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MODELLING NON-LINEAR CONTACT STI

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					                 MODELLING NON-LINEAR CONTACT STIFFNESS
                          IN TYRE/ROAD CONTACT

PACS: 43.50.Lj

Andersson, Patrik B.U.
Division of Applied Acoustics, Chalmers University of Technology, SE-41296 Göteborg, Sweden;
patrik.andersson@chalmers.se

ABSTRACT
Noise generation, rolling resistance, wear, and grip are determined by the tyre/road interaction.
The small-scale surface geometry gives an area of real contact that increases with load over the
interface leading to a non-linear relation between load and indentation depth for the interacting
objects. The small length-scales of the contact geometry have traditionally been neglected in
numerical tyre/road interaction models. This paper presents a step towards refinement of an
existing tyre/road noise model by considering the small-scale geometry via a thin interfacial layer.
A model for calculating the interfacial stiffness from road surface geometry and tread stiffness data
is proposed. The interfacial stiffness is incorporated as non-linear springs in a numerical contact
model based on a spatial discretisation of the contact geometry. Results of load-indentation
relations and contact stiffness are presented for a single tread block in contact with a road surface.
The results show that the contact stiffness of a tread block is substantially lower than the stiffness
given by the bulk modulus of the objects. The contact stiffness varies substantially for typical
variations in load and indentation depths that appear in the tyre/road contact.

INTRODUCTION
The interaction between a rolling automotive tyre and a road influence noise emissions, rolling
resistance, and wear. It is of interest to optimise the tyre and road to reduce these outcomes.
The tyre/road interaction problem is delicate due to, for instance, the large dimensions of the
contact zone relative to the wave lengths on the tyre, the time-varying size of the contact zone
yielding a non-linear contact, and the fact that the interaction process is affected by small-scale
geometry. The two former issues have been successfully studied before and proposed models
exist [1],[2],[3], [4],[5], while the latter is still an open question.

The small-scale geometry, down to the micro-meter scale, affects the stresses and strains at
the surface of the interacting bodies. The area of real contact is often only a fraction of the
apparent contact area and is dependent on geometry, load, and indentation depth. The total
contact stiffness over a given area is dependent on the area of real contact yielding a non-linear
contact stiffness on the macroscopic scale; more and more asperities makes contact as the load
is increased, which gives more and more active junctions that contribute to transfer the force.
Stresses and strains are build up in the tyre due to the friction forces, which may give pronounced
stick-slip generated sound — also under steady state rolling conditions — for certain tyre/road
combinations. The friction forces between the tyre and road are also of importance for traction
where wet grip is the limiting case. It is of importance to have a detailed description of the stresses
and strains in the contact not only to get a good estimate of the contact stiffness but also to be
able to model friction.

This paper presents an approach to include the effects of small-scale geometry into contact mod-
els for larger contact areas. A model to estimate the non-constant contact stiffness is proposed.
The model is based on that the effects of small-scale geometry within a pair of contact elements
are modelled with non-linear constitutive relations between the force and the indentation depth,
i.e. by non-linear springs. This approach is sometimes referred to as a third body approach view-
ing the interfacial layer as a new body between the actual ones. The model is used to study load-
indentation relations and the corresponding development of contact stiffness between a block of
tread compound and a road surface as a function of load and indentation depth. As a limitation
only contact in normal direction is discussed in this paper.


CONTACT MODEL
A short description of the contact modelling approach and the model used is presented in this
section. The contact model is based on the work by Kropp and co-workers (e.g. [1] and [5]),
and has been presented by the author in [6] and [7] and is here only briefly repeated. The more
specific details on how to estimate the non-linear contact stiffness within each pair of contact
elements are presented in the next section.

