Dynamic finite element analysis of axially vibrating nonlocal rods

Abstract

Free and forced axial vibrations of damped nonlocal rods are investigated. Two types of nonlocal damping models, namely, strain-rate-dependent viscous damping and velocity-dependent viscous damping, are considered. A frequency-dependent dynamic finite element method is developed to obtain the forced vibration response. Frequency-adaptive complex-valued shape functions are proposed to obtain the dynamic stiffness matrix in closed form. The stiffness and mass matrices of the nonlocal rod are also obtained using the conventional finite element method. Results from the dynamic finite element method and conventional finite element method are compared. Using an asymptotic analysis it is shown that, unlike its local counterpart, a nonlocal rod has a maximum cut-off frequency. A closed-form exact expression for this maximum frequency as a function of the nonlocal parameter has been obtained for undamped and damped systems. The frequency response function obtained using the proposed dynamic finite element method shows extremely high modal density near the maximum frequency. This leads to clustering of resonance peaks which is not easily obtainable using classical finite element analysis.