Skip to main content
Log in

Analysis of nonlinear aeroelastic characteristics of a trapezoidal wing in hypersonic flow

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

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

IJ :

Total mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement u, respectively

ij :

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

MN :

Total mode number retained in the \(\xi \) and \(\eta \) directions for transverse deflection w, respectively

mn :

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

RS :

Total mode number retained in the \(\xi \) and \(\eta \) directions for in-plane displacement v, respectively

rs :

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

uv :

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

xyz :

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 \({d^{2}(\hbox { })}/{\mathrm{d}\eta ^{2}}\)

\(({\hbox { }\dot{ }\hbox { }})\) :

\({\mathrm{d}(\hbox { })}/{\mathrm{d}\tau }\)

(ab]:

\(\left\{ {\alpha |a<\alpha \le b} \right\} \)

References

  1. Dowell, E.H.: Nonlinear oscillations of a fluttering plate. AIAA J. 4(7), 1267–1275 (1966)

    Article  Google Scholar 

  2. Dowell, E.H.: Nonlinear oscillations of a fluttering plate. II. AIAA J. 5(10), 1856–1862 (1967)

    Article  Google Scholar 

  3. Dowell, E.H.: Panel flutter—a review of the aeroelastic stability of plates and shells. AIAA J. 8(3), 385–399 (1970)

    Article  Google Scholar 

  4. Gray, C.E.: Large-amplitude finite element flutter analysis of composite panels in hypersonic flow. AIAA J. 31(6), 1090–1099 (1993)

    Article  MATH  Google Scholar 

  5. Cheng, G., Mei, C.: Finite element modal formulation for hypersonic panel flutter analysis with thermal effects. AIAA J. 42(4), 687–695 (2004)

    Article  Google Scholar 

  6. Dowell, E.H., Ye, W.L.: Limit cycle oscillation of a fluttering cantilever plate. AIAA J. 29(11), 1929–1936 (1991)

    Article  MATH  Google Scholar 

  7. Xie, D., Xu, M., Dai, H.H., et al.: Observation and evolution of chaos for a cantilever plate in supersonic flow. J. Fluids Struct. 50, 271–291 (2014)

    Article  Google Scholar 

  8. Bakhtiari-Nejad, F., Nazari, M.: Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate. Nonlinear Dyn. 56(4), 325–356 (2009)

    Article  MATH  Google Scholar 

  9. Dai, H.H., Paik, J.K., Atluri, S.N.: The global nonlinear Galerkin method for the analysis of elastic large deflections of plates under combined loads: A scalar homotopy method for the direct solution of nonlinear algebraic equations. Comput. Mater. Contin. 23(1), 69–99 (2011)

    Google Scholar 

  10. Dai, H.H., Paik, J.K., Atluri, S.N.: The global nonlinear Galerkin method for the solution of von Karman nonlinear plate equations: an optimal & faster iterative method for the direct solution of nonlinear algebraic equations \(\mathbf{F}(\mathbf{x}) ={\bf 0}\), using \({\bf x}=\lambda [\alpha {\bf F}+(1-\alpha ){\bf B}^{T}{\bf F}]\). Comput. Mater. Contin. 23(2), 155–185 (2011)

  11. Dai, H.H., Schnoor, M., Atluri, S.N.: Solutions of the von kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix. J. Mech. Mater. Struct. 9(2), 195–226 (2014)

    Article  Google Scholar 

  12. Dai, H.H., Yue, X.K., Yuan, J.P., et al.: A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity. J. Comput. Phys. 270, 214–237 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dai, H.H., Schnoor, M., Atluri, S.N.: Analysis of internal resonance in a two-degree-of-freedom nonlinear dynamical system. Commun. Nonlinear Sci. Numer. Simul. 49, 176–191 (2017)

    Article  MathSciNet  Google Scholar 

  14. Li, P., Yang, Y., Xu, W.: Nonlinear dynamics analysis of a two-dimensional thin panel with an external forcing in incompressible subsonic flow. Nonlinear Dyn. 67(4), 2483–2503 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhou, J., Yang, Z.C., Gu, Y.S.: Aeroelastic stability analysis of heated panel with aerodynamic loading on both surfaces. Sci. China Technol. Sci. 55(10), 2720–2726 (2012)

