keeping in mind that these functions are not orthogonal.

13. Referring back to either Problem 11 or 12, starting with the virtual work of the aerodynamic forces as ft'

Jo where L' and M' are the sectional lift and pitching moment expressions used to develop Eqs. (3.76), and using the given deformation modes, find the generalized forces S/,/' = 1,2 N — Nw + Ng. As discussed in the text, generalized forces are the coefficients of the variations of the generalized coordinates in the virtual work expression.

Hint: Neglecting the weight terms on the right-hand sides of Eqs. 3.74, one finds that L' is the right-hand side of the second of those equations, whereas M' is the negative of the right-hand side of the first and equal to eL'.

14. Referring back to Problems 13 and either 11 or 12, determine the generalized equations of equilibrium in the form where q is the dimensionless dynamic pressure given by q/qn„, qD0 is the torsional divergence dynamic pressure of the unswept clamped-free wing, given by

Eq. (3.54), {f | is the column matrix of all unknowns r/,, i — 1,2 Nw, and

</>;, i — 1, 2,..., Ng, and {E0} is a column matrix containing the parts of the aerodynamic generalized forces that do not depend on any unknowns. The N x N matrices [¿T] and [A] are the stiffness and aerodynamic matrices, respectively. If Problem 11 is the basis for solution, then the stiffness matrix [/¡T] is diagonal since the normal modes used to represent the wing structural behavior are orthogonal with respect to the stiffness properties of the wing.

15. Referring back to Problem 14, perform the following numerical studies:

(a) Divergence: To determine the divergence dynamic pressure, write the homogeneous generalized equations of equilibrium in the form q which is obviously an eigenvalue problem with 1 /q as the eigenvalue. After you solve the eigenvalue problem, the largest 1 /g provides the lowest dimensionless critical divergence dynamic pressure qD — qo/qD0 at the sweep angle under consideration. By numerical experimentation, determine how many modes are needed for each of w and 6 to obtain the divergence dynamic pressure to within plotting accuracy. Plot the divergence dynamic pressure versus sweep angle for a range of values for the sweep angle —45° < A < 45° and values of the dimensionless parameters e/t (0.05 and 0.1) and EI/GJ (1 and 5). Compare your results with those obtained from Eq. (3.92). Comment on the accuracy of the approximate solution in the text versus your modal solution. Which one should be more accurate? Discuss the trends of divergence dynamic pressure that you see regarding the sweep angle, stiffness ratio, and location of the aerodynamic center.

(b) Response: For the response you will need to consider the nonhomogeneous equations, which should be put into the form

Letting ar — 1°, obtain the response by solving the linear system of equations represented in this matrix equation. Plot the response of the wing tip (i.e., w and 0 at y — I) for varying dynamic pressures up to q — 0.95qD for the above values of e/l and EI/GJ with A = -25° and 0°. Plot the lift, twist, and bending moment distributions for the case with the largest tip twist angle. Comment on this result and on the trends of static aeroelastic response that you see regarding the sweep angle, stiffness ratio, and location of the aerodynamic center.

16. Consider the divergence of an unswept composite wing with k — 0, GJ/EI — 0.2, and e/t — .025. Using Eq. (3.107), determine the value of k, as defined by Eq. (3.106), needed to keep the divergence dynamic pressure unchanged for forward-swept wings with various values of A < 0. Plot these values of k versus A.

17. Using the approximate formula found in Eq. (3.107), show that divergence is possible for backswept wings when e < 0 and that in this case sweeping the wing forward may help to avoid divergence. Discuss the situations in which one might encounter a negative value of e. What sign of k would you expect to be stabilizing in this case? Plot the divergence dynamic pressure for a swept composite wing with G J/EI — 0.2 and c/t — —.025 versus k — ±0.4 for varying A.

18. Consider the divergence of a swept composite wing. Show that the governing equation and boundary conditions found in Eqs. (3.101) and (3.102) can be written as a second-order, integro-differential equation of the form with boundary conditions 0(0) = Q'( 1) = 0 and witli r — fi/x. Determine the two simplest polynomial comparison functions for this reduced-order equation and boundary conditions. Use Galerkin's method to obtain one- and two-term approximations to the divergence dynamic pressure Xq versus r. Plot your approximate solutions for the case in which GJ/EI — 0.2, c/t — 0.02, and k — —0.4, depicted in Fig. 3.24, and compare these with the approximate solution given in the text. For the two-term approximation, determine the limit point for positive e, noting that the exact values are r — 1.59768 and r„ = 10.7090.

Answers: The one-term approximation is

282 - 105r ± v/3yi5r(197r - 1,036) + 17,408 ' The approximate limit point in the first quadrant is at r — 1.61804 and Xq — 11.2394. Within plotting accuracy, the two-term approximation is virtually indistinguishable from the exact solution when —10 < Xq < 10.

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