Abstract
In this paper, the nonlinear free vibration of a stringer shell is studied. The mathematical model of the string shell, which is the most convenient for frequency analysis, is considered. Due to the geometrical properties of the vibrating shell, strong nonlinearities are evident. Approximate analytical expressions for the nonlinear vibration are provided by introducing the extended version of the Hamiltonian approach. The method suggested in the paper gives the approximate solution for the differential equation with dissipative term for which the Lagrangian exists. The aim of this study is to provide engineers and designers with an easy method for determining the shell nonlinear vibration frequency and nonlinear behavior. The effects of different parameters on the ratio of nonlinear to linear natural frequency of shells are studied. This analytical representation gives excellent approximations to the numerical solutions for the whole range of the oscillation amplitude, reducing the respective error of the angular frequency in comparison with the Hamiltonian approach. This study shows that a first-order approximation of the Hamiltonian approach leads to highly accurate solutions that are valid for a wide range of vibration amplitudes.
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Abbreviations
- φ :
-
Airy function
- w :
-
Normal displacement
- E :
-
Young’s modulus of shell
- E 1 :
-
Young’s modulus of rib
- ν :
-
Poisson’s ratio
- h :
-
Shell thickness
- t :
-
Time
- R :
-
Shell radius
- ρ 0 :
-
Densities of shell
- ρ 1 :
-
Densities of rib
- N :
-
Number of stringer
- F :
-
Square stringer cross-section
- I :
-
Statical moment of stringer cross-section
- A :
-
Dimensionless maximum amplitude of oscillation
- ω NL :
-
Nonlinear frequency
- ω L :
-
Linear frequency
References
Roth R.S., Klosner J.M.: Nonlinear response of cylindrical shells subjected to dynamic axial loads. AIAA J. 2, 1788–1794 (1964)
Tamura Y.S., Babcock C.D.: Dynamic stability of cylindrical shells under step loading. J. Appl. Mech. 42, 190–194 (1975)
Simitses J.G.: Effect of static preloading on the dynamic stability of structures. AIAA J. 21, 1174–1180 (1982)
Liew K.M., Lim C.W., Ong L.S.: Flexural vibration of doubly-tapered cylindrical shallow shells. Int. J. Mech. Sci. 36, 547–565 (1994)
Mikulars, M., Mcelman, J.A.: On free vibrations of eccentrically stiffened cylindrical shells and flat plates. NASA 1N D-3010 (1965)
Bayat, M., Pakar, I., Domaiirry, G.: Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review. Latin Am. J. Solids Struct. 9(2), 145–234 (2012)
Cveticanin L.: Homotopy-perturbation method for pure non-linear differential equation. Chaos Solitons Fract. 30, 1221–1230 (2006)
Cveticanin L.: Application of homotopy-perturbation to non-linear partial differential equations. Chaos Solitons Fract. 40, 221–228 (2009)
Pakar I., Bayat M.: Vibration analysis of high nonlinear oscillators using accurate approximate methods. Struct. Eng. Mech. 46(1), 137–151 (2013)
Bayat M., Pakar I.: Application of he’s energy balance method for nonlinear vibration of thin circular sector cylinder. Int. J. Phys. Sci. 6(23), 5564–5570 (2011)
He J.H.: Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun. 29, 107–111 (2002)
He J.H.: Determination of limit cycles for strongly nonlinear oscillators. Phys. Rev. Lett. 90, 174–181 (2006)
Pakar I., Bayat M.: Analytical solution for strongly nonlinear oscillation systems using energy balance method. Int. J. Phys. Sci. 6(22), 5166–5170 (2011)
Bayat M., Pakar I.: On the approximate analytical solution to non-linear oscillation systems. Shock Vib. 20(1), 43–52 (2013)
Bayat M., Pakar I., Bayat M.: Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell. Steel Compos. Struct. 14(5), 511–521 (2013)
Pakar I., Bayat M.: Analytical study on the non-linear vibration of Euler–Bernoulli beams. J. Vibroeng. 14(1), 216–224 (2012)
Bayat M., Shahidi M., Bayat M.: Application of Iteration Perturbation Method for nonlinear oscillators with discontinuities. Int. J. Phys. Sci. 6(15), 3608–3612 (2011)
Pakar I., Bayat M.: An analytical study of nonlinear vibrations of buckled Euler–Bernoulli beams. Acta Physica Polonica A 123(1), 48–52 (2013)
Bayat M., Pakar I., Shahidi M.: Analysis of nonlinear vibration of coupled systems with cubic nonlinearity. Mechanika 17(6), 620–629 (2011)
Pakar I., Bayat M., Bayat M.: On the approximate analytical solution for parametrically excited nonlinear oscillators. J. vibroeng. 14(1), 423–429 (2012)
He J.H.: Hamiltonian approach to nonlinear oscillators. Phys. Lett. A 374, 2312–2314 (2010)
Bayat M., Pakar I.: Nonlinear free vibration analysis of tapered beams by Hamiltonian approach. J. Vibroeng. 13(4), 654–661 (2011)
Bayat, M., Pakar, I., Bayat, M.: On the large amplitude free vibrations of axially loaded Euler–Bernoulli beams. Steel Compos. Struct. 14(1), 73–83 (2013)
Bayat, M., Pakar, I.: Accurate analytical solution for nonlinear free vibration of beams. Struct. Eng. Mech. 43(3), 337–347 (2012)
Filippov, S.B.: Theory of Conjugated and Reinforced Shells. St. Petersburg State University (1999); in Russian
Evakin A.Yu., Kalamkarov A.: Analysis of large deflection equilibrium state of composite shells of revolution. Part 1. General model and singular perturbation analysis. Int. J. Solids Struct. 38, 8961–8974 (2001)
Grigolyuk, E.I., Kabanov, V.V.: Stability of Shells. Nauka, Moscow (1987); in Russian
Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I.: Asymptotical Mechanics of Thin-Walled Structures. Springers, Berlin (2004); Printed in Germany
Amabili M.: Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach. J. Sound Vib. 264, 1091–1125 (2003)
Amabili M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)
Leissa A.W., Kadi A.S.: Large amplitude free vibration of thick shallow shells supported by shear diaphragms. J. Sound Vib. 16(5), 173–187 (1971)
Amabili M.: Non-linear vibrations of circular cylindrical panels. J. Sound Vib. 281, 509–535 (2005)
Amiro I.Ya., Zartasky V.A.: Studies of the dynamics of ribbed shells. Soviet Appl. Mech. 17(11), 949–962 (1981)
Amiro, I.Ya., Zartasky, V.A.: The Theory of Ribbed Shells. Naukova Dumka, Kiev (1980); in Russian
Cveticanin L., Kalami-Yazdi M., Saadatnia Z., Askari H.: Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power. Int. J. Nonlinear Sci. Numer. Simul. 11(12), 997–1001 (2010)
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Bayat, M., Pakar, I. & Cveticanin, L. Nonlinear vibration of stringer shell by means of extended Hamiltonian approach. Arch Appl Mech 84, 43–50 (2014). https://doi.org/10.1007/s00419-013-0781-2
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DOI: https://doi.org/10.1007/s00419-013-0781-2