    Article  Google Scholar 

  16. Yang, Z.C., Zhou, J., Gu, Y.S.: Integrated analysis on static/dynamic aeroelasticity of curved panels based on a modified local piston theory. J. Sound Vib. 333(22), 5885–5897 (2014)

  17. Mei, C.: A finite-element approach for nonlinear panel flutter. AIAA J. 15(8), 1107–1110 (1977)

    Article  Google Scholar 

  18. Xue, D.Y., Mei, C.: Finite element nonlinear panel flutter with arbitrary temperatures in supersonic flow. AIAA J. 31(1), 154–162 (1993)

    Article  MATH  Google Scholar 

  19. Dixon, I.R., Mei, C.: Finite element analysis of large-amplitude panel flutter of thin laminates. AIAA J. 31(4), 701–707 (1993)

    Article  MATH  Google Scholar 

  20. Guo, X., Mei, C.: Using aeroelastic modes for nonlinear panel flutter at arbitrary supersonic yawed angle. AIAA J. 41(2), 272–279 (2003)

    Article  Google Scholar 

  21. Guo, X., Mei, C.: Application of aeroelastic modes on nonlinear supersonic panel flutter at elevated temperatures. Comput. Struct. 84(24), 1619–1628 (2006)

    Article  Google Scholar 

  22. Wang, X.C., Yang, Z.C., Zhou, J., et al.: Aeroelastic effect on aerothermoacoustic response of metallic panels in supersonic flow. Chin. J. Aeronaut. (2016). doi:10.1016/j.cja.2016.10.003

  23. Xie, D., Xu, M.: A simple proper orthogonal decomposition method for von Karman plate undergoing supersonic flow. Comput. Model. Eng. Sci. 93(5), 377–409 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Xie, D., Xu, M., Dai, H.H., et al.: Proper orthogonal decomposition method for analysis of nonlinear panel flutter with thermal effects in supersonic flow. J. Sound Vib. 337, 263–283 (2015)

    Article  Google Scholar 

  25. Xie, D., Xu, M.: A comparison of numerical and semi-analytical proper orthogonal decomposition methods for a fluttering plate. Nonlinear Dyn. 79, 1971–1989 (2015)

    Article  MATH  Google Scholar 

  26. Tang, D., Henry, J.K., Dowell, E.H.: Limit cycle oscillations of delta wing models in low subsonic flow. AIAA J. 37(11), 1355–1362 (1999)

    Article  Google Scholar 

  27. Tang, D., Dowell, E.H.: Effects of angle of attack on nonlinear flutter of a delta wing. AIAA J. 39(1), 15–21 (2001)

    Article  Google Scholar 

  28. Tang, D., Dowell, E.H.: Limit cycle oscillations of two-dimensional panels in low subsonic flow. Int. J. Nonlinear Mech. 37(7), 1199–1209 (2002)

    Article  MATH  Google Scholar 

  29. Shokrollahi, S., Bakhtiari-Nejad, F.: Limit cycle oscillations of swept-back trapezoidal wings at low subsonic flow. J. Aircraft. 41(4), 948–953 (2004)

    Article  Google Scholar 

  30. Wolf, A., Swift, J.B., Swinney, H.L., et al.: Determining Lyapunov exponents from a time series. Phys. D: Nonlinear Phenom. 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants Nos. 11472216 and 11672240) and 111 Project of China (B07050).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhichun Yang.

Appendix

Appendix

Elements of matrix A:

$$\begin{aligned} A_{mn}^{ij} =\frac{1}{6}\int _0^1 {\phi _m \phi _i } \mathrm{d}\xi \int _0^1 {\psi _n \psi _j {J}'\mathrm{d}\eta } \end{aligned}$$

Elements of matrix B:

