Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia. (English) Zbl 1193.74053
Summary: The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov-Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.
MSC:
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K25 | Shells |
| 74D05 | Linear constitutive equations for materials with memory |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
Keywords:
Timoshenko theory; Kirchhoff-Love theory; rotatory inertia; shear deformation; viscoelasticity; kernel of relaxation; circular cylindrical shell; nonlinear vibrations; Bubnov-Galerkin methodReferences:
| [1] | Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York (1987) · Zbl 0114.40801 |
| [2] | Donnell, L.H.: Beams, Plates, and Shells. McGraw-Hill, New York (1976) · Zbl 0375.73053 |
| [3] | Volmir, A.S.: The Nonlinear Dynamics of Plates and Shells. Nauka, Moscow (1972) (in Russian) |
| [4] | Timoshenko, S.P.: Vibration Problems in Engineering. Van Nostrand, New York (1958) · JFM 63.1305.03 |
| [5] | Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 19(1), 31–38 (1951) · Zbl 0044.40101 |
| [6] | Uflyand, Ya.S.: Distribution of waves at transverse vibrations of cores and plates. Appl. Math. Mech. 12(3), 287–300 (1948) |
| [7] | Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69–A88 (1945) · Zbl 0063.06470 |
| [8] | Ambartsumyan, S.A.: General Theory of Anisotropic Shells. Nauka, Moscow (1974) (in Russian) · Zbl 0321.60010 |
| [9] | Il’yushin, A.A., Pobedrya, B.E.: Fundamentals of the Mathematical Theory of Thermoviscoelasticity. Nauka, Moscow (1970) (in Russian) |
| [10] | Christensen, R.M.: Theory of Viscoelasticity. Academic, New York (1971) · Zbl 0215.48303 |
| [11] | Rzhanitsyn, A.R.: Theory of Creep in Russian. Stroyizdat, Moscow (1968) |
| [12] | Rabotnov, Yu.N.: Elements of the Hereditary Mechanics of Solids. Nauka, Moscow (1977) (in Russian) |
| [13] | Bogdanovich, A.E.: Nonlinear Dynamic Problems for Composite Cylindrical Shells. Elsevier Science, New York (1993) |
| [14] | Awrejcewicz, J., Krys’ko, V.A.: Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. Applications of the Bubnov–Galerkin and Finite Difference Numerical Methods. Springer-Verlag, Berlin (2003) · Zbl 1015.74001 |
| [15] | Kubenko, V.D., Koval’chuk, P.S.: Influence of initial geometric imperfections on the vibrations and dynamic stability of elastic shells. Int. Appl. Mech. 40(8), 847–877 (2004) · Zbl 1130.74394 · doi:10.1023/B:INAM.0000048679.54437.f8 |
| [16] | Kayuk, Ya.F.: Geometrically Nonlinear Problems of the Theory of Plates and Shells. Naukova Dumka, Kiev, Ukraine (1987) (in Russian) |
| [17] | Amabili, M.: Nonlinear vibrations of circular cylindrical Panels. J. Sound Vib. 281, 509–535 (2005) · doi:10.1016/j.jsv.2004.01.021 |
| [18] | Dogan, V., Vaicaitis, R.: Nonlinear response of cylindrical shells to random excitation. Nonlinear Dyn. 20, 33–53 (1999) · Zbl 0988.74033 · doi:10.1023/A:1008398007849 |
| [19] | Shirakawa, K.: Effects of shear deformation and rotatory inertia on vibration and buckling of cylindrical shells. J. Sound Vib. 91(3), 425–437 (1983) · Zbl 0526.73074 · doi:10.1016/0022-460X(83)90289-4 |
| [20] | Krysko, V.A., Awrejcewicz, J., Bruk, V.: On the solution of a coupled thermo-mechanical problem for non-homogeneous Timoshenko-type shells. J. Math. Anal. Appl. 273(2), 409–416 (2002) · Zbl 1137.74402 · doi:10.1016/S0022-247X(02)00247-0 |
