A Fast Approach to Optimize Tread Pattern Shape for Tire Noise Reduction

Abstract

Impact force induced by tread pattern is one of the major mechanisms of tire noise; thus, how to reduce tire impact noise has been an important issue in regard to tire technology. In this paper, the mechanism of tire–pavement interaction noise is briefly described, and a prediction model for pattern noise is proposed. The prediction model was validated with indoor semi-anechoic chamber and pass-by noise experiments. Then, an optimization method for the tread pattern was proposed by using the basis vector method, and the synthetic pattern shape was generated through a linear combination of basis shape vectors. Finally, a novel multi-objective function was proposed, aimed at minimizing the impact noise generated by the tire pattern, and the weight factors of the basis vectors were optimized with a genetic algorithm. The method proposed in this paper can be used to evaluate or improve pattern performance and reduce trial and error in the pattern design stage.

1. Introduction

In the context of modern urbanization, an intricate transportation network serves as a symbol of progress. Convenient transportation options have greatly enriched people’s daily lives. However, noise from the road traffic network spreads throughout the city and seriously affects people’s work, lives, and physical and mental health. According to estimates from Environment Australia, road traffic contributes to over 70% of environmental noise pollution. Tire noise serves as a major contributor to vehicle noise and is a significant source of urban noise pollution, especially at speeds exceeding 50 km/h. According to statistics presented in [1], tire–pavement noise accounts for 75% to 90% of the total noise generated by passenger cars. In recent years, with the development of electric vehicles, people have higher requirements regarding the comfort and silence of vehicles. In the development of low-noise cars, tires are the most important because tire noise accounts for a large part of the total noise [2]. In particular, tire labels institutionalized in the EU show that tire noise is important because it can amplify overall noise levels [3,4].

Tire noise, or tire–pavement interaction noise, is a complex phenomenon influenced by a range of factors [5], such as tread materials, tire structures [6], vehicle dynamics, road conditions, etc. Tire noise is generated by various mechanisms, which can be categorized into two main groups [7]: aero-acoustic and vibration-induced noise. In the former, the sound pressure fluctuation is caused by aerodynamic noise sources, such as aerodynamic effects. In the latter, the sound pressure fluctuation is caused by the vibration of the structure and the interaction with the surrounding air. In general, tire vibrations are regarded as the primary mechanism for generating tire–pavement noise over low- and mid-frequency ranges. These vibrations are induced by time-varying contact forces resulting from the road roughness and tread pattern. Two primary sources of noise can be identified: The first is related to tread pattern grooves and is known as the air pump mechanism, which arises from the variation in groove volume within the contact patch. The second is related to tread blocks and is known as the vibration mechanism, which is caused by the impact and release of tread blocks entering and leaving the contact patch. For the pattern impact mechanism, the local vibration is dominant in the attachment portion of the contact area between the tread and road surface. Evaluating and predicting tire noise has always been a challenge. Indoor and outdoor tire noise experiments, including the pass-by noise experiment, have been considered the most direct methods for investigating tire noise. However, the evaluation of tire noise through experiments is not only costly and time-consuming but also fails to provide sufficient information to the designer at the design stage [8]. In such situations, there is a preference to predict tire noise through efficient numerical analysis methods during the design stage. Analytical approaches or numerical techniques, such as the Finite Element Method [9], boundary element method [10], or computational fluid dynamics method, have become the primary means for tire noise prediction. To achieve an accurate simulation of tire noise, complex tread pattern modeling is necessary, and the simulation calculation takes a long time [11]. The tread model needs to be rebuilt every time the tread pattern is modified, which greatly increases the modeling cost.

