PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Parametric Study of the Estimation of Indoor Trolley Wheel
Stiffness for Use in a Rolling Noise Prediction Model
Matt EDWARDS1; Fabien CHEVILLOTTE2; François-Xavier BÉCOT3; Luc JAOUEN4; Nicolas
TOTARO5
1-4
5Matelys – Research Lab, France

Univ Lyon, INSA-Lyon, Laboratoire Vibrations Acoustique, France

ABSTRACT
In vehicle tyre/road contact modelling, dynamic models are typically used which incorporate the vehicle’s
suspension in their estimation: thus relying on a known stiffness to determine the movement of the wheel in
response to roughness excitation. For the case of a wheeled device rolling on a floor (such as a delivery
trolley moving merchandise inside a commercial building), there is often no suspension, yet the wheel is still
too soft to able to be considered mechanically rigid. An indoor rolling prediction model needs to provide arobust way of estimating the wheel’s effective stiffness. This work presents an original technique for
estimating the stiffness of an indoor cylindrical trolley wheel. A parametric study was conducted in order to
identify the dependence of the wheel stiffness on each of the relevant variables: including the wheel’s radius,
axle size, width, applied load, and material properties. The methodology may be used to estimate the stiffnessof new wheel types (i.e. different geometries and materials) without needing to solve a finite element model
each time.
Keywords: Wheel, Stiffness, Prediction

1. INTRODUCTION
In the world of acoustics, prediction models that estimate the sound produced in a rolling event arewidespread. Mostly existing for automotive tyre/road (1–3) and train wheel/rail contact (4–6), thesemodels calculate the interaction forces between the two bodies (the wheel and the surface upon which
the wheel rolls), which are produced by the small-scale relative roughness in the contact area (7,8).
These forces induce motion in the two bodies, and their resulting structural vibration is what radiatessound to the surrounding area. Recently, work has been done to develop a prediction model applicable
to indoor rolling scenarios as well, such as that of a delivery trolley rolling on a floor inside acommercial space (9).
For such a model, it is necessary to know the stiffness of the wheels of the trolley in order toaccurately estimate their dynamic motion for calculating their radiated sound emission. For a vehicle
with a built-in suspension system, the stiffness is straightforward to calculate: it is simply the stiffnessof the suspension spring. However, most trolleys do not have suspension systems: they are purely
made up of a wheel turning about a rigid axle. In such a scenario, the stiffness used is instead thestiffness of the wheel itself. Considering the vast range of sizes and materials that trolley wheels comein, this presents a challenge for incorporation into a sound prediction model that wishes to be widely
applicable for a range of indoor rolling scenarios.
One option may be to build a finite element (FE) model for a given wheel, but this is costly, as it
requires a new FE model to be run for every single unique geometry/material composition . An
alternative option would be to run a series of FE models in the form of a parametric study, extract
trends in the data, and develop a system for estimating the stiffness of any geometry or material
1matthew.edwards@matelys.com
fabien.chevillotte@matelys.com
3
francois-xavier.becot@matelys.com
4
luc.jaouen@matelys.com
5
nicolas.totaro@insa-lyon.fr
2514

composition, without the need to solve a new FE model each time. In this paper, a straightforward
method is proposed to characterize the compressional stiffness of a cylindrical wheel i n contact witha flat surface. This is inspired by the methodology used by Sim and Kim to estimate the elastic
properties of viscoelastic materials (10). A two-dimensional surface plot is derived from high order
FE models of a cylindrical wheel under static vertical compression, linking the wheel/axle radii ratio
and contact area half-length. This forms an abacus which may be referenced to find a given wheel ’s
stiffness. Such an abacus may be incorporated into an indoor rolling noise prediction model: allowing
for a more precise estimation of the movement of the wheel and leading to increased accuracy insound predictions.

2. PARAMETRIC STUDY
In order to identify what kind of an influence each of the various wheel parameters had on its
stiffness, a parametric study was conducted. A series of FE models were run for cylindrical wheels of
various geometries and material compositions, from which trends were identified. For the parametricstudy, four geometric variables, shown in Figure 1, were chosen to be investigated: wheel radius 𝑟𝑊 ,axle radius 𝑟𝐴 , wheel width 𝑤, and contact area half-length 𝑎. Additionally, Young’s modulus 𝐸 andPoisson’s ratio 𝜈 of the wheel material, as well as the input displacement 𝑧𝑖𝑛 , were also included.

