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机械系统信号处理 211 (2024) 111181

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测量识别 滚动轮胎 平面内动态行为

伊万诺·帕利亚 a * 卢卡·拉皮诺 a 弗朗切斯科·里帕蒙蒂 a 西蒙娜·巴罗 b Roberto Corradi a
伊万诺·帕利亚a*卢卡·拉皮诺a弗朗切斯科·里帕蒙蒂a西蒙娜·巴罗b,RobertoCorradia

a Politecnico di Milano, Department of Mechanical Engineering, Via La Masa 1, 20156 Milano, 意大利 b Pirelli Tyre S.p.A., Viale Piero e Alberto Pirelli 25, 20126 米兰, 意大利
PolitecnicodiMilano,Department ofMechanicalEngineering,Via LaMasa 1,20156Milano,意大利PirelliTyreS.p.A.,VialePieroAlbertoPirelli25,20126米兰,意大利

A R T I C L E I N F O 传达 John E. Mottershead Key的话:
ARTCLENFO传达John E.Mottershead Key的话:

滚动轮胎动力学 实验夹板测试 传播解决方案 识别算法 结构噪声

A B S T R A C T

作用在轮胎可以提高道路车辆的 NVH 性能 轮胎 起着至关重要的作用 结构噪声 振动传输到 机舱最高可达 500 Hz。 本文 旨在 研究 滚动轮胎 平面内动力学 频率范围。 轮胎径向振动测量是通过 专用的实验装置进行的 基于 激光 三角测量传感器扫描滚动轮胎 胎面 实验数据是在夹板测试期间收集的,考虑了不同的滚动速度 充气压力,研究它们轮胎动力学的影响 然后,数据处理 算法 提出的目的是 识别 传播的渐进波和退 锁片撞击后的 轮胎周长 最后,为了解释 结果,确定轮胎响应进行了比较 通过 简化平面内轮胎模型获得的分析结果
作用在轮胎可以提高道路车辆的 NVH性能轮胎起着至关重要的作用结构噪声振动传输到机舱最高可达500Hz。本文旨在研究滚动轮胎的平面内动力学频率范围。轮胎径向振动测量是通过专用的实验装置进行的基于激光三角测量传感器扫描滚动轮胎胎面实验数据是在夹板测试期间收集的,考虑了不同的滚动速度和充气压力,研究它们轮胎动力学的影响然后,数据处理算法提出的目的是识别传播的渐进波和退锁片撞击后的轮胎周长最后,为了解释结果,确定轮胎响应进行了比较通过简化平面内轮胎模型获得的分析结果

1. 引言

汽车行业 车辆舒适性的关注 显著 增加。 因此 人们开始关注 车辆 NVH(噪声、 振动声振粗糙度)性能 [1,2]。 此外, 近年来 电动汽车 越来越 受欢迎,因此 轮胎/路面相互作用当今 影响 整体噪声的最重要因素,并且 在车辆感知到的振动水平 [3]。 此外, 乘客 机舱 感知也在发生变化因为 引入 自动驾驶 功能。 在这种情况下 必须 提高道路车辆 NVH 性能尤其是 <b 上<b1197> 1198>轮胎, 结构性座舱噪声振动 着重要作用 500 Hz [4,5]。
汽车行业车辆舒适性的关注显著增加。因此人们开始关注车辆NVH(噪声、振动声振粗糙度)性能[1,2]。此外,近年来电动汽车越来越受欢迎,因此轮胎/路面相互作用当今影响整体噪声的最重要因素,并且在车辆感知到的振动水平[3]。此外,乘客机舱感知也在发生变化因为引入自动驾驶功能。在这种情况下必须提高道路车辆NVH 性能尤其是在 1198>轮胎,结构性座舱噪声振动着重要作用500Hz[4,5]。

可以 采用 不同的方法来研究 轮胎 动态行为 方面, 建模的角度来看 有几种解决方案可能是 根据所需的复杂程度 进行考虑 分析模型计算 效率高,可在短时间内 提供结果; 然而,它们 提供了这种现象的一般理解 [6]。 有限模型需要苛刻的计算工作,但实现了这种可能性 轮胎设计的影响进行预测参数分析 参数 [7]。

另一方面 可以 进行实验研究并且可以使用特定的信号处理技术 用于 识别 模型标定 NVH 有用的参数 分析。 为此文献提出了 不同的方法 固有频率、振型 模态阻尼系数可以通过 Experimental 来确定 模态分析
另一方面可以进行实验研究并且可以使用特定的信号处理技术用于识别模型标定 NVH 有用的参数 分析。为此文献提出了不同的方法固有频率、振型模态阻尼系数可以通过实验来确定模态分析

* 通讯作者。

电子邮件地址ivano.lapaglia@polimi.it (I. La Paglia)。

https://doi.org/10.1016/j.ymssp.2024.111181

接收日期 2023 年 6 月 22 ; 2023 年 12 月 20 修订接收; 接受日期 2024 年 1 月 23 日在线提供 2024 年 2 月 4
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(http://creativecommons.org/licenses/by/4.0/)。

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(EMA). This analysis can be performed in static conditions, where the tyre is excited at one location through an impact hammer or a shaker, and accelerations are measured at different positions along the tyre circumference. The Frequency Response Functions (FRFs) of the system are computed and used in the extraction of the modal parameters. Several test bench configurations have been proposed in literature to carry out the EMA of a static tyre, such as in [8,9]
(欧洲药品管理局 (EMA))。这种分析可以在静态条件下进行,其中轮胎通过冲击锤或振动器在一个位置被激发,并在沿轮胎圆周的不同位置测量加速度。计算系统的频率响应函数 (FRF) 并用于模态参数的提取。文献中已经提出了几种测试台配置来执行静态轮胎的 EMA,例如 [8,9]
.

Although tyre vibrations have been deeply studied in static conditions, difficulties arise when dealing with a rolling tyre. At first, vibration measurements cannot be easily performed with conventional accelerometers placed on the tyre outer surface. Therefore, other measuring systems such as laser Doppler vibrometer should be considered [10]. Furthermore, the dynamic response at the contact patch cannot be measured. Secondly, a reliable identification of the input force becomes challenging. This prevents the execution of the classical EMA, and other identification techniques must be considered. In literature, most of the authors rely on the Operational Modal Analysis (OMA) to identify the modal parameters of the rolling tyre [9,1114]
尽管在静态条件下对轮胎振动进行了深入研究,但在处理滚动轮胎时会出现困难。起初,使用放置在轮胎外表面的传统加速度计无法轻松进行振动测量。因此,应考虑其他测量系统,如激光多普勒测振仪[10]此外,无法测量接触贴片处的动态响应。其次,可靠地识别输入力变得具有挑战性。这会阻止经典 EMA 的执行,并且必须考虑其他识别技术。在文献中,大多数作者依靠业务模态分析(OMA)来确定滚动轮胎的模态参数[9,11\u201214]
.