| [21] | Okazaki, A., Tatemichi, A.: Damping properties of two-layered cylindrical shells with an unconstrained viscoelastic layer. J. Sound Vib. 176(2), 145–161 (1994) · Zbl 0945.74575 · doi:10.1006/jsvi.1994.1365 |
| [22] | Cheng, C.-J., Zhang, N.-H.: Dynamical behavior of viscoelastic cylindrical shells under axial pressures. Appl. Math. Mech. 22(1), 1–9 (2001) · Zbl 1050.74581 · doi:10.1023/A:1015538531246 |
| [23] | Kozlov, V.I., Karnaukhova, T.V.: Basic equations for viscoelastic laminated shells with distributed piezoelectric inclusions intended to control nonstationary vibrations. Int. Appl. Mech. 38(10), 1253–1260 (2002) · Zbl 1094.74646 · doi:10.1023/A:1022266614651 |
| [24] | Potapov, V.D.: Stability of Stochastic Elastic and Viscoelastic Systems. Wiley, Chichester, UK (1999) · Zbl 0953.74035 |
| [25] | Koltunov, M.A.: Creep and Relaxation. Visshaya Shkola, Moscow (1976) (in Russian) |
| [26] | Badalov, F.B., Eshmatov, Kh.,Yusupov, M.:Oncertain methods of solving of systems of integrodifferential equations encountered in viscoelasticity problems. Appl. Math. Mech. [Transl. Prikl. Mat. Mekh.] 51(5), 683–686 (1987) · Zbl 0681.73024 |
| [27] | Badalov, F.B., Eshmatov, Kh.: The brief review and comparison of integrated methods of mathematical modeling in problems of hereditary mechanics of rigid bodies. Int. J. Electron. Model 11(2), 81–90 (1989) |
| [28] | Badalov, F.B., Eshmatov, Kh.: Investigation of nonlinear vibrations of viscoelastic plates with initial imperfections. Sov. Appl. Mech. [Transl. Prikl. Mekh.] 26(8), 99–105 (1990) · Zbl 0736.73034 |
| [29] | Badalov, F.B., Eshmatov, Kh., Akbarov, U.I.: Stability of a viscoelastic plate under dynamic loading. Sov. Appl. Mech. [Transl. Prikl. Mekh.] 27(9), 892–899 (1991) · Zbl 0925.73225 · doi:10.1007/BF00887982 |
| [30] | Verlan, A.F., Eshmatov, B.Kh.: Mathematical simulation of oscillations of orthotropic viscoelastic plates with regards to geometric nonlinearity. Int. J. Electron. Model. 27(4), 3–17 (2005) |
| [31] | Eshmatov, B.Kh.: Mathematical modeling of problems of nonlinear vibrations viscoelastic orthotropic plates. In: Materials of International Conference on Computational and Informational Technologies for Research, Engineering, and Education, pp. 359–366, Alma-Ata, Kazakhstan, 7–9 October (2004) |
| [32] | Eshmatov, B.Kh.: Nonlinear vibrations of viscoelastic orthotropic cylindrical shells in view of propagation of elastic waves. In: Materials of the XVII Session of the International School of Models of the Mechanics of the Continuous Environment, pp. 186–191, Kazan, Russia, 4–10 July (2004) |
| [33] | Eshmatov, B.Kh.: Nonlinear vibrations of viscoelastic orthotropic plates from composite materials. In: Third M.I.T. Conference on Computational Fluid and Solid Mechanics, Boston, MA, USA, 14–17 June 2005 |
| [34] | Eshmatov, B.Kh.: Dynamic stability of viscoelastic plates at growing compressing loadings. Int. J. Appl. Mech. Tech. Phys. 47(2), 289–297 (2006) · Zbl 1115.74032 · doi:10.1007/s10808-006-0055-7 |
| [35] | Eshmatov, B.Kh.: Mathematical model of problem about nonlinear vibrations and dynamical stability of viscoelastic orthotropic shells taking into account shear deformation and rotation inertia. Republican Sci. J. Proc. Acad. Sci. Republic Uzbekistan 1, 27–30 (2005) |
| [36] | Eshmatov, B.Kh.: Nonlinear vibrations and dynamic stability of viscoelastic orthotropic rectangular plates. J. Sound Vib. (to be published in 2007) · Zbl 1193.74053 |
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