[Elearnica.ir]-Experimental Test for Measuring the Normal and Tangential Line Contact Pres

EXPERIMENTAL TEST FOR MEASURING THE NORMALAND TANGENTIAL LINE CONTACT PRESSUREBETWEEN WIRE ROPE AND SHEAVES

Figure 1 depicts schematically how a traction elevatoroperates. The system is basically composed of an ele-vator car (1), a counterweight (2), a traction sheave ordrive (3), and some ropes (4) that are disposed in par-

allel and that join the car with the counterweight passingthrough the traction drive.

In this kind of elevator, it is crucial to ensure an appropriateadherence between the traction sheave and the hoisting ropesbecause if sufficient adherence is not guaranteed, the ropecould start bodily slipping. This may cause the falling of theelevator car or of the counterweight to the pit floor of thehoistway, involving significant property damages as well asloss of human life.

Traditionally, conventional elevators are dimensioned inorder to ensure that between rope and sheave enough adher-ence region always remains during duty service. This adher-ence region represents a safeguard against bodily slippingbetween rope and sheave as was introduced by Schulz.1 Evenso, the present tendencies in the lifting sector lean toward thereduction of the traction sheave diameter since it, in turn,reduces the required nominal torque of the drive (note thatthe nominal torque can be written in a simplified manner as2max T1 2T2

D ; see Fig. 1).

With a lower requirement of torque, a smaller hoisting ma-chine is possible and thereby the drive fits in the hoistway,and there is no need for additional room in the building inorder to house the drive. These types of elevator solutionsare called machine roomless elevators and they, at present,represent a competitive advantage in the hoisting sector.

However, the reduction of the ratio between sheave and ropediameter, D/d, beyond a determined value may entail, on onehand, a drastic reduction of rope life and on the other, a reduc-tion of the adherence capacity of the hoisting drive (seeHeller2 and Nabijou3) and therefore of safeguard level in theelevator. Under these circumstances, the sector is demandinga more precise and accurate comprehension about the mecha-nical interaction between rope and traction drive in order tocombine a more reduced traction sheave diameter design withenough safeguard level.

The authors have detected in this sense that little experimen-tal work has addressed rope and traction drive interaction. Thestudies made by Wiek4 and later by Haberle5 are probably themost remarkable, and even in those articles, the study isfocused on a particular operating situation. Therefore, in order

to acquire a more profound comprehension, and in order toverify the level of the agreement between testing and the cur-rent theoretical models, a new testing procedure is discussed inthis article. In the first part of it, a brief theoretical backgroundand previous experimental work is introduced. Next, the test-ing machine and procedures are presented and, finally, somepreliminary results are discussed.

THEORY

RopeSheave Interaction Longitudinal ModelsThe longitudinal models are bidimensional theoretical modelsthat in a simplified manner characterize the interaction be-tween the rope and the sheave along the plane that containsthe principal axis of the rope when the latter is bent over thetraction sheave. The first longitudinal model was developedby Eytelwein6 but should probably be attributed to Euler (seeChaplin7) with whom Eytelwein worked for some time inSwitzerland. The most popular formula derived from thismodel is the one called Capstan formula, which is still nowa-days the most widely used expression in order to dimensiontraction drives. The Capstan formula:

qc 5T2T15 emu; 1

defines a limiting ratio, qc, between the rope tension,T1 andT2,at both sides, sides 1 and 2, of D diameter traction sheave. m iscalled apparent friction coefficient of the sheave and u is thetotal winding angle. Theoretically, if T2/T1 surpass the ratiodefined by Eytelwein, the rope slips bodily through the sheave.

Following the basis of the model, it also derived the normalline pressure distribution, p(f) (note that this model beingbidimensional, the normal and frictional contact pressure arenot written in terms of [N/m2] but in terms of [N/m]), as wellas the rope tension, T(f), for any angle f, which denotes aspecific point of the winding arc.

