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New Developments in Variable Radius Gears Using Constant Pressure Angle Teeth

G.A. Danieli*

Dipartimento di Meccanica - Università degli Studi della Calabria , D. Mundo

87030 Arcavacata di Rende (CS) – Italia

Abstract

The purpose of this work is the presentation of a new methodology which greatly increases the

contact ratio between the teeth of non-circular gears, using a pressure angle constant for any

given tooth, but variable from one to the next.

At first an algorithm is proposed to obtain the kinematic synthesis of a pair of variable radius

gears, generating a prescribed law of motion. Then, the concept of constant pressure angle non-

circular helical gears is introduced and a method to generate a CAD representation proposed.

Such gears insure a more progressive meshing, with a softer and less noisy working. In

addition, a greater load capacity and duration are obtained.

Keywords: Variable Radius Gears, Helical Gears, Constant Pressure Angle Gears

1 Introduction

The paper presents new results of a research on variable radius gears, as defined by the

equation also presented in a previous paper [1], and patented under Italian law [2], which

describes the meshing of teeth working under a constant pressure angle.

Although variable radius gears are not very diffuse, they are essential for automatic equipment

in printing presses, textile industry, packaging machines, quick-return mechanisms, pumps, flow

meters, flying shears and other applications where there is a need for speed variation control

during the working cycle if a purely mechanical system is desired [3, 4]. Recently Emura and

Arakawa extended the use of elliptical gears to steering mechanisms [5], while Dooner

implemented a software package to generate non circular gears and proposed their use to reduce

torque and speed fluctuations in rotating shafts [6, 7].

Except for the case of elliptical gears, until now, variable radius gears are limited in their

diffusion due to production costs and the lack of generalised knowledge about production

methodology, resulting in the fact that only few people have the know-how necessary for design

*e-mail address: danieli@unical.it; tel.+39-0984-494824 – Fax.+39-0984-494673; address: Dipartimento di Meccanica – Ponte P. Bucci, 87036 Arcavacata di Rende (Cs) Italy

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and manufacture. Up to the present, few studies have been devoted to this argument, and some

methods have been developed to design and to analyze undercutting in elliptical gear tooth

profiles, generated by rack cutters [8] or shaper cutters [9, 10]. All of these methods use a cutter

moving without sliding on a pitch line, but the commonly used approaches are the contact line

method [11] and the envelope method [12].

The aim of the research here presented was to acquire the ability to design and produce

strange-looking gears such as the ones designed and built by Doege et al. [13], but using the

present methodology, which should allow the entire project to be examined from every point of

view via software before actually building the gears. A brief summary of the derivation of the

differential equation describing meshing of constant pressure angle non-circular gears is then

presented.

Since the aim was generality, a criterion had first to be found to determine the pitch lines that,

coupled to a crank and rod arrangement, would cause the speed of the small end of the connecting

rod to move following a predetermined velocity law. For this reason, the first goal was to develop

methods to easily determine the pitch lines and the general criteria they have to obey to. The

kinematic analysis of non circular pitch lines, coupled to a crank and rod arrangement, has been

implemented, using methodologies previously reported by Litvin [14] and others [15, 16]. Thus a

brief summary is given in the second paragraph, where pitch lines similar to those of Doege [13]

are also shown, and by which high speed during the return stroke is obtained. This procedure is

not fully automated, since it relies on the ability of the designer to solve some continuity

problems on the pitch line determination. However, a good level of automation was reached.

Once determined the pitch lines, during tooth profile generation along these curves, severe

problems regarding the contact ratio were encountered, due to the particular shape of these pitch

lines. This led to the development of the work actually shown in the next section, where the idea

to vary the pressure angle from one tooth to the next (but keeping it constant over the entire

tooth) is presented.

In order to increase the contact ratio, particularly severe in non-circular gears, a procedure to

design variable radius helical gears was invented and patents applied for [17, 18]. This is

described, and the effect of both variable pressure angle application and helical construction on

the contact ratio is presented. Finally, two matching helical gears were designed applying the

method outlined in this paper, and built using a rapid prototyping machine.

2 Analytical Determination of Meshing of Constant Pressure angle Profiles

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Let A be a generic contact point between the teeth when these are rotated by the angle θ with respect to the origin of the individual tooth (i.e. the point at which the tooth crosses the pitch line for the given tooth, labelled CC in Fig. 1). Let the origin be placed in the centre of gear 1, with x oriented toward the centre of gear 2. Note that angle θ does not generally coincide with angle AO1C.

