Analysis of Wind Tunnel Lateral Oscillatory Data of the F-16XL Aircraft

NASA/TM-2004-213246

Analysis of Wind Tunnel Lateral Oscillatory

Data of the F-16XL Aircraft

Vladislav Klein

George Washington University

Joint Institute for Advancement of Flight Sciences

Langley Research Center, Hampton, Virginia

Patrick C. Murphy

Langley Research Center, Hampton, Virginia

Nathan M. Szyba

George Washington University

Joint Institute for Advancement of Flight Sciences

Langley Research Center, Hampton, Virginia

August 2004

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NASA/TM-2004-213246

Analysis of Wind Tunnel Lateral Oscillatory

Data of the F-16XL Aircraft

Vladislav Klein

George Washington University

Joint Institute for Advancement of Flight Sciences

Langley Research Center, Hampton, Virginia

Patrick C. Murphy

Langley Research Center, Hampton, Virginia

Nathan M. Szyba

George Washington University

Joint Institute for Advancement of Flight Sciences

Langley Research Center, Hampton, Virginia

National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-2199

August 2004

Available from: NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 7121 Standard Drive 5285 Port Royal Road Hanover, MD 21076-1320 Springfield, VA 22161-2171 (301) 621-0390 (703) 605-6000

Table of Contents

Summary iv

Symbols v

1 Introduction 1

2 Model and Tests 2

3 Static Data 3

4 Oscillatory Data 4

4.1 Rolling Oscillations 6

4.2 Yawing Oscillations 8

5 Concluding Remarks 9

5.1 Rolling Oscillations 9

5.2 Yawing Oscillations 10

6 References 11

7 Acknowledgements 11

8 Appendix A: Relationships between incidence and attitude angles 12

9 Appendix B: Harmonic Analysis 13

10 Appendix C: Aerodynamic Model Using Two-Step Linear Regression 15

11 Tables 18

12 Figures 53

iii

Summary

Static and dynamic wind tunnel tests were performed on an 18% scale model of the F-16XL aircraft. Static tests provided force and moment measurements for an angle of attack range of 0 ° to 90 ° . Data with nonzero sideslip angles were obtained by taking measurements at various roll and yaw offset angles. Offset static measurements, however, were not considered in this report. Dynamic forced oscillation tests were performed at initial angles of attack from 0 ° to 75 ° for roll and from 0 ° to 90 ° for yaw, oscillation amplitudes from 5 ° to 30 ° , offset roll and yaw angles from 0 ° to 60 ° , and reduced frequencies from 0.073 to 0.269. A limited amount of data was obtained at different Reynolds numbers.

Harmonic analysis was used to estimate Fourier coefficients and in-phase and out-of-phase components. This estimation procedure provided standard errors of the derivative estimates and the total aerodynamic coefficients, as well as the coefficients of determination associated with each model. For frequency dependent data from rolling oscillations, a two-step regression method was used to obtain unsteady models (indicial functions), and derivatives due to sideslip angle, roll rate and yaw rate from in-phase and out-of-phase components. Frequency dependence was found for angles of attack between 20 ° and 50 ° . Reduced values of coefficient of determination and increased values of fit error were found for angles of attack between 35 ° and 45 ° . An attempt to estimate model parameters from yaw oscillations failed, probably due to the low number of test cases at different frequencies.

iv

Symbols

AR aspect ratio , , A A B Fourier coefficients 0 j j a indicial function gain b wing span, m C aerodynamic coefficient a , , C C C side force, rolling-moment, yawing-moment coefficients Y l n , , C C C axial force, normal force, pitching-moment coefficients A N m , C C lift, drag coefficients L D C C in-phase component of a a β , C C out-of-phase components of C a a a p r , , C C C frequency dependent parameters a a a & p r & & β

c wing mean aerodynamic chord, m

i time index j harmonic index

k reduced frequency V k / l ω =

l characteristic length, / 2 b = l LS least squares m number of harmonics N number of measurements p, r rolling and yawing velocity, rad/sec R coefficient of determination Re Reynolds number s, s estimated standard error, estimated variance t time, sec

V airspeed, ft/sec

u,v,w velocity components along each body axis, m/sec y model output z measurement

α angle of attack, rad or deg

v β sideslip angle, rad or deg

ε measurement noise

θ pitch angle, rad or deg φ roll angle, rad or deg

, σ σ standard deviation, variance

τ indicial function non-dimensional time constant ψ yaw angle, rad or deg

ω angular frequency, rad/sec

Superscript or Superscript over variable $ estimated value . time derivative Subscript A amplitude a , , a l n Y = j harmonic index 0 nominal value Aerodynamic parameters ∂ C rb pb a

( ) = ≡ ∞ C C for β η , , =

a a η η η ∂ 2 2 V V vi

1 Introduction

In recent years a significant amount of research has been conducted in the area of unsteady aerodynamics at high angles of attack. In order to support this research, data were collected mostly from various wind tunnel tests. The models tested varied from simple delta wing layouts to that of modern fighter aircraft. One of the tested planforms was the F-16XL, cranked-delta wing aircraft. The first in a series of wind tunnel tests on the F-16XL were conducted in the NASA Langley Research Center (LaRC) 30x60 Foot Full Scale Wind Tunnel are presented in references 1 and 2. Therein static and forced-oscillation data are summarized in graphic and tabulated form, and no other modeling attempts were made.

A second set of tests on the F-16XL were conducted in the NASA LaRC 12 Foot Low Speed Wind Tunnel to obtain forces and moments during forced oscillations and ramps in pitch over a large range of angle of attack. The data and analysis for these tests are described in references 3 to 5. From the analysis linear and nonlinear longitudinal aerodynamic model equations with unsteady terms were obtained.

A third set of wind tunnel data for the F-16XL aircraft was obtained from experiments in the NASA LaRC 14x22 Foot Subsonic Wind Tunnel. In this experiment the data from forced oscillations in roll and yaw with different amplitudes, frequencies, and angles of attack were collected. A limited amount of data was obtained at different Reynolds numbers and with different offsets in mean roll and yaw angles. Some results of a preliminary analysis were reported in reference 6. The purpose of this report is to summarize most of the data, static and oscillatory, and present an analysis of these data. The main effort will be to obtain models for aerodynamic coefficients using harmonic analysis and linear regression.

After the Introduction, the report proceeds with model and test description in Chapter 2 and 3, followed by a review of static and oscillatory data in Chapter 4. The data analysis is split into two parts, the rolling oscillations and yawing oscillations. Concluding remarks complete the report.

2 Model and Tests

A three dimensional view sketch of the 18% scale F-16XL model with moveable control surfaces is shown in figure 1 together with some basic model dimensions. Static and dynamic tests were conducted in the NASA LaRC 14x22 Foot Subsonic Wind Tunnel. For both tests the model was mounted on a dynamic test rig through a six-component strain-gauge balance. The dynamic test rig is powered by an electric motor with a flywheel to give nearly sinusoidal motion in each of the three axes. The model can be sting mounted either through the rear or through the top of the aircraft. The mounting arrangement allowed the model to be rotated in each axis over an angle of attack range of 0° to 90 ° . More about the apparatus can be found in reference 7. The moment reference center was located at 0.46 c .

All data were obtained with a clean configuration and zero control deflections. For static data the dynamic pressure was set at 311 Pa (6.7 psf) resulting in a Reynolds number of 2.1 x 10 based on the o o mean aerodynamic chord. The data were obtained for pitch angles from 0 to 75 at five roll-angle o o offsets of 0 ° , 15 ° , 30 ° , 45 ° , and 60 ° , and pitch angles from 0 to 90 at four yaw-angle offsets of 0 ° , 15 ° , 30 ° , and 45 ° . For zero offset angles, pitch angle and angle of attack are equal. For nonzero offsets, angle of attack and sideslip change according to the relationships given in Appendix A.

Dynamic testing included forced oscillation about the roll and yaw body axes at angles of attack o o from 0 ° to 75 ° for roll and from 0 to 90 for yaw, at frequencies from 0.26 Hz to 1.13 Hz, and at four different amplitudes of 5 ° , 10 ° , 20 ° , and 30 ° . In addition, for both rolling and yawing oscillations with the amplitudes of 30 ° the effect of roll and yaw offset was investigated. Oscillatory data were mostly collected at the dynamic pressure of 311 Pa but some limited testing was performed at 95.8 Pa (2 psf) 6 6 and 191.5 Pa (4 psf) corresponding to Reynolds numbers of 0.6 x 10 and 1.3 x 10 . Data were sampled at 200 Hz with an inline 100 Hz anti-aliasing filter and further filtered during post processing with a 4 Hz low pass filter. Each run consisted of 40 cycles of the data. The test conditions for the roll oscillations and yaw oscillations are summarized in Table I and Table II, respectively.

3 Static Data

The force and moment coefficients for zero sideslip and zero offset in roll and yaw are presented in Table III. Drag, lift, and pitching-moment coefficients, from this table, are plotted against the angle of attack in figure 2. During the measurements with nonzero offsets in roll and yaw, the angle of attack and sideslip changed for each value of the pitch angle. The relation between the five angles is developed in Appendix A. For roll offset, 1 − tan (tan * cos ) α θ φ = (1a) 1 − sin (sin *sin ) β θ φ = (1b) and for yaw offset, 1 − tan (tan / cos ) α θ ψ = (1c) 1 − sin ( cos *sin ) β θ ψ = − . (1d) The computed value of α and β for each test point are plotted in figure 3. The non-rectangular spread in data points made it impossible to compute conventional aerodynamic derivatives with respect to sideslip angle in the vicinity of β = 0. For that reason this part of static data is not included in this report.

4 Oscillatory Data

As shown in Tables II and III, the measured data in roll and yaw oscillations were obtained at different angles of attack, frequencies, and amplitudes. In addition, some of the measurements cover the effect of initial offset in roll and yaw, and the effect of Reynolds number. Some examples of measured data are presented in figures 4 to 10. Time histories of roll angle and lateral coefficients at various angles of attack, amplitudes, and offsets in roll are shown in figures 4 to 6. Similar data from yawing oscillations are introduced in figures 7 and 8. As expected, changes in simple harmonic responses for all three lateral coefficients occurred with increased amplitude of commanded roll and yaw angles, increased angle of attack, and by introducing an offset in roll or yaw. More pronounced changes can be observed in the yawing moment and side force rather than in the rolling moment.

Figures 9 and 10 demonstrate variations of the lateral coefficients with roll and yaw angle at selected angles of attack, amplitudes, and frequencies. For a linear aerodynamic model the variation of aerodynamic forces and moments should be in the form of an ellipse, or, in some cases, in the form of a parabola (see reference 3). Distortion of the oscillatory data from the linear-model curves can be ascribed to the effect of measurement noise and aerodynamic nonlinearity. For large amplitude oscillations, kinematic nonlinearity also contributes to the distortion since the small angle assumption for the linear model is violated for large bank angles (see Appendix A). As in the previous examples, this is mostly apparent in the side force and yawing moment at high angles of attack and large amplitudes.

The method of harmonic analysis (Appendix B) was applied to measured aerodynamic coefficients using 20 cycles of data and the sampling rate of 100 Hz. A mathematical model for the aerodynamic coefficients was postulated as m ( ) ( cos sin ) C t A A t B t ω ω = + + (2) ∑ a o j j j j 1 j = where C is either , , C C or C and , , ,..., , A A B A B are the Fourier coefficients. The analysis a l n Y 1 1 o m m provided estimates of these three aerodynamic coefficients and their standard errors together with estimates of the standard error of measured aerodynamics coefficients (fit error), ( ) C s , and a coefficients of determination, R . The last term indicates how much information in the data is explained by the model (Appendix B).

A = 0, the coefficients A and B for roll-oscillatory For the model with linear aerodynamics and 0 1 1 data can be expressed as B ( )sin C C k C α = = ∞ − (3a) a a a & p β β φ A A ( ) sin C C C α = = ∞ + (3b) a a a & p p β k φ A assuming ( ) ( )sin t t β φ α = (4) o Similarly for yaw-oscillatory data, B 2 1

( ) C k C C cos + ∞ = = α

(5a) a a a 0 r & β β ψ A A ( ) cos C C C α = = ∞ − (5b) a a a & r r β k ψ A assuming ( ) ( ) cos t t β ψ α = − (6) o In equations (3) to (5) C , C , and C are the in-phase and out-of-phase components of ( ) C t , a a a a β p r where φ and ψ are the amplitudes of commanded roll and yaw angles, k is the reduced frequency, A A α is the initial value of the angle of attack, and , and ( ) C ∞ are the “quasi- ( ) C ∞ ( ) C ∞ a a a β r p steady” values of aerodynamic derivatives.

In order to explain the variation of the in-phase and out-of-phase components with frequency these components were modeled in terms of indicial functions (see reference 8). After some simplifying assumptions the models for these components have the following form 2 2 k τ ( )sin sin C C a α α = ∞ − (7a) 0 0 a a 2 2 β β 1 k τ + τ ( ) sin C C a α = ∞ − (7b) 0 a a 2 2 p p 1 k τ + for rolling oscillations and 2 2 k τ ( ) cos cos C C a α α = ∞ − (8a) 0 0 a a 2 2 β β 1 k τ +

τ

( ) a C C + ∞ =

(8b) a a 2 2 r r

1 k τ +

for yawing oscillations, where τ is the nondimensional time constant and a scales the contribution of unsteady terms to the in-phase and out-of-phase components.

In equations (7) and (8) there are four unknown parameters for each oscillation axis: ( ) C ∞ , a β ( ) C ∞ or ( ) C ∞ , a and τ . They can be obtained from measured data by various estimation a a p r techniques, reference 9. For the present report a two-step linear regression is used. This method was proposed in reference 10 and is explained in Appendix C.

4.1 Rolling Oscillations

Results of harmonic analysis for the roll-oscillatory data are presented in the form of in-phase and out-of-phase components of the rolling and yawing moments, and the side force. These components, which correspond to the first harmonics in equation (2), are summarized in figure 11 to 14, 20 to 22, and 27, and partially in tables IV to XV. The tabulated values also include standard errors of the components as a measure of their accuracy.

Figures 11 to 13 show a variation of the components with the angle of attack, frequency and amplitude. Figure 14 presents some of the data for the rolling moment coefficient re-plotted in the form ( ; , ) C k α φ and similarly for C . The dependence of the components on frequency occurs 0 A l l β p for α within the interval of 20 ° to 50 ° . For that region the damping derivatives C can be a p formulated as ( ) ( ; ) ( ; )sin C C C k α α α α = ∞ + (9) 0 a a a & p p β whereas for the derivatives outside the interval ( ) ( ; ) ( ; )sin C C C α α α α = ∞ + ∞ (10) 0 a a a & p p β where ( ) C ∞ is a “quasi-steady” value of frequency dependent parameter ( ) C k .

a a & & β β There are only relatively minor differences between the data for the amplitude of 5 ° and 10 ° . For the amplitude of 20 ° and 30 ° , however, larger differences emerge with respect to angle of attack and frequency. Variation in response with amplitude is characteristic of a nonlinear system. For the assessment of linear model adequacy in roll motion, the coefficient of determination and the fit error are plotted against the angle of attack for all four amplitudes and all available test frequencies and shown in figures 15 and 16. A significant decrease of R and increase of s( C ) is visible for the a angles of attack 35 ° to 50 ° for all amplitudes and all frequencies, especially the low frequencies. This might indicate a sudden increase in measurement error or modeling error or both. For better understanding of changes in R and ( ) C s , the measured and computed data were compared in figures a 17 to 19 for two amplitudes, two initial angles of attack, and one frequency. The differences between the sets of data are pronounced at α = 45 ° , whereas for α = 20 ° a linear model explains the measured data quite well regardless of the amplitude. Because of some degree of repeatability of ( ) C φ curves, a it may be possible that modeling errors rather than measurement errors are significant contributors to the large difference between the model and measured data at 35 ° < α < 45 ° . On the other hand, the

( )

addition of higher harmonics into model equation (2) did not significantly improve the R and C s a measures.

The in-phase and out-of-phase components for different values of roll offset, φ , are shown in figures 20 to 22. For these figures, the offset angle varies from 15 ° to 60 ° and the oscillation amplitude remains the same at 30 ° . Increased φ results in decreased dependency on frequency in all three aerodynamic coefficients. The in-phase components vary with the offset over the whole range of α but the out-of-phase components remain almost the same for 0 ° < α < 20 ° . By considering time histories of the aerodynamic coefficients adequacy of the linear model structure can be qualitatively assessed. Time histories of the coefficients are plotted in figures 23 to 25 for all four values of φ , selected values of α, and one frequency k = 0.064. Distortions of time histories are visible mostly at increased values of the angle of attack and the initial offsets in roll. In these cases it was found that models with three harmonics were adequate for explaining variations in measured data. Figure 26 shows improvement obtained in the coefficients of determination by progressively adding harmonics to the model.

In figure 27 the components were obtained from data at Reynolds number, Re = 0.6 x 10 , whereas previous results were for data at Re = 2.1 x 10 . A possible effect of Re on the components is not investigated here, however, some indication of this effect can be seen in the estimated parameters in (6), as shown in the next set of figures.

The estimates of unknown parameters in the aerodynamics model equations (6) are presented in figures 28 to 30 for C , C , and C . The results were obtained from data at l n Y 10 1 . 2 Re , 5 × = ° = φ A 10 1 . 2 Re , 10 × = ° = φ A 10 6 . 0 Re , 10 × = ° = φ A The estimates also include their 2 σ -confidence level and computed parameters C from the a & β expression ( ) a α τ ( ; ) C k α = − (11) a & 2 2 β 1 k τ + The estimated parameters from all three data sets are, in general, consistent with the exception of results at α = 45 ° and for the coefficient C . As discussed previously, a linear model at α = 45 ° is in Y doubt due to low values of R and large differences between computed and measured coefficient as demonstrated in figures 17 to 19. Different estimates of parameters τ and a for C come from data at Y two different Reynolds numbers, see figure 30.

The values of time constant for ( ) C t and ( ) C t are very similar. For the side force, however, τ l n is about one third of the previous values. This difference indicates smaller unsteady effect on C (t) Y then on the remaining two coefficients. The β -derivatives from oscillatory data are compared with those from static wind tunnel data of reference 1. Differences between these two results are mostly in ( ) C ∞ at 35 ° < α < 45 ° , and in ( ) C ∞ over the whole range of α . The best agreement exists for l Y β β the derivative ( ) C ∞ . For the p-derivatives at angles of attack between 30 ° to 40 ° , the unsteady n β term ( ) C k is the main contributor to the damping-in-roll, and to the yawing moment and side a & β force due to rolling velocity.

4.2 Yawing Oscillations

Estimation of in-phase and out-of-phase components from yaw-oscillatory data was limited to three cases only. Figures 31 to 33 show the dependence of both components on the amplitude and frequency for Reynolds number Re = 2.1 x 10 . Similar results are presented in figure 34 for Re = 0.6 x 10 . From both results no significant changes due to Reynolds number are noticeable. The data in figure 34 were used in estimation of parameters in model equation (8). Unfortunately, this attempt failed probably due to the limited number of frequencies. Variation of components with four values of yaw offset is given in figure 35. As was the case for the roll-oscillation data, increased offset seems to reduce the effect of the unsteady term on the components. In addition, changes in components due to the offset also appeared in the low α -regime, mainly for the yawing moment and the side force.

5 Concluding Remarks

A wind tunnel experiment on an 18-percent scale model of the F-16XL aircraft was conducted at NASA Langley Research Center. The experiment included static tests, roll-oscillatory tests, and yaw- oscillatory tests. Static tests investigated the effect of the angle of attack and offsets in roll and yaw.

Unfortunately, the spread of data points made it impossible to compute conventional aerodynamic derivatives with respect to sideslip angle in the vicinity of its zero value.

The measured data in roll and yaw oscillations were obtained at different angles of attack, amplitudes, and frequencies. Some of the measured data covered the effect of initial offsets in roll and yaw, and the effect of Reynolds number. The preliminary assessment of the data was obtained from time histories of the lateral coefficients and from a variation of these coefficients with either roll and yaw angles. A departure from a simple harmonic response of a linear system at a given frequency appeared with increased angles of attack, amplitudes, and initial offsets.

The harmonic analysis was applied to almost all oscillatory data available. The results contained the mean values of the Fourier coefficients and their accuracy. In addition the coefficient of determination and the fit error were also available.

5.1 Rolling Oscillations

The Fourier coefficients for a model with the first harmonic, known as the in-phase and out-of- phase components, were mostly plotted against the angle of attack with the reduced frequency as a parameter. Some of the components and their standard errors were summarized in tables. The dependency of the components on frequency occurred for angle of attack from 20 ° to 50 ° . There were only relatively minor differences between results for the amplitudes of 5 ° and 10 ° . For amplitudes of 20 ° to 30 ° , however, large differences emerged with respect to the angle of attack and frequency.

A significant decrease of coefficient of determination and increase of the fit error were visible mostly for increased values of angle of attack and initial offset in roll. Improvement was achieved by adding higher harmonics into the model.

The four unknown parameters, in an unsteady linear aerodynamic model, were estimated from the in-phase and out-of-phase components using two-step linear regression. The estimates and their confidence level were obtained for the angle of attack of 20 ° to 50 ° from data with two different amplitudes and Reynolds numbers. The estimated parameters for angle of attack less than 40 ° were quite consistent, with the exception of the Reynolds number effect on the lateral force. For angle of attack larger than 40 ° , however, there was a large scatter in the estimates, very likely due to limited information content in the data. The estimated time constant showed a strong dependence on angle of attack. The sideslip derivatives of rolling and yawing moment were in agreement with the derivatives obtained from static test. There was significant difference in the side force due to sideslip angle between oscillatory and static tests. The computed term expressing the effect of rate of change of sideslip indicated a substantial contribution of this term to the out-of-phase components at the angle of attack between 30 ° and 40 ° .

5.2 Yawing Oscillations

Estimates of the in-phase and out-of-phase components were obtained from a limited number of test data with different amplitudes, frequencies, initial offsets in yaw and Reynolds number. No significant conclusions from the results obtained could be made. One set of data was used for estimation of parameters in the aerodynamic models. This attempt failed, probably due to the low number of frequencies.

6 References

1. Grafton, Sue B. and Nguyen, Luat T.: Wind Tunnel Free-Flight Investigation of a Model of a Cranked-Arrow-Wing Fighter Configuration. NASA TP 2410, 1985.

2. Hahne, David E.: Low-Speed Aerodynamic Data for a 0.18-Scale Model of an F-16XL With Various Leading-Edge Modifications. NASA TM 1999-209703, 1999.

3. Vladislav Klein, Patrick C. Murphy, Timothy J. Curry and Jay M. Brandon: Analysis of Wind Tunnel Longitudinal Static and Oscillatory Data of the F-16XL Aircraft. NASA/TM-97- 206276, December 1997.

4. Klein, Vladislav, and Murphy, Patrick C.: Estimation of Aircraft Nonlinear Unsteady Parameters From Wind Tunnel Data. NASA/TM-1998-208969, December 1998.

