Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 2, pp. 355-367, Warsaw 2011

SPLINE DESCRIPTION OF NON-TYPICAL GEARS FOR

BELT TRANSMISSIONS

Piotr Krawiec
Poznan University of Technology, Chair of Basics of Machine Design, Poznań, Poland

e-mail: piotr.krawiec@put.poznan.pl

Adam Marlewski
Poznan University of Technology, Institute of Mathematics, Poznań, Poland

e-mail: adam.marlewski@put.poznan.pl

The usage of noncircular gears makes possible to get better characteri-
stics of the transmission. This problem is widely studied in the case of
regular noncircularity, e.g., when a gear is of an elliptical or cycloidal
shape. In this study, there are taken into account non-typical irregular
gears. The design of such gears involves more advanced techniques, in
particular there has to be applied a numerical treatment to get a ma-
thematical description of the gear profile. In the paper, we discuss a
parametric spline interpolation of the third degree andwe give an exam-
ple of such approximation. We also present a prototype drive making
that the belt transmission has changeable kinematical features.

Key words: spline, noncircular wheel, belt transmission, computer aided
design

1. Introduction

The looking for exact and cheapmanufacturingmethods becames one of tasks
whose solution would enable more common usage of uneven strand trans-
missions in controlling and driving mechanisms. These transmissions enable
obtaining changeable kinematical features such as the gear ratio and velocity.
Modern drives of CNC (computer numerical control) machine tools allow to
control the movement of the tool on the path which is described mathema-
tically by, e.g., a Bézier curve or a spline. The application of these curves to
description of turned curves allows one to shorten the preparation time and
simplify the software for the control of forming motion of non-typical belt
pulleys and gears.



356 P. Krawiec, A. Marlewski

The design of noncircular gears to work in teeth transmissions is widely
described in literature, see, e.g. Bair (2002, 2009), Bair et al. (2007), Chen
and Tsay (2004), Danieli and Mundo (2005), Hasse (2000), Li et al. (2007),
Tsay andFong (2005). Inmost cases, there are considered regular noncircular
wheels, i.e., the gears of the elliptical and cycloidal profile (and only such a
plane curve is discussed by Litvin and Fuentes (2004), a bible for gear desi-
gners). In the last years, practical experiments and theoretical considerations
on noncircular wheels in belt/chain drivers were carried out. Till now, the li-
terature dedicated to this relatively new concept of toothing is rathermodest,
e.g. Krawiec (2005), Li and Liu (2002).
The paper concerns the description of irregular noncircular wheels by spli-

nes. The use of splines in describing complicated contours is a widely applied
technique,mostly in the abovementioned cases, i.e., when regular noncircular
wheels are considered (to get the idea see, e.g., Korzeniowski (2006)). In this
paper, we deal with irregular wheels (and let us say that we did not find any
paper treating such irregular profiles via splines). First, we give a concise in-
troduction to spline curves, in particular to parametric cubic splines. Next, we
present an example of the spline interpolation determined by knots laying on
a prototype noncircular wheel. At last, we report the industrial application of
this wheel in an uneven strand transmission with cog-belt. In the conlusions,
we underline the usefulness of the considered approach in an engineer practice
and in design of non-typical belt drives.

2. Spline interpolation

In brief, a spline, or spline function, is a smooth piecewise polynomial func-
tion of the real variable. This graphically means that a spline is composed of
fragments of polynomials and adjacent polynomials are joint smoothly, i.e.,
their derivatives are continuous, including the derivative of the order 0. To
give the formal definition of a real spline, we start with given n+1 distinct
real numbers tk, k = 1,2, . . . ,n +1, increasingly ordered, tj < tj+1; in the
considered context tk is called k-th knot. The knots determine n disjoint in-
tervals Ij := 〈tj, tj+1〉, j ∈ {1,2, . . . ,n}. Let d be a natural number. We say
that g is a spline (in the variable t, and of degree d) determined by these
(n+1) knots, if g is a polynomial gj of the degree d in every interval Ij and

g
(r)
k
(tk+1)= g

(r)
k+1(tk+1) (2.1)

at every t = tk, k =1,2, . . . ,n, and for r =0,1, . . . ,d.



