angrenaje.1.2 engl rom

8/6/2019 Angrenaje.1.2 Engl Rom

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Gearing152

15. GEARING (TOOTHED GEARS)

15.1. General

The gearing is a mechanism with toothed wheels which directlyand forcefully transmits the rotary motion from a driving shaft to thatdriven. The toothed wheels are machine elements which have teeth ontheir rims, uniformly distributed on theoretical surfaces called rollingsurfaces. These rolling surfaces usually are rotational surfaces. Thetoothed wheel which is assembled on the driving shaft is called pinion

(pinion wheel) and it has the speed n1. The driven toothed wheel isassembled on the driven shaft and has the speed n2.The continuous process of contact between the teeth of the

mating (conjugated) toothed wheels in view to transmit the motion iscalled meshing. The gearing can transmit the motion in both senses(direction) because the teeth have symmetrical flanks.

The toothed gears classification is made in accordance to thefollowing criteria:I. According to the relative positions of the rotary motion axes:

- spur gearing, having parallel axes (fig. 15.1, a,b,c);- bevel gearing, having concurrent axes (fig.15.1, d,e,f);

- worm and hypoid gearing, having cross axes (fig.15.1, g);II. According to the type of the rolling surfaces:

- with rotational rolling surfaces: - spur gearing;- bevel gearing;- hypoid gearing;- toroidal gearing;

- with rolling surfaces that are not rotational;III. According to the relative position of the rolling surfaces:

- external gearing (fig. 15.1);- internal gearing;

IV. According to the teeth shape:- straight teeth (fig. 15.1, a,d);- helical teeth (fig. 15.1, b,e);- curved teeth (fig. 15.1, f);- herringbone, Z and W teeth (fig. 15.1, c);

V. According to the shape of contact zone between the teeth flanks:- with linear contact;- with point contact;


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Transmisii prin angrenaje 153

15. TRANSMISII PRIN ANGRENAJE

15.1. Generaliti

Angrenajele sunt mecanisme cu roi dinate care servesc latransmiterea direct i forat a micrii de rotaie ntre doi arbori.Roile dinate sunt organe de maini care au la periferia lor dini dispui n mod regulat pe suprafee teoretice (n general de revoluie) numitesuprafee de rostogolire. Roata dinat montat pe arborele conductor

se numete pinion i se rotete cu turaia n1, iar roata dinat condus,montat pe arborele condus, se rotete cu turaia n2.Procesul continuu de contact ntre dinii roilor conjugate dintr-un

angrenaj, n vederea asigurrii micrii nentrerupte a celor dou roidinate se numete angrenare. Angrenajul poate transmite micarea derotaie n ambele sensuri, fapt posibil prin utilizarea dinilor cu flancurisimetrice.

Clasificarea angrenajelor se face dup urmtoarele criterii:I. Poziia relativ a axelor micrii de rotaie:

- angrenaje cu axe paralele (fig. 15.1, a,b,c);- angrenaje cu axe concurente (fig.15.1, d,e,f);

- angrenaje cu axe ncruciate (fig.15.1, g);II. Felul suprafeelor de rostogolire:- angrenaje cu suprafee de rostogolire de revoluie: - cilindrice;

- conice;- hiperbolice;- toroidale;

- angrenaje cu suprafeele de rostogolire care nu sunt derevoluie;

III.Poziia relativ a suprafeelor de rostogolire:- angrenaje exterioare (fig. 15.1);- angrenaje interioare;

IV.Forma dinilor roilor dinate:- cu dini drepi (fig. 15.1, a,d);- cu dini nclinai (fig. 15.1, b,e);- cu dini curbi (fig. 15.1, f);- angrenaje cu dini n V, Z sau W (fig. 15.1, c);

V. Felul contactului ntre flancuri: - angrenaje cu contact liniar;- angrenaje cu contact punctiform;


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Gearing154

VI. According to the tooth profile:- involute gears;- cycloid gears;-arch of circle gears and other profiles;

VII.According to themotion of the shaftsgeometrical axes:- with fix axes;- with mobile axes or

planetary gearing.The gearing are

machine elements witha very large utilisationin different machinesdue to their advan-tages:-constant speedratio;-durability andworking safety ;- small gauge;- the possibility to

transmit power ina large range ofspeeds andspeed ratios ;

- high efficiency, up to 0.995.The drawbacks of the gearing

are:- they need a high processing andassembling accuracy ;- their working is noisy at great

speeds;- their speed ratios are limitedbecause the teeth number must beinteger.

b. c.

Fig. 15.1

r2O2

r1O1

a.

d. e. f.

g.


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VI. Profilul dinilor: - angrenaje cu profil n evolvent;- angrenaje cu profil n cicloid;- angrenaje cu profil n arc de cerc

sau alte profile;

VII.Modul de micare aaxelor geometrice alearborilor:- angrenaje cu axefixe (ordinare);

- angrenaje cu axe

mobile (planetare).Angrenajele formeazo categorie de organede maini foarte desutilizat n cele mai di-verse maini iutilaje ca ur-mare a avan-tajelor lor:- ra-port de trans-

mitere cons-tant; - durabili-tate i siguran- n funciona-re; - gabarit re-dus; - posibili-tatea de a transmite puteri ntr-undomeniu larg de viteze i rapoartede transmitere; - randament ridicat(ajungnd la 0,995).

Dezavantajele angrenajelor

sunt: - necesitatea unei precizii nalte de execuie i montaj; -funcionare cu zgomot la vitezeridicate; - limitarea la o serie derapoarte de transmitere deoarecenumrul de dini trebuie s fie unnumr ntreg.

b. c.

Fig. 15.1

r2

O2

r1O1

a.

d. e. f.

g.


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Gearing156

15.2. Fundamental law of meshing

The teeth flanks must satisfy certain conditions, imposed by theconstant speed ratio of the gearing. These conditions are establishedby the fundamental law of meshing.

