Nonlocal forced vibrations of rotating cantilever nano-beams. (English) Zbl 1484.74028
Summary: This study presents a dynamic analysis of a single rotating nonlocal cantilever nano-beam under external excitations. The cases of undamped and damped forced vibrations are analyzed. By employing Eringen’s nonlocal elasticity theory and based on Euler-Bernoulli’s beam theory, the governing equation of motion of the forced vibration rotating nonlocal cantilever nano-beam is derived. The mentioned equation of motion is discretized by the Galerkin method. In the paper, the standard modal analysis procedure is used for determining the forced responses. The novelty of the study lies in the transient responses of the rotating cantilever nano-beams with the nonlocality magnitude effect taken into consideration. In the parametric study the influences of the varying angular velocity and varying hub radius effects are presented. The solutions for the natural frequencies of the rotating system are determined. For the undamped vibrations are carry out for two analysis of time histories. Time histories have clearly described the impact of contemplated effects for a longer time which causes periodical response. An interesting phenomenon can be noted in the case of higher values of an angular velocity effect. Only in this case, the periodical response has not been observed in the transverse deflection. The study also includes an influence of the axial extension load. Even though the qualitative impact of the axial external load is known, the quantitative effect will be shown here.
MSC:
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
| 74M25 | Micromechanics of solids |
Keywords:
forced vibration; Eringen nonlocal elasticity; Euler-Bernoulli cantilever nano-beam; Galerkin method; modal analysisReferences:
| [1] | Babaei, A.; Yang, C. X., Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field, Microsyst. Technol., 1-9 (2018) |
| [2] | Bhat, R. B., Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method, J. Sound Vib., 105, 2, 199-210 (1986) |
| [3] | Chen, C. O.K.; Ho, S. H., Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform, Int. J. Mech. Sci., 41, 11, 1339-1356 (1999) · Zbl 0945.74029 |
| [4] | Dejin, C.; Kai, F.; Shijie, Z., Flapwise Vibration Analysis of Rotating Composite Laminated Timoshenko Microbeams with Geometric Imperfection Based on a Re-modified Couple Stress Theory and Isogeometric Analysis (2019), European Journal of Mechanics-A/Solids · Zbl 1470.74033 |
| [5] | Diken, H., Vibration control of a rotating Euler-Bernoulli beam, J. Sound Vib., 232, 3, 541-551 (2000) · Zbl 1237.74059 |
| [6] | Eringen, A. C., Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci., 10, 5, 425-435 (1972) · Zbl 0241.73005 |
| [7] | Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 9, 4703-4710 (1983) |
| [8] | Eringen, A. C., Nonlocal Continuum Field Theories (2002), Springer Science & Business · Zbl 1023.74003 |
| [9] | Eringen, A. C.; Edelen, D. G.B., On nonlocal elasticity, Int. J. Eng. Sci., 10, 3, 233-248 (1972) · Zbl 0247.73005 |
| [10] | Guo, S.; He, Y.; Liu, D.; Lei, J.; Li, Z., Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving and rotating nanobeams based on nonlocal strain gradient theory, Microsyst. Technol., 24, 2, 963-977 (2018) |
| [11] | Gurtin, M. E.; Weissmüller, J.; Larche, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. Mag. A, 78, 5, 1093-1109 (1998) |
| [12] | Han, S. M.; Benaroya, H.; Wei, T., Dynamics of transversely vibrating beams using four engineering theories, J. Sound Vib., 225, 5, 935-988 (1999) · Zbl 1235.74075 |
| [13] | Hoa, S. V., Vibration of a rotating beam with tip mass, J. Sound Vib., 67, 3, 369-381 (1979) · Zbl 0428.73059 |
| [14] | Hodges, D. Y.; Rutkowski, M. Y., Free-vibration analysis of rotating beams by a variable-order finite-element method, AIAA J., 19, 11, 1459-1466 (1981) · Zbl 0468.73093 |
