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.
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, thesevalues 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.
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 createdby 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.
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 functionswere used and the ‘Hex Dominant’ setting was enabled to ensure greater mesh uniformity, as this reduces the number of nodes used and achieves better results than other settings. ‘Weak springs’ were also used to prevent rigid behavior in certain parts of the model.
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.
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.