Ti le: STATIC STRUCTURAL BEHAVIOUR OF WIRE BEARINGS UNDER AXIAL
LOAD: COMPARISON WITH CONVENTIONAL BEARINGS AND STUDY
OF DESIGN AND OPERATIONAL PARAMETERS
Au ho s: Iñigo Ma ín1, Ike He as1, Josu Agui ebei ia1, Mikel Abasolo1, Ibai Co ia1
1 Depa men o Mechanical Enginee ing, Facul y o Enginee ing Bilbao, Uni e si y o he
Basque Coun y (UPV/EHU), Plaza Ingenie o To es Que edo, 1, 48013, Bilbao, Spain.
*Co esponding au ho :
Iñigo Ma ín
E-mail add ess: [email p o ec ed]
Tel.: +0034 606 061 117
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Mechanism and Machine Theo y 132 : 98-107 (2019), which
has been published in inal o m a h ps://doi.o g/10.1016/j.mechmach heo y.2018.10.016. © 2018 Else ie unde CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Abs ac
In wi e bea ings he olling p ocess occu s on aceways machined on s eel wi es, and he ings a e
made o ligh ma e ials such as aluminium. This pa icula a chi ec u e p o ides bo h weigh and
ine ia sa ings, bu also signi ican ly di e en beha iou wi h espec o con en ional bea ings. Fo
his eason, speci ic design and analysis ools mus be de eloped; as a i s s ep, his wo k uses Fini e
Elemen models o s udy he in luence o di e en pa ame e s on he s a ic s uc u al esponse o
wi e bea ings. Thus, bea ing s i ness, load capaci y and con ac s a us (con ac o ce and angle, and
ellipse unca ion) ha e been e alua ed o se e al combina ions o con o mi y, ic ion coe icien
and bounda y condi ions. The esul s ha e been compa ed wi h an equi alen con en ional bea ing,
shedding ligh on he main s uc u al ea u es o wi e bea ings.
Keywo ds
Wi e Bea ing; Slewing Bea ing; S i ness; Con o mi y; Capaci y; T unca ion.
1. In oduc ion
Slewing bea ings a e used in slow- u n hea y-du y wo king condi ions due o hei capaci y o ace
ex e nal axial and adial loads as well as il ing momen s. Thus, hey a e used o slowly o a e s uc u al
elemen s and ans e he loads o he main s uc u e. In con en ional slewing bea ings, widely used
in a eas such as cons uc ion machine y, enewable ene gies o machine ool, olling elemen s a e in
di ec con ac wi h he bea ing ings, which a e made o s eel. Fig. 1(a) shows he c oss-sec ion o a
ou -poin con ac slewing bea ing. Ex ensi e esea ch has been published abou he s uc u al
beha iou o hese bea ings in e ms o s a ic load ca ying capaci y [1-8], s i ness [9,10] and ic ion
o que [11-14], among o he esea ch opics.
In 1936, E ich F anke de eloped and pa en ed a new concep o bea ing, he wi e bea ing [15]. Wi e
bea ings a e a u he de elopmen o con en ional slewing bea ings, whe e he aceway is shaped in
a wi e loca ed be ween he olling elemen and he ing, as illus a ed in Fig. 1(b) o he case o ou -
poin con ac bea ings. This modi ica ion allows building he ings and he wi es wi h di e en
ma e ials; he wi es can be manu ac u ed wi h ha dened s eel and he ings wi h ligh e ma e ials
(aluminium, composi es, plas ics…). The choice o a ligh e ma e ial o he ings in ol es signi ican
weigh sa ings (up o 65% acco ding o [15]) and consequen ly an ine ia educ ion. Apa om ha ,
wi e bea ings wi h aluminium ings ha e a good pe o mance abso bing shock loads and a elling
ib a ions due o he lowe elas ici y modulus, which leads o a educ ion o b inelling and s ia ion
in he aceways [16]. Fo hese easons, wi e bea ings a e used whe e weigh and ine ia sa ings a e a
key aspec , such as in medical, ae onau ical o mili a y applica ions, among o he s. Li le esea ch has
been published abou wi e bea ings: Shan e al. [17] de eloped an analy ical model o de e mining
he p eload in wi e bea ings wi h an unusual non-con o mal con ac design; Gunia and Smolnicky
[18] s udied he in luence o ce ain geome ical pa ame e s in he s ess dis ibu ion along he con ac
wi h con o mal wi es, which is a mo e ealis ic ep esen a ion o he bea ings used in he indus y.
