Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM
Source abstract
Abstract
This study is concerned with the small-scale effect on the nonlinear flapwise bending vibration of rotating cantilever and propped cantilever nanobeams. Euler–Bernoulli beam theory is used to model the nanobeam with nonlinearity. Nonlinear strain–displacement relations are employed to account for geometric nonlinearity of the system. The axial forces are modeled as the true spatial and thermal variations due to the rotation. Hamilton’s principle is used to derive the nonlinear governing equation and nonlocal nonlinear boundary conditions based on Eringen’s nonlocal elasticity theory. Finally, the differential quadrature method is used in conjunction with the direct iterative method to derive the nonlinear vibration frequencies of the nanobeam. The effects of the angular velocity, nonlocal small-scale parameter, temperature change and nonlinear amplitude on nonlinear vibration of the rotary nanobeam are discussed. The results of this work can be used in nanosensors, nanomotors, nanoturbines and NEMS applications.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Nonlocal and gradient elasticity in micro/nano structures · Thermoelastic and Magnetoelastic Phenomena · Composite Structure Analysis and Optimization
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (69)
Show source-provided references
1. R. Ansari, M.F. Oskouie, R. Gholami, F. Sadeghi, Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Compos. B Eng. 89, 316–327 (2016)10.1016/j.compositesb.2015.12.029 ↗
2. J. Aranda-Ruiz, J. Loya, J. Fernández-Sáez, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos. Struct. 94, 2990–3001 (2012)10.1016/j.compstruct.2012.03.033 ↗
3. H. Askari, E. Esmailzadeh, D. Zhang, Nonlinear vibration analysis of nonlocal nanowires. Compos. B Eng. 67, 607–613 (2014)10.1016/j.compositesb.2014.08.017 ↗
4. A. Azrar, L. Azrar, A. Aljinaidi, Nonlinear free vibration of single walled Carbone NanoTubes conveying fluid. In MATEC Web of Conferences. EDP Sciences, 02015 (2014)10.1051/matecconf/20141102015 ↗
5. A. Bahrami, A. Teimourian, Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach. Compos. Struct. 134, 1061–1075 (2015)10.1016/j.compstruct.2015.09.007 ↗
6. L. Behera, S. Chakraverty, Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories. Comput. Math Appl. 69, 1444–1462 (2015)10.1016/j.camwa.2015.04.010 ↗
7. R. Bellman, J. Casti, Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235–238 (1971)10.1016/0022-247X(71)90110-7 ↗
8. R. Bellman, B.G. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)10.1016/0021-9991(72)90089-7 ↗
9. G.R. Bhashyam, G. Prathap, Galerkin finite element method for non-linear beam vibrations. J. Sound Vib. 72, 191–203 (1980)10.1016/0022-460X(80)90652-5 ↗
10. S. Chakraverty, L. Behera, Free vibration of non-uniform nanobeams using Rayleigh-Ritz method. Phys. E 67, 38–46 (2015)10.1016/j.physe.2014.10.039 ↗
11. T.-P. Chang, Q.-J. Yeh, Nonlinear free vibration of nanobeams subjected to magnetic field based on nonlocal elasticity theory. In 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), 1 (2014)
12. T.P. Chang, Large amplitude free vibration of nanobeams subjected to magnetic field based on nonlocal elasticity theory. Appl. Mech. Mat. 764–765, 1199–1203 (2015)10.4028/www.scientific.net/AMM.764-765.1199 ↗
13. S. El-Borgi, R. Fernandes, J.N. Reddy, Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation. Int. J. Non Linear Mech. 77, 348–363 (2015)10.1016/j.ijnonlinmec.2015.09.013 ↗
14. M. Eltaher, A. Abdelrahman, A. Al-Nabawy, M. Khater, A. Mansour, Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl. Math. Comput. 235, 512–529 (2014)10.1016/j.amc.2014.03.028 ↗
15. A.C. Eringen, Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)10.1016/0020-7225(72)90070-5 ↗
16. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)10.1063/1.332803 ↗
17. B. Fang, Y.-X. Zhen, C.-P. Zhang, Y. Tang, Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Appl. Math. Model. 37, 1096–1107 (2013)10.1016/j.apm.2012.03.032 ↗
18. Y. Feng, C. Bert, Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam. Nonlinear Dyn. 3, 13–18 (1992)10.1007/BF00045468 ↗