Spline description of non-typical gears... 357

If, moreover, there are n+1 real numbers yk given, k ∈{1,2, . . . ,n+1},
and

gk(tk)= yk (2.2)

for all k then we have a collocative spline defined by the given n+1 points
Pk =(tk,yk). Obviously, for every k =1,2, . . . ,n the graph of k-th piece, gk,
of the spline g passes through points Pk and Pk+1. More frequently we say
that g is an interpolatory spline because, in general, it is used to interpolate
between neighbouring points Pk; in this context the point Pk is called a k-th
knot of collocation, knot of interpolation.

The above definition was proposed by Isaac Jacob Schoenberg. In his first
paper dealing with splines, Schoenberg (1946) says that the concept of the
spline was known to Pierre Simon Laplace. Schumaker (1980) in his book,
which is recognised as a bible for splinists, says that probably the first syste-
matic considerations on splines were undertaken by Popoviciu (1934). Since
Schoenberg and Schumaker, there has been still noticed interest in splines. A
nice survey was given by Demongeot (2007). Most frequently used splines are
Ferguson/natural splines, Bézier curves, B-splines and NURBS (non-uniform
rational B-splines). There are very widely applied interpolatory cubic splines,
i.e. those of 3 degree, and their particular form called ”cubic natural splines”.
They were originally introduced by Ferguson (1964) and they have to fulfill
some additional conditions which are perceived as natural ones; we will tell
about them later on.

An interpolatory cubic spline, g, defined by points Pk, where k =
=1,2, . . . ,n, is such that

g(t)= gj(t) (2.3)

for t ∈ Ij, where j ∈{1,2, . . . ,n} and gj is a polynomial of the third degree.
In the Stevin (a.k.a. standard, natural) base, it is

gj(t)= aj + bjt+ cjt
2+djt

3 (2.4)

and aj, bj, cj, dj are coefficients to be determined from the following system
of equations

gk(tk)= yk k =1,2, . . . ,n

gk(tk+1)= yk+1 k =1,2, . . . ,n

gk(tk)= gk+1(tk) k =1,2, . . . ,n−1

g′
k
(tk)= g

′

k+1(tk) k =1,2, . . . ,n−1

g′′
k
(tk)= g

′′

k+1(tk) k =1,2, . . . ,n−1

(2.5)



358 P. Krawiec, A. Marlewski

where, as usually, g′ denotes the derivative of g (for details see, e.g., Capra
and Canale (1990)). As we see, there are 4n − 2 equations and 4n coeffi-
cients. Imposing two more restrictions, we can form the Cramer system and,
consequently, obtain the unique solution.

In 1973, Herriot and Reinsch (1973) gave the method to produce the re-
quired splinemuchmore quickly than via direct solving of the Cramer system
mentioned above. Thismethod, since then called a Herriot-Reinsch algorithm
(HeRA), does not work with the Stevin polynomials, Eq. (2.4), but it deals
with the polynomials of the form

gj(t)= aj + bj(t− tj)+ cj(t− tj)
2+dj(t− tj)

3 (2.6)

Thanks to it, in HeRA we do not solve the system in 4n unknowns (aj, bj,
cj, dj for j = 1,2, . . . ,n), but we find n unknowns, c1,c2, . . . ,cn, satysfying
the system of linear algebraic equations with a sparse matrix (for details see,
e.g., de Boor and Schoenberg (1976), Herriot and Reinsch (1973)).

This matrix is tridiagonal, if we deal with a natural spline, i.e., we put
c1 = cn = 0; it makes that the curvature of the spline vanishes at extreme
points t1 and tn. Having yn+1 = y1 and imposing appropriate conditions, we
get a periodic spline, i.e., a spline g such that g(t+mp) = g(t) for all real t
and for every integer m, where the difference p = tn+1−t1 is the period; now
the matrix of the system differs from the mentioned tridiagonal one at two
entries: the lowest left one and themost upper right one. The periodic splines
were investigated in Friedrich Krinzessa’s doctoral dissertation entitled Zur
periodischen Spline-interpolation (Bochum, 1969), see alsoGajda et al. (2008),
Krawiec (2009), Krinzessa (2006).

