Shaft-hub Couplings With Polygonal Profiles_Citarella-Gerbino2001

BE analysis of shafthub couplings with polygonal profiles

R. Citarella*, S. GerbinoDipartimento di Progettazione e Gestione Industriale, Universita` degli Studi di Napoli Federico II, P.le Tecchio 80, 80125 Napoli, Italy

Abstract

In the mechanical transmission field, shafthub couplings with polygonal profiles play an interesting role because of their characteristics

of self-alignment, lack of projecting elements (responsible for high stress concentration) and constructive compactness. Other

characteristics, like transmission of static/oscillating torque load, even with small overall dimensions, and easy hub interchangeability,

make such couplings competitive with the traditional ones based on keys and splined shafts.

This work concerns a study on steel made polygonal couplings, with trochoidal three-lobe profile, and is aimed to highlight the contact

stress and strain state of shafthub interface, with reference to particular profile geometric parameters.

From Mechniks and Kollmanns works, in which the analysis was performed by the Finite Element Method, this work develops a CAD/

CAE methodology for coupling design, oriented to an efficient integration between CAD systems and BEM solvers. The stress analysis is

carried out with a Boundary Element code (BEASY) well suited for this kind of contact problems while coupling geometric model is made

by Pro/Engineer, a solid parametric modeller. # 2001 Elsevier Science B.V. All rights reserved.

Keywords: CAD; BEM; Polygonal profile

1. Introduction

The interest in polygonal profile shafthub couplings is

due to common characteristics as: constructive compact-

ness; lack of projecting elements responsible for high stress

concentration; selfalignment; transmission of static-oscil-

lating torque load (even with overall small dimensions);

axial sliding capability with applied loads, for some poly-

gonal shapes, such as four-lobe couplings; easy hub inter-

changeability (useful in case of failure) [1].

Nevertheless, these joints are very complicated and costly

to manufacture because they require dedicated grinding

machines and the stress analysis is very difficult due to

triaxial stress state and lack of rotational symmetry. For a

long time, these drawbacks have limited their use, discoura-

ging design and production engineers to leave traditional

connections based on keys and splined shafts.

In recent years there has been an increasing interest in

polygonal couplings (Fig. 1), mainly with reference to three-

lobe couplings (Fig. 1c and e).

Mechanical literature presents only a few works on these

joints. The first studies date back to the 1950s in Germany

and proposed the so-called K profile (Fig. 1c). In the 1960s

epicycloidal profiles with three- and four-lobes [2] were

made and, in 1979, included in the standard DIN (P3G and

P4C) [3,4]. Fig. 2 shows a standard three-lobe profile (P3G)

and its parametric equation (previously defined by Filemon

[5] in 1959), valid for n-lobe profiles.

During that period, some analytical procedures of stress

analysis have also been developed, like that by Orlov [1],

Leroy and Viseur [6], Manhurim [7], based on very strong

approximations that are not able to clarify the real stress and

strain state of polygonal couplings (see also [8]).

Only recently, a sufficiently detailed analysis of the

biaxial stress and strain state in shafthub joints has been

proposed by Mechnik, using the Finite Element Method

(FEM) [9]. By improving the connection model, Mechnik

obtained very reliable results.

After this work, a deeper understanding of this topic has

been reached [10,11]. Despite these recent results, the

employment of polygonal couplings is still very finite.

Traditional and already consolidated couplings based on

keys and splined shafts are still strongly preferred in the

mechanical transmission field, because they are easy to

manufacture and size.

Nowadays, the development of numerical control grind-

ing machines together with the improvement of hardware

and software resources has allowed to reduce the above

disadvantages, making the future of the polygonal couplings

very interesting.

From a production point of view, many drawbacks could

be circumvented using powder metallurgy (PM), as pro-

posed in [12]. Lately, this technology has been more and

more appreciated in the automotive field. It enables to obtain

Journal of Materials Processing Technology 109 (2001) 3037

* Corresponding author.

