8
ISSN 1068798X, Russian Engineering Research, 2013, Vol. 33, No. 7, pp. 392–399. © Allerton Press, Inc., 2013. Original Russian Text © Yu.N. Drozdov, S.L. Sokolov, B.N. Ushakov, 2013, published in Vestnik Mashinostroeniya, 2013, No. 4, pp. 32–38. 392 1 Hinged joints with slip bearings are widely used in airplanes, automobiles, excavators, and elsewhere. In terms of wear and jamming, the bearing life is largely determined by the contact stress in the axis–bush pair. The basic principles for the calculation of the contact stress in hinged joints and other frictional components were developed in [1–6]. In recent years, the finite element and boundaryelement methods have been widely introduced in the computerized analysis of contact stress. However, the error of such calculations may be con siderable, although their assessment is difficult. There fore, an approach that combines experiment and calcu lation is more effective, especially in analyzing the stress–strain state of threedimensional structures. In that approach, methods from experimental mechanics are employed (for example, the photoelastic method, tensometry, the brittlecoating method, and hodo graphic interferometry). The mathematical model is analyzed by comparing the results of calculations with experimental data. Then the model is used to calculate several design options and to select the best. In the present work, we consider the contact stress in hinged joints on the basis of results obtained at Blagonravov IMASh by means of the finiteelement method and threedimensional photoelastic models. Most hinged joints are characterized by a complex configuration and are investigated by means of three dimensional models. However, twodimensional models may be used for joints with a short axis. The hinged joint in Fig. 1 consists of a central lug 1 with bushes 2, lateral lug 3, and a shaft 4. Forces P are applied along the symmetry axes of the lugs. Analysis of the stress in the contact zone is based on a polar coordinate system: r is the radial coordinate; x is the longitudinal coordinate (along the axis of rotation); and t is the azimuthal coordinate. At the contact sur faces of the shaft and bush in a slip bearing, normal stress arises along the stress radius (the contact pres 1 Adapted from a presentation at the International Conference on Mechanics Problems celebrating the centenary of L.A. Galin (Moscow, September 2012). sure), as well as the tangential contact stress τ rt = τ rx ; the t and x axes are in the directions of the transverse and longitudinal tangents to the contact surface. Beyond the contact area, σ r = τ rt = τ rx = 0. At the con tact area (1) where f is the frictional coefficient. Thus, the stress σ r is compressive, while the stresses τ rt and τ rx cannot exceed the frictional force. Normal stress arises at the shaft–bush contact surfaces along σ r 0 ; τ rt f σ r ; τ rx f σ r , < < < Contact Stress in Hinged Joints 1 Yu. N. Drozdov, S. L. Sokolov, and B. N. Ushakov Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences email: [email protected] Abstract—The contact stress in hinged joints is studied by the finiteelement method and the photoelastic method. Recommendations for its reduction are derived. DOI: 10.3103/S1068798X13070058 D l d P l co 1 2 3 4 P l co 1 2 3 4 (a) (b) Fig. 1. Hinged joints with two (a) and three (b) bearings.

Contact stress in hinged joints

Embed Size (px)

Citation preview

ISSN 1068�798X, Russian Engineering Research, 2013, Vol. 33, No. 7, pp. 392–399. © Allerton Press, Inc., 2013.Original Russian Text © Yu.N. Drozdov, S.L. Sokolov, B.N. Ushakov, 2013, published in Vestnik Mashinostroeniya, 2013, No. 4, pp. 32–38.

392

1 Hinged joints with slip bearings are widely used inairplanes, automobiles, excavators, and elsewhere. Interms of wear and jamming, the bearing life is largelydetermined by the contact stress in the axis–bush pair.The basic principles for the calculation of the contactstress in hinged joints and other frictional componentswere developed in [1–6]. In recent years, the finite�element and boundary�element methods have beenwidely introduced in the computerized analysis ofcontact stress.

