Calculating Pressure Concentration Factors In Keyless Locking Assemblies

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,  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.

The  accurate  determination  of  contact  pressure  distribution  is  no  small matter. Knowing the position and intensity  of  any  tension  peaks  facilitates  the job of the engineer, who will thus be able to make full use of the mechanical capabilities  of  the  system  in  question  while still  maintaining  adequate  safety  levels.

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  locking  assembly  is  usually mounted  on  the  shaft  and  the  screws pre-tightened  in  the  tried  and  tested criss-cross sequence to limit possible eccentricity in the final configuration. The hub is mounted outside the locking assembly and enables the transmission of torque. 

 

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

were 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. 

 

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.