Dynamics of a middle ear with fractional type of dissipation

Abstract

In this paper, a model of spatial motion of the ossicular chain described as a system of two rigid bodies connected to the temporal bone through the system of particularly chosen massless viscoelastic rods was proposed. Several assumptions regarding the relative motion between incus and malleus, external loading force, distributed pressure of the perilymph on the stapes baseplate behind the oval window, and supporting ligaments were made. In order to avoid merging with fractional partial differential equations, the dissipation of energy due to the deformation of the eardrum is taken into account through deformation of its radial fibers while simple shear deformation pattern of the stapedial annular ligament was replaced by uniaxial deformation of viscoelastic rods. By use of the Gibbs–Appel approach and the complementary constitutive axioms corresponding to the fractional Kelvin–Zener model of viscoelastic body, the equations of motion were derived. The Cauchy problem given in terms of coupled fractional differential equations was transformed in the equivalent integer order form and solved numerically by standard numerical procedures. These results, obtained by means of the Atanackovic–Stankovic expansion formula, were compared with the ones received by use of Laplace’s transform and its numerical inversion. As a principal novelty, this model uses both fractional calculus and recently reported results on mechanical tests performed on human middle ear tissues. Thus, it can be used for either predicting of the middle ear behavior in normal and pathological conditions or simulations preceding implants design within restoration of the hearing function.