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Nonlinear dynamics and flutter of plate and cavity in response to supersonic wind tunnel start

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Abstract

The transient response of a plate and a cavity is investigated in a supersonic wind tunnel start experiment where the freestream flow inside the test section reaches turbulent flow at Mach 2. Experimentally measured plate displacement time history shows flutter onset, transition to limit cycle oscillation, and stabilization at a static deformed state during the 30 s run. To analyze and interpret the measured plate response, a fully coupled aero-thermal-acousto-elastic analysis is carried out. A theoretical–computational model is formulated with a nonlinear structural plate model, acoustic pressure equation for the stationary fluid in a cavity, and the first-order Piston Theory aerodynamics. A linear stability analysis is performed that includes the nonlinear added stiffness due to an initial deformation to investigate the combined effects of freestream coupling and temperature differential on system stability. Also, direct time integration of the nonlinear fluid structural equations of motion is performed using experimentally measured flow parameters as inputs. All stability transitions are captured using the theoretical model with good agreement with experiment for transitions from no flutter to flutter/limit cycle oscillations (LCO) although the theoretical LCO amplitude is approximately \(50\%\) larger than measured. The system’s sensitivity to cavity coupling, temperature differential, thickness calibration, static pressure differential, and turbulent pressure fluctuations are investigated. Lastly, snap-through buckling analyses in response to periodic and quasi-static excitations are conducted.

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Abbreviations

\(\Delta p_s = p(x,y,t) - p_{c,\mathrm{ref}}(t)\) :

Static pressure differential (Pa)

\(\Delta T = T(x,y,t) - T_{\mathrm{ref}}\) :

Temperature differential between the plate and its support (K)

\(\hat{w}, \, \hat{P}\) :

Eigenvectors

\(\lambda \) :

Eigenvalue

\(\omega \) :

Frequency (rad/s) or (Hz)

\(\psi ^c_i(x,y,z)\) :

ith basis function for \(p_c(x,y,z,t)\) expansion

\(\psi ^w_i(x,y)\) :

ith basis function for w(x, y, t) expansion

\(\rho _{\infty }, \rho _c\) :

Freestream and cavity fluid density (\(\hbox {kg/m}^3\))

\(\zeta \) :

Damping ratio

a :

Plate length (appears without subscripts) (m)

\(A, \, V\) :

Integration area (plate) and volume (cavity) domains

\(a_{\infty }, \, a_{c}\) :

Freestream and cavity speed of sound (appears with subscripts) (m/s)

b :

Plate width (m)

\(d_c\) :

Cavity depth (m)

h :

Plate thickness (m)

L.E.:

Leading edge

\(M_{\infty }\) :

Freestream Mach number

\(N_c\) :

Number of basis functions in \(p_c(x,y,z,t)\) expansion

\(N_w\) :

Number of basis functions in w(x, y, t) expansion

p :

Freestream static pressure (Pa)

\(p_0\) :

Stagnation pressure (Pa)

\(p_c(x,y,z,t), p_{c,\mathrm{ref}}(t)\) :

Cavity static pressure (perturbation and reference) (Pa)

\(P_i(t)\) :

ith Modal cavity pressure perturbation coordinate (Pa)

\(p_{\mathrm{ref}} = 20\,(\upmu \hbox {Pa})\) :

Acoustic reference pressure

R :

Gas constant (J/kg/K)

\(\hbox {Re}_{\infty }\) :

Unit Reylonds number (1/m)

\(T_{\infty }\) :

Freestream temperature (K)

\(T_{c}\) :

Cavity fluid temperature (K)

\(U_{\infty }\) :

Freestream velocity (m/s)

\(w, \, u, \, v(x,y,t)\) :

Physical displacement components (m)

\(w_i, \, u_i, \, v_i(t)\) :

ith modal displacement coordinates (m)

References

  1. Amabili, M., Pellicano, F.: Multimode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells. J. Appl. Mech. 69(2), 117–129 (2001). https://doi.org/10.1115/1.1435366

    Article  MATH  Google Scholar 

  2. Amabili, M., Pellicano, F.: Nonlinear supersonic flutter of circular cylindrical shells. AIAA J. 39(4), 564–573 (2001). https://doi.org/10.2514/2.1365

    Article  MATH  Google Scholar 

  3. Bismarck-Nasr, M.N.: Finite elements in aeroelasticity of plates and shells. Appl. Mech. Rev. 49(10S), S17–S24 (1996)

    Article  Google Scholar 

  4. Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability. Macmillan, New York (1963)