$$\begin{aligned}&B_{mn}^{ij} = \frac{1}{6}\left\{ {\int _0^1 {{{\phi ''}_m}{{\phi ''}_i}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } } \right. \\&+ {\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\int _0^1 {{{\phi ''}_m}{{\phi ''}_i}{G^4}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \\&+ 2\nu {\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\int _0^1 {{{\phi ''}_m}{{\phi ''}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \\&+ 2(1 - \nu ){\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\int _0^1 {{{\phi ''}_m}{{\phi ''}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta }\\&+ {{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^4}\int _0^1 {{\phi _m}{\phi _i}\mathrm{d}\xi } \int _0^1 {{{\psi ''}_n}{{\psi ''}_j}J'\mathrm{d}\eta }\\&+ {\left( {2{c_2}} \right) ^2}{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\int _0^1 {{{\phi '}_m}{{\phi '}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \\&+ 2(1 - \nu ){c_2}^2{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\int _0^1 {{{\phi '}_m}{{\phi '}_i}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta }\\&+ 4{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\int _0^1 {{{\phi '}_m}{{\phi '}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{{\psi '}_j}{H^2}J'\mathrm{d}\eta } \\&+ 2(1 - \nu ){\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\int _0^1 {{{\phi '}_m}{{\phi '}_i}\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{{\psi '}_j}{H^2}J'\mathrm{d}\eta }\\&+ {\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\left[ \int _0^1 {{{\phi ''}_m}{\phi _i}{G^2}\mathrm{d}\xi } \int _0^1 {{\psi _n}{{\psi ''}_j}{H^2}J'\mathrm{d}\eta } \right. \\&\left. + \int _0^1 {{\phi _m}{{\phi ''}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{{\psi ''}_n}{\psi _j}{H^2}J'\mathrm{d}\eta } \right] \\&+ v{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\left[ \int _0^1 {{\phi _m}{{\phi ''}_i}\mathrm{d}\xi } \int _0^1 {{\psi ''}_n}{\psi _j}{H^2}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{\phi _m}{{\phi ''}_i}\mathrm{d}\xi } \int _0^1 {{{\psi ''}_n}{\psi _j}{H^2}J'\mathrm{d}\eta } \right] \\&+ 2{c_2}{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\left[ \int _0^1 {{{\phi ''}_m}{{\phi '}_i}{G^3}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi ''}_i}{G^3}\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \right] \\&+ 2{c_2}{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\left[ \int _0^1 {{{\phi ''}_m}{{\phi '}_i}G\mathrm{d}\xi } \int _0^1 {\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi ''}_i}G\mathrm{d}\xi } \int _0^1 {{\psi _n}{\psi _j}{H^4}J'\mathrm{d}\eta } \right] \\&+ 4{c_2}{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\left[ \int _0^1 {{{\phi '}_m}{{\phi '}_i}{G^2}\mathrm{d}\xi } \int _0^1 {\psi _n}{{\psi '}_j}{H^3}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi '}_i}{G^2}\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{\psi _j}{H^3}J'\mathrm{d}\eta } \right] \\&+ 2{c_2}(1 - \nu ){\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\left[ \int _0^1 {{{\phi '}_m}{{\phi '}_i}\mathrm{d}\xi } \int _0^1 {\psi _n}{{\psi '}_j}{H^4}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi '}_i}\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{\psi _j}{H^4}J'\mathrm{d}\eta } \right] \\&+ 2{c_2}{{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^4}\left[ \int _0^1 {{\phi _m}{{\phi '}_i}G\mathrm{d}\xi } \int _0^1 {{\psi ''}_n}{\psi _j}{H^2}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{\phi _i}G\mathrm{d}\xi } \int _0^1 {{\psi _n}{{\psi ''}_j}{H^2}J'\mathrm{d}\eta } \right] \\&+ 2{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^4}\left[ \int _0^1 {{{\phi ''}_m}{{\phi '}_i}{G^3}\mathrm{d}\xi } \int _0^1 {\psi _n}{{\psi '}_j}{H^3}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi ''}_i}{G^3}\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{\psi _j}{H^3}J'\mathrm{d}\eta } \right] \\&+ 2{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) ^2}\left[ \int _0^1 {{{\phi ''}_m}{{\phi '}_i}G\mathrm{d}\xi } \int _0^1 {\psi _n}{{\psi '}_j}{H^3}J'\mathrm{d}\eta \right. \\&\left. + \int _0^1 {{{\phi '}_m}{{\phi ''}_i}G\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{\psi _j}{H^3}J'\mathrm{d}\eta } \right] \\&+ 2{{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^4}\left[ \int _0^1 {{\phi _m}{{\phi '}_i}G\mathrm{d}\xi } \int _0^1 {{\psi ''}_n}{{\psi '}_j}HJ'\mathrm{d}\eta \right. \\&\left. \left. + \int _0^1 {{{\phi '}_m}{\phi _i}G\mathrm{d}\xi } \int _0^1 {{{\psi '}_n}{{\psi ''}_j}HJ'\mathrm{d}\eta } \right] \right\} \end{aligned}$$

