Electrical | Mechanical | Acoustical |
---|---|---|

voltage | force | pressure |

current | velocity | vol. velocity |

resistor | dashpot | mesh |

inductor | mass | tube |

capacitor | spring | volume |

Equations have same form.

Electrical | Mechanical | Acoustical |
---|---|---|

$v=iR$ | $f=Ru$ | $p=RU$ |

$v=L\frac{di}{dt}$ | $f=M\frac{du}{dt}$ | $p=M\frac{dU}{dt}$ |

$v=\frac{1}{C}\int idt$ | $f=k\int udt$ | $p=\frac{1}{C}\int Udt$ |

Block diagram of middle-ear model.

Applies to middle ears of most mammals.

Circuit model.

Note two components for eardrum.

More complex circuit model.

For example:

- moment of inertia changes if axis of rotation changes
- mass parameter for stapes changes if its mode of vibration changes
- eardrum parameters change if its vibration pattern changes

After http://xkcd.com/730/

In the finite-element method, a distributed physical system to be analysed is divided into a number (often large) of discrete elements.

The complete system may be complex and irregularly shaped, but
the individual elements are easy to analyse.

The division into elements may partly correspond to natural subdivisions of the structure.

For example, the eardrum may be divided into groups of elements corresponding to different material properties.

Most or all of the model parameters have very direct relationships to the structure and material properties of the system.

Relatively few *free* parameters ...

... **if** parameters are known *a priori*.

Model parameters:

- Shape
- Thickness
- Material properties:
- Young's modulus
- Poisson's ratio
- Mass density

- Damping
- Loads: pressures and/or concentrated loads

In this model, there are different material properties for the triangles in

- pars tensa
- pars flaccida
- manubrium

Qualitatively similar to experimentally observed patterns.

Varying mesh resolution to decide how fine a mesh is required:

- 10 elements/diameter
- 15 elements/diameter
- 20 elements/diameter
- 25 elements/diameter
- 60 elements/diameter

Results will converge monotonically if

- the element formulation is ‘compatible’,
or ‘conforming’

(e.g., Bathe K-J (1982):*Finite element procedures in engineering analysis*, Prentice-Hall, Englewood Cliffs, xiii+735 pp.; p. 167 etc.) - the geometry doesn’t change as the mesh is refined

Based on maximum-displacement values, there seems to be little advantage here to using a mesh resolution greater than about 15 elements/diameter.

But why not use highest resolution?

Balance between *required* accuracy and
*acceptable* (or reasonable) computation time.

Undamped natural frequencies and modes of vibration.

Increasing complexity of vibration pattern with frequency.

With damping, different areas of drum have different phases.

Recent gerbil model.

(Maftoon et al., 2015)

Frequency response of point on manubrium is smoother than those of points on eardrum.

Note very large phase lag at high frequencies.

(Maftoon et al., 2015)

Variation of ossicular stiffness and moment of inertia (with fixed axis of rotation)

- little effect on lowest natural frequencies

Funnell (1983)

Variation of stiffness, density and thickness of pars tensa

- large effects

Funnell (1983)

Variation of curvature and depth of cone

- large effects
- suggestion that shape is ‘optimal’.

Funnell (1983)

Parameter sensitivity analysis in recent gerbil model

- one parameter at a time

(Maftoon et al., 2015)

Sensitivity analysis showing interactions between parameters.

- Eardrum and ligaments:

little interaction - Incudomallear and incudostapedial joints:

strong interaction

(Qi et al., 2004)

Natural frequencies, and increasingly complicated mode shapes.

Spread of natural frequencies:

*f*_{10}/*f*_{1} = 9.1

Ellipse:

*f*_{10}/*f*_{1} = 6.0

Ellipse with symmetrical ‘manubrium’:

*f*_{10}/*f*_{1} = 3.8

Ellipse with asymmetrical ‘manubrium’:

*f*_{10}/*f*_{1} = 3.4

Cat eardrum:

*f*_{10}/*f*_{1} = 2.5

Damping smears the closely-space natural modes together.

Increasing the damping (left to right) smooths the curves more and more but leaves the overall levels and slopes unchanged.

Funnell et al. (1987)

Spread of response to point stimulus, with anisotropy.

Spread of response to point stimulus, without strong anisotropy.

Actual stimulus is a pressure.

Ossicular vibration is not (in general) around a fixed axis.

Depends on

- ligaments
- eardrum
- mass distribution
- bending of bone?

- Importance of model predictions
- Interaction between modelling and experiment

Funnell WRJ, Khanna SM & Decraemer WF (1992):
On the degree of rigidity of the manubrium
in a finite-element model of the cat eardrum.
*J. Acoust. Soc. Am.* **91**(4): 2082-2090
(doi:10.1121/1.403694)

Very thin bony connection

Shrapnell HJ (1833): On the structure of the os incus.
*London Medical Gazette* **12**: 171–173

Funnell WRJ, Siah TH, McKee MD, Daniel SJ & Decraemer WF (2005):
On the coupling between the incus and the stapes in the cat.
*JARO* **6**(1): 9-18
(doi:10.1007/s10162-004-5016-3)

Experiments by Cheng et al. (2007) on strips of eardrum:

- Uniaxial stress-strain tests
- Relaxation tests

Ogden hyperelastic model fitted to loading curves. $$W=\sum _{i=1}^{N}\frac{2{\mu}_{i}}{{\alpha}_{i}^{2}}({\lambda}_{1}^{{\alpha}_{i}}+{\lambda}_{2}^{{\alpha}_{i}}+{\lambda}_{3}^{{\alpha}_{i}}-3)$$

Motallebzadeh et al. (2013)

Prony series used to fit relaxation curves. $$G(t)=1-\sum _{i=1}^{N}{g}_{i}(1-{e}^{-t/{\tau}_{i}})$$

Motallebzadeh et al. (2013)

After iterative process to adjust parameters to fit loading curves while taking relaxation into account, model fits both loading and unloading curves with same parameters.

Motallebzadeh et al. (2013)

Simulation of tympanometry

Choukir, 2017

Maftoon et al. (2015):
Finite-element modelling of the response of the gerbil middle ear
to sound. *JARO* 16(5): 547–567

BMDE-501 Modelling middle-ear mechanics

R. Funnell Last modified: 2020-11-11 12:48:08