Structural analogies on systems of deformable bodies coupled with non-linear layers

Oscillators: Phenomenological mappings and analogies: Second part: Structural analogy and chains

“…Using the analytical results as well as numerical and graphical presentation of amplitude frequency graphs from paper [18] and analytical and numerical results presented in the previous part Amplitude frequency graphs and interactions of nonlinear modes for small nonlinearity, for nonlinear dynamics of chain system with two degrees of freedom, we can identify analogy between these listed results. Eigen time functions in each eigen amplitude form of coupled multi-bodies (plates, beams, belts or membranes) in a hybrid system are multi-frequency with accordance with a number of coupled bodies that correspond to the system dynamics of a chain with corresponding number of degrees of freedom.…”

“…The Reference [18] is addressed to the phenomenological mapping and mathematical analogies of the oscillatory regimes in the systems of coupled deformable bodies. The systems consist of coupled deformable bodies like plates, beams, belts or membranes with corresponding boundary conditions (see Figures 4, 5 and 6) that are connected through visco-elastic non-linear layer, modeled by continuously distributed elements of the Kelvin-Voigt type with nonlinearity of the third order.…”

“…Analysis gives us structural, mathematical and qualitative analogies based on the elements of mathematical phenomenology [26][27][28] and [29][30]. Then all results of the analysis of linear and nonlinear dynamics with graphical presentation in our paper [14]: Oscillators: Phenomenological mappings and analogies -First part: Mathematical analogy and chains -, are useful to use for explanation nonlinear phenomena [15][16][17][8][9] of corresponding eigen time function in corresponding eigen amplitude form of transversal hybrid system that consists of multi bodies coupled by a discrete continuum layer, as in References [20], [4][5][6], [11][12][13][14] and [18][19]. The chapter is addressed to the phenomenological mapping and mathematical analogies of oscillatory regimes in systems of coupled deformable bodies (plates, beams, belts or membranes).…”

“…For example, nowadays, in the research of non-linear dynamics, Smale's horseshoe mapping is used in the vicinity of the homoclinic unstable point. At the time of the computer and software tool expansion, Roger Penrose (1989), and James Glaick (1987), had similar ideas that were later applied in graphical -computer techniques [18].…”

“…References [18][19][20][21], by Mitropolyski, contain series of different asymptotic methods of nonlinear mechanics known as the Krilov-Bogolyubv-Mitropolyski asymptotic method that is applicable for investigation of the nonlinear phenomena around the known solutions of corresponding linear or nonlinear differential equations.…”