Elements of matrices C, D

$$\begin{aligned}&C^{ij}=-\left( {\frac{h}{c_\mathrm{r} }} \right) ^{3}\sum _m^M {\sum _n^N {\sum _p^M {\sum _l^N {q_{mn} q_{pl} } } } } \int _0^1 {{\phi }'_m {\phi }'_p {u}'_i \mathrm{d}\xi } \nonumber \\&\quad \times \int _0^1 {\psi _n \psi _l v_j H^{3}{J}'\mathrm{d}\eta } \\&\quad {-\,\nu } \left( {\frac{h}{l}} \right) ^{2}\left( {\frac{h}{c_\mathrm{r} }} \right) \sum _m^M \sum _n^N \sum _p^M \sum _l^N q_{mn} q_{pl} \nonumber \\&\quad \left[ \int _0^1 {{\phi }'_m {\phi }'_p {u}'_i G^{2}\mathrm{d}\xi } \right. \nonumber \\&\quad \times \int _0^1 {\psi _n \psi _l v_j H^{3}{J}'\mathrm{d}\eta } +2\int _0^1 {{\phi }'_m \phi _p {u}'_i G\mathrm{d}\xi } \\&\quad \times \int _0^1 {\psi _n {\psi }'_l v_j H^{2}{J}'\mathrm{d}\eta }\nonumber \\&\quad \left. +\,{\int _0^1 {\phi _m \phi _p {u}'_i \mathrm{d}\xi } \int _0^1 {{\psi }'_n {\psi }'_l v_j H{J}'\mathrm{d}\eta } } \right] \\&\quad -(1-\nu )\left( {\frac{h}{l}} \right) ^{2}\left( {\frac{h}{c_\mathrm{r} }} \right) \sum _m^M \sum _n^N \sum _p^M \sum _l^N q_{mn} q_{pl} \nonumber \\&\quad \left[ \int _0^1 {{\phi }'_m {\phi }'_p {u}'_i G^{2}\mathrm{d}\xi } \int _0^1 {\psi _n \psi _l v_j H^{3}{J}'\mathrm{d}\eta } \right. \\&\quad \left. +\,\int _0^1 {{\phi }'_m {\phi }'_p u_i G\mathrm{d}\xi } \int _0^1 {\psi _n \psi _l {v}'_j H^{2}{J}'\mathrm{d}\eta } \right. \\&\quad +\, \int _0^1 {{\phi }'_m \phi _p {u}'_i G\mathrm{d}\xi } \int _0^1 {\psi _n {\psi }'_l v_j H^{2}{J}'\mathrm{d}\eta } \\&\quad \left. +\,\int _0^1 {{\phi }'_m \phi _p u_i \mathrm{d}\xi } \int _0^1 {\psi _n {\psi }'_l {v}'_j H{J}'\mathrm{d}\eta } \right] \\&D^{rs}=-\left( {\frac{h}{l}} \right) ^{3}\sum _m^M \sum _n^N \sum _p^M \sum _l^N q_{mn} q_{pl} \\&\quad \left[ \int _0^1 {{\phi }'_m {\phi }'_p {u}'_r G^{3}\mathrm{d}\xi } \int _0^1 {\psi _n \psi _l v_s H^{3}{J}'\mathrm{d}\eta } \right. \\&\quad \left. +\int _0^1 {{\phi }'_m {\phi }'_p u_r G^{2}\mathrm{d}\xi } \int _0^1 {\psi _n \psi _l {v}'_s H^{2}{J}'\mathrm{d}\eta } \right. \\&\quad +\,2\int _0^1 {{\phi }'_m \phi _p {u}'_r G^{2}\mathrm{d}\xi } \int _0^1 {\psi _n {\psi }'_l v_s H^{2}{J}'\mathrm{d}\eta } \\&\quad +\,2\int _0^1 {{\phi }'_m \phi _p u_r G\mathrm{d}\xi } \int _0^1 {\psi _n {\psi }'_l {v}'_s H{J}'\mathrm{d}\eta } \\&\quad +\, \int _0^1 {\phi _m \phi _p {u}'_r G\mathrm{d}\xi } \int _0^1 {{\psi }'_n {\psi }'_l v_s H{J}'\mathrm{d}\eta } \\&\quad \left. +\,\int _0^1 {\phi _m \phi _p u_r \mathrm{d}\xi } \int _0^1 {{\psi }'_n {\psi }'_l {v}'_s {J}'\mathrm{d}\eta } \right] \\&\quad -\,\nu \left( {\frac{h}{l}} \right) \left( {\frac{h}{c_\mathrm{r} }} \right) ^{2}\sum _m^M {\sum _n^N {\sum _p^M {\sum _l^N {q_{mn} q_{pl} } } } } \\&\quad \left[ \int _0^1 {{\phi }'_m {\phi }'_p {u}'_r G\mathrm{d}\xi }\right. \\&\int _0^1 {\psi _n \psi _l v_s H^{3}{J}'\mathrm{d}\eta } +\int _0^1 {{\phi }'_m {\phi }'_p u_r \mathrm{d}\xi } \\&\quad \left. \int _0^1 {\psi _n \psi _l {v}'_s H^{2}{J}'\mathrm{d}\eta } \right] \\&\quad -(1-\nu )\left( {\frac{h}{l}} \right) \left( {\frac{h}{c_\mathrm{r} }} \right) ^{2}\sum _m^M {\sum _n^N {\sum _p^M {\sum _l^N {q_{mn} q_{pl} } } } } \nonumber \\&\quad \left[ \int _0^1 {{\phi }'_m {\phi }'_p {u}'_r G\mathrm{d}\xi }\right. \\&\int _0^1 {\psi _n \psi _l v_s H^{3}{J}'\mathrm{d}\eta } +\int _0^1 {{\phi }'_m \phi _p {u}'_r \mathrm{d}\xi } \nonumber \\&\quad \left. \int _0^1 {\psi _n {\psi }'_l v_s H^{2}{J}'\mathrm{d}\eta } \right] \end{aligned}$$

