Prediction of dynamic hub forces as a source of structure-borne tire/coarse
road noise using a high-fidelity simulation approach
Daniel DE GREGORIIS(1,2)∗ , Frank NAETS(1,3)† , Peter KINDT(2)‡ , Wim DESMET(1,3)§(1) KU Leuven, Belgium(2) Goodyear Innovation Center* Luxembourg, Luxembourg(3) DMMS Lab, Flanders Make, Belgium

Abstract
When optimizing the tire/road structure-borne interior noise, multiple other coupled tire performances need tobe simultaneously optimized. Typically, different tire design prototypes are built and tested. However, with therecent advent in virtual product development and the Digital Twin concept, high-fidelity numerical simulation
approaches become a viable alternative to the experimental approach. These numerical approaches allow for
design space exploration without the need for building and testing physical tire prototypes. Therefore, in thiswork the use of a fully predictive nonlinear numerical approach for the prediction of the dynamic contact- andhub forces of a tire rolling over a coarse road surface is described. These predicted hub forces can then be further
used to assess the tire/road structure-borne noise performance of a tire. The influence of incorporating tiremodel nonlinearities as well as rim flexibility on the simulated hub forces is discussed. Furthermore, nonlinear
model order reduction and hyper-reduction techniques are applied to greatly reduce the numerical simulation
time, being approximately ten times faster than the experimental approach. Finally, a comparison between thepredicted hub forces and experimental hub forces is shown, where a sufficient correspondence between both canbe observed for design engineering purposes.
Keywords: Tire, Interior noise, Simulation

1

INTRODUCTION

In this work, the prediction of dynamic hub forces arising due to a tire rolling with a constant speed (i.e.
steady-state) over a coarse road surface is of interest. These hub forces are typically a main source of structureborne interior cabin noise for the 0 – 400Hz range (1, 2). The structure-borne interior noise tends to dominates
the airborne interior noise in this frequency range for rolling over a coarse road surface (1, 3, 4). The ability
to predict the hub forces and corresponding structure-borne noise of a specific tire design is therefore valuable in an overall performance assessment during virtual tire development. More specifically, the vertical hub
force is of interest as a performance indicator of the structure-borne interior noise performance of a tire design.
As the tread pattern has limited contribution to the hub forces below 400Hz for rolling over a coarse roadsurface(1), only slick and ribbed slick tires are considered in this work. Recently, the authors have proposeda fully predictive methodology for simulating dynamic contact and hub forces for rolling over a coarse roadsurface (5). Here, a fully nonlinear structural finite element (FE) tire model in combination with the Arbitrary
Lagrangian Eulerian (ALE) formulation for bodies in rolling contact (6), an acoustic FE model of the air cavity,
a constraint-based contact formulation (7) and a hyper-reduction method are used. In this work, the methodology is extended to include the rotation of the air cavity as well. In earlier work (5) the regular Helmholtz
equation was applied which corresponds to assuming a non-rotating air cavity. In this work, the acoustic domain∗ daniel.degregoriis@kuleuven.be† frank.naets@kuleuven.be‡ peter_kindt@goodyear.com§ wim.desmet@kuleuven.be483

is modeled using the convected Helmholtz equation (8), similar to the work by e.g. Cao and Bolton (9). A
nonlinear FE tire model can be seen as a virtual replica (or Digital Twin (10)) of the real tire, thus allowing adesign engineer to directly apply the same changes to this model as would be done to a physical tire design.
Examples of such design changes are reinforcement fiber angles, compound properties etc. The use of a fully
nonlinear FE tire model is typically computationally too expensive to use in a design context, as it often takes
more time to run the simulations than to build the physical tire prototypes. Therefore, the a-priori nonlinear
Multi-Expansion Modal (MEM) hyper-reduction method (11) is applied to the nonlinear FE tire model, greatly
reducing the computational costs and enabling the use of the fully predictive method as an alternative to theexperimental testing. The remainder of this work is structured as follows: the high-fidelity tire modeling approach is discussed in section 2. Application of the MEM hyper-reduction method is discussed in section 3.
The use of the predictive methodology to estimate the structure-borne interior noise performance of a test tireis discussed in section 4. Finally, conclusions are given in section 5.

