Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution

References

• A.C. Eringen, and D.G.B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci., vol. 10, pp. 233–248, 1972.
   Web of Science ®Google Scholar
• A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., vol. 54, pp. 4703–4710, 1983.
   Web of Science ®Google Scholar
• O. Rahmani, and O. Pedram, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, Int. J. Eng. Sci., vol. 77, pp. 55–70, 2014.
   Web of Science ®Google Scholar
• R. Ansari, R. Gholami, and S. Sahmani, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Compos. Struct., vol. 94, pp. 221–228, 2011.
   Web of Science ®Google Scholar
• A.C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sci., vol. 10, pp. 1–16, 1972.
   Web of Science ®Google Scholar
• C. Wang, Y. Zhang, and X. He, Vibration of nonlocal Timoshenko beams, Nanotechnology, vol. 18, pp. 105401, 2007.
   Web of Science ®Google Scholar
• J. Loya, J. López-Puente, R. Zaera, and J. Fernández-Sáez, Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, J. Appl. Phys., vol. 105, pp. 044309, 2009.
   Web of Science ®Google Scholar
• S. C. Pradhan, and T. Murmu, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Phys. E: Low-dimensional Syst. Nanostructures, vol. 42, pp. 1944–1949, 2010.
   Web of Science ®Google Scholar
• J.-C. Hsu, H.-L. Lee, and W.-J. Chang, Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory, Curr. Appl. Phys., vol. 11, pp. 1384–1388, 2011.
   Web of Science ®Google Scholar
• C.W. Lim, Q. Yang, and J.B. Zhang, Thermal buckling of nanorod based on non-local elasticity theory, Int. J. Non-Linear Mech., vol. 47, pp. 496–505, 2012.
   Web of Science ®Google Scholar
• L.-L. Ke, Y.-S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory, Smart Mater. Struct., vol. 21, pp. 025018, 2012.
   Web of Science ®Google Scholar
• R. Ansari, M.F. Oskouie, R. Gholami, and F. Sadeghi, Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory, Compos. Part B: Eng., vol. 89, pp. 316–327, 2016.
   Web of Science ®Google Scholar
• A. Bahrami, and A. Teimourian, Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach, Compos. Struct., vol. 134, pp. 1061–1075, 2015.
   Web of Science ®Google Scholar
• O. Rahmani, and A. Jandaghian, Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory, Appl. Phys. A, vol. 119, pp. 1019–1032, 2015.
   Web of Science ®Google Scholar
• S. Chakraverty, and L. Behera, Free vibration of non-uniform nanobeams using Rayleigh–Ritz method, Phys. E: Low-dimensional Syst. Nanostructures, vol. 67, pp. 38–46, 2015.
   Web of Science ®Google Scholar
• A. Jandaghian, and O. Rahmani, Free vibration analysis of magneto-electro-thermo-elastic nanobeams resting on a Pasternak foundation, Smart Mater. Struct., vol. 25, pp. 035023, 2016.
   Web of Science ®Google Scholar
• D. Karličić, P. Kozić, and R. Pavlović, Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium, Appl. Math. Modell., vol. 40, pp. 1599–1614, 2016.
   Web of Science ®Google Scholar
• M.H. Ghayesh, Nonlinear size-dependent behaviour of single-walled carbon nanotubes, Appl. Phys. A., vol. 117, pp. 1393–1399, 2014.
   Web of Science ®Google Scholar
• Y.-Z. Wang, and F.-M. Li, Nonlinear postbuckling of double-walled carbon nanotubes induced by temperature changes, Appl. Phys. A., vol. 121, pp. 731–738, 2015.
   Web of Science ®Google Scholar
• S. El-Borgi, R. Fernandes, and J.N. Reddy, Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation, Int. J. Non-Linear Mech., vol. 77, pp. 348–363, 2015.
   Web of Science ®Google Scholar
• L. Behera, and S. Chakraverty, Application of Differential quadrature method in free vibration analysis of nanobeams based on various nonlocal theories, Comput. Math. Appl., vol. 69, pp. 1444–1462, 2015.
   Web of Science ®Google Scholar
• T.P. Chang, Large amplitude free vibration of nanobeams subjected to magnetic field based on nonlocal elasticity theory, Appl. Mech. Mater.: Trans Tech Publ, vol. 764, pp. 1199–1203, 2015.
   Google Scholar
• M. Rezaee, and S. Lotfan. Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity, Int. J. Mech. Sci., vol. 96, pp. 36–46, 2015.
   Web of Science ®Google Scholar
• N. Togun, and S. M. Bağdatlı, Nonlinear vibration of a nanobeam on a pasternak elastic foundation based on non-local Euler–Bernoulli beam theory, Math. Comput. Appl., vol. 21, pp. 3, 2016.
   Google Scholar
• N. Shafiei, M. Kazemi, and M. Ghadiri, Nonlinear vibration of axially functionally graded tapered microbeams, Int. J. Eng. Sci., vol. 102, pp. 12–26, 2016.