The model that includes non-linear springs within each pair of contact elements is first presented
for a single pair of matching points, for clarity. Consider two bodies making contact in a single con-
tact zone (Figure 1). This zone can be modelled with a pair of contact elements, with the position
of each represented by a matching point. Figure 1 shows the general idea of the modelling ap-
proach, where a non-linear spring is added between the pair of matching points. The positions of
the matching points at the surface of body 1 and 2, when the system is at rest, i.e. when no forces
act or have been acting over the interface, are given by z1 and z2 , respectively. The distance is
given relative to a reference plane perpendicular to the direction of the contact. The displace-
ments of the matching points, u1 (t) and u2 (t), not including the small-scale deformation within the
element, are due to a force, F (t), and are given by a convolution with the Green functions of the
displacement response of each matching point, g1 (t) and g2 (t), respectively. The constitutive re-
lation between the force F (t) and the compression of the spring δ(t) is generally described by the
function f (δ(t)). The relation may also be formulated in terms of a spring stiffness, i.e. given by
the derivative of the function, k(δ) = df (δ)/dδ. The contact formulation is similar to the standard
penalty method but with a non-linear penalty function, if one regards the spring as the penalty
function. To summarise, the governing equations of the contact problem are

                          u1 (t) = F (t) ∗ g1 (t)   and u2 (t) = F (t) ∗ g2 (t),              (Eq. 1)
                                                            δ(t)
                                      F (t) = f (δ(t)) =           k(x)dx,                    (Eq. 2)
                                                           −∞

                               δ(t) = −d(t) = −z2 − u2 (t) + z1 + u1 (t), ,                   (Eq. 3)
where ∗ denotes the convolution operation.




Figure 1: Two dynamic systems with rough surfaces in contact (left). Geometry and modelling
approach of the problem (right).

The system described by (Eq. 1)–(Eq. 3) can generally not be solved analytically, and numerical
methods must be employed. The problem is extended to multiple pairs of contact elements and
is discretised in time yielding the equation system,
                               M                           M−1 N −1
                 u2,e (N ) =         Fm (N )gm,e (0)∆t +               F (n)gm,e (N − n)∆t    (Eq. 4)
                               m=1                         m=1 n=1
                                                             δm (N )
                               Fm (N ) = Fm (N − 1) +                   km (x)dx              (Eq. 5)
                                                            δm (N −1)


                                                     2
                                    δe (N ) = z1,e − z2,e − u2,e (N )                           (Eq. 6)

for e = 1, . . . , M , where M is the number of equations corresponding to each pair of matching
points. The iteration index N refers to the time N · ∆t where ∆t is some chosen fixed time step.
It was here assumed that the impedance of first body is infinite, i.e. u1,e = 0 for all e = 1, . . . , M .
This body will in the following represent the road with its large impedance relative to that of the
tread layer. Note that all elements on body 2, that will represent the tread, are coupled via the
first value of the Green’s function, gm,e (0), in (Eq. 4), but the constitutive relation at element m,
described by (Eq. 5), is independent of the compressions of the springs at adjacent elements.
Notice that the last sum of (Eq. 4) is only due to previous forces and is hence already given at the
present time step. The equation system is solved and the unknowns δm (N ) are found by applying
Newton-Raphson’s iteriative scheme at each time step. Results of force-indentation relations for
a contact geometry typical for a single tread block is presented below, but the non-linear relations
within the pairs of contact elements must be estimated.


NON-LINEAR CONTACT STIFFNESS FUNCTIONS
The stiffness functions of the non-linear springs within each pair of contact elements arising from
the smallest scales must be determined. There are several approaches available, either based on
detailed scans of geometry or by the statistical properties of the surface geometry. The following
approach is suggested and was used in the presented results. There are two fundamental require-
ments that the stiffness functions must describe: The interfacial stiffness (i) should start from zero
when the first contact is made and be monotonically increasing for increased load/indentation,
and (ii) should be infinite when complete saturation occurs. The latter case means that the sur-
faces of the bulks of the interacting objects have direct contact all over the interface and thus the
deformation only occurs in the bulks.


A model of a rigid and flat circular punch indenting an elastic layer is used as an approximation to
estimate the interfacial stiffness. This is a standard contact problem for which relations between
contact pressure, total force, and indentation are available in literature [8]. The actual contact ge-
ometry has to be translated to a radius of the punch and a thickness of the elastic layer. Assuming
that the indentation is made with a flat circular punch will give an overestimation of the stiffness for
cases where in reality several disjoint patches are present to transfer the force. A more detailed
model that considers coordinates that forms separated patches may be used and is a natural path
for future refinement of the model.