Tf5 T1emf: 2

pf5 2T1emf

D: 3

On one hand, note that Eqs. 2 and 3 show that the two vari-ables obey an exponential distribution. On the other, note thatEq. 1 suggests there is not any dependence of the D/d in thedrives limiting T2/T1 ratio. However, as was mentionedbefore, the influence of the ratio D/d in the critical imbalanceratio, qc, has been reported (e.g., Nabijou

3).

TECHNIQUES by H. Usabiaga, M. Ezkurra, M.A. Madoz, and J.M. Pagalday

H. Usabiaga is a PhD student in engineering and an assistant researcher andJ.M. Pagalday is an engineer and head of the Mechanical Department at Ikerlan.S. Coop., Arrasate-Mondragon Gipuzkoa, Spain. M. Ezkurra is an engineer and as-sistant professor at the Mondragon Unibertsitatea, Arrasate-Mondragon Gipuzkoa,Spain. M.A. Madoz is a PhD and head of the Mechanical Department at Orona S.Coop., Hernani Gipuzkoa, Spain.

34 EXPERIMENTAL TECHNIQUES September/October 2008 doi: 10.1111/j.1747-1567.2007.00294.x 2007, Society for Experimental MechanicsDownloaded from http://www.elearnica.ir

Heller2 developed an equivalent expression to the Eytelweinsformula which shows a diameter ratio dependence. The mostsignificant difference between the Eytelwein and the Hellermodels is that the former only considers the axial forces in therope, whereas the latter, besides considering the axial forces,also considers radial shear forces of the rope.

The expression for T(f) derived by Heller is represented inEq. 4.

Tf5 T2e2BfB

Csinh Cf 1 cosh Cf

; 4

where

B5Dd11

2m5

and

C5

B21

D

d

r: 6

Note that, as wasmentioned before, Hellers formula is depen-dent on the ratio D/d since B and C depend on this ratio.Besides this, note that Hellers approach for tension gives aswell a nearly exponential distribution.

Figure 2 depicts the critical imbalance ratio, qc, estimated bythe Heller and Eytelwein models against D/d for three differ-ent apparent friction values. As illustrated in figure, the two

models lead to similar results if D/d is high. However, whenthis ratio decreases, according to Heller, the shear internalforces cannot be neglected and, as a result, Eytelweins modeldiverges significantly from Hellers.

Rope and Sheave Slip StagesThe previous two expressions consider only the situationwhere the sheave is about to slip bodily through the sheave.Therefore, this situation implies theoretically the maximumratio at which the traction drives could operate. However, thetraction drive system must always operate under lower T2/T1ratios. The literature describes three different interactionstages as a function of the applied imbalance ratio, that is,q 5 T2/T1. These three stages are schematically representedin Fig. 3.

Following the same assumption than Eytelwein made in hismodel, it can be derived that when an equal load, T15 T25 T0,is applied at both ends of the rope, a constant contact pressureof value

p0 52T0D

7

as well as constant tension act respectively along the contactarc and along the rope (see Fig. 3a). However, this uniformsituation is not initially attained in practice during the load-ing. Some sheave rotation is required in order to distribute theuniform tension though the whole winding arc as was shownin Usabiaga et al.8 Nabijou3 named the final uniform situationnonslip stage (NSS) because, under this stage, no slip occursbetween the rope and sheave.

D

Fig. 1: Main components of an elevator hoisting system

SHEAVEWIRE ROPE INTERACTION CONTACTPRESSURE MEASUREMENT

September/October 2008 EXPERIMENTAL TECHNIQUES 35

If we start reducing the load at one end of the rope, an imbal-ance between both ends of the rope is prescribed. When thishappens, part of the contact arc still remains under the NSSsituation. However, on the remaining arc length, slipping aswell as friction force start arising between the rope andsheave, leading, in this manner and as it is depicted in Fig. 3,to exponential rope tension and normal contact pressure pro-file. This stage is called partial slip stage (PSS).

