The Behavior of two Coupled Mechanical Oscillators in Non-linear Fields and the Possibility of Obtaining Controlled Halts
Nicuşor Nistor *
Department of Physics, Chemistry and Environment, Faculty of Science and Environment, “Dunărea de Jos” University of Galaţi, Romania.
Constantin Gheorghieş
Department of Physics, Chemistry and Environment, Faculty of Science and Environment, “Dunărea de Jos” University of Galaţi, Romania.
Nelu Cazacu
Department of Materials Science and Engineering, Faculty of Engineering, “Dunărea de Jos” University of Galaţi, Romania.
Gabriela Tudor
“Calistrat Hogaş” National College of Tecuci, Galaţi, Romania.
*Author to whom correspondence should be addressed.
Abstract
The proposed mechanical system consists of two magnetic oscillating subsystems, which are mechanically coupled. The first one consists of a spring-magnet-mass subsystem and the other one is a spring-mass subsystem. The non-linear symmetric field created by two other fixed magnets, oriented for attraction, acts only upon the first subsystem. The entire system can oscillate horizontally, without friction and without loss of energy. Oscillations occur with conservation of kinetic energy and potential energy stored in the springs. During the movement, depending on the amplitude of oscillations, the main body stops for a period and this can be controlled by the physical constants of the system. The reason for these controlled halts of oscillations is the transfer of mechanical impulse, due to both oscillators entering in the frequencies equalizing status with different initial frequencies. The frequency of the main oscillator gets synchronized with the frequency of the second oscillator, by its deeper or more superficial presence in a non-linear magnetic field, which is strongly dependent on distance. This paper will apply linear algebra transformation methods, on the R3 vector spaces, for differential equations systems, that are applicable to non-linear systems only if we consider some special conditions. The general case will be solved and the method will be verified through a numerical application.
Keywords: Nonlinear second order differential equation system, elliptic functions, 2N parametric vector space, R3 vector spaces, double oscillating mechanical system, elastic forces field, magnetic forces field, controlled halts.