5. Murphy, Patrick C. and Klein, Vladislav: Estimation of Aircraft Unsteady Aerodynamic Parameters From Dynamic Wind Tunnel Testing. AIAA Paper 2001-4016, August 2001.

6. Brandon, Jay M. and Foster, John V.: Recent Dynamic Measurements and Consideration for Aerodynamic Modeling of Fighter Airplane Configurations. AIAA Paper 98-4447, 1998.

7. Chambers, Joseph R. and Grafton, Sue B.: Static and Dynamic Longitudinal Stability Derivatives of a Powered 1/9-Scale Model of a Tilt-Wing V/STOL Transport. NASA TN-D- 3591, September 1966.

8. Klein, Vladislav and Noderer, Keith D.: Modeling of Aircraft Unsteady Aerodynamic Characteristics. Part 1 – Postulated Models. NASA TM 109120, 1994.

9. Murphy, Patrick C. and Klein, Vladislav: Validation of Methodology for Estimating Aircraft Unsteady Aerodynamic Parameters From Dynamic Wind Tunnel Tests. AIAA Paper 2003- 5397, 2003.

10. Abramov, N. B., Goman, M. G., Greenwell, D. J., and Khrabrov, A. N.: Two-Step Linear Regression for Identification of High Incidence Unsteady Aerodynamic Model. AIAA Paper 2001-4080, 2001.

7 Acknowledgements

The authors would like to thank Mr. Jay M. Brandon from NASA Langley Research Center for providing the experimental data and Mr. Kyle G. Mas, The George Washington University Graduate Research Scholar Assistant, for the development of a computer program for harmonic analysis.

8 Appendix A: Relationships between incidence and attitude angles

Let the angle of attack of the longitudinal body axis (x axis) be α . Then the velocity vector for φ =0 is cos cos α θ         0 0 V V V = = (A1)         sin sin α θ     After rotation about the x axis through φ , the velocity vector V is changed to V by the 1 2 transformation V L V φ =

( )

2 1 1 0 0 cos cos V u θ θ         (A2)         0 cos sin 0 sin sin V v φ φ θ φ = = =         0 sin cos sin sin cos V w φ φ θ θ φ         −         From (A2) the expressions for the incidence angles can be obtained as v sin sin sin β θ φ = = V (A3) w tan tan cos α θ φ = = u For small φ sin β φ θ ≈ (A4) α θ ≈ When yawing through ψ about the z axis, the velocity vector is V L V ψ =

( )

2 1 cos sin 0 cos cos cos V ψ ψ θ θ ψ       (A5)       sin cos 0 0 cos sin V ψ ψ θ ψ = − = −             0 0 1 sin sin V θ θ       Now the expressions for α and β have the form sin cos sin β θ ψ = − (A6) tan θ tan α = cos ψ For small ψ cos β ψ θ ≈ − (A7) α θ ≈

9 Appendix B: Harmonic Analysis

A periodic function ( ) ( 2 ) y t y t π = + with the period π 2 is represented by a series of discrete values ( ) y i at 2 i π ( ) , 1, 2,..., t i i N = = N It is assumed that ( ) y i can be approximated by a trigonometric series m m

( ) + + = i j B i j A A i y sin cos ω ω

(B1)

∑ ∑ j j 0 0 0

= = 1 1 j j 2 π where ω = . This series represents an orthogonal polynomial. It is further assumed that the N values of ( ) y i are obtained from measurements as ( ) ( ) ( ) z i y i i ε = + (B2) where ( ) z i are the measured values and ( ) i ε is the measurement noise which white, zero mean with variance σ . The parameters A , A and B , known as Fourier coefficients, can be estimated from 0 j j the measurement by applying the least squares (LS) criterion N min z i y i  −  →

( ) ( ) ∑

  1 i = then the LS estimates of parameters in (B1) are N ˆ

( ) = i z A

∑ 0

N = i 1 N ˆ

( ) = i j i z A cos ω

∑ j 0

(B3) N = i 1 N ˆ

( ) = i j i z B sin ω

∑ j 0

N = i 1 The variance of parameter estimates is 2 2 ˆ s A σ =

( )

N (B4) 2 2 2 ˆ ˆ s A s B σ = =

( ) ( )

j j N for all j. The estimate of variance σ is N ˆ s z i y i =  −  (B5)

( ) ( ) ∑

  N 1 i = where ( ) i y ˆ follows from (B1) by replacing the parameters by their estimates.

The adequacy of the model given by (B1) can be assessed by the coefficient of determination N

ˆ

( ) ( ) [ ] − i y i z

∑

= i 1 2

− = R 1

(B6) N

( ) [ ] − z i z

∑

= i 1 where m m ˆ ˆ ˆ

( ) + + = i j B i j A A i y sin cos ˆ ω ω

∑ ∑ j j 0 0 0

= = j j 1 1 and mean measurement, z , is estimated by ˆ . z A =

10 Appendix C: Aerodynamic Model Using Two-Step Linear Regression

Aerodynamic model of an aircraft performing a one degree-of-freedom oscillatory motion about one of its body axes can be formulated in terns of the in-phase and out-of-phase components as for the rolling motion ( )sin sin C C af α α = ∞ − 1 a a β β (C1)

( ) sin C C af α = ∞ −

0 a a p p for the pitching motion C C af = ∞ −

( )

1 a a α α (C2) C C af = ∞ −

( )

0 a a q q for the yawing motion cos cos C C af α α = ∞ −

( )

1 a a β β (C3)

( ) cos C C af α = ∞ +

0 a a r r where 2 2 k τ f = 2 2 1 k τ + τ f = 2 2 1 k τ + and subscript a denotes appropriate force or moment coefficient. Expressing 2 2

k τ

1 = −

2 2 2 2

1 1 k k τ τ + +

equations (C1), (C2) or (C3) can be rearranged into a set of equations for n different values of k as , 1, 2,..., y j a a x j j n = + = (C4)

( ) ( )

0 1 where for the rolling oscillations , x C y C = = a a p β sin a C a a C α = ∞ + +

( )

( ) 0 1 a a

p β a τ = − and similarly for the pitching and yawing oscillations. In the first step a linear regression is used in estimation of parameters a and a in (C4) from measured in-phase and out-of-phase components at n 0 1 different values of k, n>2. The corresponding regression equation is Y X ε = Θ + (C5) where T T 1 2 ... Y y y y n =   , a a Θ =

( ) ( ) ( ) [ ]

0 1   1 1 x  

( )

  1 2 x

( )

  X =   : :   1 x n

( )

    and ε is a random zero mean and white measurement noise. The LS solution to (C5) is 1 − T T ˆ

( ) Y X X X = Θ (C6)

with the parameter variance matrix 1 − 2 2 T s X X σ Θ = (C7)

( )

( )

The estimate of σ and coefficient of determination, R , are obtained from the residuals ˆ e Y X = − Θ (C8) as T e e s = (C9) 2 n − T

e e

1 R − =

(C10) 2 T

y n Y Y −

n

( ) = j y y

∑ (C11)

n

= j 1 The second step of regression follows from equations (C1) to (C3) replacing τ by its estimated value.

The resulting regression equations are y j b b x j j ε = + +

( ) ( ) ( )

1 0 1 1 1 (C12) y j c b x j j ε = + +

( ) ( ) ( )

2 0 1 2 or Y X ε = Θ + where Y   1 T , Y b b c = Θ =

[ ]

0 1 0   Y 2     1 0 X X =   0 1 X 2   For the rolling oscillations the element of vectors Y , Y , X , X , and 1 are 1 2 1 2

( ) ( ) ( ) ( ) , j C j y j C j y = =

2 1 a a β p

( ) ( ) ( ) ( ) α α sin , sin j f j x j f j x − = − =

0 2 1 1 1 is the n-dimensional vector of ones and

( ) ( ) ∞ = = ∞ = C c a b C b , sin , sin α α

a a 0 1 0 p β Similar expressions are obtained for pitching and yawing oscillations. The LS parameter estimator is given by (C6) and the properties of the estimates by (C7) to (C9).

11 Tables

Table I. Test conditions for roll oscillations.

Φ A Φ o qbar V, frequency, ω, Run deg deg psf ft/s Hz rad/s k θ, deg 3 5 0 6.7 75 0.78 4.90 0.191 0 10 20 30 35 40 50 60 70 75 4 5 0 6.7 75 0.78 4.90 0.191 0 10 20 30 35 40 50 60 70 75 5 5 0 6.7 75 0.78 4.90 0.191 0 10 20 30 35 40 50 60 70 75 7 5 0 6.7 75 1.10 6.91 0.269 0 10 20 30 35 40 50 60 70 75 8 5 0 6.7 75 0.90 5.65 0.220 0 10 20 30 35 40 50 60 70 75 9 5 0 6.7 75 0.70 4.40 0.171 0 10 20 30 35 40 50 60 70 75 10 5 0 6.7 75 0.50 3.14 0.122 0 10 20 30 35 40 50 60 70 75 11 5 0 6.7 75 0.30 1.88 0.073 0 10 20 30 35 40 50 60 70 75 12 10 0 2 41 0.51 3.20 0.228 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 13 10 0 2 41 0.42 2.64 0.188 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 14 10 0 2 41 0.72 4.52 0.322 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 15 10 0 2 41 1.13 7.10 0.505 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 16 10 0 6.7 75 0.78 4.90 0.191 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 17 10 0 6.7 75 0.95 5.97 0.232 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 20 10 0 6.7 75 1.10 6.91 0.269 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 21 10 0 6.7 75 0.90 5.65 0.220 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 22 10 0 6.7 75 0.70 4.40 0.171 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 23 10 0 6.7 75 0.50 3.14 0.122 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 24 10 0 6.7 75 0.30 1.88 0.073 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 39 20 0 6.7 75 0.30 1.88 0.073 0 10 20 30 35 40 50 60 70 75 40 20 0 6.7 75 0.50 3.14 0.122 0 10 20 30 35 40 50 60 70 75 41 20 0 6.7 75 0.70 4.40 0.171 0 10 20 30 35 40 50 60 70 75 53 20 0 6.7 75 0.90 5.65 0.220 0 10 20 30 35 40 50 54 20 0 6.7 75 0.90 5.65 0.220 60 70 75 55 20 0 6.7 75 1.10 6.91 0.269 0 10 20 30 35 40 50 60 70 75 57 20 0 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 58 20 0 6.7 75 0.39 2.45 0.095 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 Table I. Concluded.

ω, Φ A Φ o qbar V, frequency, Run deg deg psf ft/s Hz rad/s k θ, deg 59 30 0 6.7 75 0.26 1.63 0.064 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 60 30 0 6.7 75 0.30 1.88 0.073 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 63 30 0 6.7 75 0.26 1.63 0.064 30 32 34 36 38 40 45 64 30 0 6.7 75 0.26 1.63 0.064 38 40 65 30 0 6.7 75 0.26 1.63 0.064 30 32 34 36 45 68 30 0 6.7 75 0.30 1.88 0.073 28 30 32 34 36 38 40 45 69 30 0 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 70 30 0 6.7 75 0.50 3.14 0.122 0 5 10 15 25 28 30 32 34 36 38 40 45 50 60 70 75 71 30 0 6.7 75 0.70 4.40 0.171 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 72 30 0 6.7 75 0.90 5.65 0.220 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 73 30 0 6.7 75 1.10 6.91 0.269 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 74 30 0 6.7 75 0.31 1.95 0.076 0 5 10 15 20 25 28 30 32 34 36 40 45 50 60 70 75 75 30 0 6.7 75 0.69 4.34 0.169 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 79 30 15 6.7 75 0.69 4.34 0.169 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 80 30 15 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 83 30 15 6.7 75 0.26 1.63 0.064 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 85 30 30 6.7 75 0.69 4.34 0.169 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 86 30 30 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 87 30 30 6.7 75 0.26 1.63 0.064 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 89 30 45 6.7 75 0.69 4.34 0.169 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 90 30 45 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 75 91 30 45 6.7 75 0.26 1.63 0.064 0 5 10 20 25 28 30 32 34 36 38 40 45 50 60 70 75 93 30 60 6.7 75 0.69 4.34 0.169 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 94 30 60 6.7 75 0.47 2.95 0.115 0 5 10 15 20 25 28 30 32 34 36 38 40 45 50 60 70 95 30 60 6.7 75 0.26 1.63 0.064 0 5 10 15 20 25 28 30 34 36 38 40 45 50 60 70 75 Table II. Text conditions for yawing oscillations.

ω, Ψ Α Ψ o qbar V, frequency, Run deg deg psf ft/s Hz rad/s k θ, deg 101 5 0 6.7 75 0.78 4.90 0.191 0 10 20 30 35 40 50 60 70 80 90 102 5 0 6.7 75 0.78 4.90 0.191 0 10 20 30 35 40 50 60 80 90 103 5 0 6.7 75 1.10 6.91 0.269 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 104 5 0 6.7 75 0.30 1.88 0.073 0 10 20 25 30 32 34 36 38 40 45 50 60 80 90 105 30 0 6.7 75 0.69 4.34 0.169 0 10 20 25 30 32 34 36 38 106 30 0 6.7 75 0.53 3.33 0.129 40 45 50 60 70 80 90 108 30 0 4 58 0.39 2.45 0.123 0 10 20 25 30 34 36 38 40 45 50 60 70 80 90 109 30 0 4 58 0.37 2.32 0.117 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 110 30 0 6.7 75 0.31 1.95 0.076 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 111 30 0 6.7 75 0.26 1.63 0.064 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 112 30 15 2 41 0.53 3.33 0.237 38 40 45 50 60 70 80 90 113 30 15 2 41 0.69 4.34 0.308 0 10 20 25 30 32 34 36 114 30 15 2 41 0.37 2.32 0.165 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 115 30 15 6.7 75 0.24 1.51 0.059 0 10 20 30 35 40 50 60 70 80 90 119 30 30 4 58 0.53 3.33 0.167 0 10 20 30 35 40 50 60 70 80 90 121 30 45 4 58 0.53 3.33 0.167 0 10 20 30 35 40 50 60 70 80 90 122 10 0 2 41 1.13 7.10 0.505 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 123 10 0 2 41 0.72 4.52 0.322 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 124 10 0 2 41 0.51 3.20 0.228 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 125 10 0 2 41 0.42 2.64 0.188 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 126 10 0 6.7 75 0.95 5.97 0.232 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 127 10 0 6.7 75 0.78 4.90 0.191 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 128 20 0 6.7 75 0.47 2.95 0.115 0 10 20 25 30 32 34 36 38 40 45 50 60 70 80 90 129 20 0 6.7 75 0.39 2.45 0.095 0 10 20 25 30 32 34 36 38 40 45 50 60 80 90 Table III. Aerodynamic coefficients at different angles of attack and zero sideslip.

α , deg C C C C C C C C N A L D m Y n l 0.0 -0.1283 0.0230 -0.1283 0.0231 0.0135 -0.0016 -0.0010 0.0033 10.1 0.2857 0.0027 0.2808 0.0527 0.0038 -0.0005 -0.0010 0.0027 20.1 0.8125 -0.0245 0.7716 0.2558 0.0001 0.0025 -0.0016 0.0014 25.1 1.1158 -0.0393 1.0272 0.4375 0.0025 0.0042 -0.0019 0.0016 30.1 1.4328 -0.0517 1.2658 0.6731 0.0117 0.0112 0.0012 0.0000 32.0 1.5237 -0.0562 1.3224 0.7591 0.0169 0.0177 0.0035 -0.0023 34.0 1.5961 -0.0599 1.3565 0.8433 0.0239 0.0263 0.0061 -0.0056 36.1 1.6459 -0.0602 1.3658 0.9204 0.0292 0.0313 0.0064 -0.0043 38.1 1.6895 -0.0601 1.3671 0.9942 0.0306 0.0353 0.0053 -0.0002 40.0 1.7310 -0.0604 1.3651 1.0662 0.0297 0.0420 0.0035 0.0026 45.1 1.7110 -0.0521 1.2451 1.1748 -0.0028 0.0219 -0.0024 0.0049 50.1 1.5508 -0.0521 1.0244 1.1648 -0.0440 0.0203 -0.0008 0.0027 60.0 1.5292 -0.0397 0.8004 1.3037 -0.0848 0.0182 0.0039 0.0004 70.0 1.6144 -0.0416 0.6088 1.4964 -0.0933 0.0048 -0.0134 0.0016 80.0 1.5971 -0.0608 0.3312 1.5633 -0.1368 0.0179 -0.0043 0.0028 90.1 1.5983 -0.0672 0.0655 1.5983 -0.1780 0.0303 -0.0093 0.0002 Table IV. In-phase and out-of-phase components of rolling moment coefficient. Φ = 5 deg.

A k = Component deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 α, 0 0.0039 0.0038 0.0027 0.0014 0.0023 0.0023 0.0012 0.0012 10 -0.0152 -0.0158 -0.0164 -0.0193 -0.0180 -0.0186 -0.0187 -0.0196 20 -0.0605 -0.0633 -0.0678 -0.0720 -0.0708 -0.0706 -0.0713 -0.0799 30 -0.1029 -0.1268 -0.1381 -0.1471 -0.1439 -0.1448 -0.1436 -0.1550 35 -0.1060 -0.1587 -0.1742 -0.1684 -0.1726 -0.1775 -0.1886 -0.1868

C

l β 40 -0.0460 -0.1143 -0.1373 -0.1571 -0.1552 -0.1520 -0.1654 -0.1759 50 -0.0531 -0.0579 -0.0549 -0.0583 -0.0588 -0.0541 -0.0539 -0.0631 60 -0.0846 -0.0801 -0.0801 -0.0774 -0.0808 -0.0800 -0.0824 -0.0764 70 -0.0865 -0.0852 -0.0857 -0.0895 -0.0861 -0.0825 -0.0829 -0.0845 75 -0.0929 -0.0902 -0.0902 -0.0840 -0.0889 -0.0902 -0.0847 -0.0881 0 3.46E-05 4.78E-05 7.15E-05 1.02E-04 5.14E-05 5.52E-05 9.03E-05 1.55E-04 10 6.47E-05 9.37E-05 1.21E-04 1.11E-04 9.72E-05 1.09E-04 1.59E-04 2.08E-04 20 5.65E-04 7.02E-04 8.25E-04 9.37E-04 8.92E-04 9.25E-04 1.09E-03 1.13E-03 30 1.35E-03 1.75E-03 1.85E-03 2.35E-03 2.07E-03 2.10E-03 2.38E-03 2.29E-03 35 2.66E-03 3.40E-03 4.16E-03 5.40E-03 4.05E-03 4.28E-03 4.48E-03 4.76E-03

( )

C s

l β 40 4.87E-03 6.00E-03 7.73E-03 6.57E-03 8.50E-03 7.54E-03 8.90E-03 8.52E-03 50 2.01E-03 2.58E-03 2.82E-03 2.79E-03 3.18E-03 2.79E-03 3.09E-03 3.28E-03 60 1.40E-03 1.75E-03 1.95E-03 2.28E-03 2.10E-03 2.08E-03 2.31E-03 2.87E-03 70 2.18E-03 2.85E-03 2.96E-03 3.19E-03 3.39E-03 3.23E-03 3.26E-03 4.64E-03 75 2.29E-03 3.04E-03 3.39E-03 3.55E-03 3.64E-03 3.65E-03 4.50E-03 4.82E-03 k = Component deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 α, 0 -0.1983 -0.1962 -0.1964 -0.1942 -0.1915 -0.1942 -0.1939 -0.1884 10 -0.2180 -0.2157 -0.2139 -0.2125 -0.2125 -0.2110 -0.2140 -0.2153 20 -0.2874 -0.2719 -0.2305 -0.2328 -0.2171 -0.2290 -0.2191 -0.1878 30 -0.7288 -0.5017 -0.3750 -0.3470 -0.3387 -0.3522 -0.3360 -0.2701 35 -1.6359 -0.9758 -0.5616 -0.4765 -0.5207 -0.4460 -0.4079 -0.3270

C

l p 40 -1.7924 -1.2135 -0.7659 -0.6199 -0.7163 -0.7304 -0.5493 -0.5289 50 -0.1472 -0.1881 -0.1898 -0.1583 -0.1945 -0.1915 -0.2011 -0.2011 60 -0.1367 -0.1060 -0.1459 -0.1289 -0.1186 -0.1272 -0.1103 -0.1432 70 -0.1866 -0.1499 -0.1557 -0.1397 -0.1581 -0.1612 -0.1493 -0.1492 75 -0.1256 -0.1390 -0.1803 -0.1485 -0.1742 -0.1455 -0.1607 -0.1693 0 4.74E-04 3.92E-04 4.18E-04 5.36E-04 2.69E-04 2.89E-04 4.10E-04 5.78E-04 10 8.86E-04 7.68E-04 7.10E-04 5.84E-04 5.09E-04 5.69E-04 7.23E-04 7.73E-04 20 7.74E-03 5.75E-03 4.82E-03 4.91E-03 4.67E-03 4.84E-03 4.95E-03 4.20E-03 30 1.84E-02 1.43E-02 1.08E-02 1.23E-02 1.08E-02 1.10E-02 1.08E-02 8.51E-03 35 3.64E-02 2.79E-02 2.43E-02 2.83E-02 2.12E-02 2.24E-02 2.04E-02 1.77E-02

( ) C s

l p 40 6.68E-02 4.92E-02 4.52E-02 3.44E-02 4.45E-02 3.95E-02 4.04E-02 3.17E-02 50 2.75E-02 2.11E-02 1.65E-02 1.46E-02 1.66E-02 1.46E-02 1.40E-02 1.22E-02 60 1.92E-02 1.43E-02 1.14E-02 1.20E-02 1.10E-02 1.09E-02 1.05E-02 1.07E-02 70 2.99E-02 2.34E-02 1.73E-02 1.67E-02 1.78E-02 1.69E-02 1.48E-02 1.72E-02 75 3.14E-02 2.49E-02 1.99E-02 1.86E-02 1.91E-02 1.91E-02 2.05E-02 1.79E-02 Table V. In-phase and out-of-phase components of rolling moment coefficient. Φ = 10 deg.

