American Institute of Aeronautics and Astronautics1

AN APPROACH TO FEEL SYSTEM CHARACTERISTICS SELECTION

B.Lee1, V.V.Rodchenko2, L.E.Zaichik3

1 Associate Technical Fellow, Boeing, Seattle, WA2 Head of Section, Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia3 Leading Research Engineer, Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia

Abstract

Improved is a theoretical approach to selectfeel system characteristics proposed in [1]. Shown arethe on-ground simulation data on the effect of wheelfeel system characteristics on aircraft handlingqualities. Developed is a simple method to select feelsystem characteristics.

Introduction

While developing an aircraft, the selection ofmanipulator feel system characteristics presents somedifficulty, thus these characteristics are usuallyselected empirically. As feel system characteristicsare numerous and their effect on handling qualities(HQ) is very complex, the empirical way to selectfeel system characteristics does not guarantee theirbeing optimum. The present work aims to refine atheoretical approach developed in [1] to select feelsystem characteristics of a wheel-type controller for alarge jet transport aircraft.

According to our approach, we analyze staticcharacteristics (wheel displacements and forcesapplied by a pilot) and feel system dynamics to selectfeel system characteristics. The approach wassubstantiated in [1], but it has to be refined, since feelsystem characteristics in [1] were selectedindependently for static and dynamic characteristicsto be optimum. However, certain feel systemcharacteristics variation (breakout, for example)causes the level of wheel forces and feel systemdynamic performance to vary simultaneously. Thus,one of our tasks here is to formulate the approach inmore general terms to select feel systemcharacteristics according to a common criterion forboth static and dynamic performance.

In [1] the approach was based on a limitednumber of experimental data on nearly optimum feelsystem characteristics, that is why the applicability of

our approach to a wider range of feel systemcharacteristics has to be tested. Thus we are to findout if our approach is applicable to select maximumpermissible feel system characteristics for variousHQ Levels, and our second task here is to receivemore detailed experimental data on feel systemcharacteristics effect on HQ and to define moreaccurately the limitations of our approach.

According to our theoretical approach, feelsystem characteristics selection requires solvingnonlinear multi-parameter differential equations,which complicates understanding the nature of feelsystem characteristics effect on HQ and makes theirselection a difficult procedure. Thus our third taskhere is to use our theoretical method to create asimpler design method to select feel systemcharacteristics without solving nonlinear feel systemdynamic equations.

1. Setting the experiment.

In the present work we analyze the datareceived on TsAGI FS-102 flight simulator (fig.1) byBoeing and TsAGI experts. The experiments wereconducted in addition to those described in [1].

Unlike the experiments in [1], the experimentmatrix was added by widening friction and breakoutvalues range and by considering damping, wheelinertia and control sensitivity. But in the course of thepresent work we did not consider feel systemcharacteristics configurations for the friction valuesignificantly exceeding the breakout value (inpractice, the breakout value is always selected to beno less than the friction value).

The main characteristics of FS-102 and theexperimental technique were described in detail in[1], thus we mention them here only briefly.

The cockpit of FS-102 reproduces a typicalRussian airliner cockpit. The TsAGI FS-102 isequipped with a 4-window collimated visual system.

AIAA Atmospheric Flight Mechanics Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-5362

Copyright © 2004 by The Boeing Company. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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The simulator has 6 degrees-of-freedom motionsystems, but the experiments were conducted withmotion system switched off.

The column-wheel controller inceptor wasloaded with an electrical loading system. The feelsystem characteristics were reproduced in accordancewith the following equation:

pilotbrfr FFFFFm =++++ δδδδδ δδ sgnsgn !!!!!

In the experiments the aerodynamics andflight mechanics model simulated a typical fly-by-wire wide-body quad. The model was linearized.

Both in [1] and in the present work weconsidered four piloting tasks: standard landing,offset landing, gust landing and free piloting in thevicinity of the airfield.

Two Russian test pilots (from Flight ResearchInstitute) and one American test pilot (from Boeing)took part in the experiments. Pilots’ comments andtheir ratings on Cooper-Harper scale were registered.

