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Analysis of Dynamic Aircraft Landing Loads, And a Proposal for Rational Design Landing Load Requirments

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Analysis of Dynamic Aircraft Landing Loads, And a Proposal for Rational Design Landing Load Requirments
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  • Analysis of dynamic aircraft landing loads, and a proposal for rational design landing load requirements

    P R O E F S C H R I F T

    TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R.v .NAUTA L E M K E , HOOGLERAAR IN DE AFDELING DER ELECTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG 27 JUNI 1972 DES NAMIDDAGS TE 4 UUR

    DOOR

    JACOB IJFF

    Vl_l EGTUIGBOUWKUNDIG I N G E N I E U R

  • Dit proefschrift is goedgekeurd door de promotoren

    prof. dr . ing, J . Taub prof. dr . i r . A. van der Neut

  • Aan de nagedachtenis van mijn Vader.

    De schr i jver wil getuigen van zijn grote erkenteli jkheid voor de bereidwilligheid van de direkt ie van de Neder landse Vliegtuig-fabrieken "Fokker -VFW" N. V. om de totstandkoming van dit proefschrif t mogelijk te maken. Van de vele medewerke r s van Fokker-VFW die daar toe hebben bijgedragen en waarvoor de schr i jve r zijn grote dank wil ui t -drukken, moeten in het bijzonder worden genoemd :

    de heer H. J . Dannenburg, die als wetenschappelijk p r o g r a m -meur de r ea l i s e r ing van de computer p r o g r a m m a ' s heeft verzorgd .

    de heren C. J . van Veen en C. G. B r e e m s , die zowel de b a s i s -gegevens voor de berekeningen, als de i l lus t ra t i es van dit proefschrift voor hun rekening hebben genomen.

    de dames C. P . Esse lman en A. Bnig, en de heer J. M. A. Boon, die het typewerk en de lay-out van het manuscr ip t hebben verzorgd.

  • - 1 -

    C o N T E N 1^ S

    0. LIST OF SYMBOLS

    1. INTRODUCTION

    1.1. General 1. 2. Physical phenomena involved in a landing impact 1.3. Existing requirements for landing impact load cases 1. 4. The objectives of the present study

    2. SURVEY OF EXISTING LITERATURE

    3. THEORY OF LANDING IMPACT LOAD ANALYSIS

    3 . 1 . Introduction 3. 2. The coordinate system 3 . 3 . The equations of motion 3. 4. Generalized landing gear forces 3. 5. Landing gear forces

    3 . 5 . 1 . Equations of motion of unsprung mass 3 . 5 . 2 . Forces and moments acting on the t i r e 3 . 5 . 3 . Shock absorber forces

    3 .6 . Generalized aerodynamic forces 3 .7 . The solution of the equations of motion 3. 8. Calculation of loads

    4. THE INITIAL CONDITIONS FOR THE LANDING IMPACT PROBLEM

    5. DETAILS OF EXAMPLE CALCULATIONS

    5 . 1 . Choice of representa t ive aircraft 5 .2 . Dynamic schematization of the F-27 5 .3 . Some further simplifications 5. 4. The equations of motion for the F-27 calculations 5 .5 . The equations for the calculation of wing loads

    6. THE INFLUENCE OF CONSTITUTIVE PARAMETERS ON LANDING IMPACT LOADS

    6. 1. Refinements in s t ruc tura l schematization 6. 2. Landing gear schematization 6 .3 . Longitudinal drag forces 6. 4. Latera l drag forces 6. 5. Initial roll angle 6. 6. Pitch attitude 6 .7 . Forward velocity 6 .8 . Descent velocity 6. 9. Summary of resu l t s and conclusions

    7. COMPARISON OF CALCULATED AND MEASURED LANDING IMPACT QUANTITIES

  • -2-

    PROPOSAL FOR RATIONAL REQUIREMENTS FOR THE PREDICTION OF LANDING IMPACT DESIGN LOADS

    8. 1. Proposals for rational requirements

    8.1-1 Introduction 8. 1-2 Formulation of rational requirements 8. 1-3 Concluding r emarks

    8.2. Comparison of resu l t s of rational landing imipact load calculations and calculations according to existing requirements

    REFERENCES

    APPENDIX A : United States Federa l Aviation Agency Airworthiness standards for ground loads of t ranspor t category a i rplanes .

    APPENDIX B : Calculation of prescr ibed wing bending- and torsion deformation modes.

  • 3 -

    LIST OF SYMBOLS

    A - work done

    Aj^ . - coefficients defining pneumatic shock absorber spring charac te r i s t i cs Ac - aerodynamic stiffness load due to unit displacement of general ised

    ^ i c o o r d i n a t e q^

    Aj). - a e r o d y n a m i c d a m p i n g load due to uni t ve loc i t y of g e n e r a l i s e d c o o r d i n a t e q^

    B (y) - wing bend ing naoment

    C -^j - e l e m e n t of g e n e r a l i s e d s t i f fness m a t r i x D - a e r o d y n a m i c d r a g f o r c e

    E - k ine t i c e n e r g y

    F (t) - i n d i c i a l funct ion

    F j - f o r c e ac t i ng in s t a t i o n j F g - shock a b s o r b e r f o r c e

    FTT - shock a b s o r b e r f o r c e due to h y d r a u l i c r e s i s t a n c e

    F p - p n e u m a t i c f o r c e in shock a b s o r b e r

    F p - f r i c t i o n f o r c e in s h o c k a b s o r b e r

    Ffj - shock a b s o r b e r n o r m a l f o r c e Ix_ y^ z - m o m e n t s of i n e r t i a

    I j - i n e r t i a load due to unit a c c e l e r a t i o n of g e n e r a l i s e d c o o r d i n a t e qj J - whee l r o t a t i o n a l m o m e n t of i n e r t i a

    Kx. y . z . " t i r e f o r c e s

    Kw/ - w h e e l ax le f o r c e s Wx. y . z

    L - a e r o d y n a m i c lift f o r c e

    L ( X. y . z . t ) - t o t a l load ac t i ng a t point x. y . z a t t i m e t

    L Q - l and ing g e a r l oads

    Mj^ j - e l e m e n t of g e n e r a l i s e d m a s s m a t r i x Mvv - m o m e n t s a c t i ng a t whee l ax l e

    " x . y . z ^

    N(, - c o r n e r i n g p o w e r

    N - shock a b s o r b e r h y d r a u l i c d a m p i n g coeff ic ient

    Q i - i th g e n e r a l i s e d f o r c e R - t i r e r a d i u s

    S (y) - wing s h e a r f o r c e

    T (y) - wing t o r s i o n a l m o m e n t

    U - p o t e n t i a l e n e r g y

    V Q - a i r c r a f t s p e e d p a r a l l e l to e a r t h s u r f a c e V , - sk idd ing ve loc i ty

    SK

  • - 4 -

    stalling speed

    vector components

    width of t i re contact region

    rigid body degrees of freedom

    wheel excentricity

    X, Y, Z components of wheel axle displacement due to unit value of 5

    wing chord

    distance in Z direction between wing plane and aircraf t c. of g.

    distance in Z direction between wing plane and engine c. of. g.

    distance in Z direction between wing plane and wheel axle

    distance in Z direction between wing plane and pylontank c. of g.

    prescr ibed wing bending deformation mode

    prescr ibed wing torsion deformation mode

    length of wheel s t rut

    radius of gyration around x, y and z axis

    element of flexibility influence coefficient matr ix

    half t i re footprint length

    distance in x direction between engine c. of g. and pitch hinge of engine pod in wing plane

    ith mass element

    acceleration, also exponent of polytropic expansion

    p re s su re in t i re , or in shock absorber

    ith generalised coordinate

    distance between upper- and lower shock absorber bearing when E = O

    effective wheel radius

    distance traveled, expressed in half chords s = -rrhr-

    static moment of the ith mass element with respect to wing el, axis

    time

    coordinates of point i

    angle of incidence

    slip angle

    wheel angular velocity

    constants defining t i re force-deflection curve

    quantity defined by eq. 3. 5-22

    generalised coordinate for wing bending

    generalised coordinate for engine pitch

    generalised coordinate for wing torsion

    slip ratio

  • - 5 -

    Af - slip ratio for max. value of (Xx U- - friction coefficient

    V - frequency

    5 - shock absorber deflection

    P - mater ia l densitj ' 0 - t i re deflection

    X - time

    T - angle

    tp f- t i re yawing angle 0) ^ \ - Euler angles around Z, Y^ and X axes 0) 1 -A - general ised coordinate for pylontank fore and aft motion with

    respect to wing

    M a t r i c e s

    A" - matr ix of aerodynamir influence coefficients

    B - matr ix defined by eq. 3 .5-2

    C - generalised stiffness matr ix

    D - vector of drag forces

    Dg - generalised aerodynamic damping matrix

    E - unity matrix

    F - force vector acting on shock absorber

    K - force vector

    L - vector of lift forces

    M - general ised m a s s matr ix

    M - moment vector

    Q - column of general ised forces R - t ransformation matr ix

    S - generalised aerodynamic stiffness matrix

    T - t ransformation matr ix relating vector component in Euler- and in Newton axes sys tems by means of Euler angles

    T2 - matr ix relating (^ and ^Q matr ices by means of Euler angles V - vector

    W - weighting vector

    X. - position vector of any point i

    m - diagonal matr ix of mass elements

    q - column matr ix of generalised coordinates

  • - 6 -

    H

    p - column matrix of Euler angles

    ^r^i - rotation vector of any point i

    (jjj - rotational speed vector of any point i

    - column matrix

    - s q u a r e m a t r i x

    L _i - row m a t r i x

    J - d i agona l m a t r i x

    r n - t r a n s p o s e of m a t r i x

    S u b - a n d S u p e r s c r i p t s

    A - a n t i - s y m m e t r i c

    Ae - a e r o d y n a m i c

    E - quan t i ty r e l a t e d to E u l e r a x e s s y s t e m

    F - fu se l age

    L - left, o r landing

    M - p o s i t i o n of E u i e r a x e s s y s t e m with r e s p e c t to mov ing Newton a x e s s y s t e m

