Designing an RC Model for Maximum Flight Duration

W. B. Garner, January 2019

ContentsObjectives....................................................................................................................................................2

Theory.........................................................................................................................................................2

Model Physical Parameters.........................................................................................................................3

Wing........................................................................................................................................................3

Tail...........................................................................................................................................................3

Battery.....................................................................................................................................................3

Fuselage...................................................................................................................................................3

Propeller & Motor Efficiency...................................................................................................................3

Model Aerodynamic Properties...................................................................................................................4

Wing Lift and Drag Coefficients...............................................................................................................4

Tail Lift and Drag Coefficients..................................................................................................................4

Fuselage Drag Coefficient........................................................................................................................6

Plane Summary........................................................................................................................................7

Example Calculation............................................................................................................................7

Time Duration Results.............................................................................................................................8

Power System Design..................................................................................................................................8

Propeller Performance............................................................................................................................9

Motor Model.........................................................................................................................................11

Relationships.....................................................................................................................................13

Estimating Motor Specifications............................................................................................................13

Full Throttle Current and Power............................................................................................................14

Motor Parameter Summary...............................................................................................................15

Example Motor..........................................................................................................................................16

Example Motor Performance................................................................................................................16

Selected Motor at Full Throttle..............................................................................................................19

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Summary...................................................................................................................................................20

Appendix A APCE Propeller Power and Thrust Coefficients.......................................................................21

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ObjectivesDesign an electric powered sailplane for maximum flight duration.

Battery size is fixed and plane designed around it.

Show trades between non-battery weight and flight duration.

Wing span set at 4 meters

Theory

Time =

Kp∗Km∗Kb∗Wbg∗(Wa+Wb )

∗√ 0.5∗ρ∗Swg∗(Wa+Wb )∗Cl1.5

∑jCdj

Kp is propeller efficiencyKm is motor efficiencyKb is battery Whrs/KgWb is battery weight, KgWa is plane weight without battery, KgG is gravity constant, 9.81 m/sec^2ρ is air density, 1.225 Kg/m^3Sw is wing area, m^2Cl is wing lift coefficientCdj are drag coefficients of wing, tail and fuselage referenced to the wing area, SwCdw is wing drag coefficientCdi is wing induced drag coefficientCdf is fuselage drag coefficientCdt is tail drag coefficientSf is maximum fuselage cross-section area, m^2St is total tail area, m^2

Cl = g∗(Wa+Wb)

0.5∗ρ∗Sw∗Vmps2 Lift coefficient required for level flight any airspeed

Vmps is airspeed, m/sec

Cdw = F(Cl) = a∗Cl3+b∗Cl2+c∗Cl+d

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Cdi= Cl2

π∗AR

AR = b2

Sw , aspect ratio

Model Physical Parameters

WingThe graph displays the assumed geometry and dimensions of half of the wing. The shape approximates an ellipse. The wing span, b, is 4 meters and the wing area, Sw, is 1.04 m2. The aspect ratio is 15.4 and the Mean Aerodynamic Chord (MAC) is 0.266 m.

Figure 1 Wing Configuration and Dimensions

TailThe horizontal tail area is 0.0765 m2 and the vertical tail area is 0.138 m2. These values were derived using an Excel program that provides estimates of pitch, roll, and yaw performance as a function of airplane dimensions.

BatteryThe battery is a 14.8Volt, 10 AHr unit weighing 0.908Kg. It has a Watt hour capacity of 148 W-Hrs.

Kb = 148.908 = 163 WHrs/Kg

Dimensions are 168 x 38 x 69 mm.

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FuselageThe assumed fuselage cross section required to fit this battery is a circle with a diameter of 8.5 cm and a total area of 0.0054 m2. The circle is chosen for aerodynamic drag reasons.

Propeller & Motor Efficiency It is assumed that the product Kp*Km = 0.5, the total efficiency of the power system.

Model Aerodynamic Properties

Wing Lift and Drag CoefficientsThe plane will be flying at low speeds where the Reynolds Number will be relatively small, on the order of 100,000 to 200,000. It is desirable to have an airfoil that works well under these conditions. There are a number of suitable airfoils; a S3010 airfoil is chosen for this model. The graph plots the polar properties of this airfoil as a function of Reynolds Number. The RE = 100,000 Cd curve is selected for use in the analysis.