The potential health benefits of noise abatement can be found in the literature [12]. In general, tire-noise-reduction strategies can be separated into three categories [13]: noise control at the noise source, along the transmission path, and at the receiver. To mitigate tire noise, numerous attempts have been made to reduce tire–pavement interaction noise at the source and to ensure a lasting effect [14,15,16,17,18], including the implementation of noise barriers or shields and quiet pavement [19,20]. The former reduces noise mainly through sound absorption and sound insulation, and the latter mainly reduces noise by reducing the reflection and propagation of sound on the pavement, as well as by reducing the generation of sound. Compared with noise barriers or shields, quiet pavement might not be a viable option to mitigate traffic noise because the pavement can wear out and become louder with time [14]. Furthermore, the construction and maintenance costs of pavement are also quite expensive. Thus, controlling tire–pavement interaction noise at its source, i.e., quiet tires, can be a more economical and effective approach. In the tire industry, a number of design concepts to reduce tire noise have been considered [21,22,23], including tire tread [24], tread pattern, tire cavity [25], tire structure, and rims [26]. However, sound-absorbing materials attached inside the tire cavity to reduce the cavity resonance noise might be the most successful so far. Among all of the strategies for reducing tire noise, the tread pattern is considered to be the easiest to modify [27]. In the literature [28], two approaches have been proposed to reduce the tire noise related to the tread pattern. One is meant to disperse tire noise energy over a broad frequency range, which can be achieved by using variable pitches and randomly arranging them around the tire [16,29,30]. The second method reduces the exciting force caused by the discontinuity of the tread block in the direction of tire rotation. In this case, engineers have to modify the pattern’s geometric design. Nevertheless, the automatic generation of a pattern that fulfills the performance requirements poses a significant challenge.

The mathematical representation of the tread pattern is challenging due to its complex structure. In [31], a method was developed to parameterize the pattern geometry, involving over 11 parameters to describe the pattern in detail. Recently, researchers have explored the interaction between the optimal tire profile and tread pattern design by employing the basis vector method [32]. An advantage of the basis vector method lies in its capability to effectively describe complex geometric structures using only a few parameters. Another advantage of this method is that the design engineers can incorporate their design ideas into the basis vector shape [33]. The capability to predict and reduce tread pattern noise will help improve tread pattern design. The contribution of this study is the development of a new optimization method related to tread patterns for tire noise reduction.

The rest of this paper is organized as follows: Section 2 reviews the mechanism of tire noise, and a pattern impact noise prediction model based on pitch noise theory [34] is established. Section 3 describes the details of the synthesis method of pattern shape based on the basis vector method. Section 4 outlines the optimization process based on the genetic algorithm. The prediction method is verified by semi-anechoic chamber and pass-by noise experiments. Then, the optimization results of the tread pattern and leading edge are discussed in Section 5. Section 6 concludes the paper.

2. Tire–Pavement Interaction Noise

2.1. Mechanism of Tire Pattern Noise

When a patterned tire rolls on a flat surface, the impact and release of the tread blocks on the pavement cause the vibration of the tire belt and tread surface, generating noise. Due to the high damping of the rubber material, these vibrational waves primarily remain concentrated near the impact area and do not propagate far. Consequently, acoustic sources located near the contact path are then amplified by the tire–pavement geometry, resulting in far-field noise, as shown in Figure 1. Finally, it is transmitted to the human ear through the air, resulting in perceptible noise.

Tire noise is generated by various mechanisms, including tire surface vibration and the noise associated with aerodynamics, which are discussed in [35]. In general, tire surface vibration can be explained by the excitation force, transfer function, and acoustic field. The former is primarily affected by the tread pattern design, while the transfer function is related to tire structure, material, and boundary conditions. The transfer function can be mathematically represented as follows:

$$H_{ij}(f)=\frac{X_{ij}(f)}{F_{ij}(f)}$$

(1)

where H_ij(f) is the vibration response transfer function in the frequency domain, being related to the tire structural design and material composition. F_ij(f) and X_ij(f) are the inputs of the exciting force and the response of vibration, respectively. Generally, X_ij(f) can be the displacement, velocity, or acceleration of the vibration. For a patterned tire, F_ij is mainly influenced by the tread pattern geometry and distribution on the tread. If we ignore the transfer characteristics of tires, that is, H_ij is constant, and Equation (1) can be simplified as:

$$\frac{{\left[X_p\right]}_{ij}}{{\left[F_p\right]}_{ij}}=const$$

(2)

where [X_p] and [F_p] are the vibration response of the tread surface and exciting force on the tread pattern. Equation (2) shows that the vibration response of the tread surface is directly proportional to the input force.

2.2. Prediction Model of Tire Pattern Noise

Tire pattern noise is mainly generated by the interaction between the tread pattern and the road surface. The leading and trailing edges play a significant role in tread noise generation. At the leading edge, noise is mainly generated due to the impact between the tread blocks and road surface when the blocks enter the contact patch, and the vibrations are excited by time-varying impact forces. The vibration wave does not propagate very far because of tread rubber’s high damping property. For the trailing edge, the noise generation mechanism is similar to that at the leading edge. These patterns located in the contact area cause minimal vibration due to their strong adhesion to the road surface. In this study, the vibration caused by the adhesion between these blocks and the road surface is negligible.

