A. Dadalau a ∙ K. Groh a ∙ M. Reuß a ∙ A. Verl a
Modeling linear guide systems with CoFEM - Equivalent
models for rolling contact
Stuttgart, February 2011
a
Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), University of
Stuttgart, Seidenstraße 36, 70174 Stuttgart/ Germany
[email protected]
www.isw.uni-stuttgart.de
Abstract Today’s machine tools are highly complex mechatronic systems capable to exert large
translational and rotatory movements. In most cases rolling bearings are used to connect the moving
parts to each other. As full FE models of rolling bearings can consume a large amount of degrees of
freedom (DOF) efficient methods for reducing the DOF consuming rolling elements to more simple
equivalent models are needed. As an example a linear guide system is used. A special feature of the
considered linear guide is that the runner block consists of three separate parts, which are hold together
only by pretension and friction. FE simulations of such linear guide system were not reported before in
the literature. Beside the full FE model three equivalent contact models are presented. The first two
equivalent contact models feature novel characteristics. Advantages and disadvantages of the
equivalent models are discussed using as reference a slice of the full model and simulation results of
static stiffness. The validation of the numerical models is also done using the general analytical
solution of Hertz. An explicit general formula for the dependence between displacement and force is
given which was, at least in the classical literature, not noticed before.
Keywords FEM ∙ linear guide ∙ rolling contact ∙ Hertz ∙ plasticity
Preprint Series
Stuttgart Research Centre for Simulation Technology (SRC SimTech)
SimTech – Cluster of Excellence
Pfaffenwaldring 7a
70569 Stuttgart
[email protected]
www.simtech.uni-stuttgart.de
Issue No. 2009-1
Modeling linear guide systems with CoFEM –
Equivalent models for rolling contact
A. Dadalau, K. Groh, M. Reuß, A. Verl
Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW)
+49 (0)711 / 685-82463
+49 (0)711 / 685-72463
[email protected]
http://www.isw.uni-stuttgart.de/
Today’s machine tools are highly complex mechatronic systems capable to exert large translational and rotatory
movements. In most cases rolling bearings are used to connect the moving parts to each other. As full FE models
of rolling bearings can consume a large amount of degrees of freedom (DOF) efficient methods for reducing the
DOF consuming rolling elements to more simple equivalent models are needed. As an example a linear guide
system is used. A special feature of the considered linear guide is that the runner block consists of three separate
parts, which are hold together only by pretension and friction. FE simulations of such linear guide system were
not reported before in the literature. Beside the full FE model three equivalent contact models are presented. The
first two equivalent contact models feature novel characteristics. Advantages and disadvantages of the equivalent
models are discussed using as reference a slice of the full model and simulation results of static stiffness. The
validation of the numerical models is also done using the general analytical solution of Hertz.
FEM ∙ linear guide ∙ rolling contact ∙ Hertz ∙ plasticity
Introduction
Rolling bearings can have an important influence on the dynamical behavior of a machine tool.
