Boeing-747 Analysis Jazz Lindquist1 Aircraft Flight Dynamics (14:650:271), New r!nswic", N#, 0$%01 Abstract A Boeing 747 aircraft was analyzed in regards to its’ stability. First, it was determined whether the aircraft was statically stable. Then it was ascertained whether the dynamic response of the aircraft supported the prediction. The uler force and moment e!uations for an aircraft were used, along with the decoupled longitudinal and lateral forms. "atlab was used to both sol#e and graph the models. $t was found that the e!uations in their original form pro#ed difficult to simulate, with errors abounding in the results obtained. %sing the decoupled e!uations pro#ed to be largely successful, with the beha#ior pro#ing that the aircraft was indeed statically stable. & Abstract' a concise summary
$ntroduction The Boeing 747 aircraft was one of the most innovative designs of its time! "ften heralded as a master#iece of industrial design$ the Boeing 747 was twice the size of any #revious airliner! %t was the result of a four year develo#ment dev elo#ment to #roduction cycle &in com#arison to the largely successful Boeing 7'7s (-year cycle)*$ and #ushed the limits of what airliners of the time could do! %n addition to +rea,ing records$ it +ecame so #o#ular that it is one of the most easily recogniza+le air#lanes of the modern era! art of this is due to its unique u##er dec, hum#$ +ut more is owed to the success it has en.oyed en .oyed for over three decades! %ts accom#lishments include holding the #assenger ca#acity record for /7 years &107'-''7)$ +eing modified to serve as Air 2orce "ne for the commander-in-chief of the 3nited tates government7$ and serving as a shuttle carrier for 5AA(! This #a#er aims to determine the flight characteristics of this iconic aircraft! To that that end$ the sta+ility of the aircraft will +e d etermined$ +oth static and dynamic$ longitudinal and lateral! %n addition ad dition to this$ 6ATLAB 6ATLAB will +e used to model some of the aircrafts res#onse to various #ertur+ations!
1 3ndergraduate$ utgers 3niversity
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Aircraft 2light 8ynamics
(eneral )omenclature of Aircraft Before the Boeing 747 can +e adequately analyzed$ a general ,nowledge of the standards used in flight dynamics is required! Aircraft flight studies concerning aircraft are rather com#le9$ and a ,nowledge of the com#rehensive nomenclature goes a long way in understanding the ca#a+ilities of any aircraft! %t is assumed that the reader has a certain familiarity with the su+.ect$ +ut for the sa,e of sim#licity$ sim#licity$ a +rief overview will +e given here! The coordinate systems used with the res#ective forces$ moments$ and angles will +e given here as a reference! Two coordinate systems are used to descri+e the #osition and orientation of the aircraft at a given time: an inertial reference frame and the +ody frame of the aircraft! 2or an inertial reference frame$ a coordinate system fi9ed to the earth will +e used! %t is usually set so that the 9-a9is #oint north$ the y-a9is #oints to the east$ and the z-a9is #oints downward toward the earth! "n the aircraft itself$ a second coordinate system referred to as the +ody frame is em#loyed! %ts 9-a9is &also called the centerline) lies along the fuselage of the aircraft$ the y-a9is of the coordinate system is along the right wing of the aircraft$ and the z-a9is #oints towards the +ottom of the aircraft! %t is is also im#ortant to note that the 9z-#lane forms a #lane of symmetry along the aircraft! %n terms of these two coordinate systems$ the following is a list of the various varia+les and forces acting on the aircraft: a ircraft: A9ial 2orce$ the force acting on the aircraft along the 9-a9is ide 2orce$ the force acting on the aircraft along the y-a9is 5ormal 2orce$ the force acting along the aircraft along the z-a9is Lift Lift 2orce 2orce The The for force ce the the airc aircra raft ft gene genera rate tess to to ,ee# ,ee# its itsel elff in in the the air air Thrust 2orce 2orce the the aircraft aircraft generat generates es to #ro#el #ro#el itself itself in the #ositi #ositive ve 9direction! 8rag 2orce force that o##oses the thrust force due to the sha#e of the aircraft and the flow it generates ;eight ;e ight efers to the force of gravity on the aircraft$ always acts towards the earth olling 6oment$ the moment of the aircraft a+out the 9-a9is itching 6oment$ the moment of the aircraft a+out the y-a9is
Aircraft 2light 8ynamics
(eneral )omenclature of Aircraft Before the Boeing 747 can +e adequately analyzed$ a general ,nowledge of the standards used in flight dynamics is required! Aircraft flight studies concerning aircraft are rather com#le9$ and a ,nowledge of the com#rehensive nomenclature goes a long way in understanding the ca#a+ilities of any aircraft! %t is assumed that the reader has a certain familiarity with the su+.ect$ +ut for the sa,e of sim#licity$ sim#licity$ a +rief overview will +e given here! The coordinate systems used with the res#ective forces$ moments$ and angles will +e given here as a reference! Two coordinate systems are used to descri+e the #osition and orientation of the aircraft at a given time: an inertial reference frame and the +ody frame of the aircraft! 2or an inertial reference frame$ a coordinate system fi9ed to the earth will +e used! %t is usually set so that the 9-a9is #oint north$ the y-a9is #oints to the east$ and the z-a9is #oints downward toward the earth! "n the aircraft itself$ a second coordinate system referred to as the +ody frame is em#loyed! %ts 9-a9is &also called the centerline) lies along the fuselage of the aircraft$ the y-a9is of the coordinate system is along the right wing of the aircraft$ and the z-a9is #oints towards the +ottom of the aircraft! %t is is also im#ortant to note that the 9z-#lane forms a #lane of symmetry along the aircraft! %n terms of these two coordinate systems$ the following is a list of the various varia+les and forces acting on the aircraft: a ircraft: A9ial 2orce$ the force acting on the aircraft along the 9-a9is ide 2orce$ the force acting on the aircraft along the y-a9is 5ormal 2orce$ the force acting along the aircraft along the z-a9is Lift Lift 2orce 2orce The The for force ce the the airc aircra raft ft gene genera rate tess to to ,ee# ,ee# its itsel elff in in the the air air Thrust 2orce 2orce the the aircraft aircraft generat generates es to #ro#el #ro#el itself itself in the #ositi #ositive ve 9direction! 8rag 2orce force that o##oses the thrust force due to the sha#e of the aircraft and the flow it generates ;eight ;e ight efers to the force of gravity on the aircraft$ always acts towards the earth olling 6oment$ the moment of the aircraft a+out the 9-a9is itching 6oment$ the moment of the aircraft a+out the y-a9is