Regarding the test configuration, one of the typical solutions adopted to excite a rolling tyre in indoor conditions relies on a cleat test, where the tyre rolls over an obstacle providing an impulsive-like excitation. This test is suitable for understanding the tyre low frequency dynamic behaviour, given that the maximum frequency excited by the cleat impact is approximately 300 Hz [2]. For instance, a cleat test was carried out in [15] mounting the obstacle on the surface of a 2 m diameter steel drum, with a perpendicular inclination with respect to the tyre rolling plane. The tyre hub forces were measured and used to tune an in-plane ring model. A similar test setup was used in [9], where a laser Doppler vibrometer was adopted to measure the tyre radial surface vibrations. An alternative cleat test setup was proposed in [13], constituted by two identical tyres which rotate one against the other. The undriven wheel is mounted on a dynamometric hub which allows the measurement of the forces generated during the test. The excitation is provided by an aluminium cleat mounted on the driven wheel, and a laser Doppler vibrometer was used to measure the surface vibrations.
关于测试配置,室内条件下激励滚动轮胎的典型解决方案之一依赖于防滑钉测试,其中轮胎obstacle提供类似冲动激励。测试适用于了解轮胎低频动态行为,因为锁片撞击激发的最大频率约为300Hz[2]对于例如,直径2m表面进行夹板测试[15],使其垂直倾斜轮胎滚动平面。测量轮胎轮毂用于调整平面内模型。类似的测试装置用于 [9]其中采用激光多普勒测振仪测量轮胎径向表面振动。[13] 提出了一种替代的防滑钉测试装置,由两个相同的轮胎组成,它们相互旋转无驱动安装在测功轮毂,可以测量测试过程中产生的激发安装在从动上的铝制夹板提供并使用激光多普勒测振仪测量表面振动。

A phenomenon that has been observed by many authors during rolling tyre tests is the bifurcation effect [16]. In static conditions, input forces generate progressive and regressive waves that propagate along the tyre structure at the same speed. Consequently, their superposition results in standing wave vibration modes. In rolling conditions, progressive and regressive waves travel at different propagation speeds due to Coriolis accelerations. Moreover, a Doppler frequency shift is present if the tyre rotation is observed in a fixed reference system. The Coriolis effect prevents the realization of standing waves, and the resulting modes will be complex; thus, they will present a propagation velocity [9,13,1618]. The bifurcation effect depends on the tyre rolling speed, the considered mode number and the corresponding mode shape. Based on these considerations, the classical definition of mode shape cannot be applied to rotating systems, and identification techniques should be based on wave propagation models.
许多作者滚动轮胎测试观察到的一个现象分岔效应[16]静态条件下,输入会产生渐进回归它们相同的速度沿轮胎结构传播因此,它们的叠加导致振动模式。滚动条件下,由于科里奥利加速度进波回归不同的传播速度传播。此外,如果在固定参考系统中观察到轮胎旋转则存在多普勒科里奥利效应阻止驻波的实现产生的模式复杂;因此,它们呈现传播速度[9,13,1618]效应取决于轮胎滚动速度、所考虑模态相应的振型基于这些考虑,经典的振型定义不能应用于旋转系统,识别技术基于传播模型。

This paper aims at investigating the in-plane dynamic behaviour of a rolling tyre in the frequency range below 300 Hz through an indoor experimental approach. A measuring setup specifically designed to this purpose is presented. A cleat test is performed on a steel drum, and tyre radial vibration measurements are carried out through a laser triangulation sensor scanning the tread of the rolling tyre. Data are collected at different rolling speeds and inflation pressures to investigate the influence of both parameters on the tyre dy- namics. Secondly, starting from the tyre surface vibration at several positions along the tyre circumference, a strategy to identify the wave propagation field is proposed. With respect to previous publications, in this paper experimental data are fitted using a wave propagation formulation. Through this approach, no Complex Modal Testing [19] or Operational Modal Analysis techniques are
本文旨在通过室内实验方法研究滚动轮胎低于300Hz的频率范围内的平面内动力学行为提出了一种专门为此目的设计的测量装置进行夹板测试并通过扫描制胎面激光三角测量传感器进行轮胎径向振动测量轮胎。在不同滚动速度和充气压力收集数据以研究这两个参数轮胎动态的影响其次,轮胎沿轮胎圆周多个位置的轮胎表面振动出发,提出了一种识别传播的策略相对于以前的出版物,本文使用传播公式拟合实验数据通过这种方法没有复杂模态测试[19] 操作模态分析技术

Fig. 1. Measurement grid: 50 nodes regularly spaced every 5 , from 33 to +212 (their position is defined in the fixed reference system). For visualization purposes, only nodes 1, 10, 30 and 50 are highlighted.
无花果。1.测量网格:50个节点5节点从 33 +212 间隔(它们的位置固定参考系统中定义)。出于可视化目的,突出显示节点1、10、3050

2

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applied; no analytical model is involved during the identification procedure, such as in [11,12]. The discontinuity in tyre radial vi- bration due to the contact patch is taken into account enabling the identification of real circumferential mode numbers. In addition, to interpret the results, the test data have been compared to the analytical results obtained with a simplified in-plane tyre model, showing a satisfactory degree of correlation.
应用的;识别过程中涉及任何分析模型例如[11,12]中,由于接触引起的轮胎径向振动不连续性被考虑在内启用模式数的标识此外,为了解释结果测试数据使用简化的面内轮胎模型获得的分析结果进行了比较显示出令人满意的程度相关性。

The paper is organized as follows. In Section 2, the experimental setup realized to measure the tyre response during a cleat test is described. In Section 3, the data processing technique to synchronize data and extract the tyre vibration out of the recorded time histories is described; then, the wave propagation algorithm is presented and details on the fitting procedure are provided. In Section 4, different datasets are compared in order to investigate the influence of speed and inflation pressure; in addition, the results are compared with those of a tyre analytical model. Eventually, Section 5 draws the conclusions of this work.
本文的组织结构如下2 节中,描述了防滑钉测试期间测量轮胎响应实验设置3 节中,描述了同步数据并从记录的时中提取轮胎振动数据处理技术;然后,给出了传播算法,并详细介绍了拟合过程4 节中,比较不同的数据集研究速度和充气压力的影响;此外,还将结果轮胎分析模型的结果进行比较最终,5得出了这项工作结论

2. Experimental setup
2.实验装置

In order to measure the vibration of the tyre in rolling conditions, a dedicated experimental setup has been designed and installed in an existing facility, where cleat tests are carried out. The test rig is made up of a steel drum driven by an electric motor. A laser triangulation sensor (MEL Microelectronics M7L 50) is adopted to scan the tyre surface. The sensor has a measuring range of 50 mm around the focus distance of 95 mm. It is kept at a fixed distance from the tyre by the supporting structure, and it measures the tyre response after the cleat impact. The vibration measurements have been performed at 50 positions (nodes) along the tyre circumfer- ence. A measurement grid of 5 has been adopted, with a total covered angle of 245. Due to the contact patch and the contiguous regions that could not be reached by the laser beam, given the presence of the drum, no measurements could be performed over 115. For the sake of clarity, Fig. 1 shows the measurement grid: for visualization purposes, only few nodes are reported. The centre of the contact patch is located at the angle position 270, while the first measuring position is in correspondence of the angle 327. The blue arrows on Fig. 1 show the adopted convention for the reference system and the tyre rolling direction. In this work, a transversal cleat (perpendicular to the x-y plane) was used, and we refer to radial vibrations occurring in the x-y plane as the in-plane tyre response.
为了测量轮胎滚动条件下振动已经设计了一个专门的实验装置,并在现有设施安装了防滑钉测试装置执行测试电动驱动的组成采用激光三角测量传感器 (MELMicroelectronicsM7L50) 扫描轮胎表面。传感器95mm 的焦周围具有 50mm测量范围通过支撑结构轮胎保持固定距离测量轮胎在锁碰撞后的响应振动测量沿轮胎圆周50个位置(节点)进行采用了 5测量网格覆盖245由于接触贴片相邻区域可以激光束无法到达鉴于滚筒的存在无法进行超过115测量为了清晰起见,无花果。1测量网格:出于可视化目的,报告了少量节点接触的中心位于角度位置270第一测量位置角度327 相对应图 1 上的蓝色箭头1显示了参考系统和轮胎滚动方向的采用约定在这项工作使用了横向夹板(垂直x-y平面),我们将发生在x-y平面上的径向振动称为平面内轮胎响应。