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Figure 2 Airfoil Polar Coefficients

Cdw =y = -18.065*Cl3 + 17.60*Cl2 + 25.084*Cl - 0.6246

Tail Lift and Drag CoefficientsThe tail stabilizers are assumed to be symmetrical and employ the NACA 009 airfoil whose polar plots are shown below.

Figure 3 Tail Airfoil Coefficients

A value for Cdt of 0.02 is chosen as being representative of the drag coefficient under operating conditions where the surfaces are being intermittently deflected.

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The Cdt reference to the wing area is:

Cdt = .02*St/Sw = .02 *(.0765+.138)/1.04 = 0.0041

Fuselage Drag CoefficientThe following figure contains wind tunnel estimates of fuselage drag coefficients for a variety of configurations. Figure 1 is chosen as being representative of a sailplane type fuselage and a value of 0.2 is selected.

Figure 4 Fuselage Drag Coefficients

The Cdf referenced to the wing area is:

Cdf = 0.2*.0054/1.04 = .0011

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Plane Summary

Table 1 Plane Summary

Parameter Value UnitsKm*Kp 0.5Kb 163 WHrs/KgWb 0.9 Kgb 4.0 mSw 1.04 m2

Ar 15.4Cdf .0011Cdt .0041Sf .0054 m2

St 0.214 m2

PF = Cl1.5

Cdw+Cdi+cdf +cdt

Drag = Cdsum*0.5*ρ*Sw*V2

Power =Drag*V

Example CalculationTable 2 Time Calculations

Wa 1.2v, kmph v, mps Cl PF T,hours RE Drag

forcePower,

W16.2 4.5 1.626 9.545 5.96 83950 1.71 7.718.0 5.0 1.317 19.931 12.44 93278 1.21 6.019.8 5.5 1.088 22.290 13.92 102606 0.973 5.3521.6 6.0 0.914 21.627 13.50 111933 0.882 5.2923.4 6.5 0.779 19.902 12.43 121261 0.87 5.625.2 7.0 0.672 17.906 11.18 130589 0.90 6.327.0 7.5 0.585 15.955 9.96 139917 0.96 7.228.8 8.0 0.514 14.166 8.84 149244 1.04 8.330.6 8.5 0.456 12.571 7.85 158572 1.13 9.632.4 9.0 0.406 11.167 6.97 167900 1.23 11.134.2 9.5 0.365 9.938 6.20 177228 1.34 12.7

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Time Duration Results

Figure 5 Time Duration Results

The model stall airspeed increases as the value of Wa increases. The peak time occurs very close to stall, and then decreases as airspeed increases. Maximum time is obtained with the minimum value of Wa.

These results are for ideal conditions: constant airspeed level flight, no maneuvering, true battery capacity, drag & lift accurate, Kp*Km value met.

If kp*km increases from 0.5 to 0.6 the time increases by a ratio of 0.6/0.5 = 1.2.

Power System DesignThere is no single power system that is optimal for all model weights. Therefore each weight must be designed on its own. An example will be presented showing how a design might be made.

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The Wa = 1.2 Kg model will be examined and a specific operating point selected for the power system design. The design will be based on the maximum calculated time duration conditions. They are:

V = 5.5 m/secTime = 13.9 hoursThrust = 0.937 NewtonsPower = 5.35 Watts

The power system must produce the thrust of 0.937 Newtons at an exhaust power of 5.35 Watts. The motor must provide the required input power to the propeller at the required rpm.

Propeller PerformanceThe first step is to select a propeller size. The objective is to maximize the propeller efficiency at the desired thrust level and power. Figure 6 is a generic graph showing how propeller efficiency varies with pitch to diameter ratio as a function of the advance ratio, J.

J = Vn∗D

V is airspeed, m/secn is the revolution rate, rev/secD is the propeller diameter, m

Figure 6 Propeller Efficiency

The maximum efficiency occurs for the greatest pitch to diameter ratio and for a specific value range of J. It is also the case that large diameter propellers are more efficient than small diameter ones.

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An analysis requires knowledge of the specific thrust and power coefficients as a function of the advance ratio for the selected propeller. Appendix A contains a table listing these coefficients for a range of APC Thin Electric propellers. Pshaft is the power input to the propeller measured at the shaft. This is the power the motor must produce.

