This paper describes a modelling approach to the dynamics of the airplane flight aimed at designing and realizing a simple but realistic “flight simulator”, able to mimic the longitudinal behaviour of a real airplane. The model is

implemented by using the Interactive Physics simulation environment. The simulator is used to reproduce all the phases of a complete flight of a light commercial airplane. All the actions on plane controls are analyzed and explained in terms of equilibrium states of the system. The main objective of the obtained simulations is in making the physical phenomenon understandable to students with a basic knowledge of mechanics and not involved in specialized aerodynamics studies

Tarantino, G.

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DOI 10.1393/ncc/i2010-10622-1Colloquia: MPTL14IL NUOVO CIMENTO Vol. 33 C, N. 3 Maggio-Giugno 2010The interactive physics flight simulatorG. TarantinoDipartimento di Fisica e Tecnologie Relative, Universita` di Palermo - Palermo, Italy(ricevuto il 30 Novembre 2009; pubblicato online il 23 Luglio 2010)Summary. — This paper describes a modelling approach to the dynamics of theairplane flight aimed at designing and realizing a simple but realistic “flight simu-lator”, able to mimic the longitudinal behaviour of a real airplane. The model isimplemented by using the Interactive Physics simulation environment. The simu-lator is used to reproduce all the phases of a complete flight of a light commercialairplane. All the actions on plane controls are analyzed and explained in terms ofequilibrium states of the system. The main objective of the obtained simulations isin making the physical phenomenon understandable to students with a basic knowl-edge of mechanics and not involved in specialized aerodynamics studies.PACS 01.50.-i – Educational aids.PACS 01.50.F- – Audio and visual aids.1. – IntroductionTeaching and learning physics via a flight simulator is an example of a pedagogicalstrategy in which physics contents are framed in the context of a familiar and fascinatingexperience.Even if many simulators are available and most of them are very realistic and withamazing graphics, very few allows to explore the physics models inside them.Moreover, physics of aircraft pilotage involves different content fields (dynamics, elec-tromagnetism, thermodynamics etc.) and at the same time offer a chance for engagingstudents in modelling approach to a real world phenomenon and for proposing somepedagogical strategies, dealing with real-world systems and everyday problem solution,which can contribute to avoid disaffection in scientific subjects, as the physical ones,making them more attractive.The argument of this paper deals with an ICT-based modelling approach to examinethe dynamics of airplane longitudinal flight and to develop a pedagogical flight simulatorwhich can be used to improve the understanding of physics of flight and of some advancedmechanics contents.From a pedagogical point of view, we think that this kind of tool might be used tobuild up several activity-based proposals aimed to make students aware of the reasoningprocedures to describe, formalise and explain the behaviour of some real systems as theairplanes.c© Societa` Italiana di Fisica 139140 G. TARANTINOFig. 1. – Radial and tangential forces with respect to the intantaneous centre of rotation O.2. – The physics model for longitudinal airplane flightConsider the airplane a point mass subjected to the four forces (thrust T , lift L, dragD and weight W ) regulating the longitudinal flight. The motion equations of the systemwill be simpler by writing them in terms of the radial and tangential components of thenet force with respect to an instantaneous centre of rotation O (fig. 1) (Anderson, 2005).As experiments in the wind tunnel show that lift and drag are proportional to thesquared speed, we can write(1)T − kDv2 −W sin θ = mdvdt ,kLv2 −W cos θ = mv2r,where v and θ are the speed and the pitch attitude of the plane respectively, while kLand kD are two parameters depending on the aircraft geometry (wings shape, surfaceetc.), on the angle of attack and on the air density. The distance r between the centreO and the centre of mass of the airplane can be eliminated by taking into account therelation v = rdθ/dt. By doing this, eqs. (1) will become(2)T − kDv2 −W sin θ = mdvdt ,kLv − W cos θv= mdθdt.The system of equations (2) is a non-linear first order differential equations and cannotbe solved exactly. The analysis of steady-state solutions can be easily carried out and isfound elsewhere (G. Tarantino et al., 2008).THE INTERACTIVE PHYSICS FLIGHT SIMULATOR 141Fig. 2. – A screenshot of the flight simulator showing a frame just after the take-off.3. – The flight simulatorIn order to perform the numerical integration of (2), we have implemented a flightsimulator able to visualise the time dependence of the relevant kinematics variables andallowing the parameters of the model to be changed during simulation. A screen-shot ofthis simulator built up with Interactive