In order to demonstrate this law, two profiled elements,presented in figure 15.2, will be studied. The profiles of these twoelements are the same as the teeth profiles and they are tangent inpoint M, where two points M1 and M2 are superposed, rigidly connectedto elements 1 and 2. The two elements represent a pair of teeth inmeshing. Considering that upon element 1 act a driving torque T1 (inreverse clockwise) and upon elements 2 a resistant torque T 2 (also inreverseclockwise) and considering that the two elements are bodies

absolutely rigid, one can say that they can not push away and also theycan not penetrate one in the other. Consequently, the relative motion of


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15.2. Legea fundamental a angrenrii

Realizarea unui raport de transmitere constant impune caflancurile dinilor s satisfac anumite condii, stabilite de legeafundamental a angrenrii. Pentru demonstrarea acestei legi, se vorstudia dou prghii profilate, reprezentate n figura 15.2, ale crorprofile sunt identice cu profilele dinilor roilor dinate i tangente npunctul M, unde sunt suprapuse dou puncte M1i M2, rigid legate deelementele 1, respectiv 2. Cele dou prghii reprezint practic opereche de dini n angrenare. Avnd n vedere faptul c asupraprghiei 1 acioneaz un moment motor T1 (n sens antiorar), iar

asupra prghiei 2 un moment rezistent Tr (tot n sens antiorar) iconsidernd cele dou prghii ca fiind corpuri absolut rigide, se poatespune c profilele nu se pot ndeprta unele de altele i nici nu potptrunde unele n altele. n consecin, micarea relativ a profilelor se


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Gearing158

these profiles is reduced at rolling and slipping (sliding) along thecommon tangent T - T, drawn in the contact point M.

The velocities of points M1 and M2 are:

.MOv;r.v

,MOv;r.v

x

x

22222

11111

(15.1)

In considered contact point M, the two mating curves of teethprofiles have the common normal N-N and common tangent T-T.

The two velocities have the following components along thesetwo directions:

.vvv

;vvv

TN

TN

222

111

(15.2)

The teeth being rigid bodies, the motion transmission is possibleonly if the normal components of these velocities are equal:

.vvNN 21

(15.3)

If this condition is not accomplished, the teeth will be not incontact.

From figure 15. 2, it results:

),BMBMOK(BMBMKO111111

and:

).AMAMOK(AMAMKO122122

Consequently it can be written:

;r

r.vv

v

r

v

ror

AA

MO

MA

KO

;r

r.vv

v

r

v

ror

MB

MO

MB

KO

x

b

N

x

N

b

x

b

N

x

N

b

2

2

212

2

2

2

2

1

1

1

11

1

1

11

1

1

1

11

1

11

(15.4)


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reduce la rostogolire nsoit de alunecare pe tangenta comun T- T,dus n punctul de contact M.

Vitezele punctelor M1i M2 sunt:

.MOv;r.v

,MOv;r.v

x

x

22222

11111

(15.1)

In punctul de contact M, curbele profilelor conjugate au normalcomun N-N i tangent comun T-T.

Componentele vitezelor pe cele dou direcii sunt:

.vvv

;vvv

TN

TN

222

111

(15.2)

Dinii fiind corpuri rigide, transmiterea micrii este posibilnumai dac cele dou component normale ale vitezelor sunt egale:

.vvNN 21

(15.3)

Dac aceast condiie nu este respectat contactul dintre dinidispare. Dinii se deprteaz unul de altul sau patrunt unul in altul.Din figura 15. 2, se poate scrie:

),BMBMOK(BMBMKO111111

i:

).AMAMOK(AMAMKO122122

n consecin se poate scrie:

;r

r.vv

v

r

v

ror

AA

MO

MA

KO

;r

r.vv

v

r

v

ror

MB

MO

MB

KO

x

b

N

x

N

b

x

b

N

x

N

b

2

2

212

2

2

2

2

1

1

1

11

1

1

11

1

1

1

11

1

11

(15.4)


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Gearing160

Substituting the calculation relation of v1 and v2, it results:

222

111

bN

bN

r.v

;r.v (15.5)

respectively:1

2

2

1

122211

b

b

bbr

riorr.r. (15.6)

The common normal intersects the centre line in point C. From

figure 15.2 it results:

CO

CO

KO

KOCKOCKO

2

1

22

11

2211

or:

,CO

CO

r

r

b

b

2

1

2

1

therefore:.ttancons

CO

COi

1

2

2

1

12 (15.7)

In order that this ratio to be constant it is necessary that thepoint C to be a fix one. This point C is called meshing poleand it isdefined as the fix point through which passes the common normal NNto the mating profiles of the two teeth in contact.

Consequently the meshing fundamental law is: In order toobtain a constant speed ratio, the teeth flanks surfaces should be

chosen so that, in any position of the toothed wheels, the commonnormal to teeth flanks to pas through a fix point C, called meshing pole,it dividing the center line in reverse proportional parts with the absolutevalues of the angular velocitiesConclusions:

a. Because21

vv

although ,2T1T2N1N

vvvv

that is themating profiles of the teeth in contact roll with sliding;


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nlocuind relaiile de calcul pentru v1i v2, rezult:

222

111

bN

bN

r.v

;r.v (15.5)

respectiv:1

2

2

1

122211

b

b

bbr

risaur.r. (15.6)

Normala comun intersecteaz linia centrelor n punctual C. Din

figura 15.2 rezult:

CO

CO

KO

KOCKOCKO

2

1

22

11

2211

sau:

,CO

CO

r

r

b

b

2

1

2

1

deci:.ttancons

CO

COi

1

2

2

1

12 (15.7)

Pentru ca acest raport s fie constant este necesar ca punctualC s fie fix. Acespunct C se numetepolul angrenriii este definit cafiind punctual fix prin care trece normala comun N N la profileleconjugate ale celor doi dini n contact.

n consecinlegea fundamentala angrenriieste: pentru camiscarea s se transmit cu raport de transmitere constant, suprafaa

flancurilor dinilor trebuie astfel aleas nct n orice poziie a roilordinate, normal comuna la flancurile dinilor n contact s treacprintr+un punct fix C numit polul angrenrii, care mparte linia centrelorn segmente invers proporionale cu vitezele unghiulare ale roilor.Concluzii:

a. Deoarece21

vv

dei ,2T1T2N1N

vvvv

deci profileleconjugate ale dinilor n contact se rostogolesc cu alunecare;


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Gearing162

b. The locus of the contact points is a line superposed on thecommon normal N-N, therefore it passes through the meshingpole C. When M coincides with C, it can be written:

0,vv;vvvv;vv2T1T2N1N2121

that is in the meshing pole C, there is pure rolling.c. When the point M passes through the meshing pole, can be

defined two circle, tangent in this point, having the radiuses O1C= rw1i O2C = rw2, called rolling circles, that roll without sliding. In

this case can be written:

v1 = v2 i rw1. = rw2.respectively:

,ttanconsr

ri

w

w

1

2

2

1

12 (15.8)

that is in the meshing pole there is pure rolling.

