Gear Geometry and Applied Theory Episode 2 Part 2 doc

11.4 INTERFERENCE BY ASSEMBLY We consider that the internal gear with the tooth number N2 was generated by the shaper with tooth number N c and the condition of nonundercutting was obser

.

(11.3.11)

The first guess for the solution of system (11.3.11) is based on considerations similar

to those previously discussed:

Step 1: Transforming equation system (11.3.11), we obtain

We take for the first guess N c = 0.8N2and obtain (φc + ) from Eq (11.3.12)

Param-eterφ2is determined from Eq (11.3.10)

Step 2: Knowing N c, φc, and φ2 for the first guess, and using the subroutine forthe solutions of equation system (11.3.11), we can determine the exact solution for

Axial Generation Radial Generation

0 0 20 40 60 80 100 120 140 160 180 200

314 Internal Involute Gears Table 11.3.1: Maximal number of shaper teeth

Nc Computations based on the above algorithms allow us to develop charts for

de-termination of the maximal number of shaper teeth, N c , as a function of N2and thepressure angleαc An example of such a chart developed for axial generation and two-

parameter generation is shown in Fig 11.3.5 Table 11.3.1 (developed by Litvin et al.

[1994]) allows us to determine the maximal number of shaper teeth for various pressureangles

11.4 INTERFERENCE BY ASSEMBLY

We consider that the internal gear with the tooth number N2 was generated by the

shaper with tooth number N c and the condition of nonundercutting was observed.Then, we consider that the internal gear is assembled with the pinion with the tooth

number N1> Nc The question is what is the limiting tooth number N1 that allows

us to avoid interference by assembly Henceforth, we consider two possible cases ofassembly – axial and radial

Axial Assembly

Axial assembly is performed when the final center distance E(2)= (N2− N1)/2P is

initially installed and the pinion is put into mesh with the internal gear by the axialdisplacement of the pinion Radial assembly means that the pinion is put into meshwith the internal gear by translational displacement along the center distance The center

distance E by the radial displacement of the pinion is changed from (N2− N1− 4)/2P

to (N2− N1)/2P

Interference in the axially assembled drive occurs if the tip of the pinion tooth ates in relative motion a trajectory that intersects the gear involute profile The trajectory

gener-is an extended hypocycloid The solution gener-is based on the same approach that was applied

for axial generation The limiting number N1of pinion teeth is a little larger than N c

due to the lessened dimension of the pinion addendum in comparison with the shaperaddendum

angular pitch; j is the space (tooth) number; and E is the variable center distance

that is installed by the assembly The superscripts “1” and “2” indicate the pinion

and the gear, respectively; N2is considered as given

(ii) Interference of pinion and gear involute profiles occurs if

r(2)f (θ2, j δ2, N2)− r(1)

f (θ1, j δ1, E, N1)= 0. (11.4.2)(iii) Equations (11.4.2) provide a system of two scalar equations

f j(θ2, θ1, j δ2, j δ1, E, N1, N2)= 0 ( j = 0, 1, 2, , m). (11.4.3)

We consider the most unfavorable case when the point of interference lies on theaddendum circles of the pinion and the gear (Fig 11.4.1), and therefore parametersθ1andθ2are known We will determine (N1, E) if the solution of equation system (11.4.3) exists The solution for N1= N (r )

1 determines the maximal number N1(r )of the pinionthat is allowed by radial assembly

Figure 11.4.1: Interference by radial assembly.

316 Internal Involute Gears

Figure 11.4.2: Radial assembly.