2

HIGH-FIDELITY TIRE MODELING FOR HUB FORCE PREDICTION

As discussed in section 1, a high-fidelity tire modeling approach is adopted (5). The semi-discretized FE equations of motion of the structural tire domain can be expressed as:
Mẍ + GALE (x, ω) ẋ + f (x, ẋ) − fALE (x, ω) = fp (x, pa ) + fc (x, xr ) + fe

(1)Here, the current configuration x(t) ∈ Rn is defined as x(t) = x0 + u(t), where x0 is a reference configuration
and u(t) is the total displacement with respect to the reference configuration at time t. The first and second derivatives of the current configuration with respect to time are denoted as ẋ and ẍ, respectively. All
configuration-dependent terms therefore depend on time as well, but this time-dependency is dropped from notation for clarity reasons. The constant mass matrix is denoted by M ∈ Rn×n , the Gyroscopic ALE matrix isdenoted by GALE (x, ω) ∈ Rn×n , the nonlinear internal force term is denoted by f (x, ẋ) ∈ Rn (12), the ALE
rotational inertial force term is denoted by fALE (x, ω) ∈ Rn , where ω is the constant angular velocity of thetire, the nonlinear inflation pressure force term is denoted as fp (x, pa ) ∈ Rn (12), where pa denotes the tire
inflation pressure, the nonlinear contact force term is denoted as fc (x, xr ) ∈ Rn , where xr is the set of roadsurface contact constraints at time t, and the constant external force term (e.g. a vertical load) is denoted asfe ∈ Rn . As slick and ribbed slick tires rolling with a constant angular velocity over a coarse road surface areconsidered in this work, the Arbitrary Lagrangian Eulerian (ALE) formulation for bodies in rolling contact, as
proposed by Nackenhorst (6), is used. Here, the steady-state material flow of the tire material is modeled ina Eulerian (i.e. spatial) formulation (material streamlines). The deformation of the Eulerian domain (i.e. thefinite element mesh) is described using a material/Lagrange formulation. As a result, the tire mesh does not
rotate and needs only to be refined in the tire/road contact area (6). The rolling contact between the tire andcoarse road surface is modeled using a penalty method constraint-based contact mechanics approach (7). The
constraint-based computational mechanics method requires no measurements and/or tuning, thus allowing a fully
predictive numerical approach. Following the ALE approach, the road surface is modeled as flowing throughthe spatial tire domain. Given the hub velocity of the rolling tire and a scan of the road surface (or analyticalexpression of the surface), the road surface asperity heights in spatial points of the tire domain can be extracted
per discrete timestep via e.g. a projection based contact detection algorithm (7). This concept is illustrated in
Figure 1 (a). These asperity heights are applied in the tire spatial points as geometrical contact constraints xr
per discrete timestep. Figure 1 (b) shows the time history of the geometrical contact constraints in a tire spatial
point, derived from a coarse surface scan.