15.3. Spur wheels elements

n figure 15.3 a spur gear and a helical spur gear areschematically represented. The toothed wheels elements will bestudied in section AA respectively B-B, perpendicular on the wheelsaxes. The shape of the tooth cross section is called the tooth profile(fig. 15.4). The external tooth part is limited by a circle calledaddendum circle and itsradius is designnated with ra.The circle limiting the toothfrom the wheel body iscalled dedendum circleand

its radius is designated withrf. From the strengths pointof view, the tooth profile isconnected to the wheelbody through a radius, itsvalue depending on thetoothed wheel processing

A A

a.

B B

Fig.15.3b.


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b. Locul geometric al punctelor de contact dintre dini este o liniesuprapus pe normal comun N-N, i deci trece prin polulangrenrii C. Cnd punctual M coincide cu C, se poate scrie:

0,vv;vvvv;vv2T1T2N1N2121

deci n polul angrenrii este rostogolire pur.c. Cnd punctual M trece prin polul angrenrii, se pot define dou

cercuri, tangent n acest punct, avnd razele O1C = rw1i O2C =rw2, numite cercuri de rostogolire, care se rostogolesc fr

alunecare. n acestt caz se poate scrie:

v1 = v2 i rw1. = rw2.respectiv:

,ttanconsr

ri

w

w

1

2

2

1

12 (15.8)

deci n polul angrenrii exist rostogolire pur.

15.3. Elementele roilor dinate cilindrice

n figura 15.3 sunt reprezentate schematic dou roi dinatecilindrice, cu dini drepi, respectiv cu dini ncli-nai. Elementele roilorse vor studia n seciunile A-A, respectiv B-B, perpendiculare pe axeleroilor. Forma seciunii transversale a dintelui se numete profiluldintelui (fig. 15.4).

Partea exterioar a dintelui este limitat de un cerc care senumete cerc de capi a cruiraz se noteaz cu ra.

Cercul care delimi-teaz dintele de corpul roii

dinate se numete cerc depiciori raza lui se noteazcu rf.

Profilul dintelui, dinMotive de rezisten, se ra-cordeaz la baz, formaracordrii depinznd de

A A

a.

B B

Fig.15.3b.


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Gearing164

method. The circle passing through the point where starts theradiusing, is called radiusing circle and its radius is designated by rrc,

The teeth number z and the pitch p are the main elementsneeded for the strength calculation of the toothed wheels. The pitch isthe length of an arc of circle, between two neighbouring teeth flankshaving the same sense, measured on a certain circle having the radiusrx. It is designated by px and it is determined with the relation:

px =2. .

.r

zx (15.9)

The pitch on the pitch diameter is designated by p and it is equalto the distance between two successively teeth flanks, having the samesenses, measured on the pitch diameter having the radius r. This pitchis calculated with the relation:

p =2. .

.r

z (15.10)

The angular pitch is the central angle (to the wheel centre)corresponding to the circular pitch and it is determined with the relation:

=3600

z. (15.11)

The pitch circle is the circle on which the pitch and the pressureangle are equal to the pitch and the pressure angle of the standardisedreferences rack bar (toothed bar).

Through dividing of the length of the pitch circle to the teethnumber z, a size called pitch is obtained. Similarly, through the divisionof the pitch diameter to the teeth number z one obtains a length calledmoduleor diametral pitch:

Tooth profile

h

p t px

rxra

r

rf

s

rrc

OFig. 15.4


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znd de metoda de prelucrare. Cercul care trece prin punctul unde ncepe racordarea se numete cerc de racordare, a crui raz senoteaz cu rrc. Elementele de baz necesare la calculul roilor dinatesunt numrul de dini z i pasul p. Pasul care reprezint lungimea

arcului de cerc dintre flancurile de acelai sensa doi dini alturai,msurat pe un cerc oarecare de raz rx, se noteaz cu px i sedetermin cu relaia:

px =2. .

.r

zx (15.9)

Pasul unghiular este unghiul la centru corespunztor pasului circulari se determin cu relaia:

=3600

z. (15.10)

Cercul de divizare este cercul pe care pasul i unghiul de pre-siune sunt egale cu pasul i unghiul de presiune ale cremalierei dereferin standardizate.

Pasul de divizare se noteaz cu p i este distana dintreflancurile de acelai sens a doi dini consecutivi, msurat pe cercul de

divizare cu raza r. Acest pas este dat de relaia:

p =2. .

.r

z (15.11)

Prin mprirea lungimii cercului de divizare la numrul de dini z,s-a obinut o mrime numit pas. n mod asemntor, prin mprireadiametrului de divizare la numrul de dini se obine o lungime numitmodul sau pas diametral:

profilul dintelui

h

p t px

rx

ra

r

rf

s

rrc

OFig. 15.4


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Gearing166

m = dz

. (15.12)

With respect to the pitch, the module has the advantage that it isa measurable size, while the pitch, being a multiple of , is anincommensurable size. The relation between the pitch and module isobtained from the relation (15.10), in which 2.r = d is replaced and therelation (15.12) is considered. It results:

p = .m (15.13)The module is measured in millimetres and it is standardised. In

order that a gearing to work (to mesh), its toothed wheels must have

the same pitch (same module) on the base and pitch circles: p1 = p2 =p; m1 = m2 = m. Other two important elements of the toothed wheelsare tooth thickness sd or the arc of tooth (which is the length of the arcof circle between the flanks of opposite senses of the same tooth) andthe width of the empty space between two teeth tg or the tooth spacearc (which is the length of the arc of circle between the flanks ofopposite sense of two successive teeth), both measured on the pitchcircle. The tooth depth h is also an element of the toothed wheel.

The teeth profile can be: - involute (evolvente) to which the lineof action is a straight line and the meshing angle is constant; -cycloidalwhen the line of action is made of two arcs of circle and the

meshing angle is variable; - roller gearingwhich is a particular caseof the cycloid gearing, the line of action being made by a single arc ofcircle; - arc de circle etc.

