LINEAR AEROELASTIC SCALING OF A JOINED WINGAIRCRAFT

Tiago José Fernandes Pires

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Afzal Suleman

Examination CommitteeChairperson: Prof. Fernando LauSupervisor: Prof. Afzal SulemanMember of the Committee: Dr. José Vale

July 2014

Abstract

Previous work has shown the possibility of aeroelastic scaling of simple structures through thematching of non-dimensional frequencies and mode shapes between full-scale and scaled models.In this work a methodology for the aeroelastic scaling of flying structures is developed andverified. Two test subjects are chosen, a simple half-span rectangular wingbox and lastly thismethodology will be applied to Boeing’s Joined Wing SensorCraft. The matching of these non-dimensional frequencies and mode shapes is performed in two steps. First it is optimized thestiffness matching of the structure through static analysis, the structure is defined this way.Then, the mass distribution is optimized through modal analysis while varying non-structuralpoint masses added to the already defined structure. The verification of the aeroelastic scalingis performed through a flutter analysis to both full-scale and scaled models and compare theresulting flutter speed and frequency.

The Wingbox aeroelastically scaled model properly catched the target flutter velocity, fre-quency and mode shape. For the Joined Wing SensorCraft, even though the scaled modelproperly catched the target flutter frequency and mode shape, there was a noticeable error interms of flutter velocity.

Structural similarity between scaled and full scale models turned out to be the most criticalaspect when aeroelastically scaling a prototype.

Keywords: Aeroelasticity, aeroelastic, scaling, Joined Wing.

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Resumo

Trabalhos anteriores mostraram a possibilidade de escalonamento aeroelástico de estruturas sim-ples através da correspondência de frequências e modos de vibração adimensionais entre modelosà escala real e escalados. Neste trabalho uma metodologia para o escalonamento aeroelásticode veículos voadores é desenvolvida e verificada. Dois casos-teste são escolhidos, uma primeiraabordagem é feita a uma asa retangular simples e, por último esta metodologia será aplicada aoBoeing Joined Wing SensorCraft. A correspondência das frequências adimensionais e modos devibração é realizada em duas etapas. Em primeiro lugar, é optimizada a rigidez da estruturapor meio de análises estáticas, a estrutura é definida desta forma. Em seguida, a distribuiçãode massa é optimizada através de análise modal enquanto variadas massas não-estruturais sãoacrescentadas à estrutura já definida. A verificação do escalonamento aeroelástico é realizadaatravés de uma análise de flutter tanto ao modelo à escala real como aos modelos escalonadoscomparando a velocidade, modo de vibração e frequência de flutter resultantes.

A asa retangular simples capturou adequadamente a velocidade e a frequência de flutter,assim como o respectivo modo de instabilidade. Para o Joined Wing SensorCraft, o modeloescalonado apanhou corretamente a frequência de flutter alvo e o modo de instabilidade mashouve, no entanto, uma notável diferença em termos de velocidade de flutter.

Conclui-se que a semelhança estrutural entre o modelo à escala e o real é o aspecto maiscrítico aquando do escalonamento aeroelástico de protótipos.

Palavras Chave: Aeroelasticidade, aeroelástico, escalonamento, Joined Wing.

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Acknowledgements

First I would like to thank my advisor, Dr. Afzal Suleman, for his support throughout the entiretime while elaborating this work and also for the opportunity of making it abroad, it was a greatexperience. I would also like to thank to the University of Victoria staff working at the CfAR,Jenner, Kurt, Kabir, Jeff, Gord and Bariş for their advices and all the good times we spent inEOW148 and Sidney.

Secondly, I have to thank Professor Aguilar Madeira from the department of mechanicalengineering in Instituto Superior Técnico (DEM-IST) for his help with advice and resources forthe JWSC optimization, it was of paramount importance for the conclusion of this work.

I would also like to thank my friends in both Portugal and Canada for helping me spend myfree time in all of the best possible ways and making this an unforgettable experience.

Finally I have to thank my parents and brother for their constant love and support throughevery step of this work. It would simply not have been possible without their help.

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Nomenclature

Symbol or Abbreviation Description(...)w Subscript related to the full scale model(...)m Subscript related to the scaled model[...]T Transpose¯(...) Non-dimensional variable¨(...) Second temporal derivative∗∗

(...) Second derivative with respect to the non-dimensional time

b Wing spanB.C. Boundary Conditionsc Wing chordCfAR Center for Aerospace Research - UVicCFRP Carbon Fiber Reinforced PolymerC.G. Center of gravityDOF Degrees Of Freedomdω 1st vibration mode frequency absolute differenceE Young’s Modulusei Relative error between target and optimized i vari-

ableei Average relative error between target and optimized

i arrayEOM Equations Of MotionFE Finite Elementff Flutter frequencyFr Froude Numberg Gravitational accelerationHALE High Altitude, Long EnduranceJWSC Joined Wing SensorCraftκ reduced frequencyM Mach numberMAXi Maximum absolute value of array im Reference massq Dynamic pressureRPV Remotely Piloted VehicleRe Reynolds numberRMSD Root Mean Square DeviationRMSDφ Root Mean Square Deviation for the mode shapes

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RMSDω Root Mean Square Deviation for the frequenciesRi Rotation amongst the ith coordinateS Reference areaSSL Standard Sea Level atmospheric conditionsUV ic University of Victoria, BC, CanadaUi Displacement amongst the ith coordinateV Reference velocityVf Flutter velocityλl Length ratioλV Velocity ratioλω Frequency ratioλρ Air density ratioλφ Mode shape ratioλi Inertia ratioλF Force ratioν Mass ratioρ Densityτ Non-dimensional timeυ Poisson ratioω Reference frequency[K]

Stiffness matrix[M]

Mass matrix

{x} Vector of elastic deformations[A1

],[A2

]and

[A3

]Aerodynamic matrices in the time domain

{ag} Vector of gravitational accelerations[A] Aerodynamic matrix in the frequency domain[φ] Matrix of mode shapes[η] Vector of modal coordinates[Am] Aerodynamic matrix in the frequency domain as

function of the mode shapes〈...〉 Diagonal matrix[...]−1 Inverse matrix

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Contents

1 Introduction 11.1 Boeing Joined Wing SensorCraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Aeroelasticity and Scaling 42.1 Aeroelastic equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Aeroelastic Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Calculating the scaling parameters for the JWSC . . . . . . . . . . . . . . 82.2.2 Reynolds and Mach number . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Wingbox aeroelastic scaling 123.1 Wingbox characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.1 Full-scale model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Scaled model & scaling parameters . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Scaling methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Optimization procedure and constraints . . . . . . . . . . . . . . . . . . . 16

3.3 Wingbox scaling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 Scaled modal response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Aeroelastic response verification . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Joined Wing SensorCraft aeroelastic scaling 274.1 Scaled model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Scaling methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Optimization procedure and constraints . . . . . . . . . . . . . . . . . . . 294.3 Joined Wing SensorCraft scaling results . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Scaled modal response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.2 Aeroelastic response verification . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusion / Future work. 425.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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List of Figures

1.1 Flowchart of the adopted aeroelastic scaling procedure. . . . . . . . . . . . . . . . 21.2 Representation of the Boeing’s Joined Wing SensorCraft. . . . . . . . . . . . . . . 3

2.1 Representation of the forced transition according to Gibbings criterium, the areasin light-blue represent flow separation. . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Wingbox FE model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Wingbox section layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Pure bending load case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Pure torsion load case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Iterative process for the stiffness matching of the Wingbox. . . . . . . . . . . . . 173.6 Point masses added to the stiffness matched structure. . . . . . . . . . . . . . . . 173.7 Iterative process for the mass distribution matching of the Wingbox. . . . . . . . 183.8 First vibration mode shape comparison, eφ1 = 2.57%. . . . . . . . . . . . . . . . . 203.9 Second vibration mode shape comparison, eφ2 = 2.35%. . . . . . . . . . . . . . . 203.10 Third vibration mode shape comparison, eφ3 = 3.26%. . . . . . . . . . . . . . . . 203.11 Fourth vibration mode shape comparison, eφ4 = 7.50%. . . . . . . . . . . . . . . . 203.12 Fifth vibration mode shape comparison, eφ5 = 7.68%. . . . . . . . . . . . . . . . . 213.13 Sixth vibration mode shape comparison, eφ6 = 5.12%. . . . . . . . . . . . . . . . 213.14 Seventh vibration mode shape comparison, eφ7 = 7.00%. . . . . . . . . . . . . . . 213.15 Eighth vibration mode shape comparison, eφ8 = 12.56%. . . . . . . . . . . . . . . 213.16 Ninth vibration mode shape comparison, eφ9 = 21.80%. . . . . . . . . . . . . . . 223.17 Tenth vibration mode shape comparison, eφ10 = 39.60%. . . . . . . . . . . . . . . 223.18 Full-scale model flutter analysis (g-method) - Air velocity versus Damping. . . . . 233.19 Full-scale model flutter analysis (g-method) - Air velocity versus Frequency. . . . 233.20 Full-scale model flutter instability vibration mode with several time steps. . . . . 243.21 Full-scale model flutter analysis (g-method) - Air velocity versus Damping. . . . . 253.22 Full-scale model flutter analysis (g-method) - Air velocity versus Frequency. . . . 253.23 Scaled model flutter instability vibration mode with several time steps. . . . . . . 26

4.1 Final version of the JWSC internal structure FE model. . . . . . . . . . . . . . . 28

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4.2 Representation of the eight different load cases and B.C. used for the stiffnessmatching of the JWSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Point masses added to the structure in order to perform the scaled modal responseoptimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Seventh vibration mode shape comparison, eφ7 = 3.89%. . . . . . . . . . . . . . . 344.5 Eighth vibration mode shape comparison, eφ8 = 3.95%. . . . . . . . . . . . . . . . 354.6 Ninth vibration mode shape comparison, eφ9 = 4.18%. . . . . . . . . . . . . . . . 354.7 Tenth vibration mode shape comparison, eφ10 = 5.78%. . . . . . . . . . . . . . . . 354.8 Eleventh vibration mode shape comparison - switched with the twelfth. . . . . . . 354.9 Twelfth vibration mode shape comparison - switched with the eleventh. . . . . . 364.10 Thirteenth vibration mode shape comparison, eφ13 = 14.72%. . . . . . . . . . . . 364.11 Fourteenth vibration mode shape comparison, eφ14 = 17.91%. . . . . . . . . . . . 364.12 Fifteenth vibration mode shape comparison - switched with the sixteenth. . . . . 364.13 Sixteenth vibration mode shape comparison - switched with the fifteenth. . . . . 374.14 Full-scale model flutter analysis (g-method) - Air velocity versus Damping. . . . . 384.15 Full-scale model flutter analysis (g-method) - Air velocity versus Frequency. . . . 384.16 Full-scale model flutter instability vibration mode with several "time steps". . . . 394.17 Scaled model flutter analysis (g-method) - Air velocity versus Damping. . . . . . 404.18 Scaled model flutter analysis (g-method) - Air velocity versus Frequency. . . . . . 404.19 Scaled model flutter instability vibration mode with several "time steps". . . . . 41