As described in Section 2.1, the relationship between sound pressure and impact force can be expressed as [36]:

$$p\propto a\propto F$$

(3)

where a is the acceleration of the tread element, and it relates to the tread impact force at the leading edge, which is necessary for predicting tire noise. Here, the impact force (forcing function) is derived from the discrete impact of each tread element on the pavement. These repeated impacts induce vibrations, which propagate through the air and surrounding structure, resulting in noise. However, accurately calculating the impact force for an actual commercial patterned tire is a significant challenge. Therefore, it is assumed that a pulse is generated when the tread blocks make contact with the road surface. On the contrary, no force is generated by the tread voids or grooves. Accordingly, the force function can be described as follows:

$$F(x,y)=\gamma (x,y)\delta (x,y)$$

(4)

where $\gamma (x,y)$ is the amplitude of the exciting force on the tire tread surface, represented by x, y, and it is related to the tire structure, material, etc. $\delta$ is the Dirac delta function, which is written as:

$$\delta (x,y)=\left\{\begin{array}{l}0 \mathrm{for} \mathrm{groove}\\ 1 \mathrm{for} \mathrm{block}\end{array}\right.$$

(5)

Integrating Equation (4) along the meridian direction of the tire tread, the impact force between the tire and road surface during rolling can be expressed as:

$$F(\phi )={\displaystyle {\int }_{leading}\gamma (x,\phi )\cdot \delta (x,\phi )}{d}_x \mathrm{for} \mathrm{all} \phi =0\dots 2\pi$$

(6)

The spectrum of sound pressure in the frequency domain can be expressed as:

$$F(t)=\frac{a_0}{2}+{\displaystyle \sum _{n=1}^{\infty }(a_n\cos n\omega t+b_n\sin n\omega t)}$$

(7)

where $\omega =2\pi /T$ is the frequency, T is the cycle of rotation, and $a_n$ and $b_n$ can be expressed by:

$$\begin{array}{l}{a}_n=\frac{2}{T}{\displaystyle \sum _{i=1}^{N}F(t)\cos n\omega t_i}\\ b_n=\frac{2}{T}{\displaystyle \sum _{i=1}^{N}F(t)\sin n\omega t_i}\end{array}$$

(8)

$$c_n=\sqrt{a_n^2+b_n^2}$$

(9)

where c_n is the amplitude for the n-th harmonics calculated using the Fourier transform of the exciting force in Equation (7).

3. Basis Vector Method for Shape Generation

During the early stage of low-noise tire pattern design, tire designers often resort to trial and error to refine the design schemes, which is very time-consuming. Optimization is a highly effective approach for mitigating tire noise and can be considered a viable method. Various structural optimization methods, such as size, shape, and topology optimization, are available for consideration. Because of a tire’s complex geometric structure, which includes ribs, lugs, sipes, and other elements, it is difficult to represent the tread pattern shape mathematically. This study aims to propose a valuable means of designing a tread pattern. The basis vector method [32,37] may be suitable because the optimized shape could be determined by finding the best possible combination of basis vectors. When using the basis vector method, it is necessary to use several slightly different shapes, called basis vectors. The synthetic shape is generated in terms of a linear combination of vectors as follows [33,37]:

$$G=G_0+{\displaystyle \sum _{i=1}^n{w}_i(G_i-G_0)}$$

(10)

where G and G₀ are the synthetic and original shapes, respectively, G_i is the i-th basis vector, and w_i is the weighting factor. Different shapes and their weighting factors are illustrated in Figure 2b.

As illustrated in Figure 2, the principle of the basis vector method is elucidated. Figure 2a shows the original shape and the basis vector. As the initial shape is a rectangle, base shape I is generated by offsetting the upper edge by 1/2 in the lateral direction, and base shape II is generated by offsetting the left node of the upper boundary along the lateral direction. By reducing the upper-end edge of the shape by one-half along the symmetry axis, base shape III is generated.

