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Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. (English) Zbl 1423.74398

Summary: This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. The Kelvin-Voigt viscoelastic model, velocity-dependent external damping and Timoshenko beam theory are employed to establish the governing equations and boundary conditions for the bending vibration of nanotubes. Using transfer function methods (TFM), the natural frequencies and frequency response functions (FRF) are computed for beams with different boundary conditions. Unlike local structures, taking into account rotary inertia and shear deformation, the nonlocal beam has maximum frequencies, called the escape frequencies or asymptotic frequencies, which are obtained for undamped and damped nonlocal Timoshenko beams. Damped nonlocal beams are also shown to possess an asymptotic critical damping factor. Taking a carbon nanotube as a numerical example, the effects of the nonlocal parameter, viscoelastic material constants, the external damping ratio, and the beam length-to-diameter ratio on the natural frequencies and the FRF are investigated. The results demonstrate the efficiency of the proposed modeling and analysis methods for the free vibration and frequency response analysis of nonlocal viscoelastic damped Timoshenko beams.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74D05 Linear constitutive equations for materials with memory
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Adhikari, S.; Chowdhury, R., Zeptogram sensing from gigahertz vibration: graphene based nanosensor, Physica E: Low-dimensional Systems and Nanostructures, 44, 7-8, 1528-1534, (2012)
[2] Adhikari, S.; Mrumu, T.; McCarthy, M. A., Dynamic finite element analysis of axially vibrating nonlocal rods, Finite Elements in Analysis and Design., 63, 42-50, (2013) · Zbl 1282.74044
[3] Arani, A. G.; Shiravand, A.; Rahi, M.; Kolahchi, R., Nonlocal vibration of coupled DLGS systems embedded on visco-Pasternak foundation, Physica B, 407, 4123-4131, (2012)
[4] Benzair, A.; Tounsi1, A.; Besseghier, A.; Heireche, H.; Moulay, N.; Boumia, L., The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory, Journal of Physics D: Applied Physics, 41, 22, (2008), 225404-1-10
[5] Calleja, M.; Kosaka, P.; San Paulo, A.; Tamayo, J., Challenges for nanomechanical sensors in biological detection, Nanoscale, 4, 4925-4938, (2012)
[6] Chen, C.; Ma, M.; Liu, J.; Zheng, Q.; Xu, Z., Viscous damping of nanobeam resonators: humidity, thermal noise, and a paddling effect, Journal of Applied Physics, 110, 034320, (2011)
[7] Ghannadpour, S. A.M.; Mohammadi, B.; Fazilati, J., Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method, Composite Structures, 96, 584-589, (2013)
[8] Kiani, K., Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model, Applied Mathematical Modelling, 37, 4, 1836-1850, (2013) · Zbl 1349.74174
[9] Kim, Sun-Bae; Kim, Ji-Hwan, Quality factors for the nano-mechanical tubes with thermoelastic damping and initial stress, Journal of Sound and Vibration, 330, 7, 1393-1402, (2011)
[10] Lee, J.; Lin, C., The magnetic viscous damping effect on the natural frequency of a beam plate subject to an in-plane magnetic field, Journal of Applied Mechanics -Transactions of the ASME, 77, 1, 011014, (2010)
[11] Lei, Y.; Friswell, M. I.; Adhikar, S., A Galerkin method for distributed systems with non-local damping, International Journal of Solids and Structures, 43, 11-12, 3381-3400, (2006) · Zbl 1121.74376
[12] Liew, K. M.; Hu, Yanggao; He, X. Q., Flexural wave propagation in single-walled carbon nanotubes, Journal of Computational and Theoretical Nanoscience, 5, 4, 581-586, (2008)
[13] Liu, H.; Yang, J. L., Elastic wave propagation in a single-layered graphene sheet on two-parameter elastic foundation via nonlocal elasticity, Physica E, 44, 7-8, 1236-1240, (2012)