2. Theoretical Approach.

2.1. The main principles.First principle. The effect of different feel

system characteristics on HQ is considered to dependon applied forces ( F ), wheel displacements (δ ),feel system overshooting (∆) and control inceptorresponse time (tr). There exists a certain desired(optimum) combination of these parameters (F∗ ,δ∗ ,∆*, tr*) which does not depend on feel systemcharacteristics, control sensitivity and aircraftdynamics. When the generalized parameters F , δ ,∆, tr deviate from their desirable values, aircraft HQdeteriorate. If the values of some of feel systemcharacteristics are given, the values of the othercharacteristics are selected for F , δ , ∆, tr to beclose to their desirable values.

This statement is based on the integration ofnumerous experimental data on feel systemcharacteristics effect on HQ for Level 1, our currentdata presented in fig.2 substantiate the statement aswell. The data in fig.2 are the pilot ratings andcomments on roll control for different values ofbreakout (Fbr), friction (Ffr) and for the given springgradient (Fδ), damping ( δ!F ) and inertia (m). We seethat low HQ ratings were accounted for by the non-optimum values of the applied forces and feel systemdynamics. It is also obvious that the terms “forces”(“displacements” in other experiments) and“overshooting” are the main terms used by the pilotsto assess feel system characteristics. To describe theother feel system dynamic characteristics the pilotsused such terms as “too stiff centering”, “largebreakout”, “too hard or insufficient damping”, “too

quick or too slow” wheel return into the neutral; infact these terms describe feel system response time.

Second principle. While the effects ofmanipulator and control sensitivity characteristics onHQ are being estimated, aircraft state parameters (φ,p,...) may be assumed independent of manipulatorand control sensitivity characteristics and aircraftdynamics.

To prove this principle it is enough to say thatapproaches and landings are performed with the sameaccuracy for different aircraft, though differences intheir stability and controllability characteristics areconsiderable. Only pilot ratings depend on thesecharacteristics because a pilot commonly maintainsadequate piloting accuracy even when considerableeffort is required.

Our second principle idealizes real flightconditions to some extent, but it allows us to simplifythe expressions to estimate characteristic forces anddisplacements (see chapter 2.2) and thissimplification does not noticeably distort the obtainedvalues.

The mathematics of the approach. Weformulate in mathematical terms the general approachto two main aspects of feel system selection: (1) theselection of optimum feel system characteristics and(2) the determining of feel system characteristicspermissible for different HQ Levels.

(1) The optimum values of wheel feel systemcharacteristics are selected to satisfy the condition

( )∗∗∗∗ −∆−∆−− rrbrFF

ttFFJ ,,,min...,

δδδ

(1)

(2) The permissible (boundary) values of feelsystem characteristics for this or that HQ Level aredetermined to satisfy the condition

F = Fper, δ =δ per , ∆ =∆per, tr= tr per;

where:J is the monotonic function (cost function) with theminimum at F =F∗ = const, δ =δ∗ = const , ∆=∆*,t=t*;Fδ , Fbr, …are feel system characteristics (springgradient, breakout,…);F , δ , ∆, t are the values of the generalizedparameters, which are the functions of the feel systemcharacteristics:

( ) ( ),...),(,...),,(

,,..., ,,...,

brrrbr

brbrFFttFFFFFFFF

δδ

δδ δδ=∆=∆==

(2)

Cost function J can be assumed pilot ratingsincrements ( )∗∗∗∗ −∆−∆−−∆ ttFFPR ,,, δδ

caused by deviation of F , δ , ∆, t from theiroptimum values. In the vicinity of the desired

American Institute of Aeronautics and Astronautics3

optimum values of feel system characteristics thisfunction can be approximated by the quadraticfunction:

( ) ( )( ) ( )22

22

∗∗∆

∗∗

−+∆−∆+

+−+−=∆

rrt

F

ttkk

kFFkPR δδδ (3)

The above mentioned expressions could besufficient to select optimum and permissible fordifferent HQ Level feel system characteristics valuesif we could define functions (2).

2.2. Determining characteristic forces Fand wheel displacements δ .

The forces applied by a pilot and wheeldisplacements are the functions of time, which areessentially random functions. Thus, beforedetermining these characteristics we have to defineour notion of “characteristic forces anddisplacements” and the method to determine theirvalues.