Aircraft 2light 8ynamics
angular rate along the z-a9is angle of attac,$ angle +etween #ro.ected onto the 9z #lane and the centerline of the aircraft flight #ath angle$ angle +etween the direction of flight and the local horizontal sidesli# angle$ angle +etween centerline of the aircraft
#ro.ected onto the 9y #lane and the
yaw angle$ angle +etween the #ro.ection of the centerline onto the horizontal #lane and north heading angle$ angle +etween
#ro.ected onto the horizontal #lane and
north #itch angle$ angle +etween local horizontal and the centerline roll angle$ angle +etween the true vertical and the z-a9is of the aircraft flight dynamic #ressure$ equal to '!*rho=>infinity? reference area$ ta,en to +e the wing #lanform area characteristic length$ which is the wings#an for the rolling and yawing moment and the mean chord for the #itching moment 2or further understanding$ three diagrams are #rovided$ as it is often easier to visi+ly see the difference +etween various #arameters! 2igure 1 dis#lays the +ody frame of the aircraft along with the the angular rates$ velocities$ moments$ and total v elocity! @ach a9is has a force$ moment$ and angular rate associated with it! 6oving on to 2igure $ the #itch$ angle of attac,$ and flight #ath angle can +e seen! The lift$ drag$ and thrust forces are also shown$ as well as the #itching moment$ velocity$ and total velocity ! 2igure / sim#ly #rovides a visual re#resentation of the yaw$ sidesli#$ heading$ and roll angles!11
Figure *. Body +eference Frame and Associated Forces and "oments*7 3
Aircraft 2light 8ynamics
Figure . -itch, Angle of Attac, and the Flight -ath Angle*7
Figure /. 0aw, 1ideslip, 2eading, and +oll Angle*7
2or this #a#er$ all of the information o+tained a+out the Boeing 747 is listed in A##endi9 1! The format used for the dimensionless coefficients is as follows: the first su+scri#t determines the ty#e of coefficient it is and the second su+scri#t$ if there is one$ details the varia+le that the derivative of the coefficient is ta,en with res#ect to! Ta,e$ Ta,e$ for e9am#le$ ! The m denotes that it is the coefficient related to the #itching moment$ whereas the second su+scri#t su+scri#t details how the coefficient has had the #artial derivative with res#ect to al#ha ta,en!
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Aircraft 2light 8ynamics
-roblem 1tatement "ne of the most im#ortant criteria for an aircraft is how sta+le it is! This mainly means its a+ility to hold constant$ steady flight conditions after a distur+ance! teady flight conditions$ also ,nown as trim conditions$ requires that the sum of the forces on the airliner as well as the sum of the moments acting on the airliner are +oth equal to zero! This distur+ance can come from the atmos#here surrounding the air#lane or the #ilots actions$ +ut the aircraft must +e a+le to adequately res#ond to +oth distur+ances on its own! Although an aircraft does not have to +e inherently sta+le to successfully fly$ it soon +ecomes e9hausting on the #art of the #ilot to constantly correct the state of the aircraft! %t is also incredi+ly unsafe to do so$ assuming that the #ilot is in the aircraft as well! %nterestingly %nterestingly enough$ an air#lane can #erform remar,a+ly and still +e deemed of #oor quality if the sta+ility and control are su+#ar! As such$ sta+ility studies studies are$ to reiterate$ an im#ortant su+.ect! ta+ility ta+ility is com#rised of two #arts: static and dynamic sta+ility! sta+ility! An aircrafts sta+ility at equili+rium is its static sta+ility$ whereas its time res#onse after a distur+ance is ,nown as its dynamic sta+ility! A. 1tatic 1tability
"nce an aircraft is in trim condition$ it is at equili+rium! Although almost every aircraft is designed to achieve trim condition$ no aircraft can maintain trim condition! @ventually$ something will occur to alter the aircraft from its initial trim condition! A force or moment will +e induced on the aircraft$ and can +e caused +y things such as a gust of wind$ the effects of thrust generated$ a change of the angle of attac, of the aircraft$ or even a +ird flying into the aircraft! After After the aircraft is #ertur+ed$ # ertur+ed$ the +eginning tendency of the aircraft to return to its equili+rium state is ,nown as a s the static sta+ility! %f the aircraft is statically sta+le$ then it will attem#t to return to the equili+rium state it +egan at! %f the aircraft is statically unsta+le$ it will continue further from the state of equili+rium after a distur+ance! The static sta+ility +etween these two states is that of +eing statically neutral! %f the aircraft were statically neutral$ once it is distur+ed it would enter a new equili+rium equ ili+rium state with different trim conditions than the one it +egan at! The Boeing 747 will +e analyzed to determine whether it is statically sta+le$ statically unsta+le$ or statically neutral! But sta+ility of the aircraft as a whole often #rove difficult to calculate$ so first the longitudinal and then the la teral-directional sta+ility will +e loo,ed at! The longitudinal a9is is #arallel to the fuselage and divides the #lane in half$ with its #ositive direction going towards the nose of the #lane! onversely$ the lateral a9is is aligned with the wings of the aircraft ¬ e9actly #arallel +ecause the wings are usually swe#t)$ with its #ositive #ositive direction going towards the right wing of the #lane! Both a9is have their origin at the center of gravity of the aircraft! A*. 3ongitudinal 1tatic 1tability 5
Aircraft 2light 8ynamics
The #rimary criteria for an aircraft to #ossess longitudinal static sta+ility is the nature of its #itching moment with res#ect to the angle of attac,! ather than deal with the #itching moment$ it is #referred to use the dimensionless coefficient of the #itching moment instead! im#lifying the analysis$ dimensionless coefficients also allow for easy com#arison +etween aircraft!1 The relationshi# +etween the #itching moment an d the angle of attac, is ,ey in determining how statically sta+le the %f the #itching moment coefficient increases as the angle of attac, increases$ then the air#lane is statically unsta+le! Loo,ing at 2igure $ Air#lane would develo# a #ositive #itching moment that would only move the aircraft further from the equili+rium #oint at B! onversely$ Air#lane 1 would develo# a negative #itching moment that would attem#t to move the aircraft +ac, towards equili+rium! ence$ if the #itching coefficient with res#ect to al#ha is negative$ the #lane can +e considered statically sta+le!