[14] Lu, pin; Lee, H. P.; Lu, C., Dynamic properties of flexural beam using a nonlocal elasticity model, Journal of Applied Physics, 99, 7, 073510, (2006)
[15] Lu, Pin; Lee, H. P.; Lu, C.; Zhang, P. Q., Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, 44, 16, 5289-5300, (2007) · Zbl 1124.74029
[16] Ma, H. M.; Gao, X. L.; Reddy, J. N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56, 12, 3379-3391, (2008) · Zbl 1171.74367
[17] Murmu, T.; Adhikari, S., Nonlocal vibration of carbon nanotubes with attached buckyballs at tip, Mechanics Research communications, 38, 1, 62-67, (2007) · Zbl 1272.74294
[18] Murmu, T.; McCarthy, M. A.; Adhikari, S., Vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach, Journal of Sound and Vibration, 331, 23, 5069-5086, (2012)
[19] Narendar, S.; Gopalakrishnan, S., Nonlocal flexural wave propagation in an embedded graphene, International Journal of Computers, 6, 1, 29-36, (2012) · Zbl 1398.74123
[20] Payton, D.; Picco, L.; Miles, M. J.; Homer, M. E.; Champneys, A. R., Modelling oscillatory flexure modes of an atomic force microscope cantilever in contact mode whilst imaging at high speed, Nanotechnology, 23, 26, 265702, (2012)
[21] Pouresmaeeli, S.; Ghavanloo, E.; Fazelzadeh, S. A., Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium, Composite Structures, 96, 405-410, (2013)
[22] Pradhan, S. C., Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory, Finite Elements in Analysis and Design, 50, 1, 8-20, (2012)
[23] Reddy, J. N., Nonlocal continuum theories for buckling, bending and vibration of beams, International Journal of Engineering Science, 45, 2-8, 288-307, (2007) · Zbl 1213.74194
[24] Roque, C. M.C.; Ferreira, A. J.M.; Reddy, J. N., Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, International Journal of Engineering Science, 49, 9, 976-984, (2011) · Zbl 1231.74272
[25] Shen, Z. B.; Li, X. F.; Sheng, L. P.; Tang, G. J., Transverse vibration of nanotube-based micro-mass sensor via Timoshenko beam, Computational Materials Science, 53, 1, 340-346, (2012)
[26] Shen, Z. B.; Sheng, L. P.; Li, X. F.; Tang, G. J., Nonlocal Timoshenko beam theory for vibration of carbon nanotube-based biosensor, Physica E: Low-Dimensional Systems and Nanostructures, 44, 7-8, 1169-1175, (2012)
[27] Thai, Huu-Tai, A nonlocal beam theory for bending, buckling and vibration of nanobeam, International Journal of Engineering Science, 52, 3, 56-64, (2012) · Zbl 1423.74356
[28] Torabi, K.; Dastgerdi, J. Nafar, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using nonlocal elasticity model, Thin Solid Films, 520, 21, 6602-6695, (2012)
[29] Wang, Z. Y.; Li, F. M., Dynamical properties of nanotubes with nonlocal continuum theory: a review, Science China: Physics, Mechanics and Astronomy, 55, 7, 1210-1224, (2012)
[30] Wang, Q.; Liew, K. M., Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures, Physics Letter A, 363, 3, 236-242, (2007)
[31] Wang, Y. Z.; Li, F. M.; Kishimoto, K., Transactions of the ASME Journal of Vibration and Acoustics, 134, 3, (2012), 031011-1-7
[32] Wang, C. Y.; Zhang, J.; Fei, Y. Q.; Murmu, T., Circumferential nonlocal effect on vibrating nanotubules, International Journal of Mechanical Science, 58, 1, 86-90, (2012)
[33] Wang, C. M.; Zhang, Y. Y.; He, X. Q., Vibration of nonlocal Timoshenko beams, Nanotechnology, 18, 10, (2007), 105401-1-9
[34] Yadollahpour, M.; Ziaei-Rad, S.; Karimzadeh, F., Finite element modeling of damping capacity in nano-crystalline materials, International Journal of Modeling, Simulation, and Scientific Computing, 1, 3, 421-433, (2010)
[35] Yang, B.; Tan, C. A., Transfer functions of one-dimensional distributed parameter system, Translation of ASME, Journal of Applied Mechanics, 59, 4, 1009-1014, (1992) · Zbl 0825.73366
[36] Yan, Y.; Wang, W. Q.; Zhang, L. X., Free vibration of the fluid-filled single-walled carbon nanotube based on a double shell-potential flow model, Applied Mathematical Modeling, 36, 12, 6146-6153, (2012) · Zbl 1349.74189
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