0924-0136/01/$ see front matter # 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 7 7 2 - X

production time and cost savings and to manufacture parts

with complex shapes and high mechanical performances.

Some recent works [13] demonstrate the benefits of sintered

steels in applications which, up to one decade ago, were

reserved to special steels. For parts subject to fatigue,

sintered steel offer the best performance [14] and further

development on polygonal profiles implementation is likely

to involve such materials.

The authors point out a CAD/CAE methodology for

coupling design, oriented to an efficient integration between

CAD systems and BEM solvers. The aforesaid methodology

is based on a synergetic and strictly integrated usage of the

Pro/Engineer CAD system, for geometric model generation,

and of the BEASY code for stress analysis.

2. Analysis system choice

In structural analysis, the Boundary Element Methodol-

ogy offers some advantages with respect to FEM, with

particular regard to the preliminary stage of new com-

ponent design when different engineering solutions are to

be compared. In fact, FEM would require a tremendous

amount of data (because of the volume meshing) making it

difficult to quickly update the model subject to continuous

development.

Stress and strain analysis strategy of mechanical compo-

nents, up to now traditionally based on FEM, has recently

involved BEM. An advantage related to this methodologies,

is that, on the contrary of FEM, BEM allows to operate

directly on the boundary model, neglecting the volume, and

to calculate surface stress with the same accuracy as surface

displacements. That is why, it is well suited for stress

analysis in problems with high stress gradient such as

contact problems. Moreover BEMCAD coupling and the

meshing process are straightforward. This coupling is essen-

tial in shape optimisation analysis where it is necessary to

iteratively update the model geometry after stress analysis,

in an integrated and automatic process. Such an automatic

process in an FEM environment would cause increasing

mesh distortion in areas susceptible to variable geometry as

successive re-meshing goes on.

The study of geometrically complex models, by numer-

ical analysis codes, always needs a preliminary work of

model simplifying. In particular, before analysing a model, it

is convenient to delete those features (fillets, chamfers)

which do not affect the final results, but are responsible

for increasing analysis run time. Actually, by using FEM, in

order to populate the volume with elements, exact geometric

details must often be sacrificed, even if this affect the

solution accuracy [15,16]. Moreover, element shapes are

often distorted, thus causing loss of solution accuracy, whilst

the proper choice of correct simplifications requires special

skills on behalf of the designer, in such a way to correctly

point out the redundant features in non-critical areas. As a

matter of fact these features, even if not localised in remark-

able zones, require high local mesh refining, because of high

stress concentration. Since the FEM meshing process

involves the whole domain, there is a propagation effect

which starts from a local dense mesh and proceeds through-

out the volume, causing a considerable growth in the overall

element number.

On the contrary in a BEM system, only the model

boundary is affected by the meshing process, so it is easier

to decrease the mesh density around the aforesaid features,

in such a way that their simplification (even if advisable) is

no longer strictly necessary.

Moreover the possibility to use discontinuous elements

allows better fitting of the mesh elements to the model

features and the capability to independently mesh such

geometry from the rest of the model. Consequently the

presence of small features in the model is not a problem

for a BEM system because a local refined mesh does not

involve mesh propagation.

Despite recent advances in finite element auto-meshing

techniques, it is still quite difficult to obtain a satisfactory 3D

element model for a complex structure. Today the single

Fig. 1. Examples of shafthub couplings with polygonal profiles.

Fig. 2. Polygonal profile with n 3 lobes and its parametric equation.

R. Citarella, S. Gerbino / Journal of Materials Processing Technology 109 (2001) 3037 31

biggest cost of performing a typical structural analysis task

is the engineering time required for the pre- and post-

processing phase, because run times in this field are no

longer critical, at least for linear problems. That is why there

is an increasing interest in BE methodology, which is

characterised by very low pre-processing times, effectively

balancing longer run times with respect to FEM [16].