Therefore the exact determination of stiffness and damping of rolling bearings is crucial. As computer
resources are increasing, simulations of rolling contact bearings become even more interesting. In
general linear guides are simulated with the help of FEM (Finite Element Method). The level of detail
can be very different, reaching from simple linear springs [10] up to full 3D FE models with accurate
contact representation [9]. Simple linear or nonlinear spring models of linear guides have the
advantage of fast solution times. However finding exact stiffness parameters is an issue. On the
contrary, 3D FE model of linear guides implicitly contain accurate stiffness information but the
computational time is enormous. In conclusion more efficient methods of reducing model size by
simultaneously preserving the original properties are needed. In this work a linear guide system with
rolling contact is considered. A special feature of the considered linear guide is that the wagon consists
of three separate parts (runner block and rails), which are hold together only by pretension and
friction. Due to the system complexity a new method for component oriented FE (finite element)
modeling is used to model the linear guide in ANSYS. The concept of component oriented modeling
is already being successfully applied in commercial CAD (Computer Aided Design) programs as a
standard modeling method. By dividing a very complex model in smaller and simpler subcomponents,
the modeling task is easier to manage, the error probability is decreasing and the model is more
flexible regarding changes of subcomponents. In contrast component oriented modeling for MBS
(Multi Body System) or FEM is far from being a standard modeling method. Fortunately some
researchers recognized the potential and are developing the method in most cases for MBS, [1]. In this
work the concept is being developed for FEM under the name “Component Oriented Finite Element
Modeling” or shortly CoFEM, [2]. For the considered linear guide system the modeling process with
CoFEM is depicted in Fig. 1. The assembly of subcomponents is done automatically with the help of
implemented algorithms and graphical user interface in the FEM software ANSYS. The assembled
model is complex due to the 114 contacting surfaces (e.g. contact between balls and rails, runner block
and rail, screws and linear guide). Nevertheless due to the modular modeling concept repeating
components such as balls need to be modeled only once, whereat the multiple assembling of the same
component at different positions and with different orientations is done automatically by the
implemented algorithms.
Fig. 1 Modeling of a linear guide with CoFEM
The aim of this work is to find more efficient ways of simulating the linear guide system shown in Fig.
1. As for this purpose many tests are required, simulating the full model in Fig. 1 would be very
inefficient. A slice model of the full model is used instead. Then mesh convergence testing is
performed in order to provide an accuracy of computed displacement below 1%. The converged mesh,
boundary conditions and applied load are depicted in Fig. 2.
Fig. 2 Slice model of linear guide system, converged mesh shown.
From the mesh convergence testing on the slice model it can be concluded that already 1% accuracy
requires a large number of DOF for the full model. The highest amount of DOF is consumed by the
rolling elements, in this case by the balls. A single ball model would require 235000 DOF. Simulating
a quarter of the linear bearing in Fig. 1 would require 12 balls, thus leading to 2.82 millions DOF just
for the balls. Therefore new methods for reducing the amount of DOF needed for the balls are
proposed and presented here. Some of the presented methods preserve the capability of modeling
rolling contact of the full model. As it will be shown this capability plays an important role for the
overall stiffness of the considered linear guide system.
Stiffness of linear guide system
In Fig. 2 the slice model of the linear guide system with within 1% converged mesh is depicted. Since
the load plate and the support are not available in this model, simplified boundary conditions are
assumed. Thus the upper surface of the runner block is modeled as rigid using coupling equations on
all translational DOF. Similarly the lower surface of the guide is constrained in all translational DOF.
Furthermore the specific normal contact stiffness of the joint patches between runner rail and runner
block, balls and runner rail, balls and guide rail is expected to be high, due to the polished contact
surfaces. In the literature there is little information on the contact stiffness of polished surfaces. In [3]
the stiffness of contact pair milled-polished is measured and for high normal pressures (
a value of 3.
)
is identified. Newer experiments [4] revealed a normal stiffness for grit-
blasted surfaces with roughness
of about
, which is 430 times higher
than the stiffness found in [3]. This difference could reside not only in different surface roughness but
also in different surface ondulation, which is not captured by the average roughness. In general the
stiffness of fixed joints is very hard to predict and must be measured for each special case. For the
linear guide system with smooth polished-polished contact surfaces the normal contact stiffness should
be higher than found in [4]. In the slice model of linear guide system a normal contact stiffness of
(
) is set up. This value is found by trials aiming minimization of contact penetration
and simulation time at the same time. Furthermore choosing a high value minimizes the influence of
the contact stiffness on the overall stiffness of the linear guide. So the slice model is expected to
provide the upper limit for the reachable stiffness of the full linear guide system. All the contact
surfaces are set to have a constant friction coefficient of 0.08. Setting up friction in the FE model is
more important for the numerical convergence rather than for the results, since no macro-sliding in the
contact is expected.