The roughness is assumed to yield a thin interfacial layer determined by the difference between
maximum and minimum height, zmax and zmin , of the road surface within the element under con-
sideration. For each element, the area of contact is estimated by considering the area of heights
above a given indentation (compression of the spring) δ,

                                  A(δ) =         H(zi − (zmax − δ))dA,                          (Eq. 7)
                                             i


where H(·) is the Heaviside step function and dA the area associated with each surface point.
The estimated area of real contact is used to calculate the equivalent radius of the circular punch

                                           req (δ) =       A(δ)/π.                              (Eq. 8)

The layer thickness is assumed to depend on the indentation δ as

                                       h(δ) = (zmax − δ) − zmin ,                               (Eq. 9)

where zmin is the lowest hight within the present contact element. This choice of layer thickness
assures that the interfacial contact stiffness becomes infinite for complete saturation (which in
practice never occurs). The relations for the flat circular punch or radius req (δ) indenting an
elastic layer of thickness h(δ) is used to numerically determine the stiffness functions. The typical
character of these stiffness functions was presented in [6].

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SIMULATIONS
Among the properties of the tyre/road interaction that may be studied with the proposed model the
following questions are discussed and answered: (i) How do the load-indentation relations for a
typical tread block / road surface contact look like? (ii) Are there large variations in load-indentation
relations between different positions on one given road surface? (iii) How do the results depend
on the chosen discretisation in the proposed model?

A wearing course of road built according to the standard ISO-10844 was used for the study. The
surface geometry is taken from the Sperenberg project, published in [9], where the wearing course
was scanned by means of laser technique. The surface geometry is given with a resolution of 38
µm. A tread layer of 1 cm thickness and an apparent contact area of 2 cm × 2 cm was considered,
which is a typical tread block size. The Green functions of the tread are calculated by using a
numerical model solving the equations in the frequency wavenumber domian as described in [10]
and [11]. Briefly, the Young’s modulus of the tread considered is on the order of 100 MPa with
a slight increase with frequency, the loss factor varies between 0.23 at low frequencies and up
to 0.80 at high frequencies. The Poisson ratio is 0.499, i.e. an nearly incompressible material,
and the density 1300 kg/m3 . Frequencies up to 51200 Hz was considered and which also is the
sampling frequency used in the contact algorithm.

The contact area are divided into elements, each represented with a matching point in the centre
of the element. Different resolutions with 4, 16, 36, 64, 100, 144, 196, 256, 324, and 400 elements
for the 2 cm × 2 cm contact area have been studied. Thus, the size to the elements was varied
between 1 cm × 1 cm to 1 mm × 1 mm. For each case stiffness functions describing the prop-
erties within the elements was calculated for each element using the 38 µm resolution between
considered heights. For instance in the 1 mm × 1 mm elements 690 surface heights are use. In
this preliminary investigation a static loading case was studied where the tread layer was pressed
0.1 mm into the wearing course. Notice, that both the contact model and the model for the Green
functions of the tread layer are generally formulated and allows also for dynamic loading cases.


RESULTS
As presented in [6], the presented contact modelling including the small-scale roughess gives a
considerable smoother and softer contact than a standard formulation with Lagrange multipliers,
which brutally states either no contact or full contact at each pair of contact elements. The differ-
ence is also seen as the number of active matching points is increased and the forces decreases
compared to the standard formulation.

Figure 2 (left) shows the the load-indentation relation for four different positions on the road sur-
face. As expected, there is clearly a non-linear relation between the total force over the interface
and the indentation of the tread into the wearing course. Figure 2 (right) shows that the corre-
sponding contact stiffness is lower when the first contact is made and increases as more and
more junctions make contact. The contact stiffness varies substantially between the different po-
sitions indicating that the spatial variations in the contact geometry are important to consider. The
contact stiffness for the case of a perfectly flat contact geometry giving direct contact of the bulks
would be on the order of 15 MN/m, which is substantially above the one when considering the
smaller length-scales in the geometry as the real contact geometry causes only a partial contact.