Elements of matrices \(\mathbf{Q}_{\mathbf{L1}}\), \(\mathbf{Q}_{\mathbf{L2}}\), \(\mathbf{Q}_{\mathbf{N}}\):

$$\begin{aligned}&Q_{kl}^{ij} =\frac{\lambda }{6}\frac{\gamma }{M_\infty }\int _0^1 {{\phi }'_k \phi _i \mathrm{d}\xi } \int _0^1 {\psi _l \psi _j H{J}'\mathrm{d}\eta }\\&Q_{\mathrm{mn}}^{ij} =\frac{\lambda }{6}\frac{\gamma }{M_\infty }\sqrt{\frac{\mu }{\lambda }}\int _0^1 {\phi _m \phi _i \mathrm{d}\xi } \int _0^1 {\psi _n \psi _j {J}'\mathrm{d}\eta }\\&Q_{\mathrm{N}}^{ij} =\frac{\lambda }{6}\left\{ \frac{\kappa +1}{4}\gamma ^{2}\frac{h}{c_\mathrm{r} }\left[ \sum _m^M \sum _n^N \sum _k^M \sum _l^N q_{mn} q_{kl} \right. \right. \nonumber \\&\quad \left. \left. \int _0^1 {{\phi }'_m {\phi }'_k \phi _i \mathrm{d}\xi \int _0^1 {\psi _n \psi _l \psi _j H^{2}{J}'} } \mathrm{d}\eta \right. \right. \\&\quad +\,2\sqrt{\frac{\mu }{\lambda }}\sum _m^M \sum _n^N \sum _k^M \sum _l^N q_{mn} \dot{q}_{kl} \int _0^1 {\phi }'_m \phi _k \phi _i \mathrm{d}\xi \nonumber \\&\int _0^1 {\psi _n \psi _l \psi _j H{J}'} \mathrm{d}\eta \\&\quad +\,\frac{\mu }{\lambda }\sum _m^M \sum _n^N \sum _k^M \sum _l^N \dot{q}_{mn} \dot{q}_{kl} \int _0^1 \phi _m \phi _k \phi _i \mathrm{d}\xi \nonumber \\&\quad \left. \times \int _0^1 {\psi _n \psi _l \psi _j {J}'} \mathrm{d}\eta \right] \\&\quad +\,\frac{\kappa +1}{12}M_\infty \gamma ^{3}\left( {\frac{h}{c_\mathrm{r} }} \right) ^{2}\\&\quad \times \left[ \sum _k^M \sum _l^N \sum _m^M \sum _n^N \sum _r^M \sum _s^N q_{kl} q_{mn} q_{rs} \right. \nonumber \\&\int _0^1 {{\phi }'_k {\phi }'_m {\phi }'_r \phi _i \mathrm{d}\xi \int _0^1 {\psi _l \psi _n \psi _s \psi _j H^{3}{J}'} } \mathrm{d}\eta \\&\quad +\,3\sqrt{\frac{\mu }{\lambda }}\sum _k^M \sum _l^N \sum _m^M \sum _n^N \sum _r^M \sum _s^N q_{kl} q_{mn} \dot{q}_{rs} \nonumber \\&\quad \times \int _0^1 {{\phi }'_k {\phi }'_m \phi _r \phi _i \mathrm{d}\xi \int _0^1 {\psi _l \psi _n \psi _s \psi _j H^{2}{J}'} } \mathrm{d}\eta \\&\quad +\,3\frac{\mu }{\lambda }\sum _k^M \sum _l^N \sum _m^M \sum _n^N \sum _r^M \sum _s^N q_{kl} \dot{q}_{mn} \dot{q}_{rs} \nonumber \\&\int _0^1 {{\phi }'_k \phi _m \phi _r \phi _i \mathrm{d}\xi \int _0^1 {\psi _l \psi _n \psi _s \psi _j H{J}'} } \mathrm{d}\eta \\&\quad +\,\left( {\sqrt{\frac{\mu }{\lambda }}} \right) ^{3}\sum _k^M \sum _l^N \sum _m^M \sum _n^N \sum _r^M \sum _s^N \dot{q}_{kl} \dot{q}_{mn} \dot{q}_{rs} \\&\left. \left. \int _0^1 {\phi _k \phi _m \phi _r \phi _i \mathrm{d}\xi \int _0^1 {\psi _l \psi _n \psi _s \psi _j {J}'} } \mathrm{d}\eta \right] \right\} \end{aligned}$$

