Exact Solution for the Free Vibration Response of Beams Resting on Viscoelastic Foundations, Taking Flexoelectricity and Temperature into Account

PurposeThis is the first investigation to employ an exact solution to explore the free vibrational behavior of beams under the impact of flexoelectricity and temperature when the beam is maintained by a viscoelastic foundation and subjected to axial compression.MethodsThis study makes use of the analytical solution.ResultsThis study provided a solution in the form of an equation for estimating the natural frequency of oscillations of a beam resting on a viscoelastic substrate while accounting for the flexoelectric effect. This is the equation used to verify the accuracy of various approximation methods (FEM, isogeometric approach, etc.), and it is simple to use in reality to estimate the nanobeams' natural frequency quickly. In addition, the study findings provide a foundation for tackling more difficult issues and aid engineers in the design, production, and practical use of nanobeams.ConclusionThe article presented the exact solution to study the specific vibration of the beam, while it was resting on theviscoelastic medium while considering influences of the flexoelectricity. In this solution, effects of temperature and the influence of mechanical compressive force acting along the beam axis are both taken into account. The calculation formulae for beams are taken from the classical beam theory. When viscoelastic foundations are included, the particular vibrational response of the beam will comprise both real and imaginary parts. This discovery is quite fascinating, and it provides a platform for additional research into the mechanical reaction of beams when they are supported by a viscoelastic foundation.