Free vibrations of elastic beams by modified nonlocal strain gradient theory

In this article, size-dependent vibrations and the stability of moving viscoelastic axially functionally graded (AFG) nanobeams were investigated numerically and analytically, aiming at the stability enhancement of translating nanosystems. Additionally, a parametric investigation is presented to elucidate the influence of various key factors such as axial gradation of the material, viscosity coefficient, and nonlocal parameter on the stability boundaries of the system. Material characteristics of the system vary smoothly along the axial direction based on a power-law distribution function. Laplace transformation in conjunction with the Galerkin discretization scheme was implemented to obtain the natural frequencies, dynamical configuration, divergence, and flutter instability thresholds of the system. Furthermore, the critical velocity of the system was evaluated analytically. Stability maps of the system were examined, and it can be concluded that the nonlocal effect in the system can be significantly dampened by fine-tuning of axial material distribution. It was demonstrated that AFG materials can profoundly enhance the stability and dynamical response of axially moving nanosystems in comparison to homogeneous materials. The results indicate that for low and high values of the nonlocal parameter, the power index plays an opposite role in the dynamical behavior of the system. Meanwhile, it was shown that the qualitative stability of axially moving nanobeams depends on the effect of viscoelastic properties in the system, while axial grading of material has a significant role in determining the critical velocity and natural frequencies of the system.

https://www.mdpi.com/1996-1944/13/7/1707/pdf

On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams

Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model

On the dynamic response of porous functionally graded microbeam under moving load

On vibrations of FG nanobeams

Eringen’s nonlocal differential model has been widely used in the literature to predict the size effect in nanostructures. However, this model often gives rise to paradoxes, such as the cantilever ...

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model:

A study of the critical strain of hyperelastic materials: A new kinematic frame and the leading order term

A novel material parameter-dependent model is proposed in this work to investigate the nonlinear vibration of a beam under a moving load within a finite deformation framework. For the planar vibration problem, the Lagrange strain is adopted and the resulting model equations for the beam are established by the Hamilton principle. Under appropriate assumptions, the coupled model equations are simplified into a nonlinear integro-partial differential equation which incorporates a material parameter and a geometrical parameter. The dynamic response of the beam is determined with the help of the Galerkin method. The solutions show that both the material parameter and geometrical parameter have the effect of reducing the amplitude of the forced vibration, increasing the speed of the fluctuation and the critical velocity for the moving load. In comparison with small deformation formulations, the effect of finite deformation herein is to reduce the vibration amplitude, especially for slender beams. If the vibration amplitude is relatively small, the nonlinear model may be replaced by the corresponding higher-order linear model while preserving the main features of the vibration problem.

Dynamic response of beams under moving loads with finite deformation

In this paper, vibration of axially moving nanobeams is studied using Eringen’s two-phase nonlocal integral model. Geometric nonlinearity is taken into account for the integral model for the first time. Equations of motion for the beam with simply supported and fixed–fixed boundary conditions are obtained by Hamilton’s Principle, which turns out to be nonlinear integro-differential equations. For the free vibration of the nanobeam, the critical velocity and the natural frequencies are obtained numerically. Furthermore, the effects of parameters on critical velocity and natural frequency are analyzed. We have found that, for the two-phase nonlocal integral model, regardless of the boundary conditions considered, both the critical velocity and the natural frequency increase with the nonlocal parameter and the geometric parameter.

Xiaowu Zhu

Papers

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model:

A study of the critical strain of hyperelastic materials: A new kinematic frame and the leading order term

Dynamic response of beams under moving loads with finite deformation

Size-Dependent Free Vibration of Axially Moving Nanobeams Using Eringen’s Two-Phase Integral Model