Nonlinear Free Vibration of Postbuckled PFG Micro/nanotubes in Presence of Internal Resonances via Nonlocal Strain and Velocity Gradient Theory

This study investigates the nonlinear response of buckled porous functionally graded (PFG) size-dependent tube in the presence of internal resonances, considering 1:2 and 1:3 internal resonances between the first and second vibration modes.

The formulation utilizes the nonlocal strain and velocity gradient theory, along with the Euler–Bernoulli beam hypothesis, to derive the nonlinear partial differential equation of motion. It is assumed that material properties are gradually graded in the radial direction, and two different porosity distribution patterns are used in the radial direction. The Galerkin method is employed to obtain the system of nonlinear ordinary differential equations, which are then solved using the multiple scale method.

The study examines the effects of various parameters such as length scale parameters, porosity distribution, and material composition on the vibration behavior of postbuckled porous functionally graded size-dependent tubes in presence internal resonances.

Observation shows that in presence 2:1 internal resonance, if the damping coefficient is very small the dominant mode of vibration will be the second mode, even though the first mode is also excited. However, the presence of porosity can decrease the maximum value of the second mode response amplitude and dissipate the energy of vibration faster.