| [15] | Hu, J.; Odom, T. W.; Lieber, C. M., Chemistry and physics in one dimension: synthesis and properties of nanowires and nanotubes, Acc. Chem. Res., 32, 5, 435-445 (1999) |
| [16] | Iijima, S., Helical microtubules of graphitic carbon, Nature, 354, 6348, 56 (1991) |
| [17] | Iijima, S.; Brabec, C.; Maiti, A.; Bernholc, J., Structural flexibility of carbon nanotubes, J. Chem. Phys., 104, 5, 2089-2092 (1996) |
| [18] | Kelly, S. G., Mechanical Vibrations: Theory and Applications (2012), Cengage learning |
| [19] | Koiter, W., Couple stresses in the theory of elasticity, I and II. Koninklijke Nederlandse Akademie van Wetenschappen, Proc. Roy. Soc. B, 67, 1, 17-44 (1964) · Zbl 0119.39504 |
| [20] | Lam, D. C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, 51, 8, 1477-1508 (2003) · Zbl 1077.74517 |
| [21] | Lu, J. P., Elastic properties of carbon nanotubes and nanoropes, Phys. Rev. Lett., 79, 7, 1297 (1997) |
| [22] | Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 1, 415-448 (1962) · Zbl 0112.38906 |
| [23] | Mohammadi, M.; Safarabadi, M.; Rastgoo, A.; Farajpour, A., Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mech., 227, 8, 2207-2232 (2016) · Zbl 1349.74180 |
| [24] | Murmu, T.; Adhikari, S., Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation, J. Appl. Phys., 108, 12, 123507 (2010) |
| [25] | Narendar, S., Mathematical modelling of rotating single-walled carbon nanotubes used in nanoscale rotational actuators, Def. Sci. J., 61, 4, 317 (2011) |
| [26] | Narendar, S.; Gopalakrishnan, S., Nonlocal wave propagation in rotating nanotube, Results Phys., 1, 1, 17-25 (2011) |
| [27] | Nayak, B.; Dwivedy, S. K.; Murthy, K. S.R. K., Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core, Eur. J. Mech. A Solid., 47, 143-155 (2014) · Zbl 1406.74399 |
| [28] | Ni, Q.; Li, M.; Tang, M.; Wang, L., Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid, J. Sound Vib., 333, 9, 2543-2555 (2014) |
| [29] | Pradhan, S. C.; Murmu, T., Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Phys. E Low-dimens. Syst. Nanostruct., 42, 7, 1944-1949 (2010) |
| [30] | Reich, S.; Thomsen, C.; Maultzsch, J., Carbon Nanotubes: Basic Concepts and Physical Properties (2008), John Wiley & Sons |
| [31] | Schilhansl, M., Bending frequency of a rotating cantilever beam, J. Appl. Mech., 25, 1 (1958) · Zbl 0079.39903 |
| [32] | Shafiei, N.; Kazemi, M.; Ghadiri, M., Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler-Bernoulli microbeams, Phys. E Low-dimens. Syst. Nanostruct., 83, 74-87 (2016) |
| [33] | Shafiei, N.; Mousavi, A.; Ghadiri, M., Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM, Compos. Struct., 149, 157-169 (2016) |
| [34] | Stamenković, M.; Karličić, D.; Goran, J.; Kozić, P., Nonlocal forced vibration of a double single-walled carbon nanotube system under the influence of an axial magnetic field, J. Mech. Mater. Struct., 11, 3, 279-307 (2016) |
| [35] | Stojanović, V.; Kozić, P., Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load, Int. J. Mech. Sci., 60, 1, 59-71 (2012) |
| [36] | Stojanović, V.; Kozić, P., Vibrations and Stability of Complex Beam Systems (2015), Springer International Publishing: Springer International Publishing Berlin · Zbl 1307.74004 |
| [37] | Toupin, R., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 1, 385-414 (1962) · Zbl 0112.16805 |
| [38] | Younesian, D.; Esmailzadeh, E., Vibration suppression of rotating beams using time-varying internal tensile force, J. Sound Vib., 330, 2, 308-320 (2011) |
| [39] | Zhang, B.; Chen, J.; Jin, L.; Deng, W.; Zhang, L.; Zhang, H.; Wang, Z. L., Rotating-disk-based hybridized electromagnetic-triboelectric nanogenerator for sustainably powering wireless traffic volume sensors, ACS Nano, 10, 6 (2016), 6241-6247.6247 |
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