   Web of Science ®Google Scholar
• M. Şimşek, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Compos. Part B: Eng., vol. 56, pp. 621–628, 2014.
   Web of Science ®Google Scholar
• R. Nazemnezhad, and S. Hosseini-Hashemi, Nonlocal nonlinear free vibration of functionally graded nanobeams, Compos. Struct., vol. 110, pp. 192–199, 2014.
   Web of Science ®Google Scholar
• Y.-Z. Wang, and F.-M. Li, Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory, Int. J. Non-Linear Mech., vol. 61, pp. 74–79, 2014.
   Web of Science ®Google Scholar
• Y.-L. Kuo, Nonlinear finite element analysis of nonlocal elastic nanobeams with large-amplitude vibrations, J. Comput. Theor. Nanosci., vol. 10, pp. 488–495, 2013.
   Google Scholar
• T.-P. Chang, and Q.-J. Yeh, Nonlinear free vibration of nanobeams subjected to magnetic field based on nonlocal elasticity theory, Proceedings of the International Conference on Scientific Computing (CSC): The Steering Committee of The World Congress in Computer Science, Computer Engineering and Applied Computing (WorldComp), July, Las Vegas, 2014.
   Google Scholar
• S. Pradhan, and T. Murmu. Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Phys. E: Low-dimensional Syst. Nanostructures, vol. 42, pp. 1944–1949, 2010.
   Web of Science ®Google Scholar
• A. Pourasghar, M. Homauni, and S. Kamarian. Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the eringen nonlocal elasticity theory under axial load, Polym. Compos., vol. 7, pp. 3175–3180, 2016.
   Google Scholar
• J. Aranda-Ruiz, J. Loya, and J. Fernández-Sáez, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory, Compos. Struct., vol. 94, pp. 2990–3001, 2012.
   Web of Science ®Google Scholar
• S. Narendar. Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia, Appl. Math. Comput., vol. 219, pp. 1232–1243, 2012.
   Web of Science ®Google Scholar
• T. Murmu, and S. Adhikari, Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation, J. Appl. Phys., vol. 108, no. 12, p. 123507, 2010.
   Web of Science ®Google Scholar
• S. Narendar, Mathematical modelling of rotating single-walled carbon nanotubes used in nanoscale rotational actuators, Defence Sci. J., vol. 61, pp. 317–324, 2011.
   Web of Science ®Google Scholar
• C. Lim, C. Li, and J. Yu, The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams, Interact. Multiscale Mech. Int. J., vol. 2, pp. 223–233, 2009.
   Google Scholar
• M. Ghadiri, and N. Shafiei, Vibration analysis of rotating nanoplate based on eringen nonlocal elasticity appling differential quadrature method, J. Vib. Control, doi: 10.1177/1077546315627723, 2016.
   PubMed Web of Science ®Google Scholar
• Y. S. Touloukian, and C. Ho, Thermal expansion. Nonmetallic solids. In: Thermophysical Properties of Matter-The TPRC Data Series, Y.S. Touloukian ( series ed.) and C.Y. Ho ( series tech. ed.). vol. 1, IFI/Plenum, New York, 1970.
   Google Scholar
• F. Fazzolari, and E. Carrera, Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions, J. Therm. Stresses, vol. 37, pp. 1449–1481, 2014.
   Web of Science ®Google Scholar
• R. Bellman, and J. Casti, Differential quadrature and long-term integration, J. Math. Anal. Appl., vol. 34, pp. 235–238, 1971.
   Web of Science ®Google Scholar
• R. Bellman, B. G. Kashef, and J. Casti, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys., vol. 10, pp. 40–52, 1972.
   Web of Science ®Google Scholar
• C. Shu, Differential Quadrature and its Application in Engineering, Springer-Verlag, London, 2000.
   Google Scholar
• N. Shafiei, M. Kazemi, and M. Ghadiri, Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams, Phys. E: Low-dimensional Syst. Nanostructures, vol. 83, pp. 74–87, 2016.
   Web of Science ®Google Scholar
• C. M. Wang, Y. Y. Zhang, and X. Q. He, Vibration of nonlocal Timoshenko beams, Nanotechnology, vol. 18, pp. 105401, 2007.
   Web of Science ®Google Scholar
• F. Ebrahimi, and E. Salari, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions, Compos. Part B: Eng., vol. 78, pp. 272–290, 2015.
   Web of Science ®Google Scholar
• N. Shafiei, M. Kazemi, and L. Fatahi. Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method, Mech. Adv. Mater. Struct., vol. 24, no. 3, pp. 240–252, 2017.
   Web of Science ®Google Scholar
• N. Shafiei, M. Kazemi, and M. Ghadiri, On size-dependent vibration of rotary axially functionally graded microbeam, Int. J. Eng. Sci., vol. 101, pp. 29–44, 2016.
   Web of Science ®Google Scholar
• J. Yang, and H.-S. Shen, Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments, J. Sound Vib., vol. 255, pp. 579–602, 2002.
   Web of Science ®Google Scholar