Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 1, pp. 117-133, Warsaw 2011

EFFECTS OF GEOMETRICAL AND DIMENSIONAL ERRORS

ON KINEMATICS AND DYNAMICS OF TRACTA COUPLING

Pier Paolo Valentini

Eugenio Pezzuti

University of Rome ”Tor Vergata”, Department of Mechanical Engineering, Rome, Italy

e-mail: valentini@ing.uniroma2.it

Thepaperdealswith the investigationof the influence of geometrical and
dimensional errorsonperformanceof theTracta coupling. It is oneof the
most simple constant-velocity joints which has been used in front-wheel
drive automobiles, all-wheel drive military vehicles, heavy vehicles and
various other applications. The proposed study is based on building and
simulating a parametric spatial multibody model of the full joint. Both
kinematic and dynamic behaviour have been investigated. Five different
linear and angular errors have been studied. The sensitivity of kinema-
tics and dynamics irregularities to these errors has been computed and
summarized in design charts.The results canbe used in optimal toleran-
ce allocation and accurate vehicle system simulation with the coupling
joint sub-model.

Key words: Tracta joint, tolerance, mechanical error, multibody model

1. Introduction

Coupling joints are widely used in mechanical and industrial applications in
order to transmit torque andmotion between twomisaligned shafts (Wagner,
1991; Schmelz et al., 1991). Scientific literature, patents and commercial cata-
logues include a lot of different types of joints with different structure, perfor-
mance and complexity. Themost important features that a designer wants in
coupling joints are two. The first is the exact kinematic coupling (i.e. homo-
kinetic or constant velocity joint). This condition is fulfilled when the angular
velocity of the inputshaft is equal to thatof theoutputone,whatever theangu-
larmisalignment is. The second important feature is the capability to transfer
an adequate amount of torque between the two shafts without irregularities.



118 P.P. Valentini, E. Pezzuti

It means that the structure has to be designed to support loads, including
transient phases and impulsive forces due to impacts. Among different joints,
there is theTracta coupling.Originally developed by theFrench inventor Jean
Pierre Gregoire in 1924, this type of joint was the first constant velocity joint
and enabled the development of the first frontwheel drive cars in the 30s. Due
to its robustness and simplicity, the Tracta joint gained widespread usage all
over the world for civil and military vehicles. On the other hand, it presents
some engineering problems. Due to the occurence of sliding contact between
cooperating parts, its mechanical efficiency is lower than other coupling joints
with rolling contacts. Moreover, it is affected by a considerable wear rate, and
the presence of mechanical errors affects its performance in sensible way.
In Figures 1 and 2 a typical Tracta assembly is shown. The entire joint

is composed of two shafts, two intermediate links and a housing system. At
the ends of the shafts, there are two forks that are connected to intermediate
links by means of revolute joints. The two intermediate links are connected
together with a planar slot with an axis which is perpendicular to those of the
revolute joints. This contact between planar surfaces ensures the transmission
of torque. The housing system, connected with two revolute joints to the two
shafts, ensures correct relative positioning of the forks and the sealing for the
lubricant.

Fig. 1. Tracta joint assembly

Due to thepresence of thehousing system, the axis of the two shafts always
intersect in the midpoint O (see Fig.3). Moreover, the distances between the
axes of the revolute joints and point O have to be the same. This accurate
positioning ensures that the intersection point K between the axes of the re-
volute joints is always placed in the bisector plane between the twomisaligned
shafts. According toMyard’s theorem (Myard, 1931; Pennestŕı andValentini,
2008), the fulfilment of this geometrical condition is sufficient to ensure an



Effects of geometrical and dimensional errors... 119

Fig. 2. Tracta joint exploded view

Fig. 3. Geometrical condition for a homokinetic Tracta joint

homokinetic joint, where the input and output rotations occur at the same
speeds.

The scientific literature includes some investigations about features and
kinematics of the Tracta joint. The structure was investigated by Freudenste-
in andMaki (1979), the general theory by Hunt (1979) and a deep kinematic
investigation including transmission errors and internal displacement between
links by Fisher (1994, 1999). Static forces in planar joints have been investi-
gated by Fisher (2007).

