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The effects of various parameters on an aeroelastic optimization problem

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Abstract

The optimal design of a panel flutter problem is investigated in this paper. A semi-infinite flat panel with either a homogeneous or sandwich cross section is considered. The thickness distribution of the panel is allowed to vary while the total weight is held fixed, and the distribution which maximizes the critical flutter parameter for stability is chosen as the optimal design. This design is calculated here by means of a generalized Ritz procedure, with the panel thickness assumed to have a certain form.

Variations in the following parameters are then considered: a minimum allowable thickness, aerodynamic damping, in-plane loading, and nonstructural stiffness and mass for the case of a sandwich panel. It is shown that the optimal design may be significantly affected by changes in these parameters.

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Abbreviations

a i :

modal amplitudes

D(X),D 0 :

panel stiffness

E c ,E f :

Young modulus of core and face sheets, respectively

g :

=ρ UL 2(M 2−2)/[(M 2−1)3/2 (D 0 μ 0)1/2], aerodynamic damping parameter

H(X),H 0 :

thickness of homogeneous panel

H c :

thickness of sandwich panel core

h(x),h i :

nondimensional panel thickness,h=H/H 0

L :

panel length

M :

Mach number

m(x):

=μ/μ 0, nondimensional mass

p i :

coefficients of characteristic equation

R x :

midplane compressive load

r :

=R x L 2/D 0, nondimensional load

s(x):

=D/D 0, nondimensional stiffness

T(X),T 0 :

thickness of sandwich panel face sheets

t(x),t i :

nondimensional face sheet thickness,t=T/T 0

U :

speed of supersonic flow

V :

functional in Ritz method

W 0 :

total weight of reference uniform panel

W(X):

panel deflection amplitude

w(x):

=W/L, nondimensional deflection amplitude

w i (x):

deflection modes

X :

coordinate along length in airflow direction

x :

=X/L, nondimensional length coordinate

α i ,β i ,ε i ,ν ij :

integrals in Ritz equations

γ :

frequency parameter

δ :

=(1+E c H c /6E f T 0)−1, nonstructural stiffness parameter

η :

=(1+ρ c H c /2ρ f T 0)−1, nonstructural mass parameter

λ :

=ρ U 2 L 3/[D 0(M 2−1)1/2], dynamic pressure parameter

λ*,λ 0*:

critical value ofλ for stability

μ(X),μ 0 :

panel mass per unit area

ρ :

density of air

ρ c ,ρ f :

density of core and face sheets, respectively

σ :

time

τ :

=σ(D 0/μ 0)1/2/L 2, nondimensional time

0:

values for reference uniform panel

References

  1. 1.

    Ashley, H., andMcIntosh, S. C., Jr.,Application of Aeroelastic Constraints in Structural Optimization, Proceedings of the 12th International Congress of Applied Mechanics, Springer, Berlin, 1969.

  2. 2.

    Turner, M. J.,Optimization of Structures to Satisfy Flutter Requirements, AIAA Journal, Vol. 7, No. 5, 1969.

  3. 3.

    McIntosh, S. C., Jr., Weisshaar, T. A., andAshley, H.,Progress in Aeroelastic Optimization—Analytical Versus Numerical Approaches, Stanford University, Department of Aeronautics and Astronautics, Report No. SUDAAR 383, 1969.

  4. 4.

    Armand, J. L., andVitte, W. J.,Foundations of Aeroelastic Optimization and Some Applications to Continuous Systems, Stanford University, Department of Aeronautics and Astronautics, Report No. SUDAAR 390, 1970.

  5. 5.

    Weisshaar, T. A.,An Application of Control Theory Methods to the Optimization of Structures Having Dynamic or Aeroelastic Constraints, Stanford University, Department of Aeronautics and Astronautics, Report No. SUDAAR 412, 1970.

  6. 6.

    Craig, R. R., JR.,Optimization of a Supersonic Panel Subject to a Flutter Constraint-A Finite Element Solution, Paper presented at AIAA/ASME 12th Structures, Structural Dynamics, and Materials Conference, Anaheim, California, 1971.

  7. 7.

    Plaut, R. H.,Approximate Solutions to Some Static and Dynamic Optimal Structural Design Problems (to appear).

  8. 8.

    Levinson, M.,Application of the Galerkin and Ritz Methods to Nonconservative Problems of Elastic Stability, Zeitschrift für Angewandte Mathematik und Physik, Vol. 17, No. 3, 1966.

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Additional information

This work was supported in part by the US Army Research Office-Durham and in part by the US Navy under Grant No. NONR-N00014-67-A-0191-0009.

Communicated by W. Prager

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Plaut, R.H. The effects of various parameters on an aeroelastic optimization problem. J Optim Theory Appl 10, 321–330 (1972). https://doi.org/10.1007/BF00934804

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Keywords

  • Optimal Design
  • Total Weight
  • Thickness Distribution
  • Sandwich Panel
  • Flat Panel