5.1 Flight testing of the geometrically scaled JWSC in Alberta in October 15, 2011.Courtesy of Jenner Richards, CfAR (UVic). . . . . . . . . . . . . . . . . . . . . . 43

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List of Tables

1.1 JWSC physical and flight properties. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Defined scaling ratios for the JWSC. . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Aluminum mechanical properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Wingbox characteristic dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Wingbox full-scale modal analysis results. . . . . . . . . . . . . . . . . . . . . . . 143.4 Defined scaling ratios for the Wingbox. . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Scaled Wingbox target frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Matching of the modal response with 6 modes. . . . . . . . . . . . . . . . . . . . 193.7 Matching of the modal response with all 10 modes. . . . . . . . . . . . . . . . . . 193.8 Flutter response characteristics for the full-scale model as obtained through the

g-method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.9 Target values for the flutter analysis of the scaled model. . . . . . . . . . . . . . . 243.10 Flutter response characteristics for the scaled model as obtained through the g-

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.11 Overview of the aeroelastic response matching for the Wingbox. . . . . . . . . . . 26

4.1 CFRP mechanical properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Optimal values for the objectives in the stiffness matching optimization procedure

for the JWSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Matching of the modal response with 10 modes for the JWSC. . . . . . . . . . . . 344.4 Flutter response characteristics for the full-scale model as obtained through the

g-method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5 Full scale (non-scaled) flutter characteristics of the JWSC. . . . . . . . . . . . . . 394.6 Flutter response characteristics for the scaled model as obtained through the g-

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.7 Overview of the aeroelastic response matching for the JWSC. . . . . . . . . . . . 41

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1 | Introduction

The flight testing of scaled remotely piloted vehicles (RPV’s) have been an increasingly inves-tigated option in order to reduce time, costs and risk related to the flight testing of prototypes.The concept of aeroelastic scaling has been used since the early 60’s and is based on the premisethat the scaled model will present the same scaled aeroelastic response as the full scale targetvehicle.

The basis of all scaling theories is the non-dimensionalization of the system’s governing equa-tions of motion (EOM) according to Buckingham’s π-theorem [1]. In the early 60’s, Bisplinghoffet al. first applied this scaling concept to the aeroelastic EOM in its Principles of Aeroelastic-ity [2] where he emphasized the importance of matching the scaled stiffness and mass betweenfull-scale and scaled models but with not much in-depth investigation on the different scalingparameters and their influence on the scaled model aeroelastic response. It was only in the late70’s that Wolowicz wrote Similitude Requirements and Scaling Relationships as Applied to ModelTesting [3], an in-depth study on the influence of the different scaling parameters on the scaledaeroelastic response of the scaled model. Following these pioneers, several other authors madea big effort on adding control laws to the classical aeroelastic scaling theory which includedservoelastic effects into the problem. With further developments into finite element analysisand computational tools, optimization of FE models became a possibility and the path towardsaeroelastic scaling procedures converged.

Lately, there has been an increasing effort from the scientific community on the validationof different methodologies and assumptions used to aeroelastically scale a model through FEmodels optimization procedures. Pereira et al. [4] optimized both stiffness and mass distribu-tions through the same optimization routine in order to match the scaled natural frequenciesof the full-scale model. Richards et al. [5](2010) and Eger et al. [6] [7] (2013) used two sepa-rate optimization routines in order to efficiently optimize the stiffness and mass distributions ofthe scaled model while matching the scaled natural frequencies and the non-dimensional modeshapes of the full-scale model with fairly good results. Bond et al. [8] (2012) added non-lineareffects (buckling eigenvalues and mode shapes) to the two-step optimization routine previouslystated however, the non-linear static response only matched up to 60% of the buckling load eventhough she proved the necessity of matching non-dimensional mode shapes in addition to thescaled natural frequencies in order to have a matching scaled aeroelastic response. Finally, Ric-

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ciardi [9] further developed Bond’s research producing methods that closely replicate the target’snon-linear aeroelastic behavior while identifying new sources of local optima.

This work will also be performed using the two-step finite element model optimization routinein order to properly aeroelastically scale a simple rectangular half-span Wingbox and finally, theBoeing’s Joined Wing SensorCraft. A flutter analysis will be performed to both case studies inorder to verify the matching critical aeroelastic response i.e. flutter velocity and frequency, aswell as the instability mode shape. The scaling procedure flowchart is represented in Figure 1.1.

Figure 1.1: Flowchart of the adopted aeroelastic scaling procedure.

Must be noticed that the type and amount of load cases included for the stiffness matchingoptimization routine will be different from case to case, the same to the amount of mode-shapesto be included in the mass distribution optimization routine, depending on the characteristics ofthe problem.

1.1 Boeing Joined Wing SensorCraft

The Joined Wing is Boeing’s conceptual approach to the next generation of unmanned aerialvehicles known as SensorCraft. SensorCraft configurations are categorized as HALE (High Alti-tude, Long Endurance) aircrafts and their main purpose is to perform intelligence, surveillanceand reconnaissance of areas of interest. Originally proposed by Wolkovich [10] in 1985, the joinedwing configuration has numerous potential benefits over conventional designs such as the reduc-tion of the wing+tail structural weight and a lower induced drag. Also, by having 2 forwardand 2 aft wings, the Joined Wing SensorCraft has an advantage over the remaining conceptualapproaches for it provides an unobstructed 360 degree sensor field-of-vision capability when sen-sors are mounted in all four wing panels. Several authors have been studying the benefits ofusing load bearing antenna arrays with structural importance for the airplane’s integrity [11] asto save weight and add stiffness. Further research and development has shown that this designtends to develop geometric nonlinearities though. HALE aircrafts are usually characterized byhaving slender wings (high aspect ratio) which are very flexible and under heavy loads this highflexibility and aspect ratios will develop large deflections [12] [13] that linear theories struggle to

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represent also, the aft wing is prone to Buckling. These large static deflections can change thenatural frequencies of the wing which, in turn, can produce significant changes to the aircraft’saeroelastic behavior. This behavior can be accounted for only by using a rigorous nonlinearaeroelastic analysis [14] [15]. However, these geometric nonlinearities and their effect on theresults represent a whole field of studies by themselves and will not take part in this thesis, onlythe linear aspect of the aeroelastic scaling procedure will be analysed.

Figure 1.2 represents the full-scale JWSC for a better understanding of its geometry, itsphysical and flight properties are also presented in Table 1.1.

Figure 1.2: Representation of the Boeing’s Joined Wing SensorCraft.

JWSC property ValueWing span (b) 45.71 [m]

Overall length 31.40 [m]

Overall height 7.92 [m]

Design TOGW 70472.40 [kg]

Design fuel load 34019.40 [kg]

Cruise Mach number 0.8

Maximum ferry range 16000 [nm]

Take-off field length <2438 [m]

Maximum altitude 21945 [m]

Maximum endurance 32 [hours]

On-station time 20 [hours]

Table 1.1: JWSC physical and flight properties.

The main objective of this work is to develop a 5 meter span RPV that has the same scaledcritical aeroelastic response that of the full scale i.e. flutter speed and frequency. The method-ology used for the scaling procedure is a variation of the one used by Richards et al. [5], Bondet al. [8] and Eger et al. [6] and some other authors and relies on the structural optimization ofa finite element model in order to match the scaled natural frequencies and mode shapes of thefull scale model, as previously shown in Figure 1.1, as to aeroelastically scale it.

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2 | Aeroelasticity and Scaling

Aeroelasticity is the science that studies the interactions between aerodynamic, elastic andinertial forces acting on a elastic structure in a fluid medium. In the early stages of aviationaeroelasticity wasn’t a design factor for the aircraft structure due to their low operation speed.However, as the airplane’s speed increased with little or no increase in the load requirements andin the absence of rational stiffness criteria for the design, airplane designers came across a widevariety of problems that we now characterize as aeroelastic phenomena. These phenomena canbe categorized as static aeroelastic phenomena where we find control system reversal, divergenceand load distribution and as dynamic aeroelastic phenomena e.g. flutter, buffeting and dynamicresponse.

Even though there are numerous areas of study under static or dynamic Aeroelasticity, thiswork will focus in the flutter characteristics of the test subjects in order to verify aeroelasticsimilitude.

Flutter is one of the most important aeroelastic phenomena and also one of the most difficultto predict. It occurs when two (or more) natural vibration modes of the structure couple due toaerodynamic forces resulting in an unstable self-excited vibration where the structure extractsenergy from the air stream and usually results in catastrophic structural failure. NowadaysFlutter speed/frequency is a necessary flight characteristic for airplane certification for it providesan estimation for the airplane’s maximum operational speed, often designated as dive speed.

2.1 Aeroelastic equations of motion

The system of equations that describe the motion for the dynamic analysis of linear aeroelasticphenomena in the time domain is the following:[

M]{

x}

+[K]{

x}

=[A1

]{x}

+[A2

]{x}

+[A3

]{x}

+[M]{

ag

}(2.1.1)

where {x} is the column vector of the elastic deformations, [M ] is the mass matrix, [K] is thestiffness matrix, [A1], [A2] and [A3] are the aerodynamic matrices and {ag} is a column vectorwith the gravitational accelerations for each degree of freedom. The dot over the generalizedcoordinates, (), represents the derivative with respect to time - double dot represents the secondtemporal derivative, as well. Transforming 2.1.1 from time to frequency domain, we get a system

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of equations where equivalence and the scaling (non-dimensionalization) are easier to see.[M]{

x}

+[K]{

x}

=[A]{

x}

+[M]{

ag

}(2.1.2)

where [A] is the aerodynamic matrix in the frequency domain composed of complex numbersrepresenting the phase and magnitude of the aerodynamic response. To be noted that the lastelement on this equation, the gravitational influence, is not a necessary component for the scalingof the EOM if there is no significant coupling between flight dynamics and aeroelasticity but isrequired to design a scaled model with correct flight dynamic scaling therefore will be used.