Pilot’s comments and time histories analysishave shown that when a pilot assesses the level ofwheel forces, the forces created for deflecting thewheel from the neutral play a decisive role (fig.3,area I). This is accounted for by the fact that whiledeflecting the wheel a pilot has to overcome feelsystem forces, but returning the wheel into the neutralcan not take a pilot any effort, since feel systemforces themselves tend to return it into the neutral.Thus, we determine the applied forces values for thewheel deflected from the neutral position.

Pilots are known to use step-wise wheeldeflections to assess aircraft characteristics, and thistype of deflections creates step-wise roll rate.Generally, roll rate amplitudes can vary. To simplifythe calculations, we assume p* a characteristic rollrate value. In accordance with “the second principle”of our approach, p* does not depend on aircraftdynamics, feel system characteristics or controlsensitivity, i.e. p*=const. Thus, we assume the valuesof forces and displacements to be characteristic ifthey create p = p* .

We should mention that this definition ofcharacteristic forces does not take into accountdamping and inertia forces, which would require usto consider another type of aircraft (or manipulator)motion. But for the case in question the forces ofdamping and inertia are usually small in comparisonto the other feel system forces (spring, breakout andfriction forces).

We assume also that control surfacesdeflection is proportionate to manipulatordisplacement and aircraft dynamics are defined bylinear differential equations.

Thus we have:

++=

= ∗

δ

δδ

δFFFF

p

frbr

p(4)

where p*=const, δp is the roll control sensitivity(wheel deflection per unit of steady-state roll rate).

The value of constant p* can be tuned forbetter agreement between the estimated andexperimental values of feel system and controlsensitivity characteristics.

2.3. Determining the feel system dynamicperformance.

Unlike the above mentioned assessment ofapplied forces and displacements, when a pilotassesses feel system dynamics, the dynamics ofwheel motion back to the neutral is decisive for apilot (fig.3, area II). Pilots normally appreciate such atype of wheel dynamics, which allows them tomanipulate the wheel without having “to draw” it intothe neutral if the wheel is sluggish while returning tothe neutral or to “hold it back” if the wheel tends toreturn to the neutral too quickly. These facts allow usto specify the general approach, described in 2.1, forthe selection of feel system dynamic characteristics.

Feel system dynamic characteristics are assessedin terms of natural motion of “relaxed limb – wheel”system. The relaxed limb is assumed to have a certainmass mpilot. The motion of the manipulator with thegiven characteristics is defined as follows (5):

( ) 0sgnsgn =+++++ δδδδδ δδ brfrpilot FFFFmm !!!!!

In [1] the limb inertia was measured for a few pilotsubjects.

Fig.4 illustrates the notions of “overshooting”and “response time” that we use here.

If a feel system is described by second orderlinear differential equations (Ffr and Fbr are zero), wecan arrive at the expressions for the overshooting andresponse time, see for example [1]. If the system isnonlinear, it is impossible to have the expressions forparameters ∆ and tr. To determine their values wehave to solve nonlinear equation (5) with 7parameters (5 feel system characteristics, limb inertiaand initial wheel deflection). The analysis of feelsystem dynamics is considerably simplified ifequation (5) is presented in dimensionless form.

Dimensionless equations of feel systemdynamics. We introduce dimensionless time anddimensionless wheel displacement:

0

~ ,δδδτ δ =⋅

+= t

mmF

pilot

American Institute of Aeronautics and Astronautics4

and substitute them for the variables in (5). We arriveat:

)0)0(~ ,1)0(~(

0~sgn~~sgn~~~2~

==

=++++

δδ

δδδδςδ!

!!!!frbr FF

(6)

where

00

)

~ ,~

,(2

δδ

ς

δδ

δ

δ

FF

FFF

F

mmF

F

frfr

brbr

pilot

==

+=

!

.

Using dimensionless equations we can reduce thenumber of parameters to 3 (compare 7 parameters in(5)), the analysis of feel system characteristics effecton feel system dynamics is also simplified. We do nothave to solve equation (6) to see that overshootingdoes not depend on the wheel inertia and that theresponse time is inversely proportionate to m .

Peculiarities of determining the feel systemdynamic performance for Level 2. Experimental datain fig.2 show that at the great values of Fbr and Ffrpilots have to hold the wheel back while its returninginto the neutral. Thus our method based on “relaxedlimb – manipulator” analysis is not applicable forLevel 2.