This wo k ies o shed ligh on he pe o mance o wi e bea ings h ough a compa ison wi h well-
known ou -poin con ac slewing bea ings. Fo ha pu pose, wo pa ame ic Fini e Elemen models
we e c ea ed and equally loaded. As a esul , di ec compa isons o he s a ic axial load capaci y,
bea ing axial s i ness and con ac beha iou we e made, hus p o iding a global iew o he
ad an ages and sho comings o his kind o bea ings. The e ec o di e en design and ope a ional
pa ame e s such as he con ac con o mi y, he ic ion coe icien and he s i ness o he suppo ing
s uc u es we e conside ed in he s udy.
(a)
(b)
Fig. 1. C oss-sec ion o a ou -poin con ac slewing bea ing: (a) con en ional bea ing (b) wi e
bea ing (dashed lines a e he con ac lines).
2. Me hodology
Among all he possible load cases, pu e axial load case is s udied. Unde axial load, all he ele an
phenomena suscep ible o analysis appea , as wi e wis ing, con ac ellipse unca ion, e ec o
lub ican p esence, wo-poin con ac condi ion, amongs o he s ha will be la e explained. Ball
p eload is no conside ed in his wo k because complex in e ela ions we e o eseen in combina ion
wi h axial load. In his sense, he simplici y o he load case enables a easonable compa ison be ween
he pe o mances o he wo bea ing ypes. Fo his pu pose, wo pa ame ic Fini e Elemen (FE)
models we e c ea ed and se e al cases we e aised in o de o s udy he in luence o di e en ac o s
and assump ions in he pe o mance o he bea ings.
The main ma e unde s udy in he wi e bea ing is he ball-wi e con ac , because he ex en o he
aceway is smalle han he one in con en ional slewing bea ings, which educes he su ace whe e
he he zian ellip ical con ac su ace can be placed. The wi e- ing con ac is no so c i ical because,
as illus a ed in Fig. 1(b), he con ac o ces a e dis ibu ed along wo con ac lines, gene a ing a lowe
con ac p essu e. Ne e heless, his wo k will p o e ha he beha iou o he wi e- ing con ac has a
g ea in luence on he pe o mance o he wi e bea ing.
2.1. Desc ip ion o he case s udies
Since he aim o he wo k was o s udy and compa e he pe o mance o wi e bea ings wi h
con en ional bea ings, di e en analyses we e ca ied ou a ying wo cha ac e is ic pa ame e s: he
oscula ion a io (s) and he ic ion coe icien (μ). The oscula ion a io (s) is he main geome ical
ac o ha de ines he con ac be ween ball and aceway; ypical alues close o 0.943 a e used o
con en ional bea ings [15] and be ween 0.87 and 0.96 o wi e bea ings in he indus y, i.e. wi e
bea ings end o ha e a less con o mal con ac han con en ional bea ings. Rega ding he ic ion
coe icien , 0.1 is a ypical alue o he ball- aceway lub ica ed s eel-s eel con ac pai [12,20,21]; o
wi e bea ings, 0.1 was also used o ball-wi e con ac , and o wi e- ing aluminium-s eel con ac wo
alues we e s udied, 0.1 and 0.3, o e alua e he e ec o he p esence o absence o lub ica ion in
he pe o mance o he wi e bea ing. The i s columns o Table 1 summa ize he i e cases analysed
in his wo k, wi h hei co esponding pa ame e alues.
E en o he same ype o bea ing (con en ional o wi e), he axial s i ness is no he same o
di e en alues o s and μ. Fo his eason, o make he compa ison easible, a di e en displacemen
p o iding he same axial eac ion o ce had o be applied o each model. This axial o ce was chosen
o be he axial s a ic load capaci y as calcula ed by he analy ical model p oposed by Agui ebei ia e
al. [7], summa ized in he las column o Table 1. The analy ical model, deeply explained and alida ed
in [7], is based on he calcula ion o he ball- aceway in e e ence ield caused by axial, adial and
il ing displacemen s o he ings due o ex e nal loads (in addi ion o ball p eload), assuming igid
ings. As he s i ness o he adjacen s uc u es has a ele an in luence in he beha iou o he
bea ings, wo ex eme si ua ions we e aken in o accoun o each case s udy in Table 1: on he one
hand, clamped ings, assuming ha he ings a e ixed o igid suppo ing s uc u es; on he o he
hand, unclamped ings, assuming ha he suppo ing s uc u es a e igid bu he bea ing ings can
eely de o m in he adial de o ma ion). O cou se, eal sys ems beha iou is placed be ween hese
wo ex eme condi ions.
