Analytical solutions of a nonlinear two degrees of freedom model of a human middle ear with SMA prosthesis

A nonlinear two degrees of freedom system with shape memory element (SMA) is presented in the paper. Thermomechanical properties, shape memory effect and pseudoelasticity of the SMA joint, are modelled by phenomenological approach with the help of a fifth order polynomial approximation. The analytical solution of the mathematical model is obtained by means of the multiple time scales method. Next, a numerical study of a biomechanical system of human middle ear is performed. It has been demonstrated that for a selected rod type prothesis the cubic nonlinearity has the most essential effect on the system dynamics in the first order approximation, which converges with direct numerical simulations.