Avaliação do método de Rayleigh na determinação de velocidades críticas de eixos para geometrias iniciais de projeto

Abstract

The vibration of machine elements and structures can originate from the application of external loads, generating oscillatory movements in the system. A mechanical system has an intrinsic property known as natural frequency, which is directly related to its distribution of stiffness and inertia. If, after an initial disturbance, a system continues to vibrate without the action of external forces, the frequency that it oscillates is known as natural frequency. In shafts, the natural frequency can also be called critical speed, and whenever it coincides with a certain external excitation frequency, a phenomenon known as resonance occurs, resulting in excessive deflections and possible failures. For this reason, it is essential to determine the critical speed of shafts in machine designs, in order to identify if the working rotation is adequate. However, a complete analysis of the natural frequencies of shafts is a complicate problem, especially if the geometry is complex. By this way, the Rayleigh Method in its discrete formulation for multiple degrees of freedom proposes a simplified form for the approximate determination of the critical speed of shafts, knowing that this method is based on the conservation of mechanical energy of the system. However, it is important to evaluate the error that the Rayleigh Method provides, and for this, the Finite Element Method (FEM) or, if existing, the analytical solution was taken as a reference. FEM is a numerical method that allows the approximate solution of partial differential equations through the discretization of the system in finite elements and can be used to solve the differential equation of motion. Problems were proposed with shafts in initial design geometries, supported by rolling bearings and containing arbitrarily arranged flywheels. The resolution of the proposed problems using Rayleigh Method involves the use of different mass and stiffness considerations for the same analyzed problem. Initially, simplified models were used and, after that, complete models, that take into account all the mass and stiffness of the system. Furthermore, the same problems were analyzed using the FEM or analytical solution (if existing), considering complete models. After that, a comparison was made between the critical velocity results obtained via Rayleigh and via MEF or analytical method. It was observed that the shaft modeling influences the critical speed value obtained via Rayleigh, by the way that the values found showed differences of a few percent in relation to the reference (consistent with the expectation cited in classical literature). It is worth noting that the reference values were determined using the FEM or analytical solution.