3. CADCAE integration

CAD and BEM systems work extremely well together

because each uses identical geometric entities, such as

points, lines, splines, circles and patches (3D only). This

is particularly relevant in an industrial environment char-

acterised by numerous efforts aimed at the integration of

analysis systems with CAD systems.

Model transfer among different Computer Aided Every-

thing (CAx) systems often involves a lot of problems, related

to the correct model data exchange because differences in

database structure of CAx systems do not allow the direct

data transfer to other engineering systems.

The most used approach for CAD data transfer is the one

based on neutral format. This approach is based on a

common neutral format for exchanging data, and on a pair

of translators (read and write) between each application and

the common neutral format.

Initial graphics exchange specification (IGES) appears to

be the most used method of transferring geometric entities,

although this approach has a drawback. IGES scrambles line

and patch direction, whilst, inside a BEM solver, consistent

entity direction is critical in determining the type of analysis,

finite or semi-infinite, to be performed. Therefore the user

must spend time adjusting the data prior to attempting an

analysis [15].

In general, however, moving data among engineering

systems, by IGES or other translators, may cause loss or

alteration of data, independently from CAD user responsi-

bility [17]. For example, accuracy errors may occur, because

intersection curves between non-planar surfaces are

approximated in most solid modellers. Approximations

are required when the precise intersection between two

curved surfaces is too complex to be exactly computed.

The best a programmer can hope to do is to limit the

maximum deviation between the curve approximating the

intersection between two surfaces and the actual intersection

itself. CAD modellers employ different tolerances when

computing the maximum deviation allowed between topo-

logical entities such as faces, curves of intersection, and the

vertex where intersection curves meet.

When deviations between faces, edges, and vertex

become too large, processes using solid geometry may fail.

Translation between systems may also fail if the maximum

allowable differences between surfaces and edges in the

receiving system are smaller than those of the sending

system. Thus moving from a system using double precision

data representation, to another using single precision, will,

result in a loss of accuracy.

Preserving surface accuracy in the translation between

different representations, such as Bezier, B-spline or

NURBS, will often require mapping a single surface into

a collection of different surfaces in another representation.

Certain approaches, such as topology mapping, can mini-

mise the impact of this mapping, but the quantity of under-

lying data will increase too.

With regard to the aforesaid problems the following

remarks are due: it is necessary to set up the same geometric

accuracy in both engineering systems involved in the trans-

lation and to choose the proper parameters responsible for a

correct surface/line translation from CAD to IGES format.

In this work, for the IGES model generation, all surfaces and

curves have been converted in B-spline entities (respectively

128 and 126 IGES entities).

4. Problem analysis

A polygonal coupling, which undergoes a static torsional

torque, has been studied with reference to the induced stress

state.

The main part of this work concerns modelling the contact

surface between hub and shaft, because, from a structural

point of view, the interface is the critical area in this kind of

problem.

4.1. Contact analysis with BEM

BEM contact analysis [1821] preliminarily requires the

calculation of interface stresses and displacements, impos-

ing compatibility and equilibrium conditions.

In zero friction contact analysis only normal stresses

are transferred through the interface boundary, in such a

way that compatibility conditions are satisfied in normal

direction.

In particular, for a conservative analysis the friction action

has been neglected because it reduces the amplitude of the

stress and strain on the contact surface.

To determine the stress and strain behaviour, special

contact elements are used. Contact modelling by means

of BEASY code is implemented through corresponding

pairs of elements along the interface of the two zones in

which the overall domain is subdivided. The pair of corre-

sponding elements, called normal gap elements, even if

coincident in the pre-processing model, can be separated by

a positive (interference) or negative (clearance) gap. By the

sign of the calculated normal stress it is possible to deter-

mine the contact state: negative indicates contact, positive

clearance.

Contact problems usually are non-linear because of the

contact area variation with increasing load and eventual

sliding, so the stress state in the interface area depends on

load, geometry, material and contact area extension.

