Abstract
The large amplitude free vibration of a laminated composite parabolic plate with parabolically orthotropic plies is investigated for the first time. The effects of out-of-plane shear deformations, rotary inertia, and geometrical nonlinearity are taken into account. The geometry of the plate is described, and the analysis performed in the parabolic coordinate system. The problem is solved numerically using a new parabolic hierarchical finite element. The nonlinear equations of free motion are mapped from the time domain into the frequency domain using the harmonic balance method. The resultant nonlinear equations are solved iteratively using the linearized updated mode method. Results for the fundamental linear and nonlinear frequencies are obtained for symmetric and antisymmetric laminates with clamped and simply supported edges. Comparisons are made with the finite element method for clamped and free isotropic parabolic plates and show excellent agreement. The aspect ratio, thickness ratio, moduli ratio, number of plies, layup sequence, and boundary conditions are shown to affect the hardening behavior.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- x, y :
-
Cartesian coordinates
- α, β :
-
Parabolic coordinates
- ξ, η :
-
Non-dimensional coordinates
- h :
-
Thickness
- t :
-
Time
- \({P_j^\ast}\) :
-
Shifted Legendre orthogonal polynomial of order j
- u, v :
-
In-plane displacements
- w :
-
Out-of-plane displacement
- ψ α :
-
Rotation about α axis
- ψ β :
-
Rotation about β axis
- \({\varepsilon_{\alpha\alpha},\varepsilon_{\beta \beta}, \gamma_{\alpha \beta},\gamma_{\alpha z}, \gamma_{\beta z}}\) :
-
Strains
- χ α , χ β , χ α β :
-
Curvatures
- p :
-
Degree of interpolating polynomial
- ρ :
-
Averaged mass density
- E α , E β :
-
Elastic moduli in α, β directions
- G α β , G α z , G β z :
-
Shear moduli in α β, α z, β z planes
- ν α β :
-
Poisson’s ratio in α β plane
- Q 11, Q 22, Q 12, Q 66 :
-
Plane stress reduced stiffnesses
- Q 44, Q 55 :
-
Out-of-plane shear reduced stiffnesses
- N :
-
Number of plies
- κ :
-
= π 2/12 (shear correction factor)
- U :
-
Strain energy
- T :
-
Kinetic energy
- \({\overline{{\bf K}}}\) :
-
In-plane stiffness matrix
- K :
-
Out-of-plane stiffness matrix
- K͝ :
-
Coupling stiffness matrix
- \({\hat{{\bf K}}_1, \tilde {{\bf K}}_{11}}\) :
-
Nonlinear stiffness matrices
- \({\overline{{\bf M}}}\) :
-
In-plane mass matrix
- M :
-
Out-of-plane mass matrix
- \({\overline{{\bf q}}}\) :
-
In-plane displacement vector
- q :
-
Out-of-plane displacement vector
- w max :
-
Maximum amplitude
- ξ 0, η 0 :
-
Non-dimensional coordinates of point of maximum amplitude
- ω :
-
Natural frequency
- Ω:
-
\({=\omega \alpha _0^2 \sqrt{\rho /{E_\beta}}}\) (non-dimensional frequency parameter)
References
Leissa A.W.: Plate vibration research, 1976–1980: classical theory. Shock Vib. Dig. 13, 11–22 (1981)
Leissa A.W.: Plate vibration research, 1976–1980: complicating effects. Shock Vib. Dig. 13, 19–36 (1981)
Bert C.W.: Research on dynamic behavior of sandwich and composite plates-IV. Shock Vib. Dig. 17, 3–15 (1985)
Chia C.Y.: Geometrically nonlinear behavior of composite plates: a review. Appl. Mech. Rev. 41, 439–451 (1988)
Kapania R.K., Raciti S.: Recent advances in analysis of laminated beams and plates, part II: Vibration and wave propagation. AIAA J. 27, 935–946 (1989)
Yu Y.Y.: Some recent advances in linear and nonlinear dynamical modeling of elastic and piezoelectric plates. J. Intell. Mater. Syst. Struct. 6, 237–254 (1995)
Sathyamoorthy M.: Nonlinear vibrations of plates: an update of recent research developments. Appl. Mech. Rev. 49, S55–S62 (1996)
Houmat A.: Nonlinear free vibration of a shear deformable laminated composite annular elliptical plate. Acta Mech. 208, 281–297 (2009)
Pal A., Bera R.K.: Large amplitude free vibration of sandwich parabolic plates-revisited. Comput. Math. Appl. 41, 513–522 (2001)
Honda S., Oonishi Y., Narita Y., Sasaki K.: Vibration analysis of composite rectangular plates reinforced along curved lines. J. Syst. Des. Dyn. 2, 76–86 (2008)
Han W., Petyt M.: Geometrically nonlinear vibration analysis of thin rectangular plates using the hierarchical finite element method—II: 1st mode of laminated plates and higher modes of isotropic and laminated plates. Comput. Struct. 63, 309–318 (1997)
Ribeiro P., Petyt M.: Nonlinear vibration of composite laminated plates by the hierarchical finite element method. Compos. Struct. 46, 197–208 (1999)
Houmat A.: Large amplitude free vibration of shear deformable laminated composite annular sector plates by a sector p-element. Int. J. Non Linear Mech. 43, 834–843 (2008)
Han W., Petyt M.: Geometrically nonlinear vibration analysis of thin rectangular plates using the hierarchical finite element method—I: the fundamental mode of isotropic plates. Comput. Struct. 63, 295–308 (1997)
Houmat A.: Mapped infinite p-element for two-dimensional problems of unbounded domains. Comput. Geotech. 35, 608–615 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Houmat, A. Large amplitude free vibration of a shear deformable laminated composite parabolic plate with parabolically orthotropic plies. Acta Mech 223, 145–160 (2012). https://doi.org/10.1007/s00707-011-0550-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0550-7