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Dynamic Aeroelasticity

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A Modern Course in Aeroelasticity

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 264))

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Abstract

Dynamic aeroelasticty is considered and the dynamic stability (Flutter) of linear aeroelastic systems is considered as well as the response to external disturbances including atmospheric turbulence (Gusts). The discussion proceeds from simpler physical models and mathematical methods to more complex ones. An introduction to the modeling of aerodynamics forces is also provided to prepare the reader for the material in chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’.

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Notes

  1. 1.

    See, for example, Meirovitvh [1].

  2. 2.

    Bisplinghoff, Mar, and pian [2], Timoshenko and Goodier [3].

  3. 3.

    And necessary, i.e., they are independent.

  4. 4.

    Meirvovitch [4].

  5. 5.

    BA, pp. 201–246.

  6. 6.

    Note \(\Delta n=1\) since any n is an integer.

  7. 7.

    Meirovicth [4].

  8. 8.

    Meirovitch [4].

  9. 9.

    Crandall and Mark [5].

  10. 10.

    \(\frac{\dot{h}}{U}+\frac{\omega _{G}}{U}\) is an effective angle of attack, \(\alpha \).

  11. 11.

    Here we choose to use a dimensional rather than a dimensionless transfer function.

  12. 12.

    We ignore a subtlety here in the interest of brevity. For a ‘frozen gust’, we must take \(\omega _{G}=\bar{\omega }_{G}\exp i\omega (t-x/U_{\infty })\) in determining this transfer function. See later discussion in Sects. 6, 2 and 3.

  13. 13.

    Crandell and Mark; the essence of the approximation is that for small \(\zeta , \Phi _{w_{G}w_{G}}(\omega )\cong \Phi {w_{G}w_{G}}(\omega _{h})\) and maybe taken outside the integral. See the subsequent discussion of a graphical analysis.

  14. 14.

    Houbolt, Steiner and Pratt [6]. Also see later discussion in Sect. 6.

  15. 15.

    Acum [7].

  16. 16.

    Acum [7].

  17. 17.

    Pines [8].

  18. 18.

    Ashley, and Zartarian [9]. Also see chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’.

  19. 19.

    See chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’.

  20. 20.

    For light bodies or heavy fluids, e.g., lighter-than-airships or submarines, they may be important.

  21. 21.

    For a clear, concise discussion of transient, two-dimensional, incompressible aerodynamics, see Sears [10], and the discussion of Sears’ work in BAH, pp. 288–293.

  22. 22.

    See BAH, pp. 367–375, for a traditional approach and chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’ for an approach via Laplace and Fourier Transforms.

  23. 23.

    Jones [11].

  24. 24.

    BAH, p. 418.

  25. 25.

    Dowell and Widnall [12], Widnall and Dowell [13], Dowell [14].

  26. 26.

    Hamming [15].

  27. 27.

    Houbolt [16].

  28. 28.

    Savant [18].

  29. 29.

    Garrick [19].

  30. 30.

    (For each complex root of the polynomial.)

  31. 31.

    Hassig [20].

  32. 32.

    Eckhaus [21], Landahl [22], Lambourne [23].

  33. 33.

    See chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’.

  34. 34.

    Abramson [24]. Viscous fluid effects are cited as the source of the difficulty.

  35. 35.

    Crisp [25].

  36. 36.

    Houbolt, Steiner and Pratt [6].

  37. 37.

    Houbolt, Steiner and Pratt [6]. The basis for the frozen gust assumption is that in the time interval for any part of the gust field to pass over the flight vehicle (the length/\(U_{\infty }\))the gust field does not significantly change its (random) spatial distribution. Clearly this becomes inaccurate as \(U_{\infty }\) becomes small.

  38. 38.

    Houbolt, Steiner and Pratt [6].

  39. 39.

    These particular examples were collected and discussed in Ashley, Dugundji and Rainey [24].

  40. 40.

    Recall Sect. 2.

  41. 41.

    Recall Sect. 2.

  42. 42.

    Meirovitch [4].

  43. 43.

    See chapter ‘Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces’, and earlier discussion in Sect. 4.

  44. 44.

    Provided \(S_{\alpha }\equiv 0\) so that \(h, \alpha \) are normal mode coordinates for the typical section.

  45. 45.

    By fixed we mean ‘clamped’ in the sense of the structural engineer, i.e., zero displacement and slope. It is sufficient to use a static influence function, since invoking by D’Alambert’s Principle the inertial contributions are treated as equivalent forces.

  46. 46.

    Bisplinghoff, Mar and Pian [2].

  47. 47.

    Cf. (7.31).

  48. 48.

    For \(q_{1}(0)=\dot{q}(0)=0.\,\mathcal {L}^{1}\equiv \) inverse Laplace Transform.

  49. 49.

    Weaver and Paidoussis [27] Also see Daidoussis [28].

  50. 50.

    Note that slender body  aerodynamic theory is used.

  51. 51.

    Sections 5.

  52. 52.

    Gregory and Paidoussis [29].

  53. 53.

    Paidoussis and Issid [30].

  54. 54.

    Chen [31].

  55. 55.

    Dowell [32]. Also see Bolotin [33].

  56. 56.

    This was suggested by Dr. H. M. Voss.

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

    Earl H. Dowell

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    Earl H. Dowell

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Dowell, E.H. (2022). Dynamic Aeroelasticity. In: Dowell, E.H. (eds) A Modern Course in Aeroelasticity. Solid Mechanics and Its Applications, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-030-74236-2_3

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