With the increasing use of computer-aided simulation in almost all engi-
neering fields, there is the need for building accurate and complex models to
support the designer in optimization and verification of his choices. In many
cases, the presence of coupling joints in amechanical system is modelled with
simple constraint equations, relating the kinematics between input and output
shafts. For example, in the simulation of the front wheel system of a ground



120 P.P. Valentini, E. Pezzuti

vehicle, a homokinetic joint is usually simulated by introducing a velocity con-
straint which imposes that rotations of the input and output shafts are the
same. This approach is useful for building hugemodels, because it reduces the
numberof equations tobe solved.On the other hand, it is not suitable for inve-
stigating local dynamic actions because computation of reaction forces is only
approximated. Moreover, the real joints, due to inevitable assembling errors,
may present irregularities in transmitting motion and variable load transfers
which can be simulated only with accurate sub-modelling. Themotivations of
the presented study come from all these reasons.

The investigation on the effects of errors on both kinematics and dynamics
of the joints is also useful to allocate the right manufacturing tolerances. In
fact, the presence of mechanical errors may affect the performance of the
mechanism, mining correct functioning, reducing the efficiency and reliability
(Pezzuti et al., 2006).

2. Dimensional and geometrical errors

The structural parts of the Tracta joint are manufactured with a common
technological process. For this reason, the achieved precision in the coupling
dependsongeometrical anddimensional tolerances related to these operations.

The first step through the investigation of the influence of these errors is
about the choice of the functional features of each part. The kinematic and
dynamic behaviour of the joint is supposed to depend on these features. A
dimensional or geometrical error which affects them,may cause an important
modification of joint performance, mining correct functioning. According to
the observation addressed in the introduction, one of the most relevant pro-
perties of the Tracta coupling is the homokineticity. Since this condition is
obtained thanks to precise placement of the two shafts, all the features which
constrain the relative position between them can be defined as functional. In
particular, it is important to fulfil the condition OC1 =OC2 (see Fig.3). This
requirement depends on the dimensional error a which affects the position of
the lateral surface of the bearing housing with respect to the centre of the
joint (see Fig.4).

Another important error is about the misalignment between the axis of
shafts and the axis of its revolute joint at the fork. In the ideal condition, the
two axes are perpendicular. Errors may affect this requirement producing an
angle of misalignment (β, see Fig.5).



Effects of geometrical and dimensional errors... 121

Fig. 4. Position error of the lateral surface of the bearing housing

Fig. 5. Angular error of the fork axis

The geometrical features of the intermediate links are also very important
for correct functioning of the joint. The ideal condition is when the axis of the
revolute joint is perpendicular to the axis of the planar slot. This requirement
cannot be fully satisfied in the case of planarity or angular error on the surface
of the slot. This kind of error is more complex to study, because it affects two
degrees of freedom of the surface. They are two possible rotations about two
independent axes which are perpendicular to that of the revolute joint (Figs.
6 and 7).

In more general cases, the two rotations of the surface occur combined,
producing a lot of possible imprecise configurations (Fig.8).

Table 1 summarizes five possible errors which have been included in the
investigation.



122 P.P. Valentini, E. Pezzuti

Fig. 6. First degree of freedom of the slot surface due to planarity or an angular error

Fig. 7. Second degree of freedom of the slot surface due to planarity or an angular
error

Fig. 8. General angular error combines the effects of the two degrees of freedom



Effects of geometrical and dimensional errors... 123

Table 1. Investigated errors

Definition Error Type

E1 Misfit of the lateral surface of the bearing
housing

linear – α

E2 Misalignment of the fork axis angular – α

E3 Misalignment of the plane of the slot sur-
face (rotation of the plane about the shaft
axis)

angular – γ

E4 Misalignment of the plane of the slot
surface (rotation of the plane about an
axis which is perpendicular to that of the
shaft)

angular – γ

E5 Combined error E3+E4 (Misalignment of
the plane of the slot surface (rotation of
the plane about an axis which is 45◦ with
respect to that of the shaft)

angular – γ

3. Multibody model

In order to investigate both kinematic anddynamic performance of theTracta
coupling, a parametric multibodymodel has been built. The assembly, depic-
ted in Fig.2, is made of 5 rigid bodies: the housing cage (considered as the
ground), two shafts and two intermediate links.Four revolute joints andacoin-
cident planar joint have been included. They are describedwith 22 constraint
equations (Haug, 1988). Friction and lubrication effects have been neglected.
A torque of 1Nmhas been applied to the output shaft and a constant angular
velocity of 360◦/s has been imposed to the input one. Themass of each body
is 1kg. The analysis has been performed starting from a three dimensional
CAD model. Thanks to solid modelling, it has been possible to simulate the
presence of errors modifying the geometrical features defined by dimensional
parameters, configuringseveral simulation scenarios.The resultsof simulations
are presented in the next sections.