484

Figure 1. Schematic overview of applying dynamic contact constraints in the ALE-formulation (a) and an
example of geometrical contact constraints extracted in a spatial point (b)In case rim flexibility is considered as well, the structural equations of motion (1) are extended to include
the flexible rim domain. Tied contact is assumed between tire and rim in the tire/rim interface, as no relative
movement of the tire bead area with respect to the rim flanges typically occurs for steady-state rolling.
In order to include the acoustic wave propagation in the rotating air cavity, the convected Helmholtz equation (8)is used, similar to the work by Cao and Bolton (9):2∂+ v0 · ∇ p − c2 ∇2 p = 0∂t(2)Here, v0 is the uniform flow velocity (i.e. space- and time independent), ∇ is the Nabla/Del operator 1 , c isthe speed of sound and p is the acoustic pressure. Information on the weak form of the convected Helmholtz
equation and its discretization can be found in e.g. the work by Guasch and Codina (13). While the convectedHelmholtz equation assumes uniform flow, the air cavity flow velocity is not constant: it increases linearlyacross the cavity cross section with respect to the distance from the tire hub. This error is assumed to be ofminimal influence for the first acoustic eigenmodes. Similarly to the structural eigenmodes, the acoustic eigenmodes also split-up into traveling modes due to the rotation (1). Applying the vibro-acoustic coupling boundary
conditions (14) to the structural equations of motion (1) and including the convected Helmholtz equation thenyields the following vibro-acoustic equations of motion:       M0ẍGALE 0ẋKsauf − fALE
fp + fc + fe
+
+
+
=
(3)
Msa Ma p̈
0
Ga ṗ
Ka − KGa p00
Here p ∈ Rna is the air cavity acoustic pressure field, ṗ and p̈ are its first and second derivatives with respect
to time, respectively. Note that all of the dependencies of the structural domain force and matrix terms havebeen dropped for simplicity of notation. The acoustic mass matrix is denoted as Ma ∈ Rna ×na , the acoustic
Gyroscopic matrix is denoted as Ga ∈ Rna ×na , the acoustic stiffness matrix is denoted as Ka ∈ Rna ×na andthe acoustic Gyroscopic stiffness matrix is denoted as KGa ∈ Rna ×na . The vibro-acoustic coupling terms aredenoted by Ksa ∈ Rn×na and Msa ∈ Rna ×n . Additional information on the weak form and discretization of these
additional acoustic domain terms and vibro-acoustic coupling terms can be found in the work by Everstine (14).
A nonlinear vibro-acoustic FE tire model, as used in this work, is shown in Figure 2.
1∇ =



∂
∂
∂
∂ x1 , ∂ x2 , ∂ x3

, where xi represents a Cartesian coordinate direction

485

Figure 2. Nonlinear FE tire model as used in this work, where the structural tire domain is shown in green andthe acoustic domain is shown in gray

3

APPLICATION OF THE MEM METHOD

In order to reduce the numerical costs associated with solving the nonlinear vibro-acoustic equations of motion (3), which can exceed several weeks of computation time to simulate one second of rolling, the MultiConfiguration Modal (MEM) hyper-reduction method (11) is applied. Using the Galerkin method, a trial solution residing in a lower dimensional subspace of the original solution space is inserted into the equationsof motion. Using the same trial solutions, the residual arising due to the insertion of the trial solutions into
the equations of motion is eliminated via the projection (i.e. pre-multiplication) of the equations of motion
onto the subspace. This yields a lower dimensional system (typically several orders of magnitude smaller),
resulting in computational speedups (since a much smaller system needs to be solved). For the nonlinear vibroT
acoustic FE tire model, the full-order solution u p ∈ Rn+na is approximated by a reduced-order solutionT
qs qa ∈ Rm , which resides in a constant subspace of the solution space. The subspace is spanned by thecolumns of V ∈ Rn+na ×m , where:T
V = Vs T Va T
(4)
Here Vs ∈ Rn×m is defined as the structural part of the subspace basis (i.e. approximating the structural field
variables) and Va ∈ Rna ×m is defined as the acoustic part of the subspace basis (i.e. approximating the acoustic
field variables). Hereafter V is called the reduced order basis (ROB). This leads to the following approximations:x ≈ x̃ = x0 + Vs qs(5)p ≈ p̃ = Va qa(6)

Exploiting the ALE-formulation, i.e. the non-rotating mesh, nonlinear static contact configurations and eigenmodes, calculated around these nonlinear static contact configurations, are used as the ROB. The nonlinear static
contact configurations capture the gross nonlinear deformation of the tire and are calculated using equidistantsamples taken from the a-priori known geometrical road surface constraints. The eigenmodes capture the dynamic response of the tire. Using the finite element assembly procedure, the projected nonlinear terms can be
expressed and approximated as (11, 15):|E|

|E s |f̃ (x) = Vs T f (x) = ∑ Vs T,i fi (x) ≈ f̄ (x) = ∑ si Vs T,i fi (x)i=1

and si ≥ 0

(7)