If the feel system response is too quick, thepilot can not relax his hands since he has to hold thewheel back. In this case, wheel motion into theneutral is, in the first approximation, quasi-static(wheel motion velocity is constant for eachcombination of Fbr and Ffr). If the feel systemresponse is too quick, pilot ratings are determined bymainly the forces pilots apply to counteract feelsystem static forces while wheel returning into theneutral i.e. F -

= Fbr - Ffr + δFδ, and pilot ratings arenot determined by tr. Thus, for Level 2 low pilotratings are accounted for by too quick feel systemresponse, but they are determined by wheel forcesF - for wheel displacements ∗= ppδδ :

F - = Fbr - Ffr + δp p* Fδ . (7)

2.4. Determining the desirable andpermissible values of static and dynamic feelsystem characteristics.

The desirable and permissible values of staticand dynamic characteristics F , δ , ∆, tr, −F andthe values of weight coefficients kF, kδ, k∆, kt inexpression (3) depend on the type of manipulator andthe control axis. There are different ways todetermine them.

The desirable and permissible values ∆, tr andweight coefficients k∆ and kt can be determined fromexperimental curves PR(ζ), ∆PR(m) for a linear feelsystem (without friction and breakout). In this casethere is one-to-one correspondence between thevalues of feel system damping and inertia, on the onehand, and overshooting and response time, on theother hand. This allows us to determine functions∆PR(∆), ∆PR(tr) (fig.5), and then to determinedesirable and permissible values of ∆ and tr andvalues of k∆ and kt.

The desirable and permissible values ofcharacteristics forces F and weight coefficient kFcan be determined from the experimental data on anonlinear feel system. Fig.5 shows pilot ratings as afunction of characteristics forces ∆PR( F ). Thisfunction was received from expression (4), we usedthe experimental data from fig.2 for Fbr ≈ Ffr . (IfFfr=Fbr, the variation of Ffr + Fbr causes only thevariation of wheel forces, values ∆ and tr remainalmost constant, thus pilot ratings are determined byonly F ).

According to our estimations the desirablevalue of δ is about 30% of the full travel for anymanipulator, i.e. δ* = 0.3δmax ; for a wheel weightcoefficient kδ is kδ=18⋅kF (see [2,3]).

For Level 2 the permissible value of −F canbe determined from fig.2: the experimental datacorrespond to PR=6.5, the pilots’ comments includedtoo quick response.

3. Simple method of feel system characteristicsselection.

It is impossible to define the exactexpressions for optimum values of feel systemcharacteristics (Fbr, Ffr, Fδ ,…) according to ourapproach if the feel system motion is described bynonlinear differential equations, that is why we areproposing a simpler method to select feel systemcharacteristics. Here we apply it to select the valuesof friction and breakout, but in the similar way it canbe applied to select any other feel systemcharacteristics.

3.1. Definition of the method.This is a graph-analytical method to

approximately estimate optimum and/or permissiblevalues of feel system characteristics. The method isreduced to drawing the curves of equal values ofF (Fbr,Ffr,Fδ ,…)=const, tr(Fbr,Ffr,Fδ ,…)=const,∆(Fbr,Ffr,Fδ ,…)=const (and F -(Fbr,Ffr,Fδ ,…)=constinstead of tr=const and ∆=const for Level 2boundaries) in the plane of Fbr, Ffr . In chapter 3.2 weshow that all these curves are straight lines, which areeasily drawn.

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Selection of optimum values of Ffr and Fbr. Toselect the optimum values of friction and breakout wedraw curves F (Fbr, Ffr, Fδ ,…)=F*, tr (Fbr, Ffr, Fδ

,…)=tr * and ∆(Fbr, Ffr, Fδ ,…)= ∆* in the plane ofFbr and Ffr. Where all the curves cross we have theminimum of function (1), and thus, this particularpoint corresponds to the optimum values of Fbr andFfr.

If the curves do not cross at the same point,the optimum values of Fbr and Ffr are found withinthe area made up by the three crossing lines (fig.6).More exactly, the optimum Ffr and Fbr are thosewhich are close to the points where curve F =F*crosses curves tr = tr* and ∆=∆* . It is seen from fig.7that pilot ratings are practically the same in this area.Thus, to simplify the procedure of selection of Fbroptand Ffropt , we assume their optimum are equallyspaced from these three curves, i.e. Fbropt and Ffropt arein the center of the circle inscribed into the trianglemade up by lines F =F* , tr = tr* and ∆=∆* (fig.6).

