Damped vibration patterns on a finite-element model of the cat eardrum

W.R.J. Funnell, W.F.S. Decraemer & S.M. Khanna
22nd Midwinter Res. Mtg., Assoc. Res. Otolaryngol., St. Petersburg Beach (1997)


In our past work with finite-element models of the cat eardrum, we have presented vibration patterns at frequencies low enough for inertia and damping to be negligible, and at undamped natural frequencies. We have also presented point-by-point frequency responses, but not overall vibration patterns, in the presence of inertia and damping. In this work we now examine in detail the eardrum vibration patterns in a model with both inertia and damping, for comparison with newly available experimental observations. The eardrum model is essentially the same as our previous models except that a finer finite-element mesh is used. The vibration patterns are displayed primarily by means of sections through the eardrum.

Supported by MRC Canada, the Québec Ministry of Education, NSRF Belgium, the University of Antwerp (RUCA), the Emil Capita Fund, and NOSH.

1. Introduction

In our early finite-element modelling of the cat eardrum we presented vibration patterns in the form of contour plots, first for low frequencies (Funnell & Laszlo, 1978) and later

for undamped natural frequencies (Funnell, 1983). These contour plots could be compared with vibration patterns measured experimentally using time-averaged holography (Khanna & Tonndorf, 1972). Once we had added damping to our finite-element models, we could compute frequency responses (Funnell et al., 1987). These results could be compared with frequency responses measured experimentally at individual points on the eardrum (Decraemer et al., 1989). Since it was only feasible to make such experimental measurements at a very few points in any one ear, it was impossible to characterize the vibration patterns across the eardrum. Recently, however, it has become possible to make very precise frequency-response measurements at scores of precisely positioned points on a single eardrum (Decraemer et al., 1997). This poster presents finite-element simulation results for comparison with such experimental data.

2. Finite-element model

The shape of the pars tensa in this model was derived from a moiré measurement of an actual cat eardrum (Funnell & Decraemer, 1996).

The model used here is essentially the same as the one used previously except that the finite-element mesh is finer. The shape and parameters of this model have not been adjusted to match the new experimental measurements.

The pars tensa is modelled as a uniform, homogeneous curved shell with a thickness of 40 m, a Young's modulus (material stiffness) of 2 x 108 dyn cm-2, and a density of 1 g cm-3, as in our earlier models. 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. The damping in the system is represented by a mass-proportional damping coefficient of 1500 s-1 (Funnell et al., 1987). The acoustical input is a uniform sound pressure of 100 dB SPL. (It is actually modelled as a step input which is then Fourier transformed.)

3. Overall vibration patterns

Figures 2 and 3 show the overall vibration patterns on the eardrum model at 2 and 4 kHz. The instantaneous 3-D displacement at each node of the finite-element mesh is represented by an arrow. (The lengths of the arrows are greatly magnified, of course.) For each frequency, the displacements are plotted for the particular point in the cycle at which the maximal eardrum displacement occurs. Note that in general the displacement vectors are not vertical, that is, they are not parallel to the z axis.

Available experimental measurements include only a single component of the displacement, so the remaining figures in this poster include only the z component. Figure 4 presents vibration patterns for six frequencies between 2 kHz and 4.5 kHz. The green lines are contours of constant displacement magnitude; phase information is not visible. For frequencies below about 1.8 kHz, all points on the eardrum vibrate in phase. At 2.0 kHz out-of-phase displacements appear in the posterior part of the drum and the vibration patterns become progressively more complex. Displacements in the anterior region gradually become larger but remain in phase until about 4.0 kHz.

4. Displacement profiles

To permit examination of the displacement patterns in more detail, they can be plotted as profiles along individual section lines. Figure 5 presents a displacement profile along a line approximately one third of the way up the manubrium from the umbo (as shown by the gray line in Figure 1). Again, only the z component of the displacement is plotted. The green lines show the displacements of elements superior to the cross-section line. The blue line shows the displacements along the line of section; the gap indicates the location of the manubrium. These displacements are those occurring at one particular instant during the cycle.

To give an impression of the time courses of such vibrations, Figures 6, 7 and 8 show sets of displacements along the same line of section for 2, 4 and 8 kHz. For each frequency, displacements are plotted at twelve evenly spaced phase values (0, 30, 60, etc.). At 2 kHz the displacements along this line are all more or less in phase. At 4 kHz the motion in the anterior region is still approximately in phase, but the motion in the posterior region has become quite complex. The displacements at 8 kHz are even more complex and would probably require a finer mesh to be accurately modelled.

For a representation comparable with that produced by time-averaged methods, displacement profiles for multiple phase values throughout the cycle can be superimposed. Figure 9 shows such superimposed profiles for six frequencies from 3.9 to 4.4 kHz. The patterns gradually evolve as the frequency changes, with the regions of maximal displacement shifting and changing their relative amplitudes. Note that a true nodal point, that is, a point which has zero displacement throughout the cycle, is present only at 4.2 kHz, in the posterior region.

5. References

1. Decraemer, W.F., Khanna, S.M., & Funnell, W.R.J. (1989): Interferometric measurement of the amplitude and phase of tympanic membrane vibrations in cat. Hear. Res. 38: 1-17

2. Decraemer, W.F., Khanna, S.M., & Funnell, W.R.J. (1997): Vibrations of the cat tympanic membrane measured with high spatial resolution. ARO Midwinter Meeting

3. Funnell, W.R.J., & Laszlo, C.A. (1978): Modeling of the cat eardrum as a thin shell using the finite-element method. J. Acoust. Soc. Am. 63: 1461-1467

4. Funnell, W.R.J. (1983): On the undamped natural frequencies and mode shapes of a finite-element model of the cat eardrum. J. Acoust. Soc. Am. 73: 1657-1661

5. Funnell, W.R.J., Decraemer, W.F., & Khanna, S.M. (1987): On the damped frequency response of a finite-element model of the cat eardrum. J. Acoust. Soc. Am. 81: 1851-1859

6. Funnell, W.R.J., & Decraemer, W.F. (1996): On the incorporation of moiré shape measurements in finite-element models of the cat eardrum. J. Acoust. Soc. Am. 100: 925-932

7. Khanna, S.M., & Tonndorf, J. (1972): Tympanic membrane vibrations in cats studied by time-averaged holography. J. Acoust. Soc. Am. 51: 1904-1920