Noncircular gears  design and generation

Introduction 115

This chapter focuses on the design of an internal noncircular gear drive, consisting of an external driving gear and an internal noncircular gear We will refer to the external gear as gear 1, which serves as the driving component in this configuration.

The gear drive can be utilized to adjust the output speed of a gear mechanism and for function generation In this system, gears 1 and 2 rotate in the same direction, as illustrated in Fig 7.1.1.

The transformation of rotation can be analyzed through two scenarios: (i) a pair of conjugated centrodes represented by frictional disks, where the smaller external disk (disk 1) interacts with the larger internal disk (disk 2), and (ii) a system of gears, consisting of an external gear (gear 1) and an internal gear (gear 2) These frictional disks and gear centrodes play a crucial role in understanding rotational dynamics.

Derivation of Centrodes 115

Preliminary Considerations of Kinematics of Internal Gear

Figure 7.1.1(a) shows schematically that external gear 1 (centrode 1) and internal gear 2 (centrode 2) perform rotation about centersO 1andO 2in the same direction.

Rotation about center O 1of external gear 1 is represented by vector ω (1) (φ 1) Sim- ilarly, rotation about center O 2 of internal gear 2 is represented by vector ω (2) (φ 1).

Here,φ 1represents the angle of rotation of external gear 1 about centerO 1

The derivative functionm 12 (φ 1) is represented as m 12(φ 1)= ω (1) (φ 1) ω (2) (φ 1) >1 (7.2.1)

The relative angular velocity is ω (12) (φ 1)=ω (1) (φ 1)−ω (2) (φ 1)=ω (1) (φ 1)

P1: JZP cuus681-driver CUUS681/Litvin 978 0 521 76170 3 July 22, 2009 17:39

116 Design of Internal Noncircular Gears

Figure 7.1.1 Illustration of transformation of rotation in case of internal noncircular gears:

(a) O 1 and O 2 are the centers of rotation; (b) vectors ω (1) and ω (2) are directed along the respective axes of rotation. wherein

Internal gear drives effectively reduce relative angular velocity and relative velocity, leading to decreased sliding This reduction in sliding is a key factor in the enhanced efficiency of internal gear drives, making them a preferred choice for high-performance gear applications.

Basic Equations of Centrodes 116

The derivation of centrodes for internal gear drives is based on principles outlined in Chapter 2 These centrodes meet tangentially at point I, which lies on the extended line O1−O2, indicating that the velocities v(12) = 0 and v(1) = v(2).

Figure 7.2.1 Illustration of mating centrodes 1 and 2, polar axes O i A i (i=1,2), rotation anglesφ i (i=1,2), polar anglesθ i (i=1,2), and polar vectorsr i (i=1,2).

The equation \( v(12) = 0 \) indicates that the centrodes of the internal gear drive are rolling over one another, resulting in a zero sliding velocity at point I This point serves as the instantaneous center of rotation for the centrodes during their relative motion.

Considering as given the derivative functionm 12(φ 1), and taking into account thatv (12) =0, we obtain m 12(φ 1)=ω (1) (φ 1) ω (2) (φ 1) =dφ 1 dφ 2

Equation (7.2.5) allows the conjugated centrodesσ 1andσ 2to be derived and represented as functions of angle φ 1, whereφ 1 is the angle of rotation of the driving centrodeσ 1.

We may representσ 1andσ 2as polar curves in terms of polar centrode angleθ 1, taking into account thatθ 1≡φ 1, butθ 1is measured opposite toφ 1:

(i) Centrodeσ 1is represented as r 1 (φ 1)=E 1 m 12(φ 1)−1 (7.2.6) or as r 1 (θ 1)=E 1 m 12(θ 1)−1 (7.2.7)

(ii) Centrodeσ 2is represented as r 2 (φ 1)=E m 12(φ 1) m 12(φ 1)−1 (7.2.8) φ 2(φ 1) φ 1

Design of Centrodes σ 1 and σ 2 as Closed-Form Curves 118

The conditions of centrodesσ 1 andσ 2 of an internal noncircular gear drive to be closed-form curves are similar to those discussed in Section2.6for an external gear drive.

We require for an internal gear drive observation of the following conditions:

(i) The derivative functionm 12 (φ 1) must be a periodic function with period T T 1 /n 1, whereT 1is the period of revolution of driving centrodeσ 1(gear 1) and n 1is an integer number.

(ii) The ratio between the revolutionsn 1andn 2of centrodes 1 and 2, respectively, must be

(7.2.10) wheren 1andn 2are integer numbers.

(iii) The gear center distanceEmust be determined as a function ofn 1andn 2.

Derivations similar to those performed in Section2.6yield the following equation ofEfor an internal noncircular gear drive:

The polar equation of the driving centrode, denoted as r1(φ1), is essential for solving Eq (7.2.11) Typically, the solution for E can be derived through numerical methods, involving an iterative process However, an analytical solution is achievable for gear drives utilizing elliptical and modified elliptical centrodes.

Examples of Design of Internal Noncircular Gear Drives 118

Gear Drive with Elliptical Pinion 118

Step 1.Centrodeσ 1of the drive is represented in polar form (see Section4.3.2) as r 1 (φ 1)= a(1−e 2 )

7.3 Examples of Design of Internal Noncircular Gear Drives 119

Step 2.The derivative function of the drive,m 21(φ 1)= 1 m 12 (φ 1), is represented (see Eq (7.2.5)) as m 21(φ 1)= r 1 (φ 1)

Step 3 The next step is determination of center distance E=E(a,e,n) It is based on application of

The meaning of this equation is the requirement that the pinion (centrodeσ 1) will performnrevolutions for one revolution of gear 2.