Figure 4. -itching oefficient #s. Angle of Attac *5
Along with a negative
$ the aircraft also needs to have #ositive #itching moment coefficient
when the angle of attac, is zero! Cnown as $ the #itching moment at zero angle of attac, determines the trim angle of attac,! A #ositive trim angle of attac, is desired so as to generate adequate lift to ,ee# the aircraft aloft!1' onsidering @quation 1$ it can +e seen that the a #ositive will yield a the a #ositive angle of attac, &for sta+ility)!
will +e negative for the criteria of static
!uation *. -itching "oment as +elated to the Angle of Attac
A. 3ateral6irectional 1tatic 1tability
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Figure 4. 0aw "oment oefficient against 1ideslip Angle *5
6oving on to the directional sta+ility$ the requirements are similar to longitudinal sta+ility! ather than sta+ility a+out the y-a9is$ the sta+ility a+out the z-a9is is of concern! 2or the z-a9is$ the #rimary moment is that of the yaw moment$ denoted +y ! Again$ it is desira+le to have an aircraft tend to move +ac, towards its original #osition after a #ertur+ation! As shown in 2igure 4$ as the sidesli# angle increases$ the yaw moment needs to increase in a #ositive direction as well &counterintuitively$ a #ositive yaw will +e o##osite in direction to that of #ositive sidesli#)! onversely$ a negative sidesli# angle requires that the yaw moment is negative as well! Therefore$ a #ositive
is the only condition for directional static sta+ility! 5ote that the value
of is equal to zero at zero sidesli#$ +ecause at trim condition the #lane needs to fly in a straight line$ not veering to one side or the other due to a #ositive or negative sidesli# angle! This is unli,e the longitudinal static sta+ility$ where it is desired that at trim the angle of attac, is #ositives so that the nose of the aircraft is #ointing u#ward for adequate lift! B. ynamic 1tability
tatic sta+ility is relatively straightforward! 8ynamic sta+ility$ however$ is not! tatic sta+ility relies on the res#onse of various moments to changes in the orientation of the aircraft$ and the conditions can +e constrained to the derivatives of a few coefficients! 8ynamic sta+ility is much more com#le9$ as it the time res#onse of an aircraft after a distur+ance$ not .ust whether or not the aircraft will return to equili+rium conditions! The +ehaviour of an aircraft de#ends on many factors$ ranging from the moments and forces ac ting on the air#lane to its relative 7
Aircraft 2light 8ynamics
orientation an o+server! Again$ the longitudinal and then the lateral-directional dynamic sta+ility will +e analyzed! 2irst$ the state res#onse of the wh ole system will +e modelled using the ,inematic and dynamic equations! %t will out#ut +oth longitudinal and lateral res#onses! 2ollowing this$ the dynamic equations will +e deco u#led into longitudinal and lateral sets! 2inally$ the sim#lest form of analyzing the res#onse of the system will +e used: first and second order differential equations that model the system through several a##ro9imations! The full derivation of the essential equations used will not +e recounted here! %nstead$ the main results will +e listed$ with +asic e9#lanations as to how the y were o+tained! B*. 8inematic and ynamic !uations
To o+tain the ,inematic and dynamic equations$ the rigid +ody equations of motion must first +e determined for an aircraft in general! %t +egins$ with all equations involving forces$ with 5ewtons econd Law of 6otion! As the name im#lies$ it is assumed that the aircraft is com#letely rigid$ which greatly sim#lifies the mathematics! ;hile generally true$ the main violator of this assum#tion is the shifting mass of fuel within an aircraft! %ts contri+ution to the overall +ody of the airliner is relatively small though$ and really only comes into #lay when analyzing the +ehavior of roc,ets! "nce the rigid +ody equations of motion are determined$ they are then defined in terms of the @uler angles of the aircraft! This allows the force and velocity com#onents to not only +e descri+ed to the #ilot within the aircraft$ +ut also to the flight director &or any other #arty) who remains on the ground! @uler angles are divided into the #itch$ roll$ and yaw angles of the aircraft$ with the reference frame +eing that of the earth coordinate system #reviously descri+ed! They are then sim#lified using the fact that the gravitational and thrust force com#onents can +e descri+ed using the @uler angles as well! This leads to the ,inematic and dynamic equations of motion seen in Ta+le 1! A full descri#tion of the derivation can +e found in the lin, #rovided in the +i+liogra#hy!1*
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Aircraft 2light 8ynamics
Table *. 1ummary of 8inematic and ynamic !uations*5
@ven though the equations are much sim#ler than when the analysis +egan$ it is im#ossi+le to solve analytically! There are twelve equations and twelve different varia+les$ ma,ing the #rocess of an analytical solution a long and difficult one! %nstead$ the equations were solved numerically with the use of 6ATLAB software! To #lot the res#onse of the s ystem$ these equations were then rearranged so as to have the derivatives of each varia+le on one side! aving it in this form was +eneficial for using 6ATLAB to solve the system of differential equations using ode/ &an ordinary differential equation solver for non-stiff differential equations)!
Figure 9. uler Force and "oment !uations +earranged.*7
The solutions o+tained are de#endent on D different varia+les: $ $ $ $ $ and ! All of these varia+les are determined +y initial conditions +efore the distur+ance! $ $ and were set ar+itrarily$ as well as the initial #$ q$ and r angular rates!! $ $ and were then determined with the coefficients of the aircraft through the method in @quation /! Then the
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Aircraft 2light 8ynamics
#rograms listed in A##endi9 were listed were used to #lot the res#onse of the system for each state!
!uation /. alculation of the -itching "oment with oefficients*7
B. ecoupled ynamic !uations
Before the dynamic equations can +e se#arated into longitudinal and directional-lateral forms$ they are sim#lified using small distur+ance theory! The fundamental assum#tion of small distur+ance theory is that the magnitude of a distur+ance to the aircraft is much$ much smaller than the magnitude of the various varia+les at equili+rium condition! @very varia+les final condition is first defined as the initial condition #lus the change in that varia+le$ as shown in @quations !
!uation 1et 5. 1mall isturbance Theory :ariables*5
Then$ this is su+stituted into the dynamic equations$ and the resulting equation is sim#lified using several ,ey initial conditions! The initial conditions$ ,nown as the reference flight conditions$ is defined as symmetric$ and the #ro#ulsive forces are assumed to +e unchanging! This allows several reference conditions to +e equal to &@quation D)!
