Phenomenological mapping and dynamical absorptions in chain systems with multiple degrees of freedom

Using Mihailo Petrović's theory of mathematical phenomenology elements, phenomenological mapping in vibrations, signals, resonance and dynamical absorptions in models of dynamics of chain systems – the abstractions of different real dynamics of a chain system are identified and presented. Using a mathematical description of a chain mechanical system with a finite number of mass particles coupled by linear elastic springs and a finite number of degrees of freedom expressed by corresponding generalized independent coordinates, translator displacements and corresponding analysis of solutions for a free and forced vibrations series of multi-frequency regimes and resonant states as well as dynamical absorption states are identified. Using mathematical analogy and phenomenological mapping, analyses of the dynamics of other chain models are made. Phenomenological mapping is used to explain dynamics in systems with multiple deformable bodies (beams, plates, membranes or belts) through resonance and dynamical absorptions in the system and transfer of mechanical energies between bodies. Amplitude-frequency graphs for homogeneous and non-homogeneous chain systems are presented for a system with 11 degrees of freedom. Expressions for generalized coordinates of a chain non-homogeneous system in resonance regimes for a general case are derived. A theorem is defined and proven.