For the numerical analysis of tire an arbitrary Lagrangian Eulerian (ALE) description has been established [17] , resulting in a decoupling of rigid body motion and deformation. A spatially fixed mesh is introduced for the description of the rigid body motion, while the general large deformations are described relatively with respect to this reference configuration. Figure 1 shows a principle sketch of the mappings between the base, the current, and the reference configuration. This yields a multiplicative split of the deformation gradient, i.e.,

F = ∂ x ∂ X = ∂ x ∂ X ∂ x ∂ X = F ^ ⋅ R (1)

where F ^ describes the relative deformation gradient and R is the rotation tensor with properties of orthogonality. The material time derivative of a quantity splits into a relative and a convective part, i.e.,

d u d t = ∂ u ∂ t + Grad   u w (2)

where w is the guiding velocity of the rolling tire. For stationary problems the partial time derivative vanishes, resulting in a time independent description of rolling. After some basic analysis on the equations of motion regarding symmetry and C⁰-smooth representation [17] , a linearized finite element equation system is derived

[ K t − W + K c ] Δ φ = f e + f i − f σ − f c ,     φ i + 1 = φ i + Δ φ (3)

Herein, K t the stiffness matrix and W is represents the inertia matrix, which results from the linearization of the centrifugal forces. The matrix K c is obtained from the linearization of the contact forces and the right side, and f e describes the equivalent nodal forces of external loads, e.g., inflation pressure oraxial load. Here, f i are the inertia forces due to rotation and f c the contact forces. The internal forces due to the divergence of the stress tensor are summarized in f σ . The linearized finite element equation of motion has to be solved for the incremental displacements Δ φ , while nonlinearities are treated iteratively within a Newton-Raphson schema.

In mechanical excitation and modal superposition approach, measured road surface textures are analyzed by a discrete Fourier transform, resulting in a harmonic excitation function

u ( t ) = ∑ j u ^ j e i ( φ j + ω j e X t ) (4)

in terms of displacement amplitudes u ^ j and associated phase angles, while the excitation frequencies are calculated from the guiding velocity. The displacement is introduced to the contact nodes of the tire model and together with the related contact stiffness K c , this yields the corresponding excitation forces f ( t ) . For a more precise modeling these contact forces are weighted with the contact pressure distribution. The total number of Fourier series elements j is determined from the resolution of the texture analysis, velocity, and frequency range of interest. With this excitation force the inhomogeneous finite element equation of motion reads

M ϕ ¨ + G ϕ ˙ + K ϕ = f ( t ) ,     f ( t ) = − K c u ( t ) (5)

which again can be transformed into the linear form,

[ i G K K 0 ] ϕ ¯ − [ M 0 0 K ] ϕ ¯ ˙ = [ f ( t ) 0 ] ,     ϕ ¯ = Z ¯ q ¯ (6)

By this approach a modal reduction is introduced for the state-space vibration vector, described with the generalized coordinates q ¯ . With the properties of orthogonality of the eigenvectors, Equation (6) decouples into a set of first-order differential equations,

ω i q ¯ i j − q ¯ ˙ i j = f i j (7)

where i and j describe the number of eigenmode and the number of Taylor series elements for the excitation approach, respectively. The operational vibration in physical space is obtained from

ϕ ( t ) = ∑ j Z ( i Ω − i ω j e X I ) − 1 Ω Z H f ^   e i ( φ j + ω j e X t ) (8)

In order to avoid singularities at resonance frequencies, damping is introduced to the system by a constant factor δ , which modifies Equation (8) to

ϕ ( t ) = ∑ j Z ( i Ω − δ Ω − i ω j e X I ) − 1 Ω Z H f ^   e i ( φ j + ω j e X t ) (9)

It is obvious that by this first approach the damping increases linearly with frequency. However, it outlines the framework for improved constitutive modeling of rubber damping in a broad frequency domain.

In order to measure the force transmissibility of tire-cavity-wheel assembly, two measuring methods [13] can be used to measure the tire and wheel transmissibility under free-free or free-fixed condition. The tire-wheel assembly using the steel and alloy wheel are tested where the assembly is hung using a bungee cord as well as fixed at the hub on a rigid steel table. The excitation is made with the impact hammer or hand held shaker. However, the excitation with impact hammer to measure input forces (0 - 400 Hz) under a free-fixed condition is applied in this paper. The test method is appropriate to use for all passenger cars, vans and small bus tire wheel systems. The tire and wheel force transmissibility is factor in structure borne road noise performance. The test set up is shown in Figure 2 and the location of the impact is shown in Figure 3.

The dynamic transmissibility of forces is defined as the ratio between the amplitude of the force transmitted to the wheel hub center and the amplitude of the excitation force atthe tire tread center,

Force Transmissibility = OutputFRF InputFRF (1)

where FRF is frequency response function. In order to compare the result of finite element analysis, experiments are performed to measure the FRF under impact hammer excitation for the vertical mode (Z direction) under free-fixed condition.