!uation 7. $nitial condition :ariables*5
Additionally$ the 9-a9is is initially aligned so that it is along the air#lanes velocity vector so as to eliminate ! ince the distur+ance is also small$ the #roducts of the small distur+ances are are considered to +e negligi+le &of an order of magnitude much less than the rest of the varia+les) then eliminated from the resulting equations! The last assum#tion made is that of small angle 10
Aircraft 2light 8ynamics
a##ro9imation$ and since the distur+ance is small the resulting changes in angles will +e small as well! Therefore the sine values of the initial angles and d istur+ance angles will +e as follows:
!uation ;. 1in Trigonometric $dentity*5
A similar argument could +e made for the cosine identities as well! All of this sim#lifies the dynamic equations to much sim#ler linear values$ shown in 2igure *!
Figure 9. 1mall isturbance Theory 1implified ynamic !uations*5
5ow that the dynamic equations have +een linearized$ they can +e rearranged into state s#ace form! tate s#ace form will model the res#onse of each varia+le so that they are a function of in#ut$ out#ut$ and state varia+les! The change in res#onse that include the change in the normal force due to the #itch angular rate and the derivative of the z-velocity are considered to +e insignificant$ and as such are ignored in the final analysis! These equations are then organized into state s#ace form to first +e numerically solved and then analyzed for the eigenvalues of each system! 6ATLABs ode4/ differential equation solver can then +e used to solve each system of equations and then show the res#onse of the system +oth laterally and longitudinally!
!uation ;. 1tate 1pace Form
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Aircraft 2light 8ynamics
Figure 5. 3inearized 3ongitude !uations*5
Figure 7. 3inearized 3ateral6irectional !uations*5
The eigenvalues of each A matri9 give information a+out the res#onse! Through com#arison to a mass s#ring system$ the dam#ing ratio the undam#ed natural frequency of the res#onse can +e determined! The dam#ing ratio will determine how quic,ly the res#onse either increases or decreases$ while the natural frequency gives the #eriod of the oscillation! Both factors determine the time until the magnitude of the res#onse is either half or dou+le the original value through the system dam#ing! The motion associated with each dam#ing ratio is listed in 2igure (!
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Aircraft 2light 8ynamics
!uation 1et <. amping +atio = >, )atural Fre!uency =
>, amped )atural Fre!uency =
>, and
*5
1ystem amping = > fom omple? +oot = >
Figure ;. escription of amping +atio and associated "otion*5
Figure <. 3ong =phugoid> and 1hort -eriod Appro?imations of the amping +atio and )atural Fre!uency*5
@stimates using the long #eriod and short #eriod forms of the longitudinal state-s#ace equation eigenvalues will +e o+tained with the equations in 2igure (! To o+tain these estimates$ the assum#tion of fi9ed controls is made! That is to say that there is no additional control in#ut to the res#onse! Although usually this wouldnt +e the case while the aircraft is +eing o#erated$ doing so allows quic, analysis to +e made! The two a##roaches will +e com#ared to see if the estimate is an accurate one! To solve and then su+sequently model the dynamic equations$ 6ATLAB was used! The #rograms were develo#ed with the aid of the 6athwor,s we+site1($ through guides to ode solvers$ state s#ace varia+les$ and systems of ordinary differential equation solvers! All of the #rograms are listed in A##endi9 / for reference! %n all of the equations a+ove$ the constants for the Boeing 747 aircraft were o+tained from the te9t Flight &ta'ility an A!tmatic *ntrl+16 13
Aircraft 2light 8ynamics
+esults tatic sta+ility can easily +e #roved from o+servation of the data in A##endi9 1! and are +oth negative$ im#lying that the #lane will return to equili+rium after a distur+ance! All of the gra#hs out#utted +y the 6ATLAB #rogram are listed in A##endi9 1 for continuity &there are several #ages worth of gra#hs)! To +egin with$ the @uler 2orce and 6oment @quation #rogram was formed! The in#uts for the systems of equations & $ $ $ $ $ and ) were all set to ' with the e9ce#tion of $ which was set to 1''' l+f$ $ $ and whos values were determined +y @quation /! A controls fi9ed model was e m#loyed$ with the elevator$ rudder$ and aileron set to 1$ /$ and * degrees res#ectively! %nitial sidesli# and angle of attac, angles were all also set to '! everal different com+inations of initial conditions were used$ +ut the results were all equally unrelia+le! Although the #rogram was chec,ed several times$ it seems that the results o+tained from the euler force and moment equations are inaccurate! 6uch was done to eliminate as much error as #ossi+le$ including cross referencing with other #rograms$ research sources$ and mani#ulating of the varia+les$ +ut the results o+tained from running the #rogram seem to +e riddled with mista,es! The velocities are all e9hi+iting ridiculous +ehavior! and are wildly oscillating +etween #ositive and negative values while increasing in value$ which is #hysically im#ossi+le! The 9 and y #ositions are decreasing at incredi+le rates$ while the #lane soars u#wards with an increasing z #osition! 6oving on to the angles and angular rates$ for whatever reason the angular rates and are dam#ed as it they should +e! 6eanwhile$ is .ust constantly decreasing! The angles continue this trend of odd +ehavior$ with all of the angles e9ce#t for .ust oscillating around different values! %f an aircraft actually +ehaved in this manner$ it would +e #lummeting towards the ground! 6oving on to the Longitudinal and Lateral 8ecou#ling @quations$ there seems to +e a moderate amount of accuracy here! Than,fully$ the mista,es seen in the @uler 2orce and 6oment @quations seem to +e a+sent here! 2irst$ the free res#onse of the longitude were modelled! The initial #itch of the aircraft was fi9ed at *$ while $ $ and were slightly varied! 2ollowing the initial distur+ance values$ all of the varia+les dam#ed to zero! The manner of dam#ing was slightly sinusoidal in the case of $ $ and $ whereas the #itch was decreased to ' e9#onentially! 5e9t$ the elevator was fi9ed at * degrees and the #rogram was run again! According to the gra#hs$ the elevator seems to have a great effect on the varia+les! ;hen it is fi9ed at * degrees$ all of the varia+les reach the same state regardless of the initial state! As for the lateral decou#led equations$ the initial conditions only seem to increase or decrease the magnitude of the res#onse! The res#onse +ehavior remains largely the same over all four states simulated! ;hile the rudder is maintained at ' deflection$ sidesli#$ $ and all dam#en sinusoidally! There is some small variation to the sinusoidal +ehavior towards the end of the test sam#le where the values decrease very slightly e9#onentially$ with its magnitude changing in res#onse to the initial conditions! The roll e9hi+its da m#ing motion closest to 14