Simple transformation of Eq (7.3.3) yields π n = p

It follows from the tables of integrals (seeDwight, 1961, Problem 858.524), that π

Equations (7.3.4) and (7.3.5) yield the following solution forE:

It is remarkable, as shown later, that Eq (7.3.6) works as well for determination of the center distance of a gear drive with modified elliptical or oval pinion.

Step 4.Centrodeσ 2of gear 2 is represented as r 2(φ 1)=E+r 1(φ 1) (7.3.7) φ 2(φ 1) φ 1

Transmission functionφ 2(φ 1) may be determined numerically by integration of

Eq (7.3.8), or analytically (seeDwight, 1961, Problem 446.00) using dφ 1

Figures7.3.1(a) and (b) show the gear drives with elliptical pinion determined forn=2 andn=3, respectively;a9.3378 mm,e=0.5.

Figure 7.3.1 Illustration of elliptical centrode with parametersa9.3378 mm ande=0.5,and the internal noncircular centrode in case of (a)n=2 and (b)n=3.

Gear Drive with Modified Elliptical Pinion 120

Step 1.The modified elliptical centrodeσ 1of gear 1 (see Section4.3.4) consists of two branches, the upper and the lower, and is represented by r 1 (I) (φ 1)= a(1−e 2 )

Step 2.The derivative function of the drive,m 21(φ 1), is determined as m 21(φ 1)=ω (2) ω (1) = r 1 (φ 1) r 2(φ 1) = r 1 (φ 1)

E+r 1(φ 1) (7.3.12) Equations (7.3.10), (7.3.11), and (7.3.12) yield m (I) 21 (φ 1)= p p+E−Eecos(m I φ 1), 0≤φ 1 ≤ π m I (7.3.13) m (I I 21 ) (φ 1)= p p+E−Eecos(m I I (2π−φ 1)), π m I ≤φ 1 ≤2π (7.3.14)

Step 3.Gear 1 (centrodeσ 1) performsnrevolutions for one revolution of gear

We recall that centrodesσ 1andσ 2are closed curves andnis an integer number.

Step 4.We apply Eqs (7.3.13), (7.3.14), and (7.3.15) for determination ofE E(a,e,n) The derivation ofEis based on the following procedure:

(i) The variables of the integrals of Eq (7.3.15) are changed as m I φ 1=x, m I I (2π−φ 1)=y (7.3.16)

(ii) We take into account that (seeDwight, 1961, Problem 858.524) π

We then obtain the solution E=E(a,e,n) for the drive with modified elliptical gears that is the same as for the drive with conventional ellipses represented by Eq (7.3.6).

Step 5.Centrodeσ 2is represented by r 2 (I ) (φ 1)=E+r 1 (I ) (φ 1) (7.3.18) φ (I 2 ) (φ 1) φ 1

Similarly, we represent equations forr 2 (I I) (φ 1) andφ 2 (I I) (φ 1).

Determination ofφ 2 (i) (φ 1), (i=I,I I), may be obtained by using the expression for independent integral dx

Figure 7.3.2 illustrates designed internal gear drives with modified elliptical gears.

Gear Drive with Oval Pinion 121

Step 1.The centrodeσ 1of the pinion is an oval (see Fig.7.3.3), a curve obtained as the result of transformation of a conventional ellipse performed as follows:

(i) the magnitude of the radius vector of the ellipse is observed, but (ii) the position angle is increased twice.

The polar equation of the oval gear 1 is represented as r 1 (φ 1)= a(1−e 2 )

1−ecos 2φ 1 , 0≤φ 1≤2π (7.3.20) whereinaandeare the semi-length of the major axis and the eccentricity of the ellipse being transformed.

Figure 7.3.2 Illustration of modified elliptical centrode with parametersa7.1241 mm and e=0.2,m I =1.5, andm I I =0.75, and the internal noncircular centrode in case of (a)n=2 and (b)n=3.

Figure 7.3.3 Illustration of oval centrode (with parameters a7.1241 mm ande=0.2),and the internal noncircular centrode in case of (a)n=2 and (b)n=3.

Step 2.The radius vector of the internal gear 2 isr 2(φ 1)=E+r 1(φ 1) (wherein

Eis the shortest center distance) and the derivative functionm 21 (φ 1) is represented as m 21(φ 1)= r 1 (φ 1)

Step 3.Relation between rotations of gears 1 and 2 This relation is similar to the one that has been represented for an external gear drive (see Section4.3.5.3):

Step 4.Derivation of center distanceE The variableφ 1is changed forxwherein

From tables of integrals (Dwight, 1961), π

After simple derivations, we obtain equation E=E(a,e,n) that is the same as

Eq (7.3.6) represented for internal noncircular gear drives with elliptical pinion and modified elliptical pinion.

Step 5.Centrodeσ 2of internal gear is determined with r 2 (φ 1)=E+r 1 (φ 1) (7.3.26) φ 2 φ 1

0 m 21(φ 1)dφ 1 (7.3.27) wherem 21(φ 1) is represented by Eq (7.3.21).

Transmission functionφ 2(φ 1) may be determined by numerical integration of

Eq (7.3.27), and analytically using the work of Dwight (Dwight, 1961, see Prob- lem 446.00), witha 2 >b 2

Centrodesσ 1andσ 2of the drive are illustrated by Fig.7.3.3.