Determining the permissible for Level 1values of Fbr and Ffr. To determine these values weneed to draw the curves, respectively, of themaximum and minimum permissible values ofF (lines I and II in fig.6), of the maximumpermissible values of ∆ (line III in fig.6) and of theminimum permissible values of tr (line IV in fig.6).To take into account the condition of wheelcentering, we also draw the curve corresponding toFbr=Ffr (line V in fig.6).

It should be mentioned that we do not have todetermine the boundary of the minimum permissibleovershooting, since this value (∆=0) corresponds tothe pilot ratings not exceeding Level 1 boundaries(fig.5). For the feel characteristics used in practice,the curve of maximum response time is in the vicinityof the centering line and is usually below the latter,so there is no need to draw the maximum permissibletr curve. As it is seen from fig.5, the permissiblevalues of tr and ∆ for a wheel are equal trmin=0.2 sec,∆max = 0.2 (20%).

It is seen from fig.5 that when control forcesF become 1.3 times greater or less than optimumvalues F∗ , pilot ratings deteriorate. This fact allowsus to assume permissible values of applied forcesequal, respectively, Fmin1 = 0.77F*, Fmax1 = 1.3F*.

Determining the permissible for Level 2values of Fbr and Ffr. The boundary of the maximumpermissible for Level 2 forces F (line I′ in fig.6) canbe drawn similar to that for Level 1. The boundary ofthe minimum permissible forces as well as theboundaries of the permissible overshooting andresponse time are absent for the Ffr and Fbrcorresponding to Level 2. Instead of permissible tr,for Level 2 we draw the boundary of the permissible

forces required to hold the wheel back while itsreturning to the neutral F = F - (line VI in fig.6).

According to the experimental data, Fmax 2 =2.2 F*, F -=1.3 F* .

Agreement between estimations andexperimental data. Optimum and boundary values ofFfr and Fbr estimated according to the method agreewith all experimental data, which is demonstrated infig.7, for example.

3.2. Determining curves of the equal F , ∆∆∆∆ ,tr and F -.

Curves of equal F . The expression for thelines of equal F can be received from (4):

constFpFFF pfrbr ==++ ∗δδ

The equations for the lines of optimum andpermissible values of applied forces can be receivedif we substitute F*, Fmin1, Fmax1 or Fmax2 for F in thisexpression.

From this expression we see that the curves ofequal F in the plane of Fbr and Ffr are straight linesinclined at –450.

The curves of equal ∆. It is impossible to findmathematical expressions for these curves.Expression (6) shows that overshooting valuesdepend on three dimensionless feel characteristics ζ,

frbr FF ~,~ . This fact allows us to determine the curves

of equal ∆ in the plane of dimensional Fbr and Ffrwith the help of fig.8, in which the curves drawn inthe plane of dimensionless frbr FF ~ and ~ are shown.

The curves of equal tr . From expression (6)we arrive at

)1sin(1

)1sin(~~22

2

ϕτςς

ϕτςςτ

ςτ

+−−−

+−=−

−

−

rr

rrfrbr

e

eFF

Let us assume the right hand side of theequation to be K(ζ ,τr), and receive the dimensionalform of the expression. Thus we have

),,(0 rfrbr tKFFF ωςδ δ=− .

It follows from this expression that the curvesof the equal tr in the plane of parameters Fbr and Ffrare straight lines inclined at 450, i.e. they are the linesparallel to the line of Fbr = Ffr and they are shiftedrelative to this line in the vertical direction, the shiftbeing equal ),,(0 rtKF ωςδ δ .

The curves of equal F-. The expression for thelines of equal F -

can be received from (7):

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−∗ =+− perpfrbr FpFFF δδ

It is seen from this expression that in the plane ofparameters Fbr and Ffr the curves of equal F- arestraight lines inclined at 450 .

Conclusions

In the course of the present work we receivedthe experimental data on the effect of breakout,friction and spring gradient on aircraft HQ for thewhole range of their values.

The data received were used to refine thetheoretical approach proposed in [1].

The theoretical approach has been simplifiedand a graph-analytical method has been proposed toestimate the optimum and permissible feel systemcharacteristics for different HQ Levels. The

simplified method allows adequate agreement ofestimations with experimental data.

