Free vibration analysis of a laminated honeycomb sandwich panel: a suggested analytical solution and a numerical validation

Notation

u, v, w_o

= Plate middle surface displacements in x, y and z directions;

w_B, w_S

= Bending and shear components of the plate displacement in z direction;

ν_xy, ν_yx

= Poisson’s ratios;

f(z)

= Shear stress shape function;

$M_x^B$, $M_y^B$, $M_{xy}^B$

= Moment resultants due to bending;

h

= Total sandwich thickness;

ε

= Total linear strain;

ε_x, ε_y

= Normal strains in x and y directions;

γ_yz, γ_xz, γ_xy

= Shear strains in yz, xz and xy planes;

ρ

= Density of the laminate material;

${\phi }_x^S$, ${\phi }_y^S$

= Shear curvatures in x and y directions;

σ_x, σ_y

= Normal stresses in x and y directions;

Q_ij

= Composite Material stiffness;

τ_xy, τ_yz, τ_xz

= Shear stresses in xy, yz and xz planes;

E_xx, E_yy

= Yong’s modulus in x and y directions;

ω

= Frequency of vibration;

u₀, v₀

= Plate initial displacements in x and y directions;

G_xy, G_yz, G_xz

= Shear modulus in xy, yz and xz planes;

N_x, N_y, N_xy

= Force resultants in x, y and xy;

t, z

= Time and thickness coordinate;

$M_x^S$, $M_y^S$, $M_{xy}^S$

= Moment resultants due to shearing;

Q_x, Q_y

= Higher order force resultants;

A, D,

= In-plane and bending stiffnesses;

$A_{ij}^S$, $D_{ij}^S$, $H_{ij}^S$

= Higher order stiffnesses;

${\phi }_x^B$, ${\phi }_y^B$

= Bending curvatures in x and y directions;

I_i

= Plate inertia;

U, V, K

= Strain, potential and kinetic energies;

Vol, A

= Volume and area of element;

p

= Intensity of distributed load;

a_mn, b_mn, c_mn, d_mn

= Constants to be determined;

a, b

= Plate dimensions; and

θ

= Orientation angle.

Introduction

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