Forced vibrations of size-dependent rods subjected to: impulse, step, and ramp excitations

Abstract

Forced longitudinal vibration response of nonlocal strain gradient (NLSG) rods is scrutinized and compared with nonlocal (NL), strain gradient (SG), and classical (CL) rods. In this respect, size-dependent kinematics and extended Hamilton’s principle are utilized to derive governing equations of motion. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods is presented including three different types of non-harmonic forcing functions: unit impulse, unit step, and unit ramp. Forcing terms are functions of both temporal and spatial domains. Nonlocal strain gradient and nonlocal rod models take into account the Laplacian of the external forcing term as well. This implies that in models including nonlocal effects (NL, NLSG-h, s, n), forcing term is reduced by the order of its Laplacian with respect to spatial variables. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark for the sake of evaluation. This implies the necessity of adopting proper values of non-classical theory and corresponding parameters through analysis of forced oscillations of small-scaled size-dependent rods. Such external excitations are modeled using spatial Dirac delta function as a concentrated force applying at the critical point of mode shape function. Findings of this research reveal the underlying impact of non-classical theorems along with corresponding non-classical parameters over the prediction of the response of size-dependent rod elements for wide range of applications such as in musical instruments, navigating equipment, gyroscopes, control of robotics, sensors and actuators, resonators, energy harvesters, and vibration control systems.