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 Supported by Canadian Institutes of Health Research. |

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.

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 2×10^{8} 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 2×10^{8} 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 10^{7} 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 cm^{2}. 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.

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 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.

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.

**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).

**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.

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
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.

Supported by the Canadian Institutes of Health Research.

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Web site - http://audilab.bme.mcgill.ca

Copyright © 2001 Robert Funnell. All rights reserved.

R. Funnell Last modified: Sun, 2010 Nov 7 14:03:59