Fayos J., Rovira A., Baeza L., Carballeira J.

In this work, the non damped vibration of an elastic shaft that rotates about its longitudinal axis is studied. A non linear model that couples axial, torsional and flexural vibrations is obtained. The equations of motion are analytically solved for the model by applying supposed modes method after linearisation of the initial equations. The main contribution of this paper consists in that the solution is then written in an adimensional form, showing the most general shape of the free vibration response orbit to be a hypotrochoid curve. Such a curve is defined by means of five parameters that can be written as function of initial conditions and a unique characteristic constant defining the system. This constant is found to be exclusively dependent on shaft slenderness (adimensional geometry) and angular velocity to eigenfrequency ratio (adimensional angular velocity). Forced response under harmonic excitation is also obtained, showing a coupling between both planes flexural modes. The response is found to only depend on the characteristic constant and the excitation frequency to eigenfrequency ratio (adimensional excitation frequency).