Further on one considers that the teeth of the spur and helicalspur gearing have involute profile.

15.3.2 The involute equations in polar coordinates

Figure 15.5 presents two branches of the involute described bythe point M of the straight line which rolls without slippage on the circlehaving the radius rb, called base circle, which have a common pointcalled returning point M0, situated on the base circle. The points B andC, rigidly connected on the generatrix line (fig. 15.5, b), describedeformed involutes. The point B describes a lengthened (elongated)involute and the point C a shortened involute. Considering that in thecase of normal involute (fig.15.5,c), the segment MN of the generatrix


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m = dz

. (15.12)

Modulul prezint fa de pas avantajul c este o mrimecomensurabil, pe cnd pasul fiind un mu ltiplu de , esteincomensurabil. Relaia dintre pas i modul se obine dac n relaia(15.11) se nlocuiete 2.r = d i se ine seama de relaia (15.12).Rezult:

p = .m (15.13)Modulul se msoar n milimetri i este standardizat. Pentru ca

un angrenaj s poat funciona este necesar ca roile dinate

componente s aib acelai pas (acelai modul) pe cercurile dedivizare: p1 = p2 = p; m1 = m2= m. Alte dou elemente importante aleroilor dinate sunt grosimea dintelui s sau arcul dintelui (care estelungimea arcului de cerc dintre flancurile de sensuri diferite aleaceluiai dinte) i lrgimea golului dintre dini t sau arcul golului (careeste lungimea arcului de cerc dintre profilele de sens contrar a doi diniconsecutivi), ambele msurate pe cercul de divizare. nlimea dinteluih este de asemenea un element al roii dinate.

Profilul dinilor poate fi: - evolventic la care linia de angrenareeste o dreapt i unghiul de angrenare este constant; - n cicloid culinia de angrenare format din dou arce de cerc i unghiul de

angrenare variabil; - angrenajul cu rolecare este un caz particular alangrenajului cicloidal, linia de angrenare fiind format doar dintr-unsingur arc de cerc; - n arc de cercetc.

n continuare se va considera c profilul dintelui roii dinate ci-lindrice cu dini drepi sau nclinai este evolventic.

15.3.2 Ecuaiile evolventei n coordonate polare

n figura 15.5 sunt reprezentate dou ramuri ale evolventeidescrise de punctul M al dreptei care se rostogolete fr alunecarepe cercul cu raza rb numit cerc de baz. Ele se ntlnesc n punctul dentoarcere M0 situat pe cercul de baz. Punctele B i C rigid legate dedreapta generatoare (fig.15.5,b) descriu evolvente deformate. PunctulB descrie o evolvent alungit, iar punctul C o evolvent scurtat.Avnd n vedere c n cazul evolventei normale (fig.15.5,c) lungimeasegmentului dreptei generatoare MN este egal cu lungimea arculuicorespunztor al cercului de baz pe care dreapta s-a rostogolit fralunecare (rezult din generarea evolventei) se poate scrie egalitatea:


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Gearing168

line is equal to the corresponding length of the arc of circle of the basecircle on which the respective lines rolls without slippage (this resultsfrom the involute properties), the equality can be write:

MN = M0N. (15.14)

Substituting the length of the segment MN = rb.tg x and of the arc M0N= r.( x + x), determined from figure 9.5,c, in equality (9.14), oneobtains:

x = tg x - x. (15.15)

The function x of the angle x is called involute functionand it isdesignated by inv x. Therefore the relation (9.15) becomes:

inv x = tg x - x. (15.16)

From the triangle MON, the polar radius rx is determined,

function the base circle radius:

rx =r b

xcos. (15.17)

The equations (15.16) and (15.17) are the equations of thenormal involute in polar coordinates, expressed function the parameter

x, which is the pressure angle.

M

M

NOrb

M0M

N

a.

Fig.15.5

rbO

N

C

B

b. c.

rb

90

Rx x

M0x x

ON


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MN = M0N. (15.14)nlocuind lungimile segmentului MN = rb.tg xi a arcului M0N =

r.( x + x),determinate din figura 15.5,c, n egalitatea (15.14), se obine:x = tg x - x. (15.15)

Funcia x de unghiul x se numete funcie involut i senoteaz cu inv x. Deci relaia (15.15) devine:

inv x = tg x - x. (15.16)Din triunghiul MON se determin raza polar rx, n funcie de

raza cercului de baz:

rx =r b

xcos. (15.17)

Ecuaiile (15.16) i (15.17) sunt ecuaiile evolventei normale ncoordonate polare, exprimate n funcie de parametrul x, care esteunghiul de presiune.

Din triunghiul MON se determin raza polar rx, n funcie deraza cercului de baz:

rx =r b

xcos. (15.17)

Ecuaiile (15.16) i (15.17) sunt ecuaiile evolventei normale ncoordonate polare, exprimate n funcie de parametrul x, care esteunghiul de presiune.

M

M

NOrb

M0M

N

a.

Fig.15.5

rbO N

C

B

b. c.

rb

90

Rx x

M0x x

ON


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Gearing170

15.3.3. The toothed wheel with involute profile and the toothed bar

In the case when the tooth profile is an involute, anotherelementof the toothed wheel occurs: the base circle having the radiusrb. A point M on the involute profile is chosen (fig. 9.6), which in thesame time is also situated on the pitch circle having the radius r. In thispoint the pressure angle MON is drawn. This angle is standardised and

is designated by , having the value = 200. Consequently the pitchcircle is characterised not only by the standardised module, but alsothrough the standardised pressure angle. From figure 15.6 it results:

rb = r.cos . (15.18)Based on the formula (15.12) one can write:

r =m z.

.2

(15.19)

Substituting the relation (15.19) in (15.18) one obtains:

rb =

m z.

.2 cos . (15.20)One considers that the teeth number of the toothed wheel (fig.

15.6) continually increases and at a given moment becomes equal toinfinite. In this case rb = r = , that is the base and pitch circles becomestraight lines and the involute profile of the tooth is also transformatedin a straight line. Such a toothed wheel, having the teeth number z =and a linear tooth profile is called rack bar (toothed bar).