A k = Component α, deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 0 0.0035 0.0035 0.0020 0.0008 0.0007 -0.0001 -0.0016 5 -0.0029 -0.0029 -0.0040 -0.0061 -0.0051 -0.0065 -0.0068 10 -0.0156 -0.0155 -0.0169 -0.0199 -0.0177 -0.0199 -0.0212 15 -0.0379 -0.0393 -0.0419 -0.0438 -0.0421 -0.0451 -0.0457 20 -0.0614 -0.0649 -0.0700 -0.0738 -0.0716 -0.0753 -0.0758 25 -0.0932 -0.0969 -0.1000 -0.1019 -0.1026 -0.1054 -0.1079 28 -0.1087 -0.1178 -0.1278 -0.1338 -0.1321 -0.1363 -0.1367 30 -0.1162 -0.1356 -0.1494 -0.1564 -0.1539 -0.1583 -0.1630

C

l β 32 -0.1185 -0.1504 -0.1678 -0.1788 -0.1731 -0.1767 -0.1815 34 -0.1081 -0.1546 -0.1754 -0.1858 -0.1848 -0.1871 -0.1916 36 -0.0885 -0.1501 -0.1719 -0.1859 -0.1848 -0.1895 -0.1973 38 -0.0675 -0.1258 -0.1600 -0.1645 -0.1685 -0.1735 -0.1860 40 -0.0331 -0.0814 -0.1095 -0.1431 -0.1272 -0.1494 -0.1734 45 0.0044 -0.0067 -0.0375 -0.0641 -0.0532 -0.0624 -0.0999 50 -0.0578 -0.0608 -0.0618 -0.0632 -0.0611 -0.0610 -0.0695 60 -0.0809 -0.0787 -0.0786 -0.0797 -0.0802 -0.0789 -0.0786 70 -0.0897 -0.0871 -0.0861 -0.0862 -0.0854 -0.0860 -0.0844 75 -0.0921 -0.0882 -0.0881 -0.0902 -0.0915 -0.0818 -0.0832 0 2.26E-05 3.67E-05 7.87E-05 9.68E-05 7.12E-05 1.50E-04 1.82E-04 5 2.21E-05 3.69E-05 6.32E-05 1.04E-04 7.90E-05 1.47E-04 1.82E-04 10 4.38E-05 7.14E-05 1.08E-04 1.71E-04 1.33E-04 2.17E-04 2.28E-04 15 1.37E-04 1.80E-04 2.20E-04 2.69E-04 2.66E-04 3.15E-04 3.41E-04 20 3.08E-04 3.91E-04 4.60E-04 4.51E-04 4.69E-04 5.82E-04 5.64E-04 25 4.96E-04 5.04E-04 5.76E-04 7.10E-04 5.54E-04 7.29E-04 6.93E-04 28 7.00E-04 8.67E-04 8.56E-04 1.08E-03 9.58E-04 1.05E-03 1.05E-03 30 8.15E-04 1.01E-03 1.28E-03 1.37E-03 1.19E-03 1.35E-03 1.41E-03 32 1.06E-03 1.39E-03 1.46E-03 1.72E-03 1.66E-03 1.89E-03 2.02E-03

( ) C s

l β 34 1.33E-03 1.59E-03 1.95E-03 2.57E-03 2.05E-03 2.25E-03 2.34E-03 36 1.36E-03 2.30E-03 2.83E-03 3.07E-03 2.86E-03 3.20E-03 3.33E-03 38 2.07E-03 2.93E-03 3.59E-03 3.99E-03 3.77E-03 4.03E-03 4.26E-03 40 2.56E-03 3.58E-03 4.37E-03 4.59E-03 4.62E-03 5.24E-03 6.15E-03 45 2.68E-03 3.12E-03 3.89E-03 4.31E-03 3.50E-03 4.10E-03 3.97E-03 50 1.12E-03 1.45E-03 1.53E-03 1.87E-03 1.79E-03 2.16E-03 1.95E-03 60 7.32E-04 9.36E-04 1.00E-03 1.29E-03 1.17E-03 1.26E-03 1.38E-03 70 1.06E-03 1.41E-03 1.51E-03 1.95E-03 1.69E-03 1.67E-03 1.88E-03 75 1.27E-03 1.35E-03 1.69E-03 2.04E-03 1.90E-03 2.10E-03 2.21E-03 Table V. Concluded.

k = Component α, deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 0 -0.1928 -0.1907 -0.1870 -0.2123 -0.1620 -0.1834 -0.1826 5 -0.1589 -0.1556 -0.1525 -0.1739 -0.1317 -0.1494 -0.1490 10 -0.2094 -0.2080 -0.2065 -0.2331 -0.1793 -0.2024 -0.2009 15 -0.2066 -0.2058 -0.2031 -0.2307 -0.1770 -0.1987 -0.1951 20 -0.2627 -0.2371 -0.2169 -0.2265 -0.1734 -0.1879 -0.1903 25 -0.2445 -0.2119 -0.2122 -0.2280 -0.1678 -0.1932 -0.1868 28 -0.4126 -0.3209 -0.2621 -0.2617 -0.1906 -0.2029 -0.1873 30 -0.6983 -0.4695 -0.3230 -0.3083 -0.2565 -0.2543 -0.2148

C

32 -1.0376 -0.6488 -0.4382 -0.3732 -0.3025 -0.2966 -0.2312 l p 34 -1.3769 -0.7755 -0.5023 -0.4171 -0.3664 -0.3494 -0.3097 36 -1.7324 -0.9707 -0.5910 -0.5123 -0.4456 -0.3882 -0.3158 38 -1.7193 -1.0063 -0.6851 -0.5789 -0.5118 -0.4807 -0.3980 40 -1.6165 -1.1184 -0.7941 -0.7555 -0.6471 -0.6060 -0.5321 45 -0.7917 -0.7710 -0.6662 -0.7225 -0.5904 -0.6233 -0.5695 50 -0.2051 -0.2206 -0.2041 -0.2391 -0.1819 -0.2173 -0.1837 60 -0.1148 -0.1421 -0.1314 -0.1595 -0.1170 -0.1385 -0.1285 70 -0.1208 -0.1414 -0.1501 -0.1682 -0.1268 -0.1438 -0.1448 75 -0.1130 -0.1442 -0.1418 -0.1476 -0.1193 -0.1422 -0.1535 0 3.09E-04 3.01E-04 4.61E-04 5.07E-04 3.24E-04 6.45E-04 6.75E-04 5 3.03E-04 3.03E-04 3.70E-04 5.45E-04 3.59E-04 6.32E-04 6.77E-04 10 5.99E-04 5.85E-04 6.30E-04 8.98E-04 6.03E-04 9.34E-04 8.49E-04 15 1.88E-03 1.48E-03 1.29E-03 1.41E-03 1.21E-03 1.36E-03 1.27E-03 20 4.22E-03 3.20E-03 2.69E-03 2.36E-03 2.13E-03 2.51E-03 2.10E-03 25 6.79E-03 4.13E-03 3.37E-03 3.72E-03 2.52E-03 3.14E-03 2.58E-03 28 9.59E-03 7.11E-03 5.00E-03 5.65E-03 4.36E-03 4.53E-03 3.92E-03 30 1.12E-02 8.30E-03 7.50E-03 7.19E-03 5.42E-03 5.80E-03 5.24E-03 32 1.45E-02 1.14E-02 8.54E-03 9.00E-03 7.54E-03 8.13E-03 7.51E-03

( ) C s

34 1.82E-02 1.30E-02 1.14E-02 1.35E-02 9.31E-03 9.72E-03 8.70E-03 l p 36 1.86E-02 1.89E-02 1.65E-02 1.61E-02 1.30E-02 1.38E-02 1.24E-02 38 2.83E-02 2.40E-02 2.10E-02 2.09E-02 1.71E-02 1.74E-02 1.58E-02 40 3.51E-02 2.93E-02 2.56E-02 2.40E-02 2.10E-02 2.26E-02 2.29E-02 45 3.67E-02 2.56E-02 2.27E-02 2.25E-02 1.59E-02 1.77E-02 1.48E-02 50 1.53E-02 1.18E-02 8.93E-03 9.77E-03 8.12E-03 9.31E-03 7.24E-03 60 1.00E-02 7.67E-03 5.87E-03 6.73E-03 5.33E-03 5.42E-03 5.12E-03 70 1.45E-02 1.16E-02 8.81E-03 1.02E-02 7.68E-03 7.21E-03 7.01E-03 75 1.73E-02 1.11E-02 9.87E-03 1.07E-02 8.66E-03 9.07E-03 8.20E-03 Table VI. In-phase and out-of-phase components of rolling moment coefficient. Φ = 20 deg.

A k = α, deg Component 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 0 0.0030 0.0030 0.0025 0.0021 0.0004 -0.0024 -0.0027 5 -0.0037 -0.0043 10 -0.0153 -0.0177 -0.0182 -0.0166 -0.0182 -0.0220 -0.0227 15 -0.0411 -0.0412 20 -0.0593 -0.0671 -0.0694 -0.0633 -0.0684 -0.0814 -0.0842 25 -0.1004 -0.1046 28 -0.1172 -0.1237 30 -0.1077 -0.1297 -0.1394 -0.1290 -0.1416 -0.1669 -0.1681 32 -0.1370 -0.1514

C

l 34 -0.1335 -0.1513 β 35 -0.0778 -0.1458 -0.1711 -0.2010 -0.2127 36 -0.1012 -0.1329 38 -0.0501 -0.0702 40 -0.0102 -0.0296 -0.0418 -0.0431 -0.0884 -0.1357 -0.1713 45 -0.0614 -0.0635 50 -0.0546 -0.0601 -0.0627 -0.0580 -0.0617 -0.0728 -0.0762 60 -0.0787 -0.0857 -0.0863 -0.0781 -0.0781 -0.0862 -0.0852 70 -0.0888 -0.0978 -0.0970 -0.0881 -0.0884 -0.0980 -0.0957 75 -0.0926 -0.0999 -0.1002 -0.0918 -0.0917 -0.1007 -0.0971 0 3.21E-05 4.68E-05 5.50E-05 5.38E-05 8.02E-05 1.92E-04 4.58E-04 5 4.25E-05 5.44E-05 10 3.43E-05 5.79E-05 6.90E-05 7.10E-05 1.36E-04 2.68E-04 5.04E-04 15 1.30E-04 1.52E-04 20 1.78E-04 2.39E-04 2.80E-04 2.44E-04 3.10E-04 4.34E-04 6.03E-04 25 3.78E-04 3.62E-04 28 5.00E-04 5.30E-04 30 4.93E-04 5.53E-04 5.92E-04 5.67E-04 7.52E-04 1.03E-03 1.31E-03 32 6.14E-04 7.05E-04 34 8.07E-04 8.65E-04

( ) C s

l β 35 9.80E-04 9.25E-04 1.19E-03 1.56E-03 2.10E-03 36 1.41E-03 1.28E-03 38 1.73E-03 1.90E-03 40 1.50E-03 1.85E-03 1.97E-03 2.00E-03 2.38E-03 2.75E-03 3.00E-03 45 1.84E-03 1.89E-03 50 5.51E-04 7.68E-04 9.04E-04 8.28E-04 1.21E-03 1.53E-03 1.76E-03 60 3.61E-04 4.45E-04 4.95E-04 4.73E-04 6.28E-04 8.13E-04 9.07E-04 70 5.10E-04 6.48E-04 6.45E-04 6.84E-04 8.50E-04 1.04E-03 1.06E-03 75 5.84E-04 7.21E-04 8.07E-04 7.40E-04 8.95E-04 1.19E-03 1.38E-03 Table VI. Concluded.

k = Component deg 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 α, 0 -0.1905 -0.2055 -0.2025 -0.1838 -0.1806 -0.1946 -0.1885 5 -0.1692 -0.1662 10 -0.2061 -0.2250 -0.2199 -0.2009 -0.1945 -0.2087 -0.2041 15 -0.2303 -0.2225 20 -0.2501 -0.2676 -0.2555 -0.2318 -0.2091 -0.2090 -0.1934 25 -0.3138 -0.2676 28 -0.4157 -0.3540 30 -0.6553 -0.5880 -0.4641 -0.4077 -0.2877 -0.2676 -0.2412 32 -0.8490 -0.6887 34 -1.1599 -0.8955

C

l p 35 -1.7265 -0.8667 -0.5668 -0.4295 -0.3416 36 -1.5761 -1.2039 38 -1.6213 -1.3744 40 -1.2024 -1.2622 -1.1522 -1.0341 -0.8597 -0.7433 -0.6250 45 -0.4955 -0.4601 50 -0.2084 -0.2382 -0.2262 -0.2255 -0.2083 -0.2230 -0.2250 60 -0.1301 -0.1296 -0.1357 -0.1280 -0.1239 -0.1406 -0.1497 70 -0.0893 -0.1284 -0.1138 -0.1134 -0.1230 -0.1403 -0.1308 75 -0.0872 -0.1148 -0.1161 -0.1006 -0.1217 -0.1255 -0.1379 0 4.39E-04 4.93E-04 4.78E-04 4.41E-04 4.69E-04 8.71E-04 1.70E-03 5 4.47E-04 4.73E-04 10 4.70E-04 6.09E-04 6.00E-04 5.82E-04 7.94E-04 1.22E-03 1.87E-03 15 1.36E-03 1.32E-03 20 2.44E-03 2.51E-03 2.43E-03 2.00E-03 1.81E-03 1.97E-03 2.24E-03 25 3.98E-03 3.15E-03 28 5.27E-03 4.61E-03 30 6.75E-03 5.82E-03 5.15E-03 4.65E-03 4.40E-03 4.69E-03 4.89E-03 32 6.46E-03 6.13E-03 34 8.50E-03 7.52E-03

( ) C s

l 35 1.34E-02 7.58E-03 6.97E-03 7.11E-03 7.79E-03 p 36 1.48E-02 1.11E-02 38 1.82E-02 1.65E-02 40 2.05E-02 1.94E-02 1.71E-02 1.64E-02 1.39E-02 1.25E-02 1.11E-02 45 1.94E-02 1.64E-02 50 7.54E-03 8.08E-03 7.86E-03 6.78E-03 7.08E-03 6.95E-03 6.54E-03 60 4.95E-03 4.68E-03 4.30E-03 3.88E-03 3.67E-03 3.70E-03 3.37E-03 70 6.99E-03 6.82E-03 5.61E-03 5.60E-03 4.97E-03 4.73E-03 3.94E-03 75 8.00E-03 7.59E-03 7.02E-03 6.06E-03 5.23E-03 5.43E-03 5.14E-03 Table VII. In-phase and out-of-phase components of rolling moment coefficient. Φ = 30 deg.

A k = Component α, deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 0 0.0030 0.0024 0.0025 0.0015 0.0019 0.0009 0.0004 -0.0015 -0.0052 5 -0.0035 -0.0038 -0.0038 -0.0044 -0.0042 -0.0058 -0.0053 -0.0065 -0.0090 10 -0.0163 -0.0169 -0.0168 -0.0179 -0.0178 -0.0189 -0.0198 -0.0220 -0.0245 15 -0.0371 -0.0377 -0.0372 -0.0394 -0.0392 -0.0399 -0.0407 -0.0436 -0.0469 20 -0.0589 -0.0594 -0.0601 -0.0640 -0.0686 -0.0693 -0.0749 -0.0769 25 -0.0808 -0.0831 -0.0839 -0.0927 -0.0953 -0.1014 -0.1017 -0.1079 -0.1122 28 -0.0913 -0.0962 -0.0981 -0.1107 -0.1137 -0.1241 -0.1246 -0.1327 -0.1379 30 -0.0956 -0.1028 -0.1048 -0.1246 -0.1282 -0.1424 -0.1435 -0.1511 -0.1577 32 -0.0835 -0.0927 -0.0999 -0.1340 -0.1386 -0.1585 -0.1573 -0.1709 -0.1777

C

l β 34 -0.0580 -0.0730 -0.0785 -0.1306 -0.1365 -0.1677 -0.1663 -0.1819 -0.1937 36 -0.0412 -0.0527 -0.0531 -0.1017 -0.1125 -0.1523 -0.1577 -0.1830 -0.1977 38 -0.0386 -0.0455 -0.0740 -0.0808 -0.1207 -0.1249 -0.1607 -0.1885 40 -0.0496 -0.0517 -0.0546 -0.0745 -0.0772 -0.1039 -0.1041 -0.1352 -0.1633 45 -0.0726 -0.0730 -0.0743 -0.0788 -0.0815 -0.0864 -0.0892 -0.0956 -0.1136 50 -0.0645 -0.0642 -0.0647 -0.0694 -0.0696 -0.0754 -0.0755 -0.0808 -0.0850 60 -0.0830 -0.0831 -0.0833 -0.0830 -0.0831 -0.0831 -0.0837 -0.0849 -0.0871 70 -0.0974 -0.0969 -0.0961 -0.0973 -0.0979 -0.0971 -0.0953 -0.0993 75 -0.1016 -0.1001 -0.0997 -0.1006 -0.1003 -0.0996 -0.0997 -0.1010 0 4.01E-05 5.14E-05 5.27E-05 8.36E-05 8.49E-05 1.06E-04 1.10E-04 2.40E-04 4.67E-04 5 3.08E-05 4.17E-05 4.19E-05 7.75E-05 8.01E-05 1.19E-04 1.14E-04 2.29E-04 4.01E-04 10 4.24E-05 5.06E-05 5.52E-05 1.06E-04 1.14E-04 2.02E-04 2.15E-04 4.00E-04 5.97E-04 15 9.31E-05 1.07E-04 1.16E-04 1.97E-04 2.24E-04 3.54E-04 3.73E-04 6.25E-04 8.42E-04 20 1.93E-04 2.21E-04 2.27E-04 3.34E-04 5.25E-04 5.23E-04 7.93E-04 9.93E-04 25 3.72E-04 3.89E-04 3.88E-04 3.80E-04 3.90E-04 3.79E-04 4.62E-04 7.43E-04 1.18E-03 28 4.03E-04 4.03E-04 4.11E-04 4.21E-04 4.32E-04 5.58E-04 5.09E-04 8.89E-04 1.26E-03 30 4.05E-04 3.74E-04 4.30E-04 4.24E-04 4.67E-04 7.08E-04 7.25E-04 1.17E-03 1.50E-03 32 6.74E-04 6.29E-04 5.97E-04 5.42E-04 5.99E-04 8.80E-04 9.32E-04 1.46E-03 1.95E-03

34 6.83E-04 7.72E-04 7.70E-04 8.34E-04 8.51E-04 1.11E-03 9.62E-04 1.79E-03 2.35E-03 ( )

C s

l β 36 9.02E-04 9.78E-04 1.03E-03 1.16E-03 1.22E-03 1.39E-03 1.26E-03 1.80E-03 2.57E-03 38 1.24E-03 1.30E-03 1.69E-03 1.74E-03 2.04E-03 2.18E-03 2.39E-03 2.65E-03 40 1.51E-03 1.60E-03 1.62E-03 1.89E-03 2.01E-03 2.50E-03 2.34E-03 2.90E-03 3.15E-03 45 1.01E-03 1.08E-03 1.10E-03 1.26E-03 1.34E-03 1.43E-03 1.49E-03 1.79E-03 2.16E-03 50 5.01E-04 5.65E-04 5.64E-04 8.52E-04 8.82E-04 1.24E-03 1.25E-03 1.34E-03 1.37E-03 60 2.52E-04 2.85E-04 2.86E-04 3.87E-04 4.21E-04 5.56E-04 5.81E-04 8.65E-04 9.29E-04 70 3.43E-04 4.02E-04 5.31E-04 5.63E-04 6.77E-04 6.35E-04 8.66E-04 8.73E-04 75 3.95E-04 4.67E-04 6.13E-04 6.14E-04 8.02E-04 7.19E-04 9.57E-04 9.87E-04 Table VII. Concluded.

k = Component deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 α, 0 -0.2047 -0.2054 -0.2042 -0.1983 -0.2026 -0.1929 -0.1938 -0.1877 -0.1812 5 -0.1665 -0.1663 -0.1655 -0.1611 -0.1634 -0.1577 -0.1580 -0.1575 -0.1545 10 -0.2166 -0.2185 -0.2160 -0.2105 -0.2146 -0.2045 -0.2060 -0.1962 -0.1887 15 -0.2121 -0.2177 -0.2147 -0.2098 -0.2133 -0.2104 -0.2105 -0.2002 -0.1935 20 -0.2518 -0.2624 -0.2613 -0.2484 -0.2232 -0.2298 -0.1999 -0.1938 25 -0.3937 -0.3842 -0.3802 -0.3134 -0.2982 -0.2489 -0.2585 -0.2192 -0.2064 28 -0.6004 -0.5577 -0.5382 -0.3960 -0.3854 -0.2976 -0.2894 -0.2329 -0.2116 30 -0.8598 -0.7798 -0.7644 -0.5155 -0.4729 -0.3386 -0.3393 -0.2780 -0.2271 32 -1.3677 -1.1934 -1.1661 -0.6710 -0.6377 -0.4233 -0.4263 -0.3040 -0.2491

C

l p 34 -1.6621 -1.4956 -1.5113 -0.9587 -0.8845 -0.5295 -0.5489 -0.3909 -0.2778 36 -1.5149 -1.4497 -1.4193 -1.1402 -1.0764 -0.7522 -0.7294 -0.4905 -0.3669 38 -1.2051 -1.1756 -1.0076 -0.9692 -0.7939 -0.7786 -0.5967 -0.4496 40 -0.9205 -0.9068 -0.8680 -0.7759 -0.7775 -0.6873 -0.6501 -0.5450 -0.4469 45 -0.3870 -0.3936 -0.3690 -0.3571 -0.3446 -0.3051 -0.3096 -0.2959 -0.2702 50 -0.2056 -0.2269 -0.2221 -0.2345 -0.2388 -0.2116 -0.2127 -0.1928 -0.1793 60 -0.1040 -0.1157 -0.1140 -0.1250 -0.1287 -0.1300 -0.1317 -0.1308 -0.1309 70 -0.0799 -0.0841 -0.0968 -0.1188 -0.1106 -0.1025 -0.1205 -0.1010 75 -0.0620 -0.0751 -0.0916 -0.1061 -0.1051 -0.1047 -0.1092 -0.1055 0 6.27E-04 7.05E-04 6.93E-04 7.27E-04 6.96E-04 6.25E-04 6.45E-04 1.09E-03 1.74E-03 5 4.81E-04 5.71E-04 5.52E-04 6.74E-04 6.56E-04 7.07E-04 6.68E-04 1.04E-03 1.49E-03 10 6.63E-04 6.93E-04 7.26E-04 9.20E-04 9.31E-04 1.19E-03 1.26E-03 1.82E-03 2.22E-03 15 1.45E-03 1.47E-03 1.53E-03 1.72E-03 1.83E-03 2.10E-03 2.18E-03 2.84E-03 3.13E-03 20 3.01E-03 3.03E-03 2.99E-03 2.90E-03 3.11E-03 3.06E-03 3.61E-03 3.69E-03 25 5.82E-03 5.33E-03 5.11E-03 3.30E-03 3.20E-03 2.24E-03 2.70E-03 3.38E-03 4.40E-03 28 6.29E-03 5.52E-03 5.40E-03 3.66E-03 3.54E-03 3.30E-03 2.98E-03 4.04E-03 4.70E-03 30 6.32E-03 5.12E-03 5.65E-03 3.69E-03 3.82E-03 4.19E-03 4.24E-03 5.33E-03 5.56E-03 32 1.05E-02 8.62E-03 7.86E-03 4.71E-03 4.91E-03 5.21E-03 5.45E-03 6.63E-03 7.26E-03

( ) C s

l p 34 1.07E-02 1.06E-02 1.01E-02 7.25E-03 6.97E-03 6.54E-03 5.63E-03 8.14E-03 8.72E-03 36 1.41E-02 1.34E-02 1.35E-02 1.01E-02 9.98E-03 8.20E-03 7.39E-03 8.19E-03 9.55E-03 38 1.93E-02 1.78E-02 1.47E-02 1.43E-02 1.21E-02 1.27E-02 1.09E-02 9.86E-03 40 2.36E-02 2.19E-02 2.13E-02 1.64E-02 1.65E-02 1.48E-02 1.37E-02 1.32E-02 1.17E-02 45 1.58E-02 1.48E-02 1.44E-02 1.10E-02 1.10E-02 8.44E-03 8.71E-03 8.12E-03 8.04E-03 50 7.82E-03 7.74E-03 7.43E-03 7.41E-03 7.23E-03 7.35E-03 7.31E-03 6.11E-03 5.11E-03 60 3.94E-03 3.90E-03 3.76E-03 3.37E-03 3.45E-03 3.29E-03 3.40E-03 3.93E-03 3.45E-03 70 5.35E-03 5.29E-03 4.62E-03 4.61E-03 4.01E-03 3.72E-03 3.93E-03 3.25E-03 75 6.18E-03 6.14E-03 5.33E-03 5.04E-03 4.74E-03 4.21E-03 4.35E-03 3.67E-03 Table VIII. In-phase and out-of-phase components of yawing moment coefficient. Φ = 5 deg.

