Sideforce and Yawing and Rolling Moments
The fuselage will produce side forces, yawing moments, and rolling moments as a function of sideslip angle. If the calculations are being done without the benefit of wind tunnel tests, the fuselage sideforce (in wind axes) and the yawing moment can be estimated using the same procedures outlined for fuselage lift and pitching moment above. The rolling moment of the fuselage due to sideslip is caused by its dihedral effect and can be either slightly positive or slightly negative. A method for estimating this moment using the physical parameters of the fuselage and wing is given in reference 8.1, but since flight test experience has shown that the dihedral effect is strongly, but mysteriously, influenced by the interference of the main rotor wake on the tail rotor and the vertical stabilizer, the estimation of fuselage dihedral effect has a very low priority, so the method will not be repeated here.
M/q = C/w (Fuse. Volume)
Source: Harris et at., “Helicopter Performance Methodology at Bell Helicopter Textron," AHS 35th Forum, 1979.
For an airplane—which has lateral symmetry—the six equations of equilibrium can be conveniently dealt with in two groups; the longitudinal equations consisting of X, Z, and M and the lateral-directional equations consisting of Y, R, and N. A helicopter is not quite as symmetrical as an airplane, and there is some crosscoupling as described in Chapter 7. Thus, from a rigorous point of view, the trim equations should be determined from a simultaneous solution of all six equations; but from a practical point of view, they can—and will—be treated as two independent sets.
The equilibrium equations in X, Z, and M can be used to find the longitudinal trim conditions of the helicopter. In Chapter 3 an approximate method was used based only on the equivalence of the X and Z equilibrium equations (in wind axes) which was satisfactory for performance calculations. That approximation ignored the pitching moments by assuming that the tip path plane was always perpendicular to the rotor mast. Now the moment equation will be used with the other two equations (in body axes) to find the magnitude of longitudinal flapping, along with the other parameters, which must exist if the aircraft is in trim.
There are two approaches for solving for the trim conditions from the equilibrium equations. The first is to write the three equations as Unear functions of three unknowns and then to solve them simultaneously. This method gives acceptable accuracy for most engineering purposes and will be illustrated in the following discussions. The second method is not constrained to Unear functions. It is done as an iterative procedure where values of the three unknowns are chosen initially based on estimates or guesses and then refined by going through the equations several times. This is well suited to high-speed computers and can readily be expanded to include aU six equiUbrium equations. As a matter of fact, in these types of computer programs, the trim conditions are simply faUouts of computing the performance using such methods as those presented in Chapter 3.
reduce the system to three linear equations in the three unknowns: 0, TM, and aX; In the equations of Table 8.4, the physical dimensions are, of course, known and the trim—or barred—terms for the aircraft components will already be known from previous performance calculations such as those done in Chapter 3- Table 8.4 also presents the numerical results for the example helicopter at 115 knots (|1 = .3). The elements that go into the equations for this example are tabulated in Table 8.5 which can serve as a check list for any helicopter.
Solving the three equations simultaneously yields the following solutions for the example helicopter:
Тд, — 20,5 86 lb 0 = -.0165 rad = -.9°
a. =-.019 rad =-1.1° –
Besides the thrust, pitch attitude, and longitudinal flapping, the other trim value of interest is the longitudinal cyclic pitch. This can be determined from:
where the value of (Bx + ax ) is found from the performance charts of Chapter 3. For the example helicopter at 115 knots, this was determined during the illustration of the method for the "Entire Helicopter in Level Flight” to be:
(B, + %) = 7.8»
and now, knowing the trim value of longitudinal flapping calculated above:
Bx = 7.8- (-1.1) = 8.9°