Integral equations of the second kind in dynamic analysis of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material dependencies

Abstract

Introduction. Development of computational methods for nonlinear systems possesses a significant potential, considering that linear theory sometimes fails to accurately describe the properties of dynamic systems, and the linear approximation gives only a very rough idea of real processes for a number of cases.Aim. Calculating linear systems and deriving the resulting equations for nonlinear systems involves impulse response and transfer functions of linear “generating” systems of differential equations. Such an approach in comparison with the traditional method of the so-called normal forms enables the calculation algorithm to be simplified considerably, avoiding several stages and presenting the solution by means of the normal mode method for linear systems directly with respect to the generalized coordinates.Materials and methods. The paper presents a method and an algorithm developed for the calculation of nonlinear systems with a finite number of degrees of freedom under arbitrary dynamic loading and material nonlinearity. Systems of nonlinear differential equations were reduced to nonlinear integral equations of the second kind, considered as resulting equations. The solution was developed in time steps, the value of which, among other things, determines the accuracy of the solution and the nature of the computational algorithm.Results. The paper presents main computational dependencies in a generalized form, convenient for numerical simulation. The author provides solutions for a nonlinear system with one degree of freedom and a cubic reactiondisplacement relation, as well as for a system with one and two degrees of freedom with a viscous damper. In both cases, the developed solution contains all properties of nonlinear systems, including the jump (transition) from the upper ascending branch to the lower, stable one, and the associated excitation of free oscillations.Conclusions. According to the calculations, the occurrence of nonlinear effects in oscillating systems makes positive impact on their behavior, in resonant modes in particular.