Pitch lineRack bar

rbr

OOO

M

NN

N

Fig. 15.6


20/34


15.3.3. Roata dinat cu profil n evolvent i cremaliera dinat.

n cazul n care profiluldintelui are forma unei evolvente, aparenc un element al roii dinate: cercul de baz cu raza rb. Se alege unpunct M pe profilul evolventic (fig. 15.6) care n acelai timp s fiesituat i pe cercul de divizare cu raza r. Se construiete n acest punctunghiul de presiune MON. Acest unghi este standardizat i se noteaz

cu ,avnd valoarea numeric = 200. n consecin cercul de divizarese caracterizeaz nu numai prin modulul standardizat, dar i prinunghiul de presiune standardizat. Din figura 15.6 rezult:

rb = r.cos . (15.18)Pe baza formulei (15.12) se poate scrie:

r =m z.

.2

(15.19)

nlocuind relaia (15.19) n relaia (15.18) se obine:

rb =

m z.

.2 cos . (15.20)Se presupune c numrul de dini ai roii dinate (fig. 15.6) se

mrete continuu i la un moment dat devine egal cu infinit. n acestcaz rb = r = , adic cercul de baz i cel de divizare se transform ndrepte iar profilul evolventic al dintelui se transform de asemenea ntr-o dreapt. O astfel de roat dinat a crui numr de dini z = i cuprofilul dintelui liniar se numete cremalier.

dreapta de divizarecremalier

rbr

OOO

M

NN

N

Fig. 15.6


21/34

Gearing172

The pitch of the toothed bar, measured on any parallel line tothe medium line (corresponding to the pitch circle), has the same valueequal to .m. Returning to a finite number of teeth and supposing thatthe pitch line is transformed in a pitch circle so that its centre to besituated upon the pitch line (fig. 15.6), an internal toothed wheel isobtained. The internal gearing is made of an external toothed wheelsituated inside an internal toothed wheel, they having the same rotationsense. The calculation relations of the internal toothed wheel elements,respectively of the internal gearing, are similarly determined to that ofthe external toothed wheels, respectively to the external gearing.

15.3.4. The determination of the tooth thickness sx measured on acircle of radius rx

The pitch of the toothed wheel is made of the tooth thicknessand the width of the empty space between two teeth t, both measuredon the pitch circle (fig. 15.7):

p = s + t. (15.21)Generally, the toothed wheels are designed so that their pitch is

not divided in equal parts between the tooth thickness and the width ofthe empty space between two teeth. Therefore the tooth thickness,

measured on the pitch circle, is not equal to half of the pitch, that is to.m/2 and it can be expressed with the relation:

s =.

.m

m2

. (15.22)

The type of the toothed wheel is determined by the value of the

M0

sx

s

V

inv

inv x

rx

rrb

sx/2rx

s/2r

OFig. 15.7


22/34


Pasul cremalierei, msurat pe orice dreapt paralel cu dreaptamedie (corespunztoare diametrului de divizare), are aceeai valoareegal cu .m. Revenind la numr finit de dini i presupunnd cdreapta de divizare se transform n cerc de divizare astfel ca centrullui s fie situat deasupra dreptei de divizare (fig. 15.6), se obine oroat dinat cu dantura interioar. Angrenajul interior este format dintr-o roat dinat cu dantur exterioar aezat n interiorul unei roi cudantur interioar, ambele roi rotindu-se n acelai sens. Relaiilematematice, care determin elementele roii dinate cu danturinterioar ct i elementele angrenajului interior, se obin n mod similar

ca i n cazul roilor cu dantur exterioar, respectiv al angrenajului

exterior.

15.3.4. Determinarea grosimii dintelui sxmsurat pe un cerc curaza rx

Pasul roii dinate se compune din grosimea dintelui sdi lrgi-mea golului sg, ambele msurate pe cercul de divizare (fig. 15.7):

p = s + t. (15.21)n general, roile dinate se proiecteaz astfel nct pasul s nu

se mpart egal ntre grosimea dintelui i lrgimea golului. De aici

rezult c grosimea dintelui msurat pe cercul de divizare nu va fiegal cu jumtatea pasului, adic cu .m/2 , ci se poate exprima astfel:

s =.

.m

m2

. (15.22)

Felul roii dinate este determinat de valoarea coeficientului de

M0

sx

s

V

inv

inv x

rx

rrb

sx/2rx

s/2r

OFig. 15.7


23/34

Gearing174

thickening coefficient: > 0, it being called positive toothed wheel; =0, it being called zero toothed wheel; < 0, it being called negativetoothed wheel. In order to determine the tooth thickness sx (fig. 15.7),the angle (the angle made by the straight lines OM0 = rb and OV ) isexpressed function inv x and inv . One can write:

= inv +s

r2.= inv x +

s

r

x

x2.,

whence one obtains: sx = 2.rx.(s

r2.+ inv - inv x).

Taking into consideration the relation (15.18) and the figure15.7, it results: rb = r.cos = rx.cos x. Using the value of rx expressedfrom this equality and considering the equation (15.22), one obtains:

sx = m.cos

cos. .

x

xz inv inv2

(15.23)

15.3.5. The generating of the teeth flanks of the spur and helicalspur toothed wheels.

Through the rolling without sliding of the generating plane(fig.15.8,a) on the base cylinder, the straight line NN situated in this

plane and parallel to the generatrix of base cylinder, M0M 0' , generates

the surface of the tooth flank. During the rolling of the generating plane

rb

rx

M0

N

N

M

base cylinder

involute

generatingplane

generating line

ruled

involutecylindricalsurface

rb

b

base cylinder

helix

helicoidally ruledinvolute cylindricalsurface

involute

involuteFig. 15.8

N

N

M0

M3

M1

M2

a. b.