A k = Component α, deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 0 0.0027 0.0023 0.0021 0.0023 0.0023 0.0022 0.0021 0.0021 10 0.0116 0.0115 0.0113 0.0119 0.0118 0.0122 0.0118 0.0120 20 0.0206 0.0202 0.0208 0.0201 0.0199 0.0205 0.0204 0.0203 30 -0.0094 -0.0015 0.0030 0.0066 0.0063 0.0114 0.0045 0.0070 35 -0.0225 0.0004 -0.0010 0.0097 -0.0002 -0.0042 0.0062 -0.0022 40 -0.0845 -0.0372 -0.0308 -0.0120 -0.0151 -0.0334 -0.0167 -0.0131

C

n β 50 -0.0815 -0.0738 -0.0985 -0.0944 -0.1398 -0.1051 -0.0513 0.0081 60 0.0257 0.0066 0.0111 0.0140 0.0105 0.0089 -0.0040 -0.0019 70 -0.0149 -0.0195 -0.0183 -0.0027 -0.0208 -0.0283 -0.0138 -0.0377 75 -0.0480 -0.0440 -0.0425 -0.0602 -0.0538 -0.0627 -0.0519 -0.0573 0 4.11E-05 4.50E-05 5.61E-05 5.39E-05 5.18E-05 6.19E-05 6.84E-05 7.35E-05 10 4.89E-05 6.59E-05 7.95E-05 6.57E-05 7.54E-05 8.27E-05 8.78E-05 1.01E-04 20 2.82E-04 3.85E-04 3.99E-04 4.08E-04 4.31E-04 4.59E-04 4.95E-04 5.84E-04 30 1.05E-03 1.37E-03 1.58E-03 1.72E-03 1.70E-03 1.93E-03 1.80E-03 2.01E-03 35 2.05E-03 2.75E-03 3.04E-03 3.10E-03 3.07E-03 3.82E-03 3.63E-03 3.92E-03

( ) C s

n 40 5.01E-03 6.26E-03 8.12E-03 8.07E-03 8.35E-03 7.27E-03 9.25E-03 1.09E-02 β 50 6.05E-03 8.11E-03 8.30E-03 1.01E-02 9.35E-03 1.03E-02 1.00E-02 1.07E-02 60 4.27E-03 6.19E-03 7.64E-03 8.09E-03 8.06E-03 1.01E-02 1.01E-02 1.24E-02 70 1.13E-02 1.49E-02 1.71E-02 1.75E-02 1.71E-02 2.15E-02 2.05E-02 2.26E-02 75 5.83E-03 7.50E-03 9.55E-03 7.63E-03 8.32E-03 7.47E-03 1.04E-02 1.17E-02 k = Component α, deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 0 -0.0023 -0.0019 -0.0025 -0.0023 -0.0022 -0.0034 -0.0025 -0.0020 10 -0.0290 -0.0283 -0.0280 -0.0297 -0.0286 -0.0295 -0.0283 -0.0287 20 -0.0562 -0.0506 -0.0502 -0.0539 -0.0551 -0.0501 -0.0522 -0.0544 30 0.1029 0.0033 -0.0336 -0.0477 -0.0489 -0.0528 -0.0416 -0.0686 35 0.4381 0.1622 0.0087 -0.0829 -0.0143 -0.0355 -0.0663 -0.0840

C

40 1.0268 0.5156 0.2344 0.2192 0.1890 0.1436 0.0830 0.0889 n p 50 0.3165 0.1476 0.1642 0.2049 0.1723 0.1964 0.2693 0.1736 60 -0.1345 -0.1185 -0.0693 -0.0593 -0.0016 0.0591 -0.0559 0.0224 70 0.0691 0.0044 0.1029 0.1000 0.1322 0.0698 0.1206 0.0543 75 -0.0449 -0.0456 -0.0572 -0.0199 -0.0800 -0.0412 0.0179 -0.0504 0 5.63E-04 3.69E-04 3.28E-04 2.82E-04 2.71E-04 3.24E-04 3.11E-04 2.73E-04 10 6.70E-04 5.40E-04 4.65E-04 3.44E-04 3.95E-04 4.33E-04 3.99E-04 3.74E-04 20 3.87E-03 3.16E-03 2.33E-03 2.14E-03 2.26E-03 2.41E-03 2.25E-03 2.17E-03 30 1.44E-02 1.12E-02 9.22E-03 9.02E-03 8.88E-03 1.01E-02 8.18E-03 7.48E-03 35 2.81E-02 2.25E-02 1.78E-02 1.62E-02 1.61E-02 2.00E-02 1.65E-02 1.46E-02

( ) C s

n 40 6.87E-02 5.13E-02 4.75E-02 4.23E-02 4.37E-02 3.81E-02 4.20E-02 4.07E-02 p 50 8.29E-02 6.65E-02 4.85E-02 5.28E-02 4.89E-02 5.38E-02 4.57E-02 3.96E-02 60 5.85E-02 5.07E-02 4.47E-02 4.24E-02 4.22E-02 5.30E-02 4.58E-02 4.62E-02 70 1.54E-01 1.22E-01 9.98E-02 9.16E-02 8.94E-02 1.12E-01 9.30E-02 8.40E-02 75 7.98E-02 6.15E-02 5.58E-02 3.99E-02 4.36E-02 3.91E-02 4.72E-02 4.33E-02 Table IX. In-phase and out-of-phase components of yawing moment coefficient. Φ = 10 deg.

A k = Component α, deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 0 0.0024 0.0021 0.0020 0.0019 0.0019 0.0018 0.0015 5 0.0082 0.0080 0.0080 0.0081 0.0080 0.0081 0.0079 10 0.0122 0.0120 0.0121 0.0120 0.0120 0.0120 0.0116 15 0.0227 0.0227 0.0225 0.0222 0.0221 0.0225 0.0222 20 0.0220 0.0219 0.0219 0.0217 0.0217 0.0220 0.0217 25 0.0200 0.0241 0.0258 0.0264 0.0264 0.0264 0.0257 28 0.0171 0.0224 0.0264 0.0262 0.0272 0.0270 0.0269 30 0.0221 0.0285 0.0315 0.0324 0.0318 0.0332 0.0342 32 0.0239 0.0331 0.0364 0.0370 0.0394 0.0371 0.0359

C

n β 34 0.0127 0.0293 0.0342 0.0377 0.0349 0.0355 0.0368 36 -0.0134 0.0146 0.0217 0.0306 0.0273 0.0295 0.0268 38 -0.0531 -0.0182 0.0003 0.0109 0.0007 0.0086 0.0120 40 -0.1153 -0.0797 -0.0618 -0.0444 -0.0390 -0.0326 -0.0202 45 -0.2384 -0.2263 -0.2033 -0.1849 -0.1800 -0.1777 -0.1471 50 -0.1066 -0.1047 -0.0992 -0.0926 -0.0988 -0.0895 -0.0546 60 0.0290 0.0263 0.0278 0.0263 0.0223 0.0300 0.0172 70 0.0150 0.0128 -0.0005 -0.0131 -0.0219 -0.0111 -0.0177 75 -0.0588 -0.0508 -0.0551 -0.0570 -0.0605 -0.0480 -0.0527 0 2.38E-05 3.08E-05 3.64E-05 3.77E-05 4.07E-05 4.02E-05 4.49E-05 5 3.16E-05 4.21E-05 4.32E-05 4.66E-05 5.51E-05 5.80E-05 6.68E-05 10 4.30E-05 5.42E-05 6.23E-05 6.30E-05 7.19E-05 7.30E-05 7.64E-05 15 1.22E-04 1.63E-04 1.90E-04 1.96E-04 2.09E-04 2.30E-04 2.42E-04 20 1.88E-04 2.51E-04 2.88E-04 3.28E-04 2.82E-04 3.19E-04 3.18E-04 25 3.83E-04 4.92E-04 5.75E-04 5.46E-04 6.22E-04 6.23E-04 7.06E-04 28 8.46E-04 1.11E-03 1.30E-03 1.40E-03 1.56E-03 1.63E-03 1.73E-03 30 1.08E-03 1.48E-03 1.74E-03 1.86E-03 1.96E-03 2.13E-03 2.42E-03 32 1.41E-03 1.93E-03 2.17E-03 2.36E-03 2.44E-03 2.70E-03 2.85E-03

( ) C s

n 34 1.91E-03 2.41E-03 2.80E-03 3.08E-03 3.58E-03 3.44E-03 3.74E-03 β 36 2.15E-03 2.52E-03 2.83E-03 3.54E-03 3.57E-03 3.91E-03 4.27E-03 38 2.39E-03 3.06E-03 3.86E-03 4.15E-03 4.46E-03 4.82E-03 5.02E-03 40 2.86E-03 4.26E-03 5.48E-03 5.60E-03 6.15E-03 6.56E-03 8.06E-03 45 3.58E-03 4.50E-03 6.02E-03 5.31E-03 6.20E-03 6.19E-03 6.48E-03 50 3.62E-03 4.54E-03 5.98E-03 6.04E-03 6.31E-03 6.45E-03 6.29E-03 60 3.15E-03 4.46E-03 5.94E-03 6.87E-03 7.74E-03 7.97E-03 8.50E-03 70 4.80E-03 6.21E-03 7.79E-03 8.60E-03 1.05E-02 9.10E-03 1.18E-02 75 3.18E-03 3.97E-03 4.45E-03 5.42E-03 5.78E-03 5.86E-03 7.13E-03 Table IX. Concluded.

k = Component deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 α, 0 -0.0034 -0.0033 -0.0028 -0.0032 -0.0024 -0.0027 -0.0025 5 -0.0214 -0.0206 -0.0197 -0.0202 -0.0203 -0.0203 -0.0196 10 -0.0323 -0.0299 -0.0294 -0.0295 -0.0302 -0.0299 -0.0297 15 -0.0538 -0.0504 -0.0493 -0.0473 -0.0471 -0.0468 -0.0449 20 -0.0521 -0.0522 -0.0515 -0.0524 -0.0496 -0.0516 -0.0490 25 -0.0023 -0.0478 -0.0555 -0.0699 -0.0706 -0.0728 -0.0766 28 0.0402 -0.0137 -0.0549 -0.0690 -0.0700 -0.0774 -0.0870 30 0.0696 -0.0075 -0.0626 -0.0757 -0.0797 -0.0862 -0.0983 32 0.1227 0.0043 -0.0547 -0.0847 -0.0944 -0.1007 -0.1212

C

n 34 0.2726 0.0363 -0.0572 -0.0754 -0.1096 -0.1092 -0.1245 p 36 0.5811 0.1923 0.0053 -0.0294 -0.0620 -0.0900 -0.1149 38 0.7932 0.3399 0.1582 0.0763 0.0444 0.0084 -0.0269 40 1.0013 0.5949 0.3591 0.2954 0.2343 0.2016 0.1608 45 0.6555 0.5824 0.5023 0.5574 0.4723 0.5044 0.4802 50 0.1734 0.2078 0.1850 0.2257 0.2284 0.2673 0.2398 60 -0.1540 -0.0906 -0.0872 -0.0867 -0.0770 -0.0878 -0.0505 70 -0.4556 -0.3433 -0.2866 -0.2149 -0.2061 -0.1856 -0.0632 75 -0.0359 -0.0522 0.0101 -0.0244 -0.0076 -0.0051 -0.0080 0 3.27E-04 2.52E-04 2.13E-04 1.97E-04 1.85E-04 1.73E-04 1.67E-04 5 4.32E-04 3.45E-04 2.53E-04 2.44E-04 2.50E-04 2.50E-04 2.48E-04 10 5.88E-04 4.44E-04 3.64E-04 3.30E-04 3.27E-04 3.15E-04 2.84E-04 15 1.67E-03 1.33E-03 1.11E-03 1.02E-03 9.51E-04 9.89E-04 8.98E-04 20 2.57E-03 2.05E-03 1.68E-03 1.72E-03 1.28E-03 1.37E-03 1.18E-03 25 5.25E-03 4.04E-03 3.36E-03 2.86E-03 2.83E-03 2.69E-03 2.62E-03 28 1.16E-02 9.11E-03 7.62E-03 7.33E-03 7.07E-03 7.03E-03 6.43E-03 30 1.48E-02 1.22E-02 1.02E-02 9.72E-03 8.91E-03 9.17E-03 8.98E-03

32 1.93E-02 1.58E-02 1.27E-02 1.23E-02 1.11E-02 1.16E-02 1.06E-02 ( ) C s

n p 34 2.61E-02 1.97E-02 1.64E-02 1.61E-02 1.63E-02 1.48E-02 1.39E-02 36 2.95E-02 2.06E-02 1.65E-02 1.85E-02 1.62E-02 1.68E-02 1.59E-02 38 3.28E-02 2.51E-02 2.26E-02 2.17E-02 2.03E-02 2.08E-02 1.86E-02 40 3.91E-02 3.49E-02 3.21E-02 2.93E-02 2.80E-02 2.83E-02 3.00E-02 45 4.91E-02 3.69E-02 3.52E-02 2.78E-02 2.82E-02 2.67E-02 2.41E-02 50 4.96E-02 3.72E-02 3.50E-02 3.16E-02 2.87E-02 2.78E-02 2.34E-02 60 4.31E-02 3.65E-02 3.48E-02 3.59E-02 3.52E-02 3.43E-02 3.16E-02 70 6.58E-02 5.09E-02 4.56E-02 4.50E-02 4.78E-02 3.92E-02 4.40E-02 75 4.35E-02 3.25E-02 2.60E-02 2.84E-02 2.63E-02 2.52E-02 2.65E-02 Table X. In-phase and out-of-phase components of yawing moment coefficient. Φ = 20 deg.

A k = Component α, deg 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 0 0.0022 0.0025 0.0024 0.0020 0.0019 0.0017 0.0014 5 0.0080 0.0081 10 0.0134 0.0127 0.0126 0.0134 0.0132 0.0122 0.0121 15 0.0218 0.0216 20 0.0278 0.0240 0.0238 0.0278 0.0277 0.0233 0.0231 25 0.0262 0.0267 28 0.0266 0.0284 30 0.0327 0.0284 0.0306 0.0389 0.0413 0.0344 0.0338 32 0.0246 0.0294

C

34 0.0109 0.0196 n β 35 -0.0288 0.0231 0.0385 0.0307 0.0340 36 -0.0370 -0.0110 38 -0.1038 -0.0910 40 -0.1630 -0.1500 -0.1420 -0.1349 -0.0993 -0.0793 -0.0494 45 -0.1736 -0.1732 50 -0.0787 -0.0833 -0.0865 -0.0833 -0.0741 -0.0692 -0.0761 60 0.0053 -0.0007 -0.0020 0.0027 0.0010 -0.0061 -0.0114 70 -0.0171 -0.0223 -0.0276 -0.0225 -0.0215 -0.0278 -0.0286 75 -0.0473 -0.0548 -0.0522 -0.0500 -0.0521 -0.0592 -0.0537 0 1.56E-05 1.80E-05 1.84E-05 2.17E-05 2.82E-05 3.57E-05 4.69E-05 5 2.22E-05 2.68E-05 10 4.19E-05 4.96E-05 5.38E-05 5.76E-05 6.61E-05 8.44E-05 1.02E-04 15 9.18E-05 9.82E-05 20 1.78E-04 1.98E-04 2.13E-04 2.21E-04 2.53E-04 2.55E-04 2.71E-04 25 3.29E-04 3.83E-04 28 5.28E-04 6.06E-04 30 6.18E-04 6.08E-04 7.01E-04 7.82E-04 9.12E-04 9.05E-04 1.07E-03 32 8.12E-04 7.99E-04 34 1.26E-03 1.19E-03

( ) C s

n β 35 1.77E-03 1.51E-03 1.53E-03 1.51E-03 1.61E-03 36 2.01E-03 1.94E-03 38 1.86E-03 2.24E-03 40 1.66E-03 1.81E-03 1.99E-03 2.30E-03 2.64E-03 2.78E-03 3.03E-03 45 2.04E-03 2.28E-03 50 1.76E-03 2.23E-03 2.56E-03 2.66E-03 3.49E-03 4.96E-03 5.12E-03 60 1.66E-03 1.84E-03 2.06E-03 2.40E-03 3.21E-03 3.47E-03 4.02E-03 70 1.85E-03 2.14E-03 2.35E-03 2.29E-03 2.93E-03 2.57E-03 2.84E-03 75 1.95E-03 2.05E-03 2.44E-03 2.33E-03 2.82E-03 3.37E-03 3.94E-03 Table X. Concluded.

k = Component α, deg 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 0 -0.0033 -0.0158 -0.0157 -0.0033 -0.0035 -0.0155 -0.0141 5 -0.0311 -0.0316 10 -0.0331 -0.0467 -0.0463 -0.0305 -0.0317 -0.0436 -0.0420 15 -0.0633 -0.0635 20 -0.0617 -0.0758 -0.0766 -0.0561 -0.0562 -0.0714 -0.0688 25 -0.0483 -0.0562 28 -0.0457 -0.0572 30 0.0369 -0.0401 -0.0639 -0.0401 -0.0792 -0.1156 -0.1177 32 0.0310 -0.0208 34 0.2305 0.1124

C

n p 35 0.8744 0.2234 0.0432 -0.0664 -0.1084 36 0.6522 0.4160 38 0.8852 0.7104 40 0.9295 0.7483 0.6521 0.7365 0.5810 0.3832 0.2521 45 0.3183 0.2718 50 0.1095 0.1071 0.1299 0.1380 0.1624 0.1401 0.1523 60 -0.1498 -0.1288 -0.1220 -0.1105 -0.1148 -0.1195 -0.1223 70 -0.1960 -0.2094 -0.1592 -0.1272 -0.1034 -0.0870 -0.0410 75 -0.1339 -0.0967 -0.1122 -0.0793 -0.0634 -0.0301 -0.0158 0 2.13E-04 1.90E-04 1.60E-04 1.78E-04 1.65E-04 1.62E-04 1.74E-04 5 2.34E-04 2.33E-04 10 5.74E-04 5.22E-04 4.68E-04 4.72E-04 3.86E-04 3.83E-04 3.78E-04 15 9.67E-04 8.54E-04 20 2.44E-03 2.08E-03 1.86E-03 1.81E-03 1.48E-03 1.16E-03 1.01E-03 25 3.46E-03 3.33E-03 28 5.56E-03 5.27E-03 30 8.47E-03 6.40E-03 6.10E-03 6.41E-03 5.33E-03 4.11E-03 3.97E-03 32 8.54E-03 6.95E-03

34 1.33E-02 1.04E-02 ( ) C s

n p 35 2.42E-02 1.24E-02 8.96E-03 6.84E-03 5.99E-03 36 2.12E-02 1.69E-02 38 1.96E-02 1.95E-02 40 2.28E-02 1.91E-02 1.73E-02 1.89E-02 1.54E-02 1.26E-02 1.13E-02 45 2.15E-02 1.98E-02 50 2.41E-02 2.34E-02 2.23E-02 2.18E-02 2.04E-02 2.25E-02 1.91E-02 60 2.28E-02 1.94E-02 1.79E-02 1.97E-02 1.88E-02 1.58E-02 1.49E-02 70 2.54E-02 2.25E-02 2.05E-02 1.88E-02 1.71E-02 1.17E-02 1.06E-02 75 2.67E-02 2.15E-02 2.13E-02 1.91E-02 1.65E-02 1.53E-02 1.47E-02 Table XI. In-phase and out-of-phase components of yawing moment coefficient. Φ = 30 deg.