2.2 Aeroelastic Scaling

As previously said, the testing of scaled wind-tunnel or RPV models has often been morerewarding than the equivalent efforts using analytical techniques or full-scale airplanes. Thegeneralised scaling procedure is based on the non-dimensionalization of the system’s governingequations of motion using the well known Buckingham π-theorem [1]. This theorem states thatan equation with n physical variables and k primary quantities can be rewritten using p = n− knon-dimensional terms. Matching these non-dimensional terms between scaled and full-scalemodels will provide a scaled model with physical similitude to the full-scale model.

The process for the non-dimensionalization of the linear aeroelastic EOM is outlined byBisplinghoff et al. in Aeroelasticity [2], some practical examples are given as well. Categorizingthe current problem as a mechanics problem and neglecting compressibility effects, the relevantprimary quantities are mass, length and time. Note that there is no fundamental reason forchoosing these as primary quantities, but rather it is the adopted system of measurement andproblem that makes them a convenient choice.

In order to start the non-dimensionalization of the EOM, we’ll take into account the linearaerodynamic theory and represent the aerodynamic coefficient matrix as a function of threenon-dimensional parameters, the reduced frequency (κ = ωb

V ), the Mach number (M) and theReynolds number (Re) as: [

A]

=1

2ρV 2S

b

[A(κ,M,Re)

](2.2.1)

being ρ the air density, V the airspeed, S the surface area and [A] is the non-dimensionalaerodynamic matrix in the frequency domain. Substituting 2.2.1 in 2.1.2, we get the followingsystem: [

M]{

x}

+[K]{

x}

=1

2ρV 2︸ ︷︷ ︸q

S

b

[A(κ,M,Re)

]{x}

+[M]{

ag

}(2.2.2)

The next step is non-dimensionalizing the deformation vector {x} using the reference lengthb, the half-span. The non-dimensional deformations, {x}, are obtained applying x = x

b and

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substituting in 2.2.2, as:[M]{

¨x}

+[K]{

x}

= qS

b

[A(κ,M,Re)

]{x}

+[M]{

ag

}(2.2.3)

We can now write the non-dimensional deformations,{x}, in terms of non-dimensional modeshapes. {

x}

=[φ]{

η}

(2.2.4)

where [φ] is the matrix of the non-dimensional mode-shapes and {η} are the modal coordinates.Substituting 2.2.4 in 2.2.3 and pre-multiplying by [φ]T , we manage to diagonalize the mass andstiffness matrices according to the bi-orthogonality property and get the diagonal modal massmatrix, 〈mi〉,and the diagonal modal stiffness matrix, 〈ki〉.

〈mi〉 = [φ]T [M ] [φ] (2.2.5)

〈ki〉 = [φ]T [K] [φ] = 〈miω2i 〉 (2.2.6)

being 〈ω〉 the diagonal matrix of modal frequencies. The complex aerodynamic coefficient matrixis also modified trough this multiplication and is now a function of the mode shapes,

[Am]

= [φ]T[A]

[φ] (2.2.7)

Substituting 2.2.5, 2.2.6 and 2.2.7 in 2.2.3, the following relationship is obtained:

〈mi〉{η}+ 〈ki〉{η} = qS

b

[Am(φ, κ,M,Re)

]{η}+ 〈mi〉 [φ]−1 {ag} (2.2.8)

this equation can be rewritten taking into account the second equality in 2.2.6, making thestiffness term also a function of the modal mass matrix and taking out the dimension of thegravitational acceleration vector, ag.

〈mi〉{η}+ 〈miω2i 〉{η} = q

S

b

[Am(φ, κ,M,Re)

]{η}+

g

b〈mi〉 [φ]−1 {ag} (2.2.9)

Let us now divide by a reference mass, in this case we’ll use m1 = 1, the modal mass of thefirst natural vibration mode. The resulting system is:

〈mi〉{η}+ 〈miω2i 〉{η} =

qS

bm1

[Am(φ, κ,M,Re)

]{η}+

g

b〈mi〉 [φ]−1 {g} (2.2.10)

where 〈mi〉 is the diagonal matrix of the non-dimensional masses and ωi are the modal frequen-cies. In order to finalize, we can non-dimensionalize the time through dividing by a referencefrequency, in this case we’ll use the modal frequency of the first natural vibration mode, ω1 = 1.

〈mi〉{∗∗η }+ 〈miω

2i 〉{η} =

qS

bm1ω21

[Am(φ, κ,M,Re)

]{η}+

g

bω21

〈mi〉 [φ]−1 {ag} (2.2.11)

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(∗∗) denotes the second derivative with respect to the non-dimensional time, τ = tω1. Expand-ing the dynamic pressure, q = 1

2ρV2 and multiplying the aerodynamic term by b2/b2 and the

gravitational term by bV 2/bV 2, we can rewrite the system as:

〈mi〉{∗∗η }+ 〈miω

2i 〉{η} =

1

2

ρSb

m1︸︷︷︸µ1

V 2

ω21b

2︸ ︷︷ ︸1/κ21

[Am(φ, κ,M,Re)

]{η}+ 〈mi〉 [φ]−1 {ag}

gb

V 2︸︷︷︸1/Fr2

V 2

ω21b

2︸ ︷︷ ︸1/κ21

(2.2.12)

As represented, three non-dimensional parameters can be extracted from this formulation, themass ratio, µi, for the first mode, the reduced frequency, κi, for the first mode as well andthe Froude number, Fr = V√

bg. The Froude number is a serious constraint for a system with

aeroelastic-flight mechanic coupling. Such coupling is inherent in any highly flexible aircraftthat is the case of the JWSC. With this last step we’re now facing a totally non-dimensionalset of EOM, let us rewrite the system of equations as a function of the three non-dimensionalparameters.

〈mi〉{∗∗η }+ 〈miω

2i 〉{η} =

1

2

µ1κ21

[Am(φ, κ,M,Re)

]{η}+

1

Fr2κ21〈mi〉 [φ]−1 {ag} (2.2.13)

This is the final form of the non-dimensional set of EOM for an aeroelastic system and willthe basis for the aeroelastic scaling for this present work. One can now see that in order to createa scaled or "equivalent" aeroelastic system some constraints have to be applied to the model,namely:

1. Same aerodynamic shape is required for A to be the same as the full-scale airplane.

2. Same non-dimensional mode shapes are required for Am to be the same as the full-scaleairplane.

3. Same Froude number is required.

4. Same set of reduced frequencies κi is required.

5. Same set of mass ratio µi is required.

6. Same Mach number is required if compressibility effects are important.

7. Same Reynolds number is required number if viscous effects are important.

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2.2.1 Calculating the scaling parameters for the JWSC

Let us start by defining the length ratio, λl. As we know the full-scale airplane span and thedesired span for the scaled model, the relation is straightforward.

λl =bmbw

=45.62856

5≈ 1

9(2.2.14)

being the full-scale characteristics represented by the subscript w and the scaled model’s bythe subscript m. The size and aerodynamic shape (item 1) of the scaled model are completelydefined by this length ratio. The next step will be the matching of the Froude number (item 3),so that:

Fr =Vm√bmg

=Vw√bwg

(2.2.15)

this relationship will give the airspeed ratio, λV , between scaled and full-scale models,

λV =VmVw

=

√bmbw

=√λl =

1

3(2.2.16)

now, having both the length ratio and the velocity ratio set, one can extract the time scale, inthis case given by the frequency ratio, λω. Taking into account the reduced frequency (item 4),κ = bω

V , we have: (bω

V

)m

=

(bω

V

)w

→ λω =ωmωw

=bwbm

VmVw

=λVλl

= 3 (2.2.17)

From the mass ratio (item 5), µ = ρSbm , we get:(ρSb

m

)m

=

(ρSb

m

)w

(2.2.18)

and knowing that the baseline chord of both full-scale and scaled model is related by cm = bmbwcw

and that the wing area is S = bc, one can relate mass ratio, λm, and air density ratio, λρ, as:(ρb3

m

)m

=

(ρb3

m

)w

→ λρ =ρmρw

=

(bwbm

)3 mm

mw= 729λm (2.2.19)

For the present work was considered λρ = 1, therefore λm = (729)−1 = 1.37 × 10−3. Note thatif cruise conditions for the full-scale and scaled models were used for the calculation of the airdensity ratio, the resulting mass ratio would be unrealistically low and impracticable for thestructural optimization.

The non-dimensional mode shapes (item 2) between full-scale and scaled models can berelated through:

(√m{φ}

)m

=(√m{φ}

)w→ λφ =

{φ}m{φ}w

=1√λm

= 27 (2.2.20)

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We have now scaling ratios defining all primary quantities (length, time and mass), one cannow deduce all the necessary scaling ratios for the problem as their function. For the currentproblem we will be defining the force ratio, λF , and the moment of inertia ratio, λi. As known,the SI unit for force is the Newton that can be derived through the primary quantities as [kg · m

s2],

therefore: [kg · m

s2

]→ λF = λρλ

2V λ

2l = λm = (729)−1 = 1.37× 10−3 (2.2.21)

The SI unit for moment of inertia is [kg ·m2] and applying the same methodology as for the forceratio, we get: [

kg ·m2]→ λi = λρλ

5l = (59049)−1 = 1.69× 10−5 (2.2.22)

With this, all the scaling ratios necessary to the solving of the problem are now defined, in Table2.1 there is a summary of all of them for the reader comfort.

Scaling ratio ValueLength ratio (λl) 1/9

Velocity ratio (λV ) 1/3

Frequency ratio (λω) 3

Air density ratio (λρ) 1

Mass ratio (λm) 1.37× 10−3

Mode shape ratio (λφ) 27

Force ratio (λF ) 1.37× 10−3

Moment of Inertia ratio (λi) 1.69× 10−5

Table 2.1: Defined scaling ratios for the JWSC.