In Fig. 3 the vertical displacement of the linear guide system is plotted over the increasing
compressive load for two different ball pretensions. In the first case the balls diameter is 4 µm bigger
than their contact point distance on the rails. In the second case the balls diameter is 18 µm bigger than
their contact point distance on the rails. In Fig. 3 the force is scaled to an equivalent force of the full
model, which is 24 times bigger than the slice model. Qualitatively the simulation result correlates
very well with manufacturing data: The displacement curve for the lower pretension case manifest for
loads lower than 5 kN a slight progressive and for loads bigger than 5 kN a slight degressive character.
On the other hand the displacement curve for the higher pretension case manifests only slight
progressive character. This phenomenon is caused by the two different states of ball pretensioned
systems: a) with pretension and b) without pretension. In the case a) the two pretensioned balls act
together as two parallel springs. Due to the nonlinearity the resulting displacement behaves
progressive. In the case b) only one ball is active and the displacement behaves degressive, as
described by the Hertz theory. The lower pretension is already lost at 5 kN, so the displacement
behaves first progressive and then degressive.
Fig. 3 Numerically computed runner block displacement of the slice model (MS=0.5)
Another interesting result of the simulation is that the contact angle of the ball change when load
force is increased. Without external load
. In Fig. 4 the contact stress on the lower ball in Fig.
2 is shown for different external loads. By graphically connecting the middle points of the contact
surfaces the change of contact angle
become visible. This effect is important as the contact angle
directly affects the overall stiffness of the linear guide. Furthermore the contact angle influences the
amount of pretension between runner rails and runner block, which in turn affects the overall stiffness.
A more detailed explanation of this effect is given in a following chapter.
Fig. 4 Change of the ball contact angle with increasing load
On the theory of contacting ellipsoids
Considering the in core computational time for the previously presented slice model of about 14 hours
on a eight cores CPU, we can imagine that simulating the full model of the rolling guide (which would
be 24 times bigger) would be very unpractical. So a more efficient model is required. Considering that
the balls are consuming the highest amount of DOF, we analyze a more efficient modeling technique
based on the theory of contacting ellipsoids.
The Hertz theory can describe the most general case of contacting ellipsoids with elastic isotropic
material. In most of the literature the relationship between force and contact pressure can be found,
whereas for the force-displacement relationship more or less explicit information is available. In [5]
and [6] the theory of contacting ellipsoids is explained very comprehensively, but no explicit solution
is provided for the force-displacement behavior of contacting ellipsoids, which is needed when
defining a nonlinear spring for a FE model. These solutions can be found for example in [7] and [8].
For the reader’s convenience we give a summary of the relevant equations. We consider two
contacting ellipsoids of contact radii
,
and ,
respectively, Fig. 5. The material properties
are represented be the elasticity moduli ,
and Poisson ratios
, . Furthermore the relative
angle between the vertical symmetry planes of the two ellipsoids is defined as .
Fig. 5 Variables defining two contacting elastic ellipsoids
In [5] and [6] derivated variables are introduced, which are recapitulated here in a slightly changed
form:
(1)
;
(2)
( )
(3)
where
(
((
)
)
(4)
(
)
(
)(
)
The constants , , are only material dependent.
and
depending on the curvature radii of the ellipsoids. The semi axes
given by
(
) ;
(
)
)
(5)
are purely geometric constants,
and of the contact ellipse are
(6)
where is the acting force and and are coefficients depending on and which can be found with
the help of look-up tables, see [5]. Finally the approach of the two ellipsoids is given implicitly by
∫
(7)
√
and
∫
(8)
√
∫
(9)
√
In order to get an explicit solution for (7) in terms of the previously introduced constants we have first
checked that the integrals in (7), (8) and (9) are well defined. Then
∫
∫
√
(
)
√
So
(10)
√
(
)
√
results to
(11)
Special care has to be taken if
: in this case
and
must be exchanged:
(12)
It can be seen from (2) that
(
and
are pure geometric constants. In conclusion we obtain:
(13)
)
Equivalent models for rolling contact elements
In the following the previously derived explicit formula for two contacting ellipsoid is applied for the
case of a ball contacting two concave cylindrical surfaces in order to derive two different reduced
models, thus making the simulation of linear guides more efficient. The reduced models are both
capable of approximating the effect of contact angle change, which due to the contact between runner
block and runner rails greatly affects the overall stiffness of the linear guide system.
