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A finite element modeling technique is employed in this paper to predict the force transmissibility of tire-cavity-wheel assembly under a free-fixed condition. The tire and wheel force transmissibility is factor in structure borne road noise performance. In order to improve structure borne noise, it is required to lower the 1
^{st} peak frequency of force transmissibility. This paper presents an application of finite element analysis modeling along with experimental verification to predict the force transmissibility of tire and wheel assembly. The results of finite element analysis for force transmissibility are shown to be in good agreement with the results from the indoor test. In order to improve structure borne noise, it is required to lower the 1st peak frequency of force transmissibility. And, the effect of the tire design parameters such as the density and modulus of a rubber and the cord stiffness on the force transmissibility is discussed. It is found that the prediction of the force transmissibility model using finite element analysis will be useful for the improvement of the road noise performance of passenger car tire.

Noise generated by rolling vehicle tires is a topic of significant and growing concern. In the automotive industry, interior road noise is one of the main issues observed when the tire is rolling on a rough surface. Interior road noise performance can be split into two contributions according to the main physical mechanisms generating the noise, which are structure borne noise and airborne noise. Atire’s noise performance is often a high priority item for vehicle manufacturers because a quiet interior is often regarded as a mark of luxury and quality. Therefore, a quiet tire reduces the need for inserting noise insulation material, resulting in savings in vehicle manufacturing cost.

In general the tire and wheel force transmissibility is factor in structure borne road noise performance. Various applications of transmissibility approach may now be found, such as structural response estimation [

In this paper, a numerical technique for predicting the force transmissibility of tire and wheel assembly under a free-fixed condition is presented. Understanding of the tire force transmission mechanism from the wheel hub center to tire tread due to road disturbances uses the modal dynamics using finite element analysis. Tire force transmissibility is defined as the ratio between the amplitude of the force transmitted to the wheel hub center and the amplitude of the excitation force at the tire tread center using impact hammer. The test of tire force transmissibility is conducted using General Motors (GM) test method [^{st} frequency of force transmissibility of tire, the finite element model of tire and wheel assembly has been developed using a commercial finite element code [^{st} peak frequency of the force transmissibility.

For the numerical analysis of tire an arbitrary Lagrangian Eulerian (ALE) description has been established [

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 C^{0}-smooth representation [

[ 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 [

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 [

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 [

• 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 [

• 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. ^{st} peak frequency and the cavity peak frequency is applied to compare the test and the

Tire | Frequency (Hz) | |||
---|---|---|---|---|

1^{st} Peak | 1^{st} Cavity Peak | |||

Test | Simulation | Test | Simulation | |

175/65 R15 | 84.0 | 83.16 | 250.0 | 248.07 |

205/60 R16 | 79.0 | 81.90 | 222.0 | 221.21 |

225/55 R18 | 84.0 | 80.50 | 201.0 | 200.10 |

205/55 R16 | 79.0 | 81.90 | 225.0 | 226.83 |

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.

An important feature of the finite element model is that it provides a cost-effective means of carrying out parametric studies on the tire so as to optimize desired performance output. As a result of this, the properties of force transmissibility of tire-wheel assembly can be optimized to achieve a low 1^{st} peak frequency. Based on the finite element analysis results well-fitting with the test results, we will try to implement DOE-based on tire design factors. The tire design parameters for DOE of the force transmissibility are performed by using the density of rubber, the modulus of rubber, and the cord stiffness.

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

Level of density | Tire component | |||
---|---|---|---|---|

Tread rubber | Under-tread rubber | Side rubber | Apex rubber | |

Low | 95% | 95% | 95% | 95% |

High | 105% | 105% | 105% | 105% |

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

^{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 ^{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.

A numerical method for predicting the force transmissibility of tire-cavity-wheel assembly under a free-fixed condition has been introduced in this paper to reduce the time and cost that is required for indoor test of a tire at the tire design. The results of finite element analysis for force transmissibility are shown to be in good agreement with the results from the indoor test. In order to improve structure borne noise, it is required to lower the 1^{st} peak frequency of force

Level of modulus | Tire component | |||
---|---|---|---|---|

Tread rubber | Under-tread rubber | Belt-topping rubber | Ply-topping rubber | |

Low | 80% | 80% | 80% | 80% |

High | 120% | 120% | 120% | 120% |

Level of EPI | Tire component | ||
---|---|---|---|

Capply | Belt | Body ply | |

Low | 80% | 80% | 80% |

High | 120% | 120% | 120% |

transmissibility. The effect of the tire design parameters such as the density and modulus of a rubber and the cord stiffness on the 1^{st} peak frequency of force transmissibility is discussed. It is found that to increase the rubber density and to decrease the rubber modulus and the cord stiffness reduce the 1^{st} peak frequency of the force transmissibility of the tire-cavity-wheel assembly. It is found that the prediction of the force transmissibility model using finite element analysis will be useful for the improvement of the structure borne noise performance of tire. In the future, it is expected that this finite element modeling technique for predicting the force transmissibility is a step forward for the automotive industry to evaluate tire-cavity-wheel assembly characteristics and the vehicle transfer functions.

The present study was supported by the Center for Environmentally Friendly Vehicles (CEFV) under the project “Development of the global top eco-friendly tire for reduction of tire wear particles and carbon dioxide” through the Ministry of Environment (ME, Republic of Korea).

The authors declare no conflicts of interest regarding the publication of this paper.

Park, W.C., Lim, H.K., Jeong, K.M. and Kim, T.W. (2018) The Effect of Tire Design Parameters on the Force Transmissibility. Open Journal of Applied Sciences, 8, 446-458. https://doi.org/10.4236/ojapps.2018.810035