Elements of matrix F:

$$\begin{aligned}&{F^{ij}} = 2\left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _m^I \sum \limits _n^J \sum \limits _k^M \sum \limits _l^N {a_{mn}}{q_{kl}}\int _0^1 {{u'}_m}{{\phi '}_k}{{\phi '}_i}\mathrm{d}\xi \\&\times \int _0^1 {{v_n}{\psi _l}{\psi _j}{H^3}J'} \mathrm{d}\eta \\&+ \sum \limits _k^M \sum \limits _l^N \sum \limits _m^M \sum \limits _n^N \sum \limits _r^M \sum \limits _s^N {q_{kl}}{q_{mn}}{q_{rs}}\\&\times \int _0^1 {{{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{\psi _s}{\psi _j}{H^4}J'} } \mathrm{d}\eta \\&+ 2{{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^3}\left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _r^R \sum \limits _s^S \sum \limits _k^M \sum \limits _l^N {b_{rs}}{q_{kl}}\\&\left[ \int _0^1 {{{u'}_r}{{\phi '}_k}{{\phi '}_i}{G^3}\mathrm{d}\xi \int _0^1 {{v_s}{\psi _l}{\psi _j}{H^3}J'} } \mathrm{d}\eta \right. \\&+ \int _0^1 {{{u'}_r}{{\phi '}_k}{\phi _i}{G^2}\mathrm{d}\xi \int _0^1 {{v_s}{\psi _l}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&+\int _0^1 {{{u'}_r}{\phi _k}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{v_s}{{\psi '}_l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&+ \int _0^1 {{{u'}_r}{\phi _k}{\phi _i}G\mathrm{d}\xi \int _0^1 {{v_s}{{\psi '}_l}{{\psi '}_j}HJ'} } \mathrm{d}\eta \\&+\int _0^1 {{u_r}{{\phi '}_k}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{{v'}_s}{\psi _l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&+ \int _0^1 {{u_r}{{\phi '}_k}{\phi _i}G\mathrm{d}\xi \int _0^1 {{{v'}_s}{\psi _l}{{\psi '}_j}HJ'} } \mathrm{d}\eta \\&+ \int _0^1 {{u_r}{\phi _k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{{v'}_s}{{\psi '}_l}{\psi _j}HJ'} } \mathrm{d}\eta \\&\left. + \int _0^1 {{u_r}{\phi _k}{\phi _i}\mathrm{d}\xi \int _0^1 {{{v'}_s}{{\psi '}_l}{{\psi '}_j}J'} } \mathrm{d}\eta \right] \\&+ {{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^4}\sum \limits _k^M \sum \limits _l^N \sum \limits _m^M \sum \limits _n^N \sum \limits _r^M \sum \limits _s^N {q_{kl}}{q_{mn}}{q_{rs}}\\&\left[ \int _0^1 {{{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{{\phi '}_i}{G^4}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{\psi _s}{\psi _j}{H^4}J'} } \mathrm{d}\eta \right. \\&+ 3\int _0^1 {{{\phi '}_k}{{\phi '}_m}{\phi _r}{{\phi '}_i}{G^3}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{{\psi '}_s}{\psi _j}{H^3}J'} } \mathrm{d}\eta \\&+3\int _0^1 {{{\phi '}_k}{\phi _m}{\phi _r}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{{\psi '}_s}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&+ \int _0^1 {{\phi _k}{\phi _m}{\phi _r}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{{\psi '}_l}{{\psi '}_n}{{\psi '}_s}{\psi _j}HJ'} } \mathrm{d}\eta \\&+ \int _0^1 {{{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{\phi _i}{G^3}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{\psi _s}{{\psi '}_j}{H^3}J'} } \mathrm{d}\eta \end{aligned}$$