However, the error of such calculations may be con�siderable, although their assessment is difficult. There�fore, an approach that combines experiment and calcu�lation is more effective, especially in analyzing thestress–strain state of three�dimensional structures. Inthat approach, methods from experimental mechanicsare employed (for example, the photoelastic method,tensometry, the brittle�coating method, and hodo�graphic interferometry). The mathematical model isanalyzed by comparing the results of calculations withexperimental data. Then the model is used to calculateseveral design options and to select the best.

In the present work, we consider the contact stressin hinged joints on the basis of results obtained atBlagonravov IMASh by means of the finite�elementmethod and three�dimensional photoelastic models.

Most hinged joints are characterized by a complexconfiguration and are investigated by means of three�dimensional models. However, two�dimensionalmodels may be used for joints with a short axis.

The hinged joint in Fig. 1 consists of a central lug 1with bushes 2, lateral lug 3, and a shaft 4. Forces P areapplied along the symmetry axes of the lugs. Analysisof the stress in the contact zone is based on a polarcoordinate system: r is the radial coordinate; x is thelongitudinal coordinate (along the axis of rotation);and t is the azimuthal coordinate. At the contact sur�faces of the shaft and bush in a slip bearing, normalstress arises along the stress radius (the contact pres�

1 Adapted from a presentation at the International Conference onMechanics Problems celebrating the centenary of L.A. Galin(Moscow, September 2012).

sure), as well as the tangential contact stress τrt = τrx;the t and x axes are in the directions of the transverseand longitudinal tangents to the contact surface.Beyond the contact area, σr = τrt = τrx = 0. At the con�tact area

(1)

where f is the frictional coefficient.Thus, the stress σr is compressive, while the stresses

τrt and τrx cannot exceed the frictional force. Normalstress arises at the shaft–bush contact surfaces along

σr 0; τrt fσr; τrx fσr,<<<

Contact Stress in Hinged Joints1

Yu. N. Drozdov, S. L. Sokolov, and B. N. UshakovBlagonravov Institute of Mechanical Engineering, Russian Academy of Sciences

e�mail: [email protected]

Abstract—The contact stress in hinged joints is studied by the finite�element method and the photoelasticmethod. Recommendations for its reduction are derived.

DOI: 10.3103/S1068798X13070058

D

ld

P

lco

1

2

3

4

P

lco

1

2

3

4

(a)

(b)

Fig. 1. Hinged joints with two (a) and three (b) bearings.

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

CONTACT STRESS IN HINGED JOINTS 393

the t axis (in the transverse direction). The stress maybe compressive or tensile. By means of photoelasticmodels, we may determine the stress on the basis of theinterference pattern obtained by the transmission ofpolarized light through thin plates cut from differentcross sections of the frozen models. From bands in theinterference pattern, we may establish the order m ofinterference and determine the maximum tangentialstress τmax or the difference between the primarystresses σ1 and σ2 in the given cross section

(2)

where σ0 is the optical constant of the material in themodel; h is the thickness of the cut plate.

Beyond the contact area, where there is a gap, oneof the primary stresses σt or σx acts along the surface.The order m of interference determines the transverseand longitudinal stress

(3)

The order m of interference at the contact surfacedepends on the three largest stresses. In the transversecross section of the model, we observe the maximumtangential stress

(4)

in the longitudinal cross section

(5)

The stresses σr, σt, τrt, σx, and τrx are found fromphotoelastic data by integration of the differentialequilibrium equations. This is difficult and insuffi�ciently accurate. The finite�element method permitsfaster and more accurate determination of the contactstress and contact surface area. The order m of inter�ference in the bush’s longitudinal cross section, at itsend, corresponds to the radial stress σr, which, at thecontact surface, is equal to the contact pressure pco. Inthe contact zone, the stress τrz is small relative to σr,since τrx < fpco. The normal stress σx at the bush’s con�tact surface is also small, since the flexural rigidity of

2τmax σ1 σ2– σ0m/h,= =

σt σ0m/h;=

σx σ0m/h.=

τmaxt σ0m/ 2h( )=

= σr σt–( )/2[ ]2

τrt2+{ }

1/2,

τmaxx σ0m/ 2h( )=

= σr σx–( )/2[ ]2

τrx2+{ }

1/2.

the adjacent component with the bush is generallylarger than the shaft rigidity. Therefore, at the bush’ssurface in the contact zone, the order m of interferencepermits calculation of the contact pressure pco, withsufficient accuracy.