After the gears are radially assembled and the final center distance E(2)is installed, thetip of the pinion generates an extended hypocycloid while the pinion and gear performrotational motions Interference of the hypocycloid with the gear involute profile is

avoided by making the number of pinion teeth N1≤ N (a)

1 , where N1(a) is the number

of pinion teeth allowed by axial assembly The designed number of pinion teeth should

not exceed N1(a) and N1(r ).Figure 11.4.2 illustrates the computerized simulation of radial assembly of the pinion

and gear The computations were performed for a gear drive with N1= 25, N2= 40,

diametral pitch P = 8, and pressure angle α c= 20◦

We can avoid the investigation of interference by radial assembly if the pinion tooth

number N1(r ) satisfies the inequality N1(r ) ≤ N (r )

c where N c (r )is the shaper tooth numberallowed by radial–axial generation (see Table 11.3.1)

Nomenclature

Ec distance between gear and cutter axes (Fig 11.2.1)

N1 pinion teeth number

N2 gear teeth number

Nc shaper teeth number

P diametral pitch

mi j transmission ratio of gear i to gear j ra1 radius of pinion addendum circle

ra2 radius of gear addendum circle (Fig 11.3.2)

rac radius of cutter addendum circle (Fig 11.2.2)

rb1 radius of pinion base circle

rb2 radius of gear base circle (Fig 11.3.2)

rbc radius of cutter base circle (Fig 11.2.2)

r p2 radius of gear pitch circle (Fig 11.3.2)

11.4 Interference by Assembly 317

r pc radius of cutter pitch circle (Fig 11.2.2)

sac tooth thickness of the cutter on the addendum circle (Fig 11.2.2)

s pc tooth thickness of the cutter on the pitch circle (Fig 11.2.2)

wa2 space width on the gear addendum circle (Fig 11.3.2)

wp2 space width on the gear pitch circle (Fig 11.3.2)

2 angle of tooth thickness on the cutter addendum circle (Fig 11.2.2)

αc pressure angle of cutter

θi parameter of gear involute profile (i = 1, 2) (Fig 11.3.1)

φ2 angle of gear rotation (Fig 11.2.1)

φc angle of cutter rotation (Fig 11.2.1)

or the velocity function, and (ii) for the generation of a prescribed function.

Figure 12.1.1 shows the Geneva mechanism that is driven by elliptical gears Theapplication of elliptical gears enables it to change the angular velocity of the crank ofthe mechanism during the crank revolution A crank–slider linkage that is driven byelliptical gears is shown in Fig 12.1.2 A kinematical sketch of the mechanism is shown

in Fig 12.1.3(a) Application of elliptical gears enables it to modify the velocity function

v( φ) of the slider [Fig 12.1.3(b)] Oval gears (Fig 12.1.4) are applied in the Bopp and

Reuter meters for the measurement of the discharge of liquid; the oval gears are shown

in the figure in three positions Figure 12.1.5 shows noncircular gears with unclosedcentrodes that are applied in instruments for the generation of functions Figure 12.1.6shows a noncircular gear of a drive that is able to transform rotation between parallelaxes for a cycle that exceeds one gear revolution During the cycle the gears performaxial translational motions in addition to rotational motions

Noncircular gears have not yet found a broad application although modern ufacturing methods enable their makers to provide conjugate profiles using the sametools as are applied for spur circular gears The following sections are based on work

man-by Litvin [1956]

12.2 CENTRODES OF NONCIRCULAR GEARS

We consider two cases, assuming as given either (i) the gear ratio function m12(φ1), or

(ii) the function y(x) to be generated.

318

12.2 Centrodes of Noncircular Gears 319

Chain conveyor

Elliptical gears

Figure 12.1.1: Conveyor driven by the Geneva mechanism and elliptical gears.

Case 1: The gear ratio function

Figure 12.1.2: Conveyor based on application of the crank–slider linkage and elliptical gears.

Figure 12.1.3: Combination of elliptical gears with a crank–slider linkage.

12.2 Centrodes of Noncircular Gears 321

Figure 12.1.5: Noncircular gears applied in

in-struments.