Aircraft 2light 8ynamics
e9#onential overdam#ing! "nce the rudder is deflected at an angle of #ositive * degrees$ the roll and change their steady state values from ' to negative values$ and the e9#onential decay seen at the end of each res#onse gra#h is more evident! As the aileron is deflected * degrees$ the system res#onse changes drastically! All of the res#onses +ecome underdam#ed e9#onentially decaying sinusoidal motion to varying degrees when the sidesli# is negative or '! 2or the other states$ the sidesli# and return to their original +ehavior! still reaches a steady state that differs with the initial values as well! 2inally the eigenvalues were calculated with 6ATLAB for +oth the longitudinal and lateral states! ;hen the equations of motions are in state s#ace form$ the eigenvalues are determined from the 6atri9 A! om#aring these to the values o+tained from the a##ro9imations$ there seems to +e no agreement! Between the two$ the values calculated are correct! 2or the a##ro9imations$ the only values used were o+tained from the te9t$ so it is unclear where the source of error is$ other than a mista,e within the calculations themselves! According 2igure ($ the eigenvalues from the #rogram do match with the results o+tained through the gra#hs! onsidering the decou#led equations$ they su##ort the fact that the aircraft in question is a finished vehicle licensed for use! 3sing the data from the te9t yields an aircraft that dam#s distur+ances to maintain trim conditions$ #roving that it is indeed statically sta+le! This is e9actly what one would e9#ect from a commercial airliner$ where a smooth$ safe ride is desira+le! ;hen the elevator$ rudder$ andEor aileron are deflected$ the change can +e seen in the direction and moments acting on the aircraft! After an initial distur+ance$ ,ee#ing these control factors dis#laced results in changing the #osition of the aircraft$ which e9hi+its the effect that controls have on the aircraft! ath and orientation can +e ad.usted de#ending on outside stimuli with the alteration of these factors!
onclusion The goal of this #a#er was to ascertain the via+ility of the Boeing 747 aircraft! 2irst$ the requirements for static sta+ility were considered! Then through the use of the @uler force and moment equations$ along with their sim#lifications$ it was seen if the aircraft would return to equili+rium! %n their original form$ efforts to adequately model the +ehavior of the aircraft #roved largely unsuccessful! 8es#ite this$ decou#ling the equations and o+serving the eigenvalues showed that the aircraft did indeed #ossess static sta+ility! 6oving forward$ care needs to +e ta,en in regards to #ro#erly modeling the system! The com#le9ity of flight sta+ility was grossly underestimated$ +ut this study has #roved useful in #roviding insight into how aircraft modeling is done!
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Aircraft 2light 8ynamics
A##endi9 1 6ATLAB ode esults
uler Force and "oment !uations
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ecoupled 3ateral !uations, Free +esponse u@*, w@*, !@9, theta@9
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Aircraft 2light 8ynamics
u@9, w@9, !@, theta@9
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u@*9, w@, !@9, theta@9
25
Aircraft 2light 8ynamics
u@, w@9, !@*9, theta@9
26
Aircraft 2light 8ynamics
ecoupled 3ongitudinal !uations, le#ator at 9 egrees u@*, w@*, !@9, theta@9
27
Aircraft 2light 8ynamics
u@9, w@9, !@, theta@9
28
Aircraft 2light 8ynamics
u@*9, w@, !@9, theta@9
29
Aircraft 2light 8ynamics
u@, w@9, !@*9, theta@9
30
Aircraft 2light 8ynamics
ecoupled 3ateral !uations, Free +esponse sideslip@6*, p@ , r@*, roll@ 9
31
Aircraft 2light 8ynamics
sideslip@*9, p@ , r@9, roll@ *
32
Aircraft 2light 8ynamics
sideslip@9, p@ 9, r@, roll@ *9
33
Aircraft 2light 8ynamics
sideslip@, p@ 9, r@9, roll@ *9
34
Aircraft 2light 8ynamics
ecoupled 3ateral !uations, +udder at 9 egrees sideslip@6*, p@ , r@*, roll@ 9
35
Aircraft 2light 8ynamics
sideslip@*9, p@ , r@9, roll@ *
36
Aircraft 2light 8ynamics
sideslip@9, p@ 9, r@, roll@ *9
37
Aircraft 2light 8ynamics
sideslip@, p@ 9, r@9, roll@ *9
38
Aircraft 2light 8ynamics
39
Aircraft 2light 8ynamics
ecoupled 3ateral !uations, Aileron at 9 egrees sideslip@6*, p@ , r@*, roll@ 9
40
Aircraft 2light 8ynamics
sideslip@*9, p@ , r@9, roll@ *
41
Aircraft 2light 8ynamics
sideslip@9, p@ 9, r@, roll@ *9
42
Aircraft 2light 8ynamics
sideslip@, p@ 9, r@9, roll@ *9
43
Aircraft 2light 8ynamics
44
Aircraft 2light 8ynamics
igen#alue Appro?imations Longitudinal @igenvalues alculated 8am#ed es#onse
-'!/4D*
Long eriod &#hugoid)
hort eriod
-'!''0*
'!D0''
-'!/4D* -'!17' -'!17'
8am#ed 2requency
1!*4'
-'!4(D
-1!*4' '!4(D1 '!14D' -'!14D'
Lateral @igenvalues alculated 8am#ed es#onse
-/!7'00
#iral -'!1/0D
oll
8utch oll
-'!(7'*
-'!1*'0
1!/(* 1!/(* -'!1/(1
8am#ed 2requency
'
'!D'0
!07(0 -!07(0 '
45
Aircraft 2light 8ynamics
A##endi9 onstants of the Aircraft
Flight coefficients and -arameters of the Boeing 747 Aircraft &oefficients are given in terms of radians) 3ongitudinal
-&-*y'.m
"@.9 at sea le#el 11!1 '!1' *!7' '!DD -1!D D!7 -/! *!4 -'!( -'!(1 '!' '!7 '!//( -1!/4
"@.< at 4, ft. '!* '!'4 *!* '!47 -1!D '!''D -0!' D!*( -*!' '! '!* -'!1' '!/ -1!