Subsequently, the coordinates of the four shapes are arranged in the same order, represented by vectors G₀, G₁, G₂, and G₃. G₀ is the original vector, while G₁, G₂, and G₃ are the three basis vectors. To generate a new shape, each coordinate of the shape change vector, G₁ − G₀, G₂ − G₀, and G₃ − G₀, is multiplied by a weighting factor: w₁, w₂, and w₃. The new node coordinate vector is then determined as follows:

$$G=G_0+w_1(G_1-G_0)+w_2(G_2-G_0)+w_3(G_3-G_0)$$

(11)

By changing the design variables w₁, w₂, and w₃, the corresponding node coordinates of the new shape can be calculated directly using Equation (11). Considering that the design variables w₁, w₂, and w₃ are continuous, the number of shapes can be imagined to be infinite. In general, the process of basis vector generation involves the following three steps:

(1)

Create a basis vector G_i based on the original design shape G₀.

(2)

Evaluate the correlation between the two shape change vectors v_i and v_j. Once the base vector is determined, the MAC (Modal Assurance Criterion) value should be calculated [37]:

$$MAC(v_i,v_j)=\frac{{(v_i^T{v}_j)}^2}{(v_i^T{v}_j)(v_i^T{v}_j)}$$

(12)

where v_i is the shape change vectors of two shapes: v_i = G_i − G₀. The MAC represents a normalized vector dot product with values between 0 and 1. If the MAC = 1, this indicates no correlation between the two vectors. If the MAC = 0, this means that there is a proportional relation, v_i = Kv_j, between the shape change vectors v_i and v_j, which suggests that the basis vector needs further modification.

(3)

To obtain a linear combination of vectors, the synthetic shape is derived from Equation (11).

4. Genetic Optimization Algorithm

The genetic algorithm [38] is used to obtain the weighting factors of the basis vector. According to [39], the genetic algorithm is a widely popular search and optimization method for resolving highly intricate problems, which has the capability of producing good-quality solutions. This section presents a novel approach for applying the genetic algorithm to determine the optimal geometry of one tire pitch for low-impact noise. Figure 3 illustrates a flowchart detailing the implementation of the genetic algorithm throughout this study.

Once an initial population of genomes is generated, the genetic evolution of the genome population undergoes a series that includes fitness tests, selection, reproduction, crossover, and mutation [40]. In the optimization algorithm, the optimization results are significantly affected by the objective function. Typically, the optimization problem is defined as follows:

$$\begin{array}{cc}Minimize:& F(X)=(f_1(X),f_1(X),\dots ,f_n(x)){}^T\hfill \\ Subject to:\hfill & \\ & { \mathrm{g}}_j(X)\le g_{j0} j=1,2,\dots ,m\hfill \\ & X_i^L\le X_i\le X_i^U i=1,2,\dots ,n\hfill \\ & X=\left\{X_1,X_2,\dots ,X_n\right\}\hfill \end{array}$$

(13)

where F_i(X) and g_i(X) are the objective and constraint functions. $X_i^L$ and $X_i^U$ are the lower and upper bound, respectively, of each design variable. During the pattern design stage, the objective function may be defined as the contact pressure variance, pattern stiffness, or another relevant factor. Generally, F and g can be computed using the Finite Element Method (FEM). However, the direct application of numerical optimization with the FEM demands multiple analyses. The objective function is calculated using the method described in Section 2.

5. Results and Conclusions

5.1. Semi-Anechoic Chamber Noise Experiment

To evaluate the prediction results of tire noise, a tire noise experiment was conducted in a semi-anechoic chamber following the measurement standard GB/T 23789-2016 [41]. The test parameters could be controlled in this chamber, minimizing the impact of external factors on the results. Before testing, the tire was mounted to the rim and was left in the laboratory for over one hour. Then, the tire was installed on the test equipment for the experiment. The M2 microphone was positioned at a distance of 1 m from the tire center and 0.2 m above the ground, as shown in Figure 4b.

In this experiment, a radial passage car tire with a size of 225/60R18 was selected. The testing conditions were an internal tire pressure of 230 kPa and a vertical load of 4640 N. The geometric shape of each pitch is illustrated in Figure 5a. There were four pitches, and the pitch length and pitch numbers are listed in

Table 1. Parameters of pitch sequence.

Pitch Type	Pitch Len. (mm)	Pitch No.	Pitch Sequence
A	45.23	10	DCDDC AABBC ABDDC ABCDB AABAA BDDCD BDCBC CAACB D
B	51.85	10
C	58.47	10
D	65.08	11

.