By definition of the constant pressure angle, point A will have to be located along the line inclined by π/2 - α with respect to the x axis, passing through C, the instantaneous swing centre in the motion of 1 with respect to 2. yA being the ordinate of point A, the corresponding abscissa is:

xA = r(θ) + yA tan α (1)

Let A' be the next contact point between teeth, following a further infinitesimal rotation dθ. This point will have to be located at the interception between the line inclined by π/2 - α starting from C', the swing centre between the pitch lines rotated by θ + dθ with respect to the tooth origin and the line inclined by dθ - α starting from B', the position assumed by point A as a consequence of the elemental rotation dθ.

Note that the angle formed by the A’B’ line with respect to the x axis derives from the fact that the tooth surface must be orthogonal to the trust force (- α), while the new position derives from the elemental rotation (+ dθ).

The resulting equation can be written thus: (yA' - yC') / (xA' - xC') = tan (π/2 - α) = 1 / tan α (2)

With the co-ordinates of point C' given by: yC' = 0; xC' = r(θ) + dr/dθ dθ (3)

where dr/dθ is the derivative of the radius of the pitch line with respect to θ. After substituting these last two in (1), and solving it with respect to xA', one obtains

xA' = yA' tan α + r(θ) + dr/dθ dθ. (4)

The line A'B', as explained before, can be described by the equation: (yA' - yB') / (xA' - xB') = tan (dθ - α) = (tan dθ - tan α) / (1 + tan dθ tan α) (5)

In order to proceed, the identification of the co-ordinates of point B', derived from the elemental rotation of dθ from point A, is now needed. Thus the equations describing co-ordinate change due to a rigid rotation about the origin have to be used; note that in this case cos dθ = 1, while tan dθ = sin dθ = dθ; and hence:

xB' = xA cos dθ - yA sin dθ: yB' = xA sin dθ + yA cos dθ (6)

and substituting (1) in (6) and rearranging, the previous expressions are turned into:

xB' = r(θ) + yA (tan α - dθ): yB' = r(θ) dθ + yA (dθ tan α + 1) (7)

Substituting (7) and (4) in equation (5) the result is obtained: (yA' - yA - r(θ) dθ - yA dθ tan α) (1 + dθ tan α) =

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((yA' - yA) tan α + dr/dθ dθ + yA dθ) (dθ - tan α) (8)

observing that dy = yA' - yA and neglecting second order differentials, one obtains:

dy /dθ = cos2 α [r(θ) - dr/dθ tan α] (9)

This equation is the key equation of the method, allowing the generation of teeth profiles on wheel pairs of any shape, provided that r(θ) and dr/dθ are known and continuous. This, in the case of constant radius (dr/dθ = 0) reduces to:

dy = ρ cos α dθ (10)

which is the correct expression for the action line in the involute, as shown in the quoted previous paper [1]. From this point on, rotating the point by - θ about the centre of gear 1, and by - δ (the corresponding angle on gear 2) about the centre of gear 2, the points of both teeth profiles are obtained.

3 Generation of variable radius pitch lines

When generating variable radius pitch lines for gears, in order to obtain a predetermined

angular velocity curve for the driven gear, starting from a constant speed of the driving gear, it is

necessary to respect a series of constraints that will be indicated here below.

As previously mentioned, teeth profiles will be computed with the method previously

described [1-2], once the pitch lines radii and hence the variable transmission ratio is given. It is,

however, clear that in order to obtain a continuous and cyclic variation of the transmission ratio,

the ratio between the lengths of the pitch lines must be a rational number, and consequently each

tooth will have to mesh always with another specific tooth, or with a fixed number of teeth, if the

average transmission ratio is different from one.

For the sake of simplicity, at the moment only the case in which the pitch lines have an equal

length will be considered. Therefore, calling 1 and 2 the two curves, it is possible to write:

∫=∫ππ

δδθθ2

0

2

0)(2)(1 dRdR (11)

A second condition imposed is the fact that the distance between the axes of the gears must be

constant, and this can be written as:

))((1 δϑR + )(2 δR = r∆ (12) valid for corresponding values of δ and θ . Notice that once the pitch line has been assigned, δ

and θ are dependent variables.

