Equations and the Stability of Motion of a Lifting System in the Extreme Ground Effect

The analysis of the dynamics of wing-in-ground-effect vehicles provides data for assessing the stability of motion, controllability, and ride comfort of the craft under development.

A study of the linearized equations of the motion of wing-in-ground-effect vehicles was carried out by Irodov [166], Kumar [167]—[170], Zhukov [171]— [175], and Staufenbiel et al. [176]-[178]. This research revealed a significant distinction between the dynamics and the stability criteria of these vehicles and the aircraft that normally operates out of the ground effect. It was also found that one of the typical problems of the design of ground-effect machines is due to the strong coupling between their aerodynamic configuration and the flight dynamics.

Irodov[166] considered the linearized equations of perturbed longitudinal motion in terms of the variation of the angle of attack and the ground clear­ance. Assuming that the speed of the vehicle remains constant, he derived a (quartic) characteristic polynomial equation of the fourth order. Applying the Gurvitz-Ruth criteria of stability, he came to the conclusion that aperiodic static stability is ensured when the aerodynamic center of height is located upstream of that of the angle of attack. This important prac­tical conclusion signifies that if the aerodynamic configuration is not selected properly, it is impossible to secure static longitudinal stability to the motion of a wing-in-ground-effect vehicle by choosing the position of the center of gravity.

According to Irodov [166], to secure the oscillatory stability of the vehicle one has to provide an appropriate location of the center of gravity upstream of the center of the angle of attack. In the same work, Irodov indicated that account of the variation in cruise speed practically does not modify the previously mentioned condition of static aperiodic stability.

Kumar [166] studied the dynamics of a wing-in-ground-effect vehicle in both longitudinal and lateral motion, incorporating the effects connected with perturbation of the speed of forward motion. His stability analysis was based on a quintic characteristic equation.

A thorough study of the dynamics of wing-in-ground-effect vehicles ac­counting for the perturbation of speed and incorporating stability analy-

K. V. Rozhdestvensky, Aerodynamics of a Lifting System in Extreme Ground Effect © Springer-Verlag Berlin Heidelberg 2000

sis with special reference to controllability[68] and design, was carried out by Zhukov, starting in the 1970s and finalized in [175]. He revealed several dis­tinct parameters, defining static stability and dynamic behavior of wing-in­ground-effect craft. In particular, he introduced the notion of binding to the ground, as a capability of a vehicle in cruising flight to stay in ground effect after the action of controls or gusts of wind.

Staufenbiel also studied stability criteria, used the quintic characteristic equation for the analysis of the dynamics, and discussed nonlinear effects.

This section covers some linear formulations related to the longitudinal dynamics of wing-in-ground-effect vehicles. First of all, a derivation is given in terms of the perturbations of the relative ground clearance h and the pitch angle в of the linearized equations of motion without (after Irodov) and with (after Zhukov) account of perturbation of forward speed. Then, we consider an approximate derivation of an asymptotic form of the linearized equations of the longitudinal motion of a wing-in-ground effect vehicle in the extreme ground effect, i. e., for vanishing relative clearances between the lifting surface and the ground. The orders of magnitude of the terms are evaluated formally on the basis of a simplified nonlinear unsteady theory of the extreme ground effect, discussed in section 4. Eventually, an asymptotic form of the equations of motion is derived for h —> 0 and small periods of time from the moment of the action of the perturbation. It is shown that on (nondimensional) time scale t = 0(1), which corresponds to distances of the order of the chord from the moment of perturbation, the equations of motion correspond to the quartic formulation of Irodov [166], i. e., the speed of the vehicle remains almost constant. From a practical viewpoint, this signifies that Irodov’s criterion of static stability is valid, although it was derived on the basis of the somewhat restrictive assumption of no perturbation of speed. Differing from Irodov, the asymptotic form of the equation, valid for a vanishing /і, does not depend explicitly on the relative ground clearance, but rather on the reduced density Д = fih and the ratios of the design pitch and the curvature of the lower surface to /і, i. e., the number of parameters is fewer by one compared to the initial formulation. On larger time scales of the order of 1 jh and 1/h? the variation of speed is first driven by height and pitch perturbations and later is determined by the speed perturbation proper. The latter conclusion confirms the results derived by Zhukov [175].