Gear Geometry and Applied Theory Episode 2 Part 3 pot

344 Noncircular Gearsas tooth profiles of the circular involute gear with the pitch circleρ A, whereρ Ais the curvature radius of the noncircular gear centrode at point A.. This chapter c

12.11 Evolute of Tooth Profiles 343

Figure 12.11.3: Gear centrode and evolutes of left- and right-hand sides of tooth profile.

of the centrode; K is the curvature center at M; L and N are the current points of both

profile evolutes

An approximate representation of tooth profiles of a noncircular gear is based on thelocal substitution of the gear centrode by the pitch circle of a circular involute gear Figure12.11.4 shows that the profiles of tooth number 1 can be approximately represented

Figure 12.11.4: Local representation of a noncircular gear by the respective circular gear.

344 Noncircular Gears

as tooth profiles of the circular involute gear with the pitch circleρ A, whereρ Ais the

curvature radius of the noncircular gear centrode at point A Similarly, tooth profiles

of tooth number 10 are represented approximately as tooth profiles of the circular gearwith the pitch circle of radiusρ B

The number of teeth of the substituting circular gear is determined as

whereρ i is the curvature radius of the centrode of the noncircular gear, and P is the

diametral pitch of the tool applied for generation

12.12 PRESSURE ANGLE

Consider Fig 12.12.1 which shows conjugate tooth profiles of the noncircular gears

The rotation centers of the gears are O1and O2; the driving and resisting torques are

m1 and m2 The tooth profiles are in tangency at the instantaneous center of rotation I The common normal to the tooth profiles is directed along n–n The pressure angle α12

is formed by the velocity vI 2 of driven point I and the reaction R(12)n that is transmittedfrom the driving gear 1 to driven gear 2 (Friction of profiles is neglected.) The pressureangle is determined with the following equations:

(i) When the driving profile is the left-side one, as shown in Fig 12.12.1, we have

α12(θ1)= µ1(θ1)+ α c−π

Figure 12.12.1: Pressure angle of noncircular gears.

Appendix 12.A Displacement Functions for Generation by Rack-Cutter 345

(ii) When the driving profile is the right-side one, we have

α12(θ1)= µ1(θ1)− α c−π

The pressure angle α12(θ1) is varied in the process of motion because µ1 is not stant The negative sign ofα12indicates that the profile normal passes through anotherquadrant in comparison with the case shown in Fig 12.12.1

con-APPENDIX 12.A: DISPLACEMENT FUNCTIONS FOR GENERATION

and the unit normal n1are represented by the equations

x1 = ρ1(θ1) cos(θ1 − µ0), y1 = −ρ1(θ1) sin(θ1 − µ0) (12.A.2)

Figure 12.A.1: Representation of gear centrode, its unit tangent, and its unit normal.

346 Noncircular Gears

whereµ0= 90◦− ψ.

τ1= [cos(θ1+ µ − µ0) − sin(θ1+ µ − µ0) 0]T (12.A.3)

n1= τ1× k1= [− sin(θ1+ µ − µ0) − cos(θ1+ µ − µ0) 0]T (12.A.4)

Rack-Cutter Centrode

The centrode of the rack-cutter coincides with the x2axis [Fig 12.10.2(b)] The

rack-cutter centrode and its unit normal are represented in S2by the equations

Coordinate Transformation

Our next goal is to represent the gear and rack-cutter centrodes in fixed coordinate

sys-tem S f The coordinate transformation from S i (i = 1, 2) to S f is based on the followingmatrix equations [Fig 12.10.2(b)]:

r(i ) f = Mf i ρ i , n(i ) f = Lf ini (12.A.7)After transformations we obtain

Equations of Centrode Tangency

The gear and rack-cutter centrodes are in tangency at any instant Thus

Appendix 12.A Displacement Functions for Generation by Rack-Cutter 347

Equations (12.A.16) yield

whereµ = arctan(ρ1(θ1)/ρ θ),µ0 = arctan(ρ1(0)/ρ θ)