24/34


ngroare: > 0, roata dinat se numete pozitiv; = 0, roata dinatpoart numele de roat zero; < 0 , roata dinat este negativ. Pentrudeterminarea grosimii dintelui sx (fig. 15.7) se exprim unghiul(unghiul nchis de dreptele OM0 = rb i i OV ) n funcie de inv xiinv . Se poate scrie:

= inv +s

r2.= inv x +

s

r

x

x2.,

din care se obine:

sx = 2.rx.(s

r2.+ inv - inv x).

innd cont de relaia (15.18) i de figura 15.7, rezult: rb = r.cos =rx.cos x. Utiliznd valoarea lui rx exprimat din aceast egalitate ilund n considerare ecuaia (15.22), se obine:

sx = m.cos

cos. .

x

xz inv inv2

(15.23)

15.3.5. Generarea suprafeelor flancurilor dinilor roilor dinatecilindrice cu dini drepi i nclinai.

Prin rostogolirea fr alunecare a planului generator (fig. 15.8,a)

pe cilindrul de baz, dreapta NN situat n acest plan i paralel cu

generatoarea cilindrului de baz, M0M 0' , genereaz suprafaa flancului

dintelui. n timpul rostogolirii planului generator, fiecare punct al dreptei

rb

rx

M0

N

N

M

cilindru de baz

evolventa

planul generator

dreapta generatoare

suprafariglat

cilindricevolventic

rb

b

cilindru de baz

elicea

suprafa riglatelicoidal evolventic

evolventa

evolventaFig. 15.8

N

N

M0

M3

M1M2

a. b.


25/34

Gearing176

plane, each point of the straight line NN generates an involute. Allthese involutes make the tooth flank. The intersection of this flanksboth with the generating plane and with a cylinder having the radius rx(concentric with the base cylinder) is a straight line. Thus it results thatthe surface of the tooth flank is a ruled involute cylindrical surface.

During the rolling without sliding of the generating plane on thebase cylinder (fig. 15.8,b), the generating line, inclined with the angle bwith respect to the base cylinder generatrix, generates the flank of theinclined tooth and simultaneously it winds on the base cylinder makingthus an helicoidal line (helix). The inclination angle of this helicoidal line

is also b. The intersection of the tooth flank with a certain cylinderconcentric with the base cylinder is a helicoidal line. Such helices areinfinitely obtained, starting with that on the base cylinder and finishingwith the helix situated on the addendum cylinder. All these helices willhave different inclination angles, but the same pitch. The intersectionbetween the flank of the inclined tooth and the generatrix plane is astraight line. It results that the flank surface of the inclined tooth is aruled helicoidal involute surface.

15.4. The characteristics of the spur gearing

15.4.1. The elements of the involute gearingIn figure 15.9 the involute profiles in meshing, of an external

spur gear are represented. The contact line is a tangent line to thebase circles having the radius rb1 and rb2. In the case of this gearing,the contact between the two profiles can be on the contact line onlyinside the segment K1K2 which represent the actual length of thecontact line. That can be explained through the fact that outside thepoints K1 and K2 the fundamental law of the meshing is notaccomplished, because they have not a common normal andconsequently the gearing is locked and in certain cases the teeth are

broken off.The contact of the teeth along the corresponding segment(K1K2) of the contact line is assured by limiting the length of theinvolutes, namely through the addendum circles having the radius ra1and ra2 (fig. 15.9) so that the contact to be between the points K1 andK2. The intersection points of the addendum circles of the toothedwheels with the contact line are designated by A, respectively E. In thepoint A the meshing starts, the tooth dedendum of the wheel 1 gets in


26/34


NN genereaz o evolvent. Totalitatea acestor evolvente formeazflancul dintelui. Intersecia acestui flanc cu planul generatorct i cu uncilindru de raz rx(concentric cu cilindrul de baz) este o linie dreapt.Rezult deci c suprafaa lateral a flancului dintelui este o suprafariglat cilindric evolventic.

n timpul rostogolirii fr alunecare a planului generator pe cilin-drul de baz (fig. 15.8,b), dac dreapta generatoare, este nclinat cuunghiul b fa de generatoarea cilindrului de baz, formeaz flanculdintelui nclinat i concomitent cu aceasta se nfoar treptat pecilindrul de baz formnd astfel o linie elicoidal. Unghiul de nclinare

al acestei linii elicoidale este tot b. Intersecia flancului dintelui cu oricecilindru concentric cu cilindrul de baz este o linie elicoidal. Astfel seobin infinit de multe elice, ncepnd cu cea situat pe cilindrul de bazi terminnd cu elicea situat pe cilindrul exterior. Toate aceste elicevor avea unghiuri de nclinare diferite, dar acelai pas. Flancul dinteluinclinat se intersecteaz cu planul generator dup o linie dreapt. Dincele expuse rezult c suprafaa lateral a dintelui nclinat este osuprafa riglat elicoidal evolventic.

15.4. Caracteristicile angrenajelor cilindrice

15.4.1. Elementele angrenajului evolventicn figura 15.9 sunt reprezentate profilele evolventice n

angrenare pentru cazul angrenajului exterior. Linia de angrenare este odreapt tangent la cele dou cercuri de baz cu razele rb1i rb2. Laacest angrenaj , contactul dintre profilele dinilor poate avea loc pe liniade angrenare numai n intervalul K1K2 carereprezint lungimea real aliniei de angrenare. Aceasta se explic prin faptul c n exteriorulpunctelor K1 i K2 nu se respect legea fundamental a angrenrii,adic profilelenu au normal comun i ca urmare angrenajul seblocheaz, iar n anumite cazuri dinii se rup.

Asigurarea contactului pe poriunea corespunztoare a liniei deangrenare (K1K2) se realizeaz prin limitarea lungimilor evoventelor ianume prin intermediul cercurilor de cap (exterioare) cu razele ra1i ra2(fig. 15.9) n aa fel nct contactul s aib loc ntre punctele K 1i K2.Punctele de intersecie ale cercurilor de cap ale roilor dinate cu liniade angrenare se noteaz cu A, respectiv E. n punctul A ncepe


27/34

Gearing178

meshing with the of tooth top of the driven wheel 2 and continues untilthe point E in which the contact is between the extreme point of the firstwheel addendum and tooth dedendum of the wheel 2. The segment AE

is called the active part of the contact line.The gearing must be so designed that the segment AE to be

situated inside segment K1K2. The tooth profile does not entirely takepart in the meshing process. The active parts of the teeth profiles of theboth wheels are determined in the following way: the tooth profile of thewheel 1 is intersected with an arc of circle having the radius O 1A andthe lower point of this tooth dedendum is obtained, which comes inmeshing with the last point of the tooth addendum of the wheel 2.Similarly, intersecting the wheel 2 profile with an arc of circle having theradius O2E, a point on the tooth dedendum is obtained which gets inmeshing with the tooth top of the wheel 1. The arc MM, designated bye2, is called meshing arcand represents the path described by a pointof the tooth profile during the two profiles are in meshing.