Thrust = Ct(J)*ρ*n2*D4, NewtonsPshaft = CP(J)*ρ*n3*D5, Watts

A 14 x12 propeller is selected for this example. Figure 7 plots the coefficients for this propeller, including the efficiency (ratio of exhaust power to shaft power). Polynomial expressions for Ct(J), Cp(J) and Eta(J) are included in the figure.

Figure 7 APCE 14 x 12 Propeller Coefficients

The approximate peak efficiency occurs at a J = 0.6; this value will be used as a starting point in calculating the associated input values for revolution rate and shaft power need to meet the output thrust and power conditions. There are two sets of conditions to be met.

V and D are constants as are thrust and exhaust power. The variables are J and n.

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nj = V

J∗D

nt2 = Thrust

Ct (J )∗ρ∗D4

Table 3 shows the calculations involved in finding a condition where nj = nt. Figure x3 plots nt and nj as a function of J. nt = nj = 27.4 rps when J = 0.564. The shaft power is about 7.4 Watts and the efficiency is about 72%.

Table 3 Matching propeller to revolution rate required

J Ct Cp Eff nt Pshaft Eff nj0.55 0.0686 0.052 0.72 26.9 7.10 0.75 28.120.56 0.0669 0.052 0.72 27.3 7.29 0.73 27.620.57 0.0651 0.051 0.73 27.6 7.49 0.71 27.130.58 0.0633 0.050 0.73 28.0 7.70 0.69 26.670.59 0.0614 0.050 0.73 28.4 7.93 0.67 26.210.6 0.0595 0.049 0.73 28.9 8.18 0.65 25.780.61 0.0576 0.048 0.73 29.4 8.44 0.63 25.360.62 0.0557 0.047 0.73 29.9 8.72 0.61 24.950.63 0.0537 0.046 0.73 30.4 9.02 0.59 24.550.65 0.0497 0.044 0.73 31.6 9.70 0.55 23.80

Figure 8 Comparison of nj and nt as a function of J

Motor Model

The next step is to select a motor that can produce the required rpm and shaft power at high efficiency.

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To do so requires a motor electrical model, presented in Figure 9.

Figure 9 Motor Model

There are standard formulas for calculating the performance of electric motors assuming an ideal drive and control mechanism. However, ESCs are not ideal and introduce additional losses, degrading performance somewhat. These losses are accounted for in the formulas that follow by the inclusion of an empirically derived function, f(d), that takes into account the operating state of the throttle, ranging from off to full on.

The battery and the motor EMF are voltage generators and their voltages begin with the letter E. Loss voltages begin with the letter V. These are conventional electrical engineering formats used to distinguish between generator and load voltages.

Beginning on the left, the battery no-load voltage is Ebat. The battery internal resistance is Rbat, in ohms. The battery is connected to the ESC by wires and connectors whose resistance is labeled Rcab. Next is the ESC; it is characterized by resistance Resc and throttle fraction d. “d” is the relative value of the throttle setting, ranging from 0 to 1.0. When d = 0, the motor is off; when d = 1, the motor is at full on.The voltage marked Vesc is the voltage delivered to the input of the ESC. Emot is the equivalent generator voltage applied to the motor itself taking into account the throttle setting fraction, d. Emot = d * Vesc.The motor parameters are the motor resistance Rmot, the back emf Emf in volts, the motor current, Imotor, the no load current, Inl, the no load power loss, Pnl, the motor voltage constant, Kv and the torque constant Kq. The motor outputs are the shaft power, Pshaft, the torque, Q and the revolution rate, Rpm.The terms and their definitions are:

Ebat No load battery voltageRbat Battery internal resistance, ohmsRcab Cable and connector resistance, ohmsResc ESC through resistance, ohms

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Rmot Motor internal resistance, ohmsPnl Motor power loss due to internal non resistive causesEmf Motor back EMF, opposed to the battery voltage, voltsPshaft Power output through the motor shaftKv Motor rpm per voltIo no load reference current at VoVo no load reference voltageInl Motor no load current, AmpsImot Motor current transferring power to the output shaft, Ampsd Fraction of full throttle setting, range 0 to 1.0Emot The source voltage driving the motor from the ESC, Volts

Relationships

There are some relationships that are common to all working formulas that follow.f(d) = 1+d*(1-d) ESC loss factor, multiplied by the motor current.Vesc = Ebat –Itotal*(Rbat + Rcab + Resc) Voltage into ESCEmotor= d*Vesc Voltage out of ESC to motor, viewed as a generatorInl = (Io/Vo)*Emf No load current, AmpsPnl = Inl*Emf No load power, WattsItotal = Inl + Imot*f(d) Total current, AmpsEmf = Kv * Rpm Motor back voltage, VoltsPshaft = Imot* Emf Shaft power, Watts