    N - quan t i ty r e l a t e d to Newton a x e s s y s t e m

    R - r i g h t

    S - s y m m e t r i c

    St - s t a b i l i s e r

    T - d i s p l a c e m e n t s

    V - v e r t i c a l s t a b i l i s e r

    W - wing

    WM - w a t e r m e t h a n o l tank

    X - r e l a t e d to p o s i t i o n v e c t o r

    a - quan t i ty r e l a t e d to e a r t h a x e s s y s t e m

    d - d y n a m i c

    e - quan t i ty r e l a t e d to s t r u c t u r a l e l a s t i c i t y

    h - h o r i z o n t a l

    m - eng ine

    n - n o r m a l to shock a b s o r b e r d i r e c t i o n

    o - quan t i ty defining E u l e r o r a i r c r a f t a x e s s y s t e m with r e s p e c t to moving Newton a x e s s y s t e m , o r v a l u e s at t i m e t ^ 0

    p - py lon t anks

    s - in d i r e c t i o n of shock a b s o r b e r

    sk - sk idd ing

    s - s p i n - u p

  • - 7 -

    second impact

    static

    wheel axle

    X. y or z-component of vector

    related to rotation vector

    value of quantity due to shock absorber deflection

    differentiation with respect to time

    absolute value

    incremental value of quantity ( ) quantity ( ) defined with respect to a i rc raf t - , or Euler axes

    system quantity defined with respect to Newton axes system but decomposed along axes paral lel to I 'uler axes system

    A few definitions, occuring in a single paragraph only, a re locally defined. All calculations of the present study have been performed bj' using the metr ic ( m. kg. sec . ) sys tem. By kg is indicated the kilogram force. F o r the introduction of the resu l t s in the aeronautical world, however, it was considered more appropriate to use for descent velocity and landing weight the Anglo-American units of f t /sec and lbs. because all important existing airworthiness requirements are formula-ted in these units .

    SI

    St

    w

    X. y. z

    ( )

    ( )

  • - 8 -

    I N T R O D U C T I O N

    , 1 . 1 , GENERAL

    The aircraft designer is constantly confronted with the problem of designing a i r -craft s t ruc tures for conflicting requirements : requirements of adequate strength and stiffness at low weight and low cost.

    The a r t of the designer is therefore to find the optimal compromise for these con-flicting requirements i. e. to find the lightest and- from point of view of production and maintenance- mos-t economic s t ructure for a given standard of strength and stiffness.

    The keyphrase here is "a given standard of strength and stiffness". A clear defi-nition of this is necessary. Generally spoken, this standard can be formulated as to minimize the chance of damage or any permanent deformation, or of malfuntioning under all anticipated operating conditions during the entire lifetime of the s t ruc tu re .

    Although the ultimate responsibility for sufficient strength and stiffness remains with the designer, the definition of the required standard of strength is not his task. It is the task of the airworthiness authorities to define the required standard of strength and stiffness to which all aircraft designs have to conform,

    This entails that primari ly these authorities define the operating conditions and/or loads to be taken into account when designing s t ruc tu res . In doing so these autho-r i t ies tend to be conservative. This attitude may in some cases resul t in a kind of extra margin of safety hidden in the prescr ibed values of operating conditions and/or loads. This hidden margin of safety is additional to the basic safety factor of 1,5, which is at present almost universally applied in aircraft design procedures .

    According to the U. S. Federa l Aviation Agency (FAA ) regulations this basic factor of safety of 1, 5 is used to provide for the possibility of loads greater than the "limit" loads, ( limit loads being the maximum loads anticipated in normal condi-tions of operation, i. e. loads with a certain small probability of occurence ) and for uncertainties in design. Limit loads multiplied with this factor of safety are called "ult imate" loads.* The s t ructure shall be capable of supporting limit loads without suffering detr imen-tal permanent deformations, and of supporting ultimate loads without failure.

    The anticipated normal operating conditions, ( as used in the definition of limit loads) can only be determined in a s tat is t ical way. This means that there is always a certain amount of uncertainty about their quantitative values. This explaine the tendency of introducing in the Airworthiness Requirements some additional (hidden) margin of safety beyond the basic safety factor of 1,5. e .g . by prescr ibing a too high gust velocity for the flight condition or a too high sinking speed for the landing condition.

    It will be clear however that the values of these additional hidden safety margins

    * This name is in fact not very logical, since - as mentioned - the factor of safety provides not only for the possibility of loads g rea te r than the limit loads but also for uncertainties is design.

  • - 9 -

    can be smal le r if bet ter calculation schemes are used and the possible operating conditions are determined more carefully, Research in these fields is going on continually and thus the prescr ibed operating conditions/loads are in discussion continuously.

    In the quest for more economic s t ruc tures the aircraft designer will desire to lower the requirements when it is possible to prove that existing requirements a re unneces-sar i ly conservative. This possibility exiats . In the past requirements have sometimes heen formulated without a sufficient knowledge of operating conditions and/or into the pa r ame te r s determining the loads.

    When damage does not occur in s t ruc tures designed on the base of such r equ i re -ments it can be said that safety is assured , but it will not be known how much extra loading capacity is hidden in the design, On the other hand, there is a good reason to be cautious.

    Damage may be experienced with new types of s t ruc tures which were designed on the bas is of existing requirenments which have proved their value in the past with ear l ie r generations of designs. Such experience proves that the existing r equ i re -ments do not include correc t ly all physical phenomena involved in the loading case considered. Such situations are the more likely to occur since formal requi re -ments have a tendency to res t unchanged for long periods of t ime and therefore situations a r i s e in which the formal requirements lag severely behind the state of the a r t . This state of affairs exists at this moment for the requirements with respect to loads due to landing impact. As a resul t of the landing impact the s t ruc ture is exposed to loads which increase in very short t ime from zero to their maximum value. The landing impact has therefore a dynamic charac ter , which means that inert ia forces due to the e l a s -tic deformation of the s t ruc ture a r e becoming ra ther important and by this fact the s t r e s s e s in many par ts of the s t ruc ture may become higher than under quas i -s ta t ic conditions, in which the ra te of increase of loads is slow.

    The existing requi rements however a re based on an obsolete state of ar t , in which no dynamic effects were taken in account by the s t r e s s analysts. Conse-quently the values of the initial conditions- the sinking speed in par t icu lar -in the existing requirements had to be ra ther conservatively specified, as a safe-guard against the shortcomings in the state of a r t .

    This safe-guard is not more necessary , since we know that in our time a more accurate dynamic analysis can be performed. A new formulation of landing impact load requirements therefore is badly needed.

    It is the aim of the present study to investigate whether it is possible to formu-late more up to date, yet simple requi rements for the landing impact cases .

    In doing this the study will embrace an extensive investigation into the relative importance of the large number of variables involved in the landing impact problem,sthat i twil l be possible to determine the design loads ( i. e. limit and ultimate loads ) in a more rational way than due to the existing requi rements .

  • -10-

    The study is res t r ic ted to the determination of design loads due to landing impacts on main landing gears . In principle a nose gear has to be treated in the same way, and can be added to the calculation scheme without further complications when three point landings are considered. The problem of determining a complete load spectrum which can be used for a fatigue analysis of the s tructure, is not t reated in the present investigation because a landing gear load spectrum is mainly due to taxi loads. The landing impact loads are only adding a very small percentage of the total num-ber of load variations and therefore can be taken into account in the fatigue spectrum in a ra ther crude way.

    PHYSICAL PHENOMENA INVOLVED IN A LANDING IMPACT

    When an aircraft touches down, the vert ical velocity has to be reduced to zero in a very short time interval . This task is mainly performed by the shock absorbers and the t i r e s . In order to reduce the vert ical deceleration at touch-down both tire and shock absorber act as ( non-linear ) springs,

    At touch-down initially only the t i res are deflected, acting mainly as springs so that a ver t ical ground reaction force develops. When this force has become so large that the amount of preload present in the shock-absorber is exceeded, the shock absorber s t a r t s to deflect, converting part of the impact energy into heat by pushing hydraulic fluid through small orifices and another par t of the energy in potential energy of the compressed air . This damping force, generated by the hydraulic shock absorber, deminishes rapidly toward the end of the s troke.

    The total shock-absorber load which r i ses very rapidly, and levels off after a few hundredths of a second, is t ransmit ted to the aircraf t s t ruc ture . As this s t ructure is elast ic it will deform by this loading and due to the rapid applica-tion of this force, s t ructural vibrations a re excited. This so called "dynamic effect" of a landing impact can increase as well as decrease the max. s t r e s s e s at a cer tain station of the s t ruc ture . The amount by which this happens is ex-pressed by "overshoot-factors", expressing the rat io of loads which a re found by taking into account s t ructural elasticity, and the loads found when the s t r u c -ture is assumed to be completely rigid. F o r modern large aircraft these dyna-mic effects a re most important. In this respect the t ime history of the vert ical landing gear load for a given descent velocity, is of pr ime importance. The time history determines which deformation modes will be excited and to which degree.

    However, these facets of a landing impact a re certainly not the only ones which deserve careful attention. A most important r61e has also the horizontal friction force between t i re and runway. This force is due to the difference in speed between t ire and runway surface at the moment of touch-down. The difference in speed can be resolved in two components. One in the plane of the wheel disk, and one perpendicular to the first one. The first one brings the wheel in a sudden rotation. The duration of this phenomenon is a few hundredths of a second only, the time needed for a complete "spin-up". Spin-up, is the action of rotational acce le ra -

  • - 1 1 -

    tion of the wheel by ground friction forces, and is completed when the circumferential speed of the t i re equals the forward speed of the aircraf t . As with all friction forces the spin-up force is proportional to the coefficient of friction and to the vert ical force acting on the t i r e . The friction coefficient /^x however, is not a constant, but is very much dependent on the "skidding velocity", as shown in the figure 1. 2 -1 , skidding velocity being the diffe-rence between aircraft forward speed and wheel c ircumferent ial speed. The value of/ix var ies between 0,5 and 1,0 onidry surfaces, but decreases very rapidly to very small

    values when the condition of rolling contact between t i re and surfaces, thus zero skidding velocity, is ap-proached. This implies that the initial value and the-refore also the variation of the friction coefficient during the spin-up phase is also dependent on the forward speed of the aircraft at the moment of touch-down.

