Mechanics, Materials Science & Engineering, September 2016
ISSN 2412-5954
Analytical Simulation of Dynamical Process in One-Dimension Task Kravets V.V.1, Kravets T.V.2, Fedoriachenko S.A.1, Loginova A.A.1 1
National Mining University, Dnipropetrovsk, Ukraine
2
National University of Railway Transport, Dnipropetrovsk, Ukraine DOI 10.13140/RG.2.2.20337.34347
Keywords: kinematics forced vibration, mathematical model, analytical solution, dynamic design.
ABSTRACT. 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.
Introduction. The vibrations play an important role in the technics and are the subject of numerous works, generalized in the fundamental work [1]. The main tool for the study of oscillations in multidimensional nonlinear dynamical systems are physical and mathematical experiment. Particular scientific interest presents the individual analytical methods for solving the problem of non-linearity, which include the method of phase plane, a small parameter, harmonic balance, successive approximation [2, 3]. The problem of multi-dimensional linear problems is solved by classical mathematical methods using Laplace transform, residue theorem [4,5]. In this paper, a new form of analytic representation of dynamic processes, which used in [6] is applied to the problem of vibration isolation of kinematic external influence. The dynamical scheme of the vibroinsulation task while kinematical influence is provided on the fig. 1.
Fig. 1. The dynamical scheme of the kinematical influence. Where
mass; dumping coefficient; MMSE Journal. Open Access www.mmse.xyz
169