The problem for the example is to specify approximate values for the key motor parameters without knowledge of all of the details. The motor will be operating at low power and current levels, so some of the detailed parameters have relatively minor effects on performance. These parameters include the battery, cable and ESC resistances which can initially be considered to be zero. Then Vesc = Ebat and Emot = d*Vesc = d*Ebat. Since Ebat is fixed at 14.8 Volts. Emot = d*14.8 Volts.

Estimating Motor SpecificationsThe next step is to select a preliminary value for d. Note that in operation as the battery is discharged the voltage begins to decrease until it drops to a point too low for operation. Therefore it is advisable to select the initial full voltage throttle setting, d, such that the throttle can be advanced as the battery voltage decreases, keeping the motor rpm constant. It also provides throttle room for climbing or other higher power operations.

Select d = 0.25.

The motor current, Imot = Pshaft*Kv/rpm.The values of Pshaft and rpm are known so the next step is to estimate the value of Kv. Note that:

Kv = rpm/EMF

Ignoring the small voltage drop caused by the motor resistance:EMF = d*Ebat =.25*14.8 = 3.7 Volts

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Kv = n∗60d∗Ebat

Kv = 60∗27.4.25∗14.8 = 444 rpm/volt

This is an approximate value that can be used to find a candidate motor.

Given this value for Kv, the motor current is:

Imot = Pshaft∗Kv

rpm =7.4∗44460∗27.4 = 2.0 Amps

The total current Itot = Imot*f(d) +Inl

F(d) = 1+.25-.252 = 1.19

Itot = 2*1.19 + Inl = 2.38 +Inl Amps

One objective is to choose a motor with a small value of Inl. In this case assume that Inl = 0.1 *f(d)*Imot

Inl = 0.1 *1.19*2 = 0.25 amps

Itot = 2.38 + .25 = 2.63 Amps

The motor input power is equal to the sum of the shaft power plus the losses incurred in the motor.

Pmotor = Pshaft + Prm +Pnl +Pesc

To a first order the value of Pesc is negligible.

Assuming an efficiency objective of 80%, then Pbat = 7.4/0.8 = 9.25 Watts and Prm + Pnl = 9.25 – 7.4 = 1.85 W

Pnl = Inl*EMF = .25*3.7 = 0.925 Watts

Prm = 1.85- .925 = 0.925 Watts

Rm = PrmItot 2

= .9252.632

= 0.133 Ohms

To summarize the key approximate motor parameters:Kv = 444 rpm/VRm <= 0.133 OhmsInl <= .25 Amps at 3.7 Volts

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Full Throttle Current and PowerThe other parameters of interest are the maximum current and power the motor must be able to handle at full power.

In this case the motor output power must match the propeller required power at full throttle.

Prop = Cp(0)*ρ*n3*D5

Pmotor = (Ebat− rpm

Kv )∗rpmKv

Rtotal

Cp(0) is the propeller power coefficient at J=0, the condition demanding most power.Rtotal is total resistance in series.

Cp(0) = .063Ebat = 14.8 VoltsKv = 444 rpm/VoltD = .356 mRtotal = 0.133 +.02 = .153 Ohms (the 0.02 value is for battery, cable and ESC resistive losses)

Table 4 shows the comparison of the two as a function of rpm. The shaft powers match at approximately 5000 rpm, shaft power of about 260 Watts, maximum current about 23 Amps and maximum battery power about 340 Watts.

rpm n, rps Pprop Pshaft Imotor Prtot Pbat4500 75.0 185 309 18 51 3604700 78.3 211 292 20 61 3524900 81.7 239 271 22 72 3435000 83.3 254 260 23 78 3385100 85.0 270 249 23 84 3335200 86.7 286 236 24 91 327

Table 4 Maximum Throttle RPM, Power and Current Match

Motor Parameter SummaryTable 5 Motor Parameter Summary

Parameter Value UnitsKv 444 Rpm/voltRm < 0.133 OhmsIo < 0.25 at 3.7 Volts Amps

Imax 23 AmpsPmax 340 Watts

Max Voltage 4S or greater

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Example MotorA Tiger Motor T-motor U5 KV400 comes close to the requirements. The slightly lower Kv will require a slightly greater throttle setting. The values for Rm and Io are less than the requirements so the efficiency should be slightly greater than 80%.