Figure 12.1.6: Twisted noncircular gear.

m12(φ1). (12.2.3)Functionφ2(φ1)∈ C2is a monotonic increasing function, and the gear ratio function

m12(φ1)∈ C1 must be positive The difference between m12 max and m12 min is to belimited to avoid undesirable pressure angles (see Section 12.12) We have to differentiatethe angle of rotationφi of gear i from the polar angle θi that determines the position

vector of the centrode (i = 1, 2) Angles φ i andθi are equal, but they are measured inopposite directions

The orientation of the tangent with respect to the current position vector of thecentrode is designated by angleµ, where

tanµi = ri dri(φi)

Functionµi(φ1)(i = 1, 2) is used for determination of variations of the pressure angle

in the process of meshing (see Section 12.12) The upper (lower) sign in the aboveexpressions with double signs corresponds to the case of external (internal) gears Angle

µi is measured in the same direction asθi.The following discussion is limited to the case of external noncircular gears Thesubscripts “1” and “2” in expressions forµ1andµ2indicate gears 1 and 2, respectively

Case 2: Function y(x) to be generated is given

y(x) ∈ C2, x2≥ x ≥ x1.

Rotation angles of the gears are determined as

φ1= k1(x − x1), φ2= k2[y(x) − y(x1)] (12.2.7)

where k1and k2are the scale coefficients of constant values Equations (12.2.7) represent

in parametric form the displacement function of the gears

The gear ratio function is

m12=d φ1

d φ2 = k1

In the case when the derivative y x changes its sign in the area x1≤ x ≤ x2, the

di-rect generation of y(x) by noncircular gears becomes impossible This obstacle can be

overcome as follows:

(i) Consider that the noncircular gears generate instead of y(x) the function

F1(x) = y(x) + k3x (k3is constant). (12.2.11)(ii) A pair of circular gears generates simultaneously the function

(iii) Functions F1(x) and F2(x) are transmitted to a differential gear mechanism, and then the given function y(x) will be executed as the angle of rotation of the driven

shaft of the differential mechanism

The maximal values of the scale coefficients are determined by the equations

k1 max= φ1 max

x2− x1, k2 max= φ2 max

y(x2)− y(x1) (12.2.13)whereφi max= 300◦∼ 330◦ for gears with unclosed centrodes Knowing function

yx (x) and the coefficients k1and k2, we are able to determine functionµ1(φ1) andestimate the variation of the pressure angle In some cases, it becomes necessary touse a sequence of two pairs of noncircular gears to decrease the maximal value ofthe pressure angle (see Section 12.8)

12.3 CLOSED CENTRODES

Noncircular gears designated for continuous transformation of rotational motion must

be provided with cl os ed centrodes This yields the following requirement for m12(φ1)

The gear ratio function m12(φ1) must be a periodic one, and its period T is related with the periods T1and T2of the revolutions of gears 1 and 2 as

T = T1

n2 = T2

n1

(12.3.1)

where n1and n2are whole numbers

Let us now consider the following design case:

(i) The centrode of gear 1 is already designed as a closed curve

(ii) Gears 1 and 2 must perform continuous rotations, and n1and n2are the numbers

of revolutions of the gears

324 Noncircular Gears

The question is, what are the requirements to be satisfied to obtain that the gear 2centrode is a closed curve as well The solution is based on the following ideas:(i) We consider that the centrode of gear 1 is represented as a closed curve by the

periodic function r1(φ1)∈ C2and r1(2π) = r1(2π/n1)= r1(0)

(ii) The angle of rotation of gear 2, φ2= 2π/n2, must be performed while gear 1performs rotation of the angleφ1= 2π/n1

(iii) Taking into account that

2π

n2 =

 2π n1

0

r1(φ1)

Equation (12.3.4) can be satisfied with a certain value of center distance E, with

which the centrode of gear 2 will be a closed curve

Problem 12.3.1

Consider that the centrode of gear 1 is an ellipse (Fig 12.3.1) and the number of

revolutions of the gears are n1= 1 and n2= n The center of rotation of gear 1 is focus

O1of the ellipse The centrode of gear 1 is represented in polar form by the equation

r1(φ1)= p

Here, p = a(1 − e2), e = c/a (Fig 12.3.1).