$$\begin{aligned}&+3\int _0^1 {{{\phi '}_k}{{\phi '}_m}{\phi _r}{\phi _i}{G^2}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{{\psi '}_s}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&\mathrm{{ + }}3\int _0^1 {{{\phi '}_k}{\phi _m}{\phi _r}{\phi _i}G\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{{\psi '}_s}{{\psi '}_j}HJ'} } \mathrm{d}\eta \\&\left. + \int _0^1 {{\phi _k}{\phi _m}{\phi _r}{\phi _i}\mathrm{d}\xi \int _0^1 {{{\psi '}_l}{{\psi '}_n}{{\psi '}_s}{{\psi '}_j}J'} } \mathrm{d}\eta \right] \\&+ 2\nu {{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^2}\left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _m^I \sum \limits _n^J \sum \limits _k^M \sum \limits _l^N {a_{mn}}{q_{kl}}\\&\left[ \int _0^1 {{{u'}_m}{{\phi '}_k}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{v_n}{\psi _l}{\psi _j}{H^3}J'} } \mathrm{d}\eta \right. \\&+ \int _0^1 {{{u'}_m}{\phi _k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{v_n}{{\psi '}_l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&\mathrm{{ + }}\int _0^1 {{{u'}_m}{{\phi '}_k}{\phi _i}G\mathrm{d}\xi \int _0^1 {{v_n}{\psi _l}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&\left. + \int _0^1 {{{u'}_m}{\phi _k}{\phi _i}\mathrm{d}\xi \int _0^1 {{v_n}{{\psi '}_l}{{\psi '}_j}HJ'} } \mathrm{d}\eta \right] + \nu {{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^2}\\&\sum \limits _k^M \sum \limits _l^N \sum \limits _m^M \sum \limits _n^N \sum \limits _r^M \sum \limits _s^N {q_{kl}}{q_{mn}}{q_{rs}}\\&\left[ 2\int _0^1 {{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{{\phi '}_i}{G^2}\mathrm{d}\xi \right. \\&\int _0^1 {{\psi _l}{\psi _n}{\psi _s}{\psi _j}{H^4}J'} \mathrm{d}\eta + \int _0^1 {{\phi '}_k}{{\phi '}_m}{\phi _r}{{\phi '}_i}G\mathrm{d}\xi \\&\int _0^1 {{\psi _l}{\psi _n}{{\psi '}_s}{\psi _j}{H^3}J'} \mathrm{d}\eta \\&+2\int _0^1 {{{\phi '}_k}{\phi _m}{{\phi '}_r}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{\psi _s}{\psi _j}{H^3}J'} } \mathrm{d}\eta \\&+\int _0^1 {{\phi _k}{\phi _m}{{\phi '}_r}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{{\psi '}_l}{{\psi '}_n}{\psi _s}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&+\int _0^1 \phi _k^\prime \phi _m^\prime \phi _r^\prime \phi _i G\mathrm{d}\xi \int _0^1\psi _l \psi _n\psi _s\psi _j^\prime H^3J^\prime \mathrm{d}\eta \\&\left. +\int _0^1 \phi _k^\prime \phi _m^\prime \phi _r\phi _i \mathrm{d}\xi \int _0^1\psi _l \psi _n\psi _s^\prime \psi _j^\prime H^2J^\prime \mathrm{d}\eta \right] \\&+\, 2v\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) \left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _r^R \sum \limits _s^S \sum \limits _k^M \sum \limits _l^N {b_{rs}}{q_{kl}}\\&\left[ \int _0^1 {{{u'}_r}{{\phi '}_k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{v_s}{\psi _l}{\psi _j}{H^3}J'} } \mathrm{d}\eta \right. \\&\left. + \int _0^1 {{u_r}{{\phi '}_k}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{{v'}_s}{\psi _l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \right] \\&+ (1 - v)\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) \left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _r^R \sum \limits _s^S \sum \limits _k^M \sum \limits _l^N {b_{rs}}{q_{kl}}\\&\left[ 2\int _0^1 {{{u'}_r}{{\phi '}_k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{v_s}{\psi _l}{\psi _j}{H^3}J'} } \mathrm{d}\eta \right. \\&+ \int _0^1 {{{u'}_r}{{\phi '}_k}{\phi _i}\mathrm{d}\xi \int _0^1 {{v_s}{\psi _l}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&\left. {\mathrm{{ + }}\int _0^1 {{{u'}_r}{\phi _k}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{v_s}{{\psi '}_l}{\psi _j}{H^2}J'} } \mathrm{d}\eta } \right] \\&+ (1 - v){{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^2}\left( {\frac{{{c_\mathrm{r}}}}{h}} \right) \sum \limits _m^I \sum \limits _n^J \sum \limits _k^M \sum \limits _l^N {a_{mn}}{q_{kl}}\\&\left[ 2\int _0^1 {{{u'}_m}{{\phi '}_k}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{v_n}{\psi _l}{\psi _j}{H^3}J'} } \mathrm{d}\eta \right. \\&+ \int _0^1 {{{u'}_m}{{\phi '}_k}{\phi _i}G\mathrm{d}\xi \int _0^1 {{v_n}{\psi _l}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&+ 2\int _0^1 {{u_m}{{\phi '}_k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{{v'}_n}{\psi _l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&+ \int _0^1 {{u_m}{{\phi '}_k}{\phi _i}\mathrm{d}\xi \int _0^1 {{{v'}_n}{\psi _l}{{\psi '}_j}HJ'} } \mathrm{d}\eta \\&+ \int _0^1 {{{u'}_m}{\phi _k}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{v_n}{{\psi '}_l}{\psi _j}{H^2}J'} } \mathrm{d}\eta \\&\left. + \int _0^1 {{u_m}{\phi _k}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{{v'}_n}{{\psi '}_l}{\psi _j}HJ'} } \mathrm{d}\eta \right] \\&+ (1 - v){{\left( {\frac{{{c_\mathrm{r}}}}{l}} \right) }^2}\sum \limits _k^M \sum \limits _l^N \sum \limits _m^M \sum \limits _n^N \sum \limits _r^M \sum \limits _s^N {q_{kl}}{q_{mn}}{q_{rs}}\\&\left[ 2\int _0^1 {{{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{{\phi '}_i}{G^2}\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{\psi _s}{\psi _j}{H^4}J'} } \mathrm{d}\eta \right. \\&+ \int _0^1 {{{\phi '}_k}{{\phi '}_m}{{\phi '}_r}{\phi _i}G\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{\psi _s}{{\psi '}_j}{H^3}J'} } \mathrm{d}\eta \\&+2\int _0^1 {{{\phi '}_k}{\phi _m}{{\phi '}_r}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{\psi _s}{\psi _j}{H^3}J'} } \mathrm{d}\eta \\&+\int _0^1 {{{\phi '}_k}{\phi _m}{{\phi '}_r}{\phi _i}\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{\psi _s}{{\psi '}_j}{H^2}J'} } \mathrm{d}\eta \\&+ \int _0^1 {{{\phi '}_k}{{\phi '}_m}{\phi _r}{{\phi '}_i}G\mathrm{d}\xi \int _0^1 {{\psi _l}{\psi _n}{{\psi '}_s}{\psi _j}{H^3}J'} } \mathrm{d}\eta \\&\left. + \int _0^1 {{{\phi '}_k}{\phi _m}{\phi _r}{{\phi '}_i}\mathrm{d}\xi \int _0^1 {{\psi _l}{{\psi '}_n}{{\psi '}_s}{\psi _j}{H^2}J'} } \mathrm{d}\eta \right] \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3511-4

Keywords

Navigation