Vibration Analysis Of Rotating Functionally Graded Nano Annular Plates In Themial Environment Based On Different Shear Deformation Plate Theories

In recent years,many new types of material have emerged,with the rapid development of materials science.Among them,functionally graded materials and nanomaterials have been widely used in aerospace,medical equipment,vehicle engineering and other fields because of their unique physical and chemical properties.As to functionally graded materials,their properties exhibit a gradient change in a certain direction.On the other hand,nano materials have a special small-scale effect due to their small size,which can be well explained by the nonlocal theory.The rotating nano annular plates have found practical applications in the micro nano structure.In addition,their operating environment will be affected by the temperature.Therefore,the vibration characteristics of rotating functionally graded nano annular plates in the thermal environment are studied based on three different deformation plate theories.Firstly,Hamilton's principle is used to obtain the differential equations of radial and transverse coupled motion.Then,the axisymmetric internal force in the middle plane of the rotating functionally graded nano annular plates is obtained through the plane stress problems.By means of differential quadrature method,the differential equations of variable coefficient and boundary condition are discretized.Finally,the vibration frequencies of the plates under different boundary conditions are calculated and analyzed.Therefore,the following results are obtained through numerical example analysis.(1)Under the three deformation theories,the vibration frequencies of the functionally graded nano annular plate are not only related to the boundary conditions and the ratio of the inner and outer diameters of the plate,but also related to the functionally graded parameters,nonlocal parameters,rotation speed and temperature.Moreover,when the inner edge is clamped and the outer edge is free,the natural frequencies of the plates change with these parameters exhibit a different trend compared to the frequencies of other boundary conditions.(2)The rotation speed of the annular plates has a great impact on the natural frequencies.When the boundary conditions are clamped or simply supported on the inner and outer sides,the frequencies decrease with increasing rotation speed.Furthermore,when the rotation speed reaches the first order critical speed,the first-order mode of the annular plate will lose its stability.(3)The calculated results of the three deformation theories are greatly affected by the rotation speed.In details,the higher the rotation speed is,the greater the differences are among the results which are obtained by the three theories with the increased thickness of the annular plate.Therefore,in the case of high rotation speed,the model which is constructed by means of the higher order theory will obtain the numerical results with a higher accuracy.(4)Nonlocal parameters have a certain influence on the calculation results of the three theories.The larger the nonlocal parameters are,the greater the influence it will exert on the eigen values.Hence,taking into consideration of the nonlocal parameters,the results which are obtained by Reddy plate theory and Mindlin plate theory are more accurate.In addition,regarding to the annular plates with larger thickness,Reddy plate theory has certain advantages in terms of precision.