## Flutter Flight Envelope Below Sealevel

Service Ceiling

Sea Level

Figure 4.16 Flight envelope for typical Mach 2 fighter.

Stag. Temp.

Figure 4.16 Flight envelope for typical Mach 2 fighter.

flutter does not appear to happen for any combination of a and r when the mass centroid, elastic axis, and aerodynamic center all coincide (i.e., when e — a — —1/2). Even if this prediction of the analysis is correct, practically speaking, it is very difficult to achieve coincidence of these points in wing design. Remember, however, that all these statements are made with respect to simplified models. One needs to analyze real wings in a design setting using powerful tools, such as NASTRAN or ASTROS. Indeed, bending-torsion flutter is a very complicated phenomenon, and it seems to defy all our attempts at generalization. Additional discussion of these phenomena, along with a large body of solution plots, can be found in Bisplinghoff, Ashley, and Halfman (1955).

The final flutter boundary can be presented in numerous ways for any given flight vehicle. The manner in which it is illustrated depends on the engineering purpose it is intended to serve. One possible presentation of the flutter boundary is to superpose it on the vehicle's flight envelope. A typical flight envelope for a Mach 2 attack aircraft is illustrated in Fig. 4.16 with two flutter boundaries indicated by the curves marked "No. 1" and "No. 2." The shaded region above the flutter boundaries, being at higher altitudes, corresponds to stable flight conditions; below the boundaries flutter will be experienced. Flutter boundary no. 1 indicates that for a portion of the intended flight envelope, the vehicle will experience flutter. It should be noted that these conditions of instability correspond to a flight Mach number near unity (transonic flow) and high dynamic pressure. This observation can be generalized by saying that a flight vehicle is more susceptible to aeroelastic flutter for conditions of (1) lower altitude, (2) transonic flow, and (3) higher dynamic pressure.

If it is determined that the vehicle will experience flutter in any portion of its intended flight envelope, it is necessary to make appropriate design changes to eliminate the instability for such conditions. These changes may involve alteration of the inertial, elastic, or aerodynamic properties of the configuration. Many times small variations in all three provide the best compromise. Flutter boundary no. 2 is indicative of a flutter-safe vehicle. Note that at the minimum altitude transonic condition there appears to be a safety margin with respect to flutter instability. All flight vehicle specifications require such a safety factor, which is generally called the "flutter margin." Most specifications require that the margin be 15% over the limit equivalent airspeed. In other words, the minimum flutter speed at sea level should not be less that 1.15 times the airspeed for the maximum expected dynamic pressure as evaluated at sea level.

### 4.8 Epilogue

In this chapter we have considered the general problem of lifting-surface flutter. Several types of flutter analysis have been presented, including the p method, classical flutter analysis, the k method, and the p-k method. The application of classical flutter analysis to discrete one- and two-degree-of-freedom wind tunnel models has been presented. The student has been exposed to Theodorsen's unsteady thin-airfoil theory along with the more modern finite-state thin-airfoil theory of Peters et al. Application of the assumed modes method to construct a flutter analysis of a flexible wing has been demonstrated as well. Finally, some of the important parameters of the flutter problem have been discussed, along with current design practice. With a good understanding of the material presented herein, the student should be sufficiently equipped to apply these fundamentals to the design of flight vehicles.

Moreover, with appropriate graduate-level studies well beyond the scope of material presented herein, the student will be able conduct research in the exciting field of aeroelas-ticity. Current research topics are quite diverse. With the increase in the sophistication of controls technology, it has become more and more common to attack flutter problems by active control of flaps or other flight control surfaces. These so-called flutter suppression systems provide alternatives to costly design changes. One type of system for which flutter suppression systems are an excellent choice is a military aircraft that must carry weapons as stores. These aircraft must be free of flutter within their flight envelope for several different configurations. Sometimes avoidance of flutter by design changes is simply beyond the capability of the designer for such complex systems. There is also research to determine in flight when a flutter boundary is being approached. This could be of great value for situations in which damage had altered the properties of the aircraft structure, perhaps unknown to the pilot, thus shifting the flutter (or divergence) boundary and making the aircraft unsafe to operate within its original flight envelope. Other current problems of interest to aero-elasticians include improved analysis methodology for prediction of flutter, gust response, and limit-cycle oscillations, design of control systems to improve gust response and limit-cycle oscillations; and incorporation of aeroelastic analyses at an earlier stage of aircraft design.

