Effective-Spring Model of Tympanic Response in Archosaurs

Abstract

Whereas for smaller animals the eardrums are well-characterized as excitable membranes or drums, some animals such as several archosaurs feature, as a first approximation, a rather stiff elastic shell supported by an elastic ring. Mathematically, the theory of plates and shells is applicable but its governing equations overly complicate the modeling. Here the notion of tympanic structure is introduced as a generalization of "ordinary" tympanic membranes so as to account for sound perception as it occurs in archosaurs, such as birds and crocodilians. A mathematical model for the tympanic structure in many archosaurs called two-spring model implements this notion. The model is exactly soluble and solutions are presented in closed form and as a series expansion. Special emphasis is put onto offering an easy-to-apply model for describing experiments and performing numerical studies. The analytic treatment is supplemented by a discussion of the applicability of the two-spring model in auditory research. An elasticity-theoretic perspective of the two-spring model is given in the Appendix.

Model Problems on Oscillations of Mechanical and Biological Membranes

“…The general provisions connected with the middle ear and drum membrane mechanics are described in the overview work [23], where it is demonstrated that the drum membrane can be interpreted as a mechanical membrane. In [24], the authors propose the system of two ordinary differential equations as a model of drum membrane oscillations, and the solution of this system is defined by an integral operator. The drum membrane model was examined and the probabilistic estimate of breakout risk as an explosion result was calculated in [25].…”

“…In article [24], the classical equation of forced harmonic vibrations without friction is considered as a simplified vibration model. So, if in our case for Equation (1) we take Lu = F f ull and assume that the solution u depends only on time, then for α 2 (t) = m, β(t) = 0, γ 2 (t) = k, we exactly obtain Equation (8) from this work.…”