A k = Component α, deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 0 0.0022 0.0022 0.0022 0.0021 0.0020 0.0018 0.0018 0.0016 0.0011 5 0.0080 0.0079 0.0080 0.0079 0.0077 0.0079 0.0078 0.0077 0.0076 10 0.0133 0.0134 0.0134 0.0132 0.0132 0.0132 0.0132 0.0129 0.0131 15 0.0223 0.0224 0.0223 0.0223 0.0222 0.0224 0.0222 0.0221 0.0222 20 0.0265 0.0265 0.0265 0.0264 0.0263 0.0264 0.0261 0.0268 25 0.0284 0.0287 0.0286 0.0300 0.0301 0.0314 0.0319 0.0333 0.0345 28 0.0253 0.0268 0.0270 0.0304 0.0307 0.0333 0.0336 0.0349 0.0360 30 0.0177 0.0207 0.0213 0.0292 0.0304 0.0343 0.0339 0.0362 0.0376 32 -0.0083 -0.0030 -0.0006 0.0218 0.0239 0.0318 0.0318 0.0346 0.0368

C

34 -0.0481 -0.0374 -0.0356 -0.0007 0.0032 0.0221 0.0228 0.0301 0.0327 n β 36 -0.0755 -0.0688 -0.0709 -0.0405 -0.0344 -0.0091 -0.0042 0.0126 0.0221 38 -0.0946 -0.0922 -0.0755 -0.0736 -0.0486 -0.0462 -0.0223 -0.0021 40 -0.1063 -0.1046 -0.1046 -0.0926 -0.0901 -0.0729 -0.0739 -0.0543 -0.0330 45 -0.1080 -0.1074 -0.1072 -0.1033 -0.1023 -0.0990 -0.0977 -0.0931 -0.0808 50 -0.0797 -0.0766 -0.0788 -0.0798 -0.0797 -0.0767 -0.0754 -0.0699 -0.0644 60 -0.0179 -0.0177 -0.0181 -0.0190 -0.0203 -0.0220 -0.0228 -0.0280 -0.0328 70 -0.0335 -0.0359 -0.0357 -0.0370 -0.0408 -0.0401 -0.0417 -0.0458 75 -0.0475 -0.0466 -0.0514 -0.0523 -0.0514 -0.0543 -0.0587 -0.0554 0 1.76E-05 1.94E-05 1.99E-05 2.46E-05 2.60E-05 3.17E-05 3.37E-05 4.48E-05 4.98E-05 5 2.16E-05 2.43E-05 2.53E-05 3.67E-05 4.04E-05 5.77E-05 5.70E-05 8.23E-05 1.00E-04 10 4.73E-05 5.14E-05 5.33E-05 6.84E-05 7.25E-05 8.51E-05 9.22E-05 1.14E-04 1.40E-04 15 5.92E-05 6.51E-05 6.69E-05 9.13E-05 9.21E-05 1.16E-04 1.19E-04 1.61E-04 2.51E-04 20 1.34E-04 1.44E-04 1.46E-04 1.78E-04 2.18E-04 2.17E-04 2.55E-04 3.02E-04 25 1.96E-04 2.10E-04 2.20E-04 3.20E-04 3.40E-04 4.55E-04 4.66E-04 5.76E-04 7.52E-04 28 3.35E-04 3.40E-04 3.35E-04 4.30E-04 4.66E-04 6.11E-04 6.41E-04 8.00E-04 9.79E-04 30 5.47E-04 5.56E-04 5.53E-04 4.96E-04 5.23E-04 6.69E-04 7.06E-04 9.22E-04 1.07E-03 32 1.09E-03 1.08E-03 1.12E-03 7.88E-04 7.54E-04 7.73E-04 7.14E-04 9.93E-04 1.18E-03

( ) C s

n β 34 1.03E-03 1.15E-03 1.23E-03 1.47E-03 1.33E-03 1.24E-03 1.20E-03 1.15E-03 1.39E-03 36 8.69E-04 8.62E-04 8.94E-04 1.24E-03 1.38E-03 1.86E-03 1.79E-03 1.72E-03 1.75E-03 38 1.20E-03 1.24E-03 1.31E-03 1.35E-03 1.63E-03 1.64E-03 1.81E-03 2.23E-03 40 1.67E-03 1.76E-03 1.76E-03 1.97E-03 2.01E-03 2.10E-03 2.12E-03 2.16E-03 2.26E-03 45 2.06E-03 2.21E-03 2.23E-03 2.62E-03 2.74E-03 2.89E-03 2.85E-03 3.03E-03 3.87E-03 50 1.37E-03 1.55E-03 1.60E-03 2.08E-03 2.16E-03 2.62E-03 2.51E-03 3.12E-03 3.29E-03 60 1.02E-03 1.16E-03 1.21E-03 1.64E-03 1.79E-03 2.17E-03 2.27E-03 2.27E-03 1.82E-03 70 1.24E-03 1.43E-03 1.75E-03 1.77E-03 2.10E-03 2.04E-03 2.26E-03 2.30E-03 75 1.22E-03 1.43E-03 1.72E-03 1.87E-03 2.06E-03 2.18E-03 2.50E-03 3.19E-03 Table XI. Concluded.

k = Component α, deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 0 -0.0162 -0.0164 -0.0160 -0.0157 -0.0160 -0.0149 -0.0153 -0.0149 -0.0146 5 -0.0317 -0.0317 -0.0310 -0.0297 -0.0301 -0.0299 -0.0280 -0.0282 -0.0266 10 -0.0480 -0.0472 -0.0474 -0.0457 -0.0458 -0.0432 -0.0433 -0.0440 -0.0412 15 -0.0661 -0.0631 -0.0634 -0.0612 -0.0612 -0.0545 -0.0555 -0.0554 -0.0519 20 -0.0830 -0.0790 -0.0791 -0.0750 -0.0712 -0.0693 -0.0717 -0.0664 25 -0.0588 -0.0581 -0.0581 -0.0635 -0.0710 -0.0733 -0.0710 -0.0808 -0.0827 28 -0.0141 -0.0288 -0.0319 -0.0561 -0.0623 -0.0758 -0.0784 -0.0908 -0.0958 30 0.1066 0.0673 0.0629 -0.0258 -0.0445 -0.0760 -0.0756 -0.0909 -0.1025 32 0.4985 0.4014 0.3582 0.0653 0.0471 -0.0463 -0.0512 -0.0911 -0.1078

C

34 0.8323 0.7308 0.6878 0.3100 0.2550 0.0417 0.0387 -0.0406 -0.0847 n p 36 0.8477 0.8237 0.7760 0.5660 0.5086 0.2734 0.2483 0.0741 -0.0059 38 0.7082 0.6863 0.5562 0.5258 0.3917 0.3931 0.2552 0.1374 40 0.5241 0.5095 0.4898 0.4221 0.4243 0.3642 0.3429 0.2765 0.2205 45 0.1602 0.1732 0.1512 0.1397 0.1507 0.1353 0.1542 0.1435 0.1559 50 0.1898 0.1604 0.1617 0.1542 0.1489 0.1357 0.1395 0.1256 0.0996 60 -0.1132 -0.1142 -0.1187 -0.1049 -0.1061 -0.0944 -0.1004 -0.0994 -0.1058 70 -0.1416 -0.1132 -0.0832 -0.0997 -0.0472 -0.0447 -0.0380 -0.0155 75 -0.1373 -0.1062 -0.1092 -0.0906 -0.0635 -0.0564 -0.0308 0.0037 0 2.76E-04 2.66E-04 2.61E-04 2.14E-04 2.13E-04 1.88E-04 1.97E-04 2.04E-04 1.85E-04 5 3.38E-04 3.32E-04 3.33E-04 3.19E-04 3.31E-04 3.41E-04 3.33E-04 3.74E-04 3.73E-04 10 7.39E-04 7.04E-04 7.01E-04 5.95E-04 5.94E-04 5.04E-04 5.39E-04 5.16E-04 5.19E-04 15 9.25E-04 8.92E-04 8.80E-04 7.94E-04 7.55E-04 6.89E-04 6.97E-04 7.32E-04 9.35E-04 20 2.10E-03 1.98E-03 1.92E-03 1.55E-03 1.29E-03 1.27E-03 1.16E-03 1.12E-03 25 3.06E-03 2.88E-03 2.89E-03 2.78E-03 2.78E-03 2.70E-03 2.72E-03 2.62E-03 2.80E-03 28 5.24E-03 4.65E-03 4.41E-03 3.74E-03 3.82E-03 3.62E-03 3.75E-03 3.64E-03 3.64E-03 30 8.55E-03 7.61E-03 7.27E-03 4.31E-03 4.29E-03 3.96E-03 4.13E-03 4.19E-03 4.00E-03 32 1.70E-02 1.48E-02 1.48E-02 6.86E-03 6.18E-03 4.57E-03 4.18E-03 4.51E-03 4.39E-03

( ) C s

n p 34 1.62E-02 1.57E-02 1.62E-02 1.27E-02 1.09E-02 7.34E-03 7.03E-03 5.22E-03 5.16E-03 36 1.36E-02 1.18E-02 1.18E-02 1.07E-02 1.13E-02 1.10E-02 1.05E-02 7.80E-03 6.52E-03 38 1.88E-02 1.70E-02 1.14E-02 1.11E-02 9.65E-03 9.57E-03 8.22E-03 8.29E-03 40 2.61E-02 2.42E-02 2.32E-02 1.71E-02 1.64E-02 1.24E-02 1.24E-02 9.81E-03 8.39E-03 45 3.21E-02 3.02E-02 2.93E-02 2.28E-02 2.25E-02 1.71E-02 1.66E-02 1.38E-02 1.44E-02 50 2.14E-02 2.12E-02 2.10E-02 1.80E-02 1.77E-02 1.55E-02 1.47E-02 1.42E-02 1.22E-02 60 1.60E-02 1.59E-02 1.59E-02 1.43E-02 1.47E-02 1.29E-02 1.33E-02 1.03E-02 6.77E-03 70 1.94E-02 1.89E-02 1.52E-02 1.45E-02 1.24E-02 1.20E-02 1.03E-02 8.54E-03 75 1.91E-02 1.88E-02 1.49E-02 1.53E-02 1.22E-02 1.27E-02 1.13E-02 1.19E-02 Table XII. In-phase and out-of-phase components of side-force coefficient. Φ = 5 deg.

A k = Component α, deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 0 -0.0060 -0.0048 -0.0044 -0.0044 -0.0045 -0.0036 -0.0046 -0.0052 10 -0.0802 -0.0787 -0.0753 -0.0755 -0.0745 -0.0754 -0.0734 -0.0694 20 -0.1650 -0.1568 -0.1495 -0.1446 -0.1443 -0.1449 -0.1381 -0.1256 30 -0.1902 -0.1759 -0.1592 -0.1509 -0.1489 -0.1389 -0.1391 -0.1114 35 -0.1296 -0.1067 -0.0791 -0.0398 -0.0799 -0.0723 -0.0604 -0.0336

C

y β 40 -0.0074 0.0529 0.0757 0.1411 0.1393 0.0781 0.1187 0.1771 50 0.1058 0.0924 0.1497 0.1627 0.1751 0.1612 0.2427 0.3388 60 -0.0696 -0.0619 -0.0557 -0.0650 -0.0501 -0.0586 -0.0760 -0.0502 70 -0.0614 -0.0628 -0.0773 -0.0718 -0.0787 -0.0601 -0.0715 -0.0673 75 -0.1102 -0.1210 -0.1033 -0.1328 -0.1233 -0.1541 -0.1228 -0.1018 0 1.50E-04 1.73E-04 2.21E-04 2.01E-04 1.88E-04 2.19E-04 2.62E-04 2.41E-04 10 1.67E-04 2.71E-04 2.68E-04 2.92E-04 2.34E-04 2.67E-04 3.54E-04 3.37E-04 20 5.41E-04 6.71E-04 8.00E-04 7.40E-04 9.03E-04 7.89E-04 1.05E-03 1.13E-03 30 1.35E-03 1.92E-03 2.47E-03 2.58E-03 2.62E-03 2.98E-03 3.06E-03 4.02E-03 35 2.68E-03 3.43E-03 3.93E-03 4.03E-03 3.98E-03 4.05E-03 4.03E-03 4.84E-03

( ) C s

y β 40 8.12E-03 1.11E-02 1.39E-02 1.39E-02 1.45E-02 1.24E-02 1.66E-02 2.00E-02 50 1.63E-02 2.26E-02 2.64E-02 2.95E-02 3.30E-02 3.05E-02 2.53E-02 2.73E-02 60 8.36E-03 1.24E-02 1.60E-02 1.47E-02 1.49E-02 1.97E-02 2.21E-02 2.74E-02 70 2.13E-02 2.87E-02 3.25E-02 3.21E-02 3.22E-02 4.13E-02 3.83E-02 4.46E-02 75 1.18E-02 1.60E-02 1.88E-02 1.61E-02 1.84E-02 1.68E-02 2.16E-02 2.21E-02 k = Component α, deg 0.073 0.122 0.171 0.191 0.191 0.191 0.22 0.269 0 -0.0003 -0.0030 -0.0001 -0.0011 -0.0016 -0.0011 -0.0014 -0.0005 10 0.1153 0.1127 0.1119 0.1165 0.1101 0.1113 0.1102 0.1084 20 0.2579 0.2311 0.2083 0.2198 0.2204 0.2001 0.1994 0.2003 30 0.5005 0.4325 0.3929 0.4044 0.4128 0.4110 0.3780 0.3759 35 1.0029 0.7361 0.6136 0.6164 0.5679 0.5690 0.5496 0.5076

C

y 40 1.7640 1.4436 1.0978 1.0187 1.0379 0.9877 0.9362 0.8683 p 50 -0.5999 -0.0379 0.2472 0.2594 0.2177 0.2905 0.5315 0.0364 60 0.2097 -0.1482 -0.0680 -0.1195 -0.1232 -0.0895 -0.0903 -0.0628 70 0.1253 -0.1990 0.0397 -0.0456 0.1194 -0.0074 -0.0792 0.0244 75 -0.3113 -0.3605 -0.2758 -0.2704 -0.0868 -0.2485 -0.2745 -0.2315 0 2.05E-03 1.42E-03 1.29E-03 1.05E-03 9.86E-04 1.15E-03 1.19E-03 8.95E-04 10 2.28E-03 2.22E-03 1.57E-03 1.53E-03 1.22E-03 1.40E-03 1.61E-03 1.25E-03 20 7.40E-03 5.50E-03 4.68E-03 3.87E-03 4.73E-03 4.13E-03 4.78E-03 4.21E-03 30 1.85E-02 1.57E-02 1.44E-02 1.35E-02 1.37E-02 1.56E-02 1.39E-02 1.49E-02 35 3.67E-02 2.81E-02 2.30E-02 2.11E-02 2.08E-02 2.12E-02 1.83E-02 1.80E-02

( ) C s

y p 40 1.11E-01 9.12E-02 8.11E-02 7.25E-02 7.57E-02 6.51E-02 7.56E-02 7.42E-02 50 2.24E-01 1.85E-01 1.55E-01 1.54E-01 1.73E-01 1.60E-01 1.15E-01 1.02E-01 60 1.14E-01 1.01E-01 9.33E-02 7.71E-02 7.83E-02 1.03E-01 1.00E-01 1.02E-01 70 2.91E-01 2.35E-01 1.90E-01 1.68E-01 1.69E-01 2.16E-01 1.74E-01 1.66E-01 75 1.61E-01 1.31E-01 1.10E-01 8.41E-02 9.62E-02 8.77E-02 9.80E-02 8.20E-02 Table XIII. In-phase and out-of-phase components of side-force coefficient. Φ = 10 deg.

A k = Component α, deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 0 -0.0054 -0.0047 -0.0042 -0.0038 -0.0039 -0.0037 -0.0030 5 -0.0408 -0.0396 -0.0379 -0.0377 -0.0365 -0.0363 -0.0338 10 -0.0807 -0.0783 -0.0759 -0.0742 -0.0715 -0.0710 -0.0656 15 -0.1061 -0.1024 -0.0987 -0.0957 -0.0910 -0.0900 -0.0835 20 -0.1636 -0.1574 -0.1492 -0.1442 -0.1373 -0.1354 -0.1257 25 -0.1908 -0.1870 -0.1789 -0.1712 -0.1620 -0.1579 -0.1431 28 -0.1893 -0.1814 -0.1698 -0.1611 -0.1534 -0.1473 -0.1294 30 -0.1749 -0.1597 -0.1451 -0.1373 -0.1235 -0.1180 -0.0996 32 -0.1598 -0.1409 -0.1230 -0.1139 -0.1016 -0.0946 -0.0669

C

Y β 34 -0.1392 -0.1227 -0.1014 -0.0840 -0.0700 -0.0664 -0.0452 36 -0.1119 -0.0880 -0.0662 -0.0452 -0.0361 -0.0222 0.0043 38 -0.0675 -0.0384 -0.0037 0.0166 0.0268 0.0396 0.0681 40 0.0190 0.0579 0.0999 0.1174 0.1359 0.1597 0.1952 45 0.0881 0.1234 0.2034 0.2574 0.2782 0.2759 0.3854 50 0.0287 0.0390 0.1064 0.0841 0.1047 0.0856 0.1338 60 -0.0314 -0.0342 -0.0417 -0.0320 -0.0332 -0.0238 -0.0355 70 -0.0453 -0.0468 -0.0509 -0.0457 -0.0677 -0.0696 -0.0743 75 -0.1345 -0.1322 -0.1339 -0.1268 -0.1237 -0.1466 -0.1513 0 7.39E-05 1.02E-04 1.17E-04 1.16E-04 1.43E-04 1.35E-04 1.63E-04 5 9.52E-05 1.35E-04 1.41E-04 1.33E-04 1.57E-04 1.74E-04 2.11E-04 10 1.95E-04 1.79E-04 2.14E-04 2.23E-04 2.74E-04 2.90E-04 3.17E-04 15 2.30E-04 3.84E-04 3.70E-04 4.03E-04 4.04E-04 5.02E-04 4.64E-04 20 3.26E-04 4.78E-04 5.18E-04 5.35E-04 4.95E-04 6.74E-04 6.64E-04 25 4.66E-04 7.36E-04 9.12E-04 9.00E-04 1.06E-03 1.10E-03 1.07E-03 28 1.14E-03 1.59E-03 1.99E-03 2.16E-03 2.44E-03 2.50E-03 3.14E-03 30 1.60E-03 2.28E-03 2.62E-03 3.01E-03 3.42E-03 3.60E-03 4.38E-03 32 2.23E-03 3.39E-03 4.27E-03 4.29E-03 4.77E-03 5.17E-03 5.84E-03

( ) C s

34 3.13E-03 4.25E-03 5.78E-03 6.14E-03 6.29E-03 6.88E-03 7.86E-03 Y β 36 3.31E-03 4.44E-03 5.45E-03 6.37E-03 7.25E-03 7.74E-03 8.78E-03 38 3.65E-03 4.94E-03 6.53E-03 7.40E-03 8.70E-03 9.13E-03 1.05E-02 40 5.76E-03 8.17E-03 1.14E-02 1.24E-02 1.39E-02 1.50E-02 1.79E-02 45 1.16E-02 1.59E-02 2.11E-02 1.95E-02 2.30E-02 2.40E-02 2.34E-02 50 9.81E-03 1.38E-02 1.38E-02 1.36E-02 1.66E-02 1.60E-02 1.84E-02 60 5.76E-03 9.49E-03 1.17E-02 1.35E-02 1.50E-02 1.58E-02 1.68E-02 70 8.30E-03 1.24E-02 1.45E-02 1.53E-02 1.92E-02 1.67E-02 2.01E-02 75 6.52E-03 8.06E-03 8.63E-03 1.02E-02 1.10E-02 1.09E-02 1.38E-02 Table XIII. Concluded.

k = Component α, deg 0.073 0.122 0.171 0.191 0.22 0.232 0.269 0 0.0014 0.0011 0.0004 0.0015 -0.0006 0.0000 -0.0018 5 0.0866 0.0854 0.0819 0.0847 0.0847 0.0844 0.0807 10 0.1322 0.1170 0.1130 0.1138 0.1183 0.1157 0.1144 15 0.1339 0.1198 0.1216 0.1152 0.1145 0.1164 0.1084 20 0.2500 0.2300 0.2150 0.2089 0.1986 0.2003 0.1820 25 0.2879 0.2854 0.2514 0.2888 0.2672 0.2731 0.2613 28 0.3784 0.3529 0.3277 0.3475 0.3204 0.3312 0.3220 30 0.4938 0.4257 0.3920 0.3858 0.3705 0.3771 0.3613 32 0.6613 0.5461 0.4849 0.4880 0.4576 0.4420 0.4402

C

Y p 34 0.8223 0.6455 0.5756 0.5463 0.5248 0.5002 0.4901 36 1.0849 0.8057 0.6744 0.6433 0.6018 0.5986 0.5694 38 1.4133 1.0162 0.8341 0.8076 0.7275 0.7242 0.6605 40 1.6443 1.2074 0.9844 0.9085 0.8847 0.8348 0.7917 45 1.6233 1.5660 1.3549 1.2331 1.1734 1.1412 0.9706 50 0.0028 0.2304 0.1408 0.4016 0.3057 0.4462 0.4296 60 -0.0563 -0.1220 -0.0815 -0.0873 -0.1141 -0.0980 -0.0707 70 -0.8093 -0.5118 -0.4960 -0.3889 -0.3932 -0.3741 -0.3014 75 -0.2564 -0.2750 -0.2915 -0.2308 -0.4207 -0.2709 -0.2840 0 1.01E-03 8.36E-04 6.84E-04 6.09E-04 6.50E-04 5.80E-04 6.05E-04 5 1.30E-03 1.11E-03 8.27E-04 6.94E-04 7.15E-04 7.51E-04 7.85E-04 10 2.68E-03 1.47E-03 1.25E-03 1.17E-03 1.25E-03 1.25E-03 1.18E-03 15 3.15E-03 3.14E-03 2.17E-03 2.11E-03 1.84E-03 2.16E-03 1.72E-03 20 4.47E-03 3.92E-03 3.03E-03 2.80E-03 2.25E-03 2.91E-03 2.47E-03 25 6.38E-03 6.04E-03 5.33E-03 4.71E-03 4.80E-03 4.76E-03 3.99E-03 28 1.56E-02 1.30E-02 1.16E-02 1.13E-02 1.11E-02 1.08E-02 1.17E-02 30 2.19E-02 1.87E-02 1.53E-02 1.58E-02 1.56E-02 1.55E-02 1.63E-02 32 3.05E-02 2.78E-02 2.50E-02 2.24E-02 2.17E-02 2.23E-02 2.17E-02

( ) C s

Y 34 4.29E-02 3.48E-02 3.38E-02 3.21E-02 2.86E-02 2.97E-02 2.92E-02 p 36 4.54E-02 3.64E-02 3.19E-02 3.34E-02 3.30E-02 3.34E-02 3.26E-02 38 5.00E-02 4.05E-02 3.82E-02 3.88E-02 3.96E-02 3.94E-02 3.90E-02 40 7.89E-02 6.70E-02 6.65E-02 6.48E-02 6.32E-02 6.47E-02 6.64E-02 45 1.58E-01 1.30E-01 1.23E-01 1.02E-01 1.04E-01 1.03E-01 8.71E-02 50 1.34E-01 1.13E-01 8.06E-02 7.10E-02 7.56E-02 6.89E-02 6.83E-02 60 7.89E-02 7.78E-02 6.85E-02 7.09E-02 6.82E-02 6.81E-02 6.25E-02 70 1.14E-01 1.01E-01 8.48E-02 8.02E-02 8.72E-02 7.21E-02 7.48E-02 75 8.94E-02 6.61E-02 5.05E-02 5.33E-02 5.02E-02 4.69E-02 5.13E-02 Table XIV. In-phase and out-of-phase components of side-force coefficient. Φ = 20 deg.