In Fig. 2a, we show the model of a hinged joint foran excavator and its load configuration. The stress isdetermined with different directions of the load Pi:when α = 0, 45, 90, 135, and 180°. Models are madeof E�2 photoelastic sheet (thickness 4 mm; shaft diam�eter d = 50 mm). The load on the models in the tests isP = 0.5, 1.0, and 1.5 kN. For each load, we obtain aninterference pattern [7]. In Fig. 2b, we show the inter�ference pattern with α = 0 and P = 1.0 kN. On thatbasis, we may determine the distribution of the maxi�mum tangential stress τmax in different cross sections,with sufficient accuracy. In Figs. 2c and 2d, we showthe experimental stress 2τmax = m = 2τmaxh/σ0 (contin�uous curves) for the internal surface of the lug, withtwo directions of the load P: α = 0 and 90°. The max�imum value of 2τmax appears at point A. The maximumstress is seen with asymmetric loading (α = 90°), andthe minimum with α = 180°. The radial gap Δ = (db –d0)/2 varies from 0.1 to 0.6 mm and has practically noinfluence on the maximum stress and the angularcoordinate of the stress�concentration point. Thecontact angle πco between the shaft and the bushdeclines with increase in the gap.

We have developed a program based on the finite�element method for the calculation of the contactstress. It includes an iterative algorithm for determiningthe dimensions of the contact zone with stepwise load�ing [7]. The results calculated for the stress 2τmax = mat the lug’s internal surface are shown by the dashedcurves in Figs. 2c and 2d. They are in agreement withthe experimental data. However, there are some differ�ences within the contact area, on account of the fric�tion, which was disregarded in the calculation. Inaddition, the calculation assumes that the shaft isabsolutely rigid. The calculated pco values are alsoshown in Figs. 2c and 2d (continuous curves). The dis�tribution of the contact pressure pco is complex anddepends on the direction of the load P. It is most com�plex with α = 90°. To assess the life of slip bearings onaccount of wear, we use the maximum contact pres�sure pcomax and the contact angle ϕco [5]. The table pre�sents the results of our approach for different direc�

Parameter values

Parameterα, deg

0 45 90 135 180

ϕ, degpco max

K = pco max/p

155 (144)3.981.21

160 (164)7.621.52

164 (167)14.02.8

108 (140)12.52.5

103 (108) 9.14 1.83

Note: Experimental values are given in parentheses.

394

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

DROZDOV et al.

tions of the load and different values of the contact�pressure concentration coefficient K = pcomax/pm,where pm = P/ld is the mean pressure within the con�tact zone; l is the contact length and d is the shaftdiameter.

The dimensions of the contact zone are determinedby means of the photoelastic models. Beyond the con�tact area, there is a gap, which may be seen on thescreen of the polarization unit when white light ispassed through the plates (the samples obtained). Thismethod overestimates the contact angle ϕco, since the

gap close to the contact zone is very small and cannotbe visually determined.

A more effective method of determining the con�tact angle relies on the finding that, in interference,irregularities are seen within the contact zone onaccount of the surface roughness of the lug and theshaft. Beyond the contact zone, there is no interactionbetween the rough lug and shaft surfaces, and so thereare no irregularities in the interference pattern. Thismethod somewhat underestimates the dimensions ofthe contact zone.

P1

P2

P3

P4

P5

2d

α

1

2

1

0 00

13

2

3

1d

123p 2

46

0

pco

ϕco2

pco

ϕcoB

A0

24

68

2τmax

(a)

(b)

(c) (d)

Fig. 2. Load configuration of the plane model (a), interference pattern with α = 0 (b), and stress curves at the shaft surface whenα = 0° (c) and 90° (d).