Physics is reported in fig. 2.The three relevant parameters represented by lift and drag coefficients and thrustmay be associated to the aircraft longitudinal controls. In this way, the main actionsperformed during flight by pilots for manoeuvring the aircraft can be simulated by settingand changing these parameters opportunely. We can easily reproduce the action on thethrottle by changing the thrust value from 0 to Tmax and the action of flaps extension bychanging the lift and drag coefficients. Flaps are, in fact, moveable surfaces situated onthe rear edge of wings causing the increasing of aerodynamical efficiency used to reducethe take-off and landing speeds. We will analyse the behaviour of a light commercialpropeller aircraft (mass = 1300 kg and maximum thrust = 2.5 kN) whose features aresummarized in table I.Here we report the results of the flight simulation obtained by applying a traditionalRunge-Kutta 4 integration method. The flight has been divided into five different phases:1) Take off - Plane initially at rest on the runway with flaps in take-off configurationand throttle set to the maximum value of 2.5 kN.2) Climb - Throttle maintained to its maximum value during all the climb, but flapshave been retracted.3) Cruise - When the plane has reached an height of 1000 ft, thrust is reduced toTable I. – Values of airplane parameters.Mass Thrust kD kL Efficiency(kg) (kN) (kg/m) (kg/m) (kL/kD)1 9 9 (without flaps or “clean” configuration)1300 0–2.5 1.05 10 9.52 (Flaps in take-off configuration)1.1 12 10.9 (Flaps in landing configuration)142 G. TARANTINOa) b)c)Fig. 3. – (a) Trajectory drawn by the plane (b) Plot of speed (black curve) and pitch (lightcurve) vs. time. Both variables perform damped oscillations around equilibrium values. Thefive phases of the flight are well distinguishable. (c) Representation of the simulation in phasespace (v, θ). We can easily recognize five spirals wrapping around five different centers.1.416 kN (this is the thrust for horizontal flight given by condition W = L).4) Descent - Throttle is reduced to 0.7 kN.5) Landing - Flaps are extended to assume landing configuration and thrust is leftunchanged.Note that, in setting phases 3 and 4, we have applied one of the most important tipscontained in all the pilotage hand-books according to which climbs and descents must beset acting on throttle instead of pulling or pushing the control yoke as one could think(1).We have observed the flight path during the various phases of the flight and measuredthe relevant kinematics variables (speed, vertical velocity, pitch, altitude etc.) usuallymonitored by the basic instruments situated in a standard panel of a real cockpit.The analysis of flight path (fig. 3a) and of time evolution of speed and pitch (fig. 3b)has shown a relevant result: the response of the system to a perturbation represented byvariations of thrust and/or efficiency is always a damped long-period oscillation around(1) See, for example, the Microsoft Flight Simulator 2004 Tutorial about climbing and descend-ing: “At this stage of your training, it’s a good time to agree on how you’ll control the airplane.Power (throttle position) should be your means of adjusting the rate of descent. The airplane’spitch attitude (controlled by the joystick) is your means of maintaining a specific airspeed. Ina climb, you’ll always use the maximum allowable power (usually full throttle) while adjustingthe airplane’s attitude using the joystick for the airspeed desired”.THE INTERACTIVE PHYSICS FLIGHT SIMULATOR 143Fig. 4. – Plot of modulus of net force acting on the plane vs. time.equilibrium configurations corresponding to a particular setting of physical parametersof frequency f0 characteristic of the system (in this case f0 = 0.06Hz).This damped-oscillating behaviour is made clearer by the representation of flight inthe bi-dimensional phase space (v, θ) (fig. 3c). This space may be subdivided in threemain regions: θ-positive, θ-zero (v-axis) and θ-negative region in which we can findrespectively all the dynamical states corresponding to climbing, horizontal flight and de-scending. By looking at building up of the phase space trajectories during evolution offlight, one can see spiral-like trajectories as wrapping around steady-state points repre-sented by the spiral centre. When a perturbation on the system is introduced by thrustand/or efficiency variation, a transition to a different spiral is induced. In detail, onecan see that take off is represented by the vertical straight line coinciding with v-axis.When take-off speed (35.7m/s) has been reached, the trajectory moves into θ-positivehalf-space (spiral a). Retraction of flaps induces the transition to spiral b centred ona higher value of speed and lower value of pitch. When thrust is reduced to its hori-zontal value, transition to spiral c centred on v-axis (θ = 0) occurs. Further reductionof thrust induces transition to spiral d centred into θ-negative half-space and finallyflaps extension before landing allows achievement of an equilibrium