Page 1 of 10
Transactions on Machine Learning and Artificial Intelligence - Vol. 9, No. 5
Publication Date: October, 25, 2021
DOI:10.14738/tmlai.95.10922. Alpatov, A., Kravets, V., Kravets, V., & Lapkhanov, E. (2021). Analytical Modeling of the Binary Dynamic Circuit Motion. Transactions
on Machine Learning and Artificial Intelligence, 9(5). 23-32.
Services for Science and Education – United Kingdom
Analytical Modeling of the Binary Dynamic Circuit Motion
Anatolii Alpatov
The Institute of Technical Mechanics of the National Academy
of Sciences of Ukraine and the State Space
Agency of Ukraine, Dnipro, Ukraine
Victor Kravets
Department of Mechanics
Dnipro University of Technology, Dnipro, Ukraine
Volodymyr Kravets
Department of Mechanics
Dnipro State Agrarian and Economic University, Dnipro, Ukraine
Erik Lapkhanov
The Institute of Technical Mechanics of the National Academy
of Sciences of Ukraine and the State Space
Agency of Ukraine, Dnipro, Ukraine
ABSTRACT
The binary dynamic circuit, which can be a design scheme for a number of technical
systems is considered in the paper. The dynamic circuit is characterized by the
kinetic energy of the translational motion of two masses, the potential energy of
these masses’ elastic interaction and the dissipative function of energy dissipation
during their motion. The free motion of a binary dynamic circuit is found according
to a given initial phase state. A mathematical model of the binary dynamic circuit
motion in the canonical form and the corresponding characteristic equation of the
fourth degree are compiled. Analytical modeling of the binary dynamic circuit is
carried out on the basis of the proposed particular solution of the complete
algebraic equation of the fourth degree. A homogeneous dynamic circuit is
considered and the reduced coefficients of elasticity and damping are introduced.
The dependence of the reduced coefficients of elasticity and damping is established,
which provides the required class of solutions to the characteristic equation. An
ordered form of the analytical representation of a dynamic process is proposed in
symmetric determinants, which is distinguished by the conservatism of notation
with respect to the roots of the characteristic equation and phase coordinates.
Keywords: dynamic design; dynamic circuit; mathematical model; characteristic
equation; analytical solution.
INTRODUCTION
The design scheme in the form of a dynamic circuit is widely used in the development and
improvement of various types of vehicles, which include conveyor transport, multi-mass
vibratory conveyors, pneumatic transport, railway transport, road transport, etc. [1-7].
Page 2 of 10
24
Transactions on Machine Learning and Artificial Intelligence (TMLAI) Vol 9, Issue 5, October - 2021
Services for Science and Education – United Kingdom
Analytical modeling is an important stage in the dynamic design of vehicles, which precedes
computational and full-scale experiments [8]. Analytical modeling is carried out on the basis of
classical mathematical methods for solving systems of linear differential equations, operational
calculus, residue theory, modal control theory, root hodograph method, root distribution
method in the complex plane [9-13]. Dynamic processes in a single-mass system were
analytically probed in the works [14-17]. It shows the possibility of analytical modeling of the
binary dynamic circuit free motion which is described by the fourth-order canonical
mathematical model [18].
FORMULATION OF THE PROBLEM
The design scheme of the considered mechanical system is shown in Fig. 1 and is defined as a
binary dynamic circuit [19].
Fig. 1. Binary dynamic circuit
Here , are the masses of the circuit elements;
, are the elastic interaction coefficients;
, are the damping coefficients.
The initial perturbation of the binary dynamic circuit is assumed to be given:
, are the deviation of masses from natural configuration;
, are the velocities of masses deviations.
It is required to select the class of technically realizable solutions that provide analytical
modeling of the binary circuit dynamics.
Mathematical model
The motion of the binary dynamic circuit is described by a system of differential equations in
the next form:
(1)
where , are the generalized coordinates.
Assuming that
μ1 μ2 m1 m2
1c 2c
m1 m2
1c 2c
μ1 μ2
1q (0) 2 q (0)
1q! (0) 2 q! (0)
11 1 2 1 1 2 1 1 2 2 2 ( ) ( )
2 2 21 21 2 2 2 2
,
,
mq q c c q q c q
mq q cq q cq
= - μ +μ - + +μ +
=μ + -μ -
!! ! !
!! ! !
q t 1 ( ) q t 2 ( )