Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

The exact relationship between the bending solutions of functionally graded material （FGM） beams based on the Levinson beam theory and those of the correspond- ing homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The de- flection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams （FGMLBs） are given analytically in terms of the deflection of the reference ho- mogenous Euler-Bernoulli beams （HEBBs） with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily deter- mined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be eas- ily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.

热变形功能梯度梁的非线性振动

基于经典梁理论,研究了热载荷作用下功能梯度梁的大幅振动问题。首先,在经典梁理论下利用物理中面概念,导出了功能梯度梁的非线性运动方程;再利用Ritz-Kantorovich方法消去时间变量,将非线性运动方程转换成了一组关于空间变量的非线性常微分方程;最后采用打靶法数值求解所得方程,并利用数值结果研究了热载荷作用下功能梯度梁静态变形和振幅、材料梯度参数、热载荷、边界条件等对功能梯度梁固有频率的影响。研究表明:热变形的存在,使夹紧FGM梁、简支FGM梁的振动响应明显不同;另外,热载荷作用下,线性振动与非线性振动行为也有显著不同;由于热过屈曲变形,FGM梁的固有频率逐渐增大,使梁的硬化行为更显著。可见,热变形对FGM梁振动响应的影响是复杂的。

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded（FG）temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field.Uniform,linear and nonlinear temperature distributions across the thickness are investigated.Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate.The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function.The governing equations are derived by Hamilton’s principle as a function of axial force due to centrifugal stiffening and displacement.The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions.By applying an analytical solution and solving an eigenvalue problem,the dispersion relations of rotating FG nanobeam are obtained.Numerical results illustrate that various parameters including temperature change,angular velocity,nonlocality parameter,wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study.The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings,nanogears,etc.

A Supersonic Aerodynamic Energy Harvester:A Functionally Graded Material Beam with a Giant Magnetostrictive Thin Film

In this study,a simply supported functionally graded material beam with a giant magnetostrictive thin film(GMF)was selected as an energy harvester.Based on the theory of large deformation and the Villari effect of GMF,piston theory was used to simulate the dynamic equation of the whole structure under supersonic aerodynamic pressure and in a thermal environment by using Hamilton^principle,and the energy harvesting effect of GMF was simulated by using a Runge-Kutta algorithm.Below the critical flutter velocity,the maximum voltage output and energy harvesting results were discussed as they were affected by external factors such as the geometric model of structure parameters,slenderness ratio,gradient index,number of turns of an electromagnetic coil,airflow velocity,and temperature.The electromechanical coupling coefficient/C33 was 71%.The results show that this proposed harvester can achieve an optimal harvesting effect by adjusting the parameters appropriately,and collect energy in thermal and supersonic environments using the GMF,which provides power to sensors of the health monitoring system of the aircraft’s own structure.