2.2.2 Reynolds and Mach number

From the list of constraints presented in section 2.2, items 6 and 7 regarding the matching ofthe Mach and Reynolds number, respectively, for the scaled model still haven’t been discussed.This section will address this subject.

The Mach number,M , is a dimensionless quantity that represents the ratio of the speed of anobject moving through a compressible fluid medium and the local speed of sound. This numberwill provide a good estimation for the existence of compressibility effects on the flow aroundthe object e.g. shocks and expansion waves. Now, in the present case, knowing that RPVs’characteristic operational speed and altitude are quite low, 0.0 < Mm < 0.3 around sea-levelconditions, the Mach number matching is an impossibility and therefore will not be taken intoaccount. It is also expected that this neglection will develop little or no difference in the finalresults. The fact that thin airfoils are used in HALE configurations is also a favourable pointbecause unless high angles of attack are imposed to the aerodynamic structure, no noticeablecompressibility effects will be present ant the upper surface of the wing.

The Reynolds number, Re, is also a dimensionless quantity that represents the ratio of inertialforces to viscous forces on an object for the given flow conditions. Its value defines the flow regime

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and in this particular case of scaling its main importance is to guarantee that similar flow regimesexist through the full-scale and scaled model wings. It is defined by:

Re(x) =V x

ν

where V is the flow velocity, x is the stream-wise coordinate with origin at the surface stagnationpoint and ν is the kinematic viscosity of the fluid. From the scaling ratios defined in the previoussection we now have constraints in the velocity as well as in length, in order to have a matchingReynolds number, the only variable is the fluid viscosity. However, as the RPV is to be flighttested in the open air, there is no possibility of altering the fluid viscosity to our liking thereforewe can only compare the flow regimes between full-scale and scaled models.

From the scaling parameters already defined and assuming both airplanes are flying at stan-dard sea-level (SSL) conditions, as assumed for the air density ratio (λρ = 1), the ratio ofkinematic viscosity of the air between both cruise conditions will also be λν = 1, we can get therelationship between full-scale and scaled models Reynolds number as:

Rem(x) =Vmx

νm=λV Vw × xλννw

=13Vw × xνw

=1

3Rew(x) (2.2.23)

Meaning that for scaled flying conditions, the scaled model Reynolds number at any point willbe three times smaller than that for the full-scale model. Taking into account that the full scalemodel has a cruise Mach speed of Mw = 0.8, at SSL conditions one can calculate its cruisevelocity as Vcruisew = 272 [m/s]. This velocity will cause a predominantly turbulent flow aroundall surfaces for the full-scale model with the transition occurring in the vicinity of the leadingedge, a simple analysis concluded that this point would be located at a distance lower than 0.2

[m] from the wing’s leading edge. As for the scaled model, this distance will be three times biggerhowever, the use of passive flow transition control systems such as wires with diameter definedby Gibbings criterium i.e. Red = V dwire

ν ≥ 826 will force transition from laminar to turbulentflow on the scaled model [16], as represented in Figure 2.1. Applying the length ratio (λl) to the

Figure 2.1: Representation of the forced transition according to Gibbings criterium, the areas inlight-blue represent flow separation.

maximum transition distance calculated for the full-scale model, one will get that a wire with

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a diameter of dwire ≥ 0.137 [mm] (from Gibbings criterium) placed at 22 [mm] from the scaledmodel’s wing leading edge will force transition from laminar to turbulent flow providing flowsimilarity between models.

From this simple study one can assume that even though there is no matching of the Reynoldsnumber between full-scale and scaled models, the flow behaviour can be controlled in a simplemanner so that no impact in the final results is expected from the inclusion of viscous effectsin the problem. The use of variable density air tunnel testing or a different fluid with a lowerkinematic viscosity could also be an option in order to have matching Reynolds number [17]however, at a much higher cost.

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3 | Wingbox aeroelastic scaling

In this chapter the scaling methodology will be put to the test with the scaling of a simple half-span rectangular Wingbox. The scaling procedure is based on the iterative optimization of a finiteelement model in order to have matching non-dimensional mode shapes and reduced frequenciesbetween full-scale and scaled models. The FE structural analysis will be accomplished usingAnsys’ R© Mechanical APDL software that will be iteratively run through Phoenix Integration’sModelcenter R© optimization tool. The flutter analysis for the matching verification of scaledflutter speed and frequency between both models will be performed using Zona’s ZAERO R©

software.Starting by presenting the Wingbox’s full-scale physical properties, scaling objectives and

scaled model, this chapter will then introduce the scaling methodology and the adopted opti-mization routine. The results for the scaled modal and aeroelastic response will be presented inthe last section.

3.1 Wingbox characteristics

The Wingbox model used for this problem is a variation of the model used by Bond et al. [8],its physical properties will be presented in this section. To be noted that both full-scale andscaled model static and modal results were obtained through finite element analysis made by theauthor.

3.1.1 Full-scale model

The Wingbox for the current problem is a simple half-span 3-spar model with 11 evenlyspaced ribs and bottom and top skins, the central spar has top and bottom caps in order toprovide extra stiffness to the structure. The material used for its construction is Aluminiummetal and its mechanical properties are presented in Table 3.1.

The model layout is depicted in Figure 3.1 and is fixed (Ux = Uy = Uz = 0) at its root,y = 0, where x is the chordwise coordinate, y is the spanwise coordinate and z completes theorthogonal coordinate system. The cross section of the Wingbox structure is also represented inFigure 3.2 for a better understanding.

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Property ValueYoung’s Modulus (E) 70 [GPa]

Density (ρ) 2700 [kg/m3]

Poisson ratio (υ) 0.35

Table 3.1: Aluminum mechanical properties.

Figure 3.1: Wingbox FE model. Figure 3.2: Wingbox section layout.

The Wingbox’s characteristic dimensions are presented in Table 3.2 making the structurenow completely defined.

Property ValueHalf-span (b) 6096 [mm]

Chord (c) 1219.2 [mm]

Wingbox hight 149.04 [mm]

Central spar thickness 27.09 [mm]

Secondary spars thickness 0.18 [mm]

Ribs thickness 10.58 [mm]

Skin thickness 4.72 [mm]

Table 3.2: Wingbox characteristic dimensions.

As previously said, the testing of the full-scale model was performed through a FE analysisusing Ansys’ R© Mechanical APDL software. For the modeling of the secondary spars, ribs andskins SHELL93 elements were used. This element is an element characterized by having 8 nodes,4 at the corners and 4 intermediate having each node six degrees of freedom, Ux, Uy, Uz, Rx, Ryand Rz being Ui the translational and Ri the rotational. The central spar was modeled usingSHELL93 and BEAM189 elements, being the beam element used on the top and bottom for themodeling of the spar caps and the shell element as the web. The BEAM189 is a quadratic (3node) finite strain beam element with 6 degrees of freedom at each node, the same DOF as theshell element.

Before the modal analysis was run, a simple mesh convergence analysis was performed andthe element size was selected. The modal analysis extracted results for the first 15 modes butonly the first 10 will be of importance and presented in Table 3.3. From the analysis was alsoextracted the total mass of the model, mw = 330.55 [kg].

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Mode number Frequency [Hz]1st 4, 6417

2nd 20, 086

3rd 23, 393

4th 26, 621

5th 66, 699

Mode number Frequency [Hz]6th 70, 421

7th 108, 77

8th 115, 14

9th 117, 56

10th 163, 50

Table 3.3: Wingbox full-scale modal analysis results.

The mode shapes will not be presented here but in the results section of this current chapterfor a matter of comparison between the scaled and full-scale models.

3.1.2 Scaled model & scaling parameters

The design constraint for the scaled model was to create a model that could fit in a wind-tunnel test chamber with a 1 meter diameter. The half-span of the scaled model was chosenfulfilling this pre-requisite and set to b = 0.75 [m].

Having the reference length, b, defined and following the same procedure as presented insection 2.2.1 for the JWSC, one can calculate all the scaling parameters needed for the scalingof the Wingbox. Note that, once again, the air density ratio was set to λρ = 1. In Table 3.4 arepresented all the relevant scaling ratios for the Wingbox.

Scaling ratio ValueLength ratio (λl) 0.123

Velocity ratio (λV ) 0.351

Frequency ratio (λω) 2.851

Air density ratio (λρ) 1

Mass ratio (λm) 1.86× 10−3

Mode shape ratio (λφ) 23.173

Force ratio (λF ) 1.86× 10−3

Moment of Inertia ratio (λi) 2.82× 10−5

Table 3.4: Defined scaling ratios for the Wingbox.

The initial design point for the scaled model has a similar structure layout as the full-scalemodel but, of course, was scaled down to 12.3% of its size. The length ratio was also used forscaling down the characteristic dimensions of the full-scale model such as spars, ribs and skinthickness and hight.

Taking into account the frequencies for the full-scale model presented in Table 3.3 and thefrequency ratio (λω) already calculated and shown in Table 3.4, the scaled target frequenciesthat have to be matched through the optimization procedure can already be calculated and arepresented in Table 3.5.

The target mass for the scaled model can also be calculated, mm = 0.6156 [kg] and of coursethat all the other scaled target quantities can already be calculated too but will not be presented

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Mode number Frequency [Hz]1st 13, 233

2nd 57, 264

3rd 66, 693

4th 75, 896

5th 190, 156

Mode number Frequency [Hz]6th 200, 768

7th 310, 099

8th 328, 260

9th 335, 159

10th 466, 133

Table 3.5: Scaled Wingbox target frequencies.

due to no relevant showing purpose at the time e.g. mode shapes.

3.2 Scaling methodology

Scaling down the full-scale model according to the length ratio is a start for the scalingprocedure but not enough to match the target scaled frequencies and mode shapes, a structuraloptimization routine is needed to correctly scale the model. In order to correctly approach theproblem and the optimization, a better understanding of the physics behind a modal analysis isneeded. For the free vibration analysis the system can be categorized as an unforced, undampedsystem and the EOM that describe its motion are:

[M ] {x}+ [K] {x} = {0} (3.2.1)

being [M ] the mass matrix, [K] the stiffness matrix and {x} a column vector of the generalizedcoordinates, as before. Usually, for finite element analysis, the previous equation is convertedinto an eigenproblem. Introducing xi(t) = XiT (t), i = 1, 2, ..., n where Xi is a constant and T isa function of time, t, and substituting this relationship into 3.0.1 while rearranging the equationso that the left-hand side is independent of the index i and the right-hand side is independentof time, t, we get that both sides must be equal to a constant. By assuming this constant as ω2,we can reach the following relationship:

[[K]− ω2 [M ]

]{X} = {0} (3.2.2)

Equation 3.0.2 represents what is known as the eigenvalue or characteristic value problem where{X} is the mode shape and ω is the respective natural frequency of the vibration system.

