Approximate Analytical Solutions of Creeper Type Fractional Maxwell–Faraday Systems with Piezoelectric Effects

We present newly derived approximate analytical solutions of the motion, free and forced modes of dynamics of two viscoelastic rheological Maxwell-Faraday discrete dynamic systems, creeper type, and fractional type, with piezoelectric polarization property of the Faraday piezoelectric element. New theoretical analytical scientific results of research into the creep dynamics of basic rheological Maxwell-Faraday discrete dynamic systems of fractional type, with piezoelectric efects, and with two degrees of freedom of motion, one external and one internal degree of freedom of motion, are presented. Rheological complex Maxwell-Faraday discrete dynamic systems of fractional type with piezoelectric polarization effect consist of standard light-binding rheological Maxwell-Faraday ensembles fractional type and rigid bodies moving in translation. They always occur in pairs, rheological discrete dynamic systems depending on the order of sparse coupling of rheological basic light elements in standard light-binding structures and their connections with the rigid body and the fixed point. They have one external degree of freedom related to the degree of freedom of motion of the rigid body and one internal degree of freedom of motion related to the internal degree of freedom of motion of the light-binding rheological assembly itself. For the dynamics of two models of a rheological discrete dynamic system, fractional type, with the opposite order of the order of the basic rheological elements, Hooke's ideally elastic, Faraday's piezoelectric and Newton's viscous fluid of fractional order, corresponding systems of ordinary differential equations of fractional order have been written. The systems have a fractional type of dissipation of the total mechanical energy of the system and the properties of normal stress relaxation. They also have the property of electric polarization, during mechanical deformation, of a rheological Faraday piezoelectric element. Approximate analytical expressions are determined, in the time domain, for independent generalized system coordinates, which correspond to the external and internal degrees of freedom of rheological creep of each of the discrete dynamic systems with opposite binding of the elements of the standard light rheological binding basic complex Maxwell- Faraday model. To obtain the inverse Laplace transforms of the independent generalized coordinates of the creep dynamics of each of the models of rheological discrete dynamic systems, the expansion in power order by the parameters of the Laplace transform was used, and with their help the transition to the time domain.