    Due to the friction force the landing gear is deflected r ea rwards as is indicated in fig. 1. 2-2. But after a

    , . , ,. , while the friction force reduces strongly, due to skidding velocity ^ " ^ ^

    p- 2 2-1 establishment of zero slip condition, and the landing gear springs back. Both the spin-up and the spring-back phenomena introduce severe dynamic effects in the s t ruc ture , due to the short duration of the friction force and the sudden

    Horizontal wing bending and tor-sion especially are excited to ra ther high frequencies by these phenomena. Spin-up and spring-back a re of par t icular importance for the dynamic loading of heavy concentrated masses which are connected to wings such as , for example, engines and pylon tanks. Additional effects and loads, can

    be expected due to a - symmet r ica l aircraft attitudes at the moment of touch-down. In such cases the right (left) side landing gear contacts the ground before the left (right) side one. The impact on the side of the first contact introduces a rolling moment by which the vert ical touch-down velocity on the opposite side may become higher than tlie ver t ical velocity of the a i rc ra f t . Such a touch-down can,for instance, take place when a c ross wind component is present . The lining up with the runway then is ob-tained by such an initial roll angle that the la tera l aerodynamic force is compensated by a weight component. The other possible procedure for c ro s s wind landings is to s t ee r a drift angle in such a way that the result ing aircraft ground speed vector is in the runway direction. Unless a perfect decrab maneuver is performed another asymmetr ic effect occurs which should be taken into account,viz. a touchdown with a certain initial l a t e r a l velocity of the aircraft with respect to the runway^ less than the c ro s s wind velocity. This produces la tera l friction forces on the t i re .

    At the moment of touch-down when the t i r e skids along the runway the friction force //t

  • - 1 2 -

    M k

    (af ter spin-up)

    Fig. 1.2-3

    experiments it follows however that for a yawed rolling t i re this is already t rue for much smal le r values of ip Due to the la teral friction force the landing gear also is deflected lateral ly. However, as there is no spin-up laterally, also the spring-back is absent and therefore also the excitation of high frequency modes is absent. The la te ra l friction force therefore only ex-cites the lower wing bending frequencies. As generally the la te ra l landing gear bending frequencies a re much higher, the la te ra l landing gear elasticity influences the landing impact phenomena to a l e s se r extent than does the fore and aft stiffness of the landing gear in relation to the spin-up phenomena.

    Finally there is sti l l another problem : the vibrations of the elastic aircraft s t ruc ture have not only a direct effect on the s t r e s s e s at different stations of the s t ruc ture , but they also influence the time history of the ver t ical landing gear load, for they convert a part of the impact energy in potential energy of the elastic deformed s t ruc ture . How accura te has to be taken into account this interaction effect ?.

    A quite different facet of a landing impact is the generation of aerodynamic forces on the a i rcraf t due to motions resulting from the landing impact. These forces can con-tribute to the total loading due to landing impact and therefore deserve our attention.

    Of course if all the physical effects mentioned above have to be taken into account, it is not longer possible to describe the landing by two initial conditions only, viz. the sinking- and the forward speed. The landing impact then depends on a number of initial conditions which, to a large extent, a re independent of each other. Also a i rc ra f t la tera l velocity, rolling angular velocity, angle of ro l l , angle of yaw, yawing angular velocity, and pitch attitude ( angle and angular velocity ) should be specified.

    A possibili ty not yet mentioned is the rebound landing. It has to be considered whether such a landing with a partially deflected shock absorber can give r i se to cr i t ical loa-ding c a s e s ,

    1. 3. EXISTING REQUIREMENTS FOR LANDING IMPACT LOAD CASES

    The most important existing requirements a re the American ones for t ransport a i r -craft a s given by the FAA ( Federa l Aviation Agency) in FAR-25, The with respec t to this study relevant part of these requirements are reproduced in Appendix A, They can best be illustrated by refer r ingto fig, 1 ,3-1 . This figure shows a typical time history of ver t ical and horizontal longitudinal wheel loads. Three combinations of these vert ical and horizontal loads, indicated in the fig, as 1, 2 and 3, have to be applied to the wheel axis, and have to be placed in equilibrium with the l inear and angular inertia forces of the aircraft as a whole.

  • -13 -

    J^ig. 1.3-1

    The ver t ical forces have to be derived f rom or checked with, drop tes t s . In these drop tests it may be assumed that at the moment of touch-down the total lift equals the aircraft weight, so that no initial ve r t i ca l accele-ration exis ts . This is in agreement with the available s ta t is t ical data as will be shown in chapter 2.

    The drag is specified by a fixed value in case 2. In case 1 a friction coefficient has to be assumed permitt ing the time required for complete spin-up, to be calculated. A simple integral formula exists which will be derived in chapter 3, for calculation of the spin-up time t . The coefficient of friction rnay be established by con-

    sidering the effects of skidding velocity and t i re p r e s su re . However this coefficient of friction need not be more than 0, 8. In the third case, as is shown in fig. 1 .3-1, the maximum forward acting horizontal force depends very strongly on the dynamic behaviour of the s t ructure and this case can only be defined properly when a complete dynamic analysis is performed. An alternative method would be a drop test on a rotating drum, or with a prerotat ing wheel, in which, however, only the elasticity of the landing gear itself will be simu-lated, and not the elast ici ty of the r e s t of aircraft s t ruc tu re . Both a level attitude and a taildown attitude have to be investigated for these three cases . The forces defined by tlie first and third case have to be applied only to the landing gear and its directly affected attaching s t ruc ture , and to large m a s s items attached to the wing such as external fuel tanks, nacelles etc,

    Forward speed at touch-down has to be varied between VT and 1,25 VT in the Li 1-2

    level attitude, in which V L - = VgQ ( T. A. S. )*, the stalling speed with respect to undisturbed a i r at sea level in standard atmosphere conditions, and V^^ = ^SO'^ ' ' ^ ' *^ '^ the stalling speed at the altitude to be considered under ** ISA + 41 F conditions. F o r taildown attitude forward speed is prescr ibed as V^p. Forward speed only influences the value of the friction coefficient and the time for complete spin-up. The ver t ical force depends on the sinking speed, which, in the American requi rements , is prescr ibed as 10 f t / sec . for the maxiraium landing weight and 6 ft/sec. for the max. take-off weight. The take-off weight case with a reduced sinking speed is defined in o rde r to t reat also the emergency case that an aircraft has to land just after take-off. As the probability of a combination of such an emergency case and an extreme sinking speed is extremely small a reduced sinking speed is prescr ibed. In practice usually the landing weight case is cr i t ical . In addition to these tliree symmetr ica l load cases , there a re two a-symmetr ic ones prescr ibed, which are also indicated in fig, 1. 3 -1 . Only main wheels a r e assumed to be in contact with the ground whilst the aircraft is in the level attitude. The liorizontal forces a re related directly to the vert ical forces and are well defined.

    In addition to the requirement that the undercar r iage- and ai rcraf t s t ruc ture should withstand the given loads, there is also a separate requirement prescribing t e s t s to prove shock absorbing ability of the landing gear under the prescr ibed landing condi-tions, i. e. that neither t i re nor shock absorber will bottom. * T . A . S . = True Air Speed

    ** ISA = International Standard Atmosphere

  • -14-

    The shock absorbing ability must be further demonstrated by a test with an ultimate descent velocity of 12 f t / sec , instead of the normal limit d velocity of 10 f t / sec , ."requiring the landing gear not to fail.

    escent

    RIGHT LANDING LEFT LANDING

    RIGHTLANOING LEFT LANDING

    max-drag force case

    GEAR ^MAX ^ M

    .4K MAX.

    GEAR AX

    .tK MAX.

    / D R A G / / D R A G

    / FORCEy / FORCE r^MAX / ^ " ^ M A X

    LATERAL FORCE max-ver t ical force

    case

    Fig . 1.3-2

    LATERAL FORCE

    'MAX

    .(K MAX. DRAG FORCE

    /

    one wheel landing case

    The present B r i t i s h regulations, the BCAR ( Bri t ish Civil Airworthiness Requirements) , issued by the Air Registration Board, which have been adopted by most commonwealth countries, are based on the same approach as that of the FAA. The principal landing load cases a re indicated in fig. 1. 3-2. Only three different cases are prescr ibed which are all related to the maximum ver t ical force K

    max. to be derived from a drop test for a cer tain sinking speed.

    For the most important case of the landing weight this sinking speed is not a constant but depends on the stalling speed of the aircraft according to the express ion;

    Z^ = 7 100 m . p . h , with straight line variation with V between these two values.

  • -15 -

    Also identical with FAA regulations is the condition that at the moment of touch-down total lift is equal to the aircraft weight,

    The Bri t ish requirement for an ultimate descent velocity test is somewhat less severe than the equivalent FAA requirement , The ARB defines an ultimate descent velocity of 1,18 t imes the limit descent I

    I velocity, as compared with the 1,20 t imes the limit descent velocity of the FAA requi rements ,

    Another important set of requirements a i e the MIL-specs, which must be sa t i s -fied by all mi l i tary aircraf t in the USA. j The one related to ground loads due to landing impact for t ransport planes, viz, MIL-A-8862 dated 18-5-1960, is , however, nearly identical to the FAR requi re-ments .

    The only important differences are a friction coefficient of 0, 55 instead of 0, 8 and the fact that the one wheel landing case is not incorporated in these requi re-ments .

