Static Longitudinal Stability
Introduction :
The study of the static longitudinal stability of an airplane is very important. Longitudinal stability, as we shall see, is intimately related to the e.g. travel and the loading configuration of an airplane. In addition the handling characteristics of an airplane in longitudinal flight are also determined by the static margin.
Analysis :
The well known condition for a longitudinally stable airplane is -
Where α is the angle of attack and CM is the moment coefficient about c.g. The condition for neutral stability is then -
C= 0 (5.A.2)
i.e. the pitching moment is independent of the angle of attack. In what follows, it may be shown, that the longitudinal stability creation for a conventional airplane is closely linked to the c.g. position, and that the neutral stability condition Eq. (5.A.2) dictates the almost c.g. location.
The pitching moment has contributions from the fuselage, nacelle, wing and tail. Summed together they appear as may also be calibrated in a similar manner. The positions of both ailerons are measured and differential angular positions noted. The rudder position is measured with the help of a protractor by attaching a pointer to the rudder.
The stick force strain gages are calibrated directly by fixing the control arm and subjecting it to various loads applied through static weights.
Sample calibration curves are given in Fig. 4.5 - 4.10 However, it is necessary to do your own calibration.
Flight Test Procedure :
Stick Fixed :
The condition of stick-fixed neutral stability requires determination of the c.g. position for which dSe / dCL is zero. A simple method of doing this in flight is to fly the airplane for a given c.g. location at various speeds or CL and measure the corresponding elevator angle to trim. The procedure is repeated for different c.g. locations. The slope of the Se - CL curves are plotted against xc.g. and the intersection of this curve with the xc.g. axis gives the stick fixed neutral point.
Stick Free :
Stick free neutral stability the c.g. position is obtained from d(Fs / q) / dcL = 0. In flight this is measured by flying the airplane with a given c.g. location at different speeds (i.e. CL and q values) and measuring the stick force at each speed. The procedure is repeated for different c.g. locations. The slope of the (FS /q) - CL curve is plotted against x c.g axis gives the stick free neutral point.
An alternative method consists of measuring the tab angle required to trim the airplane at each speed.
Instrumentation :
Neutral points are determined by measuring the elevator deflection angle Se , airspeed V, stick force FS, or tab angle te.
The airspeed may be obtained from the ASI and correcting the readings using a calibration chart. The elevator angle is measured installing a potentiometer in the elevator torque tube system so that the elevator deflection may be transformed to an electrical signal. The potentiometer is calibrated on the ground by physically measuring the elevator angle with an inclinometer and plotting it against the potentiometer reading.
The stick force may be measured by placing four strain gauges in a bridge circuit on the stick and sealing it off against moisture and dust. The signals are amplified before feeding it to an indicator. Calibration is done by applying known loads to the stick and plotting it against reading.
Flight Brief :
Here the flight procedure suggested will enable collection of both the stick fixed and stick free neutral points together.
1. A turbulence free procedure and a stable prescribed altitude is chosen and the airplane trimmed at a suitable speed so that the stick forces, in the entire range of investigation, are within the pilot's control. This speed may be determined by trimming the airplane at some speed and checking for the stick forces by flying at the extreme ends of the range of investigation. Two or three trials should be sufficient to determine a comfortable trim speed. This, however, need not be done if the tab is used to trim the airplane at each speed.
2. The airplane is flown at various speed (and trimmed if t is to be measured) from rear stall t the maximum attainable speed at a chosen altitude. When the speed stablizes the readings from the ASI, elevator potentiometer and the stick forse potentiometer or the trim tab are taken.
3. This procedure is repeated for different power settings, flap deflections and various external configurations to obtain a complete picture of the longitudinal stability of the airplane.
Calculations :  
Weight of aircraft = Altitude =
Wing Area = Mean Temp. =
Advance Ratio = Power Setting =                                RMP
Flap Angle =  
Some Remarks :
In general both CM and CL are principally functions of , M, TC and ½ pV2. Therefore, the neutral point is not index of stability with respect to Se alone but relates to the entire family of trim curves.
The curves of Se - CL , Fs/q - CL may not always turn out to be linear so that the neutral points may become a function of CL also.
Sometimes to improve the stick force gradients and the longitudinal stability, a down spring or a bob weight is attached to the elevator system. This usually produces a uniform pull force on the stick independent of speed which is helpful when the airplane is flying with rearward c.g. positions. However, they are not very satisfactory in ground operations where the heavy pull force required to keep the stick neutral may be objectionable.
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