One can now clearly see that for matching a non-dimensional modal response between full-scale and scaled models, we will need both stiffness, [K], and mass, [M ], matrices to be properlyscaled.

Some trial optimization runs with both stiffness and mass distribution being matched si-multaneously were made but with no satisfactory results and a very large run time as well asnumerous variables that make the optimization routine quite complex and not effective. Wasthen decided that the best option was to follow Richards [5] procedure for the matching of the

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modal response, it consists in dividing the problem into two optimization steps by first matchingthe models stiffness and then the mass distribution. The stiffness matching is performed by thematching of scaled displacements of a set of static analysis of both models, the scaled modelstructure is defined this way. With the structure set, non-structural point masses are distributedthroughout the model in order to match the scaled modal response i.e. reduced frequencies andnon-dimensional mode shapes.

3.2.1 Optimization procedure and constraints

The structural optimization procedure was performed using Phoenix Integration’s Modelcen-ter R© optimization tool, as previously said. The program calls Ansys R© in batch mode everyiteration while changing the defined optimization variables. Other programs can also be addedto the iterative routine such as MathWorks MATLAB R© in order to retrieve and manipulatethe data obtained through the FE analysis into usable (objective) variables for the optimizationalgorithm.

In the previous section was stated that the outer shape of the scaled model is defined solely bythe full-scale model and the length ratio. This way, the scaled model hight is already constrainedand so are the section hights for the different structural elements in the scaled model. Taking thisinto account, the only variables usable for the optimization procedure were the section thicknessesof the spars, ribs and skins.

For the stiffness matching, two different load cases were used in order to properly match boththe bending and the torsional stiffness of the model. In one case the models were subjected toa point force causing pure bending while in the other two point forces were applied causing asituation of pure torsion, as depicted in Figures 3.3 and 3.4.

Figure 3.3: Pure bending load case. Figure 3.4: Pure torsion load case.

The loads applied were all of the same magnitude Pw = 5000 [N ] in the full-scale model.Having already defined the force ratio, λF , one can calculate the forces to be applied to thescaled model as Pm = 9, 311 [N ].

In Figure 3.5 it is represented the iterative optimization process for the stiffness matching.Nodal deformations of the full-scale model from both load cases were scaled down with the lengthratio (λl) creating the deformation vector {Uw} in the MATLAB R© module. A Root Mean Square

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Figure 3.5: Iterative process for the stiffness matching of the Wingbox.

Deviation variable was defined as:

RMSD =

√√√√√ n∑i=1

(Uwi − Umi)2

n(3.2.3)

where n is the number of nodal values, the subscript w refers to the full scale model and thesubscript m refers to the scaled model. An average nodal relative error, e, is also defined andserves as reference when presenting results for the matching, it is the average value between allthe nodal relative errors. The objective function for the stiffness optimization routine was thenset as to minimize the RMSD variable.

For the mass distribution matching, 33 point masses were added to the already stiffnessmatched scaled model in a symmetric manner as to match the modal response to the targetscaled values. Figure 3.6 depicts the distribution of point masses on the structure, they have nostructural function and are represented as red dots.

Figure 3.6: Point masses added to the stiffness matched structure.

Two more RMSD variables were defined, one for the scaled mode shapes and another for thescaled frequencies, lets call them RMSDφ and RMSDω, respectively, being:

RMSDφ =

k∑j=1

√√√√√ n∑

i=1(φwi − φmi)

2

n

(3.2.4)

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and

RMSDω =

√√√√√ k∑j=1

(ωwj − ωmj

)2k

(3.2.5)

where k is the number of vibration modes used for the matching of the modal response. Notethat all mode shapes were equally weighted, as well as the modal frequencies. The objectivefunction for the optimization routine was chosen equally weighting both RMSD variables as:

min (RMSDφ +RMSDω) (3.2.6)

The optimization variables were the values of the point masses, constrained symmetrically and tothe target scaled mass, mm = 0.6156± 5% [kg]. Again, in Figure 3.7 is represented the iterativeoptimization process for the mass distribution matching of the Wingbox. A genetic algorithm

Figure 3.7: Iterative process for the mass distribution matching of the Wingbox.

was used for both the stiffness and the modal response matching optimization routines. The bestresults were then run through a gradient algorithm in order to look for improvements.

3.3 Wingbox scaling results

The best result for the stiffness matching returned an average error for the pure bending testcase of ebending = 0.538% and for the torsion test case of etorsion = 1.844%. These values imply asimilarity in the scaled static response between full-scale and scaled models and set the structurefor the modal response matching.

3.3.1 Scaled modal response

Now, for the modal response matching, the number of mode-shapes to be matched is expectedto have a big influence in the final results [18]. The fewer modes to be matched, the easier theminimization of the objective variables will be but at the cost of worse overall results. It wasthen decided to run two optimizations, one where the first 6 modes were matched and anotherwhere all 10 previously presented modes were matched.

In Table 3.6 are represented the results obtained for the optimization run where 6 modes werematched, eω represents the relative error in terms of frequency and eφ represents the average

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nodal error for the modal displacements.

Mode # Scaled target frequency Optimized value eω eφ1st 12.233 [Hz] 13.381 [Hz] 1.11% 3.22%

2nd 57.264 [Hz] 51.907 [Hz] 9.35% 7.03%

3rd 66.693 [Hz] 68.549 [Hz] 2.78% 8.97%

4th 75.896 [Hz] 76.070 [Hz] 0.23% 6.01%

5th 190.156 [Hz] 187.660 [Hz] 1.31% 15.67%

6th 200.768 [Hz] 198.920 [Hz] 0.92% 55.5%

Table 3.6: Matching of the modal response with 6 modes.

One can see that in terms of frequency matching, 6 modes are almost sufficient however, theaverage nodal error for the modal displacements grows steadily from mode to mode and reachesa value of 15.67% for the 5th mode and 55.5% for 6th mode which represents almost no matchat all. The total mass of the scaled model after modal response matching is of mm = 0.6144

[kg] that represents a relative error of emass = 0.193% to the target value presented in section3.2.1. These results are expected to be improved with the inclusion of all 10 previously presentedmodes, in Table 3.7 are presented the results for this case.

Mode # Scaled target frequency Optimized value eω eφ1st 12.233 [Hz] 13.369 [Hz] 1.02% 2.57%

2nd 57.264 [Hz] 52.818 [Hz] 7.76% 2.35%

3rd 66.693 [Hz] 66.942 [Hz] 0.37% 3.26%

4th 75.896 [Hz] 76.910 [Hz] 1.33% 7.50%

5th 190.156 [Hz] 188.480 [Hz] 0.88% 7.68%

6th 200.768 [Hz] 201.560 [Hz] 0.39% 5.12%

7th 310.099 [Hz] 277.720 [Hz] 10.44% 7.00%

8th 328.260 [Hz] 326.970 [Hz] 0.39% 12.56%

9th 335.159 [Hz] 339.510 [Hz] 1.30% 21.80%

10th 466.133 [Hz] 458.860 [Hz] 1.56% 39.60%

Table 3.7: Matching of the modal response with all 10 modes.

The improvement in the results with all 10 modes matched is visible, the total mass of thismodel after the application of the point masses is mm = 0.62334 [kg] that represents a relativeerror of emass = 1.26% to the target scaled value.

Once again there is a steadily increase in the average nodal error for the mode shapes however,if we compare these values with the ones presented in Table 3.6 the improvement is noticeable.One can thus assume that the mode shape matching performed through structural optimizationdevelops better overall results with the increase of mode shapes to be matched/included in theanalysis.

Let us now visually compare the resulting mode shapes from the full-scale and the scaledmodel according to the values presented in Table 3.7.

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(a) Target first mode shape (b) Scaled first mode shape

Figure 3.8: First vibration mode shape comparison, eφ1 = 2.57%.

(a) Target second mode shape (b) Scaled second mode shape

Figure 3.9: Second vibration mode shape comparison, eφ2 = 2.35%.

(a) Target third mode shape (b) Scaled third mode shape

Figure 3.10: Third vibration mode shape comparison, eφ3 = 3.26%.

(a) Target fourth mode shape (b) Scaled fourth mode shape

Figure 3.11: Fourth vibration mode shape comparison, eφ4 = 7.50%.

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(a) Target fifth mode shape (b) Scaled fifth mode shape

Figure 3.12: Fifth vibration mode shape comparison, eφ5 = 7.68%.

(a) Target sixth mode shape (b) Scaled sixth mode shape

Figure 3.13: Sixth vibration mode shape comparison, eφ6 = 5.12%.

(a) Target seventh mode shape (b) Scaled seventh mode shape

Figure 3.14: Seventh vibration mode shape comparison, eφ7 = 7.00%.

(a) Target eighth mode shape (b) Scaled eighth mode shape

Figure 3.15: Eighth vibration mode shape comparison, eφ8 = 12.56%.

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(a) Target ninth mode shape (b) Scaled ninth mode shape

Figure 3.16: Ninth vibration mode shape comparison, eφ9 = 21.80%.

(a) Target tenth mode shape (b) Scaled tenth mode shape

Figure 3.17: Tenth vibration mode shape comparison, eφ10 = 39.60%.

Quantitatively analysing the presented mode shape comparison, one can see the similarityof the scaled mode shapes obtained through the scaling process. However, two of the obtainedmodes, the 2nd and 7th, even though they represented the same mode shape, they were out ofphase to the target full scale and presented a bigger relative error in terms of frequency, noexplanation was found for this phenomenon.

3.3.2 Aeroelastic response verification

The last step to this aeroelastic scaling procedure is the aeroelastic response verification inorder to see if the scaled flutter velocity and frequency as well as the instability mode wereproperly matched. For the flutter analysis of the structure Zona Tech’s ZAERO R© software wasused. For subsonic flows, which is the case, this program uses a lifting surface method that adoptsa higher-order panelling scheme than the Doublet Lattice Method (DLM) to solve the flutterproblem. ZAERO R©’s flutter module contains two flutter solution techniques: the K-methodand the g-method, being that the latter generalizes the K-method and the P-K method for truedamping prediction.