A k = α, deg Component 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 0 -0.0048 -0.0056 -0.0053 -0.0043 -0.0038 -0.0025 -0.0035 5 -0.0426 -0.0429 10 -0.0818 -0.0877 -0.0866 -0.0808 -0.0766 -0.0773 -0.0729 15 -0.1265 -0.1241 20 -0.1677 -0.1758 -0.1739 -0.1609 -0.1521 -0.1515 -0.1385 25 -0.2068 -0.2041 28 -0.2085 -0.2055 30 -0.1891 -0.1980 -0.1938 -0.1779 -0.1625 -0.1589 -0.1331

C

32 -0.1781 -0.1757 Y β 34 -0.1368 -0.1359 35 -0.0825 -0.1014 -0.0935 -0.0788 -0.0499 36 -0.0409 -0.0588 38 0.0066 0.0295 40 -0.0137 0.0249 0.0445 0.0393 0.1073 0.1713 0.2027 45 0.1224 0.1260 50 0.0015 0.0151 0.0230 0.0176 0.0327 0.0325 0.0712 60 -0.0718 -0.0749 -0.0736 -0.0706 -0.0718 -0.0664 -0.0567 70 -0.1028 -0.1004 -0.1097 -0.1054 -0.1030 -0.0920 -0.0962 75 -0.1304 -0.1352 -0.1387 -0.1339 -0.1425 -0.1477 -0.1524 0 5.27E-05 6.29E-05 6.56E-05 6.49E-05 8.74E-05 1.24E-04 1.60E-04 5 8.06E-05 9.36E-05 10 1.04E-04 1.53E-04 1.63E-04 1.64E-04 1.99E-04 2.53E-04 3.33E-04 15 3.98E-04 4.16E-04 20 2.76E-04 3.45E-04 3.50E-04 3.38E-04 4.53E-04 4.35E-04 5.13E-04 25 4.68E-04 5.50E-04 28 8.00E-04 9.33E-04 30 8.98E-04 1.06E-03 1.30E-03 1.37E-03 1.82E-03 2.35E-03 3.28E-03 32 1.65E-03 1.76E-03 34 2.38E-03 2.42E-03

( ) C s

Y β 35 3.15E-03 2.91E-03 3.55E-03 4.18E-03 5.64E-03 36 4.23E-03 4.09E-03 38 4.76E-03 5.51E-03 40 3.96E-03 4.90E-03 5.57E-03 5.93E-03 7.58E-03 8.70E-03 9.56E-03 45 9.18E-03 1.01E-02 50 4.86E-03 6.59E-03 8.15E-03 8.38E-03 1.13E-02 1.43E-02 1.53E-02 60 2.99E-03 3.54E-03 4.41E-03 4.64E-03 6.76E-03 9.16E-03 9.42E-03 70 3.69E-03 4.15E-03 4.57E-03 5.06E-03 5.85E-03 6.52E-03 7.71E-03 75 3.75E-03 4.46E-03 4.77E-03 5.26E-03 5.82E-03 6.78E-03 8.85E-03 Table XIV. Concluded.

k = Component α, deg 0.073 0.095 0.115 0.122 0.171 0.22 0.22 0.269 0 0.0018 -0.0019 -0.0018 0.0010 0.0014 -0.0031 -0.0032 5 0.0879 0.0911 10 0.1302 0.1293 0.1309 0.1151 0.1212 0.1171 0.1134 15 0.1515 0.1583 20 0.2678 0.2524 0.2585 0.2293 0.2189 0.2256 0.2098 25 0.2843 0.2901 28 0.3950 0.3899 30 0.5152 0.4806 0.4746 0.4499 0.4174 0.4054 0.3737 32 0.5250 0.5295

34 0.5342 0.5550 C

Y p 35 0.6012 0.5763 0.5427 0.5250 0.5162 36 0.6620 0.6133 38 1.2395 1.0199 40 1.5419 1.4384 1.3691 1.2947 1.0371 0.8275 0.6454 45 0.8638 0.8175 50 0.1486 0.1765 0.1172 0.0969 -0.0421 -0.0824 0.0437 60 -0.1600 -0.1709 -0.1704 -0.2013 -0.1660 -0.1506 -0.1565 70 -0.4224 -0.3849 -0.4024 -0.3179 -0.2710 -0.2845 -0.3489 75 -0.5176 -0.4445 -0.3861 -0.3518 -0.3466 -0.3210 -0.3707 0 7.21E-04 6.63E-04 5.71E-04 5.32E-04 5.11E-04 5.64E-04 5.94E-04 5 8.48E-04 8.14E-04 10 1.43E-03 1.61E-03 1.42E-03 1.34E-03 1.17E-03 1.15E-03 1.24E-03 15 4.19E-03 3.62E-03 20 3.79E-03 3.63E-03 3.05E-03 2.77E-03 2.65E-03 1.98E-03 1.91E-03 25 4.92E-03 4.78E-03 28 8.42E-03 8.11E-03 30 1.23E-02 1.12E-02 1.13E-02 1.12E-02 1.07E-02 1.07E-02 1.22E-02 32 1.73E-02 1.53E-02 34 2.50E-02 2.10E-02

35 4.32E-02 2.39E-02 2.08E-02 1.90E-02 2.10E-02 ( ) C s

Y p 36 4.45E-02 3.56E-02 38 5.01E-02 4.79E-02 40 5.43E-02 5.16E-02 4.84E-02 4.86E-02 4.43E-02 3.95E-02 3.55E-02 45 9.67E-02 8.75E-02 50 6.66E-02 6.93E-02 7.09E-02 6.87E-02 6.62E-02 6.50E-02 5.68E-02 60 4.10E-02 3.73E-02 3.83E-02 3.80E-02 3.95E-02 4.16E-02 3.50E-02 70 5.05E-02 4.37E-02 3.97E-02 4.15E-02 3.42E-02 2.97E-02 2.87E-02 75 5.13E-02 4.69E-02 4.15E-02 4.31E-02 3.40E-02 3.08E-02 3.29E-02 Table XV. In-phase and out-of-phase components of side-force coefficient. Φ = 30 deg.

A k = Component α, deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 0 -0.0049 -0.0048 -0.0048 -0.0050 -0.0045 -0.0041 -0.0042 -0.0045 -0.0033 5 -0.0428 -0.0421 -0.0426 -0.0414 -0.0408 -0.0409 -0.0399 -0.0390 -0.0355 10 -0.0880 -0.0884 -0.0879 -0.0860 -0.0861 -0.0832 -0.0835 -0.0785 -0.0728 15 -0.1303 -0.1301 -0.1291 -0.1277 -0.1266 -0.1221 -0.1213 -0.1144 -0.1040 20 -0.1776 -0.1760 -0.1769 -0.1721 -0.1626 -0.1632 -0.1506 -0.1377 25 -0.2072 -0.2059 -0.2059 -0.1996 -0.1980 -0.1895 -0.1895 -0.1768 -0.1605 28 -0.2037 -0.2032 -0.2026 -0.2015 -0.1987 -0.1889 -0.1898 -0.1741 -0.1543 30 -0.1795 -0.1827 -0.1825 -0.1872 -0.1877 -0.1779 -0.1774 -0.1627 -0.1402 32 -0.1223 -0.1244 -0.1319 -0.1590 -0.1604 -0.1573 -0.1591 -0.1406 -0.1181

C

Y β 34 -0.0779 -0.0753 -0.0761 -0.0951 -0.0994 -0.1110 -0.1142 -0.1015 -0.0759 36 -0.0702 -0.0622 -0.0590 -0.0336 -0.0298 -0.0206 -0.0291 -0.0267 -0.0084 38 -0.0636 -0.0529 -0.0190 -0.0105 0.0307 0.0357 0.0640 0.0811 40 -0.0480 -0.0436 -0.0371 -0.0046 -0.0045 0.0403 0.0408 0.0908 0.1353 45 -0.0113 -0.0101 -0.0085 0.0028 0.0098 0.0312 0.0348 0.0648 0.1185 50 -0.0477 -0.0477 -0.0471 -0.0332 -0.0345 -0.0201 -0.0225 -0.0062 0.0205 60 -0.1027 -0.1045 -0.1019 -0.0990 -0.0998 -0.0924 -0.0905 -0.0764 -0.0586 70 -0.1320 -0.1339 -0.1245 -0.1225 -0.1163 -0.1204 -0.1123 -0.0927 75 -0.1402 -0.1451 -0.1476 -0.1426 -0.1407 -0.1440 -0.1363 -0.1393 0 5.18E-05 5.80E-05 6.16E-05 7.73E-05 8.14E-05 9.68E-05 1.14E-04 1.45E-04 1.67E-04 5 7.08E-05 7.59E-05 8.22E-05 1.12E-04 1.16E-04 1.59E-04 1.52E-04 1.98E-04 2.24E-04 10 1.05E-04 1.09E-04 1.09E-04 1.53E-04 1.45E-04 1.89E-04 2.31E-04 2.99E-04 3.40E-04 15 3.41E-04 3.63E-04 3.77E-04 4.79E-04 4.88E-04 5.40E-04 5.70E-04 7.01E-04 1.02E-03 20 2.47E-04 2.71E-04 2.75E-04 3.50E-04 5.37E-04 5.10E-04 7.79E-04 1.12E-03 25 3.80E-04 4.10E-04 4.49E-04 6.00E-04 6.37E-04 8.17E-04 8.53E-04 1.06E-03 1.69E-03 28 6.86E-04 6.95E-04 7.20E-04 9.39E-04 1.02E-03 1.45E-03 1.51E-03 1.95E-03 2.49E-03 30 1.33E-03 1.31E-03 1.38E-03 1.37E-03 1.38E-03 1.90E-03 2.02E-03 2.91E-03 3.85E-03 32 2.60E-03 2.73E-03 2.71E-03 2.32E-03 2.33E-03 2.52E-03 2.45E-03 3.36E-03 4.53E-03

( ) C s

Y β 34 2.39E-03 2.94E-03 3.03E-03 4.05E-03 4.10E-03 3.94E-03 3.89E-03 4.43E-03 5.20E-03 36 2.05E-03 2.34E-03 2.50E-03 3.99E-03 4.33E-03 5.91E-03 5.85E-03 6.37E-03 7.59E-03 38 2.62E-03 2.94E-03 3.93E-03 4.16E-03 5.24E-03 5.26E-03 6.69E-03 8.14E-03 40 3.41E-03 3.74E-03 3.90E-03 5.46E-03 5.54E-03 6.80E-03 7.05E-03 7.87E-03 7.96E-03 45 4.92E-03 5.19E-03 5.49E-03 6.85E-03 6.80E-03 8.62E-03 9.24E-03 1.21E-02 1.58E-02 50 4.55E-03 4.80E-03 5.09E-03 6.83E-03 7.19E-03 9.34E-03 9.54E-03 1.09E-02 1.22E-02 60 2.01E-03 2.18E-03 2.27E-03 3.21E-03 3.13E-03 5.12E-03 5.05E-03 7.46E-03 8.47E-03 70 2.42E-03 2.84E-03 3.66E-03 4.31E-03 5.65E-03 5.14E-03 6.48E-03 7.65E-03 75 2.39E-03 2.78E-03 3.63E-03 3.75E-03 4.89E-03 4.52E-03 5.82E-03 7.61E-03 Table XV. Concluded.

k = Component α, deg 0.064 0.073 0.076 0.115 0.122 0.169 0.171 0.22 0.269 0 0.0014 -0.0011 0.0007 -0.0020 -0.0014 -0.0010 -0.0004 0.0024 0.0018 5 0.0945 0.0951 0.0934 0.0870 0.0892 0.0899 0.0809 0.0847 0.0788 10 0.1454 0.1372 0.1440 0.1329 0.1319 0.1239 0.1224 0.1321 0.1192 15 0.1920 0.1750 0.1794 0.1711 0.1705 0.1410 0.1439 0.1497 0.1342 20 0.2947 0.2694 0.2752 0.2517 0.2317 0.2199 0.2316 0.2002 25 0.3824 0.3449 0.3441 0.3139 0.3291 0.2812 0.2656 0.2602 0.2408 28 0.4084 0.4123 0.4320 0.3508 0.3599 0.3211 0.3226 0.3181 0.3007 30 0.3204 0.3511 0.3592 0.3829 0.3957 0.3760 0.3739 0.3482 0.3382 32 0.0366 0.1008 0.1371 0.3535 0.3561 0.3946 0.3883 0.4053 0.3816

C

Y 34 0.3984 0.2942 0.2852 0.2519 0.2628 0.3831 0.3741 0.4009 0.4061 p 36 0.9178 0.8457 0.8673 0.5088 0.4674 0.3545 0.3565 0.3833 0.3953 38 1.2065 1.1866 0.9346 0.8743 0.6154 0.5886 0.4313 0.3629 40 1.2434 1.2076 1.1776 0.9893 1.0012 0.7932 0.7714 0.6127 0.4879 45 0.8400 0.8663 0.8186 0.7188 0.7025 0.5924 0.5891 0.5460 0.5190 50 0.0843 0.1683 0.1595 0.1639 0.1423 0.1232 0.1613 0.1474 0.1494 60 -0.0890 -0.1248 -0.0842 -0.0952 -0.1272 -0.1105 -0.0786 -0.1094 -0.1132 70 -0.3033 -0.1997 -0.1938 -0.1478 -0.1877 -0.1678 -0.1900 -0.2315 75 -0.4058 -0.3341 -0.3271 -0.2970 -0.3178 -0.3324 -0.2895 -0.3133 0 8.09E-04 7.95E-04 8.11E-04 6.72E-04 6.67E-04 5.73E-04 6.67E-04 6.57E-04 6.20E-04 5 1.11E-03 1.04E-03 1.08E-03 9.71E-04 9.55E-04 9.39E-04 8.89E-04 9.00E-04 8.31E-04 10 1.64E-03 1.49E-03 1.44E-03 1.33E-03 1.19E-03 1.12E-03 1.35E-03 1.36E-03 1.26E-03 15 5.33E-03 4.98E-03 4.96E-03 4.16E-03 4.00E-03 3.19E-03 3.33E-03 3.19E-03 3.80E-03 20 3.86E-03 3.72E-03 3.61E-03 3.04E-03 3.18E-03 2.98E-03 3.54E-03 4.17E-03 25 5.94E-03 5.61E-03 5.91E-03 5.21E-03 5.22E-03 4.84E-03 4.99E-03 4.80E-03 6.29E-03 28 1.07E-02 9.52E-03 9.48E-03 8.17E-03 8.37E-03 8.57E-03 8.80E-03 8.85E-03 9.27E-03 30 2.07E-02 1.80E-02 1.81E-02 1.19E-02 1.13E-02 1.12E-02 1.18E-02 1.32E-02 1.43E-02 32 4.06E-02 3.74E-02 3.57E-02 2.01E-02 1.91E-02 1.49E-02 1.43E-02 1.53E-02 1.68E-02

( ) C s

Y 34 3.73E-02 4.02E-02 3.99E-02 3.52E-02 3.36E-02 2.33E-02 2.28E-02 2.01E-02 1.93E-02 p 36 3.20E-02 3.20E-02 3.29E-02 3.47E-02 3.55E-02 3.50E-02 3.42E-02 2.90E-02 2.82E-02 38 4.09E-02 4.03E-02 3.42E-02 3.41E-02 3.10E-02 3.07E-02 3.04E-02 3.03E-02 40 5.33E-02 5.12E-02 5.13E-02 4.75E-02 4.54E-02 4.02E-02 4.12E-02 3.58E-02 2.96E-02 45 7.69E-02 7.11E-02 7.23E-02 5.96E-02 5.58E-02 5.10E-02 5.40E-02 5.49E-02 5.88E-02 50 7.11E-02 6.58E-02 6.69E-02 5.94E-02 5.89E-02 5.53E-02 5.58E-02 4.97E-02 4.55E-02 60 3.14E-02 2.99E-02 2.99E-02 2.79E-02 2.57E-02 3.03E-02 2.95E-02 3.39E-02 3.15E-02 70 3.78E-02 3.74E-02 3.18E-02 3.53E-02 3.34E-02 3.01E-02 2.94E-02 2.84E-02 75 3.73E-02 3.66E-02 3.16E-02 3.08E-02 2.89E-02 2.64E-02 2.65E-02 2.83E-02 Table XVI. In-phase and out-of-phase components of rolling moment coefficient. Φ = 30 deg, Φ = 15 deg.

A 0 k = Component deg 0.064 0.115 0.169 θ, 0 0.0023 0.0015 -0.0005 5 -0.0037 -0.0046 -0.0058 10 -0.0158 -0.0174 -0.0192 15 -0.0332 -0.0346 -0.0371 20 -0.0547 -0.0578 -0.0623 25 -0.0735 -0.0822 -0.0885 28 -0.0832 -0.0990 -0.1070 30 -0.0860 -0.1097 -0.1228 32 -0.0882 -0.1164 -0.1330

C

34 -0.0896 -0.1216 -0.1407 l β 36 -0.0831 -0.1180 -0.1389 38 -0.0704 -0.1037 -0.1343 40 -0.0638 -0.0921 -0.1217 45 -0.0725 -0.0827 -0.0965 50 -0.0785 -0.0791 -0.0851 60 -0.0828 -0.0811 -0.0821 70 -0.0960 -0.0938 -0.0923 75 -0.0955 -0.0939 0 4.86E-05 8.75E-05 1.27E-04 5 3.22E-05 7.13E-05 1.17E-04 10 8.17E-05 1.70E-04 2.96E-04 15 2.78E-04 4.44E-04 6.09E-04 20 4.66E-04 7.75E-04 1.05E-03 25 7.15E-04 9.90E-04 1.25E-03 28 8.26E-04 1.24E-03 1.61E-03 30 9.69E-04 1.47E-03 1.89E-03 32 1.28E-03 1.80E-03 2.21E-03

( )

C s

34 1.69E-03 2.32E-03 2.80E-03 l β 36 1.96E-03 2.70E-03 3.35E-03 38 2.24E-03 3.11E-03 3.90E-03 40 2.32E-03 3.51E-03 4.38E-03 45 1.38E-03 2.33E-03 3.59E-03 50 1.17E-03 1.63E-03 2.37E-03 60 4.81E-04 7.54E-04 1.10E-03 70 6.00E-04 9.40E-04 1.32E-03 75 9.49E-04 1.31E-03 Table XVI. Concluded.

k = θ, deg Component 0.064 0.115 0.169 0 -0.2031 -0.2024 -0.1945 5 -0.1641 -0.1629 -0.1580 10 -0.2068 -0.2092 -0.1991 15 -0.2090 -0.2155 -0.2019 20 -0.2286 -0.2189 -0.2025 25 -0.3646 -0.3217 -0.2672 28 -0.5691 -0.4096 -0.3115 30 -0.8177 -0.5168 -0.3537 32 -1.0082 -0.6410 -0.4129

C

l 34 -1.0796 -0.6917 -0.4358 p 36 -1.0440 -0.6873 -0.4523 38 -1.0170 -0.6946 -0.4922 40 -0.7984 -0.6899 -0.5170 45 -0.4085 -0.3866 -0.3349 50 -0.1160 -0.1755 -0.1885 60 -0.1008 -0.1047 -0.1133 70 -0.0249 -0.0752 -0.0842 75 -0.0598 -0.0846 0 7.59E-04 7.61E-04 7.51E-04 5 5.03E-04 6.20E-04 6.90E-04 10 1.28E-03 1.48E-03 1.75E-03 15 4.34E-03 3.86E-03 3.60E-03 20 7.28E-03 6.74E-03 6.21E-03 25 1.12E-02 8.60E-03 7.42E-03 28 1.29E-02 1.08E-02 9.51E-03 30 1.51E-02 1.27E-02 1.12E-02 32 2.00E-02 1.57E-02 1.31E-02

( ) C s

34 2.65E-02 2.02E-02 1.66E-02 l p 36 3.07E-02 2.35E-02 1.98E-02 38 3.50E-02 2.70E-02 2.31E-02 40 3.63E-02 3.05E-02 2.59E-02 45 2.16E-02 2.03E-02 2.12E-02 50 1.83E-02 1.42E-02 1.40E-02 60 7.52E-03 6.56E-03 6.52E-03 70 9.38E-03 8.17E-03 7.79E-03 75 8.25E-03 7.76E-03 Table XVII. In-phase and out-of-phase components of rolling moment coefficient. Φ = 30 deg, Φ = 30 deg.

A 0 k = θ, deg Component 0.064 0.115 0.169 0 0.0005 -0.0006 -0.0008 5 -0.0029 -0.0041 -0.0039 10 -0.0110 -0.0122 -0.0123 15 -0.0198 -0.0212 -0.0209 20 -0.0315 -0.0339 -0.0344 25 -0.0446 -0.0495 -0.0520 28 -0.0588 -0.0654 -0.0676 30 -0.0711 -0.0769 -0.0786 32 -0.0814 -0.0861 -0.0849

C

34 -0.0905 -0.0978 -0.0986 l β 36 -0.0973 -0.1075 -0.1090 38 -0.0989 -0.1091 -0.1144 40 -0.1011 -0.1087 -0.1138 45 -0.0974 -0.1023 -0.1070 50 -0.0897 -0.0870 -0.0859 60 -0.0880 -0.0793 -0.0736 70 -0.0994 -0.0874 -0.0810 75 -0.0979 -0.0859 -0.0800 0 5.32E-05 9.16E-05 1.51E-04 5 3.93E-05 7.23E-05 1.26E-04 10 1.78E-04 2.89E-04 4.23E-04 15 4.10E-04 5.97E-04 7.69E-04 20 6.97E-04 1.06E-03 1.36E-03 25 8.86E-04 1.43E-03 1.94E-03 28 1.09E-03 1.79E-03 2.38E-03 30 1.37E-03 2.10E-03 2.69E-03 32 1.64E-03 2.59E-03 3.23E-03

( ) C s

l 34 1.67E-03 2.54E-03 3.32E-03 β 36 1.88E-03 3.02E-03 3.82E-03 38 1.80E-03 3.16E-03 4.40E-03 40 1.54E-03 3.15E-03 4.56E-03 45 1.50E-03 2.88E-03 4.51E-03 50 1.41E-03 2.78E-03 4.25E-03 60 1.41E-03 2.24E-03 2.99E-03 70 1.35E-03 2.31E-03 3.19E-03 75 1.30E-03 2.13E-03 2.97E-03 Table XVII. Concluded.

k = Component deg 0.064 0.115 0.169 θ, 0 -0.1870 -0.1833 -0.1783 5 -0.1504 -0.1479 -0.1466 10 -0.1835 -0.1802 -0.1761 15 -0.1916 -0.1903 -0.1875 20 -0.1955 -0.1879 -0.1820 25 -0.2695 -0.2252 -0.2016 28 -0.3173 -0.2323 -0.1934 30 -0.3828 -0.2243 -0.1680 32 -0.3635 -0.2026 -0.1235

C

34 -0.3681 -0.1955 -0.1241 l p 36 -0.2730 -0.1654 -0.0952 38 -0.1966 -0.1304 -0.0481 40 -0.1002 -0.0870 -0.0646 45 -0.0199 -0.0300 -0.0174 50 0.1474 0.0741 0.0167 60 0.0929 0.0349 0.0073 70 0.2763 0.1492 0.0530 75 0.2403 0.1317 0.0618 0 8.32E-04 7.96E-04 8.93E-04 5 6.14E-04 6.28E-04 7.48E-04 10 2.79E-03 2.51E-03 2.51E-03 15 6.41E-03 5.20E-03 4.55E-03 20 1.09E-02 9.19E-03 8.06E-03 25 1.38E-02 1.24E-02 1.15E-02 28 1.70E-02 1.55E-02 1.41E-02 30 2.14E-02 1.82E-02 1.59E-02 32 2.57E-02 2.25E-02 1.91E-02

( ) C s

34 2.60E-02 2.21E-02 1.97E-02 l p 36 2.93E-02 2.63E-02 2.26E-02 38 2.81E-02 2.75E-02 2.60E-02 40 2.41E-02 2.74E-02 2.70E-02 45 2.34E-02 2.50E-02 2.67E-02 50 2.20E-02 2.42E-02 2.52E-02 60 2.21E-02 1.95E-02 1.77E-02 70 2.10E-02 2.01E-02 1.89E-02 75 2.03E-02 1.85E-02 1.75E-02 Table XVIII. In-phase and out-of-phase components of rolling moment coefficient. Φ = 30 deg, Φ = 45 deg.

