Approach to Dynamic Modeling of Aircraft Landing on Moving Ships

Calhoun: The NPS Institutional Archive Faculty and Researcher Publications Faculty and Researcher Publications 2003 Approach to dynamic modeling of aircraft landing on moving ships Kolar, Ramesh Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/40373 Approach to Dynamic Modeling of Aircraft Landing on Moving Ships Ramesh Kolara Department of Aeronautics & Astronautics, Naval Postgraduate School, 699 Dyer Road, Bldg 234, Rm 245, Monterey, CA 93943, U.S.A. ABSTRACT This paper presents an application of the dynamic response and load calculations on a tricycle type landing gear of an aircraft. The formulation based on the Lagrange’s equations of motion treats the aircraft landing on a stationary deck in terms of pitching motion, roll motion, and vertical heaving motion. The landing is simulated as a single point, dual point, and three-point touch down with specified initial conditions. Effect of damping on ensuing motion is discussed. The results for a typical aircraft configuration are presented in the form of state variables and representative dynamic loads are shown in the form of graphs. In order to see the effect of motion of the deck, the equations are modified to include the deck movement into the formulation. Some preliminary results are presented in this case. Keywords: Landing Gear Dynamic Loads, Ship Motion, Aircraft Carrier Landing 1. INTRODUCTION: Aircraft landing is a challenging problem in that it involves the complex transfer of loads of the aircraft in flight with all six degrees of freedom motion to the dynamic motion of the aircraft on landing with elastic elements of the landing gear becoming engaged. The loads coming on to the aircraft during various maneuvers on the Further author information: (Send correspondence to R.K.) a R.K.: E-mail: [email protected], Telephone: 1 831 656 2854 ground are critical and are to be considered in the design of various structural elements of an aircraft. This study addresses the problem of calculating the dynamic loads associated with aircraft landing based on Lagrange’s equations of motion. In this paper, we detail an analysis for calculating the dynamic loads arising out of an aircraft landing on a deck. This simplified modeling, useful in the preliminary design of landing gears, treats the landing as single point, two point and three point touch down with the equations of motions developed for a tri-cycle type of landing gear. The aircraft is assumed to have pitching, roll, and vertical heaving degrees of freedom. The Lagrange’s equations of motion are developed for the tri-cycle landing gear, and the resulting equations of motion are solved in a multi-phase solution to accommodate various specific initial conditions during the rollout phase. The equations are uncoupled using assumed modes approach and the resulting uncoupled equations are solved for displacements, velocities, and accelerations of the basic state variables. Some sample calculations showing the dynamic loads are presented to illustrate the use of the present approach. 2. EQUATIONS OF MOTION: The Lagrange’s equations1–4 may be written as d dt
∂T ∂ q̇j  − ∂T ∂V + − Qj = 0 ∂qj ∂qj (1) for j = 1, 2, · · · , N. In the equation (1), T and V are the kinetic and potential energies of the system; qj are the generalized coordinates and Qj are the corresponding generalized forces for an N degrees of freedom system. Consider an aircraft with its mass at the center of gravity having a tri-cycle gear configuration as shown in Figure 1. Let kLM , kRM , and kN denote the stiffnesses of the left main (LM) landing gear, right main (RM) landing gear, and nose (N) landing gear respectively of the aircraft. Let z, θ, and φ represent the vertical, pitch and roll degrees of freedom. Let ż, θ̇, and φ̇ represent the vertical, pitch and roll rates, while z̈, θ̈, and φ̈ represent the vertical, pitch and roll accelerations of the aircraft. The potential energy, the Rayleigh damping function, and the kinetic energy of the system are given by the following expressions: 1 