Infinite plate splines were used to transfer deformations and loads between the structuralfinite element model and the aerodynamic panel model. Standard Sea-Level (SSL) conditionswere used for both the full scale and the scaled models, being ρ∞ = 1.225 [kg/m3]. Note that forthis flutter analysis were only included the first five vibration modes and the boundary conditionswere the same as for the FE model. In Figures 3.18 and 3.19 is represented the evolution of thestructural damping with respect to the air stream velocity and the evolution of the structure

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frequency with respect to the air stream velocity, respectively, as obtained through the g-method.

Figure 3.18: Full-scale model flutter analysis (g-method) - Air velocity versus Damping.

Figure 3.19: Full-scale model flutter analysis (g-method) - Air velocity versus Frequency.

Analysing these graphics one can realize that the natural vibration mode responsible for theflutter instability is the 3rd mode, represented in red, and extract both the flutter velocity andfrequency for the structure. These are presented in Table 3.8.

Through the K-method was obtained a flutter velocity of Vf = 254.40[m/s] and frequency of

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Flutter characteristic Flutter speed - Vf Flutter frequency - ffg-method 261.38 [m/s] 15.34 [Hz]

Table 3.8: Flutter response characteristics for the full-scale model as obtained through the g-method.

ff = 15.73[Hz] also for a divergence of the 3rd vibration mode. These values are quite similar tothe ones obtained through the g-method, as expected, and the corresponding flutter instabilityvibration mode is represented in Figure 3.20. In this representation it is clear dominance of the3rd vibration mode, the first torsional mode.

Figure 3.20: Full-scale model flutter instability vibration mode with several time steps.

Now, looking back into section 3.1.2 and Table 3.4 and multiplying the full-scale model fluttervelocities and frequencies for the velocity ratio (λV ) and frequency ratio (λω), respectively, weget the expected/target values for the scaled model flutter response characteristics. These valuesare represented in Table 3.9.

g-method Vf 91.68 [m/s]ff 43.72 [Hz]

K-method Vf 89.23 [m/s]ff 44.85 [Hz]

Table 3.9: Target values for the flutter analysis of the scaled model.

For the scaled model the same methodology was used. In Figures 3.21 and 3.22 is, once again,represented the evolution of the structural damping with respect to the air stream velocity andthe evolution of the structure frequency with respect to the air stream velocity, respectively, asobtained through the g-method this time for the scaled model.

Observing Figures 3.21 and 3.22 it is clear that, once again, the 3rd natural vibration mode,represented in red, is the responsible for the flutter instability, as expected. The flutter responsecharacteristics for the g-method (Vf and ff ) are represented in Table 3.10. Through the K-method was obtained a flutter velocity of Vf = 89.00[m/s] and frequency of ff = 45.55[Hz]

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Figure 3.21: Full-scale model flutter analysis (g-method) - Air velocity versus Damping.

Figure 3.22: Full-scale model flutter analysis (g-method) - Air velocity versus Frequency.

Flutter characteristic Flutter speed - Vf Flutter frequency - ffg-method 91.04 [m/s] 44.69 [Hz]

Table 3.10: Flutter response characteristics for the scaled model as obtained through the g-method.

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also for a divergence of the 3rd vibration mode. These values are, once again, very close tothe ones obtained through the g-method, as expected and, once again, the corresponding flutterinstability vibration mode is represented in Figure 3.23.

Figure 3.23: Scaled model flutter instability vibration mode with several time steps.

Through visual analysis of Figure 3.23 it is also clear the dominance of the 3rd naturalvibration mode, the first torsional mode. Comparing both full-scale and scaled model’s flutterinstability vibration modes, one can see the clear similitude that also proves the correct scalingof the non-dimensional aeroelastic response. As for the aeroelastic response characteristics (Vfand ff ), both the values obtained through the g-method and the K-method are really close tothe target values previously presented in Table 3.9, this deviation is represented as relative errorsfor a better understanding in Table 3.11.

Target value Obtained value Relative error

g-method Vf 91.68 [m/s] 91.04 [m/s] 0.69%ff 43.72 [Hz] 44.69 [Hz] 2.22%

K-method Vf 89.23 [m/s] 89.00 [m/s] 0.26%ff 44.85 [Hz] 45.55 [Hz] 1.56%

Table 3.11: Overview of the aeroelastic response matching for the Wingbox.

With a maximum error of 0.69% for the matching of the flutter velocity (Vf ) and of 2.22%

for the flutter frequency (ff ), both for the g-method, one can conclude that the applied aeroe-lastic scaling methodology is a success and the Wingbox was properly scaled, retaining its non-dimensional aeroelastic response characteristics.

Knowing that this scaling methodology delivers a properly aeroelastically scaled model, thenext step is to apply it to the aeroelastic scaling of the JWSC. This subject will be approachedin the next chapter.

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4 | Joined Wing SensorCraft aeroelas-tic scaling

In this chapter, the scaling methodology presented in chapter 3 will be used once more, thistime for the linear aeroelastic scaling of the Joined Wing SensorCraft.

Differently from the Wingbox optimization presented in the previous chapter, in this casethere will be no structural similitude between full-scale and scaled models due to lack of infor-mation. Data for the stiffness matching with eight different load cases was provided by Boeingbut not the prototype’s internal structure. Boeing also provided modal analysis data for C.G.constrained and free modal response of the full-scale JWSC for the first 30 vibration modes.

Once again, Ansys’ R© Mechanical APDL software will be used for the scaled model FE staticand modal analysis. At first, the same Phoenix Integration’s Modelcenter R© optimization toolwas used but the obtained results were not good enough therefore another option was used, theDirect Multi-Search (DMS) optimization algorithm provided by professor Aguilar Madeira fromthe Department of Mechanical Engineering at Instituto Superior Técnico. This algorithm relieson a derivative-free method for multiobjective optimization problems that does not aggregate orscale any component of the objective function but uses the concept of Pareto dominance to attainand maintain a list of feasible nondominated (optimum) points [19]. This algorithm is proven toyield top of the art results and was used for both the stiffness and the mass distribution/scaledmodal response matching.

The flutter analysis for the matching verification of scaled flutter speed and frequency betweenboth models will, once more, be performed by Zona’s ZAERO R© software.

4.1 Scaled model

The elaboration of the internal structure of the scaled model was an iterative process relyingon the data provided by Boeing. It was a very time-consuming work where several differentmodels were optimized with no good results until the final version was found. The final versionof the scaled FE model is represented in Figure 4.1 with oblique and top views.

As can be seen in Figure 4.1, the developed FE scaled model doesn’t have an outer skin. It isonly composed of BEAM189 elements for the wings and central boom and SHELL93 elements for

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Figure 4.1: Final version of the JWSC internal structure FE model.

the internal structure of the fuselage. However, the Boeing JWSC prototype has a composite-plystructure with the outer skin also having an important structural significance. As the object ofstructural optimization is this "naked" model, when optimized this model will present by itself agood approximation of the static and modal behaviour of the objective full-scale prototype butskin has to be fitted in order to flight test it. This extra skin, if fitted around this optimizedstructure with normal techniques will add extra stiffness which will ruin the optimization workand develop non-reliable flight test results. Two options for this skin fitting will be presented inorder to not significantly alter the resultant stiffness of the scaled model.

The first option is to use a flexible fabric like impermeable compound such as polyurethane-coated Spandex with low stiffness around the internal structure to match the scaled aerodynamicshape of the JWSC prototype. One example of such an application is the morphing concept cardeveloped by BMW, the BMW GINA. This option is expected to add little or no stiffness andextra mass to the scaled model and is not expected to significantly alter the scaled aeroelasticresponse of the model while recreating a good approximation of the scaled JWSC outer shape.

The second option is to use sectioned skin panels. A set of low density and stiffness compositematerial panels with the scaled aerodynamic shape are fitted around the internal structure ina way that the shear flow gets disrupted from panel to panel and no stiffness is added to theoptimized model. It is however believed that the first option would be a cheaper alternative withbetter overall results as well as implemented in a simpler manner.

Going back to the scaled FE model, both forward and aft wings as well as the central boomare composed from the same Aluminium metal that was used for the Wingbox in the previouschapter therefore with its mechanical properties presented in Table 3.1. As for the fuselageinternal structure, a cabron-fiber reinforced polymer (CFRP) laminated composite was used andits mechanical properties are stated in Table 4.1.

The initial design point for the optimization was defined in a similar manner as for theWingbox case and taking into account the length ratio already presented in Table 2.1 applied tothe full-scale wings, central boom and fuselage section hights and lengths.

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Property ValueYoung’s Modulus - x-direction (Ex) 133 [GPa]

Young’s Modulus - y-direction (Ey) 95.28 [GPa]

Shear Modulus - (Gxy) 67 [GPa]

Density (ρ) 1700 [kg/m3]

Poisson ratio (υ) 0.3

Table 4.1: CFRP mechanical properties.

4.2 Scaling methodology

Once again, for the scaling of the JWSC, the adopted methodology will be the one used byRichards [5] where the stiffness and the mass distribution of the scaled model are uncoupledcreating a simpler 2-step problem for the optimization procedure as used in the previous chapterfor the Wingbox aeroelastic scaling.

4.2.1 Optimization procedure and constraints

As said in the beginning of the chapter, the DMS algorithm was used for the structuraloptimization of the initial design point of the scaled model. A MATLAB R© ".m" file implementedinto the algorithm calls Ansys R© in batch mode and computes the retrieved data into usableobjective variables for the optimization procedure.

Starting by the stiffness matching of the scaled model, eight different load cases where eachof the applied point forces had the value of Pw = 1000 [lb] or Pw = 4449.74 [N ] were includedin the optimization. Taking into account the force ratio (λF ) already defined in Table 2.1 andmultiplying it by Pw we reach the value of the point forces to be applied to the scaled modelas Pm = 5.325 [N ]. The different load cases and boundary conditions for the static analysis arerepresented into Figure 4.2.

Figure 4.2: Representation of the eight different load cases and B.C. used for the stiffness match-ing of the JWSC.

