Analytical Simulation of Dynamical Process in One-Dimension Task

We consider a one-dimensional dynamical system, characterized by the kinetic energy of the translational motion, the potential elastic energy and dissipative function. The system is subjected to external kinematic effects of a harmonic oscillation. It constitutes a mathematical model of the dynamic process. It is proposed to form an ordered analytical representation of a dynamic process, characterized by conservatism regarding the index of the root of the characteristic equation. The structure of the analytical solutions is determined by three components caused by the initial phase of the system state, kinematic influence of the initial time and the time-varying harmonic influence. The dynamic process is the superposition of the transition (the first two components) and establish a process corresponding to each of the roots. The qualitative analysis of the dynamic process provides, depending on the distribution of the roots of the characteristic equation including resonance and beat. Analytically solved parametric synthesis (dynamic design) providing the desired distribution of the roots in the complex plane, i.e. the required quality of the dynamic process.