Analytical Simulation of Dynamical Process in One-Dimension Task

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

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Mechanics, Materials Science & Engineering, September 2016

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rigidity; geometric parameters in natural configuration; dynamical process; kinematical influence.

External kinematic effects rely specified in the form of harmonic oscillations:

where

frequency of forced vibrations; amplitude of forced oscillations.

Required to find an analytical representation of a dynamic process, depending on the set and varying system parameters for harmonic kinematic exposure. Mathematical modelling.

A mathematical model of the problem is given in [6] and is as follows:

Matrix model in normal form. We introduce the variables:

, . .

Where the corresponding coefficients:

; ; In addition, force function while external kinematical effect:

Then the original differential equation of motion takes the following matrix notation in normal form: The analytical solution.

A mathematical model corresponds to the following analytical record of the dynamic process for any kinematic effects during given initial conditions and different existing roots of characteristic equation [2].

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Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Here the roots of the characteristic equation: associated with the dynamic parameters of the system in the form:

;

.

Due to this, harmonic kinematic effects: ,

,

power function and its derivatives take the form of: ; ; ; ; ; ; And so forth.

Respectively at initial time moment ; ; ; ; ; ; Etc. MMSE Journal. Open Access www.mmse.xyz

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Mechanics, Materials Science & Engineering, September 2016

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Where follows:

The resulting alternating series converges provided:

in addition, its sum finds in accordance with [4]:

Where follows:

Similarly, we get:

Etc.

Dynamic process is determined by three components corresponding to each of the two roots of the characteristic equation: MMSE Journal. Open Access www.mmse.xyz

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Where status:

ISSN 2412-5954

transient dynamic process due to perturbations of the initial phase of the system

unsteady dynamic process caused by harmonic external kinematic disturbance at the initial time

stationary dynamic process caused by harmonic external kinematic exposure at the current time.

The stability of the dynamic process.

Stability conditions for a dynamic process determined by the method of Lyapunov [2-3]: ; and taking into account the region of convergence obtained for the harmonic effects of a power series [4]:

,

.

Analytical modeling.

In the case of real and different roots and , satisfying the conditions of stability, transient time dynamic process tends asymptotically to zero at:

Fixed dynamic process should be presented in the form of: MMSE Journal. Open Access www.mmse.xyz

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Mechanics, Materials Science & Engineering, September 2016

where

ISSN 2412-5954

amplitude and -phase forced harmonic oscillations are known formulas:

,

where

,

Then appears a possible formulation of the problem of finding the optimal distribution of the roots of the characteristic equation, providing extreme value of the amplitude of forced harmonic oscillations and subsequent parametric synthesis of a dynamic system for optimal distribution of the roots, using variable parameters , c.

The occasion of the imaginary roots <1 corresponds to a dynamic system stability boundary and the region of convergence of a power series with the harmonic action:

The components of the dynamic process of the following form is used:

Dynamic process is a superposition of his own harmonic oscillations with a frequency of and forced harmonic oscillations with a frequency : .

where here

;

;

;

,

.

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Then analytical modeling of the dynamic process is conveniently carried out by vector (phase) diagrams [7], introduced by rotating vectors corresponding to forced vibrations:

and natural oscillations:

where

,

unit vectors defined as:

Vector resultant of the dynamic process is defined as the vector sum of:

where

the resulting vector unit;

where ,

unit vector:

frequency and phase of the resulting dynamic process.

The phase state of a dynamic system is located on the resulting vector using the scalar product of the form:

By this method, it is possible to analytically model the dynamic process in a wide range of natural frequencies, including the heartbeat mode ( ), the resonance mode ( ).

Summary. For analytical modeling of the dynamic process in the problem of vibration isolation uses a new form of analytical solutions of inhomogeneous systems of linear differential equations, wherein the ordering and recording relatively conservative index characteristic equation root. When a harmonic external kinematic effects of established conditions for the stability of a dynamic process linking the external oscillation frequency and magnitude of the characteristic equation roots. Analysis of stable oscillatory dynamic process proposed to carry out the method of vector diagrams. Tasks dynamic design system analytically solvable by the distribution of the roots.

References [1] Lobas, L.G., Lobas, Lyudm.G. Theoretical mechanics. Kiev: DETUT, 2009 p.407. (in Russian)

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[2] de Oliveira, L.R. & de Melo, G.P. J Braz. Soc. Mech. Sci. Eng. (2016) 38: 59. doi:10.1007/s40430015-0413-6 [3] Ovchinnikov P.P. Visha matematika [Higher mathematics], part 2, Kyiv, Technika Publ., 2000, p.797 (in Ukrainian).

Kalamkarov, A.L. Nonlinear Dyn (2013) 72: 37. doi:10.1007/s11071-012-0688-4 [5] Ervin, E.K. & Wickert, J.A. Nonlinear Dyn (2007) 50: 701. doi:10.1007/s11071-006-9180-3 [6] Kravets Victor V., Bass Konstantin M., Kravets Tamila V. & Tokar Lyudmila A. (2016). Analytical Modeling of Transient Process In Terms of One-Dimensional Problem of Dynamics With Kinematic Action. Mechanics, Materials Science & Engineering, Vol 2. doi:10.13140/RG.2.1.4017.0005 Cite the paper Kravets Victor V., Kravets Tamila V., Fedoriachenko Serhii A. & Loginova Anastasia A. (2016). Analytical Simulation of Dynamical Process in One-Dimension Task. Mechanics, Materials Science & Engineering, Vol 6. doi:10.13140/RG.2.2.20337.34347

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