4. Results and discussion

The kinematic and dynamic performance of the Tracta coupling has been
investigated considering two possible misalignments between the shafts: 10◦



124 P.P. Valentini, E. Pezzuti

and 20◦. Six error configurations have been taken into account: the ideal con-
dition (without errors) and the five cases (E1,...,E5) shown in Table 1.

Fig. 9. Kinematic irregularities due to errors when the misalignment between the
two shafts is 10◦

Fig. 10. Kinematic irregularities due to errors when the misalignment between the
two shafts is 20◦

Concerning the kinematics, Figures 9 and 10 show the computed angular
velocity irregularities of the output shaft. In the ideal condition the joint is
homokinetic and the output velocity is always equal to the input one. Due
to the presence of errors several irregularities affect the joint. In some cases
(presence of errors E2, E4) the irregularities occur with a frequency which is
the same as that of rotation. With E1 and E3 errors, the irregularities occur
with a frequency which is twice as much as the rotation. Error E5, that is a



Effects of geometrical and dimensional errors... 125

combination of E3 and E4, causes main irregularity with the same frequency
of the shaft rotation and lower irregularity with a frequency that is twice
of that of shaft rotation. With the same error amplitude (1◦), E4 shows a
lower irregularitymagnitude. Itmeans that the angular error on the slot plane
causes irregularity whose amplitude and frequency depend on the direction of
the misalignment. Its amplitude presents the maximum value when the error
occurs with rotation of the plane about an axis which is perpendicular to
that of the corresponding shaft.Consideringmediummanufacturing precision,
according to ISO 2768, the angular errors affect the kinematics more than the
linear ones.
When the shafts are more misaligned, the irregularities increase, keeping

the same trend. For a precise comparison, a dimensionless kinematic irregula-
rity factor (KIF) can be defined as

KIF [%]=
ωoutmax−ωin

ωin
·100 (4.1)

where ωoutmax is the maximum amplitude of the irregularity on the output
shaft angular velocity and ωin is the input shaft angular velocity. Table 2
summarizes the KIFs for different errors and differentmisalignments between
the shafts.

Table 2.KIFs due to errors

Misalignment
between E1=1mm E2=1◦ E3=1◦ E4=1◦ E5=1◦

shafts

10◦ 0.04% 0.15% 0.007% 0.15% 0.11%

20◦ 0.16% 0.30% 0.027% 0.30% 0.21%

Concerning the dynamics, the presence of errors causes variation of the
amplitude of reaction forces of joints. Figures 11 and 12 show plots of the
magnitude of radial reaction forces of the revolute joint of the input shaft. It
can be noted that even the ideal condition without errors presents a periodic
trend of the forcewith a frequencywhich is twice the input shaft rotation. The
presence of errors amplifies these irregularities, keeping the same frequency.
Error E2 seems to produce higher dynamic irregularity, while the errors on
planar slot have negligible effects on joint dynamics.
In order to compare their effects on the joint performance, a dimensionless

dynamic irregularity factor (DIF) can be defined as

DIF [%]=
Ferrormax −Fidealmax

Fidealmax
·100 (4.2)



126 P.P. Valentini, E. Pezzuti

Fig. 11. Irregularities of the radial force of the input shaft revolute joint due to
errors for the misalignment between shafts 10◦

Fig. 12. Irregularities of the radial force of the input shaft revolute joint due to
errors for the misalignment between shafts 20◦

where Ferrormax is the maximum amplitude of the reaction force computed
in the presence of errors and the Fidealmax is the maximum amplitude of the
reaction force computed in the ideal model (without errors). The computed
DIFs for the radial reaction force of the input shaft revolute joint are reported
in Table 3.

