APPENDIX A
APPENDIX A
MMLE3 .AIRCRAFT EQUATIONS OF MOTION
The following is a list of the equations of motion used in
the computer program MMLE3,covered in detail in reference 5.
The longitudinal state, control, observation and extra signal vectors
ate x= (aq @) u : (6e) z = (am qm em an m) extra = (q 8 p r 0 h V) The nonlinear longitudinal state equations are ._ = qS _o) q + i sin@ sina) - mV (CL + ÷ (cose cos¢ cosa + - tanB (p cosa + r sina) Iy _ = q Sc Cm + rp (Iz - Ix) + (r 2 - p:) ix z : q cos¢ - r sin@ + @o The longitudinal observation equations are = Ka qm=q @m = O
Xa n
an m _ CN ÷ -_- The erpansions of the longitudinal force and moment coefficients are Cm = Cme _ + Cmq _V ÷ Cm6 e _e + Cmo C N -- CNe a + CN_ e _e + CN o C L = CN The approximation of C L = CN is good for low angles of attack.
APPENDIX B
APPENDIX B DETERMINATION OF MOMENTS OF INERTIA The moments of inertia of the PA-30 were experimentally determined by the spring oscillation method. The moments of inertia (Ix, ly, Iz) were determined for empty and full fuel conditions with no crew, and landing gear retracted. The vertical center of gravity position was also determined from these experiments.
Vertical CG The vertical center of gravity was determined by the single- point suspension method [reference 7). In essence, this method applies known loads (Fa) to displace the suspended aircraft an angle (_) from a horizontal attitude. The following equation from reference 7 was used to determine the vertical CG position Z from the pivot support" F X a Eq. I s where Z a is the vertical distance from the supporting pivot axis to the aircraft reference line [waterline zero). The parameters on the right side of the equation were measured directly from the test setup. The displaced angle, e, was measured with an inclinometer affixed to the aircraft. The test configuration is shown in figure 8. The weight W s included all of the test equipment such as the cradle and supporting harness. The test equipment contributions were subtracted from the
experimental results. The vertical CG position was found to be about
4.33 inches below the aircraft's zero waterline reference.
Moments of Inertia The moments of inertia were obtained experimentally by testing the aircraft and supporting cradle as a single unit. The equations used to solve for the moments of inert{a were derived by summin E the moments about p in the simple model of a sprin E weight system shown in figure 9. The applicable equation listed below was derived in reference 2 and is a general second order differential equation for damped har- monic motion.
c _ + C a_k - wh
+ _o Io _ e = 0 Eq. 2
where c is a viscous retarding moment and K is the spring constant.
Equation 2 is a linear differential equation with constant coefficients and is solved by the substitution method in reference 2.
The solution of equation 2 is: a2k - Wh I o = _n 2 Eq. 5 where _n is the natural frequency of the system. The term Io is the moment of inertia about the axis of rotation. The natural frequency was determined from the time history plots of angular velocity in the following manner. The damped frequency, _d, and the damping ratio, _, were computed as follows: _d = 2_f
_,=
V'4.n ,'_ + '_= where £, the frequency of oscillation {cycles/second), is taken from the time history plots, and 6, the logarithmic decrement, is calculated using
= ! _n x°
n x n with n the number of cycles and x the magnitude of the angular velocity recorded on the time history plots.
Then the natural frequency was determined using the equation below.
_d
_n = (i -_=) ½
The moment of inertia, about the rotation axes, Io, was then transferred to the aircraft CG position using the parallel _is theorem to get Icg.
ka = - Wh W d= Eq. 4 Icg = ton _ g - Ire The equipment used in this test included the 256.S pound supporting cradle, the springs, safety cables, and cradle harnesses.
Anything that moved with the aircraft was taken into account. Only half the weight of items affixed to stationary supports was con- sidered. Generally most items were small enough to consider their 5O Thus moment of inertia about their own center of gravity to be zero.
the moment of inertia for test equipment, Ite , was simply Wte Ire : T (dte) where Wte is the weight of the item and dte is the distance from the center of gravity o£ the item to the oscillation axis.
The cradle's moment of inertia about its own CG, Icr , was large enough to be considered, and was found by using the spring oscillation method Wcradle (dte) 2 Ite (cradle) = Icr + g Equation 4 is used for both the pitch and roll axis tests. The pitch test for Iy, pictured in figure 10, used one spring attached to the tail tiedown with the center of gravity of the aircraft located forward of the pivot point. The roll test for Ix, figure Ii, used two springs affixed, one each, to the wing tiedowns and pivoted directly below the center of gravity position.
The single support cable, or pivot axis, for the yaw inertia test configuration, shown in figure 12, was directly above the center of gravity position of the aircraft so that the horizontal reference line was parallel to the floor and the single supporting cable coin- cided with the oscillation axis. This alignment results in zero values for distances d and h in equation 4. Thus the equation for yaw is ka= I z = _ " Ire The yaw inertia test used four springs horizontally attached with two at each wing tiedown point. The yaw analysis assumed the supporting cable to be free of torsional moments.
The spring oscillation method for determining the moments of inertia is relatively easy to use. However, many reference points exist for measuring distances. So it is a distinct advantage to sim- plify the procedure by recording all the vertical and horizontal distances to a reference point on the floor and knife edges.
The tests to determine the moments of inertia were made with the landing gear retracted at full and empty fuel conditions. This made it possible to interpolate the moments of inertia for different center of gravity positions. The moments of inertia are listed in Table 2.
S2
REFERENCES lo Maine, Richard E.; and Iliff, Kenneth W.; A Fortran Program for Determining Aircraft Stability and Control Derivatives from Flight Data. NASA TN D-7851, 197S.
.
Bradfield, Edward N.; Experimental Determination of the Moments of Inertia, Product of Inertia, and Inclination of the Principal ._xis of Conventional Aircraft by the Spring Oscillation Method. Air Force Flight Test Center, June 1971.
.
Fink, Marvin P.; and Freeman, Delma C. Jr.; Full-Scale Wind-Tunnel Investigation of Static Longitudinal and Lateral Characteristics of a Light Twin-Engine Airplane. NASA TN D-4985, 1969.
_o Plaetschke, E.; and Schulz, G.; Practical Input Signal Design.
AGARD Lecture Series No. 104 Parameter Identification.
, Maine, Richard E.; and Iliff, Kenneth W.; Users Manual for _LE5, A General Fortran Program for Maximum Likelihood Parameter Esti- mation. NASA Technical Paper 1565, 1980.
.
Iliff, Kenneth W.; and Taylor, Lawrence W. Jr.; Determination of Stability Derivatives from Flight Data Using a Newton-Raphson Minimization Technique. NASA TN D-6579, 1972.
_olowicz, Chester H.; and Yancey, Roxanah B.; Experimental Deter- mination of Airplane Mass and Inertial Characteristics. _ASA TR R-433, 1974.