Proceedings of the IMAC-XXVII February 9-, 009 Orlando, Florida USA 009 Society for Experimental Mechanics Inc. Automated Estimation of an Aircraft s Center of Gravity Using Static and Dynamic Measurements Nomenclature J. Cummins, A. Bering, D.E. Adams, R. Sterkenburg W k M k I cm a,b c,d A k ε k R k K t, K v ω nk {Y Θ} k T Force in landing gear Mass of aircraft and payload Mass moment of inertia about the center of mass Distances to landing gear relative to center of gravity Distances to center of gravity relative to rotors Cross-sectional area in landing gear supports Principal strain in landing gear supports Positions of centers of gravity of aircraft and payload Torsional/vertical stiffness provided by main rotor Natural frequencies in heave and pitch Modal vectors in heave and pitch Abstract The center of gravity of a heavy-lift helicopter changes throughout its operation due to fuel burn off, loading and unloading of passengers and supplies, shifting of load in flight, etc. Operators are interested in developing methods for estimating aircraft centers of gravity in such helicopters on the ground and in flight. Commercial air carriers are also interested in determining the center of gravity of an aircraft so it can be operated in the optimal center of gravity range resulting in fuel savings. Current methods of calculating the center of gravity of an aircraft are externally applied, time consuming, subject to human error, and only used on the ground. Two automated methods are discussed for estimating the center of gravity of an aircraft. The first method is static and uses load measurements in the aircraft landing gear axles or struts to estimate the center of gravity. The second method is dynamic and identifies rigid body motions through the use of accelerometers and uses the modal frequencies and vectors to track the motion of the center of gravity in near real time. The results from these methods are discussed and future work to validate the analytical findings in this paper is described. Section I: Introduction Heavy-lift military helicopters are utilized to transport large equipment, cargo, and troops. Current methods for locating the center of gravity are based on manual calculations and estimates of the payload magnitude and its distribution. Automated methods for estimating the location of the center of gravity using onboard sensors and algorithms would reduce the time required to launch these aircraft on missions to haul new payloads. When heavy-lift aircraft lift off, there is also a desire to track the location of the center of gravity relative to the stable region beneath the main rotor to avoid flight instabilities. The center of gravity can translate due to fuel burn off or shifts in payload either in the cargo bay or in tow. The objectives of this research are to (a) estimate the location of the center of gravity based on pre-flight static measurements and (b) track the location of the center of gravity based on dynamic operational response measurements. In addition, it is also desirable to quantify the uncertainty Purdue University, Center for Systems Integrity, School of Mechanical Engineering, West Lafayette, IN. Purdue University, Department of Aviation Technology, West Lafayette, IN.
in estimated center of gravity location because critical decisions related to the stability of the aircraft may be made using this estimate. A review of the literature on center of gravity estimation systems indicates that there have been three types of systems developed. The first system, which is described in the FAA handbook (FAA, 007), provides a general weight and balance procedure that involves jacking the aircraft up, weighing it at several locations and then calculating the center of gravity based on the weight readings using the computational method provided in the FAA handbook. In the second method using a Crane weight and balance system (Hughes, 005), the landing gear struts are converted into weight measuring devices by mounting pressure transducers in the landing gear struts. The strut piston area and the measured pressure are used to calculate the weight at the landing gear locations. The estimated weight measurements are then used to calculate the location of the aircraft center of gravity. The method in this paper proposes to utilize strain measurements to reduce the sensitivity to faults in the struts and other factors. The third method utilizes several accelerometers mounted throughout the airframe (Pandit and Hu, 997). The acceleration data is used in different algorithms to computer the center of gravity. Section II: Static Modeling and Analysis The simplified model shown in Figure is used to represent a heavy-lift aircraft that has an inherent weight M a, payload weight M p, and mass moment of inertia about the center of mass, I cm. The force due to gravity, (M a +M p )g, acts through the center of gravity (c.g.) and the two forces, W and W, act inside the front and rear landing gear, respectively. The longitudinal distances between the front and rear gear and the center of gravity are a and b, respectively. Based on a static free body diagram of the aircraft, the force and moment balances at equilibrium are given by: W + W = W a = W b ( M + M ) a p g. ( a,b) The distance a between the center of gravity and front landing gear can be calculated by solving these two equations simultaneously: a W L = ( ) W + W Figure : Simplified planar heavy-lift helicopter schematic diagram (static pre-flight).
