Mathematical modelling of membrane oscillatory processes in a nonlinear viscoelastic medium via the Caputo-Fabrizio fractional operator

This paper addresses the mathematical modelling of transient processes in a steel membrane oscillating in a nonlinear viscoelastic isotropic medium. For this purpose, three types of mathematical models of the object under study with different degrees of adequacy are presented. The first type of model is presented as a system with distributed mechanical parameters, the equations of state of which are derived from the modified Hamilton‒Ostrogradsky principle and represent a mixed problem. The second is presented as a system with concentrated parameters, which is a Cauchy problem. The third type of model is presented as a modification of the second type via the fractional derivative and integral theory using the Caputo Fabrizio operator. The results of computer simulations for all types of models are presented in the form of analysed figures. Comparative analysis was also carried out for all the models on a model steel membrane with a fixed tension force, which demonstrated that the application of the Caputo-Fabrizio operator to a simplified membrane model improved the degree of adequacy of the latter.