When we keep increasing the applied imbalance ratio, the arcwith exponential profile contact pressure progressivelyenlarges and, accordingly, the arc length that exhibits a con-stant profile arc length reduces. The increase of imbalanceratio finally leads to a fully sliding contact arc and, accord-ingly, to a fully exponential normal contact line pressure andrope tension distribution. This limiting ratio could be esti-mated, as mentioned before, by means of Capstan formulaand the associated situation is called full slip state (FSS)

(see Fig. 3c). Finally, a greater decrease in load at one endproduces a bodily slipping of rope around the sheave, promot-ing a very dangerous situation for the traction drive systems.

Besides these stages, another remarkable feature is thesheave turning sense influence in the placement of the adher-ence contact arc. This feature is evidenced for example inSchulz.1 After the sheave has turned in one sense for sometime in a PSS situation, the placement of the adherence con-tact arc seems to depend exclusively on the sheave turningsense. According to Schulz, if the sheave has been turning leftat constant speed, the constant pressure segment is at theright side of the sheave (see Fig. 3) and, on the contrary, ifit has been turning right, the constant pressure places on theopposite side, namely on the left. Theoretically, it can be dem-onstrated that the opposite statement can never happen usingthe reduction ad absurdum as Johnson9 made for similarissues.

10 20 30 40 50 60 70 801

2

3

4

5

6

7

Diameters ratio (D/d)

Imba

lanc

e ra

tio (q

)Heller =0.2Eytelwein =0.2Heller =0.4Eytelwein =0.4Heller =0.6Eytelwein =0.6

d=7.2mm

Fig. 2: Critical imbalance ratio, qc, versus diameters ratio, D/d, according to Eytelwein6 and Heller2

(a) NSS (b) PSS (c) FSS

Fig. 3: Slip stages


36 EXPERIMENTAL TECHNIQUES September/October 2008

Finally, the authors (see Usabiaga et al.8) have recently devel-oped a numerical model that is mainly based on Eytelweinsassumption but allows as well to analyze the mechanical con-ditions along the traction drive under any slipping stage.Moreover, the model considers the development in time ofthe mechanical conditions when the sheave turning sensechanges or when the load at one or both ends varies. Note thatthese two situations are very common in elevator suspensionsystems.

In particular, the results presented in that paper demonstratethat when the sheave is evenly loaded, a uniform contactpressure and rope tension are not initially achieved. Instead,two symmetrical exponential distributions are attained forline tension and normal contact line pressure. However, themodel also demonstrates that when the sheave starts turn-ing, this situation gives way progressively to a completelyuniform line tension and normal contact line pressure.According to the model, this last uniform situation remainsuntil load changes at one or both ends. Considering this, whenthe objective of the test is to record the contact normal linepressure under a completely uniform line tension and uniformnormal line contact situation, after installing and loading therope and before recording pressure, the sheave must turn atleast an angle equivalent to the total contact arc angle. Notethat this fact has important consequences for the design of thetesting procedure and machine because the testing machinemust allow turning the sheave and running enough rope onthe sheave in order to arrive to the uniform situation.

EXPERIMENTAL PROCEDURE

The key of the procedure introduced here lies in measuringnot the normal and frictional contact pressures, p(f) and f(f),which may be rather complicated, but the resultant contactforces along a large contact arc angle. However, the pressurescan be derived from the measured quantities by means ofsome mathematical manipulation we will introduce later.

The testing machine developed throughout this article is di-vided into

x the testing frame andx the sensing sheave.

