| Based on Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory,the governing differential equation of free vibration of variable-area nanobeam on Winkler-Pasternak elastic foundation considering the effect of temperature is deduced;Based on the classical thin plate theory and Eringen’s nonlocal elasticity theory,the governing differential equation of free vibration of in-plane compressed orthotropic nanoplate resting on Winkler-Pasternak elastic foundation is derived using Hamilton’s principle.The differential equations and boundary conditions of the nanobeam and nanoplate are transformed into equivalent algebraic equations by differential transformation method,the different parameters for natural frequency of nanobeam and nanoplate under different boundary conditions is programmed by MATLAB.The second chapter,the governing differential equation of free vibration of variable-area nanobeam on Winkler-Pasternak elastic foundation considering the effect of temperature is reduced to uniform thickness nanobeam,the results of DTM are compared with the exact solutions,the results are in good agreement with the exact solutions.Considering the effect of different parameter changes on the dimensionless natural frequency of the variable-area nanobeam on Winkler-Pasternak elastic foundation.The third chapter,the convergence p roperties of in-plane compression orthotropic rectangular nanoplate natural frequency and the influence of parameters under different boundary conditions on the dimensionless natural frequency and critical buckling load are discusse d,the front three vibration mode shape for different boundary conditions of in-plane compressed orthotropic nanoplate resting on Winkler-Pasternak elastic foundation is obtained simultaneously.The results of the study show that the DTM programming is simple and strong applicability to the solution.For the various boundary conditions,the weaker the boundary is,the faster the natural frequency of nanobeam and nanoplate converges;at the same time,the lower the order is,the faster the natural frequency of nanobeam and nanoplate converges.The natural frequency of variable-area nanobeam on Winkler-Pasternak elastic foundation increases with the increase of modulus of elastic foundation,and the natural frequency decreases with the increase of section variation coefficient,nanometer scale and dimensionless temperature rise.The frequency of the compressed orthotropic rectangular nanoplate on Winkler-Pasternak elastic foundation increases with the increase of elastic foundation modulus,load parameter and aspect ratio,and decreases with the increase of nanoscale.As the pressure intensity rise,the fundamental frequency keeps decreasing until zero.At this point,the pressure intensity equals to critical buckling load of nanoplate,and the critical buckling load increases as the modulus of elastic foundation increases.It is worth noting that when the value of the load parameter is greater than 1,there are two positive and negative cases of critical buckling load,because the orthotropic nanoplate will be subjectted to tension and compression simultaneously. |
