Static Stability and “Binding ’’Near the Ground

One of the difficult points in the design of lifting systems in the ground effect is to provide a sufficient margin for the static stability of longitudinal motion.

As shown by Irodov [166], Kumar [167], Staufenbiel [177], and Zhukov [171], the longitudinal static stability of the motion of a wing-in-ground-effect ve­hicle depends on the reciprocal location of the aerodynamic centers of height and of pitch. The reserve of static stability also depends on location of the center of gravity. We define positions of the aerodynamic centers of height and pitch, respectively, as

where the superscripts h and в are ascribed, respectively, to the derivatives of the lift and the moment coefficients with respect to the ground clearance and the angle of pitch. Through analysis of the linearized equations for the longitudinal motion of wing-in-ground-effect vehicles, Irodov [166] showed that static stability is ensured if the aerodynamic center of height is located upstream of the aerodynamic center of pitch, so that for x axis directed upstream, the corresponding static stability criterion can be written as

Xh – Xg > 0.

For the coordinate system adopted in this book, with x axis directed upstream, the formulation of the condition of the static stability of longitudinal motion, used by Zhukov and Staufenbiel, implies that the full derivative of the lift coefficient with respect to the ground clearance (for a fixed zero magnitude of longitudinal moment around the center of gravity) should be positive, i. e.,

Essentially, the latter inequality shows that for a statically stable vehicle, the stabilizing effect of dCy/dh should exceed the destabilizing influence of nose-down moment. Note that Zhukov designated the full derivative of the lift coefficient with respect to the relative ground clearance as a force stability pa­rameter and pointed out that this factor has an effect upon the controllability of the vehicle and its response to the action of wind. For a properly designed vehicle, the derivative dCy/dh should be negative, and we can rewrite (11.42) as

(11.43)

It can be seen from (11.43) that Irodov’s criterion deals with pitch stability, implicitly assuming that height stability is provided, i. e., Су < 0, whereas the criteria of Zhukov and Staufenbiel represent the combined effect of pitch and height stability.

We designate the above difference in the location of the aerodynamic centers as SSM = x^ — xg and, when SSM is positive, we refer to it as to the static stability margin. Suppose that both derivatives (of height and
of pitch) and the positions of corresponding aerodynamic centers are defined with respect to a certain reference point, say, to the trailing edge. In practice, they have to be defined with respect to the center of gravity which can be viewed as a pivotal point. Introducing the abscissa xcg of the latter and noting that changing the reference point does not affect differentiation with respect to h, whereas

d_ d__ d_

дв дв Xcgdh’

we obtain the following formulas for the new positions of the centers of height and of pitch (when the reference point coincides with the center of gravity) expressed by corresponding parameters referred to the trailing edge:

where /С = CyjСу. For a foil that possesses height stability and Су/Су < 0, factor /С is negative. It may be practical to evaluate the variation of the static stability margin as a function of the position of the center of gravity (the pivotal point). Simple calculations lead to the following equation:

= (®h – xe) -> SSMcg = – A_SSM. (11.46)

It can be seen from (11.45)-( 11.46) that if a wing is found to be statically stable with reference to the trailing edge, its static stability is ensured for any other upstream position of the reference point (center of gravity), see Irodov [166]. A wing that is statically unstable with reference to the trailing edge remains statically unstable for any other position of the reference point. Si­multaneously, equation (11.45) shows when the center of gravity is shifted upstream from the trailing edge, the center in pitch moves in the same di­rection, whereas the center in height retains its position. Consequently, the static stability margin diminishes.

At present several optional aerodynamic configurations of wing-in-ground – effect vehicles are known that enable us to secure the static stability of longitudinal motion. In a wing-tail combination, employed in Russian first – generation ekranoplans, the main wing operating in close proximity to the underlying surface was stabilized by a highly mounted tailplane, taken out of the ground effect. This measure shifted the center of pitch downstream, thus increasing the static stability margin for a practical range of design pitch angles and ground clearances. The negative effect, associated with the use of a large nonlifting tail unit consists in an increase in structural weight and a noticeable reduction of the lift-to-drag ratio.

Another possibility is connected with the use of a tandem aerodynamic scheme with both lifting elements located close to the ground. When de­veloping his first З-ton piloted SM-1 prototype, Alekseev borrowed a tandem configuration from his own designs of hydrofoil ships; see Rozhdestvensky and Sinitsyn [19]. Jorg [7, 8] applied a tandem configuration in the design of his “Aerofoil Flairboats.” Note that to provide static stability to a configuration comprising two low flying wings, one has to adjust the design parameters of these wings (pitch angles, relative ground clearances, and curvatures of pressure surfaces of the wings) in a certain way. The shortcoming of a tan­dem as an option in providing static stability of longitudinal motion to a ground-effect vehicle, consists of a somewhat narrow range of pitch angles and ground clearances for which the flight is stable [22].

A way to reduce the area of the tail stabilizer (or to get rid of it) is related to appropriate profiling of the lower surface of the main wing. It means that instead of a wing section with an almost flat lower surface, known to provide a considerable increase in lift in proximity to the ground, one has to give preference to wing sections with curved lower surfaces which secure static stability, although are less efficient aerodynamic ally. Staufenbiel and Kleineidam [177] proposed a simple way to augment the static stability of the Clark-Y foil with a flat lower surface, which consists of providing this foil with a trailing edge flap, deflected to an upward position. Later on, the same authors found that if unloading of the rear part of the foil is combined with decambering of its fore part, the range of static stability can be enhanced noticeably. A family of foils with an S-shaped mean line may be shown to possess such a property. In their stability prediction for an S-shaped foil, Staufenbiel and Kleineidam [177] used an approximation of the foil’s mean line with a cubic spline function. The parameters of this curve fitting function were selected to provide the maximum range of lift coefficient in which the foil would be stable. An experimental investigation of the influence of the form of the airfoil upon its static stability was carried out by Gadetski [183]. Based on his experimental data, the author concluded that it is possible to control the positions of the aerodynamic centers by proper design of the foil. He demonstrated experimentally that an upward deflection of the rear part of the foil moves the center of height upstream and the center of pitch downstream. A similar investigation using the method of conformal mapping and an experimental technique of fixed ground board was done by Arkhangelski and Konovalov [184].

In what follows, a qualitative analysis will be carried out of the static longitudinal stability of schematic aerodynamic configurations by using the mathematical models of the extreme ground effect, see Rozhdestvensky [185]. The simplest case involves a wing of infinite aspect ratio moving in immedi­ate proximity to the ground. In this case, within the assumptions of extreme ground effect aerodynamics, it is possible to determine the characteristic centers, whose reciprocal positions define both the static stability and the
controllabity of the configuration, in analytical form. The effect of the finite aspect ratio is estimated by applying the nonlinear one-dimensional theory of a rectangular wing with endplates in motion close to the ground. A study of the static stability of a tandem configuration of infinite aspect ratio is made, assuming that for h 0, the foils constituting the tandem work indepen­dently.