Testing Frame DescriptionThe bed frame of the testing machine is depicted in Fig. 4. Itbasically consists of what in the elevator sector is described as2:1 hoisting suspension. Two variable mass counterweights(1,2) are used, in order to prescribe an average rope tensionas well as an imbalance ratio between both ends of the rope.Note that the selected suspension scheme provides a gain oftwo in the ratio between the stroke length of the rope and thestroke length of the counterweights, so that a ls length ofcounterweight stroke involves 2ls length of rope running thesheave. This type of hoisting suspension is found particularlyuseful when the stroke of the counterweight is limited. Notethat when the counterweights start and end a stroke, thesheave must change the turning sense and therefore, for sometime in each stroke, the sheave must respectively accelerateand decelerate. If the interest of the test lies in recording theinteraction conditions under constant sheave turning speed,and if the counterweight stroke is short, the length of rope

stroke that runs the sheave under constant speed could be tooshort if a 1:1 suspension system is considered. A 2:1 suspen-sion system could represent an alternative solution in thosecases.

The two counterweights are guided by two vertical rails (7),as is illustrated in Fig. 4. Special care has been taken inorder to minimize the friction between the rails and the shoeguides (8) installed at the counterweights. Note that thisfriction generates additional imbalance force in the hoistingsystem; therefore, it must be minimized as far as possible. Tothis end, very low friction glide shoes which are commonly

Fig. 4: Bed frame scheme



used at the elevator sector have been selected for guiding thecounterweight along the rails. The stroke of the counter-weights is 12.5 m, which implies that the stroke of the rope(4) along the driving sheave (1) is 25 m. In order to turn thesheave, a permanent magnet synchronous motor wasselected. This machine together with an appropriate control-ler can govern the turning speed of the traction sheave asrequired.

Instrumented SheaveThe main goal of the experiment consists in measuring theresultant normal and tangential force acting onto an arc por-tion of the sheave. Properly postprocessed, this value can beconverted to its respective local value, namely p(f) and f(f).

Based upon this principle, a classical elevator cast iron driv-ing sheave (1) with a nominal diameter of 0.250m and 7.5-mmdiameter U-type groove was selected for the experiment (seeFig. 5). A 158 and 40-mm deep sheave portion was machinedwith an electrical discharge machining device taking specialprecaution in order to preserve the cut part.

After this, a triaxial piezoelectric load cell (reference: PCB260A01, PCB Piezotronics, Inc., Depew, NY) was selectedfor measuring the applied load on the removed part. Thesetransducers are ceramic made and therefore very stiff. Thedecision to select a very stiff transducer was made after run-ning several finite-element methodbased calculations of theassembly and afterward concluding that any displacement ofthe transducer or the detached part could generate undesiredinfluence on the measured magnitudes. The selection ofa ceramic transducer should avoid this.

Figure 6 depicts the final assembly of the sensing sheave. Thetransducer (2) was placed in the cavity left for this purpose inthe sheave. Above it, the grooved side of the removed part,which from here on we will call detachable part (3), wasmounted again. Note that this part had to be carefullymachined in order to take out from it the same volume isnow occupied by the transducer.

The transducer together with the detachable part were fixedto the sheave with a CopperBeryllium screw rod (4) which isalso very stiff and therefore should prevent any disturbance ofthe normal and tangential contact pressure distribution. Thefixing nut (5) prestresses the screw rod, avoiding in this man-ner any nonlinear behavior between forces and relative dis-placements of the sensing sheave parts.

It should be remarked that a very strong effort was madeto place the demountable part of the sheave on the samelevel as the rest of the sheave surface. High-definition geo-metric measuring machines were used and corrective meas-ures were taken until the groove was placed according tojD12D3j

the experimental conditions along the contact arc. Therefore,in the following, a procedure is introduced in order to com-pute pressure values from the transducer-measured resul-tant forces.

In this sense, note that the operation that performs the trans-ducer when it collects the resultant forces along the wholecontact arc can be reinterpreted as a mathematical opera-tion called convolution between the local pressure function,namely, p(f) and f(f), and a particular transfer function.

Mathematically the convolution is written as follows:

hgt5ht2tgtdt; 10

where hg represents, for this particular case, the resultantforce measured by the transducer, h; the local pressure; andg, the test function.