19. M. Ghadiri, N. Shafiei, Vibration analysis of rotating nanoplate based on Eringen nonlocal elasticity applying differential quadrature method. J. Vib. Control (2015). doi: 10.1177/107754631562772310.1177/1077546315627723 ↗
20. M. Ghadiri, N. Shafiei, Vibration analysis of a nano-turbine blade based on Eringen nonlocal elasticity applying the differential quadrature method. J. Vib. Control (2016). doi: 10.1177/107754631562772310.1177/1077546315627723 ↗
21. M.H. Ghayesh, Nonlinear size-dependent behaviour of single-walled carbon nanotubes. Appl. Phys. A 117, 1393–1399 (2014)10.1007/s00339-014-8561-6 ↗
22. J. Guo, K. Kim, K.W. Lei, D.E. Fan, Ultra-durable rotary micromotors assembled from nanoentities by electric fields. Nanoscale. 7, 11363–11370 (2015)10.1039/C5NR02347E ↗
23. M. Hosseini, M. Sadeghi-Goughari, S. Atashipour, M. Eftekhari, Vibration analysis of single-walled carbon nanotubes conveying nanoflow embedded in a viscoelastic medium using modified nonlocal beam model. Arch. Mech. 66, 217–244 (2014)
24. J.-C. Hsu, H.-L. Lee, W.-J. Chang, Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory. Curr. Appl. Phys. 11, 1384–1388 (2011)10.1016/j.cap.2011.04.026 ↗
25. A. Jandaghian, O. Rahmani, Free vibration analysis of magneto-electro-thermo-elastic nanobeams resting on a Pasternak foundation. Smart Mater. Struct. 25, 035023 (2016)10.1088/0964-1726/25/3/035023 ↗
26. D. Karličić, P. Kozić, R. Pavlović, Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium. Appl. Math. Model. 40, 1599–1614 (2016)10.1016/j.apm.2015.06.036 ↗
27. L.-L. Ke, Y.-S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater. Struct. 21, 025018 (2012)10.1088/0964-1726/21/2/025018 ↗
28. K. Kim, J. Guo, X. Xu, D. Fan, Micromotors with step-motor characteristics by controlled magnetic interactions among assembled components. ACS Nano 9, 548–554 (2014)10.1021/nn505798w ↗
29. K. Kim, X. Xu, J. Guo, D. Fan, Ultrahigh-speed rotating nanoelectromechanical system devices assembled from nanoscale building blocks. Nat. Commun. 5, 3632 (2014)10.1038/ncomms4632 ↗
30. K. Kima, D. Fana, Mechanism for assembling arrays of rotary nanoelectromechanical devices. Encycl. Nanotechnol. (2015). doi: 10.1007/978-94-007-6178-0_100910-110.1007/978-94-007-6178-0_100910-1 ↗
31. Y.-L. Kuo, Nonlinear vibration analysis of nonlocal elastic multi-walled carbon nanotubes. Adv. Sci. Lett. 14, 269–273 (2012)10.1166/asl.2012.4087 ↗
32. Y.-L. Kuo, Nonlinear finite element analysis of nonlocal elastic nanobeams with large-amplitude vibrations. J. Comput. Theor. Nanosci. 10, 488–495 (2013)10.1166/jctn.2013.2723 ↗
33. Y.-L. Kuo, Chaotic analysis of the geometrically nonlinear nonlocal elastic single-walled carbon nanotubes on elastic medium. J. Nanosci. Nanotechnol. 14, 2352–2360 (2014)10.1166/jnn.2014.8531 ↗
34. W. Lestari, S. Hanagud, Nonlinear vibration of buckled beams: some exact solutions. Int. J. Solids Struct. 38, 4741–4757 (2001)10.1016/S0020-7683(00)00300-0 ↗
35. J. Li, X. Wang, L. Zhao, X. Gao, Y. Zhao, R. Zhou, Rotation motion of designed nano-turbine. Sci. Rep. 4, 5846 (2014)10.1038/srep05846 ↗
36. C. Lim, C. Li, J. Yu, The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams. Interact. Multiscale Mech. Int. J. 2, 223–233 (2009)10.12989/imm.2009.2.3.223 ↗
37. C.W. Lim, Q. Yang, J.B. Zhang, Thermal buckling of nanorod based on non-local elasticity theory. Int. J. Non Linear Mech. 47, 496–505 (2012)10.1016/j.ijnonlinmec.2011.09.023 ↗
38. J. Loya, J. López-Puente, R. Zaera, J. Fernández-Sáez, Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model. J. Appl. Phys. 105, 044309 (2009)10.1063/1.3068370 ↗
39. P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99, 073510 (2006)10.1063/1.2189213 ↗
40. P. Malekzadeh, M. Shojaee, Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams. Compos. B Eng. 52, 84–92 (2013)10.1016/j.compositesb.2013.03.046 ↗
41. T. Murmu, S. Adhikari, Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation. J. Appl. Phys. 108, 123507 (2010)10.1063/1.3520404 ↗
42. S. Narendar, Mathematical modelling of rotating single-walled carbon nanotubes used in nanoscale rotational actuators. Def. Sci. J. 61, 317–324 (2011)10.14429/dsj.61.1091 ↗
