NONDIMENSIONAL PARAMETERS

Two mechanical systems are said to be similar when they are similar in geometry and in the distribution of mass and elasticity. In flutter analysis, let us consider two similar systems and assume that the motion is dependent on the following fundamental variables:

Symbol	Significance	Physical Dimensions
/	Typical linear dimension	L
V	Air speed	LT-1
P	Air density	ML-3
a	Typical density of structural material	ML-3
К	Typical torsional stiffness constant (ft-lb per rad) ML2T~2

These five variables can be combined into two independent nondimensional parameters, such as

a aPU2

Any nondimensional quantity relating to the motion can be expressed as a function of these parameters. Thus if, in a free oscillation, the deflection at a point is described by an expression e~et cos cot, the damping factor e, of dimension (Г-1), can be combined with U and /

to form a nondimensional parameter el/U, and hence satisfies a functional relation:

where F is some function of the arguments (1). Therefore, a sufficient condition for two similar systems to have the same value of el/U (in particular, to have є = 0, which corresponds to the critical flutter con­dition) is that they have the same values of the parameters pja and K/(aPU2).

The frequency of oscillation со (radians per second), with dimension (7’~1), can be expressed nondimensionally in the parameter

which is called the reduced frequency or Strouhal number (§ 1.5). Hence, there exists a functional relation

Combining Eqs. 4 and 2, we see that two similar systems having the same values of pja and Kj{aPU2) flutter at the same reduced frequency.

Since all derived concepts relating to the motion can be expressed in functional relations as above, it is clear that the equality of the values of the parameters pja and K/(alzU2) is sufficient to guarantee dynamic similarity of the two systems.

In a more careful consideration, the energy dissipation of the structure and the viscosity and compressibility of the fluid must be added to the list of fundamental variables. These can be incorporated non-dimen – sionally as the material damping coefficient g, the Reynolds number R, and the Mach number M. Dynamic similarity requires the equality offg, R, and M in addition to parameters in expression (1). In general, g is important in control-surface flutter, R is important in stall flutter, and M is important in high-speed flight; otherwise their effects are small.

The Strouhal number, or the reduced frequency k, is the most natural parameter in the consideration of unsteady aerodynamic forces. When­ever convenient, the parameters pja and k, instead of those in (1), may be taken as the fundamental parameters for dynamic similarity of flutter models. It should be noted that, if the Strouhal number is calculated for the fundamental oscillation frequency of the structure, it can be identified with the second parameter in (1): The factor VKjaP, of dimension Г-1,
represents a frequency of oscillation. Hence, the parameter K/(aPU2) can be identified with the square of the Strouhal number.

The Strouhal number, or reduced frequency, characterizes the variation of the flow with time. Its inverse, I//(a>/), is called the reduced speed. An interesting interpretation of the reduced frequency is given by von Karman as follows. Consider that a disturbance occurs at a point on a body and oscillates together with the body. The fluid influenced by the disturbance moves downstream with a mean velocity U. Let the fre­quency of oscillation of the body and the disturbance be со. Then the. spacing, or “wave length” of the disturbance, is l-nUju). Therefore, the ratio

2-nU 1(0 со 2irU

which is proportional to the reduced frequency shows that к represents a ratio of the characteristic length / of the body to the wave length of the disturbance. In other words, the reduced frequency characterizes the way a disturbance is felt at other points of the body. Since every point of an oscillating body disturbs the flow, one may say that the reduced frequency characterizes the mutual influence between the motion at various points of the body.