where L=a+b. If normal strain measurements, ε and ε, are made within the landing gear supports, which possess cross-sectional areas A and A, and it is assumed that the loads within each of those supports are aligned with the strain measurements, then the loads can be expressed as follows: W = A Eε W = A Eε. ( 3a,b ) After substituting these expressions into Eq. (), the location of the center of gravity is calculated to be: = A ε L = ε L a. ( 4 ) A ε + Aε A ε + ε A The uncertainty in this estimate is also important to estimate. Uncertainties primarily enter into the principal strain measurements, which will contain sensor errors due to variations in temperature and possibly other errors due to the effects of multi-axial states of stress. If the geometrical parameters are assumed to be known exactly, and A =A, then the uncertainty in a is a function of the uncertainties in the two principle strain measurements: a a Δa = L Δε + Δε ε ε = L ε Δε + ( ε + ε ) ( ε + ε ) ε Δε. ( 5a,b ) The uncertainty increases proportionally with the distance between the two landing gear supports. If both strains are measured in the supports within a certain percentage of the true strains (95% confidence bounds), then the bound on the uncertainty in the center of gravity is proportional to the difference between the front and rear landing gear strains, ε -ε, and inversely proportional to the square of the sum of these two strains, (ε +ε ). These results provide an indication of the degree of uncertainty in the estimated static centers of gravity. As an example, consider the case where the total weight of the aircraft and payload is 38880 lb. The uncertainty in strain measurements is ±0% (95% confidence interval) for a 36 ft span from the front to the rear landing gear supports, which are fashioned from steel (E=30 0 6 psi) and possess cross-sectional areas A=4 in. For 4 μstrain (ε ) and 30 μstrain (ε ) measurements, the estimated location of the center of gravity and associated uncertainty assuming a worst-case combination of the individual uncertainty terms are given by: 36 ft 30μstrain a = = 0.0 ft 4 + 30μstrain 30 0.0 4 + 4 0.0 30μstrain Δa = ± 36 ft = ( 4 + 30μstrain) ±.8 ft. ( 6a,b ) Note that the uncertainty is relatively large at 9% of the estimate a for this particular loading case. Figure shows the estimated locations and uncertainty bounds for 4 different loading cases that are calculated by applying Eqs. ()-(5). It evident that the largest uncertainties occur when loads are applied at the half-way point between the landing gear supports. The importance of these uncertainties could be quantified given the stability bounds.
C.G. Location and Uncertainty (ft) 40 35 30 5 0 5 0 5 0 0 5 0 5 0 5 Loading case Figure : Estimate of center of gravity location (x) and uncertainty (o) as a function of load case. Section III: Dynamic Modeling and Simulation The modeling and analysis performed in the previous section was applicable to the pre-flight estimation of the aircraft center of gravity. If the payload shifts or the aircraft burns fuel, the center of gravity shifts as a function of time. In this case, it is desirable to track the motion of the center of gravity using dynamic measurements. It is assumed in this section that the center of gravity is initially known based on the static estimation process described in the previous section. Figure 3 shows a schematic diagram of the aircraft in flight where the rotor is represented as a pair of torsional, K t, and translational, K v, restoring springs. The physical source of these elastic elements is the rotordynamics, which provide stability in flight under perturbations in the response of the aircraft (Gessow and Myers, 95). The distances between the two rotors (main and tail) and the center of gravity are denoted. The two forces, and f and f, represent the dynamic forces acting due to the fluctuations in lift and imbalance of the rotors. Static forces due to gravity, lift, and drag are not included in this analysis, which is focused on the dynamic (short period) response of the aircraft. It is the dynamic forces that produce the operational dynamic response, which will be used to track the center of gravity motion. Figure 3: Simplified planar heavy-lift helicopter schematic diagram (dynamic, in-flight).