After the test, the tire noise data from microphone M2 were collected under various rolling speed conditions: 60 km/h, 80 km/h, 100 km/h, and 120 km/h; then, the 1/3-octave-band sound pressure levels (dB(A)) were calculated [42].

Figure 6 illustrates the 1/3-octave-band sound pressure level (dB(A)). There is a significant correlation with velocity when the frequency is over 800 Hz. The tire noise mainly comes from the tread block impact, contact area block slip, and pump noise due to the grooves. With the increase in speed, the impact force increases, which results in a higher noise emission. When the frequency is below 800 Hz, the tire noise is mainly caused by structural vibration, including of the carcass cavity and sidewall; this is related to the tire vibration characteristics and exciting force, and there is no obvious correlation between tire velocity and tire noise.

5.2. The Validation of the Tire Noise Prediction Method

In this research, to calculate the tire noise accurately, the footprint profile was calculated using FE software ABAQUS 6.14. The simplified tread pattern was modeled, which has sufficient accuracy in analyzing the tire’s steady state [43]. The tire reinforcement was modeled by the rebar layer, the rubber material was modeled as incompressible hyperelasticity, and the Yeoh constitutive model was used in this simulation. The tire three-dimensional model was generated with symmetrical model generation technology, and the load and inflation pressure are consistent with those of the noise test.

Figure 7b shows the footprint profile of a 225/60 R18 tire at a speed of 60 km/h. In this study, the leading edge of the footprint profile was used. We assume that each pattern element produces a unit pulse, resulting in a weight function that remains constant along the tread meridian direction. Thus, the $\gamma$ can be simplified to 1.0. The excitation force was obtained by moving the leading edge along the tread pattern, resulting in a total of 2048 data points. The predicted and measured 1/3-octave sound pressure levels (dB(A)) are shown in Figure 8.

It is clear from Figure 8 that the predicted result is in good agreement with the measured noise for frequencies over 800 Hz. In this range, the tire/road noise is primarily determined by the local deformations occurring at the leading and trailing edges of the tire’s footprint; these local deformations arise due to the impact of tread blocks during tire rotation. In contrast to the urban road traffic noise experiment [44], and to further validate the correlation between the experiment results and our prediction of external noise, our study followed the pass-by noise testing process outlined in ECE R117 [45]. During this test, the vehicle was driven along a designated test track following the centerline denoted as C-C in Figure 9, and two target microphones were placed on both sides of the track to measure the noise levels. These microphones were placed in the middle of the track 7.5 m away from the vehicle’s centerline at a height of 1.2 m. Subsequently, the maximum sound pressure level was recorded while the test vehicle was coasting. To obtain the final result for a reference speed, we conducted a linear regression analysis. If the discrepancy between the measurements from the left and right microphones exceeded 1 dB (a), the test result was deemed invalid and necessitated retesting. Finally, the sound pressure level (dB(A)) at a velocity of 80 km/h was calculated.

Four different tread patterns, denoted as A, B, C, and D, were selected for the experiment, while the tire size and structural design of 205/60 R16 were kept the same. Figure 7c shows the contact patch at a speed of 80 km/h. The evaluation results of the pass-by noise experiment are presented in Figure 10 where the predicted results are compared with the measured results. In this analysis, the predicted results were calculated using the 1/3-octave dB(A) range from 800–2000 Hz, which specifically covers the pattern impact noise range. The predicted values agree well with the test noise in terms of overall trends. This level of agreement is highly advantageous, as it facilitates the initial evaluation of pattern design. While the specific values may not match perfectly, this is mainly due to the test results of the pass-by noise experiment being influenced by environmental factors [46], road conditions, and the motion state of vehicles [47]. It is important to note that these measurement outcomes are a culmination of the combination of various mechanisms. In addition, our prediction model does not consider the radiation characteristics [48,49,50] of noise. This could be a plausible explanation for why the predicted value exceeds the observed test value. Nevertheless, the overall trend alignment demonstrates the effectiveness of the prediction model. In order to improve the prediction accuracy, the spatial noise radiation characteristics of the tire and the acoustic interaction between the tread pattern and tire body should be considered.

5.3. GA Optimization of Tread Pattern

5.3.1. Computational Condition

The proposed designs of the basis vector shapes are presented in Figure 11. Constraint conditions for the optimization process are listed in

Table 2. Constraint conditions.