Gear Drive with Eccentric Pinion 123

Step 1.Derivation ofr 1 (φ 1)= |O 1 M| Centrodeσ 1of gear 1 is an eccentric circle of radius a and eccentricitye= |O 1 C 1 | = |O 1 C 2 | (Fig 7.3.4) Centrode σ 1

(the eccentric circle) performs rotation about center O 1 and is shown in two positions: the initial one, and the current one after rotation on angleφ 1.

Figure 7.3.4 Illustration for derivation ofr 1(φ 1)= |O 1 M|;O 1is the center of rotation of eccentric circle of radiusa;C 1 andC 2 are initial and current positions of geometric center of circlea.

The conjugated centrode σ 2 (not shown in Fig.7.3.4) is an internal curve and performs rotation about centerO 2 ;E= |O 2 O 1 |is the shortest center distance.

Centrodes σ1 and σ2 are tangent at point M, which lies on the extended line O1−O2 As the system rotates, point M moves along the line O1−O2 The relationship between the radius r1 and the angle φ1 is given by the equation r1(φ1) = |O1M| = (a2 - e2 sin² φ1)^(0.5) - e cos φ1, where 0 ≤ φ1 ≤ 2π.

Step 2 Derivative functionm 21(φ 1) The derivative functionm 21(φ 1) is determined as m 21(φ 1)= dφ 2 dφ 1

Step 3.Derivation ofc=E/a Centrodeσ 2(of internal gear) must be a closed- form curve, and this may be obtained by the observation of

The meaning of Eq (7.3.30) is that centrodeσ 1(of the pinion) performsnrevolutions for one revolution of centrodeσ 2of internal gear;nis an integer number.

Figure 7.3.5 Illustration of an eccentric centrode with parametersa6.75 mm ande=7.35 mm, and the internal noncircular centrode in case of (a)n=2 and (b)n=3.

Solution of Eq (7.3.30) forcrequire iterations and numerical integration Sim- ple transformations yield n−1 n = 1

Step 4.Derivation of centrodeσ 2 The centrodeσ 2is represented by r 2 (φ 1)=E+r 1 (φ 1) (7.3.32) φ 2(φ 1) φ 1

(7.3.33)Figure7.3.5illustrates examples of derived internal drives.

Figure 7.4.1 Illustration of movable coordinate systemsS 1 ,S 2 ,S s , and fixed coordinate sys- temS f : (a) initial position of centrodes; (b) current positions of centrodes.

Generation of Planar Internal Noncircular Gears by Shaper 126

Generation of pinion 1 with centrodeσ 1of the internal gear drive may be performed by a rack cutter, a shaper, or a hob (see Chapter 5).

The internal noncircular gear 2, with centrode σ 2, is produced using shapers The profile of gear 2 is defined as the envelope of the family of profiles s, which are in relative motion to the shaper.

2 The conjugation of pinion 1 and gear 2 is provided by considering simultaneous tangency of centrodes of 1, 2, and shapers(Fig.7.4.1).

The generation of 2by s can lead to undercutting, which is a defect in the meshing of 2 and s caused by the interference of s within the space of 2 To prevent undercutting, it is beneficial to design shapers with a larger pressure angle (α s), although maintaining appropriate geometric relationships between s, 2, and 1 is also essential for optimal performance.

Henceforth, we will consider the following cases of tangency of rolling centrodes: (i) tangency of centrodes σ 1, σ 2, and σ s (Fig 7.4.1), and (ii) tangency of centrodesσ 2andσ s (Fig.7.4.2).

TANGENCY OF CENTRODES σ 1 , σ 2 , AND σ s Figure7.4.1(a) shows that the centrodes are initially in tangency at common point I o (σ 1 ,σ 2 ,σ s ) Three movable coordinate systems

S 1, S 2, and S s are considered that are rigidly connected to pinion 1, internal gear

Coordinate systems S1 and S2 execute corresponding rotations around points O1 and O2 The points I₀ (σ₁, σ₂, σₛ) and I (σ₁, σ₂, σₛ) represent the initial and current positions of the tangency of centrodes, serving as the instantaneous centers of rotation for centrodes in relative motion.

PointI ( σ 1 ,σ 2 ,σ s ) of tangency of centrodesσ 1andσ 2moves in the process of motion along line O 2−O 1−I ( σ 1 ,σ 2 ,σ s ) (Fig 7.4.1) Tangency of σ s with σ 1 andσ 2 at

7.4 Generation of Planar Internal Noncircular Gears by Shaper 127

The illustration in Figure 7.4.2 depicts the coordinate systems S_n, S_f, S_s, and S_2, highlighting the point I (σ_1, σ_2, σ_s) where the shaper executes a complex motion This motion includes (a) translational movement defined by coordinate system S_s and (b) rotational movement around the origin O_s The positions O_s(o) and O_s represent the initial and current locations of the origin of S_s, respectively, while coordinate system S_f remains fixed.

The transformation of motion involves the interplay of centrodes σ1, σ2, and σs, which roll over one another A common normal to the profiles of these centrodes intersects at the point of tangency I (σ1, σ2, σs), establishing the conjugation condition for the three profiles.

TANGENCY OF CENTRODES σ 2 AND σ s Shaperswith profile s generates profile 2of internal gear 2 wherein coordinate systemsS s andS 2are rigidly connected tosand

2 We apply the following coordinate systems (Fig.7.4.2):

(i) Coordinate systemS s , that is rigidly connected tosand performs rotation about

The coordinate system S2 is rigidly attached to gear 2, enabling it to execute both translational motion alongside auxiliary coordinate system Sn and rotational motion around the origin On of Sn at an angle ψ2.