A tire usually consists of several rubber components, each of which is designed to contribute to some particular factors for tire driving performance in addition

to several cords and rubber composites. These components play a role in maintaining the stiffness and strength required in a tire. The roles of tire components are well described in a book by Clark [18] . It consists of a radial carcass ply, two belt plies, bead wires, and several rubber components. The material composition of most tires is distinguished largely into the fiber-reinforced rubber (FRR) parts and the remaining pure rubber part. The FRR parts of the tire model considered here are composed of a single-ply polyester carcass, two steel belt layers, and several steel bead cords. Since the FRR parts are in the highly complex structure, their material models are chosen based on the goal of the numerical simulation.

Figure 4 shows the 2D finite element mesh on the general structure of tire-cavity-wheel model. The finite element model of a tire is shown in a sectional view and is representative of a passenger car tire. The wheel is modeled as an elastic solid. In order to simplify the assembly of all components, we define the same condensed nodes on the tire than those on the rim. With an axisymmetrical meshing of the tire, a meridian plan is created every 4 degrees. A 3D finite element model of tire-cavity-wheel assembly is shown in Figure 5. A simple Mooney-Rivlin law is used to model the strain energy potential of the rubber materials.

The viscoelastic nature of rubber uses the prony series in simulation. A small amount of material damping is applied to the rubber in the form of Rayleigh damping. The plies and belts are modeled using rebar layers that are embedded in the surrounding rubber matrixby Abaqus [14] . The hypo-elastic material properties are applied to the reinforcement fibers. The vibration characteristics are determined in an analysis sequence consisting of 5 steps as shown in Figure 6.

• Step 1: Rim Mounting Analysis

Rim mounting is carried out using a 2D axisymmetric model of the tire cross-section, which saves significant analysis time. Leveraging the symmetry of the model and the loading conditions, only one half of the tire is modeled. Axisymmetric elements with twist are used to capture the out-of-plane deformation introduced by the belts.

• Step 2: Inflation Analysis

The symmetric model generation capability is used to revolve the 2D axisymmetric cross-section into a partial three-dimensional model. Inflation is carried out using a 3D model which made symmetric model generation of Abaqus [14] . The spoke of wheel is imported by 3D FE model using the tetra elements. Uniform mesh density around the circumference is used in this study.

• Step 3: Frequency Extraction

An eigenvalue analysis is performed after both the staticinflation steps to compute the natural frequencies and mode shapes of the tire in its stationary.

• Step 4: Complex Eigenvalue Extraction

The modes of the tire, computed in the referenceframe associated with the Eulerian-Lagrangian scheme, are complex due to the contributions from the gyroscopicterms. The complex frequency extraction procedure uses a subspace projection method based on the eigenmodes extracted in the frequency extraction step (Step 3).

• Step 5: Steady State Dynamics

A subspace-based steady state dynamics procedure is used to compute the frequency response of the tire to an applied excitation. The subspace projection method uses the eigenmodes extracted in the frequency extraction step (Step 3). The forced response analysis is performed for stationary conditions.

In order to validate FEA-based force transmissibility model, the four tires tested at the indoor room are checked with the model. Figure 7 shows the frequency response of tire-cavity-wheel assembly of 175/65R15 size. A simulation result is shown to be in good agreement with the results from the indoor test. The first peak of frequency response is 84 Hz and the peak corresponding to the tire cavity resonance mode at 250 Hz is located. As shown in Table 1, the 1^st peak frequency and the cavity peak frequency is applied to compare the test and the

simulation. The results of tests and simulations on four tires are compared as having a very high correlation. It is shown that the analysis model in this paper can be effectively used to identify the tire modes that play a role, and recommend design changes that can solve the problem such as the improved structure borne noise design of tire.

In order to analyze the effect of rubber density, the level of DOE is selected form 95% to 100% of rubber density of tire components as listed in Table 2. The version for the finite element analysis of tire-cavity-wheel assembly is preceded by full factorial, 16 versions in total, and analyzed by using Minitab software [16] .

We used the 175/65R15 and 225/55R18 tire models. In this finite element analysis, there was no effectiveness factors, but through Minitab software analysis, we found that tread rubber and under-tread rubber have some influence. As the density of rubber increases, the 1^st peak frequency of the force transmissibility tends to decrease as shown in Figure 8. It is found that the design parameter on the effect of rubber design on the force transmissibility of tire-cavity-wheel assembly is the density of tire tread rubber as shown in Pareto chart.

Table 3 shows the level for DOE for the effect of rubber modulus, and the level is from 80% to 120%. The versions for finite element analysis are preceded by full factorial. In Figure 9, there are two effectiveness factors; the modulus of tread and under-tread rubber. Especially, as the modulus of body ply topping rubber increases, the 1^st frequency increased significantly. The body ply topping rubber is the rubber coating that encapsulates the radial ply reinforcing cords. The topping rubber is calendered onto the body ply cords in thin sheets. It is analyzed that the reason is that the body ply provide the strength to contain the air pressure and provide for sidewall impact resistance.

The tire cords used in the tire consist of belt, body ply, capply. The cord is able to express stiffness with Ends PerInch (EPI) concept, so we used in the DOE as Table 4. EPI is the number of warp threads per inch of woven fabric. In general, the higher the ends per inch, the finer the fabric. Figure 10 shows the effect of the cord stiffness such as EPI on the 1^st frequency of force transmissibility of tire-cavity-wheel assembly. There is no effectiveness factor as shown in Pareto chart. It is concluded that the effect of capply and body ply on the 1^st peak frequency of the force transmissibility are noticeable, and the effect of the belt is minimal.