    1.4, THE OBJECTIVES OF THE PRESENT STUDY , .. I

    In par . 1, 2 all the physical effects involved in the landing impact process have been described. All these effects should be taken into account in an analysis aimed I at calculating exact loads i. e. loads due to given initial conditions taking into ac-count all physical effects involved. This \- ould involve a very complicated ana-lys i s . A large number of independent initial conditions then had to be specified. The val-ues of the initial conditions can be derived from measurements during landings with various aircraft types. The stat is t ical propert ies of all the quantities seperately then can be determined. The question remains however which c o m b i n a t i o n of initial | condition has to be specified in order to define cr i t ical design loading cases . The | problem becomes sti l l more complicated by the fact that for different sections of | the s t ruc ture different combinations of initial conditions will be cr i t ica l .

    In principle this problem can be approached in a stat is t ical way. Some of the more recent l i te ra ture ( e. g. ref. 1 ) follow this line of attack. From the stat is t ical pro-per t ies of the individual initial conditions which are assumed to be known, random combinations of these initial conditions can be formed by means of certain s ta t is t i -cal techniques such as Monte Carlo technique or a weighted paramete r method. When these random combinations are applied to the dynamic system describing the a i rcraf t - and landing gear behaviour of a specific aircraft , result ing loads, s t r e s - ' ses and accelerat ions a re obtained in the form of probability distr ibutions. In principle, all information needed for a complete fatigue, analysis then is available, From an abst ract physical point of view such an approach would be the only correc t one. I

    Unfortunately, however, in pract ice this approach does not work in providing j I

    d e s i g n l o a d s ( 1, e, limit loads and/or ultimate loads) . This is due to several i r easons , j The most important one is that the initial conditions defining limit and ultimate loads, have to be extrapolated from the available stat ist ical infornaation since for these j extreme conditions s tat is t ical information is not available, I

    I

  • - 1 6 -

    M o r e o v e r , due to the l im i t ed amount of a v a i l a b l e s t a t i s t i c a l da ta only a few m e a s u -red v a l u e s a r e ava i l ab l e for s e v e r e l a n d i n g s . T h u s the s t a t i s t i c a l a c c u r a c y of the ex t renae ( e x t r a p o l a t e d ) v a l u e s i s even m o r e doubtful . T h i s i s s t i l l m o r e val id for a c o m b i n a t i o n of the va lues which r a i s e e x t r e m e landing i m p a c t l o a d s .

    A f u r t h e r r e a s o n is that the g r e a t amoun t of s t a t i s t i c a l i n fo rma t ion needed i s not yet a v a i l a b l e for a l l r e l e v a n t in i t i a l c o n d i t i o n s . It i s even doubtful w h e t e r t h i s in -f o r m a t i o n e v e r wi l l become a v a i l a b l e , a s a c o n s t a n t upda t ing will be r e q u i r e d for each s u c c e e d i n g g e n e r a t i o n of a i r c r a f t .

    F o r t h e s e r e a s o n s a full s t a t i s t i c a l a p p r o a c h for c a l c u l a t i n g d e s i g n landing l o a d s i s i m p r a c t i c a b l e to u s e a s a b a s i s for p r o v i d i n g d e s i g n load c a s e s . One of the o b -j e c t i v e s of the p r e s e n t i nves t iga t ion t h e r e f o r e is to s tudy w h e t h e r a n o t h e r a p p r o a c h is p o s s i b l e for defining r a t i o n a l r e q u i r e m e n t s for l and ing i m p a c t des ign l o a d s .

    Such an a p p r o a c h has been found p o s s i b l e . It i s b a s e d on a s tudy of the r e l a t i v e i m p o r t a n c e of tlie v a r i o u s in i t i a l c o n d i t i o n s .

    When i t i s p o s s i b l e to d e m o n s t r a t e that s o m e of the in i t i a l cond i t ions e x e r t only a m i n o r in f luence on the l o a d s , it is no l o n g e r n e c e s s a r y to co l l ec t a g r e a t a m o u n t of s t a t i s t i c a l i n fo rma t ion to define t h e s e in i t i a l c o n d i t i o n s .

    It wi l l be shown that th is inf luence for s o m e of the in i t i a l cond i t ions i s so s m a l l that t h e y can be spec i f ied by fixed v a l u e s without too much l o s s of a c c u r a c y , tha t o the r o n e s have to be v a r i e d be tween two l i m i t s in o r d e r to d e t e c t m a x i m u m l o a d s , and tha t r e l i a b l e s t a t i s t i c a l i n fo rma t ion i s only r e q u i r e d for the main in i t i a l c o n -dition ( i . e . the d e s c e n t v e l o c i t y ) . Then the s t a t i s t i c a l t a sk i s s u r m o u n t a b l e and r a t i o n a l r e q u i r e m e n t s can be f o r m u l a t e d in which only the r e a l l y s ign i f ican t p a r a -m e t e r s wi l l have to be p r e s e n t . Howeve r , when a s imp le f o r m u l a t i o n is a i m e d at , it has to be i nves t i ga t ed a l s o how s i m p l e the p h y s i c a l s c h e m a t i s a t i o n of the landing i m p a c t can be , without i m -p a i r i n g the r e q u i r e m e n t of being able to p r e d i c t " t r u e " loads i. e . l oads which a r e p r e d i c t e d with such a d e g r e e of a c c u r a c y that a l l r e l e v a n t p h y s i c a l ef fec ts a r e r e -p r e s e n t e d and a r e such that the d i f f e r e n c e s with the exac t l o a d s , due to the a p p r o x i -m a t i o n s i n t r o d u c e d in the ca l cu l a t i on s c h e m e , a r e of a s e c o n d a r y n a t u r e only and, m o r e o v e r , a r e c o n s e r v a t i v e .

    B e s i d e s the m o r e o r l e s s a c c u r a t e d e s c r i p t i o n of a l l p h y s i c a l p h e n o m e n a involved , t h e r e i s a l s o the ques t ion which me thod of c a l c u l a t i n g s t r e s s e s f rom given e l a s t i c and r i g i d body m o t i o n s i s the m o s t a c c u r a t e one .

    The two m a i n ques t i ons to be a n s w e r e d be fo re a s i m p l e r a t i o n a l r e q u i r e m e n t can be f o r m u l a t e d t h e r e f o r e a r e :

    A) Which v a l u e s and combina t i ons of i n i t i a l cond i t ions have to be p r e s c r i b e d in o r d e r to f o r m u l a t e a se t of des ign l o a d s .

    B) Which d e g r e e of s o p h i s t i c a t i o n i s needed in a landing i m p a c t a n a l y s i s in o r d e r to p r e d i c t " t r u e " l o a d s and s t r e s s e s a s defined above .

    By p r e s e n t i n g c o m p r e h e n s i v e and s y s t e m a t i c a n s w e r s to t h e s e q u e s t i o n s t h i s i n -v e s t i g a t i o n e s t a b l i s h e s the foundation on which a r a t i o n a l s e t of s i m p l e r e q u i r e m e n t s for l a n d i n g i m p a c t des ign l o a d s can be b a s e d .

    M o r e in de t a i l , t he content of the p r e s e n t s tudy can be s u m m a r i z e d a s fo l lows .

  • -17 -

    In chapter 3 a set of equations of motion has been derived allowing to t r ea t pro-perly one point landings with initial la tera l aircraft speed with respect to the ground. Another novelty of this study is that all formulations a r e derived in a very general way without build-in res t r ic t ions such as small displacements and angular rotations allowing l inearisation, so that it would be possible to formulate the complete non-l inear equations of motion. ( Actual computations in the present investigation however, a r e performed for sys tems in which linearisation of rigid body motions and elastic deformation is applied ). Initial conditions to which aircraf t have to be subjected in o rde r to predict real is t ic design loads is the subject of chapter 4, in which the wealth of infor-mation available with regard to initial conditions of landing impacts is c r i t i -cally reviewed. The numerical computations a r e performed for an example aircraft , the Fokker F. 27 Friendship of which it has been shown in chapter 5 that it is very well suited to demonstra te the usefuUness of the proposed methods for defining lan-ding impact design loads.

    The resu l t s of the computations a r e presented in chapter 6. By analysing the resu l t s the relat ive importance of the various initial conditions and the degree of sophistication needed, can be investigated.

    The conclusions of this investigation have to be checked by comparing the calculated resu l t s with measured data during actual landings. This is accom-plished in chapter 7.

    Chapter 8 finally, formulates a proposal for a rat ional set of simple requirements for landing impact design loads. In order to study the consequences for actual a i r -craft designs of these proposed requirements , resul ts of landing calculations accor-ding to the existing and the proposed requirements a re moreover compared with each other. However, such an extensive investigation cannot be performed without a detailed knowledge of what has already been achieved by r e sea rch ca r r i ed out in the past. This investigation therefore s t a r t s in chapter 2 with an extensive l i te ra ture survey.

  • -18 -

    SURVEY OF EXISTING LITERATURE

    The present investigation is aimed at the development of rational requirements for the calculation of loads induced in elastic aircraft s t ruc tures by landing impacts .

    As it is intended that the proposed requirements should be applicable to modern and future aircraft , in this survey most of the l i te ra ture dating from before the second world war, devoted to landing gear types which a re now obsolete, can be omitted.

    However, the survey of l i tera ture devoted to the most commonly applied landing gear system, viz. landing gears equipped with hydraulic-pneumatic shock absorbers should start in the nineteen thir t ies .

    It was during this decade that the hydraulic, pneumatic and hydraulic-pneumatic shock absorbers made their entrance, replacing the old leaf spring and rubber spring undercar r iages .

    Only the hydraulic-pneumatic shock absorbers have survived and therefore only the l i tera ture devoted to this type of undercarr iage will be mentioned.

    Reviewing the l i tera ture , in par t icular attention will be paid to the way in which the dynamic proper t ies of wheel, shock absorber and aircraft s t ruc tures are taken into account, and to the way in which the loads due the landing impacts are calculated.

    Publications from before 1940 devoted to the landing impact phenomenon in general , and to the application of hydraulic-pneumatic shock absorbers in part icular , a re very limited in number and in most cases they are dealing with one part of the subject,or the whole subject is treated in a very simplified way. Typical examples are refs . 2-6.