Figure 1 – Description of the wheel geometry.
2.1. Meshing
The mesh densities of the FE models were first defined based on the width of the wheel , having amaximum edge length of 𝑤/7. They were then re-meshed to add extra nodes in the area of contact forparticularly small contact area half-lengths. This ensured the contact area had at least 4 nodes along
its length. Depending on the geometry, mesh sizes ranged between 1.0 ∙ 104 and 5.3 ∙ 104 nodes,
having degrees of freedom between 2.1 ∙ 106 and 1.1 ∙ 107.
To solve the FE model, the flat contact surface on the bottom of the wheel was fixed in place. A
uniform displacement in the downward vertical direction was imposed on the axle surface, simulating
the mass of the trolley borne by the wheel under load. A glued condition between the axle and wheel
was assumed for the purpose of FE calculations. The stiffness of each FE model was calculated byintegrating the vertical normal stress and vertical displacement across all the nodes on the axle surface,then dividing the two.
2.2. Comparison with an Equivalent Beam
The stiffness of a beam in axial compression is given by𝐴𝐸(1)
𝐾beam =
𝐿
where 𝐴 and 𝐿 are the cross-sectional area and height of the beam, respectively. If the wheel isrepresented as a beam with cross-section equal to the contact area and height equal to the distance
between the contact area and the bottom of the axle, then Equation 1 may be rewritten as

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𝐾beam =2𝑎𝑤𝐸(2)√𝑟𝑊 2 − 𝑎2 − 𝑟𝐴
which may be used as comparison with the results of the parametric study to see how the two differ
in application.
2.3. Influence of Parameters
Figure 2 shows the results of the FE simulations, which revealed patterns in how the wheel stiffness
is influenced by each of the various parameters: some expected, some not. Each plot shows theinfluence of a single parameter, displayed along the horizontal axis. Each plot also shows results for
multiple values of Poisson’s ratio (0-0.495). The parameters having a linear influence on the wheel
stiffness are Young’s modulus and the wheel width. For a given FE model, a change in either of thesevalues results in a proportional change to the wheel stiffness. Since Young’s modulus of a material isessentially a measure of that material’s intrinsic stiffness independent of shape, logic follows that it
would be linearly proportional to the actual wheel stiffness. Similarly, increasing the width of thewheel essentially acts like adding springs in parallel, thus increasing the stiffness in a linear fashion.
Both of these parameters act identically as they do in Equation 2 for the equivalent beam.

Figure 2 – Results of the Parametric Study. From left to right, top to bottom: Young’s Modulus 𝐸, relative
axle radius 𝑟𝐴 /𝑟𝑊 , wheel radius 𝑟𝑊 , relative wheel width 𝑤/𝑟𝑊 , relative contact area half-length 𝑎/𝑟𝑊 ,and input displacement 𝑧𝑖𝑛 . Points having the same x coordinate (but a different y coordinate) are forvarying values for Poisson’s ratio 𝜈. Schematics of the extreme wheel geometries on each plot are shown
for reference (drawn to scale with respect to one another).
The two parameters exhibiting unique relationships with the wheel stiffness are the axle radius andthe contact area half-length. Neither may be described by a simple expression, and will thus become
the topic of further investigation in section 3. For the axle radius: as the ratio between the axle radius
and the wheel radius approaches unity, the stiffness approaches infinity. Since the axle is consideredrigid in the FE models, as the volume of wheel material approaches zero, the compressed wheel getscloser and closer to exhibiting that of a rigid connection, thus having infinite stiffness. For the contact
area half-length: as the size of the contact area approaches zero, the stiffness begins to drop off
exponentially. Due to limitations with the meshing software, extremely small values for the contact
area half-length (below 0.3% of the wheel radius) were not able to be reliably meshed.
Finally, there are three parameters which have little or no influence on the wheel stiffness: thewheel radius, input displacement, and Poisson’s ratio. The first comes with a caveat, as the ratio
between the axle radius and the wheel radius does indeed have a strong effect on the whe el stiffness.
However, when this ratio is held constant, increasing the wheel radius does not result in any change
in wheel stiffness. So perhaps it is better to say that the wheel radius does have an effect on the
stiffness, but that this effect can be fully described by the changing axle/wheel radii ratio. In regards516