i=1

Here E is the total set of finite elements and Vs T,i and fi both have a sparse structure, only containing thecontributions of element i. A subset of elements E s ⊂ E is sampled from the finite element mesh, where |E s | |E|. Each element of the subset Es is given a positive weight s. Only a limited set of elements thus needs to beevaluated per iteration step to approximate the projected terms. This effectively results in a substantial reductionof the overall computational time. The nonlinear static contact configurations, as used for the ROB, are used486

for the element sampling as well. More specifically, a constrained L1 optimization problem is solved (11). As
both the ROB and the MEM method only require static configurations (and eigenmodes calculated around these
configurations), the overall MOR approach is feasible to use in a design context, as all static configurations canbe calculated in parallel and no dynamic simulations of the full-order model are required. Application of the
MEM hyper-reduction method to expression (3) then yields:
 
 
 
q¨s
q˙s
q
M̃
+ Ḡ
+ K̃sa s = f̄
(8)
q¨a
q˙a
qa
Here the reduced and hyper-reduced terms are: T
Vs MVs0M̃ =∈ Rm×m
Va T Msa Vs Va T Ma Va
 T
Vs GALE Vs
0G̃ =∈ Rm×m0
Va T Ga Va
Vs T Ksa Va
K̃sa =∈ Rm×m
Va T (Ka − KGa ) Va
 T
V (f + f + f − f)f̃ = s p c e
∈ Rm
0
" s
#
|E |
si Vs T,i Gi,ALE Vs,i
0
∑
i=1
Ḡ =
0
V a T G a Va
" s
#|E p ||E s |
T
T
T
T
s
V
f
+
V
f
+
V
f
−
s
V
f
−
f
∑
∑
s c
s e
i,ALE
i=1 i,p s ,i i,pi=1 i s ,i i
f̄ =
0

4

(9)
(10)
(11)
(12)
(13)

(14)

APPLICATION OF THE PREDICTIVE METHODOLOGY

A tire of size 205/55R16 mounted on a lightweight alloy rim is loaded force-controlled, using a vertical external
force ||fe || = 4000N, onto a drum. After achieving the desired static vertical load, the axle position is locked
and the drum is set into motion. The drum rotates with a fixed angular velocity and drives the tire. The drum
angular velocity is chosen such that the rolling speed of the tire is 22.22ms−1 , i.e. 80kmh−1 . A coarse roadsurface is mounted on the drum. A fully nonlinear vibro-acoustic FE model of the test tire is constructed using
geometrical properties extracted from the test tire design and material properties extracted from material tests.
This full order model (FOM) contains over 38,000 elements and 175,000 degrees of freedom (DOFs). Using
the ROB and the MEM hyper-reduction method, a hyper-reduced order model (HROM) of the FOM is created.
The number of degrees of freedom (DOFs) is reduced by a factor of 125, whereas the number of elements thathave to be evaluated per iteration step (|E| or |Es |) has been reduced by a factor 27. The total speedup factor(SF) after application of the MEM method is in the order of magnitude of 100. The full-order approach takes
around ten to eleven times longer than the experimental approach, while the MOR approach is typically tentimes faster than the experimental approach. This effectively means that the total simulation time is reduced
from months to hours. A total of one second of rolling is considered in order to achieve a relevant frequency
resolution. Since steady-state rolling is considered, spectral convergence is achieved for even this duration ofrolling.
As an application case, the influence of the inflation pressure on the vertical hub force spectrum and corresponding structure-borne interior noise is investigated. Two different inflation pressures, pa = 2 · 105 Pa andpa = 2.5 · 105 Pa are considered. The shift of certain peaks in the 50–200Hz frequency range is of interest to assess the effect of the inflation pressure. A comparison between the experimental and numerical results is shownin Figure (3). The pa = 2 · 105 Pa and pa = 2.5 · 105 Pa cases are shown in Figure (3) (a) and (b), respectively.
The vertical hub force spectra are predicted with good accuracy, and therefore allow to use the proposed predic-487

tive methodology as an alternative to experimental approaches to assess the structure-borne noise performanceof a tire design.

