Internal resonances in non-linear vibrations of a laminated circular cylindrical shell

Internal resonances in geometrically non-linear forced vibrations of laminated circular cylindrical shells are investigated by using the Amabili–Reddy higher-order shear deformation theory. A harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retain dissipation. Numerical results are obtained by using the pseudo-arc length continuation method and bifurcation analysis. A one-to-one-to-two internal resonance is identified, giving rise to pitchfork and Neimark–Sacher bifurcations of the non-linear response. A threshold level in the excitation has been observed in order to activate the internal resonance.

[1]  Marco Amabili,et al.  A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach , 2003 .

[2]  Aouni A. Lakis,et al.  Non-linear free vibration analysis of laminated orthotropic cylindrical shells , 1998 .

[3]  Eelco Jansen,et al.  Effect of Boundary Conditions on Nonlinear Vibration and Flutter of Laminated Cylindrical Shells , 2008 .

[4]  M. Toorani,et al.  Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells , 2003 .

[5]  Arcangelo Messina,et al.  Ritz-type dynamic analysis of cross-ply laminated circular cylinders subjected to different boundary conditions , 1999 .

[6]  Marco Amabili,et al.  Nonlinear Vibrations and Stability of Shells and Plates , 2008 .

[7]  Pedro Ribeiro,et al.  On the influence of membrane inertia and shear deformation on the geometrically non-linear vibrations of open, cylindrical, laminated clamped shells , 2009 .

[8]  J. N. Reddy,et al.  A higher-order shear deformation theory of laminated elastic shells , 1985 .

[9]  E. Carrera,et al.  Effects of thickness stretching in functionally graded plates and shells , 2011 .

[10]  R. A. Shenoi,et al.  Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory , 2002 .

[11]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[12]  James Martin Whitney,et al.  Theory of laminated plates , 1970 .

[13]  E. L. Jansen,et al.  A perturbation method for nonlinear vibrations of imperfect structures: Application to cylindrical shell vibrations , 2008 .

[14]  Alexander F. Vakakis,et al.  Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 2: Perturbation Analysis , 2000 .

[15]  Marco Amabili,et al.  A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells , 2010 .

[16]  Eelco Jansen,et al.  The effect of static loading and imperfections on the nonlinear vibrations of laminated cylindrical shells , 2008 .

[17]  C. Y. Chia,et al.  Non-linear vibration and postbuckling of generally laminated circular cylindrical thick shells with non-uniform boundary conditions , 1993 .

[18]  C. Chia,et al.  Effect of transverse shear on nonlinear vibration and postbuckling of anti-symmetric cross-ply imperfect cylindrical shells , 1988 .

[19]  T. K. Varadan,et al.  Application of a field-consistent shear flexible element for nonlinear dynamic analysis of laminated shells , 1992 .

[20]  A. Vakakis,et al.  Nonlinear vibrations and multiple resonances of fluid-filled, circular shells, part 1: Equations of motion and numerical results , 2000 .

[21]  T. K. Varadan,et al.  Nonlinear free flexural vibrations of laminated circular cylindrical shells , 1995 .

[22]  Erasmo Carrera,et al.  Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

[23]  J. Nayfeh,et al.  Nonlinear Vibration of Composite Shell Subjected to Resonant Excitations , 2000 .

[24]  M. Amabili.  Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories , 2011 .

[25]  H. Matsunaga.  Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory , 2007 .

[26]  C. Y. Chia,et al.  Non-linear vibration and postbuckling of unsymmetric cross-ply circular cylindrical shells , 1988 .