Table 6 Selected Motor Speifications

Parameter Value UnitsKv 400 Rpm/VoltRm .116 OhmsIo at 10 Volts 0.3 AmpsImax 30 AmpsPmax 850 WattsWeight 195 Grams

Example Motor PerformanceUp to this point the analysis has used a simplified version of the motor equations. Having selected a motor and having the output power and rpm is must meet, it is now possible to conduct a better estimate of performance using a detailed motor model.

Given: Pshaft, rpm, Kv, Rm, Io, Vo, Rbat, Rcable, Resc, Ebattery

Find: Itotal, d

The approach is to calculate the required motor current, Imotor given Pshaft, rpm and Kv.

Imotorp = Pshaft∗Kv

rpm

Then calculate a matching value of Imotor from the motor equations.

Itotal = d∗Ebattery− rpm

KvRm+d∗(Rbat+Rcable+Resc)

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Imotorm = Itotal−

Iovo

∗rpm

kv1+d−d2

The goal is to make Imotorp = Imotorm

There is no direct solution for d. One way to find the value of d is to write an iteration algorithm. Another way, used here, is to vary the value of d until a match is found.

Table 7 Given Requirements

Givens Value

Units

Pshaft 7.4 WattsRpm 1644 Rev/minuteKv 400 Rpm/VoltRm 0.116 OhmsIo 0.3 AmpsVo 10 VoltsRbat .018 OhmsRcable + Resc .005 Ohms

Imotorm = 7.4∗4001644 = 1.80 Amps

Table 8 shows the comparison as the value of d is changed.

d 1+d-d^2

Itot Imotorm Imotorp

0.28 1.20 0.25 0.11 1.800.285 1.20 0.86 0.61 1.800.29 1.21 1.47 1.11 1.800.295 1.21 2.07 1.61 1.800.3 1.21 2.68 2.11 1.800.305 1.21 3.28 2.60 1.80

Table 8: Matching motor Currents by varying d

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Figure 10 Matching d

The table and graph show that a value of about d = 2.97 creates a match. Note that the rate of change of Imotorm is steep, so a small error in d creates a large change in Imotorm. Fortunately this is not a big issue if the selected value of d is used in another version of the equation for Itotal.

This other equation is:

Inl = Io∗rpmVo∗Kv

Itotal = Inl+ Imotorp∗F (d )

Substituting values:

Itotal = 0.123 + 1.80*F(d)

F(d) = 1+d-d^2

Plotting f(d) shows that F(d) is not very sensitive to the value of d in the mid-range between 0.25 and 0.75

Figure 11 Function F(d)

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The results of the comparison indicate that d has a value of between 0.29 and 0.30. Substituting:

For d = 0.29: Itotal = 2.293

For d =0.30: Itotal = 2.301

There is no real difference between the results.

Having found the value for Itotal, the power loss for each element can be found.

Table 9 Total Power Estimate

Power Element Equation Value, WattsPshaft 7.4Pinl =Inl*RPM/Kv 0.51Prm =Itotal2 * Itotal 0.614`Pesc + Pcable _+ Pbattery =Itotal2*(Rbattery + Rcable + Resc) .99Ptotal Sum of powers 9.424Efficiency Pshaft/Ptotal 0.785

The overall efficiency is then Kp*Km =.72*.785 = 0.565 compared to the original estimate of 0.50.

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Selected Motor at Full ThrottleThe motor has current and power limits and the operation at full throttle should be checked for compliance. In this case d =1 and the no-load current and power is negligible. The goal is to match the propeller rpm and shaft power requirements with the motor’s ability to satisfy those requirements.

Pprop = Cp (0 )∗ρ∗(rpm60 )3

∗D5

Pmotor = (Ebat− rpmkv )∗rpm

kv∗Rtotal

Cp(0) is the propeller power coefficient at zero airspeed (the worst case)

Rtotal = Rm +Resc + Rcable + Rbattery

To solve for a match condition rpm is varied for both until the powers match. Table 10 lists the relevant parameters.

