Influence of Temperature Change on Modal Analysis of Rotary Functionally Graded Nano-beam in Thermal Environment

Document Type: Research Paper


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

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


The free vibration analysis of rotating functionally graded (FG) nano-beams under an in-plane thermal loading is provided for the first time in this paper. The formulation used is based on Euler-Bernoulli beam theory through Hamilton’s principle and the small scale effect has been formulated using the Eringen elasticity theory. Then, they are solved by a generalized differential quadrature method (GDQM). It is supposed that, according to the power-law form (P-FGM), the thermal distribution is non-linear and material properties are dependent to temperature and are changing continuously through the thickness. Free vibration frequencies are obtained for two types of boundary conditions; cantilever and propped cantilever. The novelty of this work is related to vibration analysis of rotating FG nano-beam under different distributions of temperature with different boundary conditions using nonlocal Euler-Bernoulli beam theory. Presented theoretical results are validated by comparing the obtained results with literature. Numerical results are presented in both cantilever and propped cantilever nano-beams and the influences of the thermal, nonlocal small-scale, angular velocity, hub radius, FG index and higher modes number on the natural frequencies of the FG nano-beams are investigated in detail. 


[1] Aranda-Ruiz J., Loya J., Fernanandez-Saez J., 2012, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory, Composite Structures 94: 2990-3001.
[2] Bath J., Turberfield A. J., 2007, DNA nanomachines, Nature Nanotechnology 2: 275-284.
[3] Bellman R., Casti J., 1971, Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications 34: 235-238.
[4] Bellman R., Kashef B., Casti J., 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics 10: 40-52.
[5] Chen L., Nakamura M., Schindler T. D., Parker D., Bryant Z., 2012, Engineering controllable bidirectional molecular motors based on myosin, Nature Nanotechnology 7: 252-256.
[6] Civalek Ö., 2004, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures 26: 171-186.
[7] Ebrahimi F., Barati M. R., 2017, Vibration analysis of viscoelastic inhomogeneous nano-beam s incorporating surface and thermal effects, Applied Physics A 123: 5.
[8] Ebrahimi F., Salari E., 2015, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nano-beam s with various boundary conditions, Composites Part B: Engineering, 78: 272-290.
[9] Eringen A. C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[10] Ghadiri M., Hosseini S., Shafiei N., 2015, A power series for vibration of a rotating nano-beam with considering thermal effect, Mechanics of Advanced Materials and Structures 2015: 1-30.
[11] Ghadiri M., Shafiei N., 2015, Nonlinear bending vibration of a rotating nano-beam based on nonlocal Eringen’s theory using differential quadrature method, Microsystem Technologies 2015: 1-15.
[12] Ghadiri M., Shafiei N., Akbarshahi A., 2016, Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nano-beam , Applied Physics A 122: 1-19.
[13] Ghadiri M., Shafiei N., Safarpour H., 2017, Influence of surface effects on vibration behavior of a rotary functionally graded nano-beam based on Eringen’s nonlocal elasticity, Microsystem Technologies 23: 1045-1065.
[14] Ghorbanpour-Arani A., Rastgoo A., Sharafi M., Kolahchi R., Arani A. G., 2016, Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nano-beam systems, Meccanica 51: 25-40.
[15] Goel A., Vogel V., 2008, Harnessing biological motors to engineer systems for nanoscale transport and assembly, Nature Nanotechnology 3: 465-475.
[16] Hu B., Ding Y., Chen W., Kulkarni D., Shen Y., Tsukruk V. V., Wang Z. L., 2010, External‐strain induced insulating phase transition in VO2 nano-beam and its application as flexible strain sensor, Advanced Materials 22: 5134-5139.
[17] Lee L. K., Ginsburg M. A., Cravace C., Donahoe M., Stock D., 2010, Structure of the torque ring of the flagellar motor and the molecular basis for rotational switching, Nature 466: 996-1000.
[18] Lu P., Lee H., Lu C., Zhang P., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics 99: 073510.
[19] Lubbe A. S., Ruangsupapichat N., Caroli G., Feringa B. L., 2011, Control of rotor function in light-driven molecular motors, The Journal of Organic Chemistry 76: 8599-8610.
[20] Maraghi Z. K., Arani A. G., Kolahchi R., Amir S., Bagheri M., 2013, Nonlocal vibration and instability of embedded DWBNNT conveying viscose fluid, Composites Part B: Engineering 45: 423-432.
[21] Mindlin R. D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16: 51-78.
[22] Mirjavadi S. S., Rabby S., Shafiei N., Afshari B. M., Kazemi M., 2017, On size-dependent free vibration and thermal buckling of axially functionally graded nano-beam s in thermal environment, Applied Physics A 123: 315.
[23] 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: 1232-1243.
[24] Pradhan S., 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: 1944-1949.
[25] Rahmani O., Pedram O., 2014, Analysis and modeling the size effect on vibration of functionally graded nano-beam s based on nonlocal Timoshenko beam theory, International Journal of Engineering Science 77: 55-70.
[26] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Springer.
[27] 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: 791-798.
[28] Tauchert T. R., 1974, Energy Principles in Structural Mechanics, McGraw-Hill Companies.
[29] Thai H.T., 2012, A nonlocal beam theory for bending, buckling, and vibration of nano-beam s, International Journal of Engineering Science 52: 56-64.
[30] Tierney H. L., Murohy C. J., Jewell A. D., Baber A. E., Iski E. V., Khodaverdian H. Y., Mcguire A. F., Klebanov N., Sykes E. C. H., 2011, Experimental demonstration of a single-molecule electric motor, Nature Nanotechnology 6: 625-629.
[31] Toupin R. A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11: 385-414.
[32] Vandelden R. A., Terwiel M. K., Pollard M. M., Vicario J., Koumura N., Feringa B. L., 2005, Unidirectional molecular motor on a gold surface, Nature 437: 1337-1340.
[33] Wang C., Zhang Y., He X., 2007, Vibration of nonlocal Timoshenko beams, Nanotechnology 18: 105401.
[34] Xing Y., Liu B., 2009, High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain, International Journal for Numerical Methods in Engineering 80: 1718-1742.
[35] Zhong H., Yue Z., 2012, Analysis of thin plates by the weak form quadrature element method, Science China Physics, Mechanics and Astronomy 55: 861-871.