To calculate the center of gravity location, the line diagram shown in Figure 4 is used. The position, c, of the center of gravity relative to the position of the main rotor shaft is calculated using the following equation: ( R ) = ( M M )c Rp M p + M a a p + a. ( 7 ) where the center of gravity of the payload is located at R p towards the rear of the aircraft and the center of gravity of the aircraft is located at R a towards the front of the aircraft. Likewise, the mass moment of inertia about the center of gravity is calculated as follows: ( R c) + M ( R c ) I cm = M p p a a +. ( 8 ) The undamped equations of motion of this two degree of freedom system for small oscillations are given by: M a + 0 M p 0 & y K + I & v cm θ ckv ck v K + c t y f = Kv θ cf + f df. ( 9 a,b) It will be assumed that onboard sensors can be utilized to infer the motions y and θ. For instance, gyroscopes or accelerometers already installed in many aircraft for health and usage monitoring could be utilized for this purpose. Given these measurements of the operational response, it is proposed that a frequency domain decomposition of the time domain response measurements can be used to estimate the natural frequencies and modal deflection shapes of the aircraft. These modal parameters can be found from Eqs. (7) based on eigenvalue analysis as explained below. If it assumed that the response vector of the system is given by {Y Θ} T e st for synchronous motions (at a mode of vibration), then the eigenvalue problem can be expressed as follows: M a + 0 M p 0 I cm Kv ck v ck K + c t v Y Y = λ Kv Θ Θ ( 0 ) where the eigenvalue is given by λ= s and the eigenvector is equivalent to the modal vector, {Y Θ} T. The modal frequencies, s, are found from -λ. Changes in these modal parameters are indicative of shifts in the center of gravity, c, relative to the main rotor. Furthermore, it is evident from Eq. (8) that changes in c cause changes in the matrix on the left hand side of the equation, which in turn causes changes to the modal frequencies and vectors. R p Main rotor M p c R a M a Figure 4: Location of center of mass (gravity) and calculation of mass moment of inertia. An example will be considered using K v =5 0 6 lb/ft, K t = 0 8 lb-ft/rad, M a =93 lbm, and M p =76 lbm. The nominal positions of the centers of gravity for the aircraft and payload are assumed to be R p =30 ft and R a =5 ft yielding c=3.0 ft from Eq. (7) and I cm =.6 0 6 lbm-ft from Eq. (8). The natural frequencies and mode shapes given these nominal parameters are found using MATLAB. These resulting nominal modal parameters are listed below:
Y.00 Y.00 ω n = 0.30rad / s and = ωn =.05rad / s and =. ( ) Θ 0.30 Θ 0.0 These results indicate that the first mode exhibits both vertical bounce (heave) and torsion whereas the second mode exhibits primarily bounce (heave). To track the change in the center of gravity location, c, the natural frequencies and modal vector coefficients were then calculated as a function of c. The calculations were performed over a range of payload center of gravity locations, R p, 30 ft to 53 ft. These calculations are meant to simulate a change in payload as a function of sway in the tow cable, repositioning of the load, or fuel burn. At each value of R a, the center of gravity of the aircraft as a whole, c, was calculated as was the mass moment of inertia, I cm. The resulting natural frequency and modal vector coefficient plots are shown in Figures 5(a,b). Both natural frequencies in Figure 5(a) for the heave (x) and pitch-heave (o) modes of vibration exhibit some sensitivity to the position of the center of gravity; however, the pitch-heave natural frequency is far more sensitive to shifts in the center of gravity over the entire range of c. Furthermore, the sensitivity to changes in c is highest in the pitch-heave modal frequency of vibration for smaller values of c when the center of gravity is nearest to the main rotor shaft. This result suggests that this method for tracking the location of the center of gravity is most sensitive as the center of gravity strays from the region of stability beneath the main rotor. By utilizing this model, the location of the center of gravity could be estimated as a function of shifts in the payload mass, M a. Figure 5(b) also indicates that the heave modal vector of vibration is insensitive to the shifts in payload inertia; however, the heave-pitch modal vector is sensitive to the changes in the center of gravity. Only the torsional degree of freedom (o) is sensitive to the changes in c; the vertical displacement degree of freedom is not sensitive to these changes. As in the case of the natural frequency of vibration in pitch-heave, the modal vector of oscillation in pitch-heave is also most sensitive to shifts in the center of gravity when those shifts are nearest to the main rotor shaft (support point for the model in Figure 3). When the payload is increased from 8880 lb to 8880 lb, the results shown in Figure 6 are obtained for the natural frequencies in heave (x) and pitch-heave (o). The dotted line (right) indicates the natural frequencies for the higher payload and the solid line (left) indicates the frequencies for the lower 8880 lb payload that was considered previously. Note that the range over which the center of gravity varies is larger whereas the corresponding percent change in the pitch-heave natural frequency is smaller in magnitude. The percent change in heave natural frequency is slightly larger in magnitude for the larger payload. When the payload is fixed at 8880 lb but the weight of the aircraft is reduced from 30000 lb to 800 lb (to simulate fuel burn), the natural frequencies shown in Figure 7 are obtained. In this scenario, there is a negligible change in the natural frequency in pitch-heave but a significant, and nearly linear, change in the natural frequency in heave as a function of the center of gravity position, c. This result suggests that variations in the center of gravity brought about by changes in the inherent weight of the aircraft mainly affect the heave mode of vibration. To demonstrate the ability to identify the natural frequencies of the aircraft in flight for the purpose of tracking the center of gravity movement, a linear time domain simulation was conducted using MATLAB. Random excitation functions, f and f, each with 00 lb RMS amplitude were prescribed for c=3 ft and d=00 ft. The state variable equations were then integrated to produce the dynamic responses, y and θ. The resulting responses are plotted in Figure 8(a) in the time domain and Figure 8(b) in the frequency domain using the discrete Fourier transform and Hanning window for signal processing. Note the strong frequency components at 0.3 Hz and Hz as expected from the previous analysis of the modal response characteristics. By tracking the 0.3 Hz resonant frequency, the center of gravity of the aircraft can be tracked as indicated in Figure 5(a).