"@.9 at sea le#el -'!0D -'!1 '!1*' -'!4* -'!11 '!1'1 -'!/' '!'4D1 '!''D4 '!17* '!''7 -'!1'0
"@.< at 4, ft. -'!(* -'!1' '!' -'!/' '!' '!' -'!/* '!'14 '!''/ '!'7* '!''* -'!'0
3ateral
46
Aircraft 2light 8ynamics
enter of (ra#ity and "ass haracteristics ;eight: enter of Fravity &F) at *G 6ean Aerodynamic hord &6A) olling 6oment of %nertia: itching 6oment of %nertia:
+eference (eometry hord Length:
ft
;ing lanform Area &eference Area): ;ing #an:
47
Aircraft 2light 8ynamics
A##endi9 6ATLAB ode
1tatic 1tability -roofs Gonstants for 6ach '!*$ sea levelG >Hinf1I(1!// GvHinfinity in f#s rho1I'!''/77 Gdensity of air in slugEft?/ K1I'!*=rho1=>Hinf1? Gdynamic #ressure
Mal#ha Nnis G!f and the slo#e of the m vs! L curve is G!fMO$!!! mal#ha$ mHL)
Gonstants for 6ach '!0'$ 4'$''' ft!= >HinfI(1!// GvHinfinity in f#s rhoI'!'''*(7 Gdensity of air in slugEft?/ KI'!*=rho=>Hinf? Gdynamic #ressure
GG@levator ontrolGG G6ach '!* at sea levelG LdelHe1I'!//( Gslo#e of coefficicent of lift vs! elevator deflection mdelHe1I-1!/4 Gslo#e of #itching coefficient vs! elevator deflection m'1I' G#itching coefficient at zero angle of attac, G6ach '!0' at 4'$'''G LdelHeI'!/ Gslo#e of coefficicent of lift vs! elevator deflection mdelHeI-1! Gslo#e of #itching coefficient vs! elevator deflection m'I' G#itching coefficient at zero angle of attac,
GAircraft 8imensions cI7!/1 Gchord length in ft I**'' Greference area &wing #laneform area) in ft? +I10*!D( Gwing s#an in ft
GGGLongitudinal tatic ta+ility and ontrolGGG GGtatic ta+ilityGG G6ach '!* at sea levelG mal#ha1I-1!D Gslo#e of the #itching moment coeffecient vs! angle of Gattac, curve
al#haI&':'!1:/')=#iE1(' deltaHeI' G6ach '!* at sea levelG L1ILal#ha1=al#haPLdelHe1=deltaHe Gcoefficient of lift
Lal#ha1I*!7' Gslo#e of the coefficient of lift mHL1Imal#ha1ELal#ha1 f#rintf&MNnAt 6ach I '!* and an altitude of sea level$the slo#e ofM!!! M m vs! Nnal#ha is G!f and the slo#e of the m vs! LM!!! M curve is G!fMO$mal#ha1$ mHL1)
m1Im'1Pmal#ha1=al#haPmdelHe1=deltaHe G#itch coefficient G6ach '!0' at 4'$'''G LILal#ha=al#haPLdelHe=deltaHe
G6ach '!0' at 4'$''' ftG mal#haI-1!D' Gslo#e of the #itching moment coeffecient vs! angle of Gattac, curve!
mIm'Pmal#ha=al#haPmdelHe=deltaHe
su+#lot&$$1) #lot &al#ha$L1$ al#ha$ L) su+#lot&$$) #lot &al#ha$m1$ al#ha$ m) hold on
Lal#haI*!*' Gslo#e of the coefficient of lift mHLImal#haELal#ha f#rintf&MNnNnAt 6ach I '!0' and 4'$''' ft!$the slo#e of m vs!M!!! 48
Aircraft 2light 8ynamics
su+#lot&$1$1) #lot&+eta$n1$+eta$n) Grudder deflection com#arisonG +etaI' deltaHrI':'!1:/'
al#haI' deltaHeI&':'!1:/')=#iE1(' G6ach '!* at sea levelG L1ILal#ha1=al#haPLdelHe1=deltaHe
n1I+eta=n+eta1PdeltaHr=ndelHr1 nI+eta=n+etaPdeltaHr=ndelHr su+#lot&$1$) #lot&deltaHr$n1$deltaHr$n)
m1Im'1Pmal#ha1=al#haPmdelHe1=deltaHe G6ach '!0' at 4'$'''G LILal#ha=al#haPLdelHe=deltaHe
GGoll ontrolGG mIm'Pmal#ha=al#haPmdelHe=deltaHe Gsidesli# angleG +etaI':'!1:/' deltaHaI' Gaileron deflection angle
su+#lot&$$/) #lot &deltaHe$L1$ deltaHe$ L) su+#lot&$$4) #lot &deltaHe$m1$ deltaHe$ m) hold off
G6ach '!* at sea levelG l+eta1I-'!1 Gslo#e of roll coefficient vs! the sidesli# angle ldelHa1I'!'4D1 G6ach '!0' at 4'$'''G l+etaI-'!1' Gslo#e of roll coefficient vs! the sidesli# angle ldelHaI'!'14
GGGLateral ta+ility and ontrolGGG GG8irectional ta+ility and ontrolGG Gsidesli# angle com#arisonG +etaI':'!1:/' Gsidesli# angle deltaHrI' Grudder deflection angle
l1I+eta=l+eta1PdeltaHa=ldelHa1 lI+eta=l+etaPdeltaHa=ldelHa su+#lot&$1$1) #lot&+eta$ l1$ +eta$ l) hold on
G6ach '!* at sea levelG n+eta1I'!1*' Gslo#e of yaw coefficient vs! the sidesli# angle G6ach '!0' at 4'$'''G n+etaI'!' Gslo#e of yaw coefficient vs! the sidesli# angle
Gaileron deflection angleG +etaI' deltaHaI':'!1:/' Gaileron deflection angle
G6ach '!* at sea levelG ndelHr1I-'!1'0 Gslo#e of yaw coefficient vs! the rudder deflection angle G6ach '!0' at 4'$'''G ndelHrI-'!'0 Gslo#e of yaw coefficient vs! the rudder deflection angle