    Ref, 5 of V. d. Neut and Plantema is a very extensive investigation in the field of symmetr ica l landing impact loads and is devoted in par t icular to the problem of defining cr i t ical loading cases for nosewheel landing gears . The t i re-shock absorber combination of the nosewheel landing gear is represented by a l inear spring, but for the main wheel landing gear the shock absorber damping charac te r i s t ics are represented by velocity squared damping forces . The unsprung mass is neglected while wheel and shock absorber spring charac ter i s t ics a re represented by l inear spr ings . Aircraft t ranslat ion and- pitch a re taken into account. The equations of motion are solved by a numerical step by step method. Simplified force-deflection charac ter i s t ics of the main landing gears are used however in the majority of the calculations.

    Besides defining cr i t ical loading cases for the landing gears also some other p rac t -ical conclusions a re derived. It is found for example that aircraft pitch is unimportant for landing gear loads. Fur ther , the drop test interpretation problem of how to take into account static lift forces, is already recognized.

    One of the few German publications which re la tes to this subject and is pr ior to 1940 is that of Michael ( ref. 7). In this work the linear spring damper system, by which the shock absorber is approxi-mated, is treated extensively. Also cases in which the shock absorber damping is proportional to velocity squared, as well as the case in which the damping consists of dry friction, a r e t reated. The influence of the t i re is touched upon but only briefly. The same approach was adopted by Schlaefke in two publications dated 1943 and 1944 ( refs. 8 and 9). In the first report again the model of the l inear spring damper system is used. In the second it is investigated whether a shock absorber damping proportional

  • -19-

    to velocity squared is a bet ter approxinnation or not . Strangely enough, the author concludes that this is not the case .

    Dating from the same period a re the investigations of Schlaefke ( ref. 10 ) and Kochanowsky ( ref, 11 ) , wherein also the combination of t i re and shock absorber was considered. In both these references it is assumed that the shock absorber beha-viour can be descr ibed by a l inear spr ing-damper sys tem. T i re damping is neglected and a l inear t i re spring charac te r i s t i cs is assumed.

    One of the r e su l t s of Kochanowsky's work was that the unsprung mass of the wheel is of minor importance for the landing gear loads and therefore can be assumed to be zero ,

    This has also been shown by Marquard and Meyer zur Capelle in ref, 12. This publication ( 1943 ) is fundamental in that respect that therein the complete set of equa-tions of motion for the rigid aircraft in a symmetr ica l landing in which the nose wheel touches down after the main gear wheels, is derived.

    Shock absorber damping proportional to velocity squared and shock absorber spring charac te r i s t i c s determined by the polytropic compression of the air taken into account.

    In addition t i re deflection charac te r i s t i c s and the unsprung mass are dealt with in an appropriate manner .

    In a second publication ( ref. 13 ), the investigation is extended to asymmetr ica l landing cases . This advanced analysis of the landing impact phenomenon ( but without spin-up phenomena and aircraf t e least ic i ty) was buried, together with the other German l i te ra ture , under the ruins of the Third Reich. When it was dug up after seven yea r s by Fltlgge ( ref, 50 ) the development in other countries was advanced so far that the German l i te ra ture could not more make any useful contribution.

    The reason for the fact that in German l i terature so much attention was paid to the shock absorber is due to a then existing German requirement which forced the designer to use the drop test data of one single specified test for all the different loading cases . How much this requirement deviated from the physical truth was in fact the question these theoret ical considerat ions attempted to answer,

    In England, Fa i r thorne investigated in 1938 the influence of wing elasticity on landing loads ( ref, 14 ). The landing gear i s represented by a mas s l e s s l inear springdamper system v.-hile the wing elasticity is approximated by a single mass connected to the fuselage by a single spring. It i s concluded that the influence of wing elastici ty on landing gear loads is ra ther smal l .

    Temple summar i ze s in 1944 in ref, 15 the important work performed at the RAE in foregoing years by Lindsay, Thorne and Makovski in the field of landing gear behaviour analysis .

    In this work a pract ica l s tep-by-s tep calculation method was developed for the cal-culation of landing gear forces . A l inear t i re force-deflection diagram is assumed,together with shock absorber charac te r i s t i c s given by velocity squared damping and polytropic compression of the a i r . Also the spin-up phenomena are fully investigated and taken into account. Temple himself extended these investigations to asymmetr ica l landing cases ,

  • -20 -

    The important problem of determining the loads in the elast ic a i rcraf t s t ruc ture due to landing impacts was treated by Williams in 1945 ( ref. 16 ) who assumed the landing gear loads to be known functions of t ime. The loads in the s t ructure a re considered to be the sum of the loads which would be experienced by a rigid s t ruc ture and the loads due to the response motion of the elastic wing. This method of approximately calculating wing loads in elastic s t ruc tures is called the "mode accelerat ion method",

    With these publications of Temple and Williams the r e sea rch in England with regard to the landing impact phenomena was already so far advanced that at that tinne all quest-ions could in principle be answered by analytical means. However, lack of computational facilities prevented the wide spread application of these methods so that an appreciation of the quantitative influence of the various pa rame te r s did not become available, Consequently l a te r Brit ish publications do not give really new information but a r e extensions of ref. 16.

    During the second world war also in the USA much attention was being paid to the landing impact problem. This interest was, to a large extent, due to the experience ob-tained from the, for that t ime, large and fast aircraft .

    The prac t ice revealed that by landing impacts , loads were excited in excess of the loads calculated according to the then existing regulations. The excessive loads could be explained by taking into account the dynanaic loads due to s t ructura l elasticity, a factor neglected by the then existing regulations,

    As is customary in the USA this problem was also investigated experimentally. The mos t important publications devoted to measurements of landing impact phenomena are r e f s . 17-21, The first one of Hootman reports the ineasurements conducted on a large number of a i rcraf t in the years 1937-1942. Ref. 19 descr ibes the extensive landing impact measurements performed on a large bomber. Besides measured accelerat ion and s t r e s s e s in different p a r t s of the s tructure it also contains data with respect to the aircraf t attitude and velocities at the moment of touch down. Ref. 21 descr ibes the extensive measurements on different a i rcraf t , performed by the Air Material Command of the USAF in the yea r s immediately after the second world war,

    With r ega rd to the analytic approach to the problem in the USA,first of all the work of Keller should be mentioned. ( ref. 22 and 23 ). This investigation formulates in an appropriate way the equations of motion of wheel and shock absorber , and calculates the ver t ica l landing gear loads of a rigid aircraft . In ref. 23 it is described how in an elast ic s t ruc ture the accelerat ions have to be calculated when the landing gear forces a re known a s a function of t ime. Rather crudely the spin-up time is assumed to be equal to the time for which the ver t ical force is at its maximum value,

    In the USA special attention has been paid to the problem of calculating the dynamic loads in elast ic s t ruc tures due to landing impacts .

    An example is the publication of Stowell, Houbolt and Schwartz ( ref, 24 ), in which this problem has been solved for a homogeneous pr ismat ic bar which is stopped instan-taneously by a res t ra in t acting in the middle of the bar . As an aircraft wing is not homo-geneous and not pr ismat ic , th is method is not applicable to actual wing s t ruc tu res .

  • - 2 1 -

    A publication of Biot and Bisplinghoff ( ref. 25 ) does not have this shortcoming. It is in this report that the concept of "normal modes" is introduced for the first t ime ever in the l i te ra ture of the aeronautical sciences,

    In this publication the loads acting in the wing a re calculated by means of a sum-mation of basic load distr ibutions, which in turn are the distr ibutions of inertia loads produced by vibration in one of the normal modes. In the same way as the total response follows from a summation of normal modes, each multiplied with flifferent participation factors , the so called generalised coordinates, the total load follows from a summation of these basic load distr ibutions, each multiplied with the momentary values of the r e s -pective generalised coordinates. This method for calculating loads i s known as the "mode summation method".

    However, Biot and Bisplinghoff multiplied these basic load distributions with "dynamic response factors" , i. e. maximum values of response factors for the appropriate normal mode. These maximum values were defined as the maximum value of the envelope of response factors determined from a large number of time h is tor ies of landing gear loads. This procedure was followed in o rder to avoid the calculation of the t ime histories of the generalised coordinates in a specific case ,

    Another method for the calculation of dynamic loads in elast ic s t ructures i s presen-ted by Shou-Ngo-Tu in ref. 26. Assuming known velocity time h is tor ies of the landing gear connection points to the wing, it is shown that the beam response can be described by a superposition of rigid modes and normal modes of the elastic s t ructure with nodal points at the landing gear pick-up. As a consequence the landing gea r connection point t ime history cannot contain components due to s t ruc tura l elasticity, a s the velocity time his tory of this point, which is assumed to be known, has been determined assuming the wing s t ruc ture to be rigid. This contradicts experimental data and therefore this method has not found wide-spread application in prac t ice ,

    Finally, Scanlan formulates once more in ref. 27 the equations of motion of a s t ruc-ture consisting of elements for which beam theory is applicable, which is excited by landing gear forces for which the time his tor ies a re known. This publication duplicates the work of Keller as presented in ref. 23.

    Fur the r r e sea rch in the USA usually is a further development of the work of Biot and Biaplinghoff and is devoted in most cases to checking the accuracy of this method and the validity of its p r e m i s e s .

    The accuracy of resul ts obtained with this method was checked experimentally by Ramberg and Mc Pherson of the NBS ( ref. 28 ), It was found that the use of the "dynamic response factors" was too conservative for design purposes. The neglect of the phase differences between the different normal modes did not cause appreciable e r r o r s in wing root bending moments . For wing tip loads however, this conclusion i s certainly not valid,

    Wasserman ' s group of the USAF Air Material Command has conducted an exJensive investigation in the field of landing loads, proceeding along the lines of Biot and Bispling-hoff. F r o m an experimental investigation ( ref, 21 ) the importance of spin-up phenomena and landing gear elast ici ty was appreciated.