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The load cases are identified with different colors and the respective number (1, 2, 3, 4, 5, 6,7 and 8) and the four purple dots are the nodes where the structure is clamped i.e. where thenodal displacements and rotations are set to zero, Ux = Uy = Uz = Rotx = Roty = Rotz = 0.Note that all forces are applied through the positive z axis direction with exception of load case6 and 8 that are applied through the positive y axis direction.

Once again, a set of strategically placed nodes (30 for half-airplane, because of symmetry)was selected in order to compare the obtained values to the ones provided by Boeing. Theoptimization variables for this phase were the forward and aft wing sectional properties (thicknessand hight at the root and at the tip, because all spar beams were defined as linearly tapered) andthe radius for all the different ribs, central boom and all the different connections (all defined ascircular beams).

It was chosen to first pair the load cases into four 2-objective optimization routines with thepairs being load case 1 and 7, 2 and 4, 3 and 5 and, finally, 6 and 8. This time, instead of usingthe root mean square deviation (RMSD) as in the previous chapter, was adopted a MIN-MAXoptimization objective, defined as:

MAXk = max

|Uw1 − Um1 ||Uw2 − Um2 |

...

|Uwn − Umn |

(4.2.1)

where the subscript k represents the load case number, n is the number of nodes for extractingdata, Uw represents the scaled nodal displacements of the full-scale model and Um the scaledmodel nodal displacements. Should be noticed that there was an objective function like thisfor each load case and for the load cases with point forces in the z direction was only takeninto account the z direction displacements as well as for the load cases with point forces inthe y direction where was only taken into account the y direction displacements. This objectivevariable will set the DMS algorithm to minimize the maximum absolute difference between targetand obtained nodal values.

No constraints were put into the optimization procedure at this time and no stopping criteriawas defined for all the four pairs of load cases but they were run until several feasible points in eachPareto dominance front had both objectives (MAXk) inferior to 1mm. With all the optimizationpairs achieving the pre-determined results, all eight load cases were run simultaneously as toretrieve the best overall result, if possible, with max(MAXk), (k = 1, 2, 3, ..., 8) inferior to 1mm.Now, this optimization procedure is expected to deliver the optimized layout for the forwardand rear wings’ spars and ribs, the central boom and all the connections (rear-forward wing,rear wing-central boom) however, the fact that the structure is clamped in the vicinity of thefuselage leads to non-realistic almost-zero-displacement nodal values in the fuselage componentsin all the load cases and, therefore, the impossibility of its structural optimization. In order toovercome this problem, an extra step for the stiffness matching was included.

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For the fuselage components optimization, the non-structural point masses used for the massdistribution matching were added to the structure as represented in Figure 4.3.

Figure 4.3: Point masses added to the structure in order to perform the scaled modal responseoptimization.

A total of 29 point masses, represented as red and orange dots, were added to the structure.The red dots were placed at mid-chord locations and the orange dot at the target center ofgravity. A new set of 54 nodes for extraction purposes was, again, strategically set into the FEmodel, this time through all the plane in order to catch anti-symmetric mode shapes, if needed.

A total of 6 variables were defined for this optimization procedure, the thicknesses for thefuselage components (5 variables) and another for all the applied masses i.e. all the masseswill have the same value for each optimization iteration. During this optimization procedure aconstraint was established in relation to the structure total weight. Knowing that the full scalestructural weight of the JWSC is of mw = 70472.40 [kg] and knowing the mass scaling ratio (λm)from Table 2.1, we get a scaled model target mass of mm = 84.34 [kg]. The weight constraintwas that the results which weight was superior to 90kg were discarded. The value obtained forthe masses at the end of this "extra" optimization step will serve as the initial design point forthe scaled modal response optimization procedure. Using only the 1st vibration mode of thefull-scale C.G. constrained modal analysis data as the target value, two objective variableswere defined, one for the target scaled frequency (dω) and another for the scaled mode shape(MAXφ), as:

dω = |ωw1 − ωm1 | (4.2.2)

MAXφ = max

|φw1 − φm1 ||φw2 − φm2 |

...

|φwn − φmn |

(4.2.3)

where n is once again the number of nodes for extraction, the subscript w represents the targetfull scale scaled values and m the scaled model FE modal analysis values. To be noticed that

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the scaled model structure for this modal analysis was also fixed (Ux = Uy = Uz = 0) at thecenter of gravity and the objective variable for the nodal displacements (MAXφ) only used thez component of displacements. The DMS optimization algorithm was set to minimize bothobjective variables.

With the beam structure and fuselage components defined, the missing step is to optimizethe mass distribution in order to have matching scaled frequencies and mode-shapes between fullscale and scaled models. The initial design point for this phase was set by both the previous stepsleaving us with 16 variables (due to symmetry) for the point masses, as seen in Figure 4.3. Thistime, the target values will be the full-scale free modal analysis data and, as in the Wingboxcase, two RMSD objective variables were defined, one for the scaled frequencies and another forthe scaled mode shapes. RMSD variables were preferred to the MIN-MAX ones because of itsglobal-like behaviour that better suits the current problem where more than one mode shape isincluded. And as any other 6-DOF free modal analysis, in both the target data and the scaledFE model there are 6 rigid body frequencies/mode shapes. These vibration modes were notincluded in the optimization procedure therefore, the 10 modes to be matched were from the7th to the 16th. Again was imposed a constraint on the structure total weight as to discard anydesign point which had a mass over 85kg. Note that was used 90kg in the previous optimizationprocedure in order to allow more results and this limit was now set at 85kg, a value closer to thetarget of mm = 84.34 [kg], as to filter all the non-wanted design points from the optimizationresults. The two RMSD objective variables for this step are then:

RMSDφ =k∑j=1

√√√√√ n∑

i=1(φwi − φmi)

2

n

(4.2.4)

and

RMSDω =

√√√√√ k∑j=1

(ωwj − ωmj

)2k

(4.2.5)

where k is the mode number, corresponding to the 7th mode k = 1 and to the 16th modek = 10 and n is the number of extracted nodal values, 54 as for the fuselage componentsoptimization. Once again the objective variable for the nodal displacements (RMSDφ) onlyused the z component of displacements. The DMS algorithm was then set to minimize these twovariables.

4.3 Joined Wing SensorCraft scaling results

The best optimization run for the stiffness matching of the JWSC with all the eight loadcases returned the results presented in Table 4.2. Note that this time there is no error or average

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error but the value for the MAXk variable, the absolute maximum difference between nodalvalues and target values for each case.

k MAXk Component1 0.4846 [mm] Uz2 0.2910 [mm] Uz3 0.7377 [mm] Uz4 0.0726 [mm] Uz5 1.0418 [mm] Uz6 0.5952 [mm] Uy7 0.7303 [mm] Uz8 0.2094 [mm] Uy

Table 4.2: Optimal values for the objectives in the stiffness matching optimization procedure forthe JWSC.

As can be seen in Table 4.2, the results were almost all bellow the 1mm objective previouslydetermined however, for the load case number 5, the maximum difference between nodal andtarget values is just above 1mm, MAX5 = 1.0418 [mm]. Despite the fact that MAX5 > 1mm,these values represent a proper similarity between the scaled model and the full scale staticresponse for the correspondent design variables define the section properties of the forward wing(spars+ribs), the aft wing (spars+ribs), the central boom (spars+ribs), the connection betweenthe forward and aft wings and the connection between the aft wings and the central boom.

For the fuselage optimization procedure, the best result present in the Pareto dominancefront returned a absolute difference for the first frequency of df = 0.0276 [Hz] and maximumdisplacement absolute difference of MAXφ = 0.0208 for the z component of displacement withan average relative nodal error of eφ = 7.54%. Once again, these values represent the propermatching of the 1st vibration mode of the C.G. constrained modal analysis data and the corre-spondent design variables define the fuselage structure and the point masses initial design pointfor the scaled modal response matching.

4.3.1 Scaled modal response

As previously said, 10 modes (from the 7th to the 16th) were included in the scaled modalresponse matching for the scaled JWSC. The first six rigid body modes have near-zero corre-spondent frequency for both the scaled and the full scale model and will not be presented here.The results for the matching of these 10 modes are presented in Table 4.3 with the relative errorfor the frequency (eω) and the average relative error for the mode shape matching (eφ).

This solution developed a structure with a total weight of mm = 74.32 [kg] which representsa relative error of emass = 11.90% to the target scaled value of mw = 84.34 [kg]. Comparingthe results presented in Table 4.3 with the ones retrieved from the Wingbox scaling (Table 3.7)one can see that both the relative errors for the frequencies and the average errors for the modeshapes are bigger. The reason for the more noticeable errors in these results resides in the fact

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Mode # Scaled target frequency Optimized value eω eφ7th 3.236 [Hz] 3.201 [Hz] 1.08% 3.89%

8th 4.251 [Hz] 4.415 [Hz] 3.86% 3.95%

9th 6.758 [Hz] 5.697 [Hz] 15.70% 4.18%

10th 7.615 [Hz] 6.480 [Hz] 14.90% 5.78%

11th 9.796 [Hz] 8.705 [Hz] 11.14% —12th 10.881 [Hz] 9.190 [Hz] 15.54% —13th 12.907 [Hz] 12.489 [Hz] 3.24% 14.72%

14th 17.804 [Hz] 15.366 [Hz] 13.69% 17.91%

15th 18.819 [Hz] 15.495 [Hz] 17.66% —16th 18.961 [Hz] 16.883 [Hz] 10.96% —

Table 4.3: Matching of the modal response with 10 modes for the JWSC.

that there is no structural layout similarity between scaled and the full scale model which makesthe matching of the modal response much more difficult and time-consuming while also deliveringworse overall results. The 2 pairs of vibration modes with no entry for the nodal average errorare not in the correct order as we’ll see in the visual comparison therefore, there is no sense inpresenting these errors.

Let us now visually compare the resulting mode-shapes from the modal response matchingfor the JWSC. All ten modes will be presented as done for the Wingbox case.

(a) Target seventh mode shape (b) Scaled seventh mode shape

Figure 4.4: Seventh vibration mode shape comparison, eφ7 = 3.89%.

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(a) Target eighth mode shape (b) Scaled eighth mode shape

Figure 4.5: Eighth vibration mode shape comparison, eφ8 = 3.95%.

