On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics

The size effects on the dynamic behaviors of rods are investigated within the framework of the nonlocal strain gradient elastic theory. The variationally consistent boundary conditions are derived by using the weighted residual method with respect to the known equation of motion of rods. Similar to the fourth-order differential equation of motion for classical Euler-Bernoulli beams, the variationally consistent boundary conditions of nonlocal strain gradient rods are clarified for the first time. The exact characteristic equations for determining the longitudinal frequencies are presented for four types of boundary value problems. To gain insight into the asymptotic behaviors of rods, the reduced boundary values problems are also analytically given. The numerical results show that both the softening effect and the stiffening effect can be captured by adjusting the two material length parameters. However, when the two material length parameters are the same, the present results cannot recover to the corresponding classical ones for rods with general boundary conditions, which are different from the previous findings reported in the literature.