior region; the response of this model to a static pressure also shows such a concentration, which is not consistent with the holographic results of Khanna & Tonndorf (1972). Although not shown in this figure, the displacements on the manubrium are actually larger than those on the eardrum for the first 5 μs or so.

Fig. 7. Evolution of displacement pattern in response to pressure on anisotropic 3-D model, 0 to 1 ms, globally normalized displacements.

4. Conclusion

Wave propagation is complex because of the presence of different modes (extensional, flexural and shear), because of dispersion in the flexural mode, and because of reflections from boundaries. In the use of acoustic emissions for detecting defects in structural components, for example, ‘so far, only simple modes can be roughly distinguished’ in plates (Huang et Fig. 8. Chaotic reverberation on a flat plate. (After Fink, 1999)al., 1998). Fink (1999) describes ‘chaotic reverberations’ on a flat plate with an asymmetrical boundary.

The situation is even more complex in an irregular 3-D curved structure like the eardrum, but it may be somewhat simplified by anisotropy, which can act to reduce coupling. The complexity of the real eardrum might be accommodated in the delay-line model of Puria & Allen (1998), which hypothesizes a constant travel time to the manubrium from different points around the tympanic ring, by invoking strong anisotropy and a careful adjustment of thicknesses and material properties.

The simulations presented here do not take into account the actual complex 3-D motions of the ossicles (e.g., Decraemer & Khanna, 2000) but those motions may not strongly affect displacement patterns on the eardrum.

5. Acknowledgements

Supported by the Canadian Institutes of Health Research.

6. References

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