to input displacement, the lack of dependence is in accordance with the use of a linear material in the
FE models, where increasing compression does not result in increased stiffness.
The lack of dependence on Poisson’s ratio was unexpected, as an elastic solid of uniform
cylindrical or rectangular prism shape, compressed in the axial direction, is known to have a highdependency on Poisson’s ratio (10). Looking across all six plots in Figure 2, a change in Poison’s
ratio resulted in essentially no change in stiffness (all else held constant). The points which exhibit
the largest deviation are all for the highest values of Poisson’s ratio (above 0.49), and they do notoccur in a consistent fashion (i.e. not always an increase and not always a decrease). In essence, thestiffness is remaining nearly unchanged for all values of Poisson’s ratio all the way up until around
0.49, where only then the modelling instability that exists when approaching 𝜈 = 0.5 starts to have aminor effect. When loaded in the vertical direction, it is the effect of the wheel shape which dominates.
The cylindrical form allows the wheel material to “move out of the way” under compression into thehorn-shaped free space between the wheel and the floor. The shape of the geometry itself is linking
the movement of the two transverse directions to that of the longitudinal direction in a way that negates
any influence that the Poisson effect may have.

3. GENERATION OF THE ABACUS
To generate the abacus, a second series of FE models were run for the two chosen nonlinearly
dependent variables: the axle radius and the contact area half-length: both normalized by the wheel
radius. Eighteen values between 0.05 and 0.98 were chosen for the axle/wheel radii ratio 𝑟𝐴 /𝑟𝑊 .
Eighteen values between 0.003 and 0.1 were chosen for the relative contact area half -length 𝑎/𝑟𝑊 .
Constant values of 𝑤/𝑟𝑊 = 0.4, 𝐸 = 1 Pa, 𝜈 = 0.3, and 𝑧𝑖𝑛 = 0.1 mm were chosen for the wheel
width, Young’s modulus, and Poisson’s ratio, respectively. Though it should be noted that, due to theirrelationship with the wheel stiffness (linear for 𝑤 and 𝐸 , and independent for 𝜐 and 𝑧𝑖𝑛 ), theirchoice is arbitrary.
3.1. Normalized wheel stiffness
Figure 3 shows the results of the second series of FE calculations. Equation 3 is first used to findthe normalized wheel stiffness by dividing by Young’s modulus and the wheel width𝐾(3)
𝐾norm =
𝑤𝐸
The normalized stiffness is then plotted as a function of the axle/wheel radii ratio and relative
contact area half-length. A two-dimensional linearly interpolated surface is overlaid to connect thedata points. The data is plotted with a logarithmic z-scale (base 10) in order to visualize how thestiffness changes for both low and high values of 𝑟𝐴 /𝑟𝑊 . As the axle/wheel radii ratio grows, so does
the influence of the size of the contact area on the stiffness. For ratios 0-0.8, there is a 1.5x factor in
the change in normalized stiffness. Above 0.8 the stiffness increases exponentially, up to a factor ofnearly 16x. Figure 3 may be used as an abacus, allowing one to find the normalized wheel stiffness
for a cylindrical wheel of any reasonable size (i.e. for 0.05 ≤ 𝑟𝐴 /𝑟𝑊 ≤ 1 and 0.003 ≤ 𝑎/𝑟𝑊 ≤ 0.1).
This encompasses nearly every type of cylindrical solid wheel which would be expected to be foundon an indoor trolley. This procedure may be implemented into an indoor rolling noise prediction model
to provide an accurate estimation of the wheel stiffness.