FE model with rigid balls
The first proposed reduced model replaces the flexible balls with rigid spheres. The novelty of the
proposed model is the use of mathematically determined, equivalent material property such that the
resulting stiffness remains theoretically unchanged. In concrete the unwanted increase in contact
stiffness is compensated by locally decreasing the elastic modulus of the runner block and rail guide.
From the theory presented in the previous section we can precisely derivate an equivalent elastic
modulus. The displacement in equation (11) can also be written as
(14)
,
where is a constant not depending on the material stiffness. Making the ball rigid, which means
, leads to a smaller displacement:
.
In order to make
(15)
the elastic modulus of the second contact body is changed to
. Now the write hand sides of (14) and (15) equate to:
, meaning
=
(16)
and the equivalent elastic modulus reads as follows
(17)
This new approach is validated against the model with flexible spheres in Fig. 2. In the slice model the
flexible balls are replaced by rigid balls, Fig. 6. The finite elements within a radius equal to twice the
ball radius around the contact points of each ball are assigned with equivalent elastic modulus ,
which in the case of two contacting bodies of same materials is 50% of the original elastic modulus
(steel,
). The new model, which we name RiBEM=Rigid Balls with Equivalent
Material, has only 45000 DOF and the solution is computed in 2 hours.
Fig. 6 Slice model of linear guide with rigid balls
Full model, 3µm pretension
RiBEM, 3µm pretension
Full model, 9µm pretension
RiBEM, 9µm pretension
50
Displacement [µm]
40
30
20
10
0
0
5
10
15
Total Force [kN]
20
25
Fig. 7 Comparison between slice model with rigid and flexible balls
From Fig. 7 we can conclude, that the proposed RiBEM model correlates very well with the full model
(flexible balls). The maximum difference of 2.8% and 5.8% respectively occurs at the maximum
loading force of 24 kN, which is acceptable considering that the full model takes approx. 7 times
longer to compute (14 hours).
FE model with nonlinear springs
Despite the radical reduction of DOF, the disadvantage of the RiBEM model is the required fine mesh
in the vicinity of the contact points on the rails. In the following a combination of one nonlinear spring
(COMBIN39, [12]) and one pretension element (PRETS179, [12]) is used to replace the flexible ball.
The nonlinear force-displacement relationship of the spring is computed with the help of equation (13)
but due to the two-point contact of the ball the displacement is doubled. The pretension element acts
trough two stiff beam elements on the spring element. Further, one spring node and one pretension
element node are coupled each with a cylindrical surface using surface based contact elements
(CONTA173 and TARGE170 as pilot node, [12]), see Fig. 8. The cylindrical contact surfaces of the
reduced ball model are then automatically coupled with the rails of the linear guide model with the
help of CoFEM using contact elements.
The novelty of the proposed equivalent model is that the nonlinear spring is set to have the working
direction defined by the locations of its nodes and by the special location of the nodes relative to the
contact surfaces. The spring nodes are positioned in such a way that after solving the pretension load
step, the nodes are at center of curvature of the rails. For the rigid contact between a ball and two
cylinders, the kinematic constraints enforce that the contact points are lying on a line parallel to the
line connecting the curvature points of the cylindrical surfaces, Fig. 9. This fact is exploited in the
proposed equivalent model of rolling contact, which we name RoCS=Rolling Contact Spring.
