Oscillators: Phenomenological Mappings and Analogies First Part: Mathematical Analogy and Chains

 

Katica Stevanović-Hedrih

Ana Ivanović-Šašić

Julijana Simonović

Ljiljana Kolar-Anić

Željko Čupić

 

New analytical and numerical results of dynamics for both linear and nonlinear system with two degrees of freedom are presented. For a mechanical chain system with two degrees of freedom, oscillations are investigated analytically and numerically with corresponding comparing between free and forced oscillatory dynamics of linear and nonlinear system. Also, energy analysis and transient energy between the mass particles in the system are discussed. Using Mihailo Petroviċ’s theory of  the mathematical phenomenology elements, phenomenological mappings in vibrations, signals, resonances and dynamical absorptions in models with two degrees of freedom – the abstractions of a different real system dynamics are identified and presented. Mathematical description of a chain mechanical system with two mass particles coupled by linear and nonlinear elastic springs and with two degrees of freedom is given. By analysis of corresponding solutions for free and forced vibrations, series of related two-frequency regimes and resonant states, as well as dynamical absorption states, are identified. Besides, by mathematical analogy and phenomenological mappings, the analysis of series of dynamics of other two degrees of freedom models dynamics (torsional system, double pendulum system, double electrical circuit) is performed

 

Key words: nonlinear dynamics, oscillations, free oscillations, forced oscillations, oscillator.