Vibration Analysis of a Rotating Nanoplate Using Nonlocal Elasticity Theory

Document Type: Research Paper


1 Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran

2 School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran


The nanostructures under rotation have high promising future to be used in nano-machines, nano-motors and nano-turbines. They are also one of the topics of interests and it is new in designing of rotating nano-systems. In this paper, the scale-dependent vibration analysis of a nanoplate with consideration of the axial force due to the rotation has been investigated. The governing equation and boundary conditions are derived using the Hamilton’s principle based on nonlocal elasticity theory. The boundary conditions of the nanoplate are considered as free-free in y direction and two clamped-free (cantilever plate) and clamped-simply (propped cantilever) in x direction. The equations have been solved using differential quadrature method to determine natural frequencies of the rotating nanoplate. For validation, in special cases, it has been shown that the obtained results coincide with literatures. The effects of the nonlocal parameter, aspect ratio, hub radius, angular velocity and different boundary conditions on the first three frequencies have been investigated. Results show that vibration behavior of the rotating nanoplate with cantilever boundary condition is different from other boundary conditions. 


[1] Van Delden R. A., Ter Wiel M. K. J., Pollard M. M., Vicario J., Koumura N., Feringa B. L., 2005, Unidirectional molecular motor on a gold surface, Nature 437(7063): 1337-1340.
[2] Li J., Wang X., Zhao L., Gao X., Zhao Y., Zhou R., 2014, Rotation motion of designed nano-turbine, Scientific Reports 4: 5846.
[3] Fleck N., Muller G.M., Ashby M.F., Hutchinson J.W. ,1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42(2): 475-487.
[4] Chong A. C. M., Yang F., Lam D. C. C., Tong P., 2001, Torsion and bending of micron-scaled structures, Journal of Materials Research 16(04): 1052-1058.
[5] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703-4710.
[6] Eringen A.C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 10(5): 425-435.
[7] Chen M., 2013, Large deflection of a cantilever nanobeam under a vertical end load, Applied Mechanics and Materials 353: 3387-3390.
[8] Murmu T., Adhikari S., 2010, Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation, Journal of Applied Physics 108(12): 123507-123514.
[9] Narendar S., Gopalakrishnan S., 2011, Nonlocal wave propagation in rotating nanotube,Results in Physics 1(1): 17-25.
[10] Narendar, S., Mathematical modelling of rotating single-walled carbon nanotubes used in nanoscale rotational actuators, Defence Science Journal 61(4): 317-324.
[11] Akgoz B.,CIvalek O., 2012, Analysis of micro-sixed beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82(3): 423-443.
[12] Challamel N., Wang C.M., 2008, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology 19(34): 345703.
[13] Lim C., Li C., Yu J., 2009, The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams, Interaction and Multiscale Mechanics: an International Journal 1(3): 223-233.
[14] Narendar S., 2012, Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia, Applied Mathematics and Computation 219(3): 1232-1243.
[15] Pradhan S.C., Murmu T., 2010, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E: Low-dimensional Systems and Nanostructures 42(7): 1944-1949.
[16] Aranda-Ruiz J., Loya J., Fernández-Sáez J., 2012, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory, Composite Structures 94(9): 2990-3001.
[17] Ghadiri M., Hosseini S., Shafiei N., 2016, A power series for vibration of a rotating nanobeam with considering thermal effect, Mechanics of Advanced Materials and Structures 23(12): 1414-1420.
[18] Ghadiri M., Shafiei N., 2016, Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method, Microsystem Technologies 22(12): 2853-2867.
[19] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part I: Theoretical Formulations, Physica E: Low-dimensional Systems and Nanostructures 44(1): 229-248.
[20] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part II: Parametric Studies, Physica E: Low-dimensional Systems and Nanostructures 44(1): 249-269.
[21] Kiani K., 2011, Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory, Journal of Sound and Vibration 330(20): 4896-4914.
[22] Kiani K., 2013,Vibrations of biaxially tensioned-embedded nanoplates for nanoparticle delivery, Indian Journal of Science and Technology 6(7): 4894-4902.
[23] Salehipour H., Nahvi H., Shahidi A., 2015, Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures 66: 350-358.
[24] Ansari R., Shahabodini A., Shojaei M.F., 2016, Nonlocal three-dimensional theory of elasticity with application to free vibration of functionally graded nanoplates on elastic foundations, Physica E: Low-dimensional Systems and Nanostructures 76: 70-81.
[25] Wang C., Reddy J.N., Lee K., 2000, Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier.
[26] Reddy J.N., El-Borgi S., 2014, Eringen’s nonlocal theories of beams accounting for moderate rotations, International Journal of Engineering Science 82(0):159-177.
[27] Wang J.S., Shaw D., Mahrenholtz O., 1987, Vibration of rotating rectangular plates, Journal of Sound and Vibration 112(3): 455-468.
[28] Shu C., Richards B.E.,1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 15(7): 791-798.
[29] Pradhan S.C., Phadikar J.K., 2009, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration 325(1-2): 206-223.
[30] Mohammadi M., Moradi A., Ghayour M., Farajpour A., 2014, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures 11: 437-458.
[31] Shen Z. B., Tang H.L., Daokui L., Tang G.J., 2012, Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory, Computational Materials Science 61(0):200-205.
[32] Ansari R., Rajabiehfard R., Arash B., 2010, Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets, Computational Materials Science 49(4): 831-838.