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

CONTACT STRESS IN HINGED JOINTS 395

The largest ϕco value is observed when α = 90°, andthe smallest when α = 180°. The greatest contact pres�sure pco max is observed when α = 90°, and the lowestwhen α = 0. Analysis of the results shows that theinteraction of the hinged components may be deter�mined with sufficient accuracy on the basis of theselected calculation scheme and the finite�elementmethod.

In hinged joints with immobile shafts, fretting corro�sion will occur at the contact surfaces of the lug andshaft in the slipping zones. That significantly reducesthe fatigue strength and life of the component. Thefatigue limit of the lugs in that case may be calculated as

(6)

where σsa is the fatigue limit of a smooth sample; Kef isthe effective stress�concentration coefficient for aplate with a concentrator in the absence of frettingcorrosion (a plate with a hole); KF characterizes thereduction in fatigue strength as a result of fretting cor�rosion.

Here KF depends on pcout, where pco is the contactpressure and ut is the relative displacement of the con�tact surfaces of the lug and the shaft. Hence, it is pro�portional to the work of the frictional forces in thecontact [8].

In Figs. 3–5, we show the finite�element results fora joint with a lug in which the width of the plate B = 2d.In the absence of fretting corrosion, failure occurs atpoint A (Fig. 3)—that is, in the zone of maximum azi�muthal stress σt at the lug’s inner surface. The maxi�mum value of pcout appears at point C. However, failureof the lug appears at point B, which corresponds to theworst combination of KF and σt.

To determine the fatigue strength of the hingedjoint with fretting corrosion, we use the followingmethod [8–11]. The dependence of the fatiguestrength on the load with and without fretting corro�sion is determined by tests that simulate the operatingconditions, such as the temperature, the surface treat�ment, and the surrounding medium (Fig. 4). On thebasis of the calculated σp and σt values, a single pointof the KF(pcout) dependence with the selected numberof loading cycles (say, N = 106) is obtained by thefinite�element method or the finite�boundarymethod. We plot KF(pcout) from test data for four typesof lugs with different d/B values (Fig. 5). This depen�dence may be used to determine the fatigue strength ofjoints made from the same material in the same oper�ating conditions but in different configurations. Thatconsiderably simplifies the tests of hinged joints withfretting corrosion. The calculations show that the con�tact stress and the dimensions of the contact zoneobtained by the finite�element method and the finite�boundary method are relatively close. A benefit of thefinite�boundary method is that fewer initial data arerequired, since the only boundary elements are the

σlug σsa/ KefKF( ),=

boundaries of contacting bodies, which considerablyreduces the calculation time. In addition, the finite�boundary method ensures precise determination ofthe stress and strain at any point. That is important incalculating stress gradients.

The stress–strain state of hinged joints includesflexural strain of the shaft and lugs. The contact stressis analyzed by the finite�element method and the pho�toelastic method [12, 13].

We now consider test results for a three�dimen�sional photoelastic model of a hinged joint with a slipbearing for an excavator arm (Fig. 6). The central lug 1has two bushes (length 55 mm), which connect itsshaft 2 to the external lug 3. The ratio of the shaftlength l between the external lugs and the shaft diam�eter d is 3.77. The model is placed between rigid metalplates 6 and 7 and subjected to compressive force P =

ut, mm

AB C

σr, N/mm2

σr

30.320

101

10

2 2.33

17.6

ut, mm

Fig. 3. Stress curves for a sample with a hole under theaction of load P1, when α = 0.

3.51.1

1.3

1.5

1.7

4.0 4.5 5.0 5.5 6.0logN

logσ

1

2

Fig. 4. Dependence of the fatigue strength σ on the num�ber N of loading cycles for samples obtained from a hingedjoint when α = 0 and the force P1 is applied, with (1) andwithout (2) fretting corrosion.

01.0

1.2

1.4

1.6

1.8

10 20 30 40 pcout

KF

Fig. 5. Dependence of KF on pcout.

396

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

DROZDOV et al.

