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72. Shu, C.: Differential Quadrature and its Application in Engineering. Springer, London (2000)
73. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Europ. J. Mech. A/Sol. 29, 591–599 (2010)
74. Krysko, A.V., Awrejcewicz, J., Zhigalov, M.V., Pavlov, S.P., Krysko, V.A.: Nonlinear behaviour
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Part 2. Chaotic dynamics of flexible beams. Int. J. Non-Lin. Mech. 93, 106–121 (2017)
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195
54. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent
functionally graded microbeams. Int. J. Eng. Sci. 50(1), 256–267 (2012)
55. Krysko, V.A., Awrejcewicz, J., Komarov, S.A.: Nonlinear deformations of spherical panels
ssubjected to transversal load action. Comput. Meth. Appl. Mech. Eng. 194(27–29), 3108–
3126 (2005)
56. Krysko, V.A., Zhigalov, M.V., Yakovleva, T.V., Papkova, I.V.: The method of establishing in
nonlinear problems of beams and plates, taking into account the locality of loading. Bullet.
STGU 2, 7–17 (2012)
57. Krysko, V.A., Komarov, S.A., Egurnov, N.V.: Flexing of flexible plates under the action of
longitudinal and transverse loads. Appl. Mech. 32(9), 80–87 (1996)
58. Miller, K.: Least squares methods for ill-posed problems with a prescribed bound. SIAM J.
Math. Anal. 1(6), 52–74 (1970)
59. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first
kind. J. Assoc. Comput. Mach. 9(6), 84–97 (1962)
60. Soltani, P., Kassaei, A., Mehdi, M.T., Farshidianfar, A.: Vibration of wavy single-walled carbon
nanotubes based on nonlocal Euler Bernoulli and Timoshenko models. Int. J. Adv. Struct. Eng.
4(3) (2012)
61. Yang, X.D., Lim, C.W.: Nonlinear vibrations of nano-beams accounting for nonlocal ef-fect
using a multiple scale method. Sci. China Ser. E-Tech. Sci. 52(3), 617–626 (2009)
62. Kong, S.L., Zhou, S.J., Nie, Z.F., Wang, K.: Static and dynamic analysis of micro beams based
on strain gradient elasticity theory. Int. J. Eng. Sci. 47, 487–498 (2009)
63. Ke, L.L., Wang, Y.S.: Size effect on dynamic stability of functionally graded microbeams based
on a modified couple stress theory. Compos. Struct. 93, 342–350 (2011)
64. Wang, D.H., Wang, G.F.: Surface effects on the vibration and buckling of double-nanobeamsystems. J. Nanomater. 2011, ID 518706 (2011)
65. Xia, R., Li, X., Qin, Q., Liu, J., Feng, X.Q.: Surface effects on the mechanical properties of
nanoporous materials. Nanotechnology 22, 265714 (2011)
66. Batra, R.C., Porfiri, M., Spinello, D.: Vibrations of narrow microbeams predeformed by an
electric field. J. Sound Vib. 309, 600–612 (2008)
67. Hassanpour, P.A., Esmailzadeh, E., Cleghorn, W.L., Mills, J.K.: Nonlinear vibration of micromachined asymmetric resonators. J. Sound Vib. 329, 2547–2564 (2010)
68. Kitipornchai, S., Ke, L.L., Yang, J., Xiang, Y.: Nonlinear vibration of edge cracked functionally
graded Timoshenko beams. J. Sound Vib. 324, 962–982 (2009)
69. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli-Euler
micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)
70. Krysko, A.V., Awrejcewicz, J., Kutepov, I.E., Zagniboroda, N.A., Dobriyan, V., Krysko, V.A.:
Chaotic dynamics of flexible Euler-Bernoulli beams. Chaos 23(4), 043130–1–043130–25
(2013)
71. Li, X.F.: A unified approach for analyzing static and dynamic behaviors of functionally graded
Timoshenko and Euler-Bernoulli beams. J. Sound Vib. 318, 1210–1229 (2008)
72. Shu, C.: Differential Quadrature and its Application in Engineering. Springer, London (2000)
73. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Europ. J. Mech. A/Sol. 29, 591–599 (2010)
74. Krysko, A.V., Awrejcewicz, J., Zhigalov, M.V., Pavlov, S.P., Krysko, V.A.: Nonlinear behaviour
of different flexible size-dependent beams models based on the modified couple stress theory.
Part 2. Chaotic dynamics of flexible beams. Int. J. Non-Lin. Mech. 93, 106–121 (2017)
75. Awrejcewicz, J., Krysko, V.A.: Feigenbaum scenario exhibited by thin plate dynamics. Nonlin.
Dyn. 24, 373–398 (2006)
76. Ruelle, D., Takehs, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1976)