Free transverse vibration analysis of Non-uniform Nano-Timoshenko beams rotating around two perpendicular axes using differential quadrature method

Abstract

Abstract This research examined the vibrational characteristics of nano-rotors using Timoshenko beam theory, Eringen’s non-local elasticity theory, and gyroscopic effects. The governing equations and boundary conditions for whirling analysis were formulated for a nano-rotor with a non-uniform cross-section revolving around two orthogonal axes. The equations were converted to a non-dimensional form by introducing dimensionless parameters and then solved numerically using the differential quadrature technique. Numerical results validated the precision and convergence of the proposed solution. The findings indicate that, in the absence of rotation, the forward and backward vibration frequencies of the nano-rotor were similar. Nevertheless, once the rotor began rotating along its longitudinal axis, this symmetry was compromised. The forward frequencies increased, whilst the backward frequencies decreased. The frequency divergence became more pronounced with increasing rotational velocity, leading to higher critical velocities. Besides rotational effects, structural characteristics considerably influenced the vibrational response. Augmenting the shaft radius elevated both forward and backward frequencies, as well as critical velocities, illustrating that geometric alterations may be used to optimize rotor dynamics. Moreover, non-local elasticity increased the scale-dependence of phenomena. Increasing non-local parameters marginally increased the frequencies of the first vibration mode, while substantially decreasing those of higher modes, highlighting the nanoscale dependence of vibrational properties. The research indicated that the changes in the rotor’s diameter profile significantly affected its vibrational characteristics. A more pronounced drop in diameter along the rotor’s length increased first-mode frequencies and critical velocities, while diminishing those of higher modes. A greater rotor diameter increased the frequency difference between forward and backward modes, underscoring the significance of geometric design in regulating vibrational properties.