Below we apply the HeRA to obtain the periodic spline collocated at
n = 24 points Pk = (xk,yk), k = 1,2, . . . ,24, the closeness is forced by
the smooth passage between the functions g24 and g1. All points lay on the
oval curve, see Figs.1-4, and, in fact, we calculate two sets of coefficients
(aj,bj,cj,dj), one for the points (tk,xk), one for the points (tk,yk), in both
cases with tk = k. Therefore, both systems of linear equations (with unk-
nowns ck) have the samematrix: its every its diagonal element is equal to 4,
other non-zero entries are equal to 1.We obtain 24 vector equations covering
every function gk, the k-th piece of this curve is described parametrically, the
parameter denoted as t runs the interval 〈k,k +1〉. This fragment joins the
nodes Pk and Pk+1; since now it is denoted by Ak and called a k-th arc of
the spline.

Togive somedetails, let usdescribe the situation concerning functionsno. 9
and no. 10 (see Figs.1-4). These functions are covered by the equations in the



Spline description of non-typical gears... 359

Fig. 1. Curves no. 1,10,13,17 drawnwhen t runs from 0 to 24

Fig. 2. Arcs of all 24 curves, (x,y)= gk(t), drawn for t ∈ 〈k,k+10〉

Fig. 3. Curves no. 9 drawnwhen t runs from 9 to 14, and no. 10 drawn for
t ∈ 〈5,16〉



360 P. Krawiec, A. Marlewski

Fig. 4. Spline passing through given 24 points (some of them are numbered) and
arcs seen in Fig.3. The appropriate pieces of these arcs, namely the arc A9 joining
nodes 9 and 10, and the arc A10 going from node 10 to node 11, make part of the

spline and they are joint smoothly at point 10

parameter t and their parts contribute in the final spline when t ∈ 〈9,10〉
and t ∈ 〈10,11〉, respectively. These equations are obtained in form (2.6) and
may be written in standard form (2.4). Let, as above, Aj denote the j-th
arc extended between the points Pj =(xj,yj) and Pj+1 =(xj+1,yj+1). Then
the 9th arc, A9, joins the point P9 = (−46.7562,−14.1921) with the point
P10 =(−41.8987,−22.2177) and its parametric description is

x(t)= ax,9+ bx,9(t−9)+ cx,9(t−9)
2+dx,9(t−9)

3

y(t)= ay,9+ by,9(t−9)+ cy,9(t−9)
2+dy,9(t−9)

3

where

ax,9 = x9 =−46.7562 bx,9 =3.31931 cx,9 =1.65761

dx,9 =−0.119423 ay,9 = y9=−14.1921 by,9 =−8.83033

cy,9 =0.596595 dy,9 =0.208135

That is why the 9-th part of the spline, A9, is governed by the equation

x(t)=−46.7562+3.31931(t−9)+1.65761(t−9)2−0.119423(t−9)3 =

=−0.119423?t3+4.88203t2 −55.5374t+144.695

y(t)=−14.1921−8.83033(t−9)+0.596595(t−9)2+0.208135(t−9)3 =

=0.208135t3 −5.02305t2 +31.0077t−38.1253

where t ∈ 〈9,10〉.



Spline description of non-typical gears... 361

Analogously, the 10th arc, A10, is determined by the coefficients

ax,10 = x10 =−41.8987 bx,10 =6.27626

cx,10 =1.29934 dx,10 =−0.260208

ay,10 = y10=−22.2177 by,10 =−7.01272

cy,10 =1.22100 dy,10 =−0.095979

(all calculations and figures are produced in the computer algebra system
Derive forWindows 5 fromTexas Instruments, Inc.; the coordinates of points
are the centers of notches in the wheel photographed in Fig.6a)

3. Multiwheelness and belt linkage

In two-wheel transmission systems, the average transmission ratio must be
equal to 1:2, 2:1, 2:3 etc. It essentially impedes the use of transmission systems
of that type. Thus, it has become necessary to conduct a feasibility analysis
whether it would be possible to design and constructmulti-wheel transmission
systems producing any transmission ratio.