Table 1
Cases unde s udy o clamped and unclamped condi ions.
Case
Bea ing ype
s
µ (ball-
wi e)
µ (wi e-
ing)
C0a (Ta ge axial
o ce [kN]) [7]
1
Con en ional
0.943
0.1
-
1213.1
2
Con en ional
0.870
0.1
-
674.24
3
Wi e
0.870
0.1
0.1
674.24
4
Wi e
0.943
0.1
0.1
1213.1
5
Wi e
0.870
0.1
0.3
674.24
Rega ding he geome y o he bea ings, he wo main geome ical pa ame e s a e he ball diame e
(Dw) and he bea ing mean diame e (Dpw), which we e chosen in such a way ha he esul ing bea ing
could be ound in bo h con en ional and wi e bea ing comme cial ca alogues. The geome y o he
c oss-sec ion was hen ob ained om [21], which p oposes a s anda d pa ame ic geome y o
con en ional ou -poin con ac slewing bea ings in e ms o (Dw) and (Dpw). E en hough wi e
bea ing c oss-sec ions a e la ge han con en ional bea ing sec ions o gi en (Dw) and (Dpw) alues,
he same c oss-sec ion was adop ed o bo h bea ing ypes in o de o make he compa ison in s ic ly
he same condi ions. Fig. 2 shows he c oss-sec ion o bo h bea ings wi h hei dimensions. The o al
numbe o balls in he bea ing is n=82, and ini ial con ac angle is α=45º.
(a)
(b)
Fig. 2. C oss-sec ions o he s udied bea ings in [mm]: (a) con en ional bea ing (b) wi e bea ing.
2.2. FE Models
Two FE models we e c ea ed in Ansys® o simula e he pe o mance o he bea ings, one o he
con en ional bea ing and he o he one o he wi e bea ing. Bol holes we e no modelled. Acco ding
o he cu en comme cial bea ings, s eel was used o con en ional bea ings (linea elas ic, E=200
GPa), and o wi e bea ings s eel was chosen o balls and wi e and aluminium (linea elas ic,
E=71GPa) o he ings. As i will be explained nex , wo ypes o FE models we e de eloped: hal
sec o models and submodels.
2.2.1. Hal sec o models
The axial load si ua ion p o ides a cyclic symme y load dis ibu ion which, oge he wi h a cyclic
symme y geome y, allows o simpli y he whole bea ing model in o a one sec o model.
Fu he mo e, he one sec o model has a symme y plane ha allows analysing only one hal ,
signi ican ly dec easing he numbe o Deg ees o F eedom (DoF) o he model.
Nex , he models we e meshed wi h he same elemen size in o de o make mo e accu a e
compa ison be ween model esul s. Fo his pu pose, se e al pa i ions we e ca ied ou in he
geome y. The pa i ions wi h a con ac ing su ace we e meshed wi h second-o de hexahed ons; he
o he pa i ions we e meshed wi h second-o de e ahed ons o enable quick size ansi ions wi h
high aspec a io elemen s. Con ac zones we e meshed wi h second-o de quad ila e al con ac - a ge
elemen s, allowing a pene a ion o 0.1 mic ons using augmen ed Lag ange o mula ion. The c oss-
sec ions illus a ing he mesh o bo h models a e shown in Fig. 3. The con en ional bea ing model
has 321.813 DoF, whe eas he wi e bea ing model has 614.529 DoF.
(a)
(b)
Fig. 3. C oss-sec ion o hal sec o models: (a) Con en ional bea ing (b) Wi e bea ing.
