Vibrations of conductive string with moving load in magnetic field

N.F.Kurilskaya

Altay State Technical University of I.I.Polzunov name

Lenin st., 46, Barnaul, 656000, Russia

It is considered transversal vibrations of absolutely flexible conductive string in homogeneous stationary magnetic field directed perpendicular to vibrations' plane. The string is manufactured of nonmagnetic conductive material (diamagnetic or paramagnetic). The ends of string are fixed and connected by ideal electrical circuit, isolated from external magnetic field. Under vibrations electromotive force appears taking into account the motion of the string inside active part (part of the string where magnetic field acts). As a result distributed forced load is created. This load acts to the string inside active part and depends from the string' motion. So we obtain equation of vibrations of the string in magnetic field as integro-differential equation in partial derivatives. Integral summand in equation of vibrations of the string characterizes action of magnetic field and depends from the function of displacement of the string.

In the paper it is investigated the character of magnetic action to the string's characteristic vibrations. Usually for solution problems linked with the motion of systems with distributed parameters as strings and beams generalized coordinates are used. But the presence of integral summand in equation of the string's vibration in magnetic field changes properties of this equation. In particular introduced by usual way generalized coordinates are not principal, so we obtain another approach for solution the problem we consider. This approach is based on using the form of initial equation of the string's vibration so for such equations it is necessary to modify classic method of division of variables a little. At that we essentially use the fact that magnetic field is stationary. It admits to obtain the ordinary differential equation with constant coefficients for determination of complementary function including in common form of solution and depending only from the time.

By using of this approach selective character of this action is demonstrated very easy. It is shown the existence of isolated vibrations not exposing to magnetic damping. Frequencies of these vibrations are determined. Restrictions to the sizes guaranteeing the opportunity of existence of isolated vibrations are found (in case when active part is only the part of the string). By modified method of division of variables expressions for function of displacement of the string are obtained in common case for non-isolated vibrations in complex form. In case of small external resistance these expressions are written in real area. Action of magnetic field leads to appearing of two-waved processes in it. Besides we can observe transformation of amplitude forms of vibrations and decreasing of amplitudes of non-isolated vibrations provided by the action of considered homogeneous stationary magnetic field.

The cases of action of magnetic field to the whole string and to the part of it are considered. It is significant necessity of consideration of the string as composite only in case of non-isolated vibrations. In last case important role for determination of the form of function of displacement of the string conditions of patch play. Imagination of the string as a composite one admits us to show the influence of magnetic field to the motion of the whole string by way of using conditions of the patch despite magnetic field acts only on active part of the string.

It is also shown that condition of existence of isolated vibrations of conductive string in magnetic field is the same also in case when hypothesis of absolutely flexibility of the string doesn't take place (in last case equation of the string's vibration essentially differs from equation љof the string's vibrations we considered above).

It is noted that analogous solutions we can obtain in cases if we have some active parts of the string. It is necessary only to take into account correspondent conditions of patch of solutions. These conclusions apply to case of different magnetic induction on different active parts under condition that magnetic field is homogeneous and stationary within every active part.

In the paper it is considered the problem of interference of vibrations of conductive string and motion mobile object on it. Mobile object moves as material point. The law of changing of coordinate of point is given. The cases of motion of mobile load with constant and variable velocity are discussed. Motion of mobile object on the string is complementary disturbing factor of vibration's equation. It is investigated the opportunity of damping of disturbances from the motion of mobile mass of vibrations of the string by magnetic field. At that equation of the motion of mobile load is considered as equation of relative motion taking into account projections of corresponding forces of inertia and external forces to perpendicular to direction of the string's vibrations.

Variants of action of disturbances from mobile load to isolated and non-isolated vibrations of the string are considered separately. Mobile load can be electrically neutral and have electric charge too. These variants are investigated separately. It is shown that if mobile mass is non-conductive then it is impossible to damp disturbances of isolated string's vibrations by magnetic field because there are no characteristics of magnetic field in equations of the string's vibrations and equation of relative motion of the load. But when mobile mass has electric charge and external resistance is small so, then in case constant relative velocity of mobile load and constant external force acting to mobile loadљ it is possible to damp disturbances if isolated vibrations љby corresponding choice of the value of magnetic induction. These conditions are right as for case when active part coincides with the whole string as when active part occupies only the part of the string.

Also conditions of whole and partial damping of disturbing action of mobile load to isolated vibrations of the string by magnetic field are founded in case when external force acting to mobile load changes on harmonic law and external resistance is small. These conditions establish a connection between frequency of external force, coefficients of equation of the string's vibration and relative velocity of mobile load (here it was considered only the case of constant relative velocity).

It is also obtained by using approximate method condition which states opportunity of damping of disturbances of non-isolated vibrations by magnetic field. It hold true only if external force acting to the mobile mass (in case the last one hasn't electric charge) or the sum this external force and force acting to mobile mass from magnetic field (in case the last one has electric charge) we can imagine in form of harmonic series on principal non-isolated vibrations with amplitudes depending from principal frequencies of non-isolated vibrations and value of magnetic induction for small external resistance. It is noted that we can consider condition founded form of function of external force as equation for determination of coefficient of initial equation of the string's vibrations playing the role of Stuart number for this problem and charactering relation of magnetic force acting to the string to the force of inertia of the string.