On oscillations of conductive string in magnetic field

N.F.Kurilskaya

Altay State Technical University of I.I.Polzunov name

Lenin st., 46, Barnaul, 656000, Russia

 

Some solutions of the problem of free and forced oscillations of conductive string in homogeneous stationary magnetic field acting on given part of string with different kinds of string's ends fixation (it is considered string with rigidly fixed ends and cases of one rigid and one elastic fixation of the ends of the string and also case of two elastic fixations) are obtained in complex form. The ends of string and connected by ideal electrical circuit. We take into account external viscous resistance and the electromagnetic force of dissipative character. Equation of oscillations describes small oscillations of the string in the perpendicular to vector of magnetic induction plate. Boundary conditions for this equation can vary in dependence from the conditions of fixation of the ends of the string.

On the oscillations of the string electromotive force appears which takes into account the motion of the string inside active part integrally. As a result distributed forced action inside active part of the string creates. It depends from the string's motion. Thus the problem leads to integro-differential equation in partial derivatives. In this equation integral member depends from unknown velocity of transversal oscillations of the string so it impossible to estimate it's value before determining of function of displacement. So presence of integral addend in initial equation of free string's oscillations transforms the problem into hidden non-linear one. However as non-linear character of the problem was coded only in the integral addend of initial equation the external form of equation of the string's vibration remains linear one. It is distinctive singularity of the problems under consideration from mathematical point of view.

We consider also forced oscillations of conductive string with fixed ends in homogeneous stationary magnetic field. In this case the generator of electrical signals with small internal resistance is switched into ideal external electrical circuit closed the ends of the string. Then except induced currents variable current of given density is created from external source of electromotive force. On interacting of this current with external magnetic field compelled electromagnetic force appears. Then it's action is exhibited as appearance of complementary (in common case given arbitrary function of time of arbitrary character) addend in initial equation of oscillations of conductive string in homogeneous stationary magnetic field taking into account external and internal factors of damping. In the process of solution it is established that no complementary restrictions to the form of function of external excitation are superimposed except boundary conditions.

It is shown that and in this case we also can obtain explicit solution in dependence from the form of function of external excitation and characteristics of external magnetic field. The only difference from solution of the problem of free oscillations of conductive string with fixed ends in stationary homogeneous magnetic is in form of addend included into required solution for function of displacement of the string depending from the time. However in this case changing of function of excitation plays very important part too. More detail studying of interaction between damping magnetic field of given character and external excitation in their influence to string's oscillations demands further investigations and can be subject of separate research. Note that in some cases choice of sizes of active part of the string and value of magnetic induction can lead to complete damping of forced oscillations and transformation forced oscillations of the string into free ones.

Modified method of separation of variables is used in the process of solution (required solution is represented as a sum of two addends, one of them depends from two variables (coordinate and time) and the other depends only from time) in all cases of oscillations of conductive string under consideration in this paper. Using complex form of writing of solution is also more convenient because it makes solution more compact. But using complex functions leads to conclusion that the processes we consider are two-waved. Last statement is in concordance with conclusions of other authors investigated such problems.

In cases of the string rigidly fixed on the left end and elastic fixed on the right end elastic fixed of both ends of the string (with the same coefficient of elasticity) in the paper we find solution for function of displacement of the string in the same form by the same method for free oscillations. Solution also is written in complex form. But changing of boundary conditions lead in these cases to transcendental form of the equations of frequencies.

In this paper it is shown that in dependence from the size of active part and boundary conditions the solutions can be find analytically as in form of unique function of two variables as in form of series. Solutions for required function of displacement have form series because according equations of frequencies give set of solutions. It is investigated the influence of magnetic field on the form of oscillations of the string and frequencies of these oscillations.

It is shown that in some cases generating solution (solution of initial equation of oscillations without integral addend characterized action of magnetic field) gives all spectrum of frequencies and at the same time action of magnetic field leads to the choice the single one and the other frequencies are damped completely. All founded solutions are obtained in suggestion of presence of some restrictions to the values of coefficients including into initial equation and sizes of active part.

Analytical form of obtained solutions is advantage of using suggested method for equations in partial derivatives contained integral addend dependent from velocity of oscillations. Usually in problems of theory of oscillations of systems with distributed parameters principal coordinates are considered. However in this paper such suggestion wasn't introduced and it expand the field of functions we can use in the process of searching solutions of equation of oscillations of conductive systems in magnetic field.