The test function can be computed bearing in mindx the measuring rate of the signal acquisition device,x the arc length that occupies the detachable part, andx the prescribed turning speed of the sheave.

The inverse of this mathematical operation is called deconvo-lution and can be used together with the previously intro-duced transfer function in order to invert the process, sothat the pressures can be computed from the resultant forcesrecorded by the transducer.

Tested Ropes and Testing ConditionsRegarding the tested ropes, two elevator rope constructionswere considered:

x a 7.2 2 8 3 19S-FC (the designation of wire rope followsISO17893:2003) rope

x a polyurethane (PU)-jacketed 6.5-mm diameter 4.8 2 63 19S wire strand core (WSC) rope.

The comparison between jacketed and conventional ropeshould contribute to determining if the literature hypothesesare also acceptable in order to model jacketed ropes.

Regarding the imbalance of the prescribed load, during thetest, the imbalance ratio was varied. For the conventionalrope, one counterweight was load with 2000 N, whereas thesecond counterweight was load with 2000 N, 1800 N, 1660 N,and 1560 N, applying in this manner respectively an imbal-ance ratio of q 5 1, q 5 1.1, q 5 1.2, and q 5 1.3.

Due to the higher adherence friction coefficients between thejacket and the sheave, higher imbalance ratios were used withthis second rope construction. In that case, one counterweightwas loaded with 2000 N, whereas the second was loaded with2000 N, 1660 N, 1430 N, and 1250 N, applying this time animbalance ratio of q 5 1, q 5 1.2, q 5 1.4, and q 5 1.6.

Finally, regarding the sheave-turning speed, all the testswere carried out under constant 0.8rad/s sheave turningspeed. The relatively low speed was selected keeping in mindthat quasistatic hypothesis are considered by Eytelwein. Theselected speed together with keeping sheave speed constantshould prevent any significant influence of inertial forces onthe traction drive interactions and should help make a muchclearer comparison between theory and tests.

RESULTS

Test Carried Out with Conventional RopeFigure 8 shows some normal contact pressure distributionresults for q 5 1, q 5 1.1, q 5 1.2, and q 5 1.3 imbalanceratios. Not including the first peaks that arise during therunning off and winding region (this phenomena will be dis-cussed later in Normal Pressure Peaks Near Winding andRunning Off Points section), the test we carried out showsthat for q 5 1 (see Fig. 8), a uniform constant pressure dis-tribution is attained along the sheave contact arc. However,

Fig. 6: Final sensing sheave assembly



when the applied imbalance ratio increases progressively, anearly exponential-like distribution starts arising on the rightside of the sheave according to what Schulz pointed out in hisarticle about sheave-turning sense influence on the adherenceregion placement. Finally, q 5 1.1 and q 5 1.2 show that theconstant pressure gives way progressively to an exponentialpressure distribution until the highest imbalance ratio, q 51.3, shows a nearly FSS situation.

Tests Carried Out with PU-Jacketed RopeSome disagreements between the theoretical model and theexperimental test arose with the jacketed rope test. In thiscase, even in a balanced situation, the experimental measuresshow not a constant distribution but an increase of pressuretoward the middle arc of the sheave (see Fig. 9a).

For the conventional rope case, the friction force between ropeand sheave should be the responsible of any rope tensionchange. For the jacketed rope case, the friction distributionsshows that friction force is distributed, in the case of q 5 1,evenly symmetrical and oriented from the outer limits of the

winding arc toward the center. Therefore, the recorded fric-tion force supports the idea that tension is not constant andthat it increases toward the center of the contact arc in thecarried out tests.

In the case of applying an imbalance between wire rope andsheave, a similar behavior is attained. However, this time thedistribution is not symmetric. Note that this lack of symmetryis required, since the resultant friction between sheave andrope must balance the prescribed imbalance.