    MATH  Google Scholar 

  5. Casper, K.M., Beresh, S.J., Henfling, J., Spillers, R.: Fluid–structure interactions using controlled disturbances on a slender cone at mach 8. In: 54th AIAA Aerospace Sciences Meeting (2016). https://doi.org/10.2514/6.2016-1126

  6. Chen, H., Virgin, L.N.: Finite element analysis of post-buckling dynamics in plates—part I: an asymptotic approach. Int. J. Solids Struct. 43(13), 3983–4007 (2006). https://doi.org/10.1016/j.ijsolstr.2005.04.036

    Article  MATH  Google Scholar 

  7. Clemens, N.T., Narayanaswamy, V.: Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions. Annu. Rev. Fluid Mech. 46, 469–492 (2014)

    Article  MathSciNet  Google Scholar 

  8. Currao, G.M.D., Freydin, M., Dowell, E.H., McQuellin, L.P., Neely, A.J.: Design of a panel flutter experiment in a short duration hypersonic facility. In: 21st Australasian Fluid Mechanics Conference. Adelaide, Australia (2018)

  9. Dowell, E., Gorman, G., Smith, D.: Acoustoelasticity: general theory, acoustic natural modes and forced response to sinusoidal excitation, including comparisons with experiment. J. Sound Vib. 52(4), 519–542 (1977). https://doi.org/10.1016/0022-460X(77)90368-6

    Article  MATH  Google Scholar 

  10. Dowell, E.H.: Nonlinear flutter of curved plates. AIAA J. 7(3), 424–431 (1969). https://doi.org/10.2514/3.5124

    Article  MATH  Google Scholar 

  11. Dowell, E.H.: Aeroelasticity of Plates and Shells. Noordhoff International Publishing, Leyden (1974). (now Springer)

    MATH  Google Scholar 

  12. Dowell, E.H.: A Modern Course in Aeroelasticity, 5th edn. Springer, Berlin (2015)

    MATH  Google Scholar 

  13. Dowell, E.H., Voss, H.M.: The effect of a cavity on panel vibration. AIAA J. 1(2), 476–477 (1963). https://doi.org/10.2514/3.1568

    Article  Google Scholar 

  14. Ehrhardt, D.A., Virgin, L.N.: Experiments on the thermal post-buckling of panels, including localized heating. J. Sound Vib. 439, 300–309 (2019). https://doi.org/10.1016/j.jsv.2018.08.043

    Article  Google Scholar 

  15. Ferguson, J.I., Spottswood, S.M., Ehrhardt, D.A., Perez, R.A., Snyder, M.P., Obenchain, M.B.: Experimental nonlinear dynamics of a post-buckled composite laminate plate. In: Nonlinear Structures and Systems, vol. 1, pp. 103–114. Springer (2020)

  16. Freydin, M., Dowell, E.H.: Nonlinear theoretical aeroelastic model of a plate: free to fixed in-plane boundaries. AIAA J. (under review), ID 2020-02-J059551 (2020)

  17. Freydin, M., Dowell, E.H.: Nonlinear theoretical/computational model of a plate in hypersonic flow with arbitrary in-plane stiffness at the boundaries. In: Second International Symposium on Flutter and Its Application. Paris, France (2020)

  18. Freydin, M., Dowell, E.H., Currao, M., Neely, A.J.: Computational study for the design of a hypersonic panel flutter experiment. In: The International Forum on Aeroelasticity and Structural Dynamics 2019. Savannah, Georgia, USA (2019)

  19. Freydin, M., Dowell, E.H., Whalen, T.J., Laurence, S.J.: A theoretical computational model of a plate in hypersonic flow. J. Fluids Struct. 93, 102858 (2020). https://doi.org/10.1016/j.jfluidstructs.2019.102858

    Article  Google Scholar 

  20. Gottwald, J., Virgin, L., Dowell, E.: Experimental mimicry of duffing’s equation. J. Sound Vib. 158(3), 447–467 (1992). https://doi.org/10.1016/0022-460X(92)90419-X

    Article  MathSciNet  Google Scholar 

  21. Kappus, H., Lemley, C., Zimmerman, N.: An experimental investigation of high amplitude panel flutter. NASA CR-1837 (1971)

  22. Kim, H.G., Wiebe, R.: Experimental nonlinear dynamics and snap-through of post-buckled composite plates. In: Nonlinear Dynamics, vol. 1, pp. 21–35. Springer (2017)