The translational motions of S n and S 2 are performed collinear to axes (x f , y f ) Figure7.4.2 shows the current point I (σ 2 ,σ s ) of tangency of centrodes σ 2 and σ s Initially, centrodesσ 2andσ s were in tangency at point I o (σ 1 ,σ 2 ,σ s ) (Fig.7.4.1(a)).

Then,σ 2 andσ s become in tangency at pointI (σ 2 ,σ s ) (Fig.7.4.2), and their common tangent forms angleà 2 with radius vector O 2 I of centrode 2 and is perpendicular toO s I.

The tangency of σ2 and σs at point I (σ2, σs) is established when the centrode σ2, within the coordinate system Sn, is translated by magnitudes x (Of2) and Es2,0 + y (Of2) along the axes (xf, yf) of Sf It is important to note that the magnitude of y (Of2) in Sf will be negative Here, Es2,0 represents the initial center distance between the centrodes σ2 and σs The magnitudes x (Of2) and y (Of2) can be expressed as functions of the polar angle θ2, where x (Of2) = -r2(θ2) cos α2 and y (Of2) = +r2(θ2) sin α2 + ρs Additionally, the relationship ψ2 = θ2 + α2 - π is defined.

The common normal at the intersection point I (σ2, σs) between centrodes σs and σ2 forms an angle à2 with the position vector O2I The common tangent to σs and σ2 passes through the tangency point I (σ2, σs), and the relative velocity v(s2) satisfies the condition v(s2) · N(s) = v(s2) · N(2) = 0, where N(s) and N(2) are the normals to the centrodes σs and σ2, respectively This observation is crucial for deriving the meshing equation of the shaper σs and the internal gear σ2.

The generating involute profiles of shaper teeth are illustrated in Figure 7.4.3 within the coordinate system S s These involute profiles can also be derived by examining the generating process of a spur gear using a rack cutter, as discussed by Litvin and colleagues.

Fuentes, 2004) In matrix form, the involute profile is represented as r s (θ s )

 ρ bs [sin(θ s −θ 0s )−θ s cos(θ s −θ 0s )] ρ bs [cos(θ s −θ 0s )+θ s sin(θ s −θ 0s )]

Here,ρ bs =ρ s cosα s is the base radius of the shaper,ρ s is its pitch radius, and α s is the pressure angle Parameterθ 0s for a standard involute shaper is determined as θ 0s = πm

Figure 7.4.3 Representation of shaper profile s in coordinate systemS s wheremis the module of the gear drive and invα s =tanα s −α s The unit normal n s (θ s ) to the profile s is represented by n s (θ s ) d r s dθ s ×k d r s dθ s

MATRIX DERIVATION OF PROFILE 2 OF INTERNAL GEAR 2 Profile 2is generated as the envelope to the family of generating profiles s determined inS 2 by r 2(θ s , θ 2)=M 2n (ψ 2)M nf (x (O f 2 ) ,y (O f 2 ) )M f s (ψ s )r s (θ s ) (7.4.8)

MatricesM 2n ,M nf , andM f s of Eq (7.4.8) describe coordinate transformation from coordinate systemS s to coordinate systemS 2 Here,

Matrix transformation Eq (7.4.8) may be expressed by matrices (3×3) as ρ 2 (θ s , θ 2)=L 2n L nf L f s ρ s (θ s )+R

Profile 2 is determined by simultaneous consideration of Eq (7.4.12) and the scalar product (seeLitvin & Fuentes, 2004): n 2ãv (s2) 2 =L 2s n s ã ρ ˙ 2= f(θ s , θ 2)=0 (7.4.14)

Here,n s is the unit normal to the shaper andv (s2) 2 is the relative velocity, which is represented inS 2 by v (s2) 2 = ρ ˙ 2=(˙L 2n L nf L f s +L 2n L nf ˙L f s ) ρ s +R˙ (7.4.15) wherein ˙L 2n 

−x˙ (O f 2 ) cosψ 2−y˙ (O f 2 ) sinψ 2+(x (O f 2 ) sinψ 2−y (O f 2 ) cosψ 2) ψ ˙ 2 ˙ x (O f 2 ) sinψ 2−y˙ (O f 2 ) cosψ 2+(x (O f 2 ) cosψ 2+y (O f 2 ) sinψ 2) ψ ˙ 2

Here, dr 2 dθ 2 is obtained as dr 2 dθ 2

Term d 2 r 2 dθ 2 2 (which is needed for dà 2 dθ 2

) is obtained by differentiation of Eq.

Termà 2is obtained as (see Eq (2.5.1)) à 2=arctanr 2 (θ 2) dr 2 dθ 2

Termdà 2 dθ 2 is obtained as (see Eq (2.8.4)) dà 2 dθ 2

Figure 7.5.1 Illustration of the trajectory described by the tip of the shaper tooth during the relative motion between the shaper and the internal noncircular gear.

Termds(θ 2) dθ 2 is obtained as (see Eq (2.8.1)) ds(θ 2) dθ 2

In all previous derivations, expressionsr 1 (θ 1),dr 1(θ 1) dθ 1

, and d 2 r 1(θ 1) dθ 1 2 are known and depend on the type of centrode (elliptical, eccentric, modified elliptical).

Conditions of Nonundercutting of Planar Internal Noncircular

Approach A 133

The limiting condition of undercutting can be expressed through a series of nonlinear equations, which are essential for determining the critical pitch radius, ρ s ∗ The methodology presented relies on a systematic algorithm to solve these equations effectively.