The structure that carries the sensor was designed to be stiff and stable in order to measure the tyre vibration, reducing as much as possible any disturbance. To this end, the structure was installed on supports that are dynamically decoupled from the test rig basement by means of a suspended foundation. Fig. 2 shows a 3D CAD model of the designed system, that is constituted by two set of elements: a fixed rigid frame (supporting structure), which carries the loads and provides the required stiffness; a moving structure connected to the fixed frame, that supports the laser sensor and which can be fixed at different angular positions around the centre of the tyre.
承载传感器的结构被设计坚硬和稳定以便测量轮胎振动,尽可能减少任何干扰。为此结构安装在通过悬空基础测试地下室动态解耦支架无花果。图 2显示了所设计系统的 3DCAD模型它由元件组成:一个承载载荷的固定刚性框架(支撑结构)并提供所需的刚度;一种所述固定框架相连移动结构,所述固定框架支撑所述激光传感器,且该结构可以固定在所述激光传感器中心周围的不同角度位置轮胎。

Considering the moving structure, it is positioned at the centre of the horizontal beam, aligned with the centre of the tyre. The
考虑到移动结构,位于水平的中心轮胎的中心对齐

rotation of the structure (red dashed line in Fig. 2b) is provided by an electric motor. At each position, the tyre vibration is acquired for
结构旋转图 1 中的红色虚线)。2b)电动提供在每个位置,获取轮胎振动

30 s at a sampling frequency of 5 kHz. Once the acquisition is completed, the sensor automatically moves to the subsequent position. To reduce the effect of the oscillations induced by the arm rotation, before to start the next acquisition, further 30 s are considered so as to reach a static condition. At the end of the test, 50 time histories are collected.
采样频率为 5 kHz 30采集完成后,传感器会自动移动到后续位置。为了减少机械臂旋转引起的振荡的影响开始下一次采集之前,再考虑30达到静态条件。在测试结束时将收集 50时间历史记录

Fig. 2. a) 3D CAD model of the laboratory setup used for the measurements on a rolling tyre. Red dashed line in b) indicates the trajectory followed by the measuring system during operation. c) Side view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
无花果。2.a用于测量滚动轮胎实验室设置3DCAD模型b) 中的红色线表示测量系统在运行过程中所遵循轨迹c)视图。(有关中对颜色的引用解释读者请参阅本文的网络版本

3

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Fig. 3 shows a picture of the test bench. In Fig. 3a an overview of the designed system is provided, with the tyre loaded against the driving drum; Fig. 3b shows a detail of the laser triangulation sensor together with the cleat (5 mm height, 10 mm length along the rolling direction) adopted to excite the tyre.
无花果。图 3显示了测试图片图 .3a 提供了设计系统的概述轮胎紧靠驱动鼓;无花果。图 3b显示了激光三角测量传感器以及用于激励轮胎夹板(高度为 5mm沿滚动方向长度10mm)的细节

3. Vibration measurements on a rolling tyre
3.滚动轮胎振动测量

In this section, the methodology developed to investigate the dynamic behaviour of the rolling tyre is presented. In Section 3.1, the processing of the time histories collected along the tyre circumference is described; in Section 3.2, the wave field identification al- gorithm is presented.
本节介绍了为研究滚动轮胎动态行为而开发的方法3.1 节中,描述了沿轮胎长收集的时间历史处理;3.2 节中,介绍了标识algorithm

3.1. Signal processing
3.1.信号处理

The experimental setup described in Section 2 allowed measuring the tyre radial vibration consequent to the cleat impact. Data
2中描述的实验设置允许测量锁片冲击引起的轮胎径向振动数据

have been collected at speeds of 50 and 90 km/h and inflation pressures of 2.2 and 2.7 bar in order to investigate the effects of these
5090公里/小时的速度以及2.22.7充气压力收集研究这些影响

changes on the tyre dynamics. Trigger signals of the tyre and drum rotations have been recorded to allow the data synchronization and
轮胎动力学的变化已记录轮胎旋转触发信号以实现数据同步

post-processing. In the following, a single time history (node 50, 212) out of the 50 available, measured at 50 km/h at a pressure of
后处理。在下文单个时间历史(节点 50,212 可用节点中的 50,21250 公里/小时的压力下以 50公里/小时的速度测量

2.2 bar, is considered as a reference case to present the experimental results and data processing.
2.2bar,作为参考案例呈现实验结果数据处理

Fig. 4a shows the displacement signal acquired by the laser triangulation sensor for two subsequent impacts. It is possible to notice that the signal is constituted by two main components: the first one, with a time periodicity of 0.15 s, is related to the tyre non- uniformity; the second and most important one is associated to the dynamic response of the tyre due to the cleat impact, and it shows a period of 0.57 s (identified by a vertical dashed line).
无花果。图 4a显示了激光三角测量传感器两次后续冲击采集位移信号可以注意到信号两个主要部分组成第一个周期0.15s,轮胎有关均匀性;第二个也是最重要的一个轮胎由于锁片冲击而产生的动态响应有关显示0.57s 的周期垂直线)。

In this paper, the analysis is focused on the investigation of the tyre free response due to the cleat excitation. Thus, two signal processing steps are required: on the one hand, the tyre non-uniformity contribution was removed; on the other, the forced response (initial time instants associated to the cleat-tyre interaction) was recognised and discarded from the data.
在本文中分析的重点是研究由于防滑钉激励引起的轮胎响应因此,需要两个信号处理步骤方面,去除轮胎不均匀性贡献;另一方面,强制响应Cleat-Tyre交互相关的初始时间时刻识别并从数据中丢弃

At first, attention is paid to the tyre non-uniformity contribution. The tyre trigger signal (periodic with the tyre rotation) was used to isolate the time windows associated to every tyre passage. Later, time-averaging was used to generate the non-uniformity signal, for a single tyre revolution. Then, this signal was replicated to have a time history of the same length of the original signal, as shown in Fig. 4b. Finally, the tyre response associated only to the cleat excitation was obtained by subtracting the time histories of Fig. 4a and b. The achieved result can be observed in Fig. 4c.
首先,关注轮胎不均匀性的贡献。轮胎触发信号(与轮胎位周期性用于隔离每次轮胎通过相关的时间窗口后来,利用时间平均生成单次轮胎旋转的非均匀性信号然后,复制信号具有原始信号相同长度的时间历史如图 1 所示4b.最后,通过减去图 1 的时间历程,获得防滑钉激励相关的轮胎响应4ab.所获得的结果可以在图 4 观察到4c.

