Nonlinear Oscillations of a Complex Discrete System of Rigid Rods with Mass Particles on an Elastic Cantilever

Abstract

We analyzed forced oscillations of a complex cantilever. The non-linearity of the system is introduced by a spring with nonlinear properties that oscillates in a vertical plane. The description and approximations of the system are given. The system oscillates in two orthogonal planes - horizontal and vertical with four degrees of freedom in each plane. In the horizontal plane, the system oscillates with eigen frequencies of free linear oscillations; in the vertical plane with forced nonlinear oscillations. For describing oscillatory behavior of this complex system under an external single-frequency force, influence coefficients of deflection of cantilever were used. Oscillatory behavior of this complex system in vertical plane can be described by subsystems of nonlinear differential equations that are solved using a newly introduced, generalized method of variation of constants and the method of averaging, as well as the Krilov-Bogolyubov-Mitropolyski asymptotic method of nonlinear mechanics of approximation. The presented generalized methods of constants variation, together with the averaging method, opens the possibility of studying the forced nonlinear oscillations, under the influence of external forces with different frequencies, each in the corresponding resonant range frequency interval.