c 2 V = kLM z + bθ − φ 2 2   c 2 1 + kRM z + bθ + φ 2 2 1 + kN (z − aθ)2 2   1 c 2 R = cLM ż + bθ̇ − φ̇ 2 2   c 2 1 + cRM ż + bθ̇ + φ̇ 2 2   2 1 + cN ż − aθ̇ 2 (2)   (3) ∂R ∂ q˙j the Lagrange’s equations of motion become mz̈ + (cN + cLM + cRM )ż + (bcLM + bcRM − acN )θ̇ c Ixx φ̈ + (cRM − cLM )ż 2 bc + (cRM − cLM )θ̇ 2  2 c + (cRM + cLM )φ̇ 2 c + (kRM − kLM )z 2 bc + (kRM − kLM )θ 2  2 c (kRM + kLM )φ = 0 + 2 (6) (7) (8) The equations of motion may also be written in a matrix form as follows: (9) (4) where M,C, and K are the mass, damping, and stiffness matrices respectively. (5) A MATLAB based program is developed to solve the equations (9). The landing of the aircraft is simulated by performing the following operations: Noting that Qj = − Iyy θ̈ + (bcLM + bcRM − acN )ż + (b2 cLM + b2 cRM + a2 cN )θ̇ bc + (cRM − cLM )φ̇ 2 + (bkLM + bkRM − akN )z + (b2 kLM + b2 kRM + a2 kN )θ bc + (kRM − kLM )φ = 0 2 Mq̈ + Cq̇ + Kq = 0 and 1 1 1 T = m(ż)2 + Iyy (θ̇)2 + Ixx (φ̇)2 2 2 2 c + (cRM − cLM )φ̇ 2 + (kLM + kRM + kN )z + (bkLM + bkRM − akN )θ c + (kRM − kLM )φ = 0 2 • The aircraft is assumed to land on one landing gear at touch down. Say, right main landing gear (RMLG) is selected arbitrarily as the first point of contact. Then, the stiffness of the RMLG, kRM , is non-zero, and other stiffnesses are assumed to be zero. Similarly, only the mass of the aircraft is invoked and the equations are solved for natural frequencies and mode shapes.5, 6 The response is a single degree of freedom system with the appropriate initial conditions. • The response from the first step is used to solve the landing gear that touches the deck next, using the geometry of the aircraft and landing gear spacing. The state variables corresponding to this state will form the initial conditions for the second phase. The time for the end of the first stage is computed graphically. The state variables at this time are calculated and used as the initial conditions for the second phase. • In the second phase, the two main landing gears are in contact with the deck, with kRM and kLM stiffnesses activated, and kN being still zero. The initial conditions are used to solve the two-degree of freedom model. Again the state variables are calculated and pitch response is plotted to observe when the nose gear touches the deck. This time is noted and the corresponding state variables are calculated, which form the initial conditions for the third phase. • In the third phase, all the three landing gear stiffnesses are activated. The three degree of freedom system with the initial conditions obtained from the termination stage of the second phase is solved for the response. • Once the three displacement variables are determined, the loads may be computed from the geometry of the layout of the tricycle landing gear. 3. RESULTS: The formulation and the method described in the previous sections are illustrated by applying to a typical aircraft. The basic parameters selected are as follows: Weight a b c Ixx Iyy kRM kLM kN vz = = = = = = = = = = 13,700 lbs. 12.2867 ft. 1.6167 ft. 12.7933 ft. 5200 slug − f t2 19,000 slug − f t2 300,000 lb/ft 300,000 lb/ft 300,000 lb/ft 24.6 ft/s where a is the distance between the c.g. of the aircraft and the nose landing gear, b is the distance between the c.g. and the left or right main landing gear, and c is the distance between the two main landing gears (Figure 1). The mass moments of inertia about x-axis and y-axis are Ixx and Iyy respectively, while the sink velocity or the vertical rate of descent is vz . Figure 1 shows the lay out of the aircraft and landing system as a three degrees of freedom system. The system is first solved for natural frequencies and the modes shapes.5, 7 Then mode superposition method is used to compute the response. The equations are uncoupled by transforming them using the modal matrix. The first phase when RMLG is active, the stiffnesses kLM = 0 and kN = 0. The eigen value problem yields one non-zero eigen value and the natural frequency is given by ω1 = 3106 rad/sec. The state variables, displacement, pitch, roll, and