Model of deformations of a Stieltjes string system with a nonlinear condition

Abstract

In the present paper we study a model of deformations for a system of n Stieltjes strings located along a geometric graph-star with a nonlinear condition at the node. The corresponding boundary value problem has the form 'Equation Presented' Here the functions ui(x) determine the deformations of each of the strings; Fi(x) describe the distribution of the external load; pi(x) characterize the elasticity of strings; Qi(x) describe the elastic response of the environment. The jump ΔFi(li) is equal to the external force concentrated at the point li; the jump ΔQi(li) coincides with the stiffness of the elastic support (spring) attached to the point li. The condition Σni=1pi(+0)u′i(+0) ∈ N[-m,m]u(0) arises due to the presence of a limiter in the node represented by the segment [-m,m], on the movement of strings under the influence of an external load, thus it is assumed that |u(0)| ≤ m. Here N[-m,m]u(0) denotes the normal cone to [-m,m] at the point u(0). In the present paper a variational derivation of the model is carried out; existence and uniqueness theorems for solutions are proved; the critical loads at which the strings come into contact with the limiter are analyzed; an explicit formula for the representation of the solution is presented.