The ex e nal axial load poin ed ou in Table 1 was in oduced by imposing an axial displacemen o
he uppe su ace o he ou e ing in o de o imp o e he con e gence o he analyses. Besides,
symme y condi ions we e applied in he symme y su aces. Finally, o he clamped con igu a ion,
he lowe ace o he inne ing was ixed, hus es aining axial and adial mo emen ; o he
unclamped con igu a ion, ic ionless con ac condi ion was imposed o he lowe ace o he inne
ing, allowing o ee adial mo emen .
2.2.2. Submodelling echnique
E en hough he global beha iou o he bea ing can be accu a ely simula ed by he hal sec o
models, ine mesh is necessa y in he con ac su aces in o de o ob ain be e local con ac esul s.
In his sense, submodelling echnique is a highly e icien ool [21]. In his case, he o iginal models
we e he hal sec o models in Fig. 3, and he submodels we e de ined as he con ac egions
illus a ed in Fig. 4 o bo h he con en ional and he wi e bea ing. Thus, a i s analysis is pe o med
applying he axial displacemen o he hal sec o model; hen, he displacemen esul s in he con ac
egion bounda ies a e ans e ed o he submodel, analysing his pa ial geome y wi h a ine mesh
and he e o e wi h mo e accu a e esul s. Due o he smalle dimensions o he submodels, ine
con ac meshes can be de ined (2.416.227 DoF in he con en ional bea ing submodel, and 1.167.117
DoF in he wi e bea ing submodel) and consequen ly mo e accu a e con ac esul s can be ob ained.
(a)
(b)
Fig. 4. Submodels: (a) Con en ional bea ing (b) Wi e bea ing.
3. Resul s and discussion
In his sec ion he FE esul s a e p esen ed and discussed o d aw he main conclusions o he wo k.
3.1. Wi e wis ing
The mos ema kable phenomenon ha akes place du ing he loading p ocess is he wis o he
wi e. When he axial displacemen is applied in con en ional bea ings, he ball climbs he aceway,
inc easing he ball- aceway con ac angle. In wi e bea ings, he ball-wi e con ac o ce gene a ed by
he axial displacemen is no aligned wi h he cen e o he wi e c oss-sec ion, and consequen ly a
wis ing momen is induced in he wi e. Depending on he ic ion coe icien o he con ac ing
su aces, his momen p omo es he wi e wis a he han ball climbing, as in a con en ional bea ing.
Fig. 5(a) shows he load case, Fig. 5(b) shows a de ailed iew o he unde o med mesh in he con ac
zone (no e he coinciden nodes along bo h wi e- ing ci cum e en ial con ac lines), and Fig. 5(c)
illus a es wi e wis as consequence o he applied load. The wi e wis ing has a huge in luence on
he beha iou o he wi e bea ing, due o i s e ec s on he s i ness o he bea ing, con ac ellipse
unca ion and con ac o ces.
(a)
(b)
(c)
Fig. 5. Wi e wis Case 3: (a) Load case (b) Unde o med model (c) De o med model (scale
x1.6).
3.2. Axial s i ness and s a ic load capaci y o he bea ing
Axial s i ness cu es o each case in Table 1 we e ob ained by means o FE analyses o he hal
sec o models in Fig. 3, as he ela ionship be ween he displacemen o he uppe su ace o he
ou e ing and he eac ion o ces in he lowe su ace o he inne ing. Mo eo e , he analy ical
model [7] used o ix he a ge axial o ce in Table 1, was also used o ob ain he s i ness cu es;
his analy ical model assumes igid ings, so la ge s i ness is expec ed. Fig. 6 shows he s i ness
cu es o he i e cases summa ized in Table 1, as well as he poin s in which he con ac ellipse
begins and comple es unca ion o unclamped (Fig. 6(a)) and clamped (Fig. 6(b)) si ua ions. To his
end, unca ion was conside ed o begin when he con ac ellipse eaches he aceway bounda ies,
and i was assumed o be comple e when he maximum con ac p essu e was loca ed a he bounda y
ins ead o in he cen e o he con ac ellipse. Nex , he esul s o Fig. 6 a e discussed.
(a)
(b)
Fig. 6. S i ness o he di e en bea ings and unca ion s a us: (a) Unclamped si ua ion (b)
Clamped si ua ion.
