Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

Abstract

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Automobile information modeling based on nonlinear multisensor coupling

“…The vehicle response is calculated by using the three-way coupling vehicle model and the traditional semi-vehicle ride comfort model. See Table 1, figure 2 for the comparison of the response amplitudes of the two models [6][7] . From the analysis of Table 1, it is known that compared with the semi-car model, the acceleration of the car calculated by the three-way coupling model is larger and there is a phase lead; In vertical vibration, the three-dimensional coupling effect has the greatest influence on the car body, the second influence on the cab and the least influence on the unsprung weight.…”

Size dependent effects of two phase viscoelastic medium on damping vibrations of smart nanobeams: an efficient implementation of GDQM

“…Thus, it should be mentioned that the axial force corresponding to the PMNB with a moveable end should be equal to the force applied through the fixed boundaries to maintain the length of the nanobeam at first state, and also, leads to compress or to tension the nanobeam by the value of 2b (eV 0 + qΩ 0 ). Hence, to achieve the local/nonlocal axial force generated from the piezoelectric and magnetic strains, it is important to use the relation between the longitudinal displacement at the end of a nanobeam and constant axial end load, whose proof is available in [59,61]. So, the sizedependent piezoelectric and magnetic loads can be respectively written on the basis of the two-phase theory as follow…”

“…In this work, they presented a local/nonlocal locking-free FEM model to compare the results. Nonlinear nanobeams, using two-phase elasticity, has been investigated, and the correct procedure of utilizing Galerkin method introduced [59]. Also, dynamic stability of Timoshenko twophase nanobeams which is made of viscoelastic FG porous materials studied by Behdad et al [60] which showed that the differential nonlocal theory is unable to perform stability analysis of thick nanobeams under a periodic axial load.…”

Computational analysis of thermoelastic vibrations of functionally graded nonlocal nanobeam excited by thermal shock

“…Hence, the nonlocal theory provides a global idea that is applicable to various realistic problems. However, only a few investigations concerning nonlocal effects have been reported by the scientific community (Abouelregal and Tiwari, 2022;Fakher and Hosseini-Hashemi, 2021;Tiwari et al, 2021bTiwari et al, , 2022Tiwari and Kumar, 2022). Peng et al (2021) provided a nonlocal thermoelastic analysis of a functionally graded material (FGM) nanobeam.…”