0.7 kN. Elements 1–4 of the model are made ofED20�MTTFA photoelastic material, with an opticalconstant σ0 = 57.4 N/cm. Together with the loadingattachment, the model is placed in a thermostattedchamber and heated to 155°C, at 20°C/h. It is held at155°C for 2 h and then subjected to the load and slowlycooled to normal temperature at 2°C/h [12, 13]. Afterunloading, samples (thickness t = 5–10 mm) are cutfrom different cross sections of the model (denoted byRoman numerals in Fig. 6b). Polarized light is passedthrough the samples, and the stress is determined fromthe interference patterns.

In Fig. 7, we show the interference patterns of thesamples cut in the longitudinal cross section (IV inFig. 6b) and in transverse cross sections (VIII–X inFig. 6b). The stress concentration in sample IV is

greatest at the inner edge of the lateral lug (mmax = 18)and at the outer edge of the central lug (m = 11). In themiddle of the shaft, we observe pure flexure: the inter�ference bands are parallel with the longitudinal axisand separated by a constant interval. From the inter�ference pattern of the transverse samples, we deter�mine the contact angle ϕco in different cross sections.On the basis of measurement results, we plot the con�tact area of the shaft and bush (Fig. 7c). The maximumtangential stress 2τmax at the contact surfaces of the lugand shaft is shown in Figs. 8a and 8b, respectively. Inthe center of the shaft, we observe pure flexure: thestress σx is constant. In the upper part, this stress iscompressive; in the lower part, it is tensile. The trans�verse variation in the stress is linear. At the edge of thecentral lug, there is considerable tangential stress con�centration.

155

20

55

220

90

40

110

65

170

10

10

I

II

III

IV

V

VI

VII

VIII

IX

55R

45

diam. 45

diam

. 55

X XI XII

XIII XIV

XVXVI

XVIIXVIII

XIX

P

1

2

3

4

6

78

(a)

(b)

Fig. 6. Model of a hinged joint (a) and position of sample cross sections I–XIX (b).

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

CONTACT STRESS IN HINGED JOINTS 397

In Fig. 8, we show the results of calculations andexperiments; they are in good agreement. Analysis ofthe contact stress in the hinged joints involves a com�plex algorithm for determining the contact areas andstresses. For example, we may use iterative algorithmssuch as the penalty method or Lagrangian multipliers.Such analysis requires extensive machine time. InFig. 9, we show a finite�element grid for calculationson the basis of BASYS+ software [14].

To establish how the shape of the end of the bushaffects the contact� stress concentration, we investi�gate longitudinal samples from a photoelastic model ofa hinged joint [15]. Models with bushes whose ends areof different shape are tested: with a plane end; with cir�cular and conical facets at the bush–shaft contact sur�face; or with grooves. Grooves at the end of the bush

reduce the stress by a factor of three. In that case, thepoint of maximum stress is shifted from the edge to theinterior of the bush, with equalization of the contactpressure over the length of the bush. That increases thelife of the joint and the limiting loads. Stress concen�trations appear at the bottom of the groove.

In hinged joints with long shafts, the stress over thecontact zone may be equalized by establishing inter�mediate bearings (Fig. 1b). In that case, the section ofthe shaft between the bearings is reduced, and conse�quently the flexural strain is reduced. This may be con�firmed by tests of a model with an intermediate bear�ing in the middle of a shaft, when the end has a straightfacet or a groove. The intermediate bearing reducesthe stress by a factor of 1.9 in the first case and 1.4 inthe second.

11

0

7

18

32

1

01

2

3

2 3

2

1

1

1

2

0 0

0

1

ϕco = 110° ϕco = 98° ϕco = 66°

ϕco8

ϕcok9

ϕco10

x

d

(a)

(b)

(c)

Fig. 7. Interference patterns in cross section IV (a) and cross sections VIII–X (b) and contact zone (c). The numbering of thecross sections is as in Fig. 6.

398

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

DROZDOV et al.

