Sandro Zamboni and
MassimilianoMargonari, MAV S.p.A.
Tradelink Services has been in the field of Mechanical Engineering for over 25 years. We have introduced some of the most highly successful and innovative products from Europe in the field of Mechanical Power Transmission. Incorporation of these products in industrial systems leads to increased efficiency, lower downtime and enhanced profits.
Our Italian
principals MAVSpA, Italy have more than 20 years of experience in the power
transmission field. With deep collaboration with customers MAVs technical
department provides innovative and alternative solutions for various problems
and applications.
Introduction
One of the most important goals
for designers and producers of
mechanical components is
to supply customers increasingly high quality products
with clearly defined performance expectations.
For this purpose, MAV S.p.A., an
established manufacturer of locking assemblies and shrink
disks, began to explore several years
ago the
methods of computer simulation to improve the components
in production and to support the design process. This has substantially reduced
laboratory testing, limiting its use to simply validating certain
configurations, resulting in a significant reduction in times and costs for the
certification of new products.
There are a wide number of
different reasons that led the MAV engineering division to make this decision. Firstly,
there is the management’s belief that investing in applied research and
numerical simulation is an effective means to ensure the best possible results
at the least possible cost, with obvious advantages; secondly, there is the
growing technical need for fast, reliable and accurate instruments to offer the
best possible response to the demands of the market.
The technical
data of most
interest to the mechanical engineer designing a locking
assembly are usually
the values for maximum transmissible
torsional and flexural moments. Once the geometry, friction coefficients
between the surfaces and the pre-tightening torque applied to the screws have
been established, these
values may be calculated using a
number of equations formulated on the
basis of
simplified models, which are
widely accepted and used by engineers. The results obtained are
always safety biased;
the mathematical models used
often provide transmissible
moment values well
below the real capabilities of the
system, as demonstrated by a series of tests conducted by MAV on a variety of
different products.
It is useful to know the
distribution of contact pressure between the locking assembly and the
shaft throughoutthe system’s life cycle
and, in particular, during assembly and certain operating conditions.
Unfortunately, there are no simplified models to help provide
indications on the distribution of tension generated by the locking assembly on
the mechanical components that it connects. In fact,no laboratory test currently
exists that is capable of giving
even partial answers to
these questions. Literature
cites the results obtained
in the early
1980s and a number
of indications on
the matter.
These results were obtained via
the finite element analysis of extremely simplified, axially symmetric models, probably because of the limitations of the hardware
and software resources
available at the time.
However,
while the data obtainable from two-dimensional simulation models may
undoubtedly be interesting, and MAV has also made use of this technique, they
do not provide any information on the behaviour of locking assemblies near the
gaps in the rings, where peak tensions are expected.
To
shed more light on this aspect, MAV has conducted a detailed study, using an
entirely numerical approach, to produce more precise and reliable data
concerning the real distribution of forces generated on a shaft by a locking
assembly. The study considered the MAV 1008, 4061, 2005 and
1061 series of
products.
For
each series, four different shaft sizes were considered to give the broadest
and most detailed vision possible.
The study also developed
three-dimensional models with
extremely accurate geometrical representation.
The resulting models were highly
complex and demanding in terms
How a Locking Assembly Works
Details below illustrate the four different types of locking assembly considered for this study. They also contain pertinent information concerning geometry, the screws used and the theoretical reference values for contact pressure obtained with simplified models.
The operating principle for a
locking assembly is rather
simple: a number
of conical section rings are brought together by tightening screws,
which generates high contact pressures between the shaft, the hub and the
locking assembly itself. This arrangement holds the components tightly together,
enabling the transmission of torque. The rings in contact
with the shaft and the hub always have a longitudinal gap to reduce their
circumferential rigidity, facilitating
and improving the elimination of
free play between components.
The hub
must be appropriately continued sized to effectively
oppose any radial deformity of the locking assembly and may, for reasons of
space but also for aesthetic or cost reasons, also incorporate drums, gear wheels
or any other
mechanical component deemed necessary.
Once the pre-tightening torque
for the screws has been applied, the system consists of parts solidly
connected to one another and may be subjected to external loads. The MAV 1008
and 2005 are defined as self-releasing locking assemblies.
If the screws are removed after fitment
of the
locking assembly, they
tend to loosen and
return to their
initial undeformed configuration. This is due to the fact that the rings
have a highly conical section (8, 10°),
and the coefficient
of friction, usually considered to be 0.12, is not high enough to keep
the components in the deformed configuration. This is an extremely desirable
characteristic, as the locking assembly
may be mounted
and removed numerous times during its life cycle. Conversely,
the MAV 4061
and 1061 are known as self-locking components. With a conical section of
less than 5°, these exhibit the
opposite behaviour of that
described above. In this case, the screws
perform no particular
structural role and serve only to deform the rings sufficiently.
As
can easily be
understood, the pressure generated
on the shaft
is not uniformly distributed
(which would be considered an ideal condition) and varies
both longitudinally and circumferentially
due to the varying rigidity of the
holes of the rings constituting the locking assembly
Furthermore, as mentioned previously, contact between the gaps in the
rings and the shaft and hub constitutes a substantial element of disturbance
and can cause undesirable tension peaks.
This is why three-dimensional modelling was chosen, -focusing
on zones of discontinuity, such as edges and gaps in the rings.
CAD Models
Autodesk
Inventor 10software was used to produce the three-dimensional models of the locking
assemblies considered. The capability of the software to parameterize the geometriesmodeled was
extensively exploited to speed up the preparation of the models
themselves. While this
required more work than usual procedures, it permitted
the analysis of
additional measurements when deemed
necessary,
Without requiring significant
extra effort. Painstaking
care went into modelling, bearing
in mind from
the start of the procedure that the geometries created would
subsequently have to be processed 
by a finite element
mesher. In particular, a substantial “defeaturing” process was applied to the
geometries to minimize the number of nodes and small surfaces that do not
contribute significantly to the definition of elements and do not influence structural response.