(1) Point Mbelongs to the addendum circle of the shaper, and this condition is represented by f 1(θ s M , ρ s ∗ )= |r (M) s (θ s M , ρ s ∗ )| −(ρ ∗ s +1.2m)=0 (7.5.2) Here,|r (M) s (θ s M , ρ s ∗ )| (r sx M ) 2 +(r sy M ) 2 , and vectorr (M) s (θ s M , ρ s ∗ ) is represented by Eq (7.4.5) whereinρ s =ρ s ∗ is considered as a variable.

(2) PointKbelongs to the profile 2of the noncircular gear tooth, and this condition implies that equation of meshing Eq (7.4.14) is satisfied at pointK, f 2(θ s K , θ 2 K , ρ s ∗ )=0 (7.5.3)

(3) PointKbelongs to the addendum curve of the noncircular gear, and this condition is represented by f 3 (θ s K , θ 2 K , ρ s ∗ , θ σ K 2 )= |r (K) 2 (θ s K , θ 2 K , ρ s ∗ )| −(r 2 (θ σ K 2 )−m)=0 (7.5.4)

Here, (i) |r (K) 2 (θ s K , θ 2 K , ρ ∗ s )| (r 2x K ) 2 +(r 2y K ) 2 where vector r (K) 2 (θ s K , θ 2 K , ρ ∗ s ) is obtained by matrix transformation as r (K) 2 (θ s K , θ 2 K , ρ s ∗ )=M 2s (θ 2 K )r s (θ s K , ρ s ∗ ) (7.5.5)(ii) r 2 (θ σ K 2 ) is the amplitude of polar vector of the centrodeσ 2corresponding to pointKandθ σ K 2 is its polar angle.

(5) The trajectory of pointMin coordinate systemS 2 may be obtained by application of matrix transformation r (M) 2 (θ s M , ρ s ∗ , θ 2)=M 2s (θ 2)r s (θ s M , ρ s ∗ ) (7.5.7) whereinθ 2is the parameter of the curve that represents the trajectory ofM.

(6) Intersection of curve represented by Eq (7.5.7) and profile 2at pointKrepre- sented by Eq (7.5.5) provides a vectorial equation r (M) 2 (θ s M , ρ s ∗ , θ 2)−r (K) 2 (θ s K , θ 2 K , ρ s ∗ )=0 (7.5.8) that yields two additional scalar equations f 5(θ s M , θ s K , θ 2 K , ρ s ∗ , θ 2)=r 2x M −r 2x K =0 (7.5.9) f 6(θ s M , θ s K , θ 2 K , ρ s ∗ , θ 2)=r 2y M −r 2y K =0 (7.5.10)

(7) A system of six nonlinear equations f 1 , f 2 , f 3 , f 4 ,f 5 , f 6is obtained as f 1 (θ s M , ρ s ∗ )=0 (7.5.11) f 2 (θ s K , θ 2 K , ρ s ∗ )=0 (7.5.12) f 3(θ s K , θ 2 K , ρ s ∗ , θ σ K 2 )=0 (7.5.13) f 4(θ s K , θ 2 K , θ σ K 2 , ρ s ∗ )=0 (7.5.14) f 5 (θ s M , θ s K , θ 2 K , ρ s ∗ , θ 2)=0 (7.5.15) f 6 (θ s M , θ s K , θ 2 K , ρ s ∗ , θ 2)=0 (7.5.16) and may be solved numerically for determination of unknowns (θ s M , ρ s ∗ , θ s K , θ 2 K , θ σ K 2 , θ 2).

Approach B 134

Approach B is based on the results provided in Table7.5.1, developed by Litvinet al.

(Litvinet al., 1994), to determine the maximal number of teeth for various pressure angles in the case of internal circular involute gears (see Chapter 11 of Litvin &

The substitution of the internal noncircular gear's centrode σ 2 with a new circular gear centrode σ 2c, which has a pitch radius equal to the minimal radius of σ 2, is illustrated in Figure 7.5.2 The centrode σ 2c is firmly connected to system S 2c, resulting in the trajectory of point M in system S 2c forming an extended hypocycloid.

7.5 Conditions of Nonundercutting of Planar Internal Noncircular Gears 135

Table 7.5.1 Maximal number of shaper teeth for internal circular involute gears.

Pressure angle Generation method Gear teeth Shaper teeth

Figure 7.5.2 Illustration of the trajectory described by the tip of the shaper tooth during the relative motion between the shaper and the equivalent internal circular gear.

Number of teeth of the elliptical pinion, N 1 31 Number of teeth of the internal noncircular gear, N 2 62 Number of revolutions of the elliptical pinion, n 2

Approach B is based on the following algorithm:

(1) The minimal curvature radius of centrodeσ 2 of the internal noncircular gear, ρ 2,mi n , is obtained by consideration of (see Eq (2.8.8)) ρ 2(θ 2) r 2(θ 2) 2 + dr 2 dθ 2

, andd 2 r 2 (θ 2) dθ 2 2 have been represented in Section7.4.

(2) The number of teeth for the equivalent internal circular gear is obtained as

(3) The limiting value of the number of shaper teeth can be obtained by application of Table7.5.1(see Chapter 11 ofLitvin & Fuentes, 2004).

Numerical Example 136

An internal noncircular gear drive based on a conventional elliptical pinion is considered The gear data are shown in Table7.5.2.

Parameteraof the elliptical centrodeσ 1is obtained as (see Eq (4.3.26)) a= N 1 mπ

The profile parameter for point M is measured at 0.517701 radians, while the limiting value of the pitch radius of the shaper is 82.541688 mm For point K, the profile parameter is 0.172244 radians, with a generalized motion parameter of -0.139680 radians Additionally, the polar angle for the polar vector of point K is -0.031313 radians Lastly, the trajectory parameter for point M is recorded at 0.620299 radians.