As the time taken by gears to cover a complete revolution must be the same, the assumption

can be made that:

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( ) δδ

θω

π πddT ∫ ∫ Ω

==2

'0

2

0

11 (13)

By manipulating this equation a basic condition for the building of variable radius gears can be

written as:

2)(

1 )(

2

0

2

0∫∫ ==

Ω

ππ

πδδτ

δδ

ω dd (14)

while it results that:

( ) πθθτπ

22

0=∫ d (15)

In addition, since a certain law of variation of the angular speed is given to gear 2 (or of the

transmission ratio τ), while assuming angular speed of the driving gear to be constant,

rearranging the equation one obtains:

Tdπ

θθ

πω π2

)(

22

0

=∫

Ω

= (16)

where T is the period of rotation of both gear 2 and gear 1. Note that, if the previous

expression is integrated only over a limited angle, the equation is obtained:

( ) θθ

ωδϑ

d∫Ω

=0

1 (17)

which represents the relation between angle δ and angle θ as shown in fig. 2.

Naturally, by definition of pitch lines, it is possible to write:

)(*)(2*))((1 δδωδθ Ω= RR (18)

and combining this with equation 12, one gets:

)()(2

δωωδ

Ω+⋅∆

=rR (19)

Having reviewed the constraints to be satisfied in the determination of the pitch lines, it is

possible to pass to the study of the equations describing the angular speed of a driven gear Ω(δ),

used to drive a crank and rod mechanism, given the speed desired for the small end of the

connecting rod.

In fact, while in the first article a particular epicyclic train was used to obtain a linearly

reciprocating law of motion, it was anticipated that a crank and rod mechanism would be

probably more efficient. In this case, however, the finite length of the rod poses some problems,

as it is well known.

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Considering fig. 3, let D be the point identifying the small end of the connecting rod.

Defining λ as the ratio between the crank radius and the connecting rod length and considering

the relation between angle ξ and angle δ, one can easily obtain that the displacement sD of this

point, starting from its off condition, when angle δ is zero, is given by:

( )

−−−−= )(sin111)cos(1)( 22 δλ

λδδ cD rs (20)

from which, by differentiation, the relation between the angular speed of the driven gear and the

speed of the small end of the connecting rod is obtained:

[ ])cos()(sin1)sin(

)(sin1)()(

22

22

δλδλδ

δλδδ

⋅−−⋅⋅

−⋅=Ω

c

D

rv (21)

As a consequence, the angular speed Ω(δ) can be obtained either analytically, defining the

function vD(δ), or numerically, imposing, in the proper angles, the desired values for the

accelerations and decelerations and deriving the requested speed profile from these.

As the speed of the small end of the connecting rod is not a monotonic function (behaving

more like a sinusoidal function), the function cannot generally be inverted all over its entire

domain, but only in some parts of it. Then it is necessary to use a numeric derivation and an

iterative process by which the curve continuity can be obtained.

It is now possible to show the results of the application of this method, assigning a sinusoidal

speed to point D. As should be clear, the procedure first requires the displacement law of the

small end of the connecting rod to be assigned, as a consequence generating the two pitch lines,

upon application of the equations described above. Fig. 4 shows the pitch lines determining a

sinusoidal motion of point D, where λ has the previously defined meaning.

The next step is to generalise this method creating a speed curve which could trace the desired

one for the small end of the connecting rod, using a method for designing the shapes of the cams,

to limit vibrations.

In order to do so, the second derivative of the acceleration (jount) is defined as constant for

given periods within the working cycle. By integrating this twice, acceleration is obtained, also

presenting the first derivative (jerk) continuous. By further integration the speed is obtained and

then the displacement as a function of θ , which, by definition, grows linearly with time. The

displacement computed in this way is supplied to equation 17 which allows the determination of

angle δ, and, as a consequence, of the transmission ratio τ as a function of time. The process is

summarised in fig. 5.

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It’s obvious that the previously defined conditions are necessary for the gears to be able to

describe the requested motion, but they do not guarantee that the gears obtained are regular and

able to mesh. Thus, all the controls explained in the first part of this paragraph have to be applied.

Figure 6 finally shows the results of the Doege-like pitch lines, as obtained by a procedure which

uses the formula described above to generate the desired velocity curve [13].

