The methods discussed in the last two sections can be extended to systems having any finite number (say ri) of degrees of freedom. Several different points of view may be used to determine the critical flutter speeds. In the forced-oscillation method (§ 6.9), we may proceed to write down the equations of motion, with an (undetermined) external excitating force acting in one of the degrees of freedom (say the first equation), and with no excitation term in the remaining (n — 1) equations. The amplitude and phase angle of one of the remaining degrees of freedom (say the nth) may be arbitrarily assumed. For a given speed of flow and given fre­quency, these (n — 1) homogeneous equations can be used to determine the amplitude and phase angle of various degrees of freedom in proportion to the nth, and the first equation can then be used to determine the magnitude and phase of the exciting force required to act in the first degree of freedom to produce the assumed motion. A variation in the speed of flow and the oscillation frequency will produce a corresponding change in the required exciting forces. By repeating the calculations for different combinations of speed and frequency, the trend of changes in the exciting force can be determined, and the speed and frequency at which the exciting force vanishes can be obtained. These are the flutter speed

* Ref. 6.8 also gives examples to show that the stability of a wing cannot be determined by measuring the work done by an exciting force in a steady forced oscillation. The reason is very simple: The amplitude and phase relationship between various compon­ents of motion vary with the frequency and the point of application of the exciting force. In particular, the amplitude and phase relationship in forced oscillation are in general different from those in a free oscillation. From § 5.4 it is clear that such variations will cause an important change in the energy exchange between the wing and the airstream. Frazer shows that in certain cases it is possible to provide a mechanism which extracts energy from a wing, yet causing an otherwise stable wing to oscillate sinusoidally.

and frequency. • This method has been applied by Duncan6-6’67 and Myklestad6,17 to systems having many degrees of freedom.

It is also possible to study the free oscillations following an initial disturbance and define flutter as a condition at which an oscillation of nondecreasing amplitude is obtained. A method of iteration for such calculation has been used by Goland and Luke.6-11 When the existence of flutter is presupposed, as in § 6.10, the airspeed and oscillation frequency are sought. The similarity of this formulation to the classical vibration problem of a mechanical system in still air indicates that flutter is a complex eigenvalue problem.* Besides Theodor – sen’s method, this problem may be solved by matrix methods as given by Duncan, Frazer, and Collar,6-5 or by the method of iteration, as expounded by Jordan,614 Kussner,5-28 Wielandt,11-24 Greidanus, and van de Vooren.6-12 A method proposed by R. A. Frazer610 regards the flutter determinant as defining an eigenvalue problem involving two real-valued eigenvalues, one related to the reduced frequency and the other to the stiffness of the structure.

The most straightforward method is to write the flutter equation as a matrix equation, form the characteristic equation, and solve the complex eigenvalues. The quickest way, however, is to use an analog computer. For analog approach we refer to the book by McNeal.6-28 The aero – elastic system is simulated by an electric network. An example of such an analog sets up the following correspondence:

Capacitors—concentrated or lumped inertia properties Inductors—lumped flexibility properties Transformers—geometric properties Voltage—velocities Current—forces

Such an electric analog can be regarded as a model of aircraft whose properties can be altered easily. A parametric study of the aircraft can be done with great rapidity. A particular advantage of the electric analog method is that tuned pulses may be used to separate two or more nearly unstable modes of motion.

Whereas the most convenient form of aerodynamic information used in the influence-functions formulation is the frequency response, in the analog method indicial responses are used: e. g., the lift due to a sudden change of angle of attack, the moment due to a sudden aileron deflection, etc. They can be approximated by a finite sum of exponential functions. The electric analog for the aerodynamic system can then be recognized

* Further references regarding the mathematical problem of complex eigenvalues can be found in Chapter 11.

relatively easily by an examination of the Laplace transforms of the indicial responses.

In all the methods mentioned above, one tries to find the actual airspeed at which flutter may occur, but often this is more than necessary from the point of view of airplane design. In fact, an airplane may be regarded as nonexistent for airspeeds exceeding the design speeds. The immediate question regarding the safety against flutter is this: Will the airplane flutter within the “design envelope” of Mach number and altitude? This question has a different content from the question of finding the flutter speeds in terms of miles per hour. We are concerned only with the sta­bility of the aeroelastic system in a given speed range. Mathematically, we are dealing with the problem of the existence of an eigenvalue within a specified range, instead of finding out its numerical value. Certain methods applicable to this problem will be discussed in § 10.6.

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