Figure 12.3.1: Elliptical centrode.

(ii) The substitution

The derivations above confirm Eq (12.3.6)

Using Eq (12.3.6), we obtain the following expression for E:

326 Noncircular Gears 12.4 ELLIPTICAL AND MODIFIED ELLIPTICAL GEARS

Modification of Elliptical Centrode

The modification of an elliptical centrode is based on the following ideas proposed byLitvin [1956]:

(i) Consider that a current point M of the elliptical centrode is determined with the

position vector [Fig 12.4.1(a)]

(iii) The same principle of centrode modification is applied for the lower part of the

ellipse [Fig 12.4.1(b)]; the modification coefficient is m I I Generally, m I I = m I.(iv) The initial elliptical centrode and the modified one are shown in Fig 12.4.1(c)

Figure 12.4.1: Modified elliptical centrode.

12.4 Elliptical and Modified Elliptical Gears 327

Figure 12.4.2: Ordinary and modified elliptical centrodes.

Figure 12.4.2 shows the modification of identical elliptical centrodes for the case

where m I = 3/2, n1= n2= 1, and e1= 0.5 Figure 12.4.2 illustrates the principle of

centrode modification Noncircular gears with modified elliptical centrodes transform

rotation with a nonsymmetrical gear ratio function m12(φ1) This function is symmetricalfor noncircular gears with elliptical centrodes

Figures 12.4.3 and 12.4.4 illustrate noncircular gear drives whose driving gear is vided with an elliptical centrode The driving gear performs two and three revolutions,respectively, while the driven gear performs one revolution It was proven by Litvin[1956] that the centrodes of driven gears are modified ellipses

pro-Figure 12.4.3: Conjugation of an elliptical centrode

and an oval centrode for two revolutions of the driving

gear.

328 Noncircular Gears

Figure 12.4.4: Conjugation of an tical centrode and a mating centrode for three revolutions of the driving gear.

ellip-Two oval gears with identical centrodes (Fig 12.4.5) are a particular case of

mod-ified elliptical gears when the coefficients of modification are m I = m I I = m = 2 (see

Fig 12.4.1) The gear centrode is represented by the equation

Figure 12.4.5: Oval centrodes.

12.5 Conditions of Centrode Convexity 329

Figure 12.4.6: Conjugation of an

ellip-tical centrode with the mating centrode

for four revolutions of the driving link.

12.5 CONDITIONS OF CENTRODE CONVEXITY

Noncircular gears with convex–concave centrodes can be generated by a shaper but not

by a hob The condition of convexity of a gear centrode means thatρ > 0, where ρ is

the centrode curvature radius In the case of concave–convex centrodes, there is a point

of the gear centrode whereρ = ∞.

The curvature radius of a gear centrode is represented by the equation

Using Eqs (12.2.2) and (12.2.3), we can represent the condition of centrode convexity

in terms of function m12(φ1) and its derivatives:

(i) For the driving gear we have

330 Noncircular Gears

In the case of generation of given function f (x), we obtain the following conditions

of convexity for the driving and driven gears, respectively:

(i)

k1k2[ f(x)]3+ k2

1[ f(x)]2+ 2[ f(x)]2− f(x) f(x) ≥ 0. (12.5.5)(ii)

Figure 12.6.1 shows that the center of rotation O1of the eccentric circular gear 1 does

not coincide with the geometric center of the circle of radius a The centrode of gear 2

must be conjugate with the eccentric circle, the centrode of gear 1 Such drives can be

applied with n = 1, 2, 3, ,n where n is the total number of revolutions of gear 2.

The centrode of the eccentric circular gear is represented by the equation

r1(φ1)= (a2− e2sin2φ1)1/2 − e cos φ1= a[(1 − ε2sin2φ1)1/2 − ε cos φ1] (12.6.1)whereε = e/a, and e is the eccentricity The gear ratio function m21(φ1) is