PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

In-plane Vibration and Tire Force Transmissibility owing to Tire
Non-uniformity Defects
Zhe LIU1; Kheirollah SEPAHVAND1; Yintao WEIi; Steffen MARBURG1
12Department of Mechanical Engineering, Technical University of Munich, 85748 Garching, Germany

Department of Automotive Engineering, State Key Lab Automotive Safety and Energy, Tsinghua University,Beijing, 100084, China

ABSTRACT
Tire non-uniformities give rise to rolling force variation at the spindle during steady state rolling. These kindsof characteristics are to some extent unavoidable. Therefore, limiting and managing the non-uniformities isof great economic importance to tire manufacturers. In this paper, an analytical model for predicting thehigh-speed uniformity and the force transmissibility using low-speed uniformity data is presented. The modelis established based on a three-dimensional ring on an elastic foundation model. Considering the tire mass
unbalance and the non-uniformity of the geometry, the variations of the force on the tire-road contacting area
and the length of the footprint are analyzed. The mass and the geometrical non-uniformities are transformed
into generalized forces and the equilibrium equations are solved in the space coordinate system. The length of
the footprint caused by the non-uniformities is expanded by Fourier series, and the deformation is integrated
to obtain the variation of the axial force. The method provides a possibility for the diagnosis and evaluation ofhigh-speed tire dynamic characteristics.
Keywords: Tire non-uniformity, Vibration of tires, Force transmissibility, Three-dimensional ring model,
Rolling kinematics

1. INTRODUCTION
Ride characteristics of passenger vehicles are affected by a variety of sources, including roadroughness, structural transmission and drive system. It will cause the vibration in a wide frequency
range. The overall driving vibration environment of the vehicle will also be influenced by the defectsof the wheel, such as unbalanced mass distribution and uneven radial run-outs. Tire is an importantcomponent which generates and transmits the excitation in contact patch with road surface. But
considering the complicated production process of tires which is inevitable to cause uncertainty ofthe tire structure, the dispersion of material and structural parameters will result in of the variationsof the tyre dynamic force at the wheel rotation frequency.
The wheel unbalance and non-uniformity have been reported in several published studies to be
the major causes of vehicle ride discomfort (1- 3). The perception of ride discomfort owing to theunbalanced vibration is predominantly in the low frequency range (<10Hz). The vibration caused by
the non-uniformity defects of tire is related to the rotation speed. The radial run-out of a tire would
yield appreciable vibration and tyre force variations at a frequency equal to twice the rotational
speed (4, 5). However, most of the researches only focus on the experiment method and results (6, 7).
Some existing analytical models (5, 8) are too simple to describe tire structural and geometricparameters.
Therefore, the purpose of this paper is to develop an analytical model for predicting thehigh-speed uniformity and the force transmissibility using low -speed uniformity data is presented.
The model is established based on a three-dimensional ring on an elastic foundation model.
Considering the tire mass unbalance and the non-uniformity of the geometry, the variations of theforce on the tire-road contacting area and the length of the footprint are analyzed. The mass and thegeometrical non-uniformities are transformed into generalized forces and the equilibrium equations
are solved in the space coordinate system. The length of the footprint caused by the non -uniformities
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zhe.liu@tum.de

is expanded by Fourier series, and the deformation is integrated to obtain the variation of the axialforce. The method provides a possibility for the diagnosis and evaluation of high-speed tire dynamiccharacteristics.

2. MODELING
2.1

Tire Ring Model
A tire was modeled in this paper. Figure 1 shows a deformable ring with a rectangular crosssection on an elastic foundation (9). The tire is described by using cylindrical coordinates (r, θ, z) in
the non-rotating coordinate system. The uniform pressure was applied on the inner wall. The side wallof the tire was equivalent to the elastic foundation. The elastic properties of the foundation aremodeled respectively by distributed springs in the radial, circumferential and axial directions (ku, k v
and kw ).The tread of a radial tire was modeled as a laminated composite ring considering theTimoshenko beam theory. A natural frequency analysis was created based on this model.