    Simple express ions for the approximate calculation of both ver t ica l and horizontal landing gear force time liistories a re developed, allowing a more accurate determination

  • -22 -

    of landing gear loads than with the "dynamic response factors" of Biot and Bisplinghoff. This is shown in ref. 29 by comparison with measurements on actual aircraft .

    Mc-Brear ty s t r e s s e s in ref. 30 also the importance of the contribution of spin-up phenomena and landing gear and/or wing elasticity by which large horizontal dynamic loads a re excited. This insight was gained from experience with the Lockheed Constellation undercarr iage .

    Woodson extends in ref. 31 the work of Biot and Bisplinghoff to a -symmetr ica l lan-ding cases and determines the dynamic response factors for a number of different given a -symmetr ica l landing gear forcing functions for a very much simplified physical model. He concludes that a - symmet r ica l landing cases can easily lead to higher wing loads as com-pared to symmetr ica l landing cases of the same intensity. Moreover, it is found that the mag-nitude of the dynamic response factors to a certain extent is independent of the forcing function shape,

    A comparison of the accuracy of the different methods for the calculation of landing loads, viz, the mode acceleration method of Williams, the mode summation method of Biot and Bisplinghoff, and the method of Levy (ref. 34), is performed by Ramberg (ref. 32), This is done by comparison of calculations with resu l t s of model drop tes t s . It was found that the mode acceleration method and the method of Levy are super ior to the mode summation method.

    With all these methods of calculating landing loads it is always assumed that the landing gear load time histories are unaffected by wing elasticity. Therefore the calcu-lation could be split in two pa r t s . F i rs t ly the calculation of the time histories of the lan-ding gear loads assuming the s t ructure to be rigid, and secondly the response calculation of the elastic s t ructure to the known external loads.

    Stowell, Houbolt and Batdorf in ref. 33 investigated in which way the problem of determining landing loads could best be divided in two. By using a model consisting of a homogeneous, pr i smat ic bar representing the wing with a concentrated mass in the center, representing the fuselage, and one single centrally mounted mass l e s s spring without damping, represent ing the undercarr iage , comparison with exact analytic solutions was possible. The method generally used, consisting of determining landing gear loads assu-ming the s t ructure rigid, and then calculating the response of, and the loads in, the elast ic s t ruc ture , on these known landing gear loads, proved to give the best correlat ion.

    The interaction between landing gear load and wing elasticity is the subject of an experimental as well as a theoretical investigation by Mc-Pherson, Evans and Levy of the NBS in 1948 ( ref.34 ) . It is found that landing gear loads are always reduced by taking into account wing elasticity. For actual s t ruc tures this reduction can amount to 10% and thus the neglect of this effect is a conservative procedure. It is however doubtful whether these conclusions a re valid in all cases ,

    Within the frame-work of this se r i e s of investigations also the publications of Jenkins and Pancu ( ref. 35 ) and of Poland, Luke and Kahn ( ref. 36 ) should be mentioned, together with ref. 37 from 1950 of Plan and Flomenhoft, summaris ing the work done at the MIT. By this work the conclusion of Mc-Pherson was affirmed. Moreover it was shown once again that the mode accelerat ion method is the most accurate ono of all approx-imate methods for the calculation of landing loads. Also an experimental investigation has been performed with regard to the influence of the aerodynamic forces induced by the landing shock on landing loads in elastic wings. The landing gear of the windtunnel model was r e p r e -sented by a l inear spring. The experiments have shown that the aerodynamic forces genera-ted by the landing shock are negligible with regard to the peak values of the landing loads , but they a re rapidly damping out the vibrations excited by the landing shocks.

  • -23 -

    The influence of wing elasticity on landing gear loads for a configuration for which a large influence of wing elasticity may be exp( cted viz, a high wing aircraft with engines far in front of the wing, is t reated in refs , 38 and 39 of ten Asbroek and Plantema. F rom comparative calculations in which shock absorber loads and spin-up phenomena a r e taken into account in a r igorous way it is deduced that the influence of elasticity on ver t ical landing gear loads is at the most 6%. The influence of landing gear elasticity was also investigated. For the configuration considered here with an unbraced main gear it proved to be r a the r important because the spin-up phenomenon induces fore and aft vibration of the landing gear .

    An extensive survey of landing load l i tera ture up to 1950 can be found in ref. 40. At that time landing gear loads could have been predicted in a ra ther rigorous fashion. This is shown in publications such as ref. 41 of Yorgiades ( 1945),in which a graphical method for predicting landing gear loads is developed,and also theoretically in ref. 38 as well as in ref, 42 of Hurty, both from 1950. In most cases , however, the landing gear loads were taken from resul t s of drop tes ts and subsequen' ly the loads in the elast ic s t ructure due to these known forces had to be calculated. It was known at that time that this could best be done by using the "mode accelerat ion" method of Williams.

    The knowledge as presented by ref. 40 was not res t r ic ted to research insti tutes. Also industry had adopted it, as is shown by publications as e. g. ref. 30 from Lockheed, ref. 35 from Convair, ref. 43 from Chance Vought, ref, 44 from Mc-Donnel and ref. 45 from Douglas.

    Nevertheless , the knowledge that spin-up forces could contribute considerably to the dynamic loads ( refs . 30 and 38), was seldom taken into account in actual cases .

    In actual cases the drop test has an innportant place in calculating landing loads. I.T most cases it provided for all different loading eases the external loads. Therefore it was not suprising that when the NACA started an extensive investigation into the field of landing impact phenomena the reliabil i ty of the drop test was the first subject for research .

    In par t icu lar the question how accurate the static aerodynamic forces acting on the wing at touch down a re represented by a reduced mass in a drop test, a procedure which was then introduced in a i rworthiness regulations, was investigated by Milwitzky and Lindquist ( ref. 46). It was found that this reduced mass method yielded a considerable improvement of accuracy compared with the old method in which the aerodynamic forces were completely neglected. This was also shown by Floor in ref. 47.

    As par t of this same NACA investigation Lindquist in ref. 48 t rea t s in a more general way the influence of static aerodynamic and inert ia forces on resulting landing gear loads.

    Another facet of the NACA investigation is treated in ref. 49 by Yntema and Milwitzky. They investigated analytically with an impulse method the ratio between a-symmet r ica l and symmet r ica l landing loads. The same subject had already been treated as long ago as 1932 by Taub ( ref. 2 ).

    As in modern aircraft s t ruc tures elastic deformations a re more and more pronoun-ced it became more and more important to predict quantitatively the dynamic loads due to landing impacts at an early stage of the design. Since drop tes ts can not be performed before the design attains a ra ther advanced stage, the analytical prediction of landing gear loads should receive more attention."

  • -24-

    In 1951 and 1952 the repor ts of Flgge ( ref. 50 ) and Milwitzky and Cook ( ref. 51 ) were published in which the equations of motion of tyre, wheel and shock absorber were derived. In this respect also refs . 52 and 53 of Walls must be mentioned which describe exper i -mental investigations of the NACA devoted to a bet ter understanding of a i r compression phenomena in shock absorbers . Ref. 51 is without doubt, the most important publication in this field because it descr ibes thorougly the landing impact process and investigates also the quantitative importance of the different phenomena. Therefore it can be judged which simplifications can be introduced in the analysis , without impairing the accuracy.

    These publications do not consider,however,drag loads and spin-up phenomena. On the other hand ref. 54 of Milwitzky, Lindquist and Pot te r and ref. 55 of Pot te r a r e com-pletely devoted to drag loads. In par t icular ref. 54 contains a fundamental and complete description of the physical phenomena connected with the building-up of drag forces at landing impact, and moreover gives resul ts of an experimental investigation. The work of Flgge ( ref. 56 ), is devoted to an analytic t reatment of the spin-up phenomenon.

    Now that the state of the art permitted the accurate analytic determination of the external loads, again the question arose how large is the influence of aircraft elasticity on the loads generated by tyre, wheel and shock absorber . In ref. 57 this problem is treated by Cook and Milwitzky with the help of two different simplified examples. Drag loads are neglected however and wing elasticity is only represented by fundamental wing bending. The resu l t s of this investigation show that in certain cases the influence of wing elasticity on landing gear loads is large . Quantitatively however the resul ts a r e not very reliable, because this influence is analysed in a ra ther crude way as it is assumed that the influence of elasticity is independent of landing gear position, a parameter whicn is varied in this investigation.

    Benthem in ref. 58 is the first investigator who takes into account all relevant landing gear forces and wing elasticity in a rational manner. His investigation is res t r ic ted , however, to a par t icular aircraft and t rea t s only symmetr ica l landing cases . As a com-plete analysis is ra ther complex most subsequent publications are devoted to a par t icular aircraft. An example is ref, 59, This recent publication from Douglas shows the long road, which this type of analysis has gone along from the early nineteen fourties up till the present t ime. Now a complete analytic investigation of loads due to landing impacts can be performed by aid of fast computers . This is also i l lustrated by e. g. ref. 60, to mention only one, demon-strating that this type of analysis has been completely adopted by the industry. In ref. 59 it is proved by comparison with extensive and accurate experimental resul ts , that a com-plete analytic t reatment of landing gear forces yields more accurate resul ts than a calcul ation based upon drop test resul ts . ( Elasticity of the s t ruc ture does not play a significant role in this investigation, a very rigid naval fighter being used for the experiments ).

    Finally, ( in 1967 ) ref. 61 became available, showing the application of this kind of analysis to the very uncommon landing gear design of the X-15. The actual calcu-lation, however, again is res t r ic ted to rigid aircraft symmetr ica l landing cases , without spin-up phenomena of the nose-wheels being taken into account. The equations of motion have been derived for all six rigid body degrees of freedom.

    Ref. 62 is another example of a modern analysis . This reference is devoted to a comparison of analytic and experimental resul ts of landing impact load time his tor ies of a dynamical scaled model of a supersonic transport with delta wing.

