Composite stacking sequence optimization for aeroelastically tailored forward-swept wings

Abstract

A method for stacking sequence optimization and aeroelastic tailoring of forward-swept composite wings is presented. It exploits bend-twist coupling to mitigate aeroelastic divergence. The method proposed here is intended for estimating potential weight savings during the preliminary aircraft design stages. A structural beam model of the composite wingbox is derived from anisotropic shell theory and the governing aeroelastic equations are presented for a spanwise discretized forward swept wing. Optimization of the system to reduce wing mass is undertaken for sweep angles of −35° to 0° and Mach numbers from 0.7 to 0.9. A subset of lamination parameters (LPs) and the number of laminate plies in each pre-defined direction (restricted to {0°,±45°, 90°}) serve as design variables. A bi-level hybrid optimization approach is employed, making use of a genetic algorithm (GA) and a subsequent gradient-based optimizer. Constraints are implemented to match lift requirements and prevent aeroelastic divergence, excessive deformations, airfoil stalling and structural failure. A permutation GA is then used to match specific composite ply stacking sequences to the optimum design variables with a limited number of manufacturing constraints considered for demonstration purposes. The optimization results in positive bend-twist coupling and a reduced structural mass. Results are compared to an uncoupled reference wing with quasi-isotropic layups and with panel thickness alone the design variables. For a typical geometry and a forward sweep of −25° at Mach 0.7, a wingbox mass reduction of 13 % was achieved.

Appendices

Appendix A: Divergence calculation (Ritz method)

The matrix $[\boldsymbol {\mathcal {M}}]$ (21) is of the size (N _𝜃 + N _w )×(N _𝜃 + N _w ), and is given by:

$$\begin{array}{@{}rcl@{}} &&[\boldsymbol{\mathcal{M}}] = - [\boldsymbol{A}_{stiff}]^{-1} [\boldsymbol{A}_{aero}] \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} &&\left[\boldsymbol{A}_{stiff}\right] = \left[\begin{array}{cccccc} -\left[\boldsymbol{D}\right] & -\left[\boldsymbol{E}\right] \\ -\left[\boldsymbol{C}\right] & -\left[\boldsymbol{D}\right] \end{array}\right], \left[\boldsymbol{A}_{aero}\right] = \left[\begin{array}{cccccccc} \left[\boldsymbol{K}\right] & \left[\boldsymbol{Q}\right] \\ \left[\boldsymbol{R}\right] & \left[\boldsymbol{S}\right] \end{array}\right] \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} &&K_{ij} = -\sin (\Lambda) \cos (\Lambda) \widetilde{W}_{ij}^{(2)} \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} &&Q_{ij} = \cos^{3} (\Lambda) \: \widetilde{W}_{ij}^{(1)} \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} &&R_{ij} = -\sin(\Lambda) \cos(\Lambda) \: \widetilde{W}_{ij}^{(4)} - \sin^{2}(\Lambda) \cos(\Lambda)\left( \widetilde{W}_{ij}^{(6)} \!+ \widetilde{W}_{ij}^{(8)}\right) \\ \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} &&S_{ij} = \cos^{2}(\Lambda) \widetilde{W}_{ij}^{(3)} + \sin(\Lambda)\cos^{2}(\Lambda)\left( \widetilde{W}_{ij}^{(5)} + \widetilde{W}_{ij}^{(7)}\right) \end{array} $$

(49)

The matrices $[\boldsymbol {C}],[\boldsymbol {D}],[\boldsymbol {E}],[\widetilde {\boldsymbol {W}}^{(k),k=1...8}]$ are obtained as integrals of the chosen shape functions:

$$\begin{array}{@{}rcl@{}} &&C_{ij} = {{\int}_{0}^{l}} \! EI \psi_{i}^{\prime\prime} \psi_{j}^{\prime\prime} \, \mathrm{d}\bar{y} \end{array} $$

(50)

$$\begin{array}{@{}rcl@{}} &&D_{ij} = {{\int}_{0}^{l}} \! K \psi_{i}^{\prime\prime} \varphi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(51)

$$\begin{array}{@{}rcl@{}} &&E_{ij} = {{\int}_{0}^{l}} \! GJ \varphi_{i}^{\prime} \varphi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(52)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(1)} = {{\int}_{0}^{l}} \! e c c_{l\alpha} \varphi_{i} \varphi_{j} \, \mathrm{d}\bar{y} \end{array} $$