32 R. Citarella, S. Gerbino / Journal of Materials Processing Technology 109 (2001) 3037

In the present work non-linear behaviour of the model has

been studied by an iterative-incremental procedure. The

incremental step is aimed to follow the contact evolution

while the iterative step allows to obtain an equilibrium

configuration for each load increment. In order to follow

the contact evolution, the load is gradually increased by few

sub-steps and for each of them an equilibrium configuration

is iteratively searched by Newton Raphson algorithm, con-

ditional upon compatibility normal displacement restraints.

The proper choice of the convergence tolerance (1.5% in

this case) and of the maximum number of load sub-steps (15)

and iterations (15), is critical in order to achieve stable results.

To this aim, a convergence study has been carried out,

with mesh refining mainly oriented to the high gradient

contact area.

In this case, it is necessary to refine the mesh in those

areas where the contact state changes, getting high stress

gradients. Actually, as mentioned before, the convergence

study should also involve load step size and normaltangen-

tial convergence tolerance values, but these parameters are

not accessible to the users in BEASY code, so default values

have been adopted.

4.2. Problem description

Three different polygonal couplings with three-lobe pro-

file, made by steel, are studied with a parametric CAD 3D

system (Pro/ENGINEER R.17 by PTC), in order to make the

coupling model, whilst the structural analysis is carried out

by a BEM solver (BEASY 7.0).

Fig. 3 shows the polygonal coupling model with para-

metric characteristic dimensions. Three models (A, B and C)

are characterised by the following common dimensions:

Ds 80 mm, L1 20 mm, L2 40 mm, R 5 mm,Dm 60 mm, and separately by: A-model: e 2:25 mm, Dh 155 mm; B-model: e 2:25 mm, Dh 120 mm; C-model: e 3:50 mm, Dh 120 mm.

These models will be used for an extensive benchmark of

the alternative FEMBEM procedures.

C-Model is shaped with the optimum profile, charac-

terised by an improved stressstrain behaviour, as obtained

by Mechnik [9] with a shape optimisation analysis.

All the models will be compared each other with refer-

ence to some parameters, representative of the elastostatic

behaviour. Material properties are based on isotropy, homo-

geneity and linear elasticity assumptions.

By correctly imposing cyclic symmetry kinematic con-

ditions on one third of the models it has been possible to

reduce considerably the computational efforts.

One cylindrical shaft end has been supposed to be

clamped and the torque (Mt 1000 N m) is introducedon hub contour by tangential forces. In Fig. 4, the BEM

A-model is shown.

The Young module and Poisson ratio for the steel con-

sidered are respectively E 206; 000 N/mm2 and n 0:3.As already mentioned a BEM code (BEASY) has been

used for the non-linear stress analysis and the convergence is

obtained through an iterative-incremental technique.

In the final A-, B- and C-models, the following total

numbers of reduced quadratic elements (eight nodes) are,

respectively, used: 526, 524 and 498.

For the B-model it has been attempted the hub subdivision

in two zones, in order to get enhancement in calculation

times, but without success (it is probably due to the non-

linearity of the problem). That is why, B- and C-models have

different number of elements, even if they are completely

similar.

High mesh grading has been adopted in the interface area

(Fig. 5), where contact between shaft and hub causes very

high stress gradients.

The pre- and post-processing phase are carried out in the

BEASY environment.

Fig. 3. Three-lobe polygonal coupling parametric dimensions. Fig. 4. BE polygonal coupling model.


4.3. Analysis results

Figs. 6 and 7 respectively show normal stresses and tangen-

tial stressesonthehubcouplinginterfaceareawith reference to

different values of z (axial axis), for the A-model.

The contact surface is the most stressed, in particular with

reference to the hub part for which the results are presented.

It is well evident that the results, obtained through an hp

convergence study, are well in agreement with FEM results

(Figs. 8, 9) from Mechnik [9] even if there is some dis-

crepancy (up to 10% in just a few control points).