This presented clear advantages during mesh
construction.
The external
surfaces of all
components were subdivided into as uniformly shaped quadrilaterals as
possible, permitting amore
uniform subsequent meshing
process, especially in contact areas.
The IGES format was chosen to facilitate
transfer of the geometries into the finite element simulation environment.
FEM Models
Ansys 10.0software was used for
the numerical simulations. In particular, the Workbench environment
was used for
the preparation
of the models
and the
visualization of
the results, whereas
the
batch launch
was conducted on a Linux
machinefor the
actual analyses. Only
elementswith quadratic form
functions
The very lowrigidity of these
springs enabled a solution to be reached without significantly altering the response of the
system. Table 3 indicates the number of nodes and elements and the degrees of
freedom for each individual model. Note that
thenumber of contact and target contact elements is always identical in each
model.
This is because symmetrical
contact definition was always employed.
Managing the contact
elements is fundamental; augmented Lagrangian
formulation methods were always
used, monitoring penetration occurring
between bodies and modifying the
parameters in successive instances where necessary. A coefficient of
friction of 0.12
was chosen, as this value was considered sufficiently low
and statistically reliable
for describing steel-steel
contact. The material, which is identical in all parts involved, was considered to
be a linearly
elastic isotropic
material, with a Young modulus of
200 GPa and a
Poisson coefficient of
0.3. The only non-linearity considered in the models is due to the
presence of contact points with friction.
Contact Pressure
Concentration Factors
The primary
objective of this
study is to identify
a simplified quantity
for use during the design stage that is sufficiently representative of
the state of tension
induced on the
shaft by the locking
assembly. The following
formula is often defined
for this purpose:
Equation 3 represents the radial
force transmitted from thelocking
assemblyto the shaft and to the hub. In this study, Equation 1 was not
used because it is not considered
sufficient representative in this context.
Instead, a modified version of the
formula was proposed, as described as follows.
It
is known that
in finite element
analysis, when
a linear elastic
behavior is attributed to the material, there are no limits to
stress values attainable
within the bodies. Where there are concentrated forces, sharp corners or
contact between parts, the values for the state of tension in some nodes may
increase indefinitely and congest the
calculation mesh.
This is a logical consequence of
the fact that elastostaticequations
permit the nondefinition of the tension state in certain
points in space; a well-known example of this is the Boussinesq problem, to
which there is an analytical solution.However, the integral of the state of
tension calculated for a fi nite domain that also includes singularities is
well defined and represents the result of the applied forces. In the case of a locking assembly,
the transmitted radial
force (Eq. 3)
always assumes fi nite values, even though there are
points in which
the contact pressure p(x) is not
defined or, in the case of an FEM model, increases indefinitely to congest the
mesh.
For
this reason, the following definition was introduced
the state of strain at the contact
surface and, therefore, cannot provide exhaustive information regarding the
effective state of tension. Another limit consists in the fact that the FCPprovides
no description of stress distribution, which is important in evaluating whether
a situation is dangerous or not. For operational purposes, it was decided to
consider the pressure
at the centroid of the contact
elements. Mean pressure is therefore:
Load Histories and Boundary
Conditions
Load histories
consist of six
different steps: the
pre-tightening torque is assigned
to the screws
during the first step, simulating the assembly stage.
During the second step, a torsional moment equal to approximately 90% of the
theoretical slip value is applied to the hub. In the subsequent steps, the
torsional moment is maintained and the flexural moment is progressively
increased to a value equal to 50% of the corresponding torsional moment
applied. Throughout all load steps, circumferential displacement
of the nodes of one of the transverse faces of the shaft is inhibited, as is
transverse displacement of the node on the axis of the opposite face. This
allows the shaft to deform freely without
interfering with its internal
state of strain.
Results and Conclusions
Tables 4 and 5 give the principle
results obtained with the study. In particular, the tables specify the mean
pressure values on the shafts, as determined with Equation 6, and the pressure
concentration factors, determined with the procedure described previously.
Note how
the mean contact
pressures for a given measurement vary little with each different load
step. This is justified, on the one hand, by the fact that the contact surface
remains unaltered (no phenomena of detachment between the locking assembly
and shaft ever
occur), and, on the other hand, by the
fact that as maximum values increase (also by little), this is compensated by an
analogous reduction in minimum
values. Minimum pressure never assumes values
below 30 MPa, ensuring satisfactory
adherence
between the
locking assembly and
the shaft, minimizing the risk of fretting.
Note also
that an apparently
high pressure concentration value, as may be seen with the MAV 1061, is
not an indicator of poor
locking assembly quality or synonymous with low mechanical performance.
The FCP must always be evaluated
alongside the mean
pressure value and maximum transmissible loads, in relation
with the effective
requisites of the project.
This article has
briefly described the results
obtained using a number
of finite element
analyses conducted on four different
types of locking assembly. The main goal of the exercise was to determine, to a
satisfactory degree of precision, the concentration factor for the contact
pressure generated on a shaft by a locking device.
The main reasons for the study are the fact that this
value cannot be determined by laboratory tests and a scarcity of information
available in literature.
The results obtained are
interesting as they
demonstrate that, unexpectedly, peak contact pressure depends very
little or not at all on the value of the flexural moment applied, even when the
latter reaches values well above those encountered in practice.
The reduction in FCP with
increasing shaft dimensions, a behavior seen practically throughout the entire
series, is also very interesting.