Transformations of Eqs (12.A.14) are based on the following considerations:(i) At the start of motion we haveφ1 = 0, θ1= 0, origins O f and O2coincide with

each other, and x (O2 )

f (0)= 0 Then, we obtain

where u0determines the initial position of the point of tangency of the centrodes

Due to pure rolling of the centrodes, we have that s1= s2and

348 Noncircular Gears

Generally, s1(θ1) can be determined by numerical integration Equation system (12.A.25)

is used for the numerical control of motions of the cutting machine (or for the design

of cams if a mechanical cutting machine is used)

APPENDIX 12.B: DISPLACEMENT FUNCTIONS FORGENERATION BY SHAPER

The applied coordinate systems S1, S2, and S f are rigidly connected to the gear beinggenerated, the shaper, and the frame of the cutting machine, respectively The orienta-tion of these coordinate systems at the initial position is as shown in Fig 12.B.1 The

gear centrode and its unit normal are represented in S1by Eqs (12.A.2) and (12.A.4),respectively The shaper centrode is a circle of radiusρ2, and this centrode and its normal

are represented in S2by the equations

Appendix 12.B Displacement Functions for Generation by Shaper 349

Figure 12.B.2: Derivation of displacement functions for generation by a shaper.

The centrodes of the gear and the shaper are in tangency with each other at point I ,

represented as [ 0 − ρ2 0 1]T Equations of centrode tangency provide the followingdisplacement functions:

13 Cycloidal Gearing

13.1 INTRODUCTION

The predecessor of involute gearing is the cycloidal gearing that has been broadly used

in watch mechanisms Involute gearing has replaced cycloidal gearing in many areasbut not in the watch industry There are several examples of the application of cycloidalgearing not only in instruments but also in machines that show the strength of positionsthat are still kept by cycloidal gearing: Root’s blower (see Section 13.8), rotors of screwcompressors (Fig 13.1.1), and pumps (Fig 13.1.2)

This chapter covers (1) generation and geometry of cycloidal curves, (2) Camus’theorem and its application for conjugation of tooth profiles, (3) the geometry and design

of pin gearing for external and internal tangency, (4) overcentrode cycloidal gearing with

a small difference of numbers of teeth, and (5) the geometry of Root’s blower

13.2 GENERATION OF CYCLOIDAL CURVES

A cycloidal curve is generated as the trajectory of a point rigidly connected to the circle

that rolls over another circle (over a straight line in a particular case) Henceforth, wedifferentiate ordinary, extended, and shortened cycloidal curves

Figure 13.2.1 shows the generation of an extended epicycloid as the trajectory of point M that is rigidly connected to the rolling circle of radius r In the case when generating point M is a point of the rolling circle, it will generate an ordinary epicycloid, but when M is inside of circle r it will generate a shortened epicycloid.

Point P of tangency of circles r and r1 is the instantaneous center of rotation The

velocity v of point M of the rolling circle is determined as

Vector v is directed along the tangent to the cycloidal curve being generated, and PM

is directed along the normal to at point M.

There is an alternative approach for generation of the same curve The generation

is performed by the rolling of circle rover the circle r1 The same tracing point M is rigidly connected now to circle r The new instantaneous center of rotation is Pwhich

is determined as the point of intersection of two straight lines: (i) the extended straight

350

Figure 13.1.1: Screw rotors of a compressor.

Figure 13.1.2: Screw rotors of a pump.

Figure 13.2.1: Generation of extended epicycloid.

351

352 Cycloidal Gearing

line PM – the normal to the generated curve, and (ii) line OPthat passes through point

O1 and is drawn parallel to OM Point O1is the common center of circles r1and r1

Point Ois a vertex of parallelogram OMOO1 and is also the center of circle r The radii

rand r1of the circles used in the alternative approach are determined with the equations

The velocity v of generating point M is the same in both approaches if the angular

velocitiesω P andω Pare related as

ω P

ω P = r1 + r

The Bobilier construction [Hall, 1966] enables us to determine the curvature center

C of the generated curve The theorem states:

Consider as known the centers of curvature of two centrodes and the centers of curvature of two conjugate profiles that are rigidly connected to the respective centrodes Draw two straight lines such as those that interconnect the curvature center of the respective centrode and the curvature center of the profile rigidly connected to the centrode These two straight lines intersect at point K of line PK that passes through the instantaneous center of rotation P and is perpendicular to the common normal to the conjugate profiles.

