Forced vibration analysis of non-local strain gradient rod subjected to harmonic excitations

Abstract

In this technical paper, forced longitudinal vibration response of non-local strain gradient (NLSG) rods is obtained and compared with that of non-local (NL), strain gradient (SG), and classical (CL) rods. To this end, size-dependent kinematics along with extended Hamilton’s principle are used to derive governing equations of motion. Classical clamped-free and non-classical clamped-forcing free-strained boundary conditions are adopted. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods subjected to harmonic external loading is presented. In investigation process, effects of non-classical parameters over mechanical impedance, resonance frequency, mechanical frequency response function (FRF) (transmissibility), time period, and time-domain response are determined. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark approving good level of accuracy, disclosing the intensifying effects of material length scale parameter over natural frequency as well as mechanical impedance and resonance frequency. Meanwhile, non-local parameter leads to decrement of natural frequency, mechanical impedance and resonance frequency. Furthermore, it is realized that crest value of transmissibility is not predictable based on frequency and/or impedance variations with respect to non-classical parameters; and requires separate analysis for individual cases. In other words, a pattern referring to gradations of natural and resonance frequencies, mechanical impedance and non-classical parameters and rod models is detectable; however, such patterns do not hold for estimation of mechanical frequency response function (transmissibility peak value). Moreover, stiffness-softening and stiffness-hardening effects are the most impressive and influential factors over transmissibility peak values which can be highly important for energy harvesting rod analysis. Findings of this research point to the importance of proper non-classical theorems along with corresponding non-classical parameters to predict the response of small-scaled rods for wide range of applications in control of robotics, musical instruments, navigating equipment, space wheels, fly wheels, gyroscopes, sensors, actuators, resonators, vibration isolators and energy harvesters.