(53)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(2)} = {{\int}_{0}^{l}} \! e c c_{l\alpha} \varphi_{i} \psi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(54)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(3)} = {{\int}_{0}^{l}} \! c c_{l\alpha} \psi_{i} \varphi_{j} \, \mathrm{d}\bar{y} \end{array} $$

(55)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(4)} = {{\int}_{0}^{l}} \! c c_{l\alpha} \psi_{i} \psi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(56)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(5)} = {{\int}_{0}^{l}} \! (e c c_{l\alpha})^{\prime} \psi_{i} \varphi_{j} \, \mathrm{d}\bar{y} \end{array} $$

(57)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(6)} = {{\int}_{0}^{l}} \! (e c c_{l\alpha})^{\prime} \psi_{i} \psi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(58)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(7)} = {{\int}_{0}^{l}} \! e c c_{l\alpha} \psi_{i} \varphi_{j}^{\prime} \, \mathrm{d}\bar{y} \end{array} $$

(59)

$$\begin{array}{@{}rcl@{}} &&\widetilde{W}_{ij}^{(8)} = {{\int}_{0}^{l}} \! e c c_{l\alpha} \psi_{i} \psi_{j}^{\prime\prime} \, \mathrm{d}\bar{y} \end{array} $$

(60)

The largest positive eigenvalue of $[\boldsymbol {\mathcal {M}}]$ is then used to obtain the divergence dynamic pressure.

Appendix B: Lamination parameters

The material invariant matrices [Γ _i ] (Van Campen 2011) and the material invariants U _i (Jones 1999) are defined as:

$$ [\boldsymbol{\Gamma}_{0}] = \left[\begin{array}{ccccccccc} U_{1} & U_{4} & 0 \\ U_{4} & U_{1} & 0 \\ 0 & 0 & U_{5} \end{array}\right] ,[\boldsymbol{\Gamma}_{1}] = \left[\begin{array}{ccccccccc} U_{2} & 0 & 0 \\ 0 & -U_{2} & 0 \\ 0 & 0 & 0 \end{array}\right] $$

(61)

$$\begin{array}{@{}rcl@{}} &&[\boldsymbol{\Gamma}_{2}] = \frac{1}{2}\left[\begin{array}{ccccccccc} 0 & 0 & U_{2} \\ 0 & 0 & U_{2} \\ U_{2} & U_{2} & 0 \end{array}\right]\!, [\boldsymbol{\Gamma}_{3}] = \left[\begin{array}{ccccccccc} U_{3} & -U_{3} & 0 \\ -U_{3} & U_{3} & 0 \\ 0 & 0 & -U_{3} \end{array}\right] \end{array} $$

(62)

$$\begin{array}{@{}rcl@{}} &&[\boldsymbol{\Gamma}_{4}] = \left[\begin{array}{ccccccccc} 0 & 0 & U_{3} \\ 0 & 0 & -U_{3} \\ U_{3} & -U_{3} & 0 \end{array}\right] \end{array} $$

(63)

$$\begin{array}{@{}rcl@{}} &&U_{1}=\frac{1}{8} (3Q_{11} + 3Q_{22} + 2Q_{12} + 4Q_{66}) \end{array} $$

(64)

$$\begin{array}{@{}rcl@{}} &&U_{2} = \frac{1}{2}(Q_{11} - Q_{22}) \end{array} $$

(65)

$$\begin{array}{@{}rcl@{}} &&U_{3}=\frac{1}{8} (Q_{11} + Q_{22} - 2Q_{12} - 4Q_{66}) \end{array} $$

(66)

$$\begin{array}{@{}rcl@{}} &&U_{4}=\frac{1}{8} (Q_{11} + Q_{22} + 6Q_{12} - 4Q_{66}) \end{array} $$

(67)

$$\begin{array}{@{}rcl@{}} &&U_{5}=\frac{1}{8} (Q_{11} + Q_{22} - 2Q_{12} + 4Q_{66}) \end{array} $$

(68)

Since 𝜃 does not vary within a single lamina, the integral expressions(28–30) can be simplified to obtain a summation over all plies within a laminate. By requiring that each ply be of the same thickness t, the following expressions can be derived, N being the total number of plies (Van Campen 2011):