Figure 3. Comparison of the normalized experimental and numerical vertical hub force PSDs, for two different
inflation pressures: pa = 2 · 105 Pa (a) and pa = 2.5 · 105 Pa (b)Effects and limitations of modeling assumptions
Including all nonlinearities in the equations of motion (3) results in the need for MOR and hyper-reduction.
An alternative could be to linearize the equations of motion (keeping only the contact force term nonlinear) inorder to reduce the numerical costs. The vertical hub force Power Spectral Density (PSD) corresponding to thiscase is shown in Figure 4 (a), along with the vertical hub force PSD for the fully nonlinear case (i.e. keeping
all nonlinear dependencies). As can be seen, the nonlinearities need to be taken into account in order to have afully predictive methodology for the entire frequency range of interest, as the linearized approach is limited tolower frequencies and therefore less suitable for a structure-borne interior noise prediction application.
For 15 – 17 inch passenger car tire sizes, alloy rims typically have a limited influence on the vertical hub force
spectrum, as the (radial) rim dynamics are out of the 0–400Hz frequency range (16). While of less interestfor structure-borne interior noise prediction, the lateral hub force spectrum typically is influenced by the rim
dynamics in the frequency range of interest, even for alloy rims in the 15–17 inch size range. The effect ofincluding a 16 inch flexible alloy rim model on the lateral hub force spectrum is shown in Figure (4) (b). As
can be seen, the coupling between the flexible alloy rim and the tire and its effect on the lateral hub force is
not negligible. When considering either steel rims of comparable size or larger size alloy rims, both of whichhave their resonance frequencies typically below 350Hz, the vertical hub force spectrum is therefore influenced
by the rim dynamics as well and a flexible rim model should then be included to capture these effects in orderto accurately predict the vertical hub forces and corresponding structure-borne interior noise.

488

Figure 4. Influence of the nonlinear terms on the simulated vertical hub force PSD (a) and the influence ofincluding a flexible rim model on the simulated lateral hub force PSD (b)5CONCLUSIONSIn this work, the use of a high-fidelity approach for predicting the tire/road contact and corresponding hub
forces for constant speed rolling over a coarse road surface is described. Vertical hub forces are typically athe main source of structure-borne interior noise when rolling over a coarse road surface, prediction of these
hub forces therefore allows to assess the interior noise performance of a tire design. A fully nonlinear FE tiremodel (which can be seen as a Digital Twin) is combined with an ALE-formulation and a constraint based
contact mechanics formulation. The acoustic domain is modeled using the convected Helmholtz equation. In
order to reduce the computational costs associated with solving the set of nonlinear equations of motion, theMEM hyper-reduction method is applied. Speedup factors of about 100 are achieved, effectively reducing thecomputation time from months to hours.
The influence of the inflation pressure on the structure-borne interior noise performance of a tire is investigatedas an application case. Both a physical test tire and numerical test tire are loaded onto a rolling drum with acoarse surface. The vertical hub force spectra are calculated and compared for both the physical and numericaltest tire for two different inflation pressures. A good correspondence can be observed between the experimental
and numerical results. The described predictive methodology can therefore be used as an alternative to theexperimental approach, in order to assess the structure-borne interior noise performance. While for the tireconsidered in this work the dynamic properties of the alloy rim have a limited influence on the vertical hub
force spectrum in the 0–400Hz frequency range of interest, it is shown that for the lateral hub force spectrum
the influence is not negligible. In general, a flexible rim model should be included for steel rims of comparable
sizes, as well as for larger sized alloy rims. Inclusion of the nonlinear terms in the equations of motion isnecessary to correctly predict the vertical hub force spectrum in the entire frequency range of interest, as alinearized approach is limited to lower frequencies.

ACKNOWLEDGEMENTS
The research of Daniel De Gregoriis is funded by an AFR-PhD grant supported by the Fonds National de laRecherche, Luxembourg (AFR PhD Grant Agreement PhD 2015-1, Project Reference 10222097). The Fund for
Scientific Research - Flanders (F.W.O.) is gratefully acknowledged for the support of the postdoctoral researchof Frank Naets. The Industrial Research Fund KU Leuven is gratefully acknowledged for its support.

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