Dynamic stability of the euler nanobeam subjected to inertial moving nanoparticles based on the nonlocal strain gradient theory

Abstract

This research studied the dynamic stability of the Euler-Bernoulli nanobeam considering the nonlocal strain gradient theory (NSGT) and surface effects. The nanobeam rests on the Pasternak foundation and a sequence of inertial nanoparticles passes above the nanobeam continuously at a fixed velocity. Surface effects have been utilized using the Gurtin-Murdoch theory. Final governing equations have been gathered implementing the energy method and Hamilton's principle alongside NSGT. Dynamic instability regions (DIRs) are drawn in the plane of mass-velocity coordinates of nanoparticles based on the incremental harmonic balance method (IHBM). A parametric study shows the effects of NSGT parameters and Pasternak foundation constants on the nanobeam's DIRs. In addition, the results exhibit the importance of 2T-period DIRs in comparison to T-period ones. According to the results, the Winkler spring constant is more effective than the Pasternak shear constant on the DIR movement of nanobeam. So, a 4 times increase of Winkler and Pasternak constants results in 102 % and 10 % of DIR movement towards higher velocity regions, respectively. Furthermore, the effect of increasing nonlocal and material length scale parameters on the DIR movement are in the same order regarding the magnitude but opposite considering the motion direction. Unlike nonlocal parameter, an increase in material length scale parameter shifts the DIR to the more stable region.