  • Since it is now possible to calculate analytically loads due to landing impacts accurately, it is very important to know also more accurately the t i re p roper t i e s , such as the deflection and friction charac te r i s t i c s . Both in England and in the USA full attention has been paid to this subject and a lot of r epor t s have been published. Two summaries of work in this field a re given in refs . 63 and 64, For la ter resul ts r e f s . 54, 65 - 67 should be mentioned,

    A last link in this chain of data needed for the accurate predict ion of loads due to landing impacts , a r e the flight conditions at the moment of touchdown. This invol-ves e. g. ver t ica l velocity, horizontal velocity, angle of pitch, roll and yaw angle, rolling velocity, amount of wing lift and friction coefficients. These data have to be collected in a s ta t is t ica l way. The accumulation of such data is preceding on a large scale. As these data a re also related to the type of aircraft and aircraft configuration, this mate r ia l has to be kept up to date. Fortunately, as will be shown l a t e r in this study, not all these data have the same importance. Many investigations have been performed a l -ready and a summary of investigations up to 1957 regarding conditions at touchdown is given in ref. 68 ; ref. 69 - 75 contain more recent information.

  • -26 -

    THEORY OF LANDING IMPACT LOAD ANALYSIS

    3. 1. INTRODUCTION

    Developing rational calculation methods for predicting loads due to landing impacts r e -quires the development of a set of mathematical equations providing an accurate decr ip-tion of the physical phenomena involved. This is an extremely complex task, The aircraft is a free moving continuous elastic system for which it is already difficult to define a proper schematisation such that displacements and deformations can be defi-ned with a not unreasonable amount of computational effort. Moreover, the external forces acting on this dynamical system a re of a very complex nature . Being dependent in a non-linear way on the initial conditions, these forces moreover a re dependent on aircraft d i s -placements and deformations in a very complicated way. A very important problem for the formulation of a set of equations therefore is a proper definition of an axes system allowing an easy description of the position and motion of any s t ructura l point, as well as the expression of these quantities in a limited number of chosen problem var iables . This is the subject of par . 3. 2. By introducing vector- and matrix notation these relations a re very well suited for computerised calculations. The equations of motion proper are derived from the Lagrange equations. As is derived in par. 3. 3the introduction of the matr ix relat ions derived in par . 3. 2 between absolute displa-cements and generalised coordinates leads automatically to the formulation of ma t r i ces of generalised masses and stiffnesses for the various possible choises of generalised coordinates As part of the formulation of the equations of motion, par . 3. 4 expresses the generalised forces as afunction of the loads acting at the landing gears . When t i r e s of separate landing gears of a single aircraft do not touch down simultaneously a special complication occurs . The moment at which a second set of generalised forces s ta r t s acting at the dynamical system, again, is very much dependent on the motions and deformations the dynamic sys -tem is performing. Also this relation is derived in par . 3. 4. Forces and moments excerted on the landing gear can be derived from the equations of motion of the unsprung mass when t i re forces and inert ia forces of the unsprung mass , a re known. When relations between displacements of unsprung mass and generalised coordina-tes a re known, these quantities can be determined. These relations a r e derived in par . 3. 4, from which it follows that t i re forces and unsprung mass inertia forces a re defined but for one parameter , shock absorber deflection. Shock absorber deflection is defined by the non-linear relationship between shock absorber deflection and shock absorber forces, to whicli par . 3. 5 is devoted. Also non-l inear is the relationship between t ire forces and t i re deflection, being the other subject of par . 3. 5. For vert ical t i re forces however l inearisation can be applied. For horizontal t i re forces this is certainly not allowed. These frictional forces are proportional to the vert ical t i re force,but the friction coefficients a re dependent on wheel motion in a very complicated manner. In par t icular the dependence of friction coefficients on wheel circumferential speed, causing the spin-up phenomenon, has to be mentioned, Besides landing gear forces, also aerodynamic forces due to motions imposed on the a i r -craft by the landing impact, a re acting on the aircraft . The l inearised expressions for these forces are derived in par . 3 .6 . Then the equations of motion are completely formulated. Methods for solving this set of equations are discussed in par 3.. 7. However, the solution obtained is res t r ic ted to t ime his tor ies of generalised coordinates. The different methods available for calculating internal loads in the aircraft s t ructure for given values of generalised coordinates ,are dealt with in par . 3. 8.

  • -27-

    3. 2. THE COORDINATE SYSTEM

    .The motions which an elast ic a i rcraf t is performing after a landing impact, can be divided into two pa r t s , viz.

    the motions the rigid aircraf t is performing with respect to the earth surface. the deformation of cer ta in pa r t s of the aircraft with respect to the rigid body s ta te .

    With regard to the description of motions due to aircraft elasticity it is normal to think of the use of an aircraf t axes system. The axes system is fixed to the aircraft as a rigid body. The motions due to a i rcraf t elasticity can thus conveniently be described with respect to it. The origin of this axes system can in principle be chosen at will. However, when the aircraf t construction is such that wing and fuselage deformation can be described as bending of and torsion about elast ic axes of these pa r t s , it will be convenient to chose the intersect ion of these elast ic axes, or the intersect ion of the perpendicular projection of one elastic axis on the other, as the origin. A different choice of the origin takes into account the fact that the motions of the aircraf t with respec t to the ear th surface also have to be described. These are entirely determined by the speed vector of the rigid aircraft center of gravity and i ts rotation vector . In order to be able to compare directly the motions of a rigid and an elast ic aircraf t it i s therefore preferable to chose the origin of the aircraft axes system in the center of gravity of the rigid aircraf t .

    As aircraft s t ruc tu res in which elastic axes can be defined are exceptions nowadays ra ther that the rule, this last definition will in general be considered the more impor-tant one. Therefore , the present investigation chooses the origin of the aircraf t , in the rigid aircraf t center of gravity.

    The elast ic deformations of any point of the aircraft s t ruc ture have to be defined with respect to this aircraft axes sys tem. There i s , in principle, an infinite number of deformation possibil i t ies of continuous s t ruc tu res . In pract ice this infinite num-ber is being reduced to a smal l number of normalised modes of deformation which conveniently schematize the behaviour of the physical system. The multiplication factors of the normalised modes which define their participation in the deformation of the aircraf t , a re called the g e n e r a l i z e d e l a s t i c c o o r d i n a t e s . The displacements of an a rb i t r a ry point i with respect to the aircraft axes system can be expressed into the generalized elast ic coordinate by,

    3. 2-1 '\'H Lm-rj.\: \^e

    in which A' Ax

    are the displacements of the point i

    is the column matrix of eeneralized elastic coordinates

    and RT is the transformation matr ix .

    E. g . , for a wing for which the linearized deformation can be described by a single bending deformation mode f (y) and a single torsional deformation mode g (y), which

  • -28 -

    are normalized at a certain value at the wing tip, the relation between displacements of a point i of the wing ( z ; 0), and the generalized elast ic coordinates is as follows

    AXj = 0 AY- = 0

    AZ j = q, f (y j ) -q j> f i ^ (y i ) Hence, for this example

    H-Fig. 3. 2-1

    f(yj)-->

  • - 2 9 -

    F i g . 3 . 2-2

    v e c t o r c o m p o n e n t s Uj^, V-, and Wj^ in t h e so ob ta ined new a x e s s y s t e m as fol lows

    U c o s {\) + Vs in ( | ) U V^ = - U s i n 4 + VcoslJ) 3. 2 - 3

    W , W

    T h i s new a x e s s y s t e m X | , Y , , Z j i s now r o t a t e d c l o c k w i s e o v e r an angle 0 a r o u n d tlie Y]^ a x i s .

    The r e l a t i o n s b e t w e e n the c o m p o n e n t s U , , V , , W-^ and U2, V2, W2 a r e o b t a i n e d by c y c l i c p e r m u t a t i o n of 3, 2-3

    W2 = U i s i n 0 -t- Wi c o s 0 U2 = U i cos 0 - Wi s in 0 3. 2-4

    The t h i r d r o t a t i o n i s o v e r an angle 0 a r o u n d the X2 a x e s . The r e l a t i o n b e t w e e n the c o m p o n e n t s U3 , Vg, W3 and U2, V2, W2 a r e ob t a ined aga in by c y c l i c p e r m u t a t i o n

    V

    w

    = Vg = V2 COS 0 -t- W2 s in 0 Wg = -V2 Sin 0 + W2 cos 0 Ug = U2

    3 . 2 - 5

    Af te r s u b s t i t u t i o n of eq . 3 . 2-3 in to 3 . 2-4 and of eq . 3 . 2-4 in to 3 . 2-5 the fo l lowing r e l a t i o n b e t w e e n v e c t o r c o m p o n e n t s in the E u l e r - and in the Newton axes s y s t e m i s ob ta ined

    W V. y

    c o s e cosy sin;^ sine cos^-cos^S.siny cosp

  • Hence [T]-' . [T]' 3.2-10 This result could have been predicted at once from the matr ix theory ( see e. g. ref. 77 ) because T is an orthogonal matr ix for which eq. 3. 2-10 is valid in general . The vector V can be e. g, a vector defining a point i in space. Then

    Ix.} = [T](..] 3.2-11 The vector \ v [ can also represent a rotation or an angular velocity in a point i. Then

    When the Euler angles ^^ , 0 and 0 are so small that l inearization is allowed, eq. 3. 2-6 yields,

    H-1

    4 8

    4;. 1

    -r

    -e 0 1

    3.2-13

    The rotation by which the Newton axes system is t ransformed into the Euler axes system can also be defined by rotation vectors along the axes of the Newton axes system instead of being defined by the three Euler angles. The relations between the Euler angles and rotation vectors ^PQ , ^Q and ^Q- along the X-, Y- and Z -

    X y z axes of the Newton axes system can be derived by decomposing the rotation vectors

    l|) , 0 and 0 along the X, Y and Z axes. ( See fig. 3 . 2 - 3 ) . The vector i^ is entirely along the Z-ax i s . The vector 0 has components 0. cos tj) and - 0. sin tjj along the Y and X axes. The vector 0 has components -i- 0. cos 0 and -0 sin 0 along the X-^ and the Z]^ = Z axes. Finally the component 0. cos 0 can be de-composed along X and Y axes with components 0 cos 0 . cos 4) and 0 cos 0 . sin 4* By adding the different components along the X, Y and Z axes the following relation is

    Fig. 3. 2-3 obtained. cos e -c o s e '

    -sin.e

    cosy sin y

    _s in y COSy

    0

    0 0

    1

    (0] 9 M 9 or t^pj -m[0]

    3.2-14

    which reduces to < tX) ^ '0] e 4^

    when linearization is allowed.