The distance measured on the centre line, between theaddendum circle of a toothed wheel and the dedendum circle of theothers is called clearance (play) to the tooth addendum and isdesignated by c. The value of this play depends on the machiningmanner and is expressed as a function of module with the relation:

c = c o* .m, (15.24)

w( )

K1

Fig. 15.9

O1

1w

w( 2

O2rw1

rb1rf

ra1

AC

rb2rf2

rw2

ra2c

E

M

M

aw(a)

K


28/34


angrenarea, piciorul dintelui roii 1 intr n angrenare cu vrful dinteluiroii conduse 2 i continu pn n punctul E n care este n contactpunctul extrem al capului dintelui primei roi cu piciorul dintelui roii 2.Segmentul AE se numete partea activ a liniei de angrenare.Angrenajul trebuie s fie astfel proiectat nct segmentul AE s fiesituat n interiorul lui K1K2.

Profilul dintelui nu particip n ntregime n procesul de angre -nare. Prile active ale profilelor dinilor celor dou roi se determin nfelul urmtor: se intersecteaz profilul dintelui roii 1 cu un arc de cerccu raza O1A i se obine punctul inferior de pe piciorul acestui dinte care intr n angrenare cu punctul extrem al capului dintelui roii 2. nmod asemntor, intersectnd profilul roii 2 cu un arc de cerc cu razaO2E, se obine pe piciorul dintelui un punct care intr n angrenare cuvrful dintelui roii 1. Arcul MM, notat cu e2, se numete arc de angre-nare i reprezint traiectoria descrisde un punct al profilului dintelui n

perioada n care cele dou profile se afl n angrenare.Distana msurat pe linia centrelor, ntre cercul exterior al uneiroi i cercul interior al celeilalte roi se numete joc la capul dintelui ise noteaz cu c. Mrimea acestui joc depinde de felul prelucrrii i seexprim n funcie de modul cu relaia:

c = c o* .m, (15.24)

w( )

K1

Fig. 15.9

O1

1w

w( 2

O2rw1 rb1

rf

ra1

AC

rb2rf2 rw2

ra2

cE

M

M

aw(a)

K


29/34

Gearing180

where: c o* is the coefficient of the play to the tooth addendum and hasthe value c o

* = 0.20...0,30.The gearing elements are: the contact line, the meshing pole,

the centre distance, the rolling circles, the active parts of the toothprofile, the meshing arc, the meshing angle etc. These elementscharacterise the gearing and they can not be distinguished on a singletoothed wheel which is not in meshing.

15.5. Characteristics of the spur toothed wheels machined

with rack-shaped cutter15.5.1. Basic dimensions of the spur toothed wheels

In order to determine the dimensions of the spur toothed wheels,the meshing of the reference rack bar and the toothed wheel is studied,that is the machining meshing (fig. 15.10).

During the machining, the pitch circle of the toothed wheel rolls withoutsliding on the rolling line of the rack bar. According to the position ofthe rack bar with respect to the processed wheel, any parallel line tothe mean line can overlap the rolling line. The distance between themean line and the rolling line (fig. 15.10) is called tooth profilecorrectionand is designated by m.x, where x is the coefficient of thetooth profile correction. The value of x can be positive, negative or

reference line

rolling line

radiusing line

radiusing linem. y c0

m.c

m. *a

h

m. *a

h

m.c0

*

900

O

rarfrb

= 0

r

m.x

0

0 0C

Ksd

M

M

Fig. 15.10


30/34


Elementele angrenajului sunt: linia de angrenare, polulangrenrii, distana dintre axe, cercurile de rostogolire, partea activ aprofilului dinte-lui, arcul de angrenare,unghiul de angrenare etc. Acesteelemente caracterizeaz angrenajul i nu pot fi distinse pe o singurroat care nu este n angrenare.

15.5. Caracteristicile roilor dinate cilindrice prelucrate cucuit pieptene

15.5.1. Dimensiunile de baz ale roilor dinate cilindrice

Pentru determinarea dimensiunilor roilor dinate cilindrice, se vastudia angrenarea cremalierei de referin cu roata dinat, adicangrenarea de prelucrare (fig. 15.10).

n timpul prelucrrii, cercul de divizare al roii se rostogolete f-r alunecare pe dreapta de rulare (rostogolire) a cremalierei. n funciede poziia cremalierei fa de roata de prelucrat, oricare din dreptele

pa-ralele cu dreapta medie poate s se suprapun cu dreapta derostogolire. Distana dintre dreapta medie i dreapta de rostogolire (fig.15.22) se numete deplasare de profil i se noteaz cu m.x, unde xeste coeficientul de deplasare a profilului dintelui. Valoarea lui x poatefi pozitiv, negativ sau egal cu zero. Distana dintre cercul de cap alequal to zero. unde: c o

* este coeficientul jocului la capul dintelui i are

valoarea c o* = 0.20...0,30.

linia de referin

dreapta de rostogolire

dreapta de racordare

dreapt de racordarem. y c0

m.c

m. *a

h

m. *a

h

m.c 0*

900

O

rarfrb

= 0

r

m.x

0

00C

Ksd

M

M

Fig. 15.10


31/34

Gearing182

equal to zero. The distance between the addendum circle and thededendum line of the tool is called the clearance (play) to the toothaddendum of the machining meshing. The distance between theaddendum circle of the toothed wheel and the radiusing line of the toolis designated by m. y, where y is the coefficient of the toothaddendum play decreasing, its signification being given further on.

From figure 15.10 one can write:

ra = r + m.x + m.h a* - m. y,

or: ra =m z.

2

+ m.x + m.h a* - m. y.

The diameter of the addendum circle da will be determined withthe relation:

da = m.(z + 2.x +2.h a* - 2. y). (15.25)

The tooth depth h is calculated with the relation:

h = m.c 0 2* *. . .m h m ya = m.(c 0 2

* *.h ya ). (15.26)

The determination of the tooth thickness on the pitch circle ismade from the figure 15.10, considering that the pitch circle rollswithout sliding on the rolling line, thus:

sd= MM =.

. . .m

m x tg2

2 . (15.27)

Taking into consideration the relation (15.22), from the equation(15.27) it results = 2.x.tg .