References

1. Lee, B.P., Rodchenko, V.V., Zaichik, L.E.,“Effect of Wheel System Characteristics onAircraft Handling Qualities,” AIAA Paper 5310,AIAA AFM Conference, Texas, Aug., 2003.

2. Zaichik, L.E., Lyasnikov, V.V., Perebatov, V.S.,Rodchenko, V.V., Saulin, V.K. “Method toEvaluate Optimum Roll Control SensitivityCharacteristics of Nonmaneuverable AircraftEquipped With a Wheel”, TsAGI Transactions,issue 2477, 1990 (in Russian).

3. Rodchenko, V.V., Zaichik, L.E., Yashin, Y.P.,“Similarity Criteria for Manipulator Loading andControl Sensitivity Characteristics”, Journal ofGuidance, Control , and Dynamics, vol.21, No2,1998.

American Institute of

Fig.1. TsAGI Flight Simulator (FS-102)

Fig.2. Some of experimental da

0 0.2 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Fbr/F*

5.2,4.5

1

o

3.7,4.8

7

F

0

F

4.8,4.5o

o

7.4 7.3 6.8 7.5 7.5

6.9 6.5 5.8 6.2 6.5 6.5

6.1 5.6 6.0 5.8 6.9

4.8 5.1 5.5 6.4

F

LF,SC

LF,Pio

LF,GC

LF,SC,Pio

LFr,LF

EFSC,Hb

LF,SC

EF,EC,Hb

LF

EF,Pio

LF,SC

LF,HbLF,SC,Hb

LF, ,Hb

B = Breakout, C = Centering, D = Damping,E=excessive, F = Forces, Fr=Friction, G = Great,I = Insufficient, Gd=Good, Hb=Hold the wheel back,L = Large, N = No, O = Overshooting, Pio = PilotInduced Oscillation, S = Stiff, W = Weak

ID,GC,LB,O

4.1

LO,LB

4.

WF,O,Pi

O

3.

O,L

3.4

O

3.

GdF,Pio

3.5

NO,GC,LF

3.1

Gd

LF,Pi

3.7

LF,Pi

EC

ID,SC,LB

LF,SC

Aer7

ta re

0.6

L

onautics and Astronautics

ceived on TsAGI FS-102.

0.8 Ffr/F*

American Institute of Aeronautics and 8

δwheel

I II

time

Fig.3. The process of forces creating (I)and releasing (II).

e

0

Figsel

0 0.2 0.6 tr, sec1

3

4

PR

Fig.5. Pilot ratings as functions of response time tapplied forces ∗FF / .

2

01

2

3

PR

1

n xtr*

0.4

0 0.5 1 1.5

2

4

6

Level1

Level2

PR

F*Fmin1 Fmax1

tr

Astronautics

.4. Time responseection of the feel s

r, overshooting ∆,

0.1

1

∆*

FF /2

Fmax2

∆

elements affectinystem characteristi

and

0.2

∆max

∗

im

t

δ

g thecs.

Level

Level

tr mi

tr ma

∆

American Instit

Fig.6. Determining the opto

Fbr/F*

0 0.20

0.2

0.4

0.6

IIV

III

V

I′′′′

VI

0.8

1.0

1.2

1.4

Level 2

Le l 1

optimum

ute of A

imum af break

0.4

I

F*

tr**

∆∆∆∆

ve

eronautics and Astronautics9

nd permissible for Levels 1 and 2 valuesout and friction.

Ffr/F*0.6

I

0.8

brF~ ∆=10%

Fig.7. Comparison of experimental data with the estimationsaccording to the method.

1.2

1.4

Fbr/F*

7.4 7.3 6.8 7.5 7.5

6.9 6.5 5.8 6.2 6.5 6.5

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Ffr/F*

6.1 5.6

4.8 5.1

5.2,4.5

5

3.7,4.8 0

1 1

6.0 5.8 6.9

5.5 6.4

4.8,4.5

Level 1

Level 2

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Fig.8. The curves of equal ∆ in the plane of dimensionbreakout and friction.

0 2 4 60

2

4

6

8

10

12

F~

ζ=0.8

ζ=0.8

0.4 0.0

0.4

0.0

∆=20%

4.1

3.7 3.

3.

4.

3.4 3.

3.7

Estimations according tothe method:- optimum- HQ

s

lessfr