7.5 Conditions of Nonundercutting of Planar Internal Noncircular Gears 137

Figure 7.5.3 Observation of undercutting whenN s B and nonundercutting whenN s A.

Center distanceEof the gear drive is obtained as (see Eq (7.3.6))

APPLICATION OF APPROACH A The solution of the set of the nonlinear equations corresponding to the Approach A provides the values of magnitudes (θ s M , ρ s ∗ , θ s K , θ 2 K , θ σ K 2 , θ 2) shown in Table7.5.3.

The limiting value of the number of shaper teeth is obtained as

In Figure 7.5.3, the trajectories of point M are illustrated for two scenarios: N s A with a radius of ρ s 82 mm and N s B with a different radius The detailed view A reveals that undercutting takes place in the case of N s B, while no undercutting is observed for N s A.

APPLICATION OF APPROACH B The minimal radius of curvature of centrode σ 2 is obtained atθ 2=0 (see Fig.7.3.1(a)) In this case, ρ 2 , mi n = 1

The number of teeth of the equivalent internal circular gear,N 2 ∗ , is obtained as

N 2 ∗ =2ρ 2 , mi n m F.794701 Applications of results shown in Table 7.5.1(see Chapter 11 ofLitvin & Fuentes,

2004) provides the following limiting value of the number of shaper teeth:

N s ∗ =1.004N 2 ∗ −9.1627.8199Approach B provides a more conservative result than the exact solution provided by Approach A.

8 Application for Design of Planetary Gear Train with Noncircular and Circular Gears

Introduction 138

A planetary gear train with noncircular and circular gears (Litvin & Ketov, 1949) may be applied in design for the following purposes:

(1) For providing a lesser pressure angle for each pair of applied gears of the discussed train.

The design of the function y(x), defined for the interval x2 ≥ x ≥ x1, is characterized by a derivative y that changes sign within this range This function will be derived as the output from a planetary gear train It is important to note that a function y(x) with a varying sign of y'(x) cannot be produced using a pair of noncircular gears, as discussed in Section 10.3.

(3) For obtaining as the output of the planetary train a product f 1 (x)ã f 2 (x) of two nonlinear functions.

Structure and Basic Kinematic Concept of Planetary Train 138

This article focuses on a planetary gear train consisting of two pairs of externally meshing gears The first pair includes gears 1 and 2, while the second pair comprises gears 3 and 4 In this setup, gear 1 is fixed in place, while gears 2 and 3, referred to as satellites, are mounted on a carrier These satellites possess two degrees of freedom: one in their motion with the carrier and the other in their relative rotation around the carrier.

The notation φ 41 (c) signifies the rotational transformation occurring from gear 1 to gear 4, with carrier c remaining stationary In this setup, gears 1 and 2, as well as 3 and 4, create a conventional gear train rather than a planetary system Thus, the relationship can be expressed as φ 41 (c) = φ 4(φ 3(φ 2(φ 1))).

In a planetary gear train, the angles of rotation for gears 3 and 2 are equal (φ 3 ≡ φ 2) since they do not rotate relative to one another Additionally, the notation φ 4c (1) represents the rotational transformation from the carrier (c) to gear 4, while gear 1 remains stationary According to the kinematics of this system, the relationship between these angles is expressed as φ 4c (1) = φ (c) 41 − φ 1 (8.2.2).

Planetary Gear Train with Elliptical Gears 139

Figure 8.2.1 Structure of planetary train with two pairs of gears and carrierc.

Equation (8.2.2) may be obtained as follows:

All four gears and the carrier rotate together as a rigid body, with the angle of rotation denoted as φ c, which defines the positions of both the gears and the carrier.

To reset gear 1 to its initial position, it must be rotated in the opposite direction of its initial movement This rotation angle is equal to the absolute value of the initial angle, |φ 1| = |φ c|.

2, the rotations of gears 1, 2, 3, and 4 are performed about their axes and the positions of the gears are determined by Eq (8.2.2).

The final positions of the gears and the carrier obtained by rotation at steps 1 and 2 are determined by Eq (8.2.2) This yields Eq (8.2.2), which may be represented as φ 4c (1) =φ (c) 41 −φ 1=φ 4(φ 3(φ 2(φ 1)))−φ 1 (8.2.3)

Equation (8.2.3) is derived under the assumption that gears 1 and 2, as well as gears 3 and 4, are engaged in external meshing This methodology can also be applied to scenarios where the gears in the planetary train exhibit a combination of external and internal meshing.

8.3 Planetary Gear Train with Elliptical Gears

The planetary gear system depicted in Figure 8.3.1(a) features two pairs of identical elliptical gears, labeled as pairs 1 and 2, and 3 and 4 This design aims to enhance the variation in the angular velocity ratio between the input and output links of the gear train.

The planetary gear train illustrated in Figure 8.3.1 features two pairs of elliptical gears, demonstrating the initial position of carrier c along with gears 1, 2, 3, and 4 After the carrier rotates by angle φc, gears 2, 3, and 4 shift positions while gear 1 remains stationary It is important to note that the pressure angle α12 is constrained for each pair of elliptical gears, as detailed in Section 3.4.

The derivation of transmission function φ 41 (c) (see Eq (8.2.1)) is performed as follows:

(i) Transmission functionφ 2(φ 1) is determined as (see Section4.3.2.1) tan φ 2

(ii) Similarly, we obtain that tan φ 4

We take into account in further derivations that φ 3 ≡φ 2 Gears 2 and 3 are mounted on the carrier and perform rotation with respect to the carrier with the same angular velocity.

