The Helicopter in Trim

EQUATIONS OF EQUILIBRIUM

Like any aircraft in steady flight, the helicopter must be in equilibrium with respect to three forces and three moments acting along and around three orthogonal axes through its center of gravity. The analysis an be based on one of three possible systems of axes: wind axes, stability axes, or body axes. These systems are distinguished from each other by relating them to the three types of balance systems that are used in wind tunnels as shown in Figure 8.1. If the balance system is always aligned with the tunnel center line, the measurements will be in the wind axes system in which the X axis points along the line of flight in both the side and top views. If the balance system is mounted on a yaw table that rotates to produce sideslip conditions, the measurements will be in the stability axes system. For this case, the X axis will be aligned with the flight path in the side view but with the body in the top view. If the balance system is contained in the body of the model, the forces and moments will be measured in the body axes system and the X axis will line up with the body in both the side and top views. Although each system is valid, there are two reasons for using the body axes system in helicopter analysis. First, the other systems lose their significance in hover. Second, many helicopters are equipped with stability augmentation systems using gyros or accelerometers
whose displacements are measured with respect to the airframe, and the analysis of the effects of these devices is easiest in the body axes system.

(One of the minor annoyances that may have to be faced with the body axis system is the reluctance of the design department to define waterlines as being perpendicular to the rotor shaft. It is more likely that the designers will lay out the fuselage using some such arbitrary line as a cabin floor for a waterline and then tilt the rotor shaft forward to obtain a level floor and a streamline fuselage attitude at

Axes System

FIGURE 8.1 (cont.)

the cruise condition. This produces unnecessary bookkeeping complications for the aerodynamicist in keeping track of center-of-gravity positions, moments of inertia, and horizontal stabilizer angle of attack. It would be helpful if the designers would use the shaft as their primary reference and then tilt the cabin floor and draw the fuselage contours for minimum drag. The finished helicopter would look the same but would be easier to analyze. This is the scheme that has been used for the example helicopter, but the analytical procedures that follow allow for those unfortunate cases where the shaft is tilted with respect to the body axis defined as a designer’s waterline.)

The aerodynamic moments and forces in the body axis system that are acting on the helicopter are shown in Figure 8.2. They are due to the main rotor, the tail rotor, the horizontal stabilizer, the vertical stabilizer, and the fuselage. (For some helicopters, add a wing and/or a propeller.)

The six equations of equilibrium are:

XM + XT + XH + Xv + XF = G. W. sin 0 YM + YT +Yv + Yf= – G. W. sin Ф Zft + Zj + Zfj + Zy + Zp = —G.^F. cos 0

Rm + YMhM + ZMyM + YThT + Yyhy + YFhF + RF = 0

+ ZMlM + MT XThT + ZTlT XHhH + ZHlH — Xvhv + Mp + Zp Ip — Xphp — 0

NM — YMlM — YTlT — Yyly + Nf — Ypip = 0

When the equilibrium equations were developed for airplanes many years ago, all engineering analysis and reports were hand-lettered, and the three moments were written in script: і? Л, and Ж Nowadays, typewriters and word processors don’t make script characters. This makes a special problem with rolling moment, since L is used for total lift and, if also used for total rolling moment, causes confusion, especially in distinguishing Ci, meaning lift coefficient, from CL meaning roll moment coefficient. For that reason, R is used in this book for rolling moment.