Once the tyre non-uniformity was removed, the average cleat response was evaluated to limit the influence of background noise. To this end, the drum trigger was used to identify the time windows related to a cleat impact (vertical dashed line in Fig. 4c), and time- domain averaging was applied to a total of 52 cleat responses. Moreover, the signal related to the first order of the drum revolution was removed since this signal component is not related to the cleat response. The result of this procedure is shown in Fig. 5 as a black line.
消除轮胎不均匀性评估平均防滑钉响应限制背景噪声的影响为此,使用触发器来识别防滑钉撞击相关的时间窗口图 1 中的垂直虚线)。4c),并将平均应用于总共52cleat响应。此外,旋转相关的信号删除因为信号分量夹板响应无关此过程的结果如图 1 所示5显示为线。

In addition, the standard deviation of the collection of 52 measurements was quantified to be of approximately 50 μm during the whole drum revolution. This value, associated to the background noise, is significantly smaller compared to the vibration levels shown in Fig. 5. Moreover, the averaging operation further reduces this measurement uncertainty.
此外,在整个滚筒旋转过程中52测量值的标准量化50μm背景噪声相关的该值与图 1 中所示振动水平相比得多。5此外,平均运算进一步降低了这种测量不确定性。

The black signal shown in Fig. 5 consists of both the forced response (when the tyre rolls over the obstacle) and the subsequent free response, which is the main focus of this work. Therefore, the next step of the signal processing is the distinction of the two contri- butions. To this aim, the duration of the cleat impact was estimated, given the size of the obstacle (5 mm height), the contact patch length and the corresponding tyre angular section influenced by the cleat excitation (0.58 rad). An example of the distinction of the time response influenced by the cleat impact (forced response) and the tyre free response can be observed in Fig. 5. Note that the duration of the free response (red curve) was identified considering a thirty-fold amplitude reduction (evaluated considering the whole
图 .5包括强制响应(当轮胎翻越障碍物时)随后自由响应,这是这项工作的主要重点因此,信号处理下一步区分两个贡献因素。为此根据障碍物的大小(5mm高)、接触长度相应的轮胎,估计了防滑钉撞击的持续时间受夹板激发影响的角度截面 (0.58rad)。图 1 可以观察到防滑钉撞击(强制响应)轮胎响应的时间响应区别的一个例子5请注意自由响应(红色曲线)持续时间考虑到减少 30 倍(考虑整体

Fig. 3. Test bench for vibration measurements on a rolling tyre. a) Overview of the laboratory with tyre installed on the test machine. b) Detailed view of the laser triangulation sensor and the cleat adopted to excite the tyre.
无花果。3.用于滚动轮胎振动测量试验一)测试机上安装了轮胎实验室概览湾)激光三角测量传感器用于激励轮胎夹板的详细视图

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Fig. 4. Laser signal measured at 50 km/h, 2.2 bar, angular position 212 (node 50). a) Measured response to the cleat excitation; b) tyre non- uniformity; c) laser signal with reduced non-uniformity component, obtained as the algebraic subtraction of signals a) and b).
无花果。4.激光信号50公里/小时,2.2巴,角度位置212(节点50)下测量一)测量防滑钉激励的反应;湾)轮胎均匀性;c)均匀性分量降低激光信号作为信号a) b代数法获得

Fig. 5. Distinction of the tyre free and forced response from the average dynamic response (50 km/h, 2.2 bar, node 50, angular position 212).
无花果。5.轮胎自由受迫响应平均动态响应(50km/h,2.2 bar,节点50,位置212的区别

measurement grid). This was necessary to limit the amount of data to be later processed, still ensuring to include all the relevant harmonic contributions of the tyre free response.
测量网格)。这对于限制以后要处理的数据必要的同时仍然确保包括轮胎自由响应所有相关谐波贡献

Finally, the described procedure was iterated for all 50 angular positions. In doing so, the tyre trigger signal allowed to synchronize
最后,对所有50角度位置迭代了所描述的过程在此过程中,轮胎触发信号允许同步

all the measured time histories.
所有测量的 timehistory。

3.2. Wave field identification algorithm
3.2.识别算法

In this section, the algorithm adopted to identify the dynamic behaviour of the tested tyre is presented. The collected experimental
本节介绍了用于识别被测轮胎动态行为算法收集的实验

signals are fitted through the following model:
信号通过以下模型进行拟合

n

i=1

Ai eαi t cos(ωi t ki θ + ϕ
Ae−cosω− kθ + φ
i
)

(1)

Eq. (1) represents the tyre free response to the cleat impact, described in the fixed reference system, in terms of radial displacement of
方程(1)表示轮胎锁片冲击的响应固定参考系统中描述径向位移

the tyre belt ur . Thus, ur is defined as a function of both angular position θ and time t. Eq. (1) relies on a wave propagation solution, that
轮胎u因此,u定义为位置θ时间Eq.(1) 函数,依赖于传播解,

models the tyre response as the superposition of both progressive and regressive waves travelling along the tyre circumference. In Eq.
轮胎响应建模沿轮胎圆周传播渐进波和回归的叠加方程中。

(1):

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ur is defined as the summation of n travelling waves, n being an even number associated to the order of the system (defined a priori,
u定义为n的总和n系统数相关的(先验定义

as later addressed) and n/2 representing the number of progressive and regressive wave pairs propagating along the tyre circumference;
稍后讨论)n2表示沿轮胎圆周e 传播回归的数量;

ωi = 2πfi is the circular frequency of the i-th wave function;
ω=2π 是第 i函数圆周频率;

ki represents the circumferential mode number, that is a real constant assuming positive values in case of progressive waves
k表示圆周态数,进波的情况下假设为正值的实常

(travelling counterclockwise, in the same direction as the tyre rotation) and negative values in case of regressive waves (travelling clockwise, against the tyre rotation);
逆时针行进轮胎旋转方向相同如果出现回归顺时针行进,与轮胎旋转相反);

Ai and ϕi are real constants respectively associated to the amplitude and the phase of the generic i-th wave;
Aφ分别是通用第 i振幅相位相关的;

αi is the damping coefficient; the damping contribution is represented by the decaying exponential function eαi t , where αi = ξiω
• α 为阻尼系数;阻尼贡献由衰减指数函数 e−α 表示,其中 α= ξω
i

and ξi is the nondimensional damping ratio.
ξ无量纲阻尼比。

Eq. (1) is evaluated for specific θ values corresponding to the measurement grid of Fig. 1. As a result, for each angular position θ, ur is a function of time t only. Considering the data collected during the experimental cleat test, the difference between the measured signal and the fitting function of Eq. (1) can be evaluated. At any time instant t and each measurement position, the squared error has been evaluated and the cost function to be minimized has been defined based on the cumulated error. In order to identify the unknown parameters of Eq. (1), a nonlinear data fitting has been performed by means of the lsqnonlin Matlab function.
方程 (1) 针对对应于图 1 测量网格的特定 θ进行评估1因此,对于每个位置θu 只是时间函数考虑到实验夹板测试期间收集的数据可以评估测量信号方程(1)拟合函数之间的差异在任何时刻每个测量位置,都评估了平方误差,并根据累积的错误。为了识别方程(1)未知参数通过lsqnonlinMatlab函数进行了非线性数据拟合

Although all the 5xn parameters are identified at the end of the minimization procedure, attention should be focused on the values of the circumferential mode number k. When the case of the unload tyre is considered, the circumferential mode number k needs to be an integer [17]. In fact, since the tyre is closed in the θ direction and no constraints are applied on the tread surface, the period of the function ur is necessarily an integer submultiple of 2π. However, in the considered case, the tyre is loaded against the drum, and therefore some influence of the contact patch is expected. Consequently, no constraints on the parameter k have been imposed, leaving it free to assume any real value.
尽管最小化过程结束时确定了所有 5xn参数,但注意模式k 的值考虑卸载轮胎的情况下圆周态数k需要是一个整数[17]事实上,由于轮胎θ 中闭合方向没有胎面施加约束函数u周期必然2整数子倍数 π但是,所考虑的情况下轮胎负载因此预计接触有一些影响因此,没有参数k 施加任何约束让它自由地假设任何值。