Actually, further convergent studies should be done with

regard to reduced values of the normal convergence tol-

erance (this parameter has yet to be customised in the

BEASY code), in order to avoid few hot spots still

existing in the contact area.

In the disconnected area, BEM normal stresses are rig-

orously zero because BEM does not need to extrapolate

results from the interior towards the surface as in FEM.

With respect to the A-model, in Fig. 10 a comparison is

made between experimental [9] and BEM results for that

concerns the equivalent stress seq calculated with the totalstrain energy criterion, on the external hub surface at

z 40 mm.Figs. 11 and 12 show, for the same A-model, the compar-

ison between experimental [9] and BEM results for that

concerns the normal displacements at z 0 and z 40 mm,respectively.

Fig. 5. BE polygonal coupling interface.

Fig. 6. Contact normal stress by BEM.

Fig. 7. Contact tangential stress by BEM.

Fig. 8. Contact normal stress by FEM [9].

Fig. 9. Contact tangential stress by FEM [9].


One of the criteria adopted in order to decide the critical

state for the coupling is based on the maximum deformation.

In particular, comparing BEM and FEM results, relative

tangential displacements between hub and shaft are pre-

sented in Fig. 13 for B- and C-models.

It is well evident that increasing the eccentricity it is

possible to obtain reduced values of coupling deformation,

as reported in more detail in Figs. 14 and 15 for that concern

hub tangential displacements and in Figs. 16 and 17 for hub

normal displacements, with reference to B- and C-model

respectively.

Moreover, there is an evident improvement from the

elastostatic point of view, getting higher Von Mises stress

for B-model (Fig. 18) with respect to C-model (Fig. 19).

In Figs. 20 and 21, deformed configurations are shown for

B- and C-model respectively, in the same deformed scale so

as to be comparable.

For a correct evaluation of a certain degree of discrepancy

still existing between BEM and FEM results, it is necessary

Fig. 10. Equivalent stresses for the BEM A-model compared with

experimental ones on the external hub surface at z 40.

Fig. 11. Hub normal displacements on the external surface at z 0.

Fig. 12. Hub normal displacements on the external surface at z 40.

Fig. 13. Relative tangential displacements between hub and shaft for

B- and C-models.

Fig. 14. Hub tangential displacements on the interface contact area for

B-model.


Fig. 15. Hub tangential displacements on the interface contact area for

C-model.

Fig. 16. Hub normal displacements on the interface contact area for the

B-model.

Fig. 17. Hub normal displacements on the interface contact area for the

C-model.

Fig. 18. Hub Von Mises stress on the interface area for B-model.

Fig. 19. Hub Von Mises stress on the interface area for C-model.

Fig. 20. Deformed polygonal coupling (B-model).


to point out that the comparison is made between two

approximated numerical solutions and in some case with

experimental results (affected by a certain degree of appro-

ximation) but never with analytical solutions. Moreover

different kind of elements order, linear in FEM analysis

and quadratic in BEM analysis, are used.

5. Conclusion

Nowadays, there is an increasing interest in BE metho-

dology, which is characterised by very low pre-processing

time with respect to FEM, effectively balancing longer run

times (about 6 h for the studied models, with a PC Pentium

Pro, 128 Mb of RAM).

The use of the boundary representation, like in BEM,

integrates well with other Computer Aided Design and

Manufacturing systems, which use a similar description

of the component geometry.

Future perspective of the present work is turned to the

study of sintered steel coupling undergoing fatigue load, and

to 3-D automatic crack propagation, for which BE metho-

dology is well suited.

Acknowledgements

The authors want to express their warmest thanks to

Prof. R. Esposito for his precious help in BE analysis and

to Prof. F. Caputo for his assistance.

References

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Fig. 21. Deformed polygonal coupling (C-model).


Shaft-hub Couplings With Polygonal Profiles_Citarella-Gerbino2001

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