Table 10 Full Throttle Parameters

Item Value UnitsEbat 14.8 VoltsΡ 1.225 Kg/m3

Kv 400 Rpm/VoltD 0.356 mRtotal 0.139 OhmsCp(0) .063

Table 11 contains calculations for finding a power match. It also provides estimates of the motor current and battery power. RPM is varied until a match is found; in this case at about 4900 rpm.

Table 11 Power Matching Calculations

rpm n, rps Pprop Pmotor Imotor Presistance Pbatttery4500 75.0 185 294 16 37 3314700 78.3 211 264 18 44 3074900 81.7 239 230 20 52 2825000 83.3 254 211 20 56 2685100 85.0 270 192 21 61 2535200 86.7 286 172 22 66 238

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The resulting current and power is within the motor limits.

The corresponding static thrust can be found where Ct(0) = .091. The result is 11.7 Newtons or 1.2 Kg. This compares to the all up weight of 2.1 Kg, so the plane will have moderate climb and maneuvering capability.

SummaryThe process of designing a model for maximum flight duration consists of the following steps.

Layout the airplane dimensions and its configuration with the goal of minimizing drag.

Model its aerodynamic properties.

Select a battery and a total weight goal or goals.

Estimate the resulting flight duration as a function of airspeed and weight.

Choose an operating point for airspeed, thrust and exhaust power.

Choose a propeller and determine the motor rpm and shaft power required to meet the thrust requirements.

Conduct an analysis to determine the approximate specifications for a motor.

Select an actual motor meeting those specifications.

Conduct a detailed performance analysis using the selected motor.

There is now a point solution. The next steps would be to use the selected propeller and motor and analyze the performance for other airspeeds. The process can be automated with the use of convergence algorithms where matching is required.

Appendix A APCE Propeller Power and Thrust CoefficientsCt = tcof4*J^4 + tcof3*J^3 + tcof2*J^2 + tcof1*J + tcof0Cp=pcof5*J^5 + pcof4*J^4 + pcof3*J^3 + pcof2*J^2 + pcof1*J + pcof0

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diameter pitch pcof5 pcof4 pcof3 pcof2 pcof1 pcof0 tcof5 tcof4 tcof3 Tcof2 tcof1 tcof08 4 0 0 0 -0.0821 0.0123 0.0411 0 0 0.3661 -0.3989 -0.0288 0.09688 6 0 0 -0.115 0.0628 -0.0198 0.0749 0 0.2603 -0.3483 -0.0561 0.0263 0.11388 8 0 0 -0.1029 0.1016 -0.0556 0.116 0.3075 -0.6425 0.29836 -0.04 -0.0206 0.13039 4.5 0 0.3639 -0.4287 0.038 0.0074 0.0398 0 0 0.403 -0.5016 0.0023 0.10139 6 0 0.4545 -0.7492 0.2672 -0.0189 0.0519 0 0.2194 -0.0838 -0.2735 0.0197 0.11329 7.5 0 0.303 -0.6682 0.3753 -0.0746 0.083 0 0.4449 -0.7575 0.2225 -0.0344 0.12419 9 0 0 -0.1195 0.1043 -0.0435 0.103 0.0098 0.4 -0.9255 0.5259 -0.1256 0.1359