Figure 1 – Schematic of the laminated composite ring on an elastic foundation
This model can be simplified as a 2D in-plane model when only the in-plane vibration and force
transmissibility is considered. The steady state response of the tire body under the concentrated
forces can be written as follow.

vb T , tkcf¦¦ ª¬ A Q  1  n QE cos n I  I  A Q sin n I  I º¼2n1vk

k0

n2

uk0

k 1 n 1

ub T , tkcfª nAn1 Qvk  1  n 2 QE k sin n I0  I  nAn 2Quk cos n I0  I º¦¦¬¼
k 1 n 1

(1)

Eq.(1) also can be transformed into a matrix form.

U = T*Q(2)
where, U is the displacements of tire body; T is the flexibility matrix; Q is the matrix of generalizedforces.
2.2 Contact Forces
The displacements of the tire body cannot be calculated in advance, because the tire body is notdirectly in contact with the rode surface. The displacements of the tire body and the tread rubber
should be satisfied the geometry compatible conditions (10). Under a given overall deformation, thenormal and tangential deformations of the tread rubber are:usRl cosK  ReI sin K  ( R  ub ) cos E  vb sin Evs

 Rl sin K  ReI cosK  ( R  ub )sin E  vb cos E(3)K IEThe forces acting on the rubber surface can be calculated.

Fns

kEs (us  h0 )Fts

kGs vs(4)The generalized forces and the tractions are obtained by using the coordination transformation.

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­ qu (I ) ½°°® qv (I ) ¾°q (I ) °¯ E ¿ªcos E« sin E««¬ 0

 sin E º
­ F (I ) ½
cos E »» ® ns ¾
F (I )
us »¼ ¯ ts ¿(5)

­V ½
® ¾
¯W ¿ªcosK«¬ sin K sin K º ­ Fns ½® ¾
cosK ¼» ¯ Fts ¿(6)

Assuming the displacements of tire body are small. Substituting Eq.(3-4) into Eq.(5) andtransforming to the matrix form, the linearized boundary equation is:
Q = F + HQ
(7)The exact contact forces are obtained by successive substitution which is treated as the input of
the rigid ring to calculate the transient responses.
2.3 Flexible-rigid Ring
Solving the partial differential equations of tire motion which is considered the large-deformationand the rotation effect will make the dynamic model complex (11). Therefore, the transient responsesof tire body are approximated using a rigid ring to describe the first modes of tire body (<100Hz).
The rigid ring represents the inertia of a tire.
The rigid ring is connected to the rim by 6-DOF spring. The contact forces generated by theflexible ring are the input load on the rigid ring to calculate the transient response. Then the position
will be frozen and calculate the local deformation and the next-step contact forces of the tread
rubber.

Figure 2 –Schematic of flexible-rigid ring model
2.4 Non-uniformity parameterIn this paper, the mass and the geometrical non-uniformities are considered. The unbalance
concentrated mass is transformed into generalized forces and the equilibrium equations are solved in
the space coordinate system. The uneven radial run-out, the geometrical non-uniformity, affects theparameter matrixes in the Eq. (7), and it also influences the contact forces.

3. PARAMETER IDENTIFICATION
To demonstrate the validity of this proposed theoretical model, a radial tire was modeled in thispaper. The parameters of the flexible ring including the geometric and physical parameters will beidentified from FEM model and engineering design parameters. As in the case study, the proposed
new three-dimensional ring model is applied to the vibration modal analysis of the radial tire,
195/70R14. The parameters are listed in Table 1.
The parameters of the rigid ring and the stiffness parameters of the 6-DOF spring are identified
by using the first modes results of the natural frequencies. The free vibration experiment should take
into account the lateral modes. The cleat test and the static stiffness experiments also can be used toidentify those parameters (11).

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Table 1 –Parameters of a 195/70R14 radial tireParameter type

Unit

Numerical value

Ring width bm
Ring thickness hm
Internal pressure p0N/m2
Effective density Ukg/m3
Mean radius rm
In-plane bending stiffness EIN m2
Radial distributed springs of sidewall ku
N/m2
Circumferential distributed springs of sidewall kv N/m2
Tread band thickness h0m
Tread normal stiffness kEs
N/mTread tangential stiffness kGs
N/m