Natural frequencies, ω n (Hz). 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 3 4 5 6 7 8 9 C.G. Location, c (ft) Modal vector coefficients, Mode 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 3 4 5 6 7 8 9 Modal vector coefficients, Mode 0.4 0. 0-0. -0.4-0.6-0.8-3 4 5 6 7 8 9 C.G. Location, c (ft) Figure 5: (a) Trends in natural frequencies with center of gravity location for heave mode (x) and pitch-heave mode (o) and, (b) trends in modal vector coefficients for heave mode (upper) and (tower) pitch-heave mode for vertical (x) and torsional (o) degrees of freedom showing sensitivity to variation in center of gravity location.
. 0.9 Natural frequencies, ω n (Hz) 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 4 6 8 0 4 6 8 C.G. Location, c (ft) Figure 6: Trends in natural frequencies with center of gravity location for heave mode (x) and pitch-heave mode (o) for 8880 lb (left, ) and 8880 lb (right,.) payloads..3.. Natural frequencies, ω n (Hz) 0.9 0.8 0.7 0.6 0.5 0.4 3 3. 3.4 3.6 3.8 4 4. 4.4 4.6 4.8 5 C.G. Location, c (ft) Figure 7: Trends in natural frequencies with center of gravity location for heave mode (x) and pitch-heave mode (o) for the case involving 0000 lb reduction in inherent aircraft weight during flight (i.e., fuel burn).
y (ft) and theta (rad) y (ft) and theta (rad) 0.5 0-0.5 0 5 0 5 0 Time (sec) 0.05 0.0 0.005 0 0 3 4 5 Freq (Hz) Figure 8: (a) Time domain responses for y ( ) and θ ( ) and (b) discrete Fourier transform of these time histories showing strong modes at 0.3 Hz and Hz. Section IV: Conclusions Static and dynamic methodologies were investigated using analytical simulation models for estimating the location of the center of gravity for pre-flight and in-flight applications involving heavy-lift helicopters. In this kind of aircraft, there is a desire to develop automated methods for estimating the center of gravity location and tracking the location of the center of gravity due to the wide variations in payload and other operational variations. The static method for estimation of the center of gravity location used a static force balance for the aircraft system along with strain calculations in the landing gear supports. It was demonstrated that the center of gravity could be easily located front to back in the aircraft and that the uncertainty in this estimate varied as a function of the center of gravity location. The uncertainty was relatively small but was highest when the center of gravity is nearest to the main rotor. The dynamic method for estimating the center of gravity location used a dynamic two degree of freedom planar model of the aircraft to demonstrate that variations in the center of gravity cause variations in the modal parameters. The natural frequency associated with the pitch-heave mode of vibration and its corresponding modal deflection shape were by far the most sensitive to the change in center of gravity location. To measure the small changes in natural frequencies that are observed in the analytical models observed in this paper, it is likely that DC accelerometers would be required on the aircraft. Future work will focus on the
experimental validation of these analytical findings and on the online identification process for modal parameter estimation using operational data. Section V: References Federal Aviation Administration, Aircraft Weight and Balance Handbook FAA-H-8083-A, 007, Washington D.C. U.S. Government Printing Office. Hughes, D., Crane Offers Aircraft Weight and Balance System, 005, Aviation Week & Space Technology, volume 6 (issue ), pg. 9. Pandit, S. M., and Hu, Z. Q., Determination of Rigid Body Characteristics from Time Domain Modal Test Data, 997, Journal of Sound and Vibration, pp. 3-4. Gessow, A. and Myers, G., Aerodynamics of the Helicopter, 95, Frederick Ungar Publishing Co., New York, NY.