Figures 13 and 14 show the axial thrust at the input shaft revolute jo-
ints. These axial reaction forces show similar behaviour with respect to the
radial ones. Even the ideal condition without errors causes a periodic trend of



Effects of geometrical and dimensional errors... 127

Table 3.DIFs of the radial reaction force of the input shaft revolute joint

Misalignment
between E1=1mm E2=1◦ E3=1◦ E4=1◦ E5=1◦

shafts

10◦ 2.40% 11.54% 0.23% 0.09% 0.10%

20◦ 2.50% 6.67% 0.23% 0.10% 0.12%

Fig. 13. Irregularities of the axial force of the input shaft revolute joint due to errors
for the misalignment between shafts 10◦

Fig. 14. Irregularities of the axial force of the input shaft revolute joint due to errors
for the misalignment between shafts 20◦



128 P.P. Valentini, E. Pezzuti

the axial force with a frequency which is twice the input shaft rotation. The
presence of errors amplifies these irregularities, keeping the same frequency.
E2 error seems to produce higher dynamic irregularity, while E3 andE4 show
anegligible effect on joint dynamics, and their irregularities are not affected by
the angular misalignment between shafts. The dynamic irregularities factors
(DIFs) for the axial reaction forces are reported in Table 4.

Table 4.DIFs of the axial reaction force of the input shaft revolute joint

Misalignment
between E1=1mm E2=1◦ E3=1◦ E4=1◦ E5=1◦

shafts

10◦ 2.40% 12.58% 0.57% 0.36% 0.72%

20◦ 2.50% 6.25% 0.56% 0.36% 0.71%

A similar trend concerning the dynamic irregularities can be observed also
for the revolute joint of the output shaft (both radial andaxial reaction forces).

5. Design charts for tolerance allocation

The results discussed in the previous section are about the presence of errors
with a specificmagnitude (1 millimetre for the linear dimension and 1 degree
for the angular ones). In order to guide the designer to allocate the right tole-
rance, the influenceof their amplitudehas tobe investigated too.The scientific
literature includes works where linear sensitivity coefficients have been intro-
duced (Valentini, inpress). Itmeans that thepresenceof anerror influences the
kinematic and dynamic performances in a linear way. It is a useful simplifica-
tionwhen the closure loop equations are deduced algebraically. Themodelling
of the Tracta joint using multibody techniques allows one to study in depth
the influence of errors and to investigate their nonlinear effects. For this pur-
pose, the investigations discussed in the previous section have been extended
to include variable errors (from 0.5 to 2mmand from 0.5◦ to 3◦) and different
misalignments between shafts (from 5◦ to 20◦). The results have been shown
in the following Figures, collected as design charts. In order to compare them,
theKIFs and theDIFs have been shown.As it can be noted fromFigs.15-20,
the KIFs and DIFs shows a slightly non linear trend with respect to the in-
crease of error amplitude. This nonlinearity is more evident with the increase
of angularmisalignment between shafts. The reaction forces of revolute joints
of the input and output shafts are quite insensitive to the amplitude of E3, E4



Effects of geometrical and dimensional errors... 129

Fig. 15. Design chart of the angular errorsKIFs of the output shaft angular velocity

Fig. 16. Design chart of the linear errorKIFs of output shaft angular velocity

Fig. 17. Design chart of the angular errorsDIFs of the radial reaction force of the
input shaft revolute joint



130 P.P. Valentini, E. Pezzuti

Fig. 18. Design chart of the angular errorsDIFs on the axial reaction force of the
input shaft revolute joint

Fig. 19. Design chart of the linear errorsDIFs on the radial reaction force of the
input shaft revolute joint

Fig. 20. Design chart of the linear errorsDIFs of the axial reaction force of the
input shaft revolute joint



Effects of geometrical and dimensional errors... 131

and E5 errors and to the misalignment between shafts (Figs.17 and 18). The
presence of E2 error affects both kinematic and dynamics in a considerable
way. Planar errors on the central slot affect more the kinematic performance
than the dynamic one. The dynamic irregularities caused by the presence of
linear error E1 are slightly influenced by the misalignment between shafts.