In our case, one of the mating profiles is the tracing point M, and the other mating profile is the generated extended epicycloid The determination of curvature center C

of the extended epicycloid in accordance with the Bobilier construction is based on thefollowing procedure (Fig 13.2.1):

Step 1: Identify as given (a) the centers of curvature O and O1 of the centrodes r and r1, (b) point M is one of the mating profiles that is rigidly connected to centrode r , (c) point P is the point of tangency of the centrodes r and r1, and (d) the normal PM

to the generated curve

Step 2: Draw straight line MO that interconnects points M and O.

Step 3: Draw through point P line PK that is perpendicular to the normal PM.

Step 4: It is evident that the sought-for center of curvature of the extended epicycloid

is point C The two straight lines OM and O1C intersect each other at point K

Step 5: The Bobilier construction can be similarly applied for the alternative method

of generation of the extended epicycloid where the rolling centrodes are the circles of

radii r1 and r(Fig 13.2.1) The two straight lines OM and O1 C intersect each other at

point Kof the straight line PK; line PKpasses through point Pand is perpendicular

to the curve normal PM.

Figure 13.2.2 shows the generation of an extended hypocycloid by two alternative approaches In this case it is necessary to take (r1− r ) instead of (r1+ r ) in equations

13.2 Generation of Cycloidal Curves 353

Figure 13.2.2: Generation of extended

hypocy-cloid.

that are similar to (13.2.2), (13.2.4), and (13.2.5) The Bobilier construction has beenapplied in this case as well to illustrate the geometric way of determining the curvature

center C for the extended hypocycloid.

Figures 13.2.3 and 13.2.4 illustrate the generation of an ordinary epicycloid and an dinary hypocycloid Again, two alternative methods for generation can be applied Here,

354 Cycloidal Gearing

Figure 13.2.4: Generation of ordinary hypocycloid.

13.3 EQUATIONS OF CYCLOIDAL CURVES Extended Epicycloid

The position vector O1M (Fig 13.3.1) is represented as

13.4 Camus’ Theorem and Its Application 355

Figure 13.3.2: For derivation of equations of

In the case of an ordinary hypocycloid, we have to take a = r

13.4 CAMUS’ THEOREM AND ITS APPLICATION

Camus’ theorem formulates the conditions of conjugation of two cycloidal curves

Con-sider that gear centrodes are given An auxiliary centrode a (Fig 13.4.1) is in tangency with centrodes 1 and 2, and P is their common instantaneous center of rotation An arbitrarily chosen point M is rigidly connected to centrode a Point M traces out in

relative motion (with respect to centrodes 1 and 2) the curves1and2, respectively.Camus’ theorem states that curves1and2may be chosen as conjugated shapes forteeth of gears 1 and 2, respectively

356 Cycloidal Gearing

Figure 13.4.1: For illustration of Camus’ theorem.

To prove this theorem, let us consider an instantaneous position of centrodes 1, 2,

and a Supposing that centrode 1 is fixed and centrode a rolls over centrode 1, we say that the motion of centrode a relative to centrode 1 is rotation about point P Assume that centrode a rotates about point P through a small angle Then point M of centrode

a traces out in this motion a small piece of curve 1 (point M moves along 1) Line

MP is the normal to 1 at point M Similarly, by rotation of centrode a about P with respect to centrode 2, point M traces out a small piece of curve 2 (M moves along

2 ) Line MP is also the normal to shape 2 at point M.