  23. Lazan, B.J.: Damping of Materials and Members in Structural Mechanics. Pergamon Press, Oxford (1968)

    Google Scholar 

  24. Leissa, A.W.: Vibration of plates. NASA (1969)

  25. Lyman, T.C., Virgin, L.N., Davis, R.B.: Application of continuation methods to uniaxially loaded postbuckled plates. J. Appl. Mech. (2014). https://doi.org/10.1115/1.4024672

  26. Marzocca, P., Fazelzadeh, S.A., Hosseini, M.: A review of nonlinear aero-thermo-elasticity of functionally graded panels. J. Therm. Stress. 34(5–6), 536–568 (2011)

    Article  Google Scholar 

  27. Mei, C., Abdel-Motagaly, K., Chen, R.: Review of nonlinear panel flutter at supersonic and hypersonic speeds. Appl. Mech. Rev. 52(10), 321–332 (1999)

    Article  Google Scholar 

  28. Miller, B., McNamara, J., Spottswood, S., Culler, A.: The impact of flow induced loads on snap-through behavior of acoustically excited, thermally buckled panels. J. Sound Vib. 330(23), 5736–5752 (2011). https://doi.org/10.1016/j.jsv.2011.06.028

    Article  Google Scholar 

  29. Nydick, I., Friedmann, P., Zhong, X.: Hypersonic panel flutter studies on curved panels. In: 36th Structures, Structural Dynamics and Materials Conference. New Orleans, LA, USA (1995)

  30. Pezeshki, C., Dowell, E.: An examination of initial condition maps for the sinusoidally excited buckled beam modeled by the duffing’s equation. J. Sound Vib. 117(2), 219–232 (1987). https://doi.org/10.1016/0022-460X(87)90535-9

    Article  MathSciNet  MATH  Google Scholar 

  31. Spottswood, S.M., Beberniss, T.J., Eason, T.G., Perez, R.A., Donbar, J.M., Ehrhardt, D.A., Riley, Z.B.: Exploring the response of a thin, flexible panel to shock-turbulent boundary-layer interactions. J. Sound Vib. 443, 74–89 (2019). https://doi.org/10.1016/j.jsv.2018.11.035

    Article  Google Scholar 

  32. Ventres, C.S., Dowell, E.H.: Comparison of theory and experiment for nonlinear flutter of loaded plates. AIAA J. 8(11), 2022–2030 (1970). https://doi.org/10.2514/3.6041

    Article  MATH  Google Scholar 

  33. Whalen, T.J., Kennedy, R.E., Laurence, S.J., Sullivan, B., Bodony, D.J., Buck, G.: Unsteady surface and flowfield measurements in ramp-induced turbulent and transitional shock-wave boundary-layer interactions at mach 6. In: AIAA Scitech 2019 Forum (2019). https://doi.org/10.2514/6.2019-1127

  34. Whalen, T.J., Schöneich, A.G., Laurence, S.J., Sullivan, B.T., Bodony, D.J., Freydin, M., Dowell, E.H., Buck, G.M.: Hypersonic fluid-structure interactions in compression corner shock-wave/boundary-layer interaction. AIAA Journal (2019). https://doi.org/10.2514/1.J059152

  35. Yuen, S.W., Lau, S.L.: Effects of in-plane load on nonlinear panel flutter by incremental harmonic balance method. AIAA J. 29(9), 1472–1479 (1991)

    Article  Google Scholar 

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Funding

This work was supported in part by funding with a grant from the Air Force Office of Scientific Research. Dr. Jaimie Tiley is the program director. The authors would like to thank Dr. Tiley and also Dr. Ivett Leyva of AFOSR for their support and direction of this work.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA

    Maxim Freydin & Earl H. Dowell

  2. Structural Sciences Center, Air Force Research Laboratory, AFRL/RQHF, Wright-Patterson AFB, Dayton, OH, 45433, USA

    S. Michael Spottswood & Ricardo A. Perez

Authors
  1. Maxim Freydin
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  2. Earl H. Dowell
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  3. S. Michael Spottswood
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  4. Ricardo A. Perez
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Contributions

MF contributed to writing of original draft, formal analysis, software, visualization. EHD contributed to conceptualization, supervision. SMS and RAP performed investigation and data curation. All authors contributed to writing, review, and editing.

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Correspondence to Maxim Freydin.

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Freydin, M., Dowell, E.H., Spottswood, S.M. et al. Nonlinear dynamics and flutter of plate and cavity in response to supersonic wind tunnel start. Nonlinear Dyn 103, 3019–3036 (2021). https://doi.org/10.1007/s11071-020-05817-x

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