A 0 k = Component θ, deg 0.064 0.115 0.169 0 -0.0002 -0.0015 -0.0020 5 -0.0009 -0.0023 -0.0029 10 -0.0027 -0.0035 -0.0041 15 -0.0040 -0.0047 20 -0.0042 -0.0052 -0.0060 25 -0.0131 -0.0140 -0.0140 28 -0.0267 -0.0251 -0.0233 30 -0.0416 -0.0347 -0.0283 32 -0.0600 -0.0494 -0.0363 34 -0.0688 -0.0610 -0.0488

C

l 36 -0.0717 -0.0672 -0.0553 β 38 -0.0698 -0.0676 -0.0571 40 -0.0664 -0.0627 -0.0540 45 -0.0538 -0.0534 -0.0450 50 -0.0507 -0.0460 -0.0376 60 -0.0507 -0.0469 -0.0418 70 -0.0473 -0.0410 -0.0305 75 -0.0456 -0.0390 -0.0286 0 4.47E-05 8.63E-05 1.41E-04 5 5.99E-05 9.12E-05 1.46E-04 10 2.72E-04 3.96E-04 5.14E-04 15 6.66E-04 8.36E-04 20 8.58E-04 1.15E-03 1.42E-03 25 1.21E-03 1.80E-03 2.33E-03 28 1.55E-03 2.19E-03 2.78E-03 30 1.69E-03 2.32E-03 2.88E-03 32 1.95E-03 2.57E-03 3.10E-03

( ) C s

34 2.11E-03 2.89E-03 3.59E-03 l β 36 2.35E-03 3.26E-03 3.94E-03 38 2.55E-03 3.64E-03 4.50E-03 40 2.67E-03 4.01E-03 5.15E-03 45 2.91E-03 4.56E-03 6.12E-03 50 2.66E-03 4.10E-03 5.80E-03 60 2.48E-03 3.71E-03 5.20E-03 70 2.79E-03 3.87E-03 5.14E-03 75 2.69E-03 3.70E-03 4.66E-03 Table XVIII. Concluded.

k = Component deg 0.064 0.115 0.169 θ, 0 -0.1913 -0.1850 -0.1795 5 -0.1582 -0.1532 -0.1500 10 -0.1787 -0.1723 -0.1680 15 -0.1891 -0.1829 20 -0.1993 -0.1908 -0.1839 25 -0.1861 -0.1754 -0.1682 28 -0.1399 -0.1271 -0.1241 30 -0.0100 -0.0305 -0.0568 32 0.0796 0.0607 0.0172

C

34 0.0878 0.0926 0.0520 l p 36 0.1233 0.1170 0.0894 38 0.1535 0.1406 0.1162 40 0.1782 0.1750 0.1506 45 0.1768 0.1747 0.1684 50 0.2659 0.2237 0.1969 60 0.2565 0.2173 0.1724 70 0.3841 0.2563 0.2133 75 0.3363 0.2695 0.2188 0 6.98E-04 7.50E-04 8.36E-04 5 9.36E-04 7.93E-04 8.64E-04 10 4.24E-03 3.44E-03 3.04E-03 15 5.79E-03 4.95E-03 20 1.34E-02 9.97E-03 8.38E-03 25 1.89E-02 1.56E-02 1.38E-02 28 2.42E-02 1.91E-02 1.64E-02 30 2.65E-02 2.02E-02 1.70E-02 32 3.04E-02 2.23E-02 1.84E-02

( ) C s

34 3.29E-02 2.51E-02 2.12E-02 l p 36 3.66E-02 2.83E-02 2.33E-02 38 3.98E-02 3.16E-02 2.66E-02 40 4.18E-02 3.49E-02 3.05E-02 45 4.54E-02 3.97E-02 3.62E-02 50 4.16E-02 3.57E-02 3.43E-02 60 3.87E-02 3.23E-02 3.08E-02 70 4.36E-02 3.36E-02 3.04E-02 75 4.21E-02 3.22E-02 2.76E-02 Table XIX. In-phase and out-of-phase components of rolling moment coefficient. Φ = 30 deg, Φ = 60 deg.

A 0 k = θ , deg Component 0.064 0.115 0.169 0 -0.0013 -0.0022 -0.0038 5 0.0008 -0.0001 -0.0015 10 0.0063 0.0055 0.0047 15 0.0157 0.0153 0.0131 20 0.0268 0.0261 0.0254 25 0.0315 0.0329 0.0335 28 0.0280 0.0333 0.0379 30 0.0201 0.0308 0.0399 32 0.0283 0.0404

C

l β 34 0.0202 0.0278 0.0383 36 0.0255 0.0310 0.0413 38 0.0345 0.0394 0.0479 40 0.0424 0.0501 0.0621 45 0.0567 0.0669 0.0807 50 0.0647 0.0751 0.0881 60 0.0625 0.0726 0.0848 70 0.0769 0.0882 0.1008 75 0.0795 0 3.66E-05 8.20E-05 1.33E-04 5 5.66E-05 9.23E-05 1.45E-04 10 2.53E-04 3.45E-04 4.23E-04 15 4.71E-04 6.35E-04 7.54E-04 20 8.07E-04 1.09E-03 1.31E-03 25 1.39E-03 1.87E-03 2.21E-03 28 1.97E-03 2.39E-03 2.71E-03 30 2.41E-03 2.75E-03 2.97E-03 32 3.41E-03 3.45E-03

( ) C s

l 34 3.11E-03 3.94E-03 4.20E-03 β 36 3.31E-03 4.24E-03 4.71E-03 38 3.53E-03 4.50E-03 5.05E-03 40 3.73E-03 4.89E-03 5.52E-03 45 4.00E-03 5.44E-03 6.22E-03 50 4.29E-03 5.76E-03 6.91E-03 60 4.71E-03 6.05E-03 7.25E-03 70 4.80E-03 6.11E-03 7.20E-03 75 4.79E-03 Table XIX. Concluded.

k = Component θ, deg 0.064 0.115 0.169 0 -0.1914 -0.1864 -0.1804 5 -0.1670 -0.1619 -0.1575 10 -0.1746 -0.1681 -0.1628 15 -0.2016 -0.1929 -0.1867 20 -0.2091 -0.1979 -0.1910 25 -0.1784 -0.1698 -0.1690 28 -0.0894 -0.1141 -0.1389 30 0.0200 -0.0255 -0.0767 32 0.0033 -0.0299

C

34 0.0225 0.0138 -0.0028 l p 36 0.0492 0.0290 0.0134 38 0.0896 0.0530 0.0329 40 0.1189 0.0898 0.0479 45 0.1545 0.1120 0.0873 50 0.1516 0.1071 0.0888 60 0.1226 0.1236 0.0965 70 0.1916 0.1226 0.1065 75 0.1458 0 5.71E-04 7.13E-04 7.89E-04 5 8.85E-04 8.02E-04 8.60E-04 10 3.96E-03 3.00E-03 2.50E-03 15 7.36E-03 5.52E-03 4.46E-03 20 1.26E-02 9.51E-03 7.74E-03 25 2.17E-02 1.63E-02 1.31E-02 28 3.08E-02 2.08E-02 1.60E-02 30 3.77E-02 2.40E-02 1.76E-02 32 2.96E-02 2.04E-02

( )

C s

34 4.86E-02 3.42E-02 2.49E-02 l p 36 5.17E-02 3.69E-02 2.79E-02 38 5.52E-02 3.92E-02 2.99E-02 40 5.83E-02 4.25E-02 3.26E-02 45 6.25E-02 4.73E-02 3.68E-02 50 6.70E-02 5.01E-02 4.09E-02 60 7.37E-02 5.26E-02 4.29E-02 70 7.50E-02 5.31E-02 4.26E-02 75 7.49E-02 Table XX. In-phase and out-of-phase components of rolling moment coefficient. Φ = 10 deg, Re = 0.6x10 .

A k = Component deg 0.188 0.228 0.322 0.505 θ, 0 0.0016 0.0023 -0.0002 -0.0052 5 -0.0050 -0.0048 -0.0057 -0.0092 10 -0.0177 -0.0176 -0.0186 -0.0092 15 -0.0415 -0.0423 -0.0441 -0.0448 20 -0.0697 -0.0730 -0.0741 -0.0718 25 -0.0946 -0.0974 -0.1016 -0.0993 28 -0.1198 -0.1242 -0.1292 -0.1284 30 -0.1439 -0.1482 -0.1489 -0.1526 32 -0.1610 -0.1660 -0.1686 -0.1734

C

l 34 -0.1726 -0.1789 -0.1834 -0.1919 β 36 -0.1713 -0.1825 -0.1882 -0.2017 38 -0.1623 -0.1634 -0.1840 -0.2016 40 -0.1317 -0.1544 -0.1765 -0.1933 45 -0.0703 -0.0881 -0.1227 -0.1158 50 -0.0536 -0.0559 -0.0571 -0.0616 60 -0.0769 -0.0765 -0.0682 -0.0583 70 -0.0836 -0.0842 -0.0718 -0.0688 75 -0.0822 -0.0860 -0.0758 -0.0691 0 6.58E-05 6.83E-05 1.43E-04 5.16E-04 5 6.53E-05 7.38E-05 1.35E-04 5.36E-04 10 9.47E-05 1.06E-04 1.98E-04 5.36E-04 15 1.98E-04 2.25E-04 2.78E-04 5.92E-04 20 3.92E-04 4.24E-04 5.37E-04 8.49E-04 25 4.95E-04 5.97E-04 7.34E-04 1.01E-03 28 5.79E-04 6.62E-04 8.27E-04 1.19E-03 30 7.21E-04 7.70E-04 8.45E-04 1.37E-03 32 9.68E-04 1.18E-03 1.14E-03 1.66E-03

( ) C s

l 34 1.26E-03 1.66E-03 1.74E-03 2.31E-03 β 36 1.82E-03 2.17E-03 2.32E-03 3.41E-03 38 2.66E-03 2.62E-03 3.03E-03 3.12E-03 40 2.60E-03 3.25E-03 3.64E-03 4.18E-03 45 3.59E-03 4.42E-03 4.18E-03 4.27E-03 50 1.50E-03 1.77E-03 2.01E-03 3.41E-03 60 9.72E-04 9.91E-04 1.14E-03 1.72E-03 70 1.22E-03 1.30E-03 1.63E-03 2.25E-03 75 1.52E-03 1.59E-03 1.77E-03 2.46E-03 Table XX. Concluded.

k = Component θ, deg 0.188 0.228 0.322 0.505 0 -0.1834 -0.1789 -0.1732 -0.1628 5 -0.1506 -0.1487 -0.1430 -0.1346 10 -0.2061 -0.2022 -0.1943 -0.1346 15 -0.2053 -0.2033 -0.1866 -0.1698 20 -0.2121 -0.2020 -0.1727 -0.1494 25 -0.2049 -0.2066 -0.1821 -0.1678 28 -0.2610 -0.2486 -0.1983 -0.1645 30 -0.2867 -0.2658 -0.2012 -0.1772 32 -0.3658 -0.3129 -0.2269 -0.1837 34 -0.4277 -0.3422 -0.2446 -0.1754

C

l p 36 -0.5086 -0.3877 -0.2916 -0.1877 38 -0.5879 -0.4852 -0.3105 -0.1971 40 -0.6364 -0.5536 -0.3439 -0.1755 45 -0.5224 -0.4730 -0.2971 -0.1716 50 -0.2493 -0.2524 -0.2346 -0.2072 60 -0.1398 -0.1384 -0.1479 -0.1376 70 -0.1362 -0.1379 -0.1431 -0.1598 75 -0.1466 -0.1320 -0.1478 -0.1722 0 3.50E-04 2.99E-04 4.45E-04 1.02E-03 5 3.47E-04 3.24E-04 4.19E-04 1.06E-03 10 5.04E-04 4.67E-04 6.14E-04 1.06E-03 15 1.05E-03 9.88E-04 8.65E-04 1.17E-03 20 2.09E-03 1.86E-03 1.67E-03 1.68E-03 25 2.63E-03 2.62E-03 2.28E-03 1.99E-03 28 3.08E-03 2.90E-03 2.57E-03 2.36E-03 30 3.83E-03 3.38E-03 2.63E-03 2.72E-03

( ) C s

32 5.15E-03 5.18E-03 3.54E-03 3.29E-03 l p 34 6.69E-03 7.28E-03 5.41E-03 4.58E-03 36 9.69E-03 9.54E-03 7.20E-03 6.75E-03 38 1.41E-02 1.15E-02 9.41E-03 6.18E-03 40 1.38E-02 1.43E-02 1.13E-02 8.28E-03 45 1.91E-02 1.94E-02 1.30E-02 8.46E-03 50 7.96E-03 7.78E-03 6.24E-03 6.76E-03 60 5.17E-03 4.35E-03 3.54E-03 3.41E-03 70 6.47E-03 5.72E-03 5.06E-03 4.46E-03 75 8.09E-03 6.98E-03 5.51E-03 4.88E-03

12 Figures

Figure 1. Three view sketch of F16XL 18% scale model (dimensions in feet).

1.5 C 0.5 L -0.5 -20 0 20 40 60 80 100 1.5 C D 0.5 -20 0 20 40 60 80 100 0.2 0.1 C m -0.1 -0.2 -20 0 20 40 60 80 100 α (deg) Figure 2. Variation of longitudinal coefficients with angle of attack.

60 o =0 φ o =15 φ o =30 φ o =45 φ o =60 φ o =15 ψ o =30 ψ o =45 ψ , deg β -20 -40 -60 -20 0 20 40 60 80 100 , deg α Figure 3. Computed values of α and β for each static measurement test point.

10 10 o o φ = 5 φ = 5 A A φ , deg , deg φ 0 0 -10 -10 0.02 0.1 o o α =0 α =30 o o =20 =40 α α C C 0 0 Y Y -0.02 -0.1 0.02 0.04 0.01 0.02 C C l l 0 0 -0.01 -0.02 .005 0.02 C C 0 0 n n -.005 -0.02 0 1 2 3 4 5 0 1 2 3 4 5 t, sec t, sec Figure 4. Time histories of roll angle and lateral coefficients, k=0.171.

10 50 = 5 = 20 φ ° φ ° A A φ , deg φ , deg φ = 10 ° φ = 30 ° A A 0 0 -10 -50 0.05 0.2 C C 0 0 Y Y -0.05 -0.2 0.05 0.1 C C 0 0 l l -0.05 -0.1 0.02 0.05 C 0 0 n C n -0.02 -0.05 0 2 4 6 0 2 4 6 t, sec t, sec o Figure 5. Time histories of roll angle and lateral coefficients, α =30 , k=0.171.

50 50 o o φ = 0 φ =30 δφ , deg , deg 0 δφ 0 o o =15 =45 φ φ 0 0 0 0 -50 -50 0.2 0 C C Y Y 0 -0.2 -0.2 -0.4 0.1 0.1 C C l 0 0 l -0.1 -0.1 0.05 0.05 C C 0 0 n n -0.05 -0.05 0 2 4 6 0 2 4 6 t, sec t, sec o o Figure 6. Time histories of roll angle, δ φ φ φ = − , and lateral coefficients, k=0.168, θ =30 , φ = 30 .

0 0 A 10 10 o o = 5 ψ ψ = 5 A A ψ , deg ψ , deg 0 0 -10 -10 0.05 0.05 = 0 α ° = 30 α ° 0 0 α = 20 = 40 ° α ° C C 0 0 Y Y -0.05 -0.05 0.05 0.05 C C 0 l l -0.05 -0.05 0.01 0.02 C C n n -0.01 -0.02 0 2 4 6 t, sec t, sec Figure 7. Time histories of yaw angle and lateral coefficients, k=0.190.

10 20 50 o o o ψ = 5 =20 =30 ψ ψ ψ , deg A A A o =10 ψ A 0 0 0 -10 -20 -50 0 2 4 0 5 10 0 2 4 6 0.1 0.1 0.1 C 0 0 0 Y -0.1 -0.1 -0.1 0 2 4 0 5 10 0 2 4 6 0.05 0.1 0.2 C 0 0 0 l -0.05 -0.1 -0.2 0 2 4 0 5 10 0 2 4 6 0.04 0.1 0.1 0.02 C 0 0 n -0.02 -0.1 -0.1 0 2 4 0 5 10 0 2 4 6 t, sec t, sec t, sec o Figure 8. Time histories of yaw angle and lateral coefficients, k = 0.190, α = 36 (for Ψ =20, 0 A k=.115; for Ψ =30, k=.168).

A 0.05 0.1 α = 20 ° α = 30 ° 0 0 0.05 C C 0 0 l l -0.05 -0.05 -0.1 -40 -20 0 20 40 -40 -20 0 20 40 0.15 0.05 α = 40 ° α = 60 ° 0.1 0.05 C C l l -0.05 -0.1 -0.05 -40 -20 0 20 40 -40 -20 0 20 40 , deg φ φ , deg Figure 9a. Effect of angle of attack and amplitude on lateral coefficient, C , for rolling oscillations l with k=0.190.

0.02 0.03 α = 30 ° α = 20 ° 0.02 0.01 0.01 C n n 0 0 C -0.01 -0.01 -0.02 -0.02 -0.03 -40 -20 0 20 40 -40 -20 0 20 40 0.15 0.02 0.1 0 C n n C = 40 α ° 0.05 -0.02 0 -0.04 = 60 α ° -0.05 -0.06 -40 -20 0 20 40 -40 -20 0 20 40 φ , deg φ , deg Figure 9b. Effect of angle of attack and amplitude on lateral coefficient, C , for rolling oscillations n with k=0.190.

0.1 0.2 α = 20 ° = 30 α ° 0.05 0.1 C C Y 0 Y 0 -0.05 -0.1 -0.1 -0.2 -40 -20 0 20 40 -40 -20 0 20 40 0.2 0.1 α = 60 ° α = 40 ° 0.1 0.05 C C 0 Y 0 Y -0.1 -0.05 -0.2 -0.1 -40 -20 0 20 40 -40 -20 0 20 40 φ , deg φ , deg Figure 9c. Effect of angle of attack and amplitude on lateral coefficient, , for rolling oscillations C Y with k=0.190.

0.1 0.2 α = 20 ° α = 30 ° 0.05 0 0.1 C C 0 0 l l -0.05 -0.1 -0.1 -0.2 -40 -20 0 20 40 -40 -20 0 20 40 0.1 0.03 α = 60 ° α = 40 ° 0.02 0.05 0.01 C C 0 l 0 l -0.01 -0.05 -0.02 -0.1 -0.03 -40 -20 0 20 40 -40 -20 0 20 40 φ , deg φ , deg Figure 10a. Effect of angle of attack and amplitude on lateral coefficient, , for yawing oscillations C l o o with k=0.073 (for Ψ =5 ) and k=0.076 (for Ψ =30 ).

A A 0.04 0.05 = 20 α ° α = 30 ° 0 0 0.02 C C 0 0 n n -0.02 -0.04 -0.05 -40 -20 0 20 40 -40 -20 0 20 40 0.1 0.02 α = 60 ° α = 40 ° 0.01 0.05 C n C n -0.01 -0.05 -0.02 -0.1 -0.03 -40 -20 0 20 40 -40 -20 0 20 40 φ , deg φ , deg Figure 10b. Effect of angle of attack and amplitude on lateral coefficient, C , for yawing oscillations n o o with k=0.073 (for Ψ =5 ) and k=0.076 (for Ψ =30 ).

A A 0.2 0.2 α = 20 ° = 30 0 α ° 0.1 0.1 Y Y 0 0 C C -0.1 -0.1 -0.2 -0.2 -40 -20 0 20 40 -40 -20 0 20 40 0.1 0.05 α = 40 ° α = 60 ° 0.05 Y Y 0 0 C C -0.05 -0.1 -0.05 -40 -20 0 20 40 -40 -20 0 20 40 φ , deg φ , deg Figure 10c. Effect of angle of attack and amplitude on lateral coefficient, C , for yawing oscillations Y o o with k=0.073 (for Ψ =5 ) and k=0.076 (for Ψ =30 ).

A A 0.05 = 5 φ ° A -0.05 _ C -0.1 l β -0.15 -0.2 -0.25 k=0.073 -0.5 k=0.122 k=0.171 _ k=0.191 -1 C k=0.191 l k=0.191 p k=0.22 -1.5 k=0.269 -2 0 10 20 30 40 50 60 70 80 , deg α 0.05 -0.05 _ -0.1 C l -0.15 β = 10 φ ° A -0.2 -0.25 k=0.073 -0.5 k=0.122 k=0.171 _ k=0.191 -1 C k=0.22 l k=0.232 p k=0.269 -1.5 -2 0 10 20 30 40 50 60 70 80 , deg α Figure 11. Variation of in-phase and out-of-phase components of rolling moment coefficient with angle of attack. Rolling oscillations.

0.05 0.0 = 20 φ ° A -0.05 _ C -0.1 l β -0.15 -0.2 -0.25 k=0.073 -0.5 k=0.095 k=0.115 k=0.122 _ k=0.171 -1 k=0.22 C l k=0.22 p k=0.269 -1.5 -2 0 10 20 30 40 50 60 70 80 , deg α 0.05 0.0 = 30 φ ° A -0.05 _ -0.1 C l -0.15 β -0.2 -0.25 k=0.064 -0.5 k=0.073 k=0.076 k=0.115 _ k=0.122 -1 k=0.169 C l k=0.171 p k=0.22 -1.5 k=0.269 -2 0 10 20 30 40 50 60 70 80 , deg α Figure 11. Concluded.

0.1 _ C -0.1 n β = 5 φ ° A -0.2 -0.3 1.5 k=0.073 k=0.122 k=0.171 k=0.191 k=0.191 _ k=0.191 0.5 C k=0.22 n k=0.269 p -0.5 0 10 20 30 40 50 60 70 80 , deg α 0.1 _ -0.1 C n = 10 β φ ° A -0.2 -0.3 1.5 k=0.073 k=0.122 k=0.171 k=0.191 k=0.22 _ k=0.232 0.5 k=0.269 C n p -0.5 0 10 20 30 40 50 60 70 80 , deg α Figure 12. Variation of in-phase and out-of-phase components of yawing-moment coefficient with angle of attack. Rolling oscillations.

0.1 _ -0.1 C n β = 20 φ ° A -0.2 -0.3 1.5 k=0.073 k=0.095 k=0.115 k=0.122 k=0.171 _ k=0.22 0.5 k=0.22 C n k=0.269 p -0.5 0 10 20 30 40 50 60 70 80 , deg α 0.1 _ -0.1 C n = 30 φ ° β A -0.2 -0.3 1.5 k=0.064 k=0.073 k=0.076 k=0.115 k=0.122 _ k=0.169 0.5 k=0.171 C k=0.22 n p k=0.269 -0.5 0 10 20 30 40 50 60 70 80 , deg α Figure 12. Concluded.