4 Generation of the teeth profiles.

As previously shown, once the r(θ) function is known, it is possible to generate both the right

and the left profile of each tooth. Fig. 7 shows the calculated points of the teeth profiles, with

respect to one of the pitch lines represented in fig. 4 and with a constant pressure angle α=26°. In

this application both meshing gears have 36 teeth.

Because of the non-circular shape of the pitch lines, it can be noted that there are critical

conditions in which the procedure gives one or a few points per tooth. In order to obtain a better

determination of teeth profiles, a subroutine was introduced that calculates the angle between the

actual normal line to the pitch line and the horizontal axis, which is equal to the angle that the

tangent line makes with the vertical axis (Fig. 8).

Evaluating the difference between this angle and the one characteristic of a circular pitch line

(βir,l and ζ in fig. 8), a value to correct, by addition or subtraction, the inclination of the pressure

line is obtained. As a consequence, the pressure angle is assumed as constant within a certain

tooth, but it varies from tooth to tooth, being instead constant the pressure angle measured from

the tangent to the pitch line at the intersection point with the tooth profile [19].

It is clear that two meshing teeth on different gears must have the same pressure angle value,

and so they will receive the same correction.

Fig. 9 shows the teeth profiles generated starting from the same pitch line, but with the

correction described above. The average pressure angle value is α=26. Fig. 10 shows an enlarged

image of some teeth presenting the integration points obtained with the two methods, and the

advantage is evident. In fact, since the integration interval is constant, more points present on a

tooth profile means more transmission duration. Since the modified pressure angle causes more

points to be present on one side of the tooth, where they were fewer, and less points on the other

side, where there were even too many, equilibrating the number of points between the two tooth

profiles, then the situation is judged preferable.

Naturally, the improvements deriving from the correction of the pressure line inclination are

particularly evident when the pitch lines are very different from a circular shape, as a greater

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number of points belonging to the teeth profile is obtained, causing therefore an increment in the

contact ratio.

5 Helical Gears

If the teeth of a gear develop according to a spiral shape around a cylinder, the gear is defined

as helical. The main characteristic of such gears is that the teeth mesh progressively with a softer

and less noisy meshing. In addition, they can transmit a greater load and have a longer life.

Helical gears can be considered as consisting of a great number of thin cylindrical gears rigidly

rotated, thus creating a spiral along the gear axis.

The previous description is not valid for variable radius gears, as the solid built starting from

the pitch line would not be a cylinder if applying the rules previously described. In fact, if the

elemental step should be rigidly rotated, an irregular volume would be obtained. Also a

theoretical error is evident, as along the same meshing line, points with different transmission

ratios would be found, thus having different tangential speeds.

The primitive volume that must generate the variable radius gear has to be obtained by

extrusion of the pitch line (fig. 11), and the teeth must envelop this volume, with a spiral shape

giving the same benefits of traditional helical gears. This is obtained by moving the point of

initial tooth integration along the pitch line, determining a series of tooth profiles, all being

characterised by the same pressure angle. In order to achieve this goal, the basic method of tooth

profile generation is repeated many times with a starting point of integration shifted of a certain

angle along the primitive. As a consequence, a suitable number of sections of each gear with a

plane normal to the axis of rotation is obtained. By enveloping these sections both sides of each

tooth are determined, thus generating profiles which develop along a helical shape.

To calculate the offset angle value (difference between the integration starting angles on two

consecutive sections), it is necessary to assign the helix angle and to establish their distance. The

former is a design parameter whose value has to be chosen according to dynamic considerations,

while the latter must guarantee the best compromise between need for definition of the teeth

lateral surface and reduced computational weight. Once chosen this angular increment on the

driving gear, the corresponding value on the driven gear must be computed. In fact, for both

traditional and variable radius gears, the meshing profiles must have opposite helical directions.

The value of rotation that the second gear must receive is not generally equal to the one of the

first gear, as in this case the ratio arc / angle is not constant like with circular gears. Using the

same angle, with the opposite value, the detachment of the primitives would be obtained. Thus

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the length of the arc was calculated, defined on the first gear pitch line by the new integration

starting point, and then the corresponding angle belonging to the second gear was evaluated.