In order to find the global minimum of the error function, the lsqnonlin function requires reliable initial values for the parameters. These have been estimated by analysing the 2D Fast Fourier Transform (2D FFT) of the available time histories. Fig. 6a shows the results of the 2D FFT processing applied to the reference dataset (50 km/h, 2.2 bar). The x and y axes respectively report the circumferential mode number and the frequency of the wave contributions. The zero-padding technique was adopted to manage the data measured along the tyre portion of 245, to increase the spatial resolution in terms of circumferential mode numbers k. As a result, it can be observed that integer circumferential mode numbers k are obtained. It is worth clarifying that this strategy was used just to compute the initial values for the minimization. Conversely, at the end of the minimization, the k values are expected to assume non- integer values, as also confirmed in [12]
为了找到 error 函数的全局最小值,lsqnonlin 函数需要可靠的参数初始值。这些是通过分析可用时间历史的 2D 快速傅里叶变换 (2D FFT) 来估计的。图 6a 显示了应用于参考数据集 (50 km/h, 2.2 bar) 的 2D FFT 处理结果。x 轴和 y 轴分别报告圆周模态数和波贡献的频率。采用零填充技术来管理沿轮胎部分 245 测量的数据,以提高圆周模态数 k的空间分辨率,因此可以观察到获得了整数圆周模态数 k。值得澄清的是,此策略仅用于计算最小化的初始值。相反,在最小化结束时,k 值应采用非整数值,这在 [12] 中也得到了证实
.

Attention is now paid to the estimation of the initial guesses. Fig. 6a allows estimating the number of waves mainly contributing to the overall tyre free response. Five progressive and five regressive waves are mostly highlighted in the colormap, thus defining the n parameter of Eq. (1) for the considered dataset (50 km/h). Note that n generally varies depending on the tyre speed.
现在注意初始猜测估计无花果。6A允许估计主要影响整体轮胎无响应5 个5回归色图大多突出显示从而所考虑数据集50km/h)定义了方程(1)n参数请注意n通常根据轮胎速度而变化

Reference is now made to a specific column of the Fig. 6a, which represents the spectrum of the signal for fixed circumferential mode number (i.e., for every wave function). For instance, Fig. 6b shows the harmonic content associated to ki = 2. The spectrum allows estimating first attempt values for the amplitude Ai, extracted considering the most relevant harmonic component, and the
现在参考Fig.6a,表示固定圆周态数(即每个函数)的信号频谱例如图 .6b显示了k=2 相关的谐波含量 该频谱允许估计考虑相关的谐波量提取振幅A首次尝试并且

Fig. 6. Signal processing for the initial guess estimation (reference dataset 50 km/h, 2.2 bar). a) results of the 2D FFT processing; b) spectrum associated with the k =−2 wave function.
无花果。6.初始猜测估计信号处理(参考数据集50km/h,2.2 bar)。一)2DFFT处理的结果;湾)K=−2函数相关的频谱

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corresponding frequency ωi. The nondimensional damping ratio ξi was estimated according to the half-power point method. Finally, the phase ϕi was set to a null value as first attempt guess.
对应频率ω无量纲阻尼ξ根据半功率估算最后,相位φnull作为首次尝试猜测。

4. Results and discussion
4.结果讨论

In this section, the outcomes of the identification process are presented and discussed to investigate the effect of tyre rolling speed and inflation pressure. At first, a reference case is analysed, with the tyre inflated at 2.2 bar and rolling at 50 km/h. The identified model parameters are listed in Table 1, with each column associated with a specific parameter. Progressive and regressive waves are considered separately and listed for increasing values of the circumferential mode number k. As expected, the k values are not integer numbers. However, it is generally possible to identify pairs of circumferential mode number with positive and negative values, and similar magnitude, although the corresponding frequencies are considerably different (this observation will be later analysed in more detail in Fig. 12). These results are in general accordance with those reported in [12]
在本节中,介绍并讨论了识别过程的结果,以研究轮胎滚动速度和充气压力的影响。首先,分析一个参考案例,轮胎在 2.2 bar 时充气并以 50 km/h 的速度滚动。表 1 中列出了已识别的模型参数,每列都与特定参数相关联。行波和回归波是分开考虑的,并列出了圆周模数 k的增加值。正如预期的那样,k 值不是整数。然而,通常可以识别出具有正值和负值以及相似幅度的圆周模态数对,尽管相应的频率差异很大(稍后将在图 12 中更详细地分析这一观察结果)。这些结果与 [12] 中报告的结果基本一致
.

The quality of the identification can be checked by plotting the identified wave function superimposed to the experimental signal. For instance, Fig. 7 shows the comparison between the identified signals and the experimental ones, corresponding to the measurement nodes 10 and 30. The same procedure can be applied to each one of the 50 measuring positions. The vertical dashed lines in Fig. 7 identify six time instants (the first occurring at t = 0 s) that will be later considered in Fig. 9
可以通过绘制叠加到实验信号的已识别波函数来检查识别质量。例如,图 7 显示了识别信号与实验信号之间的比较,对应于测量节点 10 和 30。相同的程序可以应用于 50 个测量位置中的每一个。图 7 中的垂直虚线标识了 6 个时间时刻(第一个出现在 = 0 s 时),稍后将在图 9 中讨论
.

A satisfactory degree of agreement is reached between the experimental and identified free response, respectively shown as a solid
实验鉴定自由响应之间达到令人满意一致性分别显示为固体

red line and a dashed black one.
线黑色虚线

For the sake of completeness, Fig. 8 shows the mean square error computed over the whole duration of the free response for each of
为了完整起见,8显示了自由响应的整个持续时间计算误差

the 50 measuring positions.
50测量位置。

Slightly worse performances are observed in proximity of the contact patch. As a result, an increase in the identification error can be recognised for these measurement positions. This is related to slight mismatch of the signal amplitudes at these locations during the initial portion of the time history.
接触贴片附近观察到的性能因此可以识别出这些测量位置的识别误差增加这与时间历史的初始部分这些位置的信号幅度的轻微不匹配有关

Table 1
1

Results of the identification procedure. Reference case: tyre speed of 50 km/h, inflation pressure of 2.2 bar. Influence of rolling speed: tyre speed of 90 km/h, inflation pressure of 2.2 bar. Influence of inflation pressure: tyre speed of 50 km/h, inflation pressure of 2.7 bar.
鉴定程序的结果参考案例:轮胎速度50公里/小时,充气压力2.2巴。滚动速度的影响轮胎速度90公里/小时,充气压力2.2巴。充气压力的影响轮胎速度50公里/小时,充气压力2.7巴。

Reference case:
参考案例:

50 km/h, 2.2 bar
50公里/小时,2.2

Regressive waves (clockwise)
Regressivewavesclockwise)

Progressive
进展

waves
WAV地震魔

(counterclockwise)
(逆时针方向)

k

5.06
5.06

3.99
3.99

3.38
3.38

2.42
2.42

1.25
1.25

1.19 2.03 2.72 3.32 4.07
1.19 2.032.723.324.07

f[Hz] 149.20 125.81 115.01 95.59 80.23 82.97 110.29 132.08 154.96 185.57
[Hz]149.20125.81115.0195.5980.2382.97110.29132.08154.96185.57

ξ[%] 2.41 2.89 2.38 4.12 12.10 6.35 3.82 3.29 3.59 4.33

A[mm] 0.22 0.27 0.33 0.40 0.53 0.34 0.46 0.32 0.37 0.26
一个mm]0.220.270.330.400.530.340.460.320.370.26

ϕ[rad]
φrad]

1.49
1.49

2.18
2.18

1.36
1.36

2.62
2.62

0.55
0.55

1.25
1,25%

1.61
1,61%

1.36
1.36

1.51
1.51

2.00
2.00

Influence of rolling speed:
轧制速度的影响

90 km/h, 2.2 bar
90公里/小时,2.2

Regressive waves (clockwise)
Regressivewavesclockwise)