(a) Target ninth mode shape (b) Scaled ninth mode shape

Figure 4.6: Ninth vibration mode shape comparison, eφ9 = 4.18%.

(a) Target tenth mode shape (b) Scaled tenth mode shape

Figure 4.7: Tenth vibration mode shape comparison, eφ10 = 5.78%.

(a) Target eleventh mode shape (b) Scaled eleventh mode shape

Figure 4.8: Eleventh vibration mode shape comparison - switched with the twelfth.

Despite the bigger average relative errors in terms of the nodal displacements when com-

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(a) Target twelfth mode shape (b) Scaled twelfth mode shape

Figure 4.9: Twelfth vibration mode shape comparison - switched with the eleventh.

(a) Target thirteenth mode shape (b) Scaled thirteenth mode shape

Figure 4.10: Thirteenth vibration mode shape comparison, eφ13 = 14.72%.

(a) Target fourteenth mode shape (b) Scaled fourteenth mode shape

Figure 4.11: Fourteenth vibration mode shape comparison, eφ14 = 17.91%.

(a) Target fifteenth mode shape (b) Scaled fifteenth mode shape

Figure 4.12: Fifteenth vibration mode shape comparison - switched with the sixteenth.

paring with the Wingbox case, through visual comparison of the resulting scaled mode shapesrepresented from Figure 4.4 to 4.14 one can see that in a general way they are the same as the

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(a) Target sixteenth mode shape (b) Scaled sixteenth mode shape

Figure 4.13: Sixteenth vibration mode shape comparison - switched with the fifteenth.

target. Two pairs of modes are in an incorrect order though. The eleventh and twelfth and thefifteenth and sixteenth mode shapes of the scaled model represent the twelfth and eleventh andthe sixteenth and fifteenth, respectively, of the target full scale modes. The fact that the naturalfrequencies for these pairs of vibration modes are really close to each other is the main cause forthis discrepancy besides the fact that there is no structural similarity between the scaled andthe full scale models. At first sight it might seem that there is no correlation between the scaledand target thirteenth mode however, they are the same mode shape only out of phase, as wasthe case for the second and seventh modes of the Wingbox in the previous chapter.

Even though the matching for the JWSC returned relative errors for the frequencies and thedisplacements considerably bigger than the ones for the Wingbox scaling procedure, this was,by far, the best achieved design and will be flutter tested in the next section as to verify theaeroelastic response matching.

4.3.2 Aeroelastic response verification

For the aeroelastic response verification of the JWSC was once again used Zona Tech’sZAERO R© software. As no flutter data (flutter mode, velocity and frequency) was providedby Boeing for the full scale (target) aircraft, it was necessary to run both the full scale and thescaled models in order to have a proper comparison.

To transfer the deformations and loads between the structural FE model and the aerodynamicmodel were used infinite plate splines, as for the Wingbox case. The properly meshed computa-tional model of the full scale airplane for the flutter testing turned out to be far more heavy incomputational terms than expected. As in the previous chapter was proven, for flutter testing,that the geometrically scaled model with the scaled frequencies and mode shapes will have thesame scaled aeroelastic response as the full scale model, the target aeroelastic response will beachieved testing a geometrically scaled aerodynamic model with the scaled target frequenciesand mode shapes.

The flutter testing was performed in SSL conditions for both the full scale and the scaledmodels and in Figures 4.14 and 4.15 is represented the evolution of the structural damping withrespect to the air stream velocity and the evolution of the frequency with respect to the air

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stream velocity, respectively, as obtained through the g-method for the full scale model.

Figure 4.14: Full-scale model flutter analysis (g-method) - Air velocity versus Damping.

Figure 4.15: Full-scale model flutter analysis (g-method) - Air velocity versus Frequency.

Through the analysis of these two graphics one can see that the vibration mode responsiblefor the flutter instability is the 11th mode, represented in magenta. Both the flutter velocity (Vf )and frequency (ff ) for this case are presented in Table 4.4.

Flutter characteristic Flutter speed - Vf Flutter frequency - ffg-method 101.39 [m/s] 8.49 [Hz]

Table 4.4: Flutter response characteristics for the full-scale model as obtained through the g-method.

Through the K-method, also for a divergence of the 11th vibration mode, the obtained fluttercharacteristics were Vf = 105.10[m/s] and ff = 8.71[Hz] which are in accordance with the values

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obtained through the g-method. The resulting flutter instability vibration mode is representedin Figure 4.16.

Figure 4.16: Full-scale model flutter instability vibration mode with several "time steps".

As stated in the beginning of this section, these flutter response characteristics obtainedthrough both the g-method and the K-method for the full scale model are already scaled valuesand therefore, the target values for the scaled model flutter analysis. The real (non-scaled) fullscale values are obtained by dividing Vf and ff by the velocity ratio (λV ) and the frequencyratio (λω) from Table 2.1, respectively, and are presented in the following Table.

g-method Vf 304.17 [m/s]ff 2.83 [Hz]

K-method Vf 315.30 [m/s]ff 2.90 [Hz]

Table 4.5: Full scale (non-scaled) flutter characteristics of the JWSC.

Now, for the flutter analysis of the scaled model a similar aerodynamic model and methodol-ogy were used. The resulting evolution of the structural damping and frequency to the sir streamvelocity as obtained through the g-method are represented in Figures 4.17 and 4.18.

One can see that the vibration mode responsible for the flutter instability of the scaled modelis the 12th mode, represented in black, and its flutter response characteristics are represented inTable 4.6.

Flutter characteristic Flutter speed - Vf Flutter frequency - ffg-method 68.80 [m/s] 8.99 [Hz]

Table 4.6: Flutter response characteristics for the scaled model as obtained through the g-method.

Through the K-method was obtained a flutter velocity, also for a divergence of the 12th

vibration mode, of Vf = 70.00[m/s] and a flutter frequency of ff = 8.99[Hz] that, once again,

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Figure 4.17: Scaled model flutter analysis (g-method) - Air velocity versus Damping.

Figure 4.18: Scaled model flutter analysis (g-method) - Air velocity versus Frequency.

are close to the values obtained through the g-method. The resulting flutter instability vibrationmode is represented in Figure 4.19.

Should be noticed that the optimized FE scaled model had two pairs of switched vibrationmodes, being one of them the eleventh and the twelfth, this fact is the reason behind the similaritybetween the scaled and full scale flutter instability vibration modes. In Table 4.7 there is acomparison between the obtained flutter response characteristics for the scaled model and thefull scale.

Even though the relative error in terms of flutter frequency is low, there is a notoriousdiscrepancy between the target and the obtained velocities for the scaled model. The reasonbehind such disparity in the flutter velocities is the poor matching of the mode shape that comesfrom the scaled model eleventh and twelfth being switched. The smaller relative error for theflutter frequency comes from the similarity between the scaled eleventh natural frequency for thefull scale model (9.796 [Hz]) and the twelfth natural frequency for the scaled model (9.190 [Hz])

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Figure 4.19: Scaled model flutter instability vibration mode with several "time steps".

Target value Obtained value Relative error

g-method Vf 101.39 [m/s] 68.80 [m/s] 32.14%ff 8.49 [Hz] 8.99 [Hz] 5.89%

K-method Vf 105.10 [m/s] 70.00 [m/s] 33.3%ff 8.71 [Hz] 8.99 [Hz] 3.21%

Table 4.7: Overview of the aeroelastic response matching for the JWSC.

that are characteristic of the same vibration mode.When comparing these results with the ones obtained for the Wingbox in the previous chapter

one can see the importance of structural similarity when matching scaled aeroelastic responsesbetween computational models.

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5 | Conclusion / Future work.

The results obtained in this thesis prove that a aeroelastically scaled model retaining thescaled aeroelastic response i.e. flutter velocity and frequency as well the correspondent instabilitymode, is achievable through the matching of the structure scaled modal response.

For the modal response matching the stiffness and the mass distribution were successfullydecoupled and separately optimized. This decoupling proved to be a more efficient approach interms of computational effort than the matching of the scaled modal response (stiffness+massdistribution) in a single step.

During the modal response matching optimization routine it was also proven that when morenatural frequencies and mode shapes are included into the modal response matching, the overallresults are better. The scaled inertia and center of gravity of similar structures also tend to thetarget scaled value with a good similarity of the modal response.

For the Wingbox case there was a really good matching of the scaled aeroelastic responsewith the correct flutter mode, frequency and velocity being captured with both the g and theK-method. However, for the Joined Wing SensorCraft, even though the correct flutter mode andfrequency were caught, there was a big disparity in terms of the flutter velocity with relativeerrors around 33% for both the g and the K-method. These big errors come from the fact thatthere was no structural similarity between the scaled and the full scale models which made theoptimization procedure much more time-consuming and with worse overall results than that ofthe Wingbox where the scaled model structure layout was previously defined. This fact provesthe importance of using a structurally similar scaled model as the initial design point whenaeroelastically scaling a prototype.

This procedure of aeroelastically scaling a prototype is then a cost-effective manner of flighttesting large span such as HALE aircrafts and commercial airliners with the risks associatedwith piloting a small RPV. Despite the increase in cost, the use of variable density wind tunnels(in order to have a air density ratio of λρ = 1) to test aeroelastically scaled models can deliverreliable flight test results for operational conditions i.e. cruise conditions in order to have apre-prototype airworthiness evaluation.

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5.1 Future work

A geometrically scaled model of the JWSC was already built and flight tested in 2011 byCfAR (UVic), as seen in Figure 5.1.

Figure 5.1: Flight testing of the geometrically scaled JWSC in Alberta in October 15, 2011.Courtesy of Jenner Richards, CfAR (UVic).

The construction and flight testing of the aeroelastically scaled model presented in this workwill provide a better understanding of the flight behaviour of the JWSC than the geometricallyscaled model. As the flight testing of an aeroelastically scaled JWSC with structure similaritywould provide top notch predictions in terms of aeroelastic response and flight characteristics.

The flight testing of aeroelastically scaled models is still in its first steps and there is a lotof potential in its use. It has been gaining momentum in the last few years with the interestof various renown scientists however in a more scientific basis, as to prove theories and buildmethodologies. It is the opinion of the author that a lot more work in terms of testing realaeroelastically scaled vehicles and publicise these results to big aircraft manufacturers is neededas to make it a profitable business.

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