Fig. 12 shows forty-three sections spaced out in a uniform way along the gear thickness. The

distance between the extreme sections identifies the gear thickness. Since a helix angle of 25

degrees and a thickness of 20 mm were chosen as design parameters, an offset angle of 0.00567

radiants between the first tooth starting points on two consecutive sections was imposed. The

three-dimensional representation of the gears was generated by a CAD program, PRO-

ENGINEER, and the resulting solid was supplied to a rapid prototyping machine that "created"

the gears shown in fig. 13.

In order to show the improvements obtained with the methods described in the previous

paragraphs, an ad hoc formula has been introduced to evaluate the contact ratio:

( )2

1,,1 +− ∆+∆−∆

∆=

iiiii

irc

θθθ

θ (22)

where ∆θi is the primitive rotational angle variation from the beginning to the end of the contact

along the ith tooth, while ∆θi-1,i corresponds to variation from the beginning of meshing on the ith

tooth and the end on the i-1th one (zone in which there is a double contact between teeth) and

similarly ∆θi,i+1 from beginning on the i+1th tooth and the end on the ith one.

Fig. 14 shows the comparison of the contact ratio relative to the gears depicted in figg. 7, 9

and 13. As can be seen, the new approach, characterised by the variable pressure angle seems less

satisfactory, in terms of contact ratio, than the previous one in a small portion of the gear, thus

proving that the chosen basic pressure angle value is not optimal under this point of view in that

region. The helical gear is, as obvious, the best solution, as the contact ratio pattern is shifted

upwards of an about constant value of one with respect to corresponding spur gears.

CONCLUSION

The paper shows the actual results of the research developed at Calabria University on non-

circular gears. Though the study case presented in this paper is relative to a pair of elliptical pitch

lines, the method allows to build any helical gear, thanks to the procedure which varies the

pressure angle according to the local primitive shape. Moreover, if it is very stimulating to find a

solution for the problem of generating helical gears of non circular shape, on the other hand a

long way is expecting the researcher before he is able to instruct a numerical control milling

machine to actually build them. Also the effect of mounting the gears at the wrong distance, the

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resistance of the teeth, and the effect of vibrations will have to be investigated, both theoretically

and experimentally.

However, if the work ahead is impressive, the work done so far is equally important, since

now it is clear that the building of helical non-circular gears is feasible on primitives of any

shape.

Nomenclature:

α pressure angle

βil,r value of θ at which the left or right profile of the i-th tooth crosses the pitch line

γil,r value of δ at which the left or right profile of the i-th tooth crosses the pitch line

δ rotational angle on gear 2

θ rotational angle on gear 1

ζ angle between the normal to the pitch line and the horizontal axis

λ ratio between the crank radius and the connecting rod length

ξ angle of inclination of the rod with respect to the line joining the centres

ρ radius of the base circle of a constant radius gear

τ instantaneous value of the transmission ratio between (constant or non-constant) radius

gears

ω speed of the driving gear

Ω speed of the driven gear

A actual mesh point

A' next mesh point

B' point obtained from a rigid rotation dθ of point A

Cc point in which the profile of the tooth crosses the pitch line (initial point of tooth profile

generation)

C actual contact point between pitch lines when teeth are in contact in A

C' actual contact point between pitch lines when teeth are in contact in A'

cr contact ratio

D point representing the centre of the small end of the rod

∆r distance between the axes of the two gears

O1 absolute centre of rotation of the pitch line of gear 1

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O2 absolute centre of rotation of the pitch line of gear 2

nz number of teeth

r radius of the pitch line

rc crank radius

REFERENCES

1. Danieli, G.A., 2000, "Analytical Description of Meshing of Constant Pressure Angle Teeth

Profiles on a Variable Radius Gear and its Applications", Journal of Mechanical Design,

ASME Transactions,Vol. 122, pp. 123-129

2. Danieli, G.A. "Profilo di un Dente ad Angolo di Spinta Costante per Ruote a Primitiva

Generalizzata e Relative Applicazioni", Italian Patent Application No. CS96A00010,

deposited 25/7/96 granted 23/10/00, No. 1303048.