1. Compute the flutter speed for the incompressible, one-degree-of-freedom flutter problem with

Answer. UF = 405.6 ft/s 2. According to Theodorsen's theory, the circulatory lift is proportional to a quantity that, for simple harmonic motion, can be shown to be equal to the effective angle of attack given by

### Problems k k

For a — —1/2 and simple harmonic motion such that 0—1 and h — bz[cos(<p) + i sin(</>)], plot a as a function of time for five periods for the following four cases:

Comment on the behavior of a for increasing k, changing the phase angle from 0° to 90°, and increasing the plunge magnitude. You may approximate Theodorsen's function as

3. Show that the coefficients used in a classical flutter analysis, if based on Theodorsen's theory, are

4. Consider an incompressible, two-degree-of-freedom flutter problem in which a — —1/5, e — —1/10, [i — 20, r2 — 6/25, and a = 2/5. Compute the flutter speed and the flutter frequency using the classical flutter approach. For the aerodynamic coefficients use those of Theodorsen's theory with C(k) approximated as in Problem 2.

Answers: Uf — 2.170 btog and u>f = 0.6443 u>g

5. Consider an incompressible, two-degree-of-freedom flutter problem in which a — —1/5, )JL = 3, and r — 1/2. Compute the flutter speed and flutter frequency for two cases x„ = e — a — 1/5 and.v0 — e — <7 = 1/10, and let a = 0.2, 0.4, 0.6, 0.8, and 1.0. Use the classical flutter approach, and for the aerodynamic coefficients use those of Theodorsen's theory with C(k) approximated as in Problem 2. Compare with the results in Fig. 4.14.

6. Set up the complete set of equations for flutter analysis by the p method using the unsteady aerodynamic theory of Peters et al. (1995), nondimensionalizing Eqs. (4.97) and redefining a, as bcogXj.

7. Write a computer program using MATLAB or Mathematica to set up the solution of the equations derived in Problem 6.

8. Using the computer program written in Problem 7, solve for the dimensionless flutter speed and flutter frequency for an incompressible, two-degree-of-freedom flutter problem in which a — —1/3, e — —1/10, [i — 50, r — 2/5, and a = 2/5.

Answers: Uf — 2.807ba>g and u>f — 0.5952wg

9. Write a computer program using MATLAB or Mathematica to set up the solution of a two-degree-of-freedom flutter problem using the k method.

10. Use the computer program written in Problem 9 to solve a flutter problem in which a — —1/5, e — —1/10, = 20, r2 — 6/25, and a = 2/5. Plot the values of (On/oe and 8 versus U / (b to g) and compare your results with the quantities plotted in Figs. 4.11 and 4.12. Noting how the quantities plotted in these two sets of figures are different, comment on the similarities and differences you observe in these plots and why those differences are there. Finally, explain why your predicted flutter speed is the same as that determined with the classical method.

Figure 4.17 Plot of a)i,2/tog versus U/(bwg) using the k method and Theodorsen aerodynamics with a = -1/5, e = -1/10, fjL = 20, r2 = 6/25, and cr =2/5.

11. Show that the flutter determinant for the p—k method applied to the typical section using Theodorsen aerodynamics can be expressed as

2 _ , £1 , 2ikCjk) k(i+ak)+[2+ik(\-2aj\C(k)+p2n.xe, p n "T" k2 + n n ak2-i k(l+2a)C(k)+p2 nxg 4ik2( l+2«)[2/-A( l-2a)]C(k )+k\4i-Sia-k-Sa2k)+>S(n+k2 p2 n )r2 ¡1 8 k2 n

12. Write a computer program using MAT LAB or Mathematica to set up the solution of a two-degree-of-freedom flutter problem using the p—k method and Theodorsen aerodynamics.

13. U se the computer program written in Problem 12 to solve a flutter problem in which a — —1/5, e — —1/10, [i — 20, r2 — 6/25, and a = 2/5. Plot the values of the estimates of Qi^/cog and Ti^/cog versus l!/(ho>,,) and compare your results with the quantities plotted in Figs. 4.11 and 4.12. Explain why the estimated damping from the p—k method sometimes differs from that of the p method.

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