Progressive
进展

waves
WAV地震魔

(counterclockwise)
(逆时针方向)

k

6.17
6.17

5.54
5.54

4.36
4.36

3.67
3.67

3.02
3.02

2.23
2.23

1.33
1.33

1.14 2.12 3.06 4.10 5.05 5.91 6.76
1.14 2.123.064.105.055.916.76

f[Hz] 148.17 136.88 118.63 107.90 96.61 87.60 76.36 82.44 122.47 164.97 210.02 252.00 292.80 329.36
[Hz]148.17136.88118.63107.9096.6187.6076.3682.44122.47164.97210.02252.00292.80329.36

ξ[%] 3.40 4.18 2.27 2.74 4.08 4.45 6.24 10.61 7.98 6.41 4.62 3.51 3.13 2.77

A[mm] 0.29 0.36 0.21 0.42 0.25 0.27 0.43 0.55 0.74 0.70 0.51 0.35 0.24 0.12
一个mm]0.290.360.210.420.250.270.430.550.740.700.510.350.240.12

ϕ[rad] 1.05
φ[rad]1.05

0.32
0.32

1.06
1.06

2.80
2.80

2.05
2,05

1.42
1.42

2.13
2.13

0.71
0.71

0.90
0.90

1.15
1.15

1.05
1.05

0.97
0.97

1.04
1.04

1.14
1.14

Influence of inflation pressure:
充气压力的影响

50 km/h, 2.7 bar
50公里/小时,2.7

Regressive waves (clockwise)
Regressivewavesclockwise)

Progressive
进展

waves
WAV地震魔

(counterclockwise)
(逆时针方向)

k

5.02
5.02

3.92
3.92

3.46
3.46

2.42
2.42

1.08
1.08

1.16 2.17 2.93 3.54 4.14
1.16 2.172.933.544.14

f[Hz] 157.54 134.28 123.11 101.74 81.93 87.38 117.39 144.64 165.06 193.74
[Hz]157.54134.28123.11101.7481.9387.38117.39144.64165.06193.74

ξ[%] 2.92 2.14 2.35 4.16 15.52 5.82 4.30 3.85 2.08 4.80

A[mm] 0.22 0.20 0.29 0.42 0.54 0.30 0.49 0.34 0.17 0.19
一个mm]0.220.200.290.420.540.300.490.340.170.19

ϕ[rad]
φrad]

0.39
0.39

1.39
1.39

1.92 0.84 0.07
1.92 0.840.07

1.68
1.68

0.89
0.89

1.13
1.13

0.55
0.55

1.31
1.31

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Fig. 7. Comparison of the experimental and identified time histories of the free response of a) node 10; b) node 30. Tyre speed of 50 km/h, inflation pressure of 2.2 bar.
无花果。7.a节点10 自由响应实验鉴定时间历程的比较;湾)节点30.轮胎速度50公里/小时,充气压力2.2巴。

Fig. 8. Mean square error between the experimental and identified signals as a function of the measuring position. Tyre speed of 50 km/h, inflation pressure of 2.2 bar.
无花果。8.实验信号识别信号之间的误差测量位置的函数关系。轮胎速度50公里/小时,充气压力2.2巴。

Finally, in Fig. 9 the tyre deformed configurations about the mean radius at different time instants of the free response are pre- sented: the red solid curves stand for the experimental signal, while the identified signals are reported as black dashed lines. For simplicity, the tyre rolling direction is also reported as a blue arrow. The red dots identify nodes 10 and 30, whose time histories were presented and discussed in Fig. 7. Good correspondence can be observed between the two signals at most of the instants and angular positions, meaning that the algorithm provides reliable results to identify the free response of the rolling tyre. Minor amplitude mismatches are visible at the inlet and outlet regions at the initial time instants, confirming the considerations drawn from Fig. 8
最后,在图 9 中,预先显示了轮胎在自由响应的不同时刻关于平均半径的变形配置:红色实线代表实验信号,而识别出的信号报告为黑色虚线。为简单起见,轮胎滚动方向也报告为蓝色箭头。红点标识节点 10 和 30,其时程如图 7 所示和讨论。7在大多数时刻和角度位置都可以观察到两个信号之间的良好对应关系,这意味着该算法提供了可靠的结果来识别滚动轮胎的自由响应。在初始时间时刻,在入口和出口区域可以看到轻微的幅度失配,证实了图 8 中得出的注意事项
.

The analysis can be extended to other dataset in order to verify the effectiveness of the algorithm. To this end, the influence of speed
分析可以扩展到其他数据集,以验证算法的有效性为此,速度的影响

(experimental data gathered at 90 km/h at the same inflation pressure of 2.2 bar) and of pressure (considering an inflation pressure of
2.2bar 的相同充气压力90km/h的速度收集的实验数据压力(考虑到

2.7 bar at the nominal speed of 50 km/h) will be analysed in the following sections.
2.7bar标称速度50km/h)将在以下部分中进行分析

To summarize the results of the identification procedure, Table 1 lists all the parameters for the considered test cases, distinguishing between the reference case (50 km/h, 2.2 bar), the dataset adopted to investigate the influence of the rolling speed (90 km/h, 2.2 bar) and of the inflation pressure (50 km/h, 2.7 bar) on the tyre dynamics.
为了总结识别程序的结果1列出了所考虑测试用例的所有参数区分了参考案例(50km/h,2.2 bar),采用数据集研究了滚动速度 (90km/h,2.2 bar) 充气压力 (50km/h,2.7 bar) 轮胎动力学的影响

4.1. Influence of rolling speed
4.1.轧制速度的影响

Considering the data collected at a higher speed of 90 km/h, the propagation of the waves on the tyre surface will be different with
考虑到90km/h较高速度下收集的数据轮胎表面的传播将与

respect to the previous case. Given the higher speed, the obstacle impact will excite a higher number of waves, so that 14 contributions
尊重前一种情况。给定较高的速度,障碍物撞击激发更多的波数,因此14贡献

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Fig. 9. Comparison of the experimental and identified tyre deformation at different time instants of the free response. Tyre speed of 50 km/h (42 rad/s), inflation pressure of 2.2 bar.
无花果。9.比较自由响应的不同时刻实验确定轮胎变形轮胎速度50公里/小时(42rad/s),充气压力2.2bar。

(thus, n = 14 in Eq. (1) evenly split between progressive and regressive waves have been considered. The outcomes of the identifi- cation procedure are reported in Table 1. Regarding the identification error, the results show a trend similar to the one presented in Fig. 8
(因此,考虑了方程 (1) 中的 n = 14)在渐进波和回归波之间平均分配。识别程序的结果如表 1 所示关于识别误差,结果显示出类似于图 8 所示的趋势
.