Normal Pressure Peaks Near Winding andRunning Off PointsIn all the tests that we carried out, near the winding on andrunning off points, the pressure peaks always rise divergingfrom what was expected from the theoretical point of view.Wiek4 and later Haberle5 describe the same peaks in the teststhat they carried out.

As it is depicted in Fig. 10a, due to its bending stiffness, theropes need certain transition length before taking the same

Fig. 7: Conversion to local quantities

60 80 100 120 140 160 180 200 220 240 260

0

5

5

10

15

20

25

Nor

mal

line

pre

ssur

e (p)

[N/m

m]

Winding angle ( ) [degrees]

q= 1.0q= 1.1q=1.2q= 1.3

D=200mmU groove (dg=7.5mm)

p0 p0

Fig. 8: Normal line contact pressure, p(f), versus sheave winding angle, f, for q 5 1, q 5 1.1, q 5 1.2, and q 5 1.3 withconventional rope



radius of curvature as the sheave. This transition curvaturewas theoretically estimated by Feyrer.10 As it is depicted inFig. 10a, the transition length makes the rope meet thesheave curvature beyond the point it would, if such transitionlength would not exist when winding on and previously whenrunning off, which creates, in turn, additional moments andincreasing peaks at the external regions of the sheave contactarc (see Fig. 10b).

Haberle5 has made thorough considerations with respect tothis phenomena. By means of a regression analysis, the

experimental test that he carried out provides expressionsfor estimating the peak level as well as the delay and leadangle, nA and nB, of, respectively, the winding on and run-ning off contact points. Feyrer11 gives some theoreticalexpressions as well in order to estimate n and the reactionforce Q.

The test that we carried out shows also a slight windingangle difference with respect to the theoretical 1808, so thereason argued by the former authors is consistent in ourtests.

50 0 50 100 150 200 2505

0

5

10

15

20

25

Nor

mal

line

pre

ssur

e (p)

[N/m

m]

Nor

mal

line

pre

ssur

e (p)

[N/m

m]

5

0

5

10

15

20

25

Tang

entia

l line

pre

ssur

e (f)

[N/m

m]

Tang

entia

l line

pre

ssur

e (f)

[N/m

m]

Tangential line pressureNormal line pressure

p0p0

4.8619 SWSC (jacketed dj=6.5mm) D=200mmU groove(dg=7.5mm)

50 0 50 100 150 200 250

5

0

5

10

15

20

25

30

5

0

5

10

15

20

25

30

Tangential line pressureNormal line pressure

4.8619 SWSC (jacketed dj=6.5mm) D=200mmU groove(dg=7.5mm)

(a)

(b)Winding angle ( ) [degrees]

Winding angle ( ) [degrees]

Fig. 9: Normal and tangential normal line contact pressure, p(f) and f(f), versus winding angle, f, for (a) q 5 1 and (b) q 5 1.6with jacketed rope



DISCUSSION

According to the test performed during this study, the hypoth-esis assumed by the theory characterize correctly the phe-nomena, at least under constant sheave-turning speed andfor conventional (not jacketed) ropes.

Inparticular, for theq51case, theresultsattained in those testsare very similar from the ones obtained by Haberle.5 In bothcases, the agreement between the experimental value and thetheoretical value estimated from Chaplin7 is very remarkable.

However, the tests performed for the jacketed rope case showremarkable differences between test and theory. This mayindicate that, for the jacketed rope, Eytelweins or Hellershypothesis is not appropriate for measuring the phenomenon.

In this sense, note that, although frictional distributionmatches the normal pressure distribution, suggesting thatrope is not constant, as it is depicted in Figs. 9a and b, thefrictional pressure distribution is not proportional to the nor-mal pressure distribution as would be expected if Amontonslaw was fulfilled and sliding between the polymer jacket andthe sheave would occur.