  • -.^1.

    The relat ion between the Euler angles and the rotation vectors along the axes X, Y and Z of the Euler axes sys tem is , according to eq. 3.2-12, 3. 2-10 and 3. 2-14.

    {*^H-'W"H'W-H'NM 3,2-15 Deriving the relation directly and in the same way as eq, 3. 2-14 has been derived, the resul t is

    r.

    ^r

    0 COS gl

    -s in 0

    - s i n jef cos e sin 0 COS e c o s ^ H " K: 0 3,2-16

    Since T2, and in the same way T ' T 2 = T2 . relate vectors along the intermediate positions of the axes to the components of these vectors along the axes of the Newton-or Euler axes sys tems , the same transformation mat r i ces T2 and T2 relate vector components of other quantities in those axis sys tems. Therefore the relations

    W,

    are valid.

    If.] r_i r 1 r 1 T, 0 and CO - T, J

    J L J I J L J L , 3.2-17

    Finally it must also be possible to express the transformation matr ix [ T ] itself in the rotation vector [vp ]

    The projection of the vector component of U and W, in the Newton axes system, on OZ is given by : ( see fig. 3. 2-4 )

    W. cosii) + U.sin'S)^ Oy ' Oy

    The projection of this vector along OZ^ on axis OZ3 of the Euler axis system is given by

    ( W c o s i P o y -I- Usin^Poy ). cos ( Zg O Z ^

    As moreover the projection of V on OZ3 is given by V. sin (ZgOZ^), the final expression for the vector component Wg is obtained by substituting the expression for sin ZgOZ^ and cos ZgOZ^. The resul t is :

    W: / , ' , = [ c o s ^ P . s i n P^ .U_sin^5 ,cos% V+cos^l cos^) .w]

    fty

    The expression for U3 and Vg can be ob-tained by cyclic permutation of the indices X, y and z in 3. 2-18. When linearization is allowed eq. 3, 2-18

    , reduces to

    '-*"^ox^zr/;xao^^ fz , 1

    cos ( Z o O Z ^ ) = 1 , ^ 3 ' \Vco?V^

    1 -ton'^'-Pov ^ s i n { Z Q O Z M = , /r- y-

    3 ' Vtar?^5. cos^Pn ! x y

    Fig . 3, 2-4

    W3 = ^ 0 y U - 9 o ^ V + W

    and in the same way

    Po,

    3,2-18a

    U3 =^02 V - iPoy W + U V3 =^0x W - ^ 0 2 " + V

  • - 32 -

    o r

    U3 V3 W,

    I ^n

    S -^ox

    U vl w

    3.2-18b

    JV which is in accordance with eqs . 3. 2-9, 3. 2-13 and 3. 2-14.

    The origin of the aircraft axes system with respect to the Newton axes system fixed in space is defined by the coordinates XQ , y^ and ZQ. The position of any other point of the aircraf t with respect to the Newton axes is then defined by the matr ix expression.

    W^Hi^ ^K 3.2-19 The position of the same point with respect to a Newton axes system which is moving with a constant speed V^ is given by

    f 1 r T

    3.2-20 [xi}-[T][xi]+[xo|-[vo]'

    \ M[M4 Choosing Vg to be equal to the forward horizontal speed of the aircraft at touch down, Xjy[ r ep resen t s the displacements due to landing forces of the origin of the Euler axis system, with respec t to the origin of the Newton axes sys tem. The p o s i t i o n of a point of the d e f o r m e d aircraft i s defined by adding A^Xij given by eq. 3 .2-1 to [Xj] as given by eq. 3 .2-20.

    [4 X|1 + 'ai][se}+[x} 3,2-21 Decomposed along axis paral le l to the axis of the Euler axis system eq. 3. 2-21 be -comes

    [^^e)-H*NK]^H'lxH] 3.2-22 Hence the total d i s p l a c e m e n t of a point of the deformed aircraft , thereby taking into account that at t = 0 the two axes sys tems coincide and that f Xj| is independent of t ime, is given by

    es and along the X axes

    A X , J -

    3.2-23

    3.2-24

    Upon linearization, that is neglecting products of var iables assumed to remain small , this reduces to

    A X, = 0.-4). e 4J. 0.-0

    -6. 0 . 0 + .''^ IH^Kl 3 . 2 - 2 4 a

  • . 3 3 -

    a n d t h u s [ ^ A X i J - [ A X , J In a slightly different way eq. 3. 2-24a then can be written as

    [AXiJ-[AXi^}-[x] o.2i.-yi -Zj, 0 . xj yj-xi. 0

    e LYJ ^ Nb]

    in which

    and

    Fil^n is ciel defined by I 0 0 0 z j -yi 0 I 0 -?| 0 Xj G O - I yi -5^ i 0 MN

    3 .2 -25

    3 .2 -26

    N- I 0 0 0 Z j . ^ i 0 1 0 -2j 0 Xj O O I - 71 -Xj 0 and

    l - ' 'J-LXoYo.Zo.0.e.4).%.--%J 3 .2 -27 In the same way the r o t a t i o n v e c t o r of a mass element (*Pj] can be wr i t ten as a superposition of the rotation of the rigid aircraft f^ ^o] ^nd the rotation due to e las -tic deformation tip j j

    According to eqs. 3. 2-14,-15,-2 and-12,

    and l ^ j f -

    ^ 1

    3. 2-28

    which in l inearized form become

    or

    in which W-W-NW ;''.]-[[c].[E]f>,j]

    3 . 2 - 2 9

    3. 2-30

    Thus for describing the position of the aircraf t with respect to the ear th a Newton axes system is always necessary . However, for the derivations of the equations of motion of the aircraf t both axes sys tems can still be used. In the field of a i rc ra f t stability and control when the aircraf t i s considered to be a rigid body, usually the equations of motion a re derived with respect to the a i rcraf t - or Euler axes sys t em. This i s because for rigid a i rcraf t with this axes system the m a s s quantities a r e constants . F o r an elast ic aircraft this i s no longer t rue , though the deviations a r e

  • - 3 4-

    second order effects.

    Fo r aeroelast ic investigations therefore the equations of motion are generally derived with respect to a Newton axes system which is moving with the constant speed of the aircraft . With respect to such an axes system the equations of motion are much s impler . Also in this axes system the inert ia quantities deviate from constants with second order effects when the values of all generalized coordinates a re of the f irs t order of smal lness .

    With flutter problems this is always so as in fact then only the stability of the physi-cal system against infinitesimal deformations is considered. With response calcula-tions, such as gust- and landing impact calculations, the generalized coordinates are finite and it has to be checked for any calculation, also the present ones, whether the assumption to neglect second order t e rms , is valid, and thus if l inearization is allowed. Experience thus far indicates that for landing impact calculations l inear i -zation is allowed and therefore will be assumed to hold true for the present invest i-gation. Because in that case all basic mass data are identical whith that for a e roe l a s -tic investigations, for convenience the equations of motion for the present invest i -gation, on landing impacts of elast ic aircraft , a re derived with respect to the same axes system as used for aeroelast ic investigations, i. e. a Newton axes system moving with constant speed, coinciding with the aircraft axes system at t = 0.

    THE EQUATIONS OF MOTION

    The equations of motion of a dynamical system with a finite number of degrees of freedom can be derived most conveniently from the Lagrange equations

    d t 'li f^l 'li 3 .3-1 in which

    E = kinetic energy of the system relative to the chosen reference frame U = potential energy Q; = generalized forces, i. e, the work done by external forces when a unit dis-

    placement i s performed by the i th generalized coordinate qj = generalized coordinates, i. e, independent coordinates giving the possible

    displacements of the dynamical sys tem.

    The kinetic energy E is of a system of n discrete masses m, is

    E= I [i(xV y^ -f i') m + u)^ ^ l^ + i U)^ , ly + 1 lx)\ |^ 1 3. 3 -2 in which the coordinates x, y and z and the angular velocities tO . ) . and ijj have to be expressed in the generalized coordinates q.. These generalized coordi-nates qj^ can, according to the choice made in par . 3. 2, be split in the generalized elastic coordinates q and the 6 cyclic coordinates viz, the 3 linear coordinates of

  • -35-

    the center of gravity together with the three Euler angles defining the position of the aircraft reference frame in space with respect to the moving Newton axis system.

    In matr ix notation the contribution to the total kinetic energy of a single mass ele-ment nij^ of the s t ruc ture can be written .;is

    dE-T^i(L>

  • - 3 G -

    It depends on the choice of the generalized coordinates q which form the submatr ix C^ will take.

    When the deformation modes to wjiich the elast ic generalized coordinates apply a re defined as the displacements and rotations of a finite number of segments of the s t ruc ture , the matrix Cj^ is the stiffness influence coefficient matr ix which is the inverse of the flexibility influence coefficient matr ix .

    When the amplitudes q of uncoupled deformation modes f of cer ta in pa r t s of the s t ruc ture , e, g, wing or fuselage, between which no elastic coupling exis ts , a re chosen as the elastic generalized coordinates, the generalized stiffness matr ix is a diagonal mat r ix of which the diagonal elements can be determined as follows : Assume the deformation q f to be the only possible deformation. Eq. 3 .3-11 than takes the form

    {t ^ n i i f ' ( i ) j S e + C c , g - 0 3.3-12 from which can be derived

    z n ? . C - + V Y_ mj f (