The comparison between the measured and identified tyre free responses is shown in Fig. 10. Specifically, in Fig. 10a the results are presented in terms of time histories in correspondence of the measuring position denoted as nodes 10. Three time instants are identified by vertical red lines, for which the comparison of the measured and identified tyre deformation is presented in Fig. 10b. The proposed modelling approach proves to be effective, being the red solid and black dashed curves in good agreement at any time instant. If reference is made to Fig. 10b, also the tyre deformation confirms the model capability to reproduce the dynamic response of the whole tyre to the cleat excitation.
测量识别轮胎响应之间的比较如图 2 所示10具体来说,图 1 中。10a结果时间历史表示,与表示为节点10测量位置相对应三个时刻垂直线标识,其测量识别轮胎变形的比较如图 1 所示10b.所提出的建模方法被证明有效的,红色线黑色虚线在任何时间都很好地一致如果参考图 1。10b,轮胎变形也证实模型再现整个轮胎锁片激励动态响应的能力

The increase of rolling speed causes several variations of the identified parameters with respect to the reference case. Considering Fig. 10a, higher vibration levels (up to 2 mm) are reached compared to those measured at a rolling speed of 50 km/h (see Fig. 7). Higher vibrations are visible also throughout the other measurement positions. From an identification point of view, the amplitudes of the progressive waves show higher values, whereas no clear trend can be extrapolated from regressive waves. The overall higher amplitude of vibration is justified also by the higher number of identified contributions, meaning that most of the signal energy is spread over a higher number of propagating waves (n = 14).
滚动速度的增加会导致已识别参数相对于参考情况多种变化考虑到10a,50km/h 的滚动速度测得振动水平(高达2mm)相比(见图 D)。7).在其他测量位置也可以看到更高的振动识别的角度来看进波振幅显示出更高的值,回归波中无法推断明显的趋势总体上较高的振动振幅较多的已识别贡献数量相这意味着大部分信号能量都分布在传播的数量更多 n=14)。

Moreover, the effect of rolling speed is related the well-known bifurcation effect [9,13,1618]. The increase of the tyre rolling speed modifies the waves propagation, causing progressive waves to be faster and regressive waves to be slower. For the test case under analysis, this can be confirmed by computing the ratio between the frequency fi of each wave and the circumferential mode numbers ki. The results are shown in Fig. 12 as dispersion diagram (ki fi plot).
此外,滚动速度的影响众所周知的分岔效应有关[9,13,1618]轮胎滚动速度的增加改变了的传播,导致进行更快回归慢。对于分析测试用例可以通过计算每个的频率圆周k之间的比率确认 结果如图所示12作为色散k 图)。

Focusing on the identified ki and fi values, they are both affected by the rolling speed. The identification results show that at higher speed, in general, both quantities increase for progressive waves and decrease for regressive waves. It can be concluded that the waves frequency fi changes more than ki when the rolling speed changes.
关注识别出的k值,它们滚动速度的影响识别结果表明较高速度,通常增加回归波的量减小可以得出结论滚动速度发生变化频率的变化大于k

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Fig. 10. Comparison of the experimental and identified results, considering the tyre speed of 90 km/h (76 rad/s) and inflation pressure of 2.2 bar. a) Comparison of the time histories of the free response for node 10. b) Comparison of the tyre deformation at different time instants of the free response.
无花果。10.实验结果和确定结果的比较考虑到轮胎速度90公里/小时(76拉德/秒)2.2的充气压力一)节点10 freeresponse的时比较。湾)自由响应在不同时刻轮胎变形比较

Fig. 11. Comparison of the experimental and identified results, considering the tyre speed of 50 km/h (42 rad/s) and inflation pressure of 2.7 bar. a) Comparison of the time histories of the free response for node 10. b) Comparison of the tyre deformation at different time instants of the free response.
无花果。11.实验结果和确定结果的比较考虑到50公里/小时(42rad/s)轮胎速度和2.7的充气压力一)节点10 freeresponse的时比较。湾)自由响应在不同时刻轮胎变形比较

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I.La Paglia人。

4.2. Influence of inflation pressure
4.2.充气压力的影响

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As a second analysis, data gathered at 50 km/h at a higher inflation pressure of 2.7 bar are considered. The identification procedure relies on the same steps adopted at 2.2 bar, thus including the contribution of n = 10 travelling waves to the tyre free response. The time domain comparison between the signals measured by the laser and those identified by the model is proposed in Fig. 11. In Fig. 11a the comparison is carried out considering the time histories of node 10, whilst in Fig. 11b the animation of the tyre deformation is shown. A significant degree of agreement can be recognised comparing the measured and the identified tyre vibration, both in terms of amplitudes and frequency components.
作为第二项分析,考虑了50公里/小时2.7的较高充气压力收集的数据识别程序依赖于2.2bar 时采用的相同步骤因此包括n=10个行轮胎响应的贡献激光器测量信号模型识别的信号之间的比较如图 2 所示11图 11 中。11a 比较考虑了节点10 图 1 中。11b 显示了轮胎变形动画比较测量识别轮胎振动,在振幅频率分量方面,可以识别很大程度一致性

The influence of the inflation pressure on the dynamics is driven by changes of the tension of the tyre reinforcements, which in- creases at higher pressures [2]. Therefore, a stiffening effect is observed, which causes waves to propagate at higher speed. This is mainly visible as an increment of the wave frequencies, for both progressive and regressive waves. For the test case under analysis, an average variation of +6 % is observed. Slight modifications of the circumferential mode numbers can be observed too, but no clear trend can be extrapolated from the identified data. It is worth mentioning that the modifications of the circumferential mode numbers might be related to changes in the footprint dimensions due to the different inflation pressure.
充气压力动力学的影响是由轮胎增强件的张力变化驱动的较高压力会增加[2]因此,a观察到硬化效应导致更高的速度传播主要表现为频率的增加无论是进波还是回归对于分析测试用例,观察到 +6% 的平均变化也可以观察到圆周态数轻微变化无法识别的数据中推断明显的趋势值得一提的是,圆周态编号修改可能与封装尺寸的变化有关因为通货膨胀压力。

4.3. Comparison with in-plane tyre flexible ring model
4.3.平面内轮胎柔性模型的比较

In this section, the relationship between the k and f values identified by the proposed methodology is compared with the results of a simplified in-plane flexible ring tyre model. The model is presented in [17], and its parameters have been tuned based on an Experimental Modal Analysis carried out on the same tyre considered in this manuscript. The considered analytical model, although simplified and reproducing an unloaded rotating tyre, was chosen since it directly allows evaluating the dispersion curve of the rotating tyre [16,20]. This can be regarded as a major advantage compared to other modelling strategies. For instance, Finite Element Models would allow computing the natural frequencies and mode shapes, but it would not be possible to directly derive the corre- sponding circumferential mode number.
本节所提出的方法确定k之间的关系简化面内柔性轮胎模型的结果进行了比较模型 [17] 介绍,其参数根据手稿中考虑相同轮胎进行的实验模态分析进行了调整所考虑解析模型虽然简化再现了空载旋转轮胎,但之所以选择,是因为它可以直接评估旋转轮胎的色散曲线[16,20]与其他建模策略相比可以被视为一个主要优势例如有限模型允许计算固有频率振型不可能直接推导出相应的倾斜的圆周模式数。

The comparison between the identified and analytical results has been performed considering different inflation pressures and tyre rotating speeds, to match the experimental data. The results are reported in Fig. 12 in terms of dispersion diagrams, showing the circumferential mode number k along the x-axis and the frequency f of the considered wave contribution along the y-axis. It is remined that the tyre is rolling in counterclockwise direction, and that waves characterized by positive k values rotate together with the tyre, while negative k values are associated to clockwise travelling direction. It is worth noting that while real circumferential mode number k are considered in this paper (reported as dot markers), integer values are adopted by the analytical model (reported as dashed lines with cross markers). This difference is consistent with the tyre conditions adopted in the two cases: when the tyre is rolling during the experimental cleat test, the contact patch introduces a discontinuity in the wave field propagating along the tyre belt. Conversely, the analytical results of the simplified model consider a rotating tyre (unloaded tyre, no contact patch).
考虑到不同的充气压力轮胎转速,对鉴定结果和分析结果进行了比较匹配实验数据。结果报告图 1 中。12色散为例显示沿x 轴圆周态数k沿y 轴的所考虑贡献的频率结果表明轮胎沿逆时针方向滚动k为特征波浪轮胎一起旋转k顺时针行驶方向相关联。值得注意的是虽然本文考虑了实际的圆周态数k(报告标记),整数分析模型(报告带有交叉标记的虚线)。这种差异两种情况下采用的轮胎条件一致:当轮胎实验夹板测试期间滚动接触Patch沿轮胎传播中引入了不连续性。相反,简化模型的解析结果考虑了旋转轮胎(空载轮胎,接触面)。