Design variable	Base 1	Base 2	Base 3	Base 4
Range of design variables (wi)	−0.4~2.0	−0.5~4.0	−2.0~2.0	−0.8~1.2

, which involve the weighting factors of Base 1, Base 2, Base 3, and Base 4. The design variables for the i-th shape are defined as w_i.

In general, the leading edge can be defined by a straight line, two arcs [31], or a hyperelliptic equation [51]. For simplicity, the leading edge is described by the parabolic equation:

$$x^2=-2py (p>0)$$

(14)

where x and y are the coordinate values along the tread meridian and longitudinal direction, the parameter p can be obtained from the right-end point of the leading edge, and the maximum width is FW. Figure 12 shows the leading edges of different Y_c values. In the process of shape optimization, we set the leading edge to have Y_c = 25 mm and FW = 190.5 mm.

5.3.2. The Objective Function

The objective function plays a crucial role in determining the optimization result. According to the literature [52], any reduction in H_ij(f) can reduce the radiated noise. An exciting force between the tread pattern and the road that is around its mean value will lead to similar acoustic pressure variations inside the vehicle. According to the literature [28,42,53], a simple measure to quantify this change is the standard deviation; the standard deviation of the tread impact force function is considered to be a measure of the sound pressure level inside the vehicle:

$${\tilde{F}}_{STD}=\sqrt{{\displaystyle {\int }_0^{2\pi }{(\tilde{F}(\phi )-{\tilde{F}}_{mean})}^2}}$$

(15)

where F_mean is the average value of the exciting force. Furthermore, during the collision, the tread block impacts the road and pulsive noise is generated due to the deformation of the tread surface in the radial direction. Acoustic sources located near the contact path are then amplified by the tire–pavement geometry, resulting in far-field noise, which is then perceived by humans. The sound pressure level (SPL) experienced by pedestrians is important [54]. Here, we define the 1/3-octave sound pressure level curve, as shown in Figure 13.

f_s and f_e are the start and end frequencies, and S is the difference between the area of the original curve and that of the optimized curve. The objective function can be defined as:

$${\tilde{F}}_{spl}=\max (S)$$

(16)

When taking both objectives into account, the problem transforms into a multi-objective optimization problem. Generally, two methods can be used to solve multi-objective optimization problems: the Pareto method and the scalarization method [55]. In this study, NSGA-II was applied [56].

5.3.3. Computational Results

Based on the optimization method proposed above, the experimental tire pattern was optimized.

Figure 14 shows the original and optimized results where the noise has been significantly reduced in the target frequency range. It can be seen from the results that the tread pattern shape optimization is effective at reducing noise above 500 Hz. The low-frequency component of the pattern impact is mainly transmitted to the interior of the vehicle through the suspension system, which causes the vibration of the vehicle structures. In this paper, only pattern noise was considered, and low-frequency components were reduced by modifying the objective function.

The reduced values at each target frequency are 6.4 dB(A), 9.22 dB(A), 9.08 dB(A), 7.79 dB(A), and 6.55 dB(A), and the maximum reduction rate is 12.1% at the frequency of 1250 Hz. These specific values may differ from those of the experiment. This is mainly because the acoustic radiation characteristics and the coupling effect between the pattern blocks and tire body are not considered in our prediction model. However, these values can provide us with a reference to improve our design sketch in the pattern design stage.

In terms of optimization time, it takes only 5.5 s to obtain the first optimization solution and only 76 s to produce the final result. Different from regression methods, which include the support vector machine (SVM), relevance vector machine (RVM), ANN [57], and convolutional neural network (CNN) [58] (the RVM, ANN, and CNN take 66,297, 32,428, and 205,694 s for data training [59]), the optimization method proposed in this paper does not need to collect and train data. In addition, there are fewer parameters for pattern parameterization [31]. Last but not least, engineers’ design ideas can be integrated into the design of basis vector shapes.

The optimized pattern shapes are shown in Figure 15. The standard deviations of the exciting force of the original and optimized patterns are 2.62, 2.03, 2.00, and 1.99. The optimized pattern has lower variance and a better Sound Quality Preference Index [53]. Compared with other methods presented in the literature [31], the proposed method, which is based on the linear combination of the basis vector shapes, has fewer optimization parameters when needing to change the pattern shape.