(iii) Equations (8.3.1) and (8.3.2) yield φ 41 (c) =2 arctan

(8.3.3)The meaning of Eq (8.3.3) is that application of two pairs of elliptical gears is equivalent to application of one pair of elliptical gears with a very large eccentricity.

Planetary Gear Train with Noncircular and Circular Gears 141

Figure 8.3.2 Illustration of (a) transmission functionφ 4(φ c ); (b) derivative functionm 4c ω 4 /ω c

However, large eccentricity is accompanied with a large pressure angle (see Section

3.4), but this is avoided by application of a planetary gear train with two pairs of elliptical gears.

The output of the planetary train, represented as φ 4(φ c ), is influenced by the reverse motion of gear 4 while the carrier rotates in the same direction, as illustrated in Fig 8.3.2(a) Additionally, Fig 8.3.2(b) indicates that the derivative function m 4c (φ c ) changes its sign within the generation interval, a phenomenon that cannot occur when generating a function using a single pair of noncircular gears.

8.4 Planetary Gear Train with Noncircular and Circular Gears

In Chapter 10, it is established that for the function y(x) to be generated, its derivative y'(x) must remain positive within the generation interval To overcome this limitation, an alternative function can be created: y₁(x) = y(x) + bx, ensuring that y₁(x) remains greater than zero throughout the specified interval.

To derive the output function y(x) with a varying sign of its derivative, it is essential to subtract the linear function y2(x) = bx from the function y1(x) This subtraction is achieved through the application of a spiral bevel gear differential, leading to the function y(x) = sin(x).

Similar generation may be accomplished by application of a planetary gear train as follows:

(1) Assume that noncircular gears 1 and 2 (Fig 8.2.1) generate the function Eq.

(8.4.1) Circular gears 3 and 4 are designed with gear ratio 1, andφ (c) 41 is proportional toy 1(x)=y(x)+bx.

(2) Taking into account Eq (8.2.2), we obtain that the output of the planetary train, φ 4c (1) , is proportional to functiony(x)=sinx.

9 Transformation of Rotation into Translation with Variation of Gear Ratio

Introduction 143

This book explores the meshing of noncircular gears with racks in two distinct contexts: first, where the rack acts as a generating tool, and second, where the combination of the rack and noncircular gear creates a mechanism for motion transformation.

Figure 9.1.1illustrates that for previous case (ii), rotation with angular velocity ω 1 is transformed into translation with velocity v 2 and the ratio ω 1 /v 2 varies.

The mechanism applied for such transformation of motion (Fig.9.1.1) is formed by a noncircular gear 1 and rack 2 The centrodes 1 and 2 are curves determined by functions 2(φ 1), denoted asF(φ 1).

The instantaneous pointIof tangency of centrodes (Fig.9.1.1) is determined as follows:

(1) The derivativeF (φ 1) of functions 2 (φ 1)=F(φ 1) is obtained as

(9.1.1) wherev 2= ds 2 dt is the instantaneous velocity of translation of rack 2, and dφ 1 dt is the instantaneous velocity of rotation of noncircular gear 1.

(2) The instantaneous center I of rotation of gear 1 and rack 2 (in relative motion) is located at a point of line O 1 n Here, O 1 is the center of rotation of the gear

1, and O 1 n is drawn perpendicular to velocity v 2 of translation The relative velocityv 12 =v 1−v 2is equal to zero at point I.

(3) The location|O 1 I|of the instantaneous center of rotation on lineO 1 nis determined as v 2 =v 1 = ω 1×O 1 I

Figure 9.1.1 Centrodes 1 and 2 of a noncircular gear and a rack.

Determination of Centrodes of Noncircular Gear and Rack 144

Centrodes 1 and 2 are traced out by the instantaneous centerIof rotation in coordinate systemsS 1 andS 2 that are rigidly connected to gear 1 and rack 2, respectively.

(i) Centrode 1 is a polar curve determined as r 1(φ 1)=F (φ 1) (9.2.1)

The polar axis is O 1 A 1 (Fig.9.2.1(a)), andr 1(φ 1) is the current position vector of centrode 1.

(ii) Centrode 2 (Fig.9.2.1(b)) is represented in coordinate system (x 2 ,y 2 ) by x 2 φ 1

Application of Mechanism Formed by a Noncircular Gear and

Consider that the mechanism formed by a noncircular gear and a rack shown in Fig.

9.1.1is applied for generation of functiony= f(x),x 2 ≥x ≥x 1 as follows:

(i) Angleφ 1of rotation of noncircular gear and translation s 2of rack are represented as φ 1 =κ 1(x−x 1 ) (9.3.1) s 2 =κ 2(y−y 1 )=κ 2[f(x)− f(x 1 )] (9.3.2) whereκ 1andκ 2are scalar coefficients Equations (9.3.1) and (9.3.2) determine functions 2(φ 1) parametrically.

9.3 Application of Mechanism Formed by a Noncircular Gear and Rack 145

Figure 9.2.1 Toward derivation of centrodes 1 and 2: (a) centrode 1 of noncircular gear;

(ii) Function of gear ratio is obtained as

(iii) Gear centrode (Fig.9.2.1(a)) is determined parametrically as φ 1=κ 1(x−x 1) (9.3.4) r 1(x)=κ 2 κ 1 f (x) (9.3.5)

(iv) The centrode of the rack is determined in coordinate system (x 2 ,y 2 ) as x 2(x)=κ 2[f(x)− f(x 1)] (9.3.6) y 2(x)=r 1−r 10= κ 2 κ 1

Axisx 2 (x) coincides with the direction of translational motion of the rack We recall that derivative f (x) of the generated function y(x) is to be a smooth and differentiable function.