3. Anonymous, 1972, “Elliptical Gears are Back and More Versatile than Ever”, Product

Engineering, pp.34-38

4. Chironis, N.P., 1965, “Mechanisms, linkages, and mechanical controls”, New York,

McGraw-Hill

5. Emura, T. and Arakawa, A., 1992, “A New Steering Mechanism Using Non Circular

Gears” JSME International Journal, Series III, Vol.35, No.4, pp.604-610

6. Dooner, D. B.,1997, “Use of Non-circular Gears to Reduce Torque and Speed Fluctuations

in Rotating Shafts” ASME Journal of Mechanical Design, Vol.119, pp.299-306

7. Dooner, D. B., and Seireg, A., 1995, “The Kinematic Geometry of Gearing”, John Wiley

and Sons, Inc., New York

8. Chang, S.L., and Tsay, C.B., and Wu, L.I., 1996, “Mathematical Model and Undercutting

Analysis of Elliptical Gears Generated by Rack Cutters”, Mechanism and Machine Theory,

Vol.31, No.37, pp.879-890

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9. Chang, S.L., and Tsay, C.B., 1998, “Computerized Tooth Profile Generation and Undercut

Analysis of Noncircular Gears Manufactured With Shaper Cutters”, Journal of Mechanical

Design, ASME Transactions,Vol. 120, pp. 92-99

10. Bair, B.W., 2002, “Computerized Tooth Profile Generation of Elliptical Gears

Manufactured by Shaper Cutters”, Journal of Materials Processing Technology, Vol. 122,

pp. 139-147

11. Litvin, F.L., and Tsay, C.B., 1985, ”Helical Gears with Circular Arc Teeth: Simulation of

Conditions of Meshing and Bearing Contact”, ASME Journal of Mechanisms,

Transmissions and Automation in Design, Vol.107, pp.556-564

12. Tsay, C.B., and Hwang, G:S:, 1994, “Application of the Theory of the Envelope to the

Determination of Camoid Profiles with Translating Followers”, ASME Journal of

Mechanical Design, Vol.116, pp.320-325

13. Doege, E. and Hindersmann, M., "Optimised Kinematics of Mechanical presses with

Noncircular Gears", Annals of the CIBP, Vol. 46/1/97, pp.213-216

14. Litvin, F., 1994, "Gears Geometry and Applied Theory", Prentice Hall, New Jersey

15. Figliolini, G., Lanni, C., and Ceccarelli, M., 2000, "On Kinematic Synthesis of Non-

Circular Gears", Proceedings of the International Conference on Gearing, Transmission,

and Mechanical Systems, Nottingham 2000, pp.283-292

16. Tong, S.H., and Yang, D.C.H., 1998, “Generation of Noncircular Pitch Curves”, ASME

Journal of Mechanical Design, Vol.120, pp.337-341

17. Danieli, G.A. "Ruote dentate elicoidali a raggio variabile e ad angolo di spinta costante",

Italian Patent Application No. CS99A0019, deposited 15/12/99

18. Danieli, G.A. Variable radius helical gears", PCT/IT00/00524 of 15/12/2000, WO

01/4469/3 of June the 6th 2001

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19. Lozzi, A., 2000, “Non-circular gears-graphic generation of involutes and base outlines”,

IMechE, Vol.214, pp.411-422

Figure Captions

Fig. 1 Schematic representation of the tooth mesh for constant pressure angle on variable

radius gears

Fig. 2 Schematic representation of the two pitch lines

Fig. 3 Schematic representation of the investigated mechanism

Fig. 4 Pitch lines for λ=0.5 and r=100 mm, generating sinusoidal motion of point D

Fig. 5 Velocity profile determination through integration of the second derivative of the

acceleration

Fig. 6 Pitch lines of the Doege – like gears

Fig. 7 Teeth profiles with constant pressure angle (nz=36, =26° and r=100 mm)

Fig. 8 Schematic representation of the scheme to calculate correction

Fig. 9 Teeth profiles with pressure angle varying from tooth to tooth

Fig. 10 Blow up of the tooth profiles in the case of constant pressure angle and variable

pressure angle

Fig. 11 Three dimensional primitive for a variable radius gear

Fig. 12 Representation of tooth profiles for helical gears

Fig. 13 Variable radius helical gears built by a rapid prototyping machine

Fig. 14 Contact ratio for the gears represented in figg. 7, 9 and 13

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Figures:

Fig. 1

Fig. 2

Fig. 3

15

Fig. 4

Fig. 5

Fig. 6

16

Fig. 7

Fig. 8

Fig. 9

17

Fig. 10

Fig. 11

Fig. 12

18

Fig. 13

Fig. 14