the corresponding rates are shown in Figure 2 and 3 respectively. Using the geometry of the main landing gear spacing, as shown in Figure 1, the end of phase I is characterized by the solving the geometric constraint problem when the angle of tilt (roll) becomes zero. This signifies that the left main landing gear touches the deck. The time τ1 to achieve this is read from the Figure 4, where the y-ordinate crosses the x-axis. The initial conditions at time τ1 = 0.008 is prescribed affect the initial conditions of various phases. Results from such studies are underway and will be reported elsewhere at a later date. found to be z θ φ ż θ̇ φ̇ = = = = = = 1.9215 in 0.1211 rad 0.0517 rad 219.84 in/s -0.1984 rad/sec -3.2858 rad/sec (10) The equations of motion for the phase II is now solved with kLM and kRM active and the initial conditions given by (10). Figures 5 and 6 give the response displacement and rate quantities as computed using the developed program. The end of phase II is characterized by the pitch angle variation such that the nose gear touches the deck. This is calculated again by the geometry of the aircraft and landing gear layout given in Figure 1. The time τ2 at which the phase II ends, signifying the nose gear touch down is obtained from the graph shown in Figure 7. The state variables corresponding to this value of τ2 becomes the initial conditions for the third and final phase, and is given by z θ φ ż θ̇ φ̇ = = = = = = 1.0069 in -0.0419 rad 0.000 rad 12.368 in/sec -0.8243 rad/sec -0.00609 rad/sec (11) The initial conditions (11) are used in computing the phase III response of the system. The three displacements and there rates are shown in Figures 8 and 9. Once these displacements are known, the loads on the landing gear may be computed in a straight forward manner. The loads calculated in each of the three phases are shown in Figures 10-12, from which the maximum loads are used in the design of the landing gears. The method developed here may easily be changed to accommodate the motion of the ship. There are several ways of incorporating the ship motion descibed by the pitch, roll, and heaving degrees of freedom. Each of these quantities 4. CONCLUSIONS An approach to compute dynamic loads on the tricycle type landing gear of an aircraft is presented. The equations of motion are derived from Lagrange’s equations. The landing is accomplished in three phases and the final stages of each phase corresponds to the initial conditions of the following phase. Calculation of the landing gear loads are described, which may prove to be of use in the preliminary design and sizing of the landing gears. ACKNOWLEDGMENTS The work reported here is in part supported by the Naval Air Systems Command, Patuxent River. The author expresses his appreciation to Professor Jerry Lindsey for his suggestions and discussions during the development of the computer program in this research. REFERENCES 1. D. Greenwood, Classical Dynamics, PrenticeHall, Englewood Cliffs, NJ, 1977. 2. H. Goldstein, Classical Mechanics, AddisonWelsley, Reading, MA, 1959. 3. G. Warburton, The Dynamic Behavior of Structures, Pergamon Press, Oxford, 1976. 4. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, London, 1967. 5. R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, Wiley, New York, 1981. 6. W. Thompson, Theory of Vibrations with Applications, Prentice-Hall, Englewood Cliffs, 1964. 7. A. Jennings, Matrix Computatoin for Engineers and Scientists, Wiley, Chichester, England, 1977. Figure 1 Aircraft Landing Gear Layout Figure 2 Displacement, Pitch, and Roll in Phase I Figure 3 Displacement, Pitch, and Roll Rate – Phase I Figure 4 Roll Response in Phase I(t1=0.008 secs.) Figure 5 Displacement, Pitch, and Roll – Phase II Figure 6 Displacement, Pitch, and Roll Rate – Phase II Figure 7 Pitch Response – Phase II; zero cross-over at t = 0.186 sec Figure 8 Displacement, Pitch, and Roll – Final phase Figure 9 Displacement, Pitch and Roll Rates – Final Phase Figure 10 Loads History in Phase I Figure 11 Loads History – Phase II Figure 12 Loads History – Phase III