6. Conclusion

In this paper, an investigation about the influence of geometrical and dimen-
sional errors on the kinematic and dynamic performance of the Tracta joint
has been presented.Due to its plain architecture, theTracta joint is one of the
simplest homokinetic coupling. On the other hand, its usage is often limited
by the influence of construction errors on its performances. In this paper a
multibody virtual model has been proposed. The presence of one linear error
and four different angular errors has been simulated. The influence of these
errors has been computed and summarized in design charts. The results show
that some errors cause kinematic irregularities with a frequency that is equal
or twice of that of the input shaft rotation. This information is useful to ad-
dress possible vibration phenomena.The presence of errors influences reaction
forces, too, by amplifying their magnitude without modifying the frequency
which is always twice that of the input shaft rotation. The correct assessment
of internal loads is useful to optimize the structural dimensioning, preventing
damage and fatigue phenomena. The discussedmodel and simulations can be
useful not only for analysing the presence of possible errors, but also for the
synthesis and allocation of adequate tolerance for every feature. In this case,
the designer has to find the admissible error combination that allows the per-
formance of the joint to be within an acceptable range. Given a threshold on
the irregularities, he has to find the maximum value of possible errors, using
the presented design charts.

References

1. Fischer I.S., Remington P.M., 1994, Errors in constant-velocity shaft co-
uplings,ASME Journal of Mechanical Design, 116, 204-209

2. Fischer I.S., 1999, Numerical analysis of displacements in a Tracta coupling,
Engineering with Computers, 15, 334-344



132 P.P. Valentini, E. Pezzuti

3. Fischer I.S., 2007, Static forces and torques in mechanisms with plane jo-
ints, with application to the Tracta coupling, Proceedings of the Institution of
Mechanical Engineers, Part K: Journal of Multi-Body Dynamics, 221, 599-614

4. Freudenstein F., Maki R.R., 1979, The creation of mechanisms according
to kinematic structure and function,Environment and Planning B, 6, 375-391

5. Haug E.J., 1988, Computer-Aided Kinematics and Dynamics of Mechanical
Systems, Allyn and Bacon, Boston, USA, Vol. 1, 48-104

6. Hunt K.H., 1979, Constant-velocity shaft couplings: a general theory,ASME
Journal of Engineering for Industry, 95, 445-464

7. MyardF.E., 1931,Contribution à la géometriedes systémesarticulés,Bulletin
de la Société Mathématique de France, 59, 183-210

8. Pennestŕı E.,ValentiniP.P., 2008,Kinematicdesignandmultibodyanaly-
sis of theRzeppa pilot-lever joint, Institution ofMechanical Engineers, Part K,
Journal of Multi-Body Dynamics, 222, 135-142

9. Pezzuti E., Stefanelli R., Valentini P.P., Vita L., 2006, Computer
aided simulation and testing of spatial linkages with joint mechanical errors,
International Journal For Numerical Methods in Engineering, 65, 1735-1748

10. Schmelz F., Seherr-Thoss H., Aucktor E., 1991, Universal Joints and
Driveshafts, Springer Verlag, Berlin

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Wpływ błędów geometrycznych i wymiarowych na kinematykę

i dynamikę przegubu równobieżnego typu Tracta

Streszczenie

Wpracy przedstawiono badania wpływu błędów geometrycznych i wymiarowych
na pracę przegubu typuTracta. To jedno z najprostszych rozwiązań konstrukcyjnych
przegubu równobieżnego, który szeroko stosowany jest w przednich osiach napędo-
wych samochodów osobowych, pojazdach wojskowych z napędem na wszystkie osie,
ciężkich pojazdach i wielu innych maszynach. Przedstawione badania oparto na sy-
mulacjach wielobryłowego i przestrzennego modelu przegubu. Przeanalizowano wła-
ściwości kinematyczne i dynamiczne modelu uwzględniającego pięć różnych błędów



Effects of geometrical and dimensional errors... 133

geometrycznych odnoszących się do wymiarów liniowych i kątowych.Wrażliwośćmo-
delu na błędy opisano za pomocą charakterystyk kinematycznych i dynamicznych
układu w funkcji wielkości tych odchyłek. Otrzymane wykresy mogą stanowić pod-
stawę do optymalizacji obszaru tolerancji w projektowaniu przegubuTracta oraz być
pomocne przy właściwej symulacji zachowania się pojazdów wyposażonych w takie
przeguby i opisanychmodelem omówionymw pracy.

Manuscript received June 9, 2010; accepted for print July 30, 2010