0.16
0.01
2.5u105
2.28u1030.285
2.0
6.3*105
1.89*105
0.01254.3*105
2.53*105

4. SIMULATION RESULTS
4.1 Contact Pressure Distribution
The results of contact force distribution under five different vertical load conditions are shown inFigure 3. It shows that the center part of the contact patch has a trend of buckling.
500Tranction forces(N/m)

1.031(kN)
2.138(kN)
4.525(kN)
7.319(kN)
5.858(kN)

0-0.050
Contact nodes distance (m)0.05

Figure 3 –The results of contact force distribution
4.2 Numerical Simulation Results
The results show that the vibration response caused by the mass imbalance mainly appears near therotation angular frequency of the wheel, while the vibration response caused by the geometric
non-uniformity of the tire appears near twice the rotation angular frequency of the wheel. At higherspeeds, the effect of mass imbalance is more obvious, because the generalized force is directly related
to the angular velocity of the wheel. The results suggest that under the excitation of flat road surface,
the influence of wheel unbalance is the most pronounced, while on rough road surface, the influence
gradually diminishes.

5. CONCLUSIONS
Tire non-uniformities give rise to rolling force variation at the spindle during steady state rolling.
These kinds of characteristics are to some extent unavoidable. Therefore, limiting and managing thenon-uniformities is of great economic importance to tire manufacturers. In this paper, an analyticalmodel for predicting the high-speed uniformity and the force transmissibility using low-speed
uniformity data is presented. The model is established based on a three-dimensional ring on an elastic
foundation model. Considering the tire mass unbalance and the non -uniformity of the geometry, the3203

variations of the force on the tire-road contacting area and the length of the footprint are analyzed. The
mass and the geometrical non-uniformities are transformed into generalized forces and the equilibrium
equations are solved in the space coordinate system. The length of the footprint caused by thenon-uniformities is expanded by Fourier series, and the deformation is integrated to obtain thevariation of the axial force. The method provides a possibility for the diagnosi s and evaluation ofhigh-speed tire dynamic characteristics.

ACKNOWLEDGEMENTS
The authors acknowledge the support from the China Scholarship Council (CSC).

REFERENCES
1.

Long WC. Factors Affecting Performance of Hubs, Drums, Wheels, and Rims on Truck Tractors andTrailers. SAE Technical Paper; 1965 Feb 1.
2. Kenny TM. Quantifying tire, Rim, and vehicle effects on ride quality. SAE Technical Paper; 1989 Feb
1.
3. Stutts DS, Krousgrill CM, Soedel W. Parametric excitation of tire-wheel assemblies by a stiffness
non-uniformity. Journal of sound and vibration. 1995 Jan 19;179(3):499-512.
4. Cebon D. Interaction between heavy vehicles and roads. SAE Technical Paper; 1993 Mar 1.
5. Deodhar A, Rakheja S, Bhat RB. Vibration and tyre force transmissibility of commercial vehicles
owing to wheel unbalance and non-uniformity defects. International Journal of Heavy Vehicle Systems.
2006 Jan 1;13(3):212-40.
6. Kim KW, Park JB, Lee SJ. Tire mass imbalance, rolling phase difference, non-uniformity inducedforce difference, and inflation pressure change effects on steering wheel vibration. SAE TechnicalPaper; 2005 May 16.
7. Dorfi HR. Tyre non-uniformities: comparison of analytical and numerical tyre models and correlation
to experimentally measured data. Vehicle system dynamics. 2005 Jan 1;43(sup1):223-40.
8. Dillinger BL, Jalili N, Ul Haque I. Analytical modelling and experimental verification of tyre
non-uniformity. International journal of vehicle design. 2008 Jan 1;46(1):1-22.
9. Liu Z, Zhou F, Oertel C, Wei Y. Three-dimensional vibration of a ring with a noncircular cross-section
on an elastic foundation. Proceedings of the Institution of Mechanical Engineers, Part C: Journal ofMechanical Engineering Science. 2018 Jul;232(13):2381-93.
10. Kim SJ, Savkoor AR. The contact problem of in-plane rolling of tires on a flat road. Vehicle System
Dynamics. 1997 Jan 1;27(S1):189-206.
11. Kim S, Nikravesh PE, Gim G. A two-dimensional tire model on uneven roads for vehicle dynamic
simulation. Vehicle system dynamics. 2008 Oct 1;46(10):913-30.

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