Despite the differences in terms of tyre conditions (loaded and unloaded), the comparison shown in Fig. 12 allows observing that the (ki, fi) couples experimentally identified (reported as dots) are respecting the trend of the analytical rotating flexible ring (dashed lines with cross markers), both in case of rolling speed (Fig. 12a) and inflation pressure (Fig. 12b) increase.
尽管轮胎状况(装载卸载)存在差异,但比较如图 2 所示12允许观察实验识别k(报告点)是否遵循解析旋转柔性带有交叉标记的虚线趋势滚动速度的情况下12a)充气压力12b)增加。

Focussing on the effect of tyre speed, Fig. 12a shows that an increase in the rolling speed leads to an increase in the frequency of the waves characterized by positive circumferential mode numbers. Conversely, waves with negative circumferential mode number are subjected to a decrease in the frequency. This behaviour can be related to the well-known bifurcation effect observed in a fixed reference frame [17]. If reference in now made to the effect of the inflation pressure, Fig. 12b shows an average increase of the fre- quencies of +6 %, which is particularly visible at higher circumferential mode numbers. This behaviour is characteristic of both progressive and regressive waves, that is associated to the tyre structure stiffening effect. Some differences can be recognised, for instance in Fig. 12a on the branch representing the positive k values, that can be associated to the uncertainties associated to the experimental data and to the simplifying assumptions of the analytical model.
关注轮胎速度的影响12a表明滚动速度的增加导致的频率增加其特征圆周模式数。相反,圆周为负频率会降低这种行为可能与固定参考中观察到众所周知的分岔效应有关[17]如果现在参考膨胀的影响pressure,12b表示频率的平均增加 +6%,这在较高的圆周态数尤其明显这种行为渐进退行波特征这与轮胎结构的硬化效应有关可以识别出一些差异例如图 1 中。12a在分支表示k值,该值可能与实验数据相关的不确定性简化分析模型的假设

The comparison between the experimental data for the loaded tyre and the analytical model predictions for the unloaded tyre
装载轮胎实验数据卸载轮胎分析模型预测之间的比较

shows that the operating conditions have a minor effect on the relationship between k and
表示操作条件k
f .

5. Conclusions
5.结论

This paper aims at investigating the in-plane dynamic behaviour of a rolling tyre in the frequency range below 300 Hz. To this end, at first the rolling tyre response to the cleat impact has been experimentally characterized. The measuring setup specifically designed to this purpose has been described. It relies on a completely automated system, constituted by a fixed and a moving structure which carries a laser triangulation sensor. The system allows scanning the tyre radial vibration after the cleat impact, along a predetermined measurement grid of 5. Experimental data have been gathered at different rolling speeds and inflation pressures, so as to investigate the influence of both parameters on the tyre dynamics.
本文旨在研究滚动轮胎低于 300Hz 的频率范围内的面内动态行为,为此首先滚动轮胎锁片冲击已经进行了实验表征。已经描述了专门为此目的设计的测量设置依赖于一个完全自动化的系统,该系统由一个固定移动的结构构成该系统带有一个激光三角测量传感器。该系统允许扫描轮胎在锁碰撞后的径向振动沿着预定测量网格5在不同滚动收集实验数据速度和充气压力,从而研究这两个参数轮胎动力学的影响

Secondly, a data processing algorithm to identify the travelling waves that contribute to the response of the tyre has been presented. The model relies on a wave propagation solution, and it purposely allows considering real circumferential mode numbers on account of the contact patch, that introduces a discontinuity in the tyre vibration. A significant agreement between the measured and identified signals has been achieved, both in terms of time histories (at a fixed position along the tyre circumference) and of dynamic defor- mation. The performance of the identification algorithm has been tested considering experimental data collected at different rolling speeds and inflation pressures.
其次,提出了一种数据处理算法来识别有助于轮胎响应模型依赖于传播解,并且由于接触贴片,它特意允许考虑实际圆周态数引入了不连续性轮胎振动中。测量信号和识别信号之间时间历史(沿轮胎圆周固定位置动态defor-mation考虑到在不同轧制速度和充气压力下收集实验数据已经识别算法的性能进行了测试

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Fig. 12. Dispersion diagrams comparing the identified results with those achieved with an in-plane tyre flexible ring model, considering the fixed reference frame. a) Influence of rolling speed; b) influence of inflation pressure.
无花果。12.考虑到固定参考,将确定的结果使用平面内轮胎柔性模型获得的结果进行比较色散一)轧制速度的影响;湾)通货膨胀压力的影响

Finally, the identified data have been compared with the analytical results obtained with a simplified in-plane tyre model, showing a satisfactory degree of agreement. Both the bifurcation effect related to the speed increase and the stiffening effect related to the pressure increase are well observed and replicated. In addition, the comparison suggests a minor effect of the tyre loading conditions on the dispersion diagrams.
最后,鉴定数据使用简化的面内轮胎模型获得的分析结果进行比较显示出令人满意一致性。速度增加相关的效应压力增加相关的硬化效应都得到了很好的观察复制。 此外,比较表明轮胎负载条件色散的影响很小

In the end, the designed measurement system and wave field identification algorithm may represent useful tools to investigate the
最后所设计的测量系统和波场识别算法可能研究

dynamic behaviour of the rolling tyre, to possibly support the product development and to mitigate structure-borne noise contribution. CRediT authorship contribution statement
滚动轮胎动态行为可能支持产品开发减轻结构噪声的贡献。CRediT作者贡献声明

Ivano La Paglia: Conceptualization, Methodology, Writing original draft, Validation. Luca Rapino: Conceptualization, Meth- odology, Validation, Writing original draft. Francesco Ripamonti: Formal analysis, Methodology, Supervision, Validation, Writing review & editing. Simone Baro: Resources, Validation, Visualization. Roberto Corradi: Funding acquisition, Project administration, Supervision, Writing review & editing.
IvanoLaPaglia:概念化,方法论,写作-稿,验证。卢卡·拉皮诺:概念化,冰毒颂扬,验证,写作——稿。FrancescoRipamonti:形式分析,方法论,监督,验证,写作-审查编辑。SimoneBaro:资源,验证,可视化。RobertoCorradi:资金获取,项目管理监督,写作-审查编辑。

Declaration of competing interest
利益争夺声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
作者声明他们没有已知的相互竞争的经济利益个人关系,这些关系可能看起来

influence the work reported in this paper.
影响本文报道的工作

Data availability
数据可用性

The data that has been used is confidential.
已使用的数据机密的。

Acknowledgments

This study is part of the Structure Borne Tyre Noise (SBN), carried out in the framework of the Joint Labs cooperation agreement between Politecnico di Milano and Pirelli. The authors gratefully acknowledge Pirelli for providing the support and data necessary to this work.
这项研究结构轮胎噪声 (SBN) 的一部分,在米兰理工大学倍耐力之间的联合实验室合作协议框架进行作者非常感谢倍耐力这项工作提供必要的支持和数据

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
这项研究没有公共、商业非营利部门资助机构获得任何具体资助

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