Fig. 8 Equivalent model with nonlinear spring and pretension element
Fig. 9 Kinematic constraints of rigid rolling contact with friction (left) and frictionless (right)
The RoCS model is compared with what we call a FiCS=Fixed Contact Spring. The FiCS model is
similar to the RoCS model, but the work direction of the spring is defined by its nodal coordinate
systems. A FiCS-similar model is used in [7, 10], but there the pretension is applied by using a
additional layer of elements in the runner and the effect of thermal dilatation, rather than using
3µm
pretensionfor the RoCS and FiCS models the slice model
pretension elements directly inFull
themodel,
ball. As
a reference
pretension displacements for increasing load are shown in Fig.
with flexible balls in Fig. 2 is FiCS,
used. 3µm
The computed
RoCS,
3µm11
pretension
10 for both pretension cases and
in Fig.
only for the low pretension case. The displacement field in
Full
model,
9µm
pretension
Fig. 11 is scaled by factor 50.
FiCS, 9µm pretension
Full model, 3µm pretension
RoCS, 9µm pretension
FiCS, 3µm pretension
50
RoCS, 3µm pretension
Full model, 9µm pretension
FiCS, 9µm pretension
RoCS, 9µm pretension
Displacement [µm]
40
50
Displacement [µm]
40
30
30
20
20
10
10
0
0
0
0
5
5
10
15
Total Force [kN]
10
15
Total Force [kN]
20
20
25
25
Fig. 10 Influence of rolling contact representation on the overall stiffness
Both Fig. 10 and Fig. 11 show that the RoCS model can better approximate the overall stiffness than
the FiCS model. But still, the maximum difference between the RoCS model and the reference model
amounts to 23% for the higher pretension case. Replacing the rolling elements in linear guide models
by Hertzian springs inevitably induced an error related to the fact, that the second contact surface does
not belong to a half-space as stated in the Hertz contact theory [11]. Another source of error can be
seen in Fig. 10, where the runner rail displacement is very sensitive to the ball contact angle. This can
be explained by the orthogonality between initial contact normal of the ball and contact surfaces of the
runner rail. Small changes in the contact angle of the ball are causing a sliding of the runner rail along
one of its contact surface. Since the FiCS model underestimates the contact angle change, no sliding
occurs. On the other side the fixed contact angle leads also to a lower ball stiffness, so the overall
stiffness is also lower for the FiCS model. From these considerations we can also conclude that sliding
of the runner rail could be avoided if its contact surfaces would build an angle smaller than 90°, which
inevitably would increase the overall stiffness of the linear guide system.
Fig. 11 Influence of rolling contact representation on the displacement field of the runner
Conclusion
In this work we presented finite element simulation results of a linear guide, which is modeled with
the help of a new efficient component oriented method called CoFEM. A special feature of the
considered linear guide is that the runner block consists of three separate parts, which are hold together
only by pretension and friction. For the first time FE simulations of such linear guide system are
presented. It is shown that the correct representation of rolling contact and thereby of the changing
contact angle is important for the computed overall stiffness of the linear guide system. In order to
account for these affects and still provide efficient simulation times, the DOF consuming balls are
replaced by two new equivalent models called RiBEM and RoCS. In the case of the RiBEM model
simulation time is reduced by factor 7, whereat the accuracy of the computed displacement remains
almost unchanged. In the case of the RoCS model, the simulation time is further reduced, but due to
insufficient rolling contact representation the induced error amounts to 23%. RiBEM provides the
optimal trade-off between computation time and accuracy. With the help of the theory of contacting
ellipsoids, which is briefly introduced de the equivalent Joung modulus for the RiBEM model can be
determined exactly. Further work will be the experimental validation of the presented RiBEM and
RoCS model and comparison with manufacturer’s stiffness data.
Acknowledgements
This research was supported by the Institute of Control of Manufacturing Units Stuttgart (ISW), by the
excellence cluster SimTech Stuttgart and by GSaME Stuttgart. This support is highly appreciated.
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