configuration char-acterised by lower speed and pitch. We can summarize these results by stating thatincreasing of efficiency induces transitions to higher pitch and lower speed. On the otherhand, increasing of thrust causes transitions to higher pitch leaving speed approximatelyunchanged.It is also very interesting to consider the time evolution of tangential and radialcomponents of the net force acting on the aircraft. We find that both componentsperform damped oscillations around zero with frequency f0. Composition of the twoforces gives raise to a vector rotating with the characteristic frequency f0 of the systemwhose modulus decreases exponentially to zero with a modulating frequency f = 2f0(fig. 4). Variations of parameters induce steep variations of net force.These long-period oscillations we found are well-known in aerodynamics. The cor-responding motion is named “phugoid motion” (Lancester, 1910). This is one of the144 G. TARANTINOa) b)Fig. 5. – The action of trim in the phase space (a) and on the real trajectory (b). In both caseswe have plotted trajectories of the flight in the same conditions but without trim (light curves).airplane characteristics tested by pilots for dynamical stability(2). It is established byperturbing the horizontal flight by acting on the elevators and leaving the plane establishthe equilibrium state again. However, usually, if we watch at an aircraft just after thetake-off, for example, we do not observe a “phugoid” motion. This is essentially dueto the possibility the aircraft to be longitudinally “trimmed”. This trimming action onpitch is generated by regulating the elevators placed on the horizontal tail of the planein order to lower or to lift up the aircraft tail inducing variations of the angle of attack.If the tail goes down (pulling the control yoke) the angle of attack increases and so doeslift causing climb. If the trim is opportunely regulated during flight, the oscillations aredamped much more quickly and we do not observe the “phugoid” motion. We have alsoupgraded our flight simulator by introducing the effect of trim. The results of simulationobtained by acting on the trim in such a way to contrast the plane tendency to oscillateare represented in the graphs of fig. 5a and b. The action of trim is well representedin the phase space as a “push” of the trajectory towards the equilibrium point. This istranslated in the real trajectory as a quicker achievement of stable configuration. Com-pare, for instance, the climb of the airplane in a “untrimmed” situation with the trimmedone (fig. 5a, b).4. – ConclusionsThe flight simulator was employed with a group of 15 students enrolled in a graduatePhysics Course for Secondary School Physics Teacher Preparation. All the students weregraduated in mathematics and their university curricula included two physics courses.These consisted in lectures concerning the theory and applications aimed at the solutionof quantitative problems without any laboratory activity. However, their initial ideasabout the physics of flight were very vague. Students attended dedicated laboratorysessions where they worked in groups of two or three by analyzing the model and runningthe simulations to answer to problems, reported in individual work-sheets, outliningspecific questions that would be discussed. Two teaching assistants co-operated with thecourse instructor in each laboratory session.(2) See for example the NASA technical paper downloadable at http://www.nasa.gov/centers/dryden/pdf/87880main H-953.pdfTHE INTERACTIVE PHYSICS FLIGHT SIMULATOR 145The analysis of student worksheets as well as of discussions with the students allowedto draw some conclusion about the pedagogical usefulness of the materials. The studentsshowed a positive attitude toward such a didactical approach and a real interest in theconcepts, models and approximations lying behind the process of simulating real-worldsystems. Student answers revealed a good level of understanding of the main conceptsrelated to the physics of flight and a notable ability to differentiate the effects of thedifferent physical variables. Moreover, the possibility of visualizing these effects played arelevant role in the student process of understanding of the dynamic properties of flight.Finally, we have verified that simulation environments, as the one we have used,can bridge the gap between graphical, symbolic, and visual representations and thesemultiple representations contribute to meaningful learning.REFERENCESAnderson J.D. (2005) Introduction to flight (5th edition) Mc Graw-Hill New York chap. 6p. 436Lancester F. W. (1910) Aerodonetics Constable & Co. LondonTarantino G., Fazio C. and Sperandeo-Mineo R. M. (2008) A Pedagogical Flight Simu-lator for Longitudinal Airplane Flight Computer Applications in Engineering Education(in press) DOI: 10.1002/cae.20177

The interactive physics flight simulator

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The interactive physics flight simulator

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