43. 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. 219, 1232–1243 (2012)10.1016/j.amc.2012.07.032 ↗
44. R. Nazemnezhad, S. Hosseini-Hashemi, Nonlocal nonlinear free vibration of functionally graded nanobeams. Compos. Struct. 110, 192–199 (2014)10.1016/j.compstruct.2013.12.006 ↗
45. J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)10.1016/S0020-7225(02)00210-0 ↗
46. A. Pourasghar, M. Homauni, S. Kamarian, Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the Eringen nonlocal elasticity theory under axial load. Polym. Compos. (2015). doi: 10.1002/pc.2351510.1002/pc.23515 ↗
47. S.C. Pradhan, T. Murmu, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Phys. E 42, 1944–1949 (2010)10.1016/j.physe.2010.03.004 ↗
48. O. Rahmani, A. Jandaghian, Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory. Appl. Phys. A 119, 1019–1032 (2015)10.1007/s00339-015-9061-z ↗
49. J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)10.1016/j.ijengsci.2007.04.004 ↗
50. J.N. Reddy, S. El-Borgi, J. Romanoff, Non-linear analysis of functionally graded microbeams using Eringen's non-local differential model. Int. J. Non Linear Mech. 67, 308–318 (2014)10.1016/j.ijnonlinmec.2014.09.014 ↗
51. M. Rezaee, S. Lotfan, Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity. Int. J. Mech. Sci. 96, 36–46 (2015)10.1016/j.ijmecsci.2015.03.017 ↗
52. N. Shafiei, M. Kazemi, L. Fatahi, Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method. Mech. Adv. Mater. Struct. (2015). doi: 10.1080/15376494.2015.112802510.1080/15376494.2015.1128025 ↗
53. N. Shafiei, M. Kazemi, M. Ghadiri, On size-dependent vibration of rotary axially functionally graded microbeam. Int. J. Eng. Sci. 101, 29–44 (2015)10.1016/j.ijengsci.2015.12.008 ↗
54. N. Shafiei, M. Kazemi, M. Ghadiri, Nonlinear vibration of axially functionally graded tapered microbeams. Int. J. Eng. Sci. 102, 12–26 (2016)10.1016/j.ijengsci.2016.02.007 ↗
55. C. Shu, Differential Quadrature and Its Application in Engineering (Springer, New York, 2000)10.1007/978-1-4471-0407-0 ↗
56. M. Şimşek, Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Compos. B Eng. 56, 621–628 (2014)10.1016/j.compositesb.2013.08.082 ↗
57. G. Singh, A.K. Sharma, G. Venkateswara Rao, Large-amplitude free vibrations of beams—a discussion on various formulations and assumptions. J. Sound Vib. 142, 77–85 (1990)10.1016/0022-460X(90)90583-L ↗
58. H.-T. Thai, A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)10.1016/j.ijengsci.2011.11.011 ↗
59. H.-T. Thai, T.P. Vo, A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54, 58–66 (2012)10.1016/j.ijengsci.2012.01.009 ↗
60. N. Togun, S.M. Bağdatlı, Nonlinear vibration of a nanobeam on a Pasternak elastic foundation based on non-local Euler–Bernoulli beam theory. Math. Comput. Appl. 21, 3 (2016)
61. C.M. Wang, Y.Y. Zhang, X.Q. He, Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007)10.1088/0957-4484/18/10/105401 ↗
62. Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 124301 (2005)10.1063/1.2141648 ↗
63. Q. Wang, V. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15, 659 (2006)10.1088/0964-1726/15/2/050 ↗
64. Y.-Z. Wang, F.-M. Li, Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory. Int. J. Non Linear Mech. 61, 74–79 (2014)10.1016/j.ijnonlinmec.2014.01.008 ↗
65. Y.-Z. Wang, F.-M. Li, Nonlinear postbuckling of double-walled carbon nanotubes induced by temperature changes. Appl. Phys. A 121, 731–738 (2015)10.1007/s00339-015-9471-y ↗
66. X. Xu, K. Kim, C. Liu, D. Fan, Fabrication and robotization of ultrasensitive plasmonic nanosensors for molecule detection with Raman scattering. Sensors 15, 10422–10451 (2015)10.3390/s150510422 ↗
67. X. Yao, Q. Han, Buckling analysis of multiwalled carbon nanotubes under torsional load coupling with temperature change. J. Eng. Mater. Technol. 128, 419–427 (2005)10.1115/1.2203102 ↗
68. Y. Zhang, G. Liu, J. Wang, Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys. Rev. B 70, 205430 (2004)10.1103/PhysRevB.70.205430 ↗
69. Y.-X. Zhen, B. Fang, Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation. Int. J. Non Linear Mech. 76, 48–55 (2015)10.1016/j.ijnonlinmec.2015.05.005 ↗