Thus, to reduce the stress in hinged joints and toensure uniform stress distribution over the length ofthe lug, we must select rational geometry of the endand also an optimal combination of flexural rigidity ofthe adjacent part and the shaft rigidity and use inter�mediate bearings. For this purpose, it is expedient touse an approach that combines calculations and

experiments—specifically, the finite�element methodand experimental study of models.

REFERENCES

1. Galin, L.A., Kontaktnye zadachi teorii uprugosti (Con�tact problems in elasticity theory), Moscow: Gos�tekhizdat, 1953.

2. Galin, L.A., Kontaktnye zadachi teorii uprugosti i vyaz�kouprugosti (Contact problems in elasticity and vis�coelasticity theory), Moscow: Nauka, 1980.

3. Podgornyi, A.N., Gontarovskii, P.P., et al., Zadachikontaktnogo vzaimodeistviya elementov konstruktsii(Contact interactions of structural elements), Kiev:Naukova Dumka, 1989.

4. Goryacheva, I.G., Mekhanika friktsionnogo vzaimode�istviya (Mechanics of frictional interactions), Moscow:Nauka, 2001.

5. Drozdov, Yu.N., Pavlov, V.G., and Puchkov, V.N., Tre�nie i iznos v ekstremal’nykh usloviyakh (Friction andwear in extreme conditions), Moscow: Mashinostroe�nie, 1986.

6. Drozdov, Yu.N., Yudin, E.G., and Belov, A.N.,Prikladnaya tribologiya (Applied tribology), Eko�Press,2010.

7. Drozdov, Yu.N., Naumova, N.M., and Ushakov, B.N.,Contact stress in hinged joints, Probl. Mashinostr.Nadezhn. Mash., 1997, no. 3, pp. 52–57.

8. Drozdov, Yu.N., Sil’vestrov, I.N., and Ushakov, B.N.,Analysis of contact stress in wear and fretting fatigue,

0

4

8

12

4

0

12

16

20

2τmax

0

4

8

12

4

0

12

16

20

2τmax

12

2

1

242

1

2

σx

σx

4 2 0

(a) (b)

Contact

Contact length

length

Fig. 8. Curves of the contact stress 2τmax obtained by the photoelastic method (continuous curves) and by calculation (dashedcurves): a) for the bush (cross section III in Fig. 6); b) for the shaft (cross section IV).

Fig. 9. Finite�element grid.

RUSSIAN ENGINEERING RESEARCH Vol. 33 No. 7 2013

CONTACT STRESS IN HINGED JOINTS 399

Probl. Mashinostr. Nadezhn. Mash., 2002, no. 2,pp. 50–54.

9. Drozdov, Y.N. and Ushakov, B.N., Contact stress–strain analysis in wear and fretting fatigue, Proceedingsof second world tribology congress, Vienna, 2001.

10. Drozdov, Yu.N. and Ushakov, B.N., Analysis of contactstress and strain in wear and fretting fatigue, 8�i Vseros.s’ezd po teoreticheskoi i prikladnoi mekhanike (EighthRussian congress on theoretical and applied mechan�ics), Perm, 2001, pp. 237–246.

11. Drozdov, Y.N., Ushakov, B.N., and Silvestrov, I.N.,Contact stress–strain analysis in problems of wear andfretting fatigue, Proceedings of third international sym�posium on tribo�fatigue, Beijing, 2000.

12. Drozdov, Yu.N. and Ushakov, B.N., Contact stress inthree�dimensional hinged joints, Probl. Mashinostr.Nadezhn. Mash., 1997, no. 5, pp. 50–55.

13. Drozdov, Yu.N. and Ushakov, B.N., Contact stress inslip bearings under complex load, Tren. Iznos, 1997,no. 4, pp. 429–437.

14. Sokolov, S.L., Finite�element calculation of the stress–strain state of pneumatic tires, Probl. Mashinostr.Nadezhn. Mash., 2007, no. 1, pp. 57–62.

15. Drozdov, Yu.N. and Ushakov, B.N., Contact stressconcentration in hinged joints, Vestn. Mashinostr.,1997, no. 12, pp. 10–11.

Translated by B. Gilbert