In figure 15.10 one can see that through the increase of thedistance m.x, the tooth thickness on the pitch circle increases too, thatis the signs of the and x coincide. Consequently if x > 0, the toothedwheel is positive, if x < 0, the wheel is negative and for x = 0 it resultsa zero toothed wheel. From the relations (15.26) and (15.27) it resultsthat the toothed wheels dimensions depends on the coefficient of thetooth profile correction. That is at the same teeth number z and modulem, the wheel diameter and the tooth thickness can be increased ordecreased by changing x.


32/34


roii i dreapta de fund a sculei se numete joc la capul dintelui alangrenrii de prelucrare. Distana dintre cercul de cap al roii i dreaptade racordare se noteaz cu m. y, unde y este coeficientul demicorare a jocului la capul dintelui, semnificaia acestuia precizndu-se n continuare.

Din figura 15.10 se poate scrie:

ra = r + m.x + m.h a* - m. y,

sau: ra =m z.

2+ m.x + m.h a

* - m. y.

Diametrul cercului de cap da se va determina cu relaia:

da = m.(z + 2.x +2.h a* - 2. y). (15.25)

nlimea dintelui h se calculeaz cu relaia:

h = m.c 0 2* *. . .m h m ya = m.(c 0 2

* *.h ya ). (15.26)

Determinarea grosimii dintelui pe cercul de divizare se face por-

nind de la figura 15.10 i innd seama de faptul c cercul de divizarese rostogolete fr alunecare pe dreapta de rulare, astfel:

sd= MM =.

. . .m

m x tg2

2 . (15.27)

Lund n considerare relaia (15.22), din ecuaia (15.27) rezult= 2.x.tg .

Din figura 15.10 se vede c prin mrirea distanei m.x, semrete i grosimea dintelui pe cercul de divizare, adic semnele luii x coin-cid. Prin urmare dac x > 0, roata este pozitiv, dac x < 0,roata este negativ i la x = 0 rezult o roat dinat zero. Din relaiile(15.26) i (15.27) rezult c dimensiunile roilor dinate depind decoeficientul de deplasare de profil. Cu alte cuvinte la aceleai valori alelui z i m se pot mri sau micora diametrul roii i grosimea dinteluiprin schimbarea lui x.


33/34

Gearing184

15.6. The purpose of the toothed wheel correction

The toothed gears are made with profile correction in thefollowing purposes:- to improve the contact pressure strength - in this purpose they havepositive profile correction, resulting thus a greater curvature radius ofthe flanks;- a maximum bending strength realisation and the equalising of thebending strength of the pinion base with driven wheel (also positiveprofile correction);- the equalising of the relative maximum sliding speed at the tooth baseof the pinion and driven wheel;- to realise an imposed centre distance;- to realise toothed wheels with a teeth number smaller than 17 withoutthe interference occurrence.

To improve the gearing working, the coefficients of the profilecorrection should be chosen different for the two mating wheels,namely:- xs = x1 + x2 = 0 (x1 = -x2), it is considered that the gearing is made withcompensated teeth, because in this case only the ratio between theaddendum and dedendum changes, the working pressure angle

(meshing angle) is equal to the machining pressure angle and thecentre distance is aw = a (the reference centre distance);- xs = x1 + x2 0, the teeth is made with specific coefficients of theprofile correction x1 x2, positive or negative, according to thedemands imposed to the gearing. The centre distance aw a and themeshing angle w .

The values of the profile correction coefficients should respectthe following conditions:- to avoid the teeth interference during operation or during machining;- to avoid the teeth sharpening on the outside diameter;- the contact ratio (engagement factor) should not be smaller than 1.2.

By the teeth correction, an adequate portion of the involute ischosen, changing thus the curvature radius and consequently thefatigue strength and the tooth thickness on the base increases thusobtaining an increased bending strength of the tooth.

For a better choosing of the coefficients x1 and x2, the blockingoutlines (locking) are used.


34/34


15.6. Scopul corijrii roilor dinate

Roile dinate cu deplasri de profil se mai numesc i roi dinatecorijate. Corijarea roilor dinate are drept scop mbuntirea compor-trii n funcionare a angrenajului din care fac parte, prin intermediul eiputndu-se realiza:- creterea rezistenei la presiune de contact, n acest scop, se adoptdeplasri de profil pzitive, obinndu-se astfel o raz de curbur a flan-curilor mai mare;- realizarea unei rezistene maxime la ncovoiere i egalizarearezistenei la ncovoiere la baza dintelui pinionului i a roii conduse (totdeplasri de profil pozitive);- egalizarea alunecrii relative maxime la baza dintelui roii i pinionu -lui, micorarea acestor alunecri pe ct posibil;- realizarea unei distane dintre axe impuse;- utilizarea unor roi dinate cu un numr de dini mai mic dect 17 frpericolul apariiei interferenei.

Depasarea profilului dinilor se poate face diferit la cele dou roidinate i anume:- xs = x1 + x2 = 0 (x1 = -x2), se spune c angrenajuleste cu dantur compensat, deoarece n acest caz se schimb doarraportul dintre nlimile capului i piciorului dintelui, unghiul de

angrenare n funciona-re rmne egal cu cel de la prelucrare iardistana dintre axe aw= a (distana dintre axe de referin);- xs = x1 + x20, dantura este executat cu deplasri specifice x1 x2, pozitive sau

negative, funcie de cerinele impuse angrenajului . Distana dintre axeaw a i unghiul de angrenare w .

Mrimea deplasrilor de profil nu poate fi aleas arbitrar ci curespectarea urmtoarelor condiii:- evitarea interferenei i a subtierii n timpul funcionrii respectiv a prelucrrii; - evitarea ascuirii dinilorpe cercul exterior; - meninerea gradului de acoperire 1,2.

Necesitatea deplasrii profilului dintelui este legat dembuntirea condiiilor de lucru a angrenajului. Alegnd pentru profilul

dintelui o poriune corespunztoare a evolventei, se modific razele decurbur ale flancurilor i comportarea la oboseal a acestora se mbuntete, n acelai timp, se modific grosimea dintelui la bazasa, ceea ce duce la creterea rezistenei la ncovoiere a dintelui.

Pentru stabilirea ct mai corect a valorilor coeficienilor dedeplasare a profilului, x1i x2, care s ndeplineasc cerinele pretinseangrenajului, se utilizeaz conturele de blocare.

angrenaje.1.2 engl rom

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