Rheologic Fractional Oscillators or Creepers

Abstract

Using the newly introduced, by the author, basic complex and hybrid complex rheologic models of the fractional type, the dynamics of a series of mechanical rheologic discrete dynamical systems of the fractional type (RDDSFT) of rheologic oscillators (ROFTs) or creepers (RCFTs), with corresponding independent generalized coordinates (IGCs) and external (IGCEDF) and internal (IGCIGF) degrees of freedom of movement, were studied. Laplace transformations of solutions for independent generalized coordinates (IGCs), as well as external (IGCEDFs) and internal (IGCIDF) degrees of freedom of system dynamics, were determined. On the studied specimens, it was shown that rheologic complex models of the fractional type introduce internal degrees of freedom into the dynamics of rheologic discrete dynamical systems. New challenges appear for mathematicians, such as translating the Laplace transformations of solutions for independent generalized coordinates (LTIGCs) into the time domain. A number of translations of Laplace transformations of solutions into the time domain were performed by the author of this paper. A series of characteristic surfaces of elongations of Laplace transformations of independent generalized coordinates (IGCs) of the dynamics of rheologic discrete dynamic systems of the rheologic oscillator type, i.e., the rheologic creeper type, is shown as a function of fractional order differentiation exponent and Laplace transformation parameter. This manuscript presents the scientific results of theoretical research on the dynamics of rheologic discrete dynamic systems of the fractional type that was conducted using new models and a rigorous mathematical analytical analysis with fractional-order differential equations (DEFOs) and Laplace transformations (LTs). These results can contribute to new experimental research and materials technologies. A separate section presents the theoretical foundations of the methods and methodologies used in this research, without the details that can be found in the author’s previously published works.