G6ach '!* at sea levelG ldelHa1I'!'4D1 Gslo#e of roll coefficient vs! aileron deflection angle G6ach '!0' at 4'$'''G ldelHaI'!'14 Gslo#e of roll coefficient vs! aileron deflection angle
n1I+eta=n+eta1PdeltaHr=ndelHr1 nI+eta=n+etaPdeltaHr=ndelHr
l1I+eta=l+eta1PdeltaHa=ldelHa1 lI+eta=l+etaPdeltaHa=ldelHa 49
Aircraft 2light 8ynamics
#lot&deltaHa$ l1$ deltaHa$ l) hold off
su+#lot&$1$)
+igid Body !uations of "otion F%)T$) 1+$B$)( C%AT$)1 function d9dt I B@6&t$9) GAircraft onstants c I 7!/1 I **'' + I 10*!D( g I /! mID/DD''E/! %99 I 1(!=1'?D %yy I //!1=1'?D %zz I 40!7=1'?D %9z I '!07=1'?D Goefficients mHal# I -1!D mHde I -1!/4 mHq I -'!( lH+et I -'!1 lHdr I '!''7 lHda I '!''D4 lH# I -'!4* lHr I '!1'1 nH+et I '!1*' nHdr I -'!1'0 nHda I '!''D4 nH# I -'!11 nHr I -'!/' GAtmos#heric onditions rhoI'!''/77 Gdensity of air in slugEft?/ G#ecified %nitial onditions dHele I 1=#iE1(' dHail I /=#iE1(' dHrud I *=#iE1(' al#ha I '=#iE1(' +eta I '=#iE1(' v'I(1!// GvH' in f#s KI'!*=rho=v'? Gdynamic #ressure # I ' q I ' r I ' Q I 1''' < I ' R I ' L I &lH+et=+eta P lHdr=dHrud P lHda=dHail P &lH#=KPlHr=K)=+E&=v'))='!*=rho==v'? 6 I &mHal#=al#ha P mHde=dHele P mHq=K=cE&=v'))='!*=rho==v'? 5 I &nH+et=+eta P nHdr=dHrud P nHda=dHail P&nH#=KPnHr=K)=+E&=v'))='!*=rho==v'? d9dt I zeros&1$1) d9dt&1) I QEm - g=sin&9&11))P 9&0)=9&) - 9&()=9&/) Gudot
50
Aircraft 2light 8ynamics
d9dt&) I
ecoupled 3ongitudinal !uations F%)T$) 1+$B$)( C%AT$)1 function dQ I LzedLong&t$Q)
%yy I //!1=1'?D 51
Aircraft 2light 8ynamics
c I 7!/1 I **'' mID/DD''E/! gI/! rhoI'!''/77 u'I(1!// KI'!*=rho=u'?
mdeI-1!/4
QuI -&8uP=8')=K=E&m=u') QwI -&8a-L')=K=E&m=u') RuI -&LuP=L')=K=E&m=u') RwI -&La-8')=K=E&m=u') 6uI mu=&K==c)E&u'=%yy) 6wI ma=&K==c)E&u'=%yy) 6qI mq=&K==c?)E&=u'=%yy) 6wdI mad=&K==c?)E&=u'?=%yy)
8uI' 8'I'!1' 8aI'!DD LuI' L'I11!1 LaI*!7' muI' maI-1!D madI-/! mqI-'!(
AIQu Qw ' -gRu Rw u' '&6uP6wd=Ru) &6wP6wd=Rw) 6qP6wd=u' '' ' 1 'O BI' ' ' ' &mde=&K==c))E%yy ' ' 'O 5I*'O dQIA=QPB=5 end
13:+ A1 D33 A1 -3TT$)( GuI1'$ wI 1'$ qI*$ #itchI* iI1' 1' * *O
GuI1*$ wI '$ qI*$ #itchI* iI1* ' * *O
figure #lot&ts$Q&:$1)) 9la+el&Mtime&sec)M) yla+el&MuM) grid on figure #lot&ts$Q&:$)) 9la+el&Mtime&sec)M) yla+el&MwM) grid on figure #lot&ts$Q&:$/)) 9la+el&Mtime&sec)M) yla+el&MqM) grid on figure #lot&ts$Q&:$4)) 9la+el&Mtime&sec)M) yla+el&M#itchM) grid on
GuI*$ wI *$ qI'$ #itchI* iI* * ' *O
Aircraft 2light 8ynamics
figure #lot&ts$Q&:$1)) 9la+el&Mtime&sec)M) yla+el&MuM) grid on figure #lot&ts$Q&:$)) 9la+el&Mtime&sec)M) yla+el&MwM) grid on figure #lot&ts$Q&:$/)) 9la+el&Mtime&sec)M) yla+el&MqM) grid on figure #lot&ts$Q&:$4)) 9la+el&Mtime&sec)M) yla+el&M#itchM) grid on
GuI'$ wI *$ qI1*$ #itchI* iI' * 1* *O
ecoupled 3ateral !uations F%)T$) 1+$B$)( C%AT$)1 function dQ I LzedLat&t$Q) G#itchI*$ rudder %zz I 40!7=1'?D %99 I 1(!=1'?D %yy I //!1=1'?D c I 7!/1 I **'' + I 10*!D( mID/DD''E/! gI/! rhoI'!''/77 u'I(1!// KI'!*=rho=u'? theta'I*
ldaI'!'4D1 ldrI'!''7 ndaI'!''D4 ndrI-'!1'0 <+IK==y+Em <#IK==+=y#E&=m=u')
y+I-'!0D y#I ' yrI ' l+I-'!1 l#I-'!4* lrI '!1'1 n+I '!1*' n#I-'!11 nrI-'!/'
ydrI'!17*
53
Aircraft 2light 8ynamics
dQIA=QPB=5
end
13:+ A1 D33 A1 -3TT$)( iI-1' ' 1' *O
figure #lot&ts$Q&:$4)) 9la+el&Mtime&sec)M) yla+el&MrollM) grid on iI* * ' 1*O
figure #lot&ts$Q&:$1)) 9la+el&Mtime&sec)M) yla+el&Msidesli#M) grid on figure #lot&ts$Q&:$)) 9la+el&Mtime&sec)M) yla+el&M#M) grid on figure #lot&ts$Q&:$/)) 9la+el&Mtime&sec)M) yla+el&MrM) grid on figure #lot&ts$Q&:$4)) 9la+el&Mtime&sec)M) yla+el&MrollM) grid on iI1* ' * 1'O
iI' * 1* 1*O
ts$QO I ode4*&SLzedLat$ts#an$Q') figure #lot&ts$Q&:$1)) 9la+el&Mtime&sec)M) yla+el&Msidesli#M) grid on figure #lot&ts$Q&:$)) 9la+el&Mtime&sec)M) yla+el&M#M) grid on figure #lot&ts$Q&:$/)) 9la+el&Mtime&sec)M) yla+el&MrM) grid on 54
Aircraft 2light 8ynamics
figure #lot&ts$Q&:$4)) 9la+el&Mtime&sec)M)
yla+el&MrollM) grid on
igen#alues and amping of ecoupled !uations L"5F%T38%5AL %yy I //!1=1'?D c I 7!/1 I **'' mID/DD''E/! gI/! rhoI'!''/77 u'I(1!// KI'!*=rho=u'?
