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Nonlinear dynamic instability analysis of laminated composite thin plates subjected to periodic in-plane loads

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Abstract

In this paper, the dynamic instability of thin laminated composite plates subjected to harmonic in-plane loading is studied based on nonlinear analysis. The equations of motion of the plate are developed using von Karman-type of plate equation including geometric nonlinearity. The nonlinear large deflection plate equations of motion are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu-Hill equations. Dynamically unstable regions, and both stable- and unstable-solution amplitudes of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both antisymmetric and symmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the orthotropy, magnitude of both tensile and compressive longitudinal loads, aspect ratios of the plate including length-to-width and length-to-thickness ratios, and in-plane transverse wave number on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.

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Abbreviations

\(A_{ij}\) , \(B_{ij}\), \(D_{ij}\) :

Extensional, coupling, bending stiffness

\(\epsilon _{xx}^{(0)}\), \(\epsilon _{yy}^{(0)}\) , \(\gamma _{xy}^{(0)}\) :

Membrane strains

\(\epsilon _{xx}^{(1)}\), \(\epsilon _{yy}^{(1)}\) , \(\gamma _{xy}^{(1)}\) :

Flexural (bending) strains

\(E_{1}\) , \(E_{2}\), \(G_{12}\) , \(\upsilon _{12}\) , \(\upsilon _{21}\) :

Engineering constants of orthotropic composite ply

h :

Plate thickness

a :

Plate length

b :

Plate width

m :

Longitudinal half-wave number

n :

In-plane transverse half-wave number

\(\lambda _{m}\) :

\(m\pi / a\)

\(\lambda _{n}\) :

\(n\pi b\)

\(F_{xx}\left( t \right) \) :

Pulsating longitudinal load

\(F_{s}\) :

Static component of \(F_{xx}\left( t \right) \)

\(F_{d}\) :

Harmonic component of \(F_{xx}\left( t \right) \)

P :

Excitation frequency

p :

Nondimensionalized P

\(q_{mn}\) :

Generalized coordinate

t :

Time

\(u_{0}\) , \(v_{0}\) , \(w_{0}\) :

Orthogonal components of mid-plane displacement functions

(X, Y, Z):

Plate coordinates

\(\rho \) :

Mass density

\(\rho _{t}\) :

Mass density per unit length

(\(\mathrm{N}_{\mathrm{xx}}\) , \(\mathrm{N}_{\mathrm{yy}}\), \(\mathrm{N}_{\mathrm{xy}})\) :

The total in-plane force resultants

(\({\mathrm{M}}_{{\mathrm{xx}}}\) , \({\mathrm{M}}_{{\mathrm{yy}}}\), \({\mathrm{M}}_{{\mathrm{xy}}})\) :

The total moment resultants

\(N_{\mathrm{cr}}\) :

Static buckling load

A :

Amplitude of steady-state vibrations

\({\Omega }_{mn}\) :

Frequency of the free vibration

\({\upmu }_{mn}\) :

Excitation parameter

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Acknowledgements

The authors thank the Faculty of Engineering and Computer Science at Concordia University, and Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support for the research work conducted. The NSERC support was provided through the Discovery Grant # 172848-2012 to the second author of the present paper.

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Authors and Affiliations

  1. Department of Mechanical, Industrial and Aerospace Engineering, Concordia Center for Composites, Concordia University, 1455 de Maisonneuve Blvd. W, Montreal, QC, Canada

    M. Darabi & R. Ganesan

Authors
  1. M. Darabi
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  2. R. Ganesan
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Corresponding author

Correspondence to R. Ganesan.

Appendix A

Appendix A

$$\begin{aligned} M_{mn}= & {} -16 {\uppi } a^{4} \,A22\,{\uprho }_{t}A11(A22\,A66\,a^{4}\,n^{4}+n^{2}(A11 A22\\&-\,A12(A12+2A66)) m^{2}b^{2}a^{2}+A11 \,A66 \,b^4 m^{4})mb^{4}\\ Q_{mn}= & {} 16 \,A11 \,A33 \,a^{2} m^{3} (A22 \,A66 \,a^{4}n^{4}+ n^{2}(A11\,A22\\&-\,A12 (A12+2 A66))m^{2}\,b^{2}\,a^{2} +A11 \,A66 \,b^{4}m^{4}){\uppi }^{3}b^{4} \\ {\upeta }_{mn}= & {} (A11 b^{4} \,m^{4}+ A22\, a^{4}n^{4}) {\uppi }^{5}m (A11 \,A22\\&-\,A12^2)(A22 \,A66 \,a^{4}\,n^{4}+n^2(A11 \,A22-A12 (A12\\&+\,2 A66)) m^2 b^2a^2+A11 \,A66 b^{4} m^{4}) \\ K_{mn}= & {} 16m \,A22 \,A11{\uppi }^{5}(b^{8}\,A66 (-A11 \,D11 +B11^{2})m^{8}\\&+\,(D11\, A12^{2}+ (2 D11 \,A66-2B11 (B12\\&+\,2B66))A12+(-D11 \,A22 + (-2 D12-4 D66)A66\\&+\,(B12+2B66)^2)A11+B11^{1}\, A22) a^{2}\,n^{2}\,b^{6}\,m^{6}\\&+\,2\,a^{4} n^{4}\,b^{4}((D12+2D66)A12^{2}\\&+\,((2D12+4D66)A66-B22 \,B11-B12^{2}\\&-\,4B66\,B12-4B66^2)A12+ \left( (-2D66-D12)A22 \right. \\&\left. -\,\frac{1}{2}D22\,A66+B22(B12+2 B66)\right) A11 \\&+\left( \frac{1}{2}D11 A66+ B11 (B12+2B66)\right) A22\\&-\,B11\,B22\,A66) m^4+(D22 \,A12^{2} \\&+\,(2 D22 \,A66 -2 B22(B12+2B66))A12\\&+\,(-A22\,D22+B22^2)A11+A22((-2D12\\&-\,4D66)A66+(B12+2B66)^2))a^{6}\,n^{6}b^{2}m^{2}\\&+\,a^{8}n^{8}A66(-A22 \,D22 +B22^{2})) \end{aligned}$$

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Darabi, M., Ganesan, R. Nonlinear dynamic instability analysis of laminated composite thin plates subjected to periodic in-plane loads. Nonlinear Dyn 91, 187–215 (2018). https://doi.org/10.1007/s11071-017-3863-9

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