The static load on a tyre is on the order of 4 kN and numerical simulation shows that the dynamic
pressure varies ± 10% around the static one [12]. A typical contact patch has dimension of 10
cm × 15 cm yielding an average contact pressure of about 300 kPa. Thus the expected force
transfered over patch of 2 cm × 2 cm is 120 N. Figure 3 (left) shows that the contact stiffness for
a block of these dimensions are 3–5 MN/m for the expected maximum pressure. However, at the
leading edge the load starts from zero and increases as the block enters the contact zone. Thus,
the contact stiffness varies substantially not only between different contact geometries but also
during the run through the contact.

Figure 3 shows force-indentation relation for different chosen discretisations corresponding to
elements of size 1 mm × 1 mm, 1.1111 mm × 1.1111 mm, 1.25 mm × 1.25 mm, and 1.4286
mm × 1.4286 mm. Different sizes of the element means that different amount of the geometry is

                                                   4
                                     200                                                                                             10




                                                                                                 Apparent contact stiffness [MN/m]
                                                                                                                                      8
                                     150

                                                                                                                                      6
   Force [N]




                                     100
                                                                                                                                      4

                                      50
                                                                                                                                      2


                                          0                                                                                           0
                                           0   0.02    0.04       0.06         0.08    0.1                                             0   0.02    0.04       0.06   0.08   0.1
                                                      Indentation [mm]                                                                            Indentation [mm]



Figure 2: Force as a function of indentation depth for four different positions on the same wearing
course (left). Contact as a function of indentation depth for four different positions on the same
wearing course (right).

                                     10                                                                                   200
 Apparent contact Stiffness [MN/m]




                                     8
                                                                                                                          150

                                     6
                                                                                             Force [N]




                                                                                                                          100
                                     4

                                                                                                                                     50
                                     2


                                     0                                                                                                0
                                      0          50       100            150          200                                              0   0.02    0.04       0.06   0.08   0.1
                                                        Force [N]                                                                                 Indentation [mm]



Figure 3: Contact stiffness as a function of force for four different positions on the same wearing
course (left). Force as a function of indentation depth for discretisation of 20×20, 18×18, 16×16,
14×14 elements (right).


included in the spatial discretisation and in the non-linear contact springs. It is obvious that the
model is rather robust with respect to the chosen discretiation for the presented cases. However,
differences in the calculated results starts to be significant when a spatial discretisation below
12×12, i.e. with elements larger than 1.5 mm × 1.5 mm, is used for the studied cases.


CONCLUSIONS
A numerical contact model based on discretisation of the contact geometry and time was pre-
sented and a methodology for assessing individual and non-linear force-indentation relations
within each pair of contact elements was proposed. Results from the model show that the con-
tact stiffness when a tread is indenting a wearing course is substantially lower than the apparent
contact stiffness given for full contact between the bulk stiffness of the interacting objects. The
relation between the indentation depth and force shows a non-linear behaviour, i.e. the stiffness
varies. It is important to consider the effect of the small scale geometry that will give a lower
contact stiffness compared to an ideally flat surface.

The effect of small-scale geometry has not been explicit considered in previous tyre/road noise
prediction models based on Winkler bedding models with springs of constant stiffness, elastic
half-spaces, or rough discretisation of the geometry. Aiming at friction and wear prediction is

                                                                                            5
likely to demand more details than noise predictions. However, more investigations are needed to
assess to which degree simplifications can be made: Is it possible to reduce the amount of details
while still capturing the behaviour of interest for the optimisation with respect to noise emissions,
rolling resistance, etc.? One could imagine using a linearisation of the stiffness functions around
a working point or working with a limited set of force-indentation relations for the tread blocks of
the tyre. A notice: It is actually easier to work with artificially computer generated surfaces as
for existing roads often only a limited amount of information about the geometry of the wearing
course is available; detailed laser scans of surface geometry is tedious and is often only available
for areas in the order of 10 cm × 10 cm.

The model may be further developed by including more information about how disjoint patches
within the pairs of contact elements forms and contributes to the total contact stiffness within
the elements. Whether including these details are of importance for optimisation purposes is an
interesting question. The next natural step is to include the calculated force-indentation relations
for single blocks in a global tyre/road interaction model.


ACKNOWLEDGEMENT

The work presented was financially supported by the Swedish Research Council (Project n:o
621-2005-5790) and the European Commission (ITARI, FP6-PL-0506437) and (SILENCE, TIP4-
CT-2005-516288).


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