Thus, curves1and2 have a common point M, they are in tangency at M, and their common normal MP passes through point P , the instantaneous center of rotation of

centrodes 1 and 2 According to the general theorem of planar gearing (see Section 6.1),the generated curves1and2are the conjugate ones

Tooth Addendum–Dedendum Profiles

Considering the synthesis of planar cycloidal gears, the Camus’ theorem should beapplied twice, for conjugation of profiles for the gear addendum and dedendum

Figure 13.4.2 shows the gear centrodes 1 and 2 with radii r1 and r2 To generatethe profiles of the gear addendum and dedendum, two auxiliary centrodes 3 and 3, of

radii r and r, are used The generation of conjugate profiles for gears 1 and 2 may berepresented as follows:

Step 1: Consider that the auxiliary centrode 3 rolls over the gear centrodes 1 and

2 Centrodes 3 and 1 are in external tangency and centrodes 3 and 2 are in internal

tangency Point P of auxiliary centrode 3 generates in coordinate system S1 rigidly

connected to gear 1 the epicycloid P α as the profile of the addendum of gear 1.

13.4 Camus’ Theorem and Its Application 357

Figure 13.4.2: Generation of conjugate profiles for cycloidal gears.

Respectively, point P of auxiliary centrode 3 generates in coordinate system S2rigidly

connected to gear 2 the hypocycloid P β as the profile of the dedendum of gear 2 In

accordance to Camus’ theorem, curves P α and Pβ are the conjugate profiles for gears

1 and 2

Step 2: We consider now that the other auxiliary centrode, circle 3, rolls over thegear centrodes 1 and 2 Centrodes 1 and 3are in internal tangency, and centrodes 2 and

3are in external tangency Point P of circle 3generates in S1the hypocycloid P βas

the profile of the tooth dedendum of gear 1 Respectively, point P of circle 3generates

in S2the epicycloid P αas the profile of the addendum of gear 2.

Unlike involute planar gears, the addendum and dedendum profiles of a cycloidal gearare represented by two different curves, an epicycloid and a hypocycloid The change ofthe center distance of cycloidal gears is accompanied with the breaking of conjugation

of tooth profiles

The line of action of cycloidal gears is a combination of two circular arcs that belong to

auxiliary centrodes 3 and 3as shown in Fig 13.4.3 Here, L1PK1 and L1PK1represent

the lines of action for both sides of tooth profiles The tangent T to the line of action

at P is perpendicular to the center distance O1O2 Point P of the gear tooth profiles is

a singular point However, normal N to the tooth profiles at P can be determined The line of action of N is the same as of T Thus, the pressure angle at P is equal to zero.

(i) Driving gear 1 in a watch gear mechanism is provided with a larger number N1

of teeth than the number of teeth N2 of the driven gear Therefore, the angular

Figure 13.4.4: Watch gearing.

13.5 External Pin Gearing 359

Figure 13.4.5: Centrodes of gears 1 and

2, rack-cutter 3, and auxiliary centrodes

The watch gear mechanism is designated to multiply the angular velocity because

the rotation is provided to the watch arrows Recall that the main purpose of a

reducer with involute gears is to reduce the angular velocity of the driven gear In

a reducer, we have that N1is less than N2, and the gear ratio is m21< 1.

(ii) The profile of the tooth dedendum is a straight line directed from P to O i (i = 1, 2) Such straight lines are particular cases of hypocycloids P β and Pβ (Fig 13.4.2)

that are generated when r= r1/2 and r = r2/2 This statement can be proven with

the analysis of Eqs (13.3.4) for an extended hypocycloid

Rack-Cutter Profiles for Watch Gears

Figure 13.4.5 shows gear centrodes 1 and 2 and two auxiliary centrodes a and a∗that

are used for generation of conjugate profiles of watch gears The radii of the circles a and a∗ are

r a =r2

2, r∗

a =r12

The rack-cutter centrode 3 is a straight line that is tangent to the gear centrodes at P

The rack-cutter profiles,3and∗

3, are ordinary cycloids that are generated in S3by

point P of auxiliary centrodes a and a∗(Fig 13.4.6)

13.5 EXTERNAL PIN GEARING

Pin gearing (Fig 13.5.1) is a particular case of cycloidal gearing The teeth of the pinionare cylinders and the gear tooth surface is conjugate to the cylinder surface Pin gearing

is used in reducers for cranes, in some planetary trains, and is still used as watch gearing.The main advantage of pin gearing is the possibility of avoiding the generation of the