In order to ensure the cyclicity of the four-wheel transmissions system,
there must be taken an appropriate shape of the noncircular wheel. Circum-
ferences of all wheels must be equal to the multiplicity of the belt pitch. To
avoid the belt skipping on teeth of the wheels, the belt has to be properly
pre-tightened.

In most cases, the shape of the rim of the noncircular wheel had to be
computed iteratively, in a way to ensure the constant tension of the belt and
the cyclicity of the transmission system. Noncircular gears aremostly applied
as the drivingmechanism for a linkage tomodify displacements and/or veloci-
ties (e.g., in conveyors driven by theMaltese cross; Litvin and Fuentes (2004)
calls it aGenevamechanism). Noncircular gear toothing is well described and
used, e.g., in cycle hydraulic engines.

An example of the considered linkage is an uneven strand transmission, see
Fig.5. There are four wheels in it: drivingwheel, drivenwheel and twowheels
used to ensure the fixed length of the envelope of the system (Fig.6). Two of
these wheels are circular, one wheel is circular and eccentrically, the other is
noncircular.



362 P. Krawiec, A. Marlewski

Fig. 5. Four-wheel transmission systems installed in LEDM (Laboratory for
Experimental Design andManufacturing at the Faculty of Machines and Transport,
PoznanUniversity of Technology); the system has two circular wheels, one

noncircular wheel and one eccentrically mounted wheel

Fig. 6.Wheels of the variable speed transmission system: (a) noncircular wheel,
(b) circular eccentrically mounted wheel

4. Design, forming and verification

In the transmission system with noncircular gears, the process of shape-
frictional contact is characterised by essential differences, which result from
different geometrical features of gears (elliptical, oval, cycloidal and, ingeneral,
irregular) and different kinematic features. In each case, in themanufacturing
process it is necessary to have the description of rolling lines by appropriate
equations. These equations are of various forms and there exist some techni-
ques to obtain them. In this paper, we describe noncircular gears via splines.
The advantage of this approach is the direct consequence of the properties
of the mathematical representation: the arc Aj is covered by the parametric
equation

x = x(t) y = y(t) j ¬ t ¬ j +1



Spline description of non-typical gears... 363

Hence, at the endpoints, there hold the equalities

(xj(j),yj(j)) = Pj (x(j +1),y(j +1))= Pj+1

an arbitrary point P(t) laying on the arc Aj is identified by the certain value
t ∈ 〈j,j+1〉 and, in the considered case (when the length |Aj| of the j-th arc
is relatively small, |Aj = 9.525mm), this identification is practically linear.
In consequence, it is easy to indicate, up to a certain accuracy (and it is big
enough in manufacturing practice), the value t which produces the desired
point P(t). Clearly, it is trivial to say exactly which values of t produce the
endpoints of the arc Aj. In consequence, the use of spline curves reduces the
setup time of software driven on numerically controlled machines.
Noncircular gears may be formed, e.g., on profile cutters, on CNC gas

laser cutting machines, by cutting out on CNC electrical discharge machines,
by water cutting with abrasive and sintered metal powders, and the precision
of their manufacturing was analysed by Krawiec (2009).
Our irregular wheel, represented by the parametric spline (see Fig.4), was

manufacturedonCNCmachinewith the set of endmills (seeFig.7).Theaccu-
racy of forming geometrical parameters and stereometry of the wheel surface
was experimentally verified on CMM (coordinate measuring machine) Con-
tura from Zeiss and the profilograph from Taylor Hobson (see Fig.8). Till
now, there are no regulations concerning the systems composed of noncircular
wheels and timing belts, so we can only refer to the requirements which are
obligatory to circular wheels. This examination shows that the manufactured
wheel has desired values of the parameters (such as tolerance of the outsi-
de rim, tolerance of the tooth shape with respect to the axle, errors on the
pitch).

