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Vibration analysis of a tapered laminated thick composite plate with ply drop-offs

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Abstract

In this study, vibration characteristics of a tapered laminated thick composite plate have been investigated using finite element method by including the shear deformation and rotary inertia effects. The governing differential equations of motion of a tapered laminated thick composite plate are presented in the finite element formulation based on first-order shear deformation theory for three types of taper configurations. The effectiveness of the developed finite element formulation in identifying the various dynamic properties of a tapered laminated thick composite plate is demonstrated by comparing natural frequencies evaluated using the present FEM with those obtained from the experimental measurements and presented in the available literature. Various parametric studies are also performed to investigate the effect of taper configurations, aspect ratio, taper angle, angle ply orientation and boundary conditions on free and forced vibration responses of the structures. The comparison of the transverse free vibration mode shapes of the uniform and tapered composite plates under various boundary conditions is also presented. The forced vibration response of a composite plate is investigated to study the dynamic response of tapered composite plate under the harmonic force excitation in various tapered configurations. It is concluded that the dynamic properties of laminated thick composite plates could be tailored by dropping off the plies to yield various tapered composite plate.

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Acknowledgments

Authors are grateful to Aeronautics Research and Development Board, Defence Research and Development Organisation, India, for providing financial support through the project entitled ‘Vibration-based structural health monitoring and progressive Failure Analysis of a Rotating Tapered Composite Plate’ under the Grant No. DARO/08/1051682/M/I to carry out this work.

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Correspondence to R. Vasudevan.

Appendix

Appendix

$$\begin{aligned} {[}{k^{b}}]= & {} \int \limits _{-b}^b {{\int \limits _{-a}^a} {{[{B_{j}}^b (x,y)]}^\mathrm{T}\;\left[ {\begin{array}{l} A\;B \\ B\;D \\ \end{array}} \right] }} \;[B_j^b (x,y)]\;dx\;dy \end{aligned}$$
(24)
$$\begin{aligned} {[}k^{s}]= & {} \int \limits _{-b}^{b} {\int \limits _{-a}^a {[B_j^s (x,y)]^\mathrm{T}\;[H]}} \;[{B_{j}}^{s} (x,y)]\;\;dx\;dy \end{aligned}$$
(25)
$$\begin{aligned} {[}k^{e}]= & {} [k^{b}]+[k^{s}] \end{aligned}$$
(26)
$$\begin{aligned} {[}B_j^b (x,y)]= & {} \left[ {{\begin{array}{lllll} {N_{1,x}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {N_{1,y}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ {N_{1,y}}&{}\quad {N_{1,x}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_{1,x}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {N_{1,y}} \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_{1,y}}&{}\quad {N_{1,x}} \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ \end{array}} \;{\begin{array}{l} {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ \end{array}}\; {\begin{array}{lllll} {N_{4,x}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {N_{4,y}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ {N_{4,y}}&{}\quad {N_{4,x}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_{4,x}}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {N_{4,y}} \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_{4,y}}&{}\quad {N_{4,x}} \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ \end{array}}} \right] \end{aligned}$$
(27)
$$\begin{aligned} {[}B_j^s (x,y)]= & {} \left[ {{\begin{array}{lllll} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {N_{1,y}}&{}\quad 0&{}\quad {N_1 } \\ 0&{}\quad 0&{}\quad {N_{1,x}}&{}\quad {N_1 }&{}\quad 0 \\ \end{array}}\;{\begin{array}{l} {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ \end{array}}\; {\begin{array}{lllll} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {N_{4,y}}&{}\quad 0&{}\quad {N_4 } \\ 0&{}\quad 0&{}\quad {N_{4,x}}&{}\quad {N_4 }&{}\quad 0 \\ \end{array}}} \right] \end{aligned}$$
(28)
$$\begin{aligned} {[}m^{e}]= & {} \int \limits _{-1}^1 {\int \limits _{-1}^1 {[N_j (x,y)]^\mathrm{T}\;[I]}} \;[N_j (x,y)]\;\;\hbox {d}x\;\hbox {d}y \end{aligned}$$
(29)
$$\begin{aligned} {[}N_j (x,y)]= & {} \left[ {{\begin{array}{lllll} {N_1 }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {N_1 }&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {N_1 }&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_1 }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {N_1 } \\ \end{array}}\;{\begin{array}{l} {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ {\cdots } \\ \end{array}}\;{\begin{array}{lllll} {\;N_4 }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {N_4 }&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {N_4 }&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {N_4 }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {N_4 } \\ \end{array}}} \right] \end{aligned}$$
(30)
$$\begin{aligned} \left[ {I } \right]= & {} \left[ {{\begin{array}{lllll} {I_T }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {I_T }&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad {I_T }&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {I_R }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {I_R } \\ \end{array}}} \right] \end{aligned}$$
(31)

where \(\;[H]=\left[ {{\begin{array}{ll} {kA_{44}}&{}\quad {kA_{45}} \\ {kA_{45}}&{}\quad {kA_{55}} \\ \end{array}}} \right] \) is the transverse shear stiffness matrix, \([k^{b}], [k^{s}][k^{e}]\) and \([m^{e}]\) are bending stiffness matrix, shear stiffness matrix, element stiffness matrix and element mass matrix. Bending stiffness matrix and shear stiffness matrix are evaluated by 2 \(\times \) 2 points and 1 \(\times \) 1 point Gaussian quadrature numerical integrations, respectively.

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Edwin Sudhagar, P., Ananda Babu, A., Vasudevan, R. et al. Vibration analysis of a tapered laminated thick composite plate with ply drop-offs. Arch Appl Mech 85, 969–990 (2015). https://doi.org/10.1007/s00419-015-1004-9

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