These disagreements may be attributed to the local compres-sion of the polyurethane jacket. Due to the winding and run-ning off peaks, the jacket may be unevenly compressedbetween the outer regions and the mid-length region of thewinding arc. Note that this difference of compression existsalso for the conventional rope case; however, the compressiondeformation level is not as significant for the jacketed ropecase because the jacket stiffness is remarkably lower thanthe stiffness of rope cross-section.

The higher compression of the polyurethane jacket along theouter region of the winding arc might locally reduce the effec-tive sheave diameter and this in turn might stretch the ropein the mid length of the winding arc.

The relation between the recorded frictional and the normalpressure distribution could also be consistent with the exist-

ing theories. The stretch of the rope in the mid-length arcshould not obligatorily mean that the polymer surface in con-tact with sheave should be as well stretched in the same way.Therefore, it does not necessary imply slipping between jacketand sheave.

Thanks to the polyurethane flexibility, the polymer jacketcould be subjected to shear deformation in such a way thatthe inner surface of the sheave is deformed according tothe rope deformation (which in turn might be producedby the effective sheave diameter variation), and on the con-trary, the outer part remains in adherence with regardsto the sheave. Note that if the sheave is in adherence withthe jacket, the recorded values are still compatible withAmontons law and that in this case, the tangential forcewe are measuring is strictly the applied shear force inthe jacket.

However, note that the thickness of the jacket is very thin,0.85 mm. Therefore, there might be reasonable doubtsfor this particular case whether the effective diameterchange promoted by the jacket uneven compression alongthe winding arc could produce such significant effects inthe interaction.

CONCLUSIONS

The experimental procedure described in this article is dem-onstrated to be particularly robust and useful in order todetermine experimentally the interaction conditionsbetween the rope and the sheave. This is particularly truefor the ease of analysis for certain rope and sheave configu-rations to determine the effect of the

x prescribed mean load,x prescribed imbalance load, andx sheave turning speed

in the mechanical conditions of the hoisting drive.

The carried out preliminary tests show that conventional ropecould exhibit a very close behavior from what was theoreti-cally expected. However, further studies should be carried outin order to certify the explanation that is given for under-standing the disagreements between the theoretical hypoth-esis and the result we attained.

References1. Schulz, S., Braking Equipment for Friction Hoists, The

South African Mechanical Engineer 28:426433 (1978).

2. Heller, S.R., The Contact Pressure Between Rope andSheave, Naval Engineering Journal 4957 (1970).

3. Nabijou, S., Frictional Behaviour and Fatigue Perfor-mance of Wire Ropes Bent Over Small Diameter Sheaves, PhDThesis, Imperial College of Science, Technology and Medicine,London, UK (1990).

4. Wiek, L., The Distribution of the Contact Forces on SteelWire Ropes, OIPEEC Bulletin 44:1025 (1982).

5. Haberle, B., Pressung zwischen Seil und Seilrille, PhDThesis, Universitat Stuttgart, Stuttgart, Germany (1995).

6. Eytelwein, J.A., Ausgaben grobtentheils aus der angen-wandten mathematik, Freidrich Manrer, Berlin, Germany (1793).

(a) (b)Fig. 10: Source of the apparition of the peaks at the windingon and running off points of the winding contact arc



7. Chaplin, C., Rope Tension in Double Drum TractionWinches, OIPEEC Conference 2003: Experiences with Ropes,pp. 201212, Lenzburg, Switzerland (2003).

8. Usabiaga, H., Ezkurra, M., Madoz, M., and Pagalday, J.,Mechanical Interaction between Wire Rope and Sheaves, OipeecConference 2006: Trends for Rope, pp. 157166, Athens, Greece (2006).

9. Johnson, K.L., Contact Mechanics, Cambridge UniversityPress, Cambridge, UK (1985).

10. Feyrer, K., Effect of Bending Length on Endurance of WireRopes, Wire World International 23:115119 (1981).

11. Feyrer, K., Wire Ropes: Tension, Endurance, Reliability, 1stEdition, Springer-Verlag, Berlin, Germany (2006). n



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