On the external and internal resonance phenomena of the elastic bodies with the complex oscillations

Abstract

Complex nonlinear oscillations in the elastic bodies are studied using a priori information about the oscillations form and taking into account a refined mathematical model of the second (other) form of oscillations. Application of existing methods or development of the new ones for the analysis of received non-autonomous boundary value problems is proposed. The effectiveness of the practical implementation of the discussed methodology significantly increases in cases where the magnitude of the elastic body displacements due to the one form of oscillations is much higher than the other one. To analyze the problem one can use the well-known tested analytical methods for the systems with the small nonlinearity. Torsional and bending oscillations of the elastic body are shown as the example. It is also demonstrated that especially dangerous resonant processes can be caused not only by the external perturbations but also by the internal influence between some forms of oscillations. The obtained results allow to choose the basic technological and operational parameters of the machine oscillating elements in order to avoid the resonance phenomena.

Method of normal oscillations and substantiation of the choice of parameters for certain nonlinear systems with two degrees of freedom

“…The solution of the nonlinear system of differential equations (4) will be sought using the method of normal oscillations [23][24][25][26][27][28]. According to its main idea, the unknown functions ϕ(t) and z(t) must be connected by the relation z(t) = λϕ(t).…”