10 5 0 0 0.0412 -0.1432 0.0322 0.0394 0 0 0 -0.1845 -0.0544 0.099210 6 0 0 0.0394 -0.1155 0.0131 0.0465 0 0 0 -0.1628 -0.0429 0.102810 7 0.613 -0.8638 0.1852 0.0306 0.002 0.0516 0 0.6348 -0.9515 0.2753 -0.072 0.114710 8 1.149 -2.5054 1.716 -0.4508 0.0384 0.0558 0 0 0 -0.1552 0.0033 0.117510 10 0 0 -0.1385 0.1114 -0.0202 0.0943 0 0 -0.1822 0.1135 -0.0112 0.110611 5.5 -2.6283 4.0669 -2.0935 0.2922 0.0032 0.0318 0 0 0.3679 -0.4142 -0.0028 0.0911 7 0 0.5539 -0.9178 0.3512 -0.0245 0.0451 0 0.799 -1.1005 0.2788 -0.0639 0.10911 8 0 0.529 -0.9605 0.4429 -0.0579 0.0521 0 0.693 -1.0471 0.3216 -0.0738 0.111511 10 1.0459 -2.5085 1.918 -0.5313 0.0082 0.083 0 0.4767 -0.9278 0.3949 -0.0454 0.106412 6 0 0 0.1259 -0.1933 0.0297 0.0304 0 0 0 -0.105 -0.0889 0.082112 8 0 0 -0.0477 -0.0367 0.0067 0.0459 0 0 0 -0.1385 -0.0257 0.097812 10 0 0 -0.0518 -0.0358 0.0168 0.0609 0 0 0.105 -0.2693 0.0451 0.104312 12 0 -0.0931 0.1169 -0.0839 0.0091 0.082 0 0 -0.0965 0.0413 -0.0395 0.1213 4 0 0 0.0443 -0.0938 0.0048 0.0175 0 0 0 -0.1098 -0.1021 0.055813 6.5 0 0.3402 -0.4831 0.1313 -0.0137 0.0301 0 0 0 -0.1581 -0.0407 0.082213 8 0 0 -0.156 0.0783 -0.012 0.0346 0 0 0 -0.1469 -0.0319 0.090613 10 0 0.1893 -0.3923 0.1638 -0.0204 0.0494 0 0.2551 -0.347 0.0428 0.021 0.090614 12 0.5014 -0.6723 -0.0847 0.2983 -0.0927 0.0634 0 0.5751 -0.9241 0.262 -0.003 0.091214 6 0 0 0.0305 -0.0743 0.0013 0.0235 0 0 0 -0.1096 -0.1027 0.078314 7 0 0 -0.1534 0.0533 -0.0112 0.0274 0 0 0 -0.0905 -0.0909 0.086514 8.5 0 0 -0.1418 0.0582 -0.0058 0.0355 0 0 0 -0.098 -0.0613 0.097914 10 0 0 -0.077 -0.0007 0.0033 0.0409 0 0 0 -0.0835 -0.0546 0.101515 4 0 0 0 -0.0002 -0.009 0.0139 0 0 0 -0.1182 -0.0837 0.049115 6 0 0 0 -0.0368 -0.0046 0.0227 0 0 0 -0.0994 -0.0845 0.070215 7 0 0 0 -0.052 0.0026 0.0272 0 0 0 -0.0994 -0.0779 0.079515 8 0 0 0 -0.0593 0.0089 0.0317 0 0 0 -0.0907 -0.074 0.08815 10 0 0 -0.0708 -0.0136 0.0117 0.0416 0 0 0 -0.1188 -0.0347 0.098816 4 0 0 0.0443 -0.0194 -0.0052 0.0129 0 0 0 -0.1104 -0.0857 0.046516 6 0 0 0.0831 -0.0917 0.0044 0.0208 0 0 0 -0.1038 -0.0845 0.066416 8 0 0 0.096 -0.1405 0.02318 0.0296 0 0 0 -0.0796 -0.0842 0.084416 10 0 0 -0.0167 -0.0641 0.0195 0.0385 0 0 0 -0.1086 -0.0495 0.095516 12 0 0 -0.0136 -0.0615 0.019 0.0424 0 0 0 -0.1109 -0.0364 0.09817 6 0 0 0.0832 -0.0813 0.0016 0.0193 0 0 0 -0.1018 -0.0862 0.06317 7 0 0 0.1238 -0.1329 0.0133 0.0232 0 0 0 -0.0786 -0.0923 0.072417 8 0 0 0.0815 -0.1289 0.0206 0.0269 0 0 0 -0.1017 -0.0765 0.079917 10 0 0 0.0085 -0.0864 0.0222 0.0355 0 0 0 -0.107 -0.0572 0.093617 12 0 0 -0.0117 -0.0676 0.0214 0.0394 0 0 0 -0.1141 -0.0403 0.09618 8 0 0 0.0952 -0.127 0.0163 0.0253 0 0 0 -0.0941 -0.0824 0.076518 10 0 0 -0.0064 -0.0695 0.0142 0.0338 0 0 0 -0.1088 -0.0612 0.089818 12 0 0 -0.0083 -0.0699 0.0214 0.0424 0 0 0 -0.1101 -0.0397 0.099119 8 0 0 0.1131 -0.1288 0.0133 0.0239 0 0 0 -0.0829 -0.0897 0.073619 10 0 0 0.0173 -0.082 0.013 0.0317 0 0 0 -0.1026 -0.0693 0.086919 12 0 0 -0.1108 0.0165 0.0021 0.0399 0 0 0 -0.1217 -0.0415 0.0965

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