Axial s i ness beha iou
Analy ical Model s=0.943
Analy ical Model s=0.87
Case 1
Case 2
Case 3
Case 4
Case 5
bo a
T unca ion S a Case 1
T unca ion S a Case 3
T unca ion S a Case 4
T unca ion Comple e Case 4
T unca ion S a Case 5
T unca ion Comple e Case 5
I can be obse ed ha he s i ness p o ided by he analy ical model i s e y well wi h he
con en ional bea ing models wi h clamped con igu a ion (Fig. 6(b)), and is sligh ly la ge o he
unclamped condi ion (cases 1 and 2 in Fig. 6(a)). This is because he clamped condi ion es ic s he
adial de o ma ion o he ings, hus being close o he igid ings assump ion o he analy ical model.
Wi e bea ing ings a e buil wi h ligh e and mo e complian ma e ials. This ac , oge he wi h he
wi e wis ing, makes hem mo e lexible han he ings o con en ional bea ings. The wi e wis ing
e ec on he s i ness can be clea ly app ecia ed obse ing he wi e bea ings wi h he same con o mi y
ac o alue (0.87) bu di e en wi e- ing ic ion coe icien s (0.1 and 0.3), i.e. cases 3 and 5; a la ge
ic ion coe icien dec eases he wi e wis ing and consequen ly inc eases he axial s i ness o he
bea ing.
Ano he phenomenon o ake in o accoun is he di e en s i ness beha iou o each bea ing ype
unde di e en bounda y condi ions. Con en ional bea ings ha e exponen ial s i ness beha iou
due o he exponen ial na u e o he ball- aceway con ac de o ma ion and he a ia ion o he con ac
angle. Wi e bea ings p o ide almos linea s i ness beha iou o unclamped condi ion, mainly
caused by he low s i ness o he ings and he sligh a ia ion o he con ac angle due o he wi e
wis ing, as i will be explained in he ollowing sec ion; o he clamped con igu a ion, he lexibili y
o he ings does no play such an impo an ole, and he e o e he esponse is exponen ial.
S a ic axial capaci y and con ac ellipse unca ion
Acco ding o Table 1, and as illus a ed in Fig. 6, he s a ic axial load capaci y ob ained om he
analy ical model [7] highly depends on he con ac con o mi y: he mos con o mal bea ings (s=0.943)
ha e app oxima ely wice he heo e ical capaci y o he less con o mal ones (s=0.87). Howe e , he
analy ical model does no conside he unca ion o he con ac ellipse, which can ha e a huge e ec
in he s a ic capaci y. F om his poin o iew, Fig. 6(a) shows ha o unclamped con igu a ion, cases
1 and 3 ha e a simila beha iou : case 1, he con en ional bea ing wi h s=0.943, s a ed su e ing
unca ion a 69.5% o he heo e ical s a ic load capaci y and did no each he comple e unca ion,
whe eas case 3, he wi e bea ing wi h s=0.87 and μ=0.1 in he aceway- ing con ac , s a ed su e ing
unca ion a 89.9% o i s heo e ical s a ic load capaci y and nei he eached he comple e unca ion.
As i has been men ioned in he p e ious sec ion, as he wi e- ing ic ion coe icien inc eases, so
does he axial s i ness because he wi e wis dec eases, especially in he unclamped con igu a ion;
wi e- ing ic ion also a ec s he con ac ellipse unca ion. The wi e bea ing o case 5 (µ=0.3), s a s
unca ion a 17%C0a and comple es i a 67.6%C0a, which is clea ly wo se han he esponse o case
3 (µ=0.1). This ac demons a es ha in e ms o s a ic load capaci y, he wi e wis ing phenomenon
imp o es he pe o mance since i p e en s he unca ion o he con ac ellipse. In he clamped
con igu a ion, wi e wis ing is mo e es ic ed, so his phenomenon is no so c i ical.
Analysing he esul s o cases 2 and 4, bo h ha e clea disad an ages. On he one hand, he low
con o mi y con en ional bea ing (case 2, wi h s=0.87) is less op imal han he high con o mi y one
(case 1, wi h s=0.943) because i has hal he s a ic load capaci y. On he o he hand, he high
con o mi y wi e bea ing (case 4, wi h s=0.943) s a ed and comple ed he unca ion o he con ac
ellipse a e y low pe cen ages o he s a ic load capaci y, bo h o clamped and unclamped
condi ions.