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Figure 3 – Normalized wheel stiffness as a function of axle/wheel radii ratio and relative contact area halflength (black points). A two-dimensional linearly-interpolated surface (shown in blue) has been overlaid toform the abacus. Plotted on a logarithmic z-scale (base 10).
3.2. Example Estimation
The process of using the abacus is relatively straightforward. As an example, let us take a wheelwith the following properties: 𝐸 = 200 MPa, 𝜈 = 0.3, 𝑟𝑊 = 60 mm, 𝑟𝐴 = 5 mm, 𝑤 = 35 mm. Fora cylindrical wheel, Hertzian contact theory states the size of the contact area to be
4𝑄𝑟𝑊
(4)
𝑎=√
𝜋𝑤𝐸 ∗
where Q is the applied load, and 𝐸 ∗ is the apparent Young’s Modulus between the wheel and floor−11 − 𝜈𝑤ℎ𝑒𝑒𝑙 2 1 − 𝜈𝑓𝑙𝑜𝑜𝑟 2
∗
(5)
𝐸 =(
+
)
𝐸𝑤ℎ𝑒𝑒𝑙𝐸𝑓𝑙𝑜𝑜𝑟
For the example, we will use a concrete floor (𝐸 = 3.3 GPa, 𝜈 = 0.2). If the wheel is put under astatic load of 𝑄 = 200 N, Equations 3-4 estimate a contact area half-length of 𝑎 = 1.45 mm.
Plugging these values into the abacus presented in section 3.1, we estimate a normalized wheel
stiffness of 𝐾𝑛𝑜𝑟𝑚 = 0.087. Finally, using Equation 3, we obtain a wheel stiffness of 𝐾 = 607 kN/m.

4. SCOPE OF THE METHOD’S APPLICABILITY
In development of both the parametric study and the abacus, care was taken to run FE models fora wide range of parameter values. Thus, for the most part, any kind of trolley wheel which may bereasonably expected to be found in the real world can have this method applied. Due to their linear
dependence, for the wheel width, radius, and Young’s modulus, any value may be used. Axle radii of
5-98% of the wheel radius are valid, which for all intents and purposes is comprehensive, as a wheel
outside this range would be of no practical use. For the contact area half-length, the method is assumed
to be valid for any value below 10% of the wheel radius. Again, the wheel softness and applied load
would need to be so high to achieve a contact area half-length above this limit, that in reality it is ofno concern. For values below 0.3% of the wheel radius, the wheel stiffness is assumed to continue tofollow the linear profile extrapolated from between the values at 0.003𝑟𝑊 and 0.0036𝑟𝑊 . This results
in an overestimation of the stiffness for these extremely small contact area half-lengths. However, thisis considered acceptable in order to allow the model to allow for loss of contact between the wheel
and the floor: a possibility in the presence of wheel flats and/or floor joints. In this type of situation,
the size of the contact area tends to zero before disappearing at the moment of loss of contact.
On a larger scale, the results show in this paper (particularly in regards to the discovery of the lackof influence of the Poisson effect) demonstrate that a similar modeling technique may be used to518

characterize the properties of other shapes as well. Characterizations have been performed for constantcircular cross sections (10,11) (for which the Poisson effect plays a large role), and characterizationsof materials with constant square cross sections yield results which are nearly identical to those of aconstant circular cross section below Poisson’s ratios of about 0.47. However, analysis of other nonconstant cross sections, or even more complex shapes, could potentially provide beneficial insights inapplications and industries beyond rolling contact modeling.
In a future work, it will be shown how the abacus may be used to derive polynomial r elationships
between the stiffness, axle/wheel radii ratio, and relative contact area half -length. It is possible for
either a single two-dimensional expression or a series of one-dimensional expressions to be developed
to fully characterize the wheel stiffness. This eliminates the need for interpolation, leading toincreased accuracy and applicability in the estimation technique.

5. CONCLUSIONSThis work presents an original technique for estimating the stiffness of a cylindrical trolley wheel.
A parametric study was conducted to identify the dependence of the wheel stiffness on each of therelevant variables. The stiffness is linearly dependent on the wheel width and Young’s modulus, andlargely independent from Poisson’s ratio. A unique dependency exists for the axle/wheel radii ratio
and contact area half-length. Using the information from these relationships, an abacus was createdby generating a two-dimensional linearly interpolated surface for estimating the stiffness of virtually
any cylindrical wheel. The abacus and can be implemented into a rolling noise model to provide an
accurate estimation of the wheel stiffness. In a future work, it will be shown how the abacus may bereplaced by one or two-dimensional polynomial relations to further improve the estimation process.

ACKNOWLEDGEMENTS
The work was done as part of Acoutect: an innovative training network composed of 5 academic
and 7 non-academic participants. This consortium comprises various disciplines and sectors withinbuilding acoustics and beyond, promoting intersectoral, interdisciplinary and innovative training andmobility of the researchers within the project. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement No 721536 .

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