5.4. Optimization of Footprint Leading Edge

5.4.1. The Process for Optimization of Leading Edge

The tire–pavement noise is influenced not only by the exciting force from the tread pattern but also significantly by the shape of the leading edge [22]. In the stage of pattern design, a well-designed pattern should exhibit superior performance over various footprint contours, which is a challenging task. Once the pattern design scheme is determined, the most suitable contact patch should be determined.

The contact patch can be controlled by the tire structure [60]. In this subsection, we further discuss the optimization of the leading edge to reduce pattern noise. The leading edge is described in the same way as in Section 5.3.1 where the equation coefficients are determined by the Y_c coordinate on the right end. In the optimization process, the leading edge where Y_c = −25 mm was selected as the original design scheme. The design variable was the y coordinate on the right end of the leading edge, which ranges from −50 mm to 0 mm.

5.4.2. Result of Optimization of Leading Edge

Figure 16 shows the 1/3-octave result of the optimized leading edges where a result similar to that of the pattern optimization was obtained. The reduced values at each target frequency are 5.19 dB(A), 4.40 dB(A), 4.88 dB(A), 4.56 dB(A), and 3.01 dB(A) within the optimized target frequency of 800–2000 Hz. The maximum reduction rate is 7.02% at the frequency of 800 Hz. Under similar time conditions, the pattern shape optimization has a higher reduction effect and takes less time.

The optimized shape of the leading edge is shown in Figure 17. It can be seen from the results that the leading edge has a significant effect on pattern noise, as reported in the literature [31], and the change in footprint shape may have an impact on other tire properties. This suggests that for a certain tread pattern, the leading edge should maintain minimal change under various conditions for a better tire noise performance.

6. Conclusions

This paper aimed to contribute to reducing tire pattern impact noise. The tire pattern noise prediction method was introduced and validated through indoor anechoic chambers and pass-by noise experiments. Then, the pattern shape optimization method was proposed, and the pattern shape was generated using the basis vector method. Finally, the potential of leading edge and pattern shape optimization to reduce pattern noise was discussed.

The main conclusions of this work are summarized below:

1.

A predictive model for tire–pavement interaction noise was developed, and noise experiments were conducted in a semi-anechoic chamber by measuring pass-by noise. The prediction results are in good agreement with the experiment results above 800 Hz.

2.

The basis vector method was applied to generate complex tire pattern structures. The new pattern shape was synthesized with linear weighting parameters among these base shapes, and it was found that the optimization parameters can be reduced by this method.

3.

The novel multi-objective function that was proposed aims to minimize the impact noise generated by the tire pattern. The optimization parameters were obtained using a genetic algorithm. This method can be used to improve the design scheme at the pattern design stage.

4.

Noise can be reduced through optimization of the pattern shape or leading edge, but changing the pattern may be a better choice when considering other performance factors.

In summary, the tire noise prediction and optimization method proposed in this paper offers the potential to effectively reduce noise, minimize trial and error costs in the early stage of pattern design, and significantly enhance development efficiency. However, low noise is a tire design criterion that may conflict with other objectives [61], such as wet traction, handling, hydroplaning, or safety. For example, when considering the wet skid performance of a tire, the water flow characteristics around a tire are largely influenced by lateral grooves [62]. The model proposed in [63] can be used to determine the parameters of the tread pattern, such as the width and angle of the lateral groove, for better wet skid performance. By keeping these parameters with minimal changes in each base shape vector, the optimization tread pattern can combine maximum safety with the lowest noise level.

In addition, compared to traditional fuel vehicles, electric vehicles (Evs) have lower engine noise and lower inlet and exhaust noise, and the tire–pavement interaction noise is the main source of vehicle noise at higher velocities [64]. The motion states of Evs (i.e., the constant speed state, the acceleration state, and the deceleration state) have a significant influence on vehicle noise [47]. When considering the optimization of Ev noise, several crucial steps must be followed. Firstly, it is essential to accurately determine the real leading edge through experiments or Finite Element Method (FEM) simulations at various motion states. Secondly, the obtained real leading edge should be used in the optimization process, as demonstrated in Figure 3. Subsequently, the objective function for reducing electric vehicle noise should be established by Equations (15) and (16) at each motion state. Finally, the optimization process of Ev noise can be carried out by employing the method proposed in this research.