Abstract
Nonlinear aeroelastic behavior of a trapezoidal wing in hypersonic flow is investigated. The aeroelastic governing equations are built by von Karman large deformation theory and the third-order piston theory. The Rayleigh–Ritz approach combined with the affine transformation is formulated and employed to transform the equations of a trapezoidal wing structure, modeled as a cantilevered wing-like plate, into modal coordinates. And then the modal equations are solved by numerical integrations. Several typical cases are studied to validate the capability of the proposed method for linear and nonlinear aeroelastic analysis of trapezoidal cantilever plate in hypersonic flow. The effects of Rayleigh–Ritz mode truncation for various wing-plate geometrical characteristics, i.e., sweep angle of leading edge, taper ratio and span, are examined to determine the appropriate mode number for accurate modeling and fast calculation. Meanwhile, the effects of various geometries of trapezoidal cantilever plates on the flutter stability are investigated. The nonlinear dynamic behaviors of the model with three typical geometries, namely, the rectangular, parallelogram and trapezoidal wing-like plate, are simulated numerically. Furthermore, complex dynamic behaviors are observed and identified via the phase plot, the Poincare map and the largest Lyapunov exponent. The results demonstrate that geometrical parameters of trapezoidal wing have significant effects on the nonlinear aeroelastic behaviors of wing structure. In particular, the evolution processes of chaos exhibit remarkable difference for these three wing configurations.
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Abbreviations
- AR:
-
Aspect ratio
- \(a_{ij} ,b_{rs} \) :
-
Mode coordinate for in-plane displacement u and v, respectively
- \(c_\mathrm{r} ,c_\mathrm{t} \) :
-
Root chord length and tip chord length, respectively
- D :
-
Plate stiffness, \(D={Eh^{3}}/{12(1-\nu ^{2})}\)
- E :
-
Young’s modulus
- h :
-
Plate thickness
- I, J :
-
Total mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement u, respectively
- i, j :
-
Mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement u, respectively
- l :
-
Semi span
- \(L=T-U\) :
-
Lagrangian
- Ma :
-
Mach number
- M, N :
-
Total mode number retained in the \(\xi \) and \(\eta \) directions for transverse deflection w, respectively
- m, n :
-
Mode number retained in the \(\xi \) and \(\eta \) directions for transverse deflection w, respectively
- \(\Delta p\) :
-
Aerodynamic pressure
- \(q_\infty \) :
-
Dynamic pressure, \(q_\infty ={\rho _\infty V_\infty ^2 }/2\)
- \(q_{mn} \) :
-
Mode coordinate for transverse deflection w
- R, S :
-
Total mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement v, respectively
- r, s :
-
Mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement v, respectively
- T :
-
Kinetic energy
- TR:
-
Taper ratio, \(\hbox {TR}={c_\mathrm{t} }/{c_\mathrm{r} }\)
- t :
-
Time
- U :
-
Elastic energy
- u, v :
-
In-plane displacement in the \(\xi \) and \(\eta \) directions, respectively
- \(\bar{{u}},\bar{{v}}\) :
-
Non-dimensional in-plane displacement in the \(\xi \) and \(\eta \) directions, respectively
- \(u_{i(r)} ,v_{j(s)} \) :
-
Mode in the \(\xi \) and \(\eta \) directions for in-plane displacement u(v), respectively
- \(V_\infty \) :
-
Flow velocity
- w :
-
Transverse deflection
- \(\bar{{w}} \) :
-
Non-dimensional transverse deflection
- x, y, z :
-
Chordwise, spanwise, and normal coordinate, respectively
- \(\alpha \) :
-
Sweep angle of leading edge, positive backswept
- \(\gamma \) :
-
Glauert’s aeroelastic correction factor
- \(\kappa \) :
-
Isentropic gas coefficient
- \(\lambda \) :
-
Non-dimensional pressure, \(\lambda ={2q_\infty c_\mathrm{r}^3 }/D\)
- \(\mu \) :
-
Non-dimensional mass ratio, \(\mu ={\rho _\infty c_\mathrm{r} }/{\rho _m h}\)
- \(\nu \) :
-
Poisson ratio
- \(\rho _\infty \) :
-
Air density
- \(\rho _m \) :
-
Plate density
- \(\xi ,\eta \) :
-
Non-dimensional coordinates
- \(\tau \) :
-
Non-dimensional time, \(\tau =t\left( {D/{\rho _m hc_\mathrm{r}^4 }} \right) ^{1/2}\)
- \(\varphi _m ,\psi _n \) :
-
Mode in the \(\xi \) and \(\eta \) directions for transverse deflection w, respectively
- \((\hbox { })^{\prime }\) :
-
\({\mathrm{d}(\hbox { })}/{\mathrm{d}\xi }\) or \({\mathrm{d}(\hbox { })}/{\mathrm{d}\eta }\)
- \((\hbox { }{)}''\) :
-
\({\mathrm{d}^{2}(\hbox { })}/{\mathrm{d}\xi ^{2}}\) or \({d^{2}(\hbox { })}/{\mathrm{d}\eta ^{2}}\)
- \(({\hbox { }\dot{ }\hbox { }})\) :
-
\({\mathrm{d}(\hbox { })}/{\mathrm{d}\tau }\)
- (a, b]:
-
\(\left\{ {\alpha |a<\alpha \le b} \right\} \)
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants Nos. 11472216 and 11672240) and 111 Project of China (B07050).
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Appendix
Appendix
Elements of matrix A:
Elements of matrix B:
Elements of matrices C, D
Elements of matrices \(\mathbf{Q}_{\mathbf{L1}}\), \(\mathbf{Q}_{\mathbf{L2}}\), \(\mathbf{Q}_{\mathbf{N}}\):
Elements of matrix F:
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Tian, W., Yang, Z., Gu, Y. et al. Analysis of nonlinear aeroelastic characteristics of a trapezoidal wing in hypersonic flow. Nonlinear Dyn 89, 1205–1232 (2017). https://doi.org/10.1007/s11071-017-3511-4
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DOI: https://doi.org/10.1007/s11071-017-3511-4