0.6 0.4 = 5 φ ° A 0.2 _ C 0 Y β -0.2 -0.4 k=0.073 k=0.122 k=0.171 k=0.19 k=0.19 _ k=0.19 C k=0.219 Y 0 p k=0.269 -1 0 10 20 30 40 50 60 70 80 , deg α 0.6 0.4 = 10 φ ° A 0.2 _ C Y β -0.2 -0.4 k=0.073 k=0.122 k=0.171 k=0.19 _ k=0.22 C k=0.232 Y k=0.268 p -1 0 10 20 30 40 50 60 70 80 , deg α Figure 13. Variation of in-phase and out-of-phase components of side-force coefficient with angle of attack. Rolling oscillations.

0.6 0.4 = 20 φ ° A 0.2 _ C Y β -0.2 -0.4 k=0.073 k=0.095 k=0.114 k=0.122 k=0.171 _ k=0.219 C k=0.219 Y k=0.269 0 p -1 0 10 20 30 40 50 60 70 80 , deg α 0.6 0.4 = 30 φ ° A _ 0.2 C Y β -0.2 -0.4 k=0.063 k=0.073 k=0.076 k=0.115 k=0.123 _ k=0.168 C Y k=0.171 p k=0.22 k=0.268 -1 0 10 20 30 40 50 60 70 80 , deg α Figure 13. Concluded.

= 5 φ ° A -0.05 _ C -0.1 l β -0.15 -0.2 -0.5 _ =30 α ° =40 C α ° -1 l =50 p α ° -1.5 -2 0.05 0.1 0.15 0.2 0.25 0.3 k 0.05 = 10 φ ° A _ -0.05 C l -0.1 β -0.15 -0.2 -0.5 _ =30 α ° -1 C α =40 ° l p =45 α ° =50 α ° -1.5 -2 0.05 0.1 0.15 0.2 0.25 0.3 k Figure 14. Variation of in-phase and out-of-phase components of rolling moment coefficient with frequency. Rolling oscillations.

= 20 φ ° A -0.05 _ -0.1 C l β -0.15 -0.2 -0.5 =30 α ° _ -1 =40 α ° C =45 α ° l p =50 α ° -1.5 -2 0.05 0.1 0.15 0.2 0.25 0.3 k = 30 φ ° A -0.05 _ -0.1 C l β -0.15 -0.2 -0.5 =30 α ° =40 α ° _ -1 =45 α ° C l =50 α ° p -1.5 -2 0.05 0.1 0.15 0.2 0.25 0.3 k Figure 14. Continued.

-0.08 = 30 α ° -0.1 -0.12 _ C l -0.14 β -0.16 -0.18 -0.2 =5 -0.4 φ ° A =10 φ ° A φ =20 ° _ A -0.6 =30 φ ° C A l p -0.8 -1 0.05 0.1 0.15 0.2 0.25 0.3 k = 40 α ° -0.05 _ -0.1 C l β -0.15 -0.2 -0.5 _ =5 φ ° A -1 =10 C φ ° A l φ =20 ° p A =30 φ ° A -1.5 -2 0.05 0.1 0.15 0.2 0.25 0.3 k Figure 14. Concluded.

k=0.073 k=0.122 0.75 k=0.171 k=0.191 k=0.191 R ( C ) k=0.191 0.5 l k=0.22 k=0.269 = 5 φ ° A 0.25 k=0.073 0.75 k=0.122 k=0.171 k=0.191 R ( C ) l k=0.22 0.5 k=0.232 k=0.269 = 10 φ ° A 0.25 k=0.073 k=0.095 0.75 k=0.115 k=0.122 k=0.171 R ( C ) k=0.22 0.5 l k=0.22 = 20 φ ° k=0.269 A 0.25 k=0.064 k=0.073 0.75 k=0.076 k=0.115 k=0.122 R ( C ) k=0.169 0.5 l k=0.171 k=0.22 = 30 φ ° A k=0.269 0.25 0 10 20 30 40 50 60 70 80 , deg α Figure 15. Variation of coefficient of determination for rolling-moment coefficient with angle of attack. Rolling oscillations.

0.06 k=0.073 k=0.122 k=0.171 0.04 k=0.191 = 5 φ ° k=0.191 s ( C ) A l k=0.191 k=0.22 k=0.269 0.02 0.06 k=0.073 k=0.122 k=0.171 = 10 φ ° 0.04 A k=0.191 k=0.22 s ( C ) k=0.232 l k=0.269 0.02 0.06 k=0.073 = 20 φ ° k=0.095 A k=0.115 0.04 k=0.122 k=0.171 s ( C ) l k=0.22 k=0.22 k=0.269 0.02 0.06 k=0.064 = 30 φ ° A k=0.073 k=0.076 0.04 k=0.115 k=0.122 k=0.169 s ( C ) l k=0.171 k=0.22 0.02 k=0.269 0 10 20 30 40 50 60 70 80 , deg α Figure 16. Variation of fit error of rolling-moment coefficient with angle of attack. Rolling oscillations.

0.03 0.02 = 10 , = 45 φ ° α ° A 0 = 10 , = 20 φ ° α ° A 0 0.02 0.01 C l 0.01 0 0 -0.01 -0.01 -0.02 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 , deg , deg φ φ 0.04 0.05 measured computed 0.02 C 0 0 l -0.02 = 30 , = 20 φ ° α ° A 0 = 30 , = 45 φ ° α ° A 0 -0.04 -0.05 -40 -20 0 20 40 -40 -20 0 20 40 , deg , deg φ φ Figure 17. Comparison of measured and computed rolling-moment coefficient. Rolling oscillations, k=0.073.

0.01 0.1 = 10 , = 20 φ ° α ° A 0 0.005 0.05 n 0 0 C -0.005 -0.05 = 10 , = 45 φ ° α ° A 0 -0.01 -0.1 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 , deg , deg φ φ 0.02 0.1 measured φ = 30 ° , α = 20 ° computed A 0 0.01 0.05 n 0 0 C -0.01 -0.05 = 30 , = 45 φ ° α ° A 0 -0.02 -0.1 -40 -20 0 20 40 -40 -20 0 20 40 , deg , deg φ φ Figure 18. Comparison of measured and computed yawing-moment coefficient. Rolling oscillations, k=0.073.

0.04 0.15 = 10 , = 45 φ ° α ° A 0 0.02 0.1 C 0 Y 0.05 -0.02 0 = 10 , = 20 φ ° α ° A 0 -0.04 -0.05 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 , deg , deg φ φ 0.1 0.15 measured computed 0.05 0.1 = 30 , = 45 φ ° α ° A 0 0 0.05 C Y -0.05 0 -0.1 -0.05 = 30 , = 20 φ ° α ° A 0 -0.15 -0.1 -40 -20 0 20 40 -40 -20 0 20 40 , deg , deg φ φ Figure 19. Comparison of measured and computed side-force coefficient. Rolling oscillations, k=0.073.

0.15 0.1 = 15 φ ° o 0.05 _ C l β -0.05 -0.1 -0.15 0.4 _ k=0.064 C k=0.115 -0.4 l k=0.169 p -0.8 -1.2 0 10 20 30 40 50 60 70 80 , deg θ 0.15 0.1 = 30 φ ° o 0.05 _ C l β -0.05 -0.1 -0.15 0.4 k=0.064 _ k=0.115 C k=0.169 -0.4 l p -0.8 -1.2 0 10 20 30 40 50 60 70 80 , deg θ Figure 20. Effect of offset in roll angle on in-phase and out-of-phase components of rolling-moment o coefficient. Rolling oscillations, φ =30 .

A 0.15 0.1 0.05 _ C l β -0.05 = 45 φ ° o -0.1 -0.15 0.4 k=0.064 k=0.115 _ k=0.169 -0.4 C l p -0.8 -1.2 0 10 20 30 40 50 60 70 80 , deg θ 0.15 0.1 = 60 φ ° o 0.05 _ C l β -0.05 -0.1 -0.15 0.4 k=0.064 _ k=0.115 -0.4 k=0.169 C l p -0.8 -1.2 0 10 20 30 40 50 60 70 80 , deg θ Figure 20. Concluded.

0.1 0.05 _ C n = 15 β φ ° o -0.05 -0.1 0.6 k=0.064 k=0.115 k=0.169 0.3 _ C n p -0.3 0 10 20 30 40 50 60 70 80 , deg θ 0.1 0.05 _ C n β = 30 φ ° o -0.05 -0.1 0.6 k=0.064 k=0.115 k=0.169 0.3 _ C n p -0.3 0 10 20 30 40 50 60 70 80 , deg θ Figure 21. Effect of offset in roll angle on in-phase and out-of-phase components of yawing-moment o coefficient. Rolling oscillations, φ =30 .

A 0.1 0.05 _ C n β = 45 φ ° o -0.05 -0.1 0.6 k=0.064 k=0.115 k=0.169 0.3 _ C n p -0.3 0 10 20 30 40 50 60 70 80 , deg θ 0.1 0.05 _ C n β = 60 φ ° o -0.05 -0.1 0.6 k=0.064 k=0.115 k=0.169 _ 0.3 C n p -0.3 0 10 20 30 40 50 60 70 80 , deg θ Figure 21. Concluded.

0.1 -0.1 _ C Y -0.2 β -0.3 = 15 φ ° -0.4 1.5 k=0.062 k=0.115 k=0.168 _ 0.5 C Y p -0.5 0 10 20 30 40 50 60 70 80 , deg θ 0.1 -0.1 _ C Y -0.2 β = 30 φ ° -0.3 -0.4 1.5 k=0.063 k=0.115 k=0.17 _ 0.5 C Y p -0.5 0 10 20 30 40 50 60 70 80 , deg θ Figure 22. Effect of offset in roll angle on in-phase and out-of-phase components of rolling-moment o coefficient. Rolling oscillations, φ =30 .

A 0.1 -0.1 _ C k=0.064 Y -0.2 β k=0.114 k=0.169 -0.3 -0.4 1.5 = 45 φ ° _ 0.5 C Y p -0.5 0 10 20 30 40 50 60 70 80 , deg θ 0.1 -0.1 _ k=0.064 C Y k=0.115 -0.2 β k=0.169 -0.3 -0.4 1.5 = 60 φ ° _ 0.5 C Y p -0.5 0 10 20 30 40 50 60 70 80 , deg θ Figure 22. Concluded.

=15 =30 =45 =60 θ ° θ ° θ ° θ ° 0 0 0 0 0.05 0.05 0.05 0.05 0 0 0 0 =15 φ ° -0.05 -0.05 -0.05 -0.05 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0.05 0.05 0.05 0.05 0 0 0 0 =30 φ ° -0.05 -0.05 -0.05 -0.05 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 C l 0.05 0.05 0.05 0 0 0 =45 φ ° -0.05 -0.05 -0.05 o -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0.05 0.05 0.05 0.05 0 0 0 0 =60 φ ° -0.05 -0.05 -0.05 -0.05 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 time, sec o Figure 23. Time histories of rolling-moment coefficient. Rolling oscillations, φ =30 , k=0.064.

A =15 =30 =45 =60 θ ° θ ° θ ° θ ° 0 0 0 0 0.1 0.1 0.1 0.1 =15 φ ° 0 0 0 0 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0.1 0.1 0.1 0.1 =30 φ ° 0 0 0 0 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 C 0.1 0.1 0.1 n =45 φ ° 0 0 0 o -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0.1 0.1 0.1 0.1 =60 φ ° 0 0 0 0 o -0.1 -0.1 -0.1 -0.1 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 time, sec o Figure 24. Time histories of yawing-moment coefficient. Rolling oscillations. φ =30 , k=0.064.

A =15 =30 =45 =60 θ ° θ ° θ ° θ ° 0 0 0 0 0.2 0.2 0.2 0.2 0 0 0 0 =15 φ ° o -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0.2 0.2 0.2 0.2 0 0 0 0 =30 φ ° o -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 C 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 Y 0.2 0.2 0.2 0 0 0 =45 φ ° o -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 0 5 10 15 0 5 10 15 0 5 10 15 0.2 0.2 0.2 0.2 0 0 0 0 =60 φ ° o -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 time, sec o Figure 25. Time histories of side-force coefficient. Rolling oscillations, φ =30 , k=0.064.

A 0.8 0.6 R (C ) l 0.4 0.2 0.8 0.6 R (C ) n Order 1 0.4 Order 2 Order 3 0.2 0.8 0.6 2 R (C ) Y 0.4 0.2 0 10 20 30 40 50 60 70 80 , deg θ Figure 26. Variation of coefficient of determination for rolling-moment coefficient with angle of o o attack. Rolling oscillations, k=0.064, φ =30 , φ =45 .

A 0 0.1 _ -0.1 C l β -0.2 -0.3 -0.2 k=0.188 _ k=0.228 -0.4 k=0.322 C l k=0.505 p -0.6 -0.8 0 10 20 30 40 50 60 70 80 , deg α 0.05 -0.05 _ C n -0.1 β -0.15 -0.2 0.4 k=0.188 k=0.228 k=0.322 0.2 k=0.505 _ C n p -0.2 0 10 20 30 40 50 60 70 80 , deg α Figure 27. Variation of in-phase and out-of-phase components of lateral coefficients with angle of o 6 attack. Rolling oscillations, φ =10 , Re=0.6x10 .

A 0.4 0.2 _ C Y β -0.2 -0.4 1.5 k=0.187 k=0.227 k=0.322 k=0.505 _ 0.5 C Y p -0.5 0 10 20 30 40 50 60 70 80 , deg α Figure 27. Concluded.

τ

0 10 20 30 40 50 60 , deg α 0.7 =5 , Re=2.1x10 φ ° A 0.6 =10 , Re=2.1x10 φ ° A =10 , Re=0.6x10 φ ° A 0.5 0.4

a

0.3 0.2 0.1 0 10 20 30 40 50 60 , deg α Figure 28. Estimated parameters, their 2- σ confidence intervals and computed unsteady term.

Rolling-moment coefficient. Rolling oscillations.

0.1 static data 0.05 =5 , Re=2.1x10 φ ° A φ =10 ° , Re=2.1x10 A 0 =10 , Re=0.6x10 φ ° A C ( ) ∞ -0.05 l β -0.1 -0.15 -0.2 -0.25 0.3 0.2 0.1 C ( ∞ ) l p -0.1 -0.2 -0.3 -0.4 0 10 20 30 40 50 60 , deg α k=0.073 -0.5 k=0.122 k=0.191 -1 k=0.269 C (k) -1.5 l .

β -2 =10 , Re=2.1x10 φ ° A -2.5 -3 0 10 20 30 40 50 60 , deg α Figure 28. Concluded.

10 τ

0 10 20 30 40 50 60 , deg α 0.4 0.2 -0.2 φ =5 ° , Re=2.1x10 A

a 6

=10 , Re=2.1x10 φ ° A -0.4 =10 , Re=0.6x10 φ ° A -0.6 -0.8 -1 0 10 20 30 40 50 60 , deg α Figure 29. Estimated parameters, their 2- σ confidence intervals and computed unsteady term.

Yawing-moment coefficient. Rolling oscillations.

0.1 static data φ =5 ° , Re=2.1x10 C ( ∞ ) -0.1 A n =10 , Re=2.1x10 φ ° β A =10 , Re=0.6x10 φ ° A -0.2 -0.3 -0.4 -0.1 -0.2 C ( ) ∞ n p -0.3 -0.4 -0.5 0 10 20 30 40 50 60 , deg α k=0.073 k=0.122 1.5 k=0.191 k=0.269 C (k) n .

β =10 , Re=2.1x10 φ ° A 0.5 0 10 20 30 40 50 60 , deg α Figure 29. Concluded

τ

0 10 20 30 40 50 60 , deg α -0.5 -1 -1.5 a =5 , Re=2.1x10 φ ° -2 A φ =10 ° , Re=2.1x10 A =10 , Re=0.6x10 φ ° A -2.5 -3 -3.5 0 10 20 30 40 50 60 , deg α Figure 30. Estimated parameters, their 2- σ confidence intervals and computed unsteady term. Side- force coefficient. Rolling oscillations.

0.1 static data =5 , Re=2.1x10 φ ° A =10 , Re=2.1x10 φ ° A -0.1 φ =10 ° , Re=0.6x10 A -0.2 C ( ∞ ) Y β -0.3 -0.4 -0.5 -0.6 0.5 C ( ∞ ) Y -0.5 p -1 -1.5 0 10 20 30 40 50 60 , deg α k=0.073 3.5 k=0.122 k=0.191 k=0.269 2.5 C (k) Y .

β 1.5 =10 , Re=2.1x10 φ ° A 0.5 0 10 20 30 40 50 60 , deg α Figure 30. Concluded.

0.1 = 5 ψ ° A _ -0.1 C l -0.2 β -0.3 -0.4 2.5 k=0.073 k=0.191 k=0.191 k=0.269 1.5 _ C l r 0.5 -0.5 0 10 20 30 40 50 60 70 80 90 , deg α 0.1 = 10 ψ ° A -0.1 _ C -0.2 l β -0.3 -0.4 2.5 k=0.191 k=0.232 1.5 _ C l r 0.5 -0.5 0 10 20 30 40 50 60 70 80 90 , deg α Figure 31. Variation of in-phase and out-of-phase components of rolling-moment coefficient with angle of attack. Yawing oscillations.

0.2 0.1 _ C ψ = 5 ° n A β -0.1 -0.2 -0.3 0.5 k=0.073 -0.5 _ k=0.191 k=0.191 C n k=0.269 -1 r -1.5 -2 0 10 20 30 40 50 60 70 80 90 , deg α 0.2 0.1 _ = 10 ψ ° C A n -0.1 β -0.2 -0.3 0.5 -0.5 _ k=0.191 C k=0.232 n -1 r -1.5 -2 0 10 20 30 40 50 60 70 80 90 , deg α Figure 32. Variation of in-phase and out-of-phase components of yawing-moment coefficient with angle of attack. Yawing oscillations.

0.6 k=0.073 0.4 k=0.191 k=0.191 0.2 _ k=0.269 C Y β -0.2 -0.4 -0.6 _ -1 C Y r -2 = 5 ψ ° -3 A -4 0 10 20 30 40 50 60 70 80 90 , deg α 0.6 0.4 k=0.19 k=0.232 0.2 _ C Y β -0.2 -0.4 -0.6 _ -1 C Y r -2 -3 = 10 ψ ° A -4 0 10 20 30 40 50 60 70 80 90 , deg α Figure 33. Variation of in-phase and out-of-phase components of side-force coefficient with angle of attack. Yawing oscillations.

0.1 _ -0.1 C l β -0.2 -0.3 -0.4 2.5 k=0.188 k=0.228 k=0.322 1.5 k=0.505 _ C l r 0.5 -0.5 0 10 20 30 40 50 60 70 80 90 , deg α 0.2 0.1 _ C n -0.1 β -0.2 -0.3 0.5 -0.5 _ k=0.188 C n k=0.228 -1 r k=0.322 k=0.505 -1.5 -2 0 10 20 30 40 50 60 70 80 90 , deg α Figure 34. Variation of in-phase and out-of-phase components of lateral coefficients with angle of o 6 attack. Yawing oscillations, ψ =10 , Re=0.6x10 .

A 0.6 0.4 k=0.189 k=0.227 0.2 k=0.322 _ k=0.504 0 C Y β -0.2 -0.4 -0.6 -1 _ C Y -2 r -3 -4 0 10 20 30 40 50 60 70 80 90 , deg α Figure 34. Concluded.

0.1 _ -0.1 C l β -0.2 -0.3 0.8 k=0.169, ψ =0 ° o k=0.165, ψ =15 ° 0.6 o k=0.168, =30 ψ ° o k=0.167, =45 ψ ° o 0.4 _ C l 0.2 r -0.2 0 10 20 30 40 50 60 70 80 90 , deg θ 0.1 0.05 _ C n β -0.05 -0.1 0.2 -0.2 k=0.169, ψ =0 ° _ o k=0.165, ψ =15 ° o C -0.4 k=0.168, =30 ψ ° o n k=0.167, =45 ψ ° r o -0.6 -0.8 0 10 20 30 40 50 60 70 80 90 , deg θ Figure 35. Effect of offset in yaw angle on in-phase and out-of-phase components. Yawing o oscillations, ψ =30 .

A 0.2 _ -0.2 C Y β -0.4 -0.6 k=0.169, =0 ψ ° k=0.165, =15 ψ ° k=0.168, =30 ψ ° 0.5 _ k=0.167, =45 ψ ° C Y r -0.5 0 10 20 30 40 50 60 70 80 90 , deg θ Figure 35. Concluded.

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1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To) 08 - 2004 01- Technical Memorandum 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Analysis of Wind Tunnel Lateral Oscillatory Data of the F-16XL Aircraft 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER Klein, Vladislav; Murphy, Patrick C.; and Szyba, Nathan M.

5e. TASK NUMBER 5f. WORK UNIT NUMBER 23-762-45-DB 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER NASA Langley Research Center Hampton, VA 23681-2199 L-18387 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) National Aeronautics and Space Administration NASA Washington, DC 20546-0001 11. SPONSOR/MONITOR'S REPORT NUMBER(S) NASA/TM-2004-213246 12. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified - Unlimited Subject Category 05 Availability: NASA CASI (301) 621-0390 Distribution: Nonstandard 13. SUPPLEMENTARY NOTES An electronic version can be found at http://techreports.larc.nasa.gov/ltrs/ or http://ntrs.nasa.gov 14. ABSTRACT Static and dynamic wind tunnel tests were performed on an 18% scale model of the F-16XL aircraft. These tests were performed over a wide range of angles of attack and sideslip with oscillation amplitudes from 5 ° to 30 ° and reduced frequencies from 0.073 to 0.269. Harmonic analysis was used to estimate Fourier coefficients and in-phase and out-of-phase components.

For frequency dependent data from rolling oscillations, a two-step regression method was used to obtain unsteady models (indicial functions), and derivatives due to sideslip angle, roll rate and yaw rate from in-phase and out-of-phase components.

Frequency dependence was found for angles of attack between 20 ° and 50 ° . Reduced values of coefficient of determination and increased values of fit error were found for angles of attack between 35 ° and 45 ° . An attempt to estimate model parameters from yaw oscillations failed, probably due to the low number of test cases at different frequencies.

15. SUBJECT TERMS Dynamic wind tunnel tests; Indicial models; Unsteady aerodynamics; Harmonic analysis; Regression; Parameter estimation; System identification 19a. NAME OF RESPONSIBLE PERSON 18. NUMBER 17. LIMITATION OF 16. SECURITY CLASSIFICATION OF: OF ABSTRACT STI Help Desk (email: help@sti.nasa.gov) a. REPORT c. THIS PAGE b. ABSTRACT PAGES 19b. TELEPHONE NUMBER (Include area code) (301) 621-0390 U U U UU Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39.18