Fig. 7. Forming noncircular pulleys



364 P. Krawiec, A. Marlewski

Fig. 8. Measurement of geometrical features (a) and roughness features (b) of
noncircular wheels executed in LEDM

Our irregular wheel has been mounted in the prototype stand shown in
Fig.5. In this stand there are pulleys with the noncircular envelope in the
uneven transmission with cog-belt. It makes possible to change kinematical
features such as the gear ratio and velocity. For the stand presented in Fig.5,
the changes in the transmission ratio are shown in Fig.9.

Fig. 9. Changes in the transmission ratio of: wheel 2 in relation to wheel 1 (u2/1),
wheel 3 in relation to wheel 1 (u3/1), wheel 4 in relation to wheel 1 (u4/1). Along
the horizontal axis there are marked values of the angle (ψ, measured in degrees) of

rotation of the driving wheel

5. Conclusions

The paper deals with the cubic spline interpolation of non-typical irregular
wheels. The discussion concerns theoretical and practical aspects of this ap-
proximation, in both mathematical and technical issues of the problem, and
reveals the advantages of this approach in both design andmanufacturing sta-



Spline description of non-typical gears... 365

ges. An example of a prototype 4-wheel drive installed in a stand at LEDM
(see Fig.5) is also presented. Drives like that have a changeable kinematical
characteristic of the velocity of the elementdrivenby themotorworkingwith a
constant speed. The feature is obtained by the use of an appropriately shaped
noncircular toothed-wheel rims when the motion is transmitted by a timing
belt.

The importance of the presented problem for industrial practice is deter-
mined by kinematic features of uneven transmission drives. In the designing
of a drive having required parameters there can be applied various techniques.
In the case of regular noncircular gears, we can refer to well-know formulas
such as the equation of an ellipse. When we study irregular shapes (like that
presented in this paper), we have to find their mathematical description. An
example of such an investigation is given by Gajda et al. (2008), where the
Bézier approximation is analysed. Another attempt is discussed in this paper.
Here we deal with the spline interpolation. In relation to theBézier approach,
the method of splines is more effective, because we can indicate more easily
the points which are crucial in the design and forming processes (e.g., the
points at which a designer has to get the perpendicular lines to the profile
at hand) and to determine the angles at which the projected gear has to be
situatedwith respect to the cutting tool, e.g.WEDM(wire electrical discharge
machine), CNCmachine. Another positive feature of the spline approximation
is its friendliness as seen from the numerical point of view. Indeed, the spline
is a piecewise polynomial function, the equation of its every arc is composed
of two formulas of the form given by expression (2.6). This form is stable and
fast in computation (when rewritten in the Horner scheme). All the features
discussed above let to conclude that the spline approximation is highly re-
commended to be used in the mathematical description of wheels exhibiting
significant irregularity of shape.

Acknowledgments

Authors express their thanks to an anonymous Reviewer for all remarks which

improved this paper in its substantial and linguistic aspects.

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Opis krzywymi sklejanymi nietypowych kół zębatych stosowanych

w przekładniach cięgnowych

Streszczenie

Zastosowanie nieokrągłych kół zębatych powala uzyskać lepsze charakterystyki
przełożenia. Przypadki, gdy koła nieokrągłe mają kształty regularne, np. eliptycz-
ny lub cykloidalny, zostały już szeroko przebadane. W tej pracy są poddane anali-
zie nietypowe nieregularne koła zębate. Projektowanie takich kół wymaga bardziej
zaawansowanych technik, w szczególności podejścia numerycznego w celu uzyskania
matematycznego opisu profilu koła.W pracy omawiamy interpolację parametryczną
za pomocą funkcji sklejanych stopnia trzeciego i stosujemy ją do uzyskania odpowied-
niego opisu. Ponadto prezentujemy prototypowynapęd, w którymzostało zastosowa-
ne przedmiotowe nieregularne koło nieokrągłe. Zastosowanie tego koła w przekładni
cięgnowej sprawia, że ma ona zmienne cechy kinematyczne.

Manuscript received December 17, 2009; accepted for print October 15, 2010