3.3. Con ac o ces and con ac angles in he ball- aceway con ac
The con ac no mal o ce (Q) is commonly used in analy ical models o ob ain he con ac p essu e
and shea s ess dis ibu ion [7,11,19]. The con ac angle (𝛼) is de ined as he angle o (Q) wi h he
ho izon al axis. In his wo k, he con ac no mal o ce was ob ained by means o a pos p ocessing
mac o in Ansys®, based on he assump ion ha he no mal con ac o ce is he ec o sum o he
no mal o ces in each node; in o de o alida e his p ocedu e, he angle be ween he poin o he
con ac ellipse wi h he maximum p essu e and he ho izon al axis was measu ed, and i was ound
o be iden ical o he one ob ained by means o o ces. Thus, Fig. 7 illus a es he e olu ion o he
ball- aceway con ac angle wi h he no mal o ce o each bea ing unde clamped and unclamped
si ua ions.
(a)
(b)
Fig. 7. Con ac o ces and angles a) Unclamped si ua ion b) Clamped si ua ion.
Fig. 7 shows ha , in con en ional bea ings (cases 1 and 2), he con ac angle inc eases wi h he axial
load due o ball climbing, which can inally esul in ball- aceway con ac ellipse unca ion. Thus, i
he con ac no mal o ce and he con ac angle a e known, he dimensions o he con ac ellipse can
be calcula ed; i i eaches he limi o he aceway, unca ion occu s. In wi e bea ings, wi e wis ing
in ol es ha he con ac angle no always inc eases wi h he axial load, as illus a ed in cases 3, 4 and
5 in Fig. 7: o low load alues, he con ac angle inc eases as in con en ional bea ings, because ball
climbing is a o ed a he han wi e wis ing; howe e , om a gi en axial load on, wi e wis ing s a s
and consequen ly con ac angle dec eases. Due o his complex beha io , he s udy o he con ac
ellipse unca ion in wi e bea ings is no as s aigh o wa d as in con en ional bea ings. As he
schema ic illus a ion in Fig. 8 shows, wi e wis ing in ol es con ac angle dec ease bu no con ac
ellipse unca ion, because he con ac ellipse emains cen e ed in he wi e aceway. This s a emen is
demons a ed by he plo s in Fig. 9, which shows he con ac p essu e dis ibu ion along he majo
semi-axis o he ellipse o inc easing load alues: o con en ional bea ings (case 1 in Fig. 9(a), and
case 2 in Fig. 9(b)), con ac ellipse mo es owa ds he aceway limi s as he load and consequen ly he
con ac angle inc eases (see cases 1 and 2 in Fig. 7); on he con a y, o he wi e bea ing o case 3
wi h unclamped condi ion, Fig. 9(c) e inces ha he con ac ellipse emains cen e ed, e en hough
Fig. 7(a) shows ha he con ac angle clea ly dec eases acco ding o Fig. 8; inally, i he wi e- ing
ic ion coe icien is inc eased (case 5), Figs. 7(a) and 9(d) show ha a la ge load is needed o s a
wi e wis ing, which ini ially a o s ball climbing and he e o e con ac ellipse unca ion. As a
consequence, ball- aceway con ac angle alone is no enough o s udy con ac ellipse unca ion in
wi e bea ings, wi e wis ing mus also be aken in o accoun ; his aspec is especially c i ical i
simpli ied analy ical models a e o be de eloped. Fig. 10 shows he con ac ellipses o he ou plo s
in Fig. 9 o he 100% o he a ge axial load (s a ic load capaci y). No e ha , acco ding o Fig. 9, he
maximum p essu e is no exac ly 4200 MPa o ha load alue, as i should be. The s a ic load capaci y
in Table 1 was calcula ed using he analy ical model, which conside s igid ings; as he ings a e
lexible in he FE model, he con ac angles and no mal o ces a e sligh ly di e en om hose
p edic ed by he analy ical model, and so is he maximum con ac p essu e.
40
45
50
55
60
65
0 5 10 15 20 25
Con ac Angle [deg]
Con ac No mal Fo ce (Q) [kN]
40
45
50
55
60
65
0 5 10 15 20 25
Con ac Angle [deg]
Con ac No mal Fo ce (Q) [kN]
Analy ical Model s=0.943
Analy ical Model s=0.87
Case 1
Case 2
Case 3
Case 4
Case 5