$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccccccccc} {V_{1}^{A}}\\ {V_{2}^{A}}\\ {V_{3}^{A}}\\ {V_{4}^{A}} \end{array}\right] \!= \frac{1}{N} \sum\limits_{k=1}^{N} \left[\begin{array}{ccccccccc}\cos2\theta_{k} \\ \sin2\theta_{k} \\ \cos4\theta_{k} \\ \sin4\theta_{k} \end{array}\right] \end{array} $$

(69)

$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccccccccc} {V_{1}^{B}} \\ {V_{2}^{B}}\\ {V_{3}^{B}}\\ {V_{4}^{B}} \end{array}\right] \,=\, \frac{2}{N^{2}} \sum\limits_{k=1}^{N} \left( \left( \frac{N}{2}\!-k\,+\,1\right)^{2}\,-\,\left( \frac{N}{2}-k\right)^{2} \right) \left[\begin{array}{ccccccccc}\cos2\theta_{k} \\ \sin2\theta_{k} \\ \cos4\theta_{k} \\ \sin4\theta_{k} \end{array}\right]\\ \end{array} $$

(70)

$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccccccccc} {V_{1}^{D}} \\ {V_{2}^{D}} \\ {V_{3}^{D}} \\ {V_{4}^{D}} \end{array}\right] \!= \frac{4}{N^{3}} {\sum}_{k=1}^{N} \left( \left( \frac{N}{2}\,-\,k\,+\,1\right)^{3}\,-\,\left( \frac{N}{2}-k\right)^{3} \right) \left[\begin{array}{ccccccccc} \cos2\theta_{k} \\ \sin2\theta_{k} \\ \cos4\theta_{k} \\ \sin4\theta_{k} \end{array}\right]\\ \end{array} $$

(71)

Appendix C: Reduced stiffness matrix

For each panel, a reduced stiffness matrix is obtained by neglecting the normal stress N _s and bending moment M _s about s. The laminates considered here are symmetric. The following relations exist between panel resultant forces and mid-plane strains and between moments and curvatures (Barbero et al. 1993; Jones 1999):

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccccccccc} N_{z} \\ N_{s} \\ N_{sz} \end{array}\right] = \left[\begin{array}{ccccccccc} \boldsymbol{A} \end{array}\right] \left[\begin{array}{ccccccccc} \epsilon_{z} \\ \epsilon_{s} \\ \gamma_{sz} \end{array}\right] , \left[\begin{array}{ccccccccc} M_{z} \\ M_{s} \\ M_{sz} \end{array}\right] = \left[\begin{array}{ccccccccc} \boldsymbol{D} \end{array}\right] \left[\begin{array}{ccccccccc} \kappa_{z} \\ \kappa_{s} \\ \kappa_{sz} \end{array}\right] \end{array} $$

(72)

Where [A] and [D] are the extensional and bending stiffness matrices of the panel considered, as defined by Jones (1999). The two matrices are inverted, their second row and column are neglected, and the 4×4 resulting matrices are inverted again. The expressions (72) hence become:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccccccccc} N_{z} \\ N_{sz} \end{array}\right] \,=\, \left[\begin{array}{ccccccccc} A_{z} A_{sz}\\A_{sz} A_{s} \end{array}\right] \left[\begin{array}{ccccccccc} \epsilon_{z} \\ \gamma_{sz} \end{array}\right] , \left[\begin{array}{ccccccccc} M_{z} \\ M_{sz} \end{array}\right] \,=\, \left[\begin{array}{ccccccccc} D_{z} D_{sz}\\D_{sz} D_{s} \end{array}\right]\! \left[\begin{array}{ccccccccc} \kappa_{z} \\ \kappa_{sz} \end{array}\right]\\ \end{array} $$

(73)

And the reduced matrices are:

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccccccccc} X_{z} X_{sz}\\ X_{sz} X_{s} \end{array}\right] = \left[\begin{array}{ccccccccc} X_{11}-\frac{X_{12}^{2}}{X_{22}} & X_{13}-\frac{X_{12}X_{23}}{X_{22}} \\ X_{13}-\frac{X_{12}X_{23}}{X_{22}} & X_{33}-\frac{X_{23}^{2}}{X_{22}} \end{array}\right] \end{array} $$

(74)

Where X can be replaced by A or D.