The Motion Equation of a Spring-Magnet-Mass System Placed in Nonlinear Magnetic Field. An Analytical Solution of Elliptic Sine form Functions

In this work, we are studying about a special oscillator system, which consists of one spring and a magnet-mass. The system is placed in nonlinear magnetic field, produced by two other permanent magnets, which are oriented for attraction, where can appear different types of oscillations. The magnet-body is simultaneously the subject of the linear field of spring and also of the nonlinear magnetic field of permanent magnets which has inverse quadratic dependence on distance. We are studying the ideal case, without friction, where the oscillations are produced with energy conservation, the oscillator system is started by applying the initial impulse and we consider the hypothesis that magnetic field produced by the permanent magnets is conservative and there is no loss of energy in the magnetic interactions. We are going to find the law of motion for the general case of study and a typically numerical application will be done.