6wdI mad=&K==c?)E&=u'?=%yy) RaI' 6aIu'=6w 6adIu'=6wd
AIQu Qw ' -gRu Rw u' '&6uP6wd=Ru) &6wP6wd=Rw) 6qP6wd=u' '' ' 1 'O e$vOIeig&A) eigenIdiag&v) 8am#es#Ireal&eigen) 8am#2reqIimag&eigen) eriodI=#i!E8am#2req tHhdI'!D0/!Ea+s&8am#es#) 5cycI'!11=a+s&8am#2req)!Ea+s&8am#es#)
8uI' 8'I'!1' 8aI'!DD LuI' L'I11!1 LaI*!7' muI' maI-1!D madI-/! mqI-'!(
f#rintf&MA##ro9imationsNnLong eriod &#hugoid)NnM) 5at2reqLIsqrt&-Ru=gEu') 8am#atioLI-QuE&=5at2reqL) 8am#es#LI-8am#atioL=5at2reqL 8am#2reqLI5at2reqL=sqrt&1-8am#atioL?)
QuI -&8uP=8')=K=E&m=u') QwI -&8a-L')=K=E&m=u') RuI -&LuP=L')=K=E&m=u') RwI -&La-8')=K=E&m=u') 6uI mu=&K==c)E&u'=%yy) 6wI ma=&K==c)E&u'=%yy) 6qI mq=&K==c?)E&=u'=%yy)
f#rintf&MA##ro9imationsNnhort eriodNnM) 5at2reqIsqrt&Ra=6qEu'-6a) 8am#atioI-&6qP6adPRaEu')E&=5at2req) 8am#es#I-8am#atio=5at2req 8am#2reqI5at2req=sqrt&1-8am#atio?)
LAT@AL %zz I 40!7=1'?D %99 I 1(!=1'?D %yy I //!1=1'?D c I 7!/1 I **'' + I 10*!D( mID/DD''E/! gI/! rhoI'!''/77 u'I(1!// KI'!*=rho=u'? theta'I'
y#I ' yrI '!17* l+I-'!1 l#I-'!4* lrI '!1'1 n+I '!1*' n#I-'!11 nrI-'!/' ydrI'!17* ldaI'!'4D1 ldrI'!''7 ndaI'!''D4 ndrI-'!1'0
y+I-'!0D 55
Aircraft 2light 8ynamics
eigenIdiag&v) 8am#es#Ireal&eigen) 8am#2reqIimag&eigen) eriodI=#i!E8am#2req tHhdI'!D0/!Ea+s&8am#es#) 5cycI'!11=a+s&8am#2req)!Ea+s&8am#es#)
<+IK==y+Em <#IK==+=y#E&=m=u')
G#iral and oll 6odes Ls#iralI&L+=5r-Lr=5+)EL+ LrollIL#
G8utch oll 8am#es#@Ireal&eigen&)) 8am#2req@Iimag&eigen&)) 5at2req8Isqrt&&<+=5r-5+=
AI
56
Aircraft 2light 8ynamics
+eferences 1 Boeing! &'1D)! /istrical &nasht+ etrieved from htt#:EEwww!+oeing!comEhistoryE#roductsE7'7!#age
lar,e$ hris! &'1*$ e#tem+er 1)! 0 st 3mrtant Airlanes f All ime! etrieved from htt#:EEwww!#o#ularmechanics!comEflightEg14Ethe-/'-most-im#ortant-air#lanes-of-all-timeE /
Boeing! &'1D)! /istrical &nasht+ etrieved from htt#:EEwww!+oeing!comEhistoryE#roductsE747!#age 4
Cull$ Cris! &'1D$ "cto+er )! he egenary en ehin the eing 747 ! etrieved from htt#:EEwww!airlinere#orter!comE'14E1'Elegendary-men-+ehind-historic-+oeing-747E *
Aviation@9#lorer! &'1D)! eing 747 Aircraft Airliner Facts, Dates, ict!res, an /istry fr all eing 747 ariants! etrieved from htt#:EEwww!aviatione9#lorer!comE747Hfacts!htm D
van itte$ @d! &'1D)! he 8ne9ecte &!ccess f the eing 747 ! etrieved from htt#s:EEwor,sthatwor,!comEE+oeing-747 7
&'1D)! Air Frce ne+ etrieved from htt#s:EEwww!whitehouse!govE1D''Eair-force-one
(
Fi++s$
Lednicer$ 8avid! &'1'$ e#tem+er 1*)! he 3ncmlete ;!ie t Airfil 8sage htt#:EEmselig!ae!illinois!eduEadsEaircraft!html 1'
aughy$ 8avid! &'11)! 3ntr!ctin t Aircraft &ta'ility an *ntrl *!rse Ntes fr
Arninc! &'1*$ A#ril 0)! A>iatin ?eference aterial+etrieved from htt#:EEarninc!+logs#ot!comEsearchEla+elEAircraftu#dated-ma9I'1*-'4-'0T'(:/':'''7:''Uma9-resultsI'UstartI*U+y-dateIfalse 1
tanford! Nnimensinali@tin+ etrieve from htt#:EEadg!stanford!eduEaa'(EdynamicsEnondimen!html 1/
The @ngineering Tool+o9! he 8+&+ &tanar Atmshere+ etrieved from htt#:EEwww!engineeringtool+o9!comEstandard-atmos#here-dHD'4!html