Figure 9.3.1 Illustration of centrodes 1 and 2 determined for generation of functiony(x) κ 3 x 2

NUMERICAL EXAMPLE 9.3.1 Consider generation of function y(x)=κ 3 x 2 ,x a ≥x≥ x b , wherex a 0 m,x b =1,200 m,κ 3=2.5×10 −6 s/m Centrode 1 is an unclosed curve Coefficientκ 1=0.4ã(π/360) 1/m;κ 2=1/12 1/s The units ofx andyare in meters (m) and seconds (s), respectively.

The angle of rotation of gear and displacements 2 of the rack are represented as φ 1(x)=κ 1(x−x a ) (9.3.8) s 2(y)=κ 2(y−y(x a )) (9.3.9) whereκ 1andκ 2are the assigned coefficients The gear centrode is represented by φ 1(x)=κ 1(x−x a ) (9.3.10) r 1(x)= ds 2 dφ 1

It is easy to verify thatr 1(φ 1) is an Archimedes spiral.

The centrode of the rack is represented in coordinate system (x 2,y 2) by x 2 =κ 2 κ 3(x 2 −x a 2 ) (9.3.12) y 2 =r 1(x)−r(x a )=2κ 2 κ 3 κ 1

Centrodes 1 and 2 are represented in Fig.9.3.1.

10 Tandem Design of Mechanisms for Function

Generation and Output Speed Variation

Historically, the synthesis of planar linkages has been a focal point of research for numerous esteemed scholars, including Artobolevski et al (1959), Burmester (1888), Chebyshev (1955), and Freudenstein.

1955) The success of modern technology for the manufacture of noncircular gears has inspired researchers to apply noncircular gears for generation of functions

(Dooner, 2001;Litvinet al., 2008;Ottavianoet al., 2008;Modleret al., 2009;Litvin et al., 2009).

This chapter discusses the generation of functions using a single pair of noncircular gears, as well as a novel method involving a multigear drive with noncircular gears It highlights that the function y(x), where x ranges from x1 to x2, must be a monotonically increasing function with a positive derivative, y'(x) > 0 Additionally, if the derivative y'(x) varies in sign within the derivation interval, an algorithm for function generation is presented in Section 10.3.

The use of a multigear drive featuring noncircular gears for function generation offers several key benefits: it delivers a cohesive impact on output results, ensures optimal force transmission conditions, and enhances performance by significantly increasing various design parameters.

The specific characteristic of the proposed approach for application of a multigear drive is illustrated as follows:

(a) Figure10.1.1shows the centrodes of a gear drive wherein only a single pair of gears is applied (Fig.10.1.1(a)) for generation of functionψ(α) (Fig.10.1.1(b)).

In the analysis presented in Figure 10.1.2(a), two pairs of gears are utilized to generate the function ψ(α) depicted in 10.1.2(b) This results in the transmission function of centrodes 1 and 2 being defined as β(α) ≡ g1(α), while the transmission function of the gears is also established.

The shape of a pair of centrodes that generate the function g i (α) is influenced by the derivative dg i /dα This relationship is illustrated in Figures 10.1.3(a), (b), and (c), which depict the derivatives of transmission functions based on the design parameters.

148 Tandem Design of Mechanisms for Function Generation and Output Speed Variation

Figure 10.1.1 (a) Illustration of centrodes of a single pair of noncircular gears; (b) function β(α)≡ψ(α) generated by centrodes. on application of (i) one pair of gears (Fig.10.1.3(a)), (ii) two pairs of gears (Fig.

10.1.3(b)), and (iii) three pairs of gears (Fig.10.1.3(c)) Centrodes of a multigear drive with three pairs of gears are shown in Fig.10.1.4.

The illustrations in Fig 10.1.3 demonstrate that the derivative of the transmission function significantly decreases in a multigear drive Additionally, utilizing a multigear drive enhances the overall effectiveness of the final output.

The theory of design of a multigear drive has been developed and is represented here This chapter also covers the tandem design of a planar linkage coupled with

The generation of the function \( z = \ln u \) for \( 1 < u < 100 \) is achieved using two pairs of noncircular gears, allowing for a wider range of output speeds Additionally, the function \( \psi(\alpha) \) is utilized for generation, with \( g_1(\alpha) \equiv \beta(\alpha) \), and the relationship \( \delta(\gamma) \approx \beta(\alpha) \) is established This design approach is exemplified through various case studies.

(a) Tandem design of a double-crank mechanism coupled with two pairs of noncircular gears.

150 Tandem Design of Mechanisms for Function Generation and Output Speed Variation

Figure 10.1.3 Illustration of derivative function for (a) a single gear pair; (b) two pairs of gears; (c) three pairs of gears.

Figure 10.1.4 (a) For generation of functionz=lnu, 1 0 is evident since b > 1 The convexity criteria for centrodes 1 and 2 are outlined in Section 2.9, specifically through inequalities in Eq (2.9.2) and (2.9.3) Notably, the convexity condition for centrode 1 is established as follows.

Tiêu đề	Noncircular Gears: Design and Generation
Tác giả	Faydor L. Litvin, Alfonso Fuentes-Aznar, Ignacio Gonzalez-Perez, Kenichi Hayasaka
Trường học	University of Illinois at Chicago
Chuyên ngành	Mechanical Engineering
Thể loại	Book
Năm xuất bản	2009
Thành phố	Chicago

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Số trang	222
Dung lượng	3,43 MB