机械系统与信号处理 211 （2024） 111181

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机械系统和 信号处理

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测量和识别 滚动轮胎的 平面内动态行为

伊万诺·拉帕利亚 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·拉帕利亚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 传达者 John E. Mottershead Key的话：
ARTCLENFO传达者John E.Mottershead Key的话：

滚动轮胎动力学 实验夹板测试 波传播解决方案 识别算法 结构噪声

A B S T R A C T

作用在轮胎上可以提高道路车辆的 NVH 性能 ，轮胎 起着至关重要的作用 将结构噪声 和振动传输到 机舱中，最高可达 500 Hz。 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 在激光 三角测量传感器上扫描滚动轮胎的 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，以研究它们对轮胎动力学的影响。 然后，数据处理 算法为 提出的目的是 识别 传播的渐进波和退行波 锁片撞击后的 轮胎周长。 最后，为了解释 结果，对确定的轮胎响应进行了比较 通过 简化的平面内轮胎模型获得的分析结果。
作用在轮胎上可以提高道路车辆的 NVH性能，轮胎起着至关重要的作用将结构噪声和振动传输到机舱中，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，基于在激光三角测量传感器上扫描滚动轮胎的胎面。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，以研究它们对轮胎动力学的影响。然后，数据处理算法为提出的目的是识别传播的渐进波和退行波锁片撞击后的轮胎周长。最后，为了解释结果，对确定的轮胎响应进行了比较通过简化的平面内轮胎模型获得的分析结果。

1. 引言

汽车行业 对车辆舒适性的关注 度显著 增加。 因此， 人们开始关注 车辆 NVH（噪声、 振动和声振粗糙度）性能 [1,2]。 此外， 近年来 ，电动汽车 越来越 受欢迎，因此 轮胎/路面相互作用是当今 影响 整体噪声的最重要因素，并且 在车辆中感知到的振动水平 [3]。 此外， 乘客对 机舱 的感知也在发生变化，因为 引入 自动驾驶 功能。 在这种情况下 ，必须 提高道路车辆的 NVH 性能，尤其是 在 <b 上<b1197> 1198>轮胎， 在结构性座舱噪声和振动中起 着重要作用 到 500 Hz [4,5]。
汽车行业对车辆舒适性的关注度显著增加。因此，人们开始关注车辆NVH（噪声、振动和声振粗糙度）性能[1,2]。此外，近年来，电动汽车越来越受欢迎，因此轮胎/路面相互作用是当今影响整体噪声的最重要因素，并且在车辆中感知到的振动水平[3]。此外，乘客对机舱的感知也在发生变化，因为引入自动驾驶功能。在这种情况下，必须提高道路车辆的NVH 性能，尤其是在 上 1198>轮胎，在结构性座舱噪声和振动中起着重要作用到500Hz[4,5]。

可以 采用 不同的方法来研究 轮胎的 动态行为。 一方面，从 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，它们仅 提供了对这种现象的一般理解 [6]。 有限元模型需要更苛刻的计算工作，但实现了这种可能性 对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 对模型标定和 NVH 有用的参数 分析。 为此，文献中提出了 不同的方法 。 固有频率、振型 和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对模型标定和 NVH 有用的参数 分析。为此，文献中提出了不同的方法。固有频率、振型和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

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I. La Paglia et al. Mechanical Systems and Signal Processing 211 (2024) 111181
I.La Paglia等人机械系统与信号处理211（2024）111181

(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 [9,11–14]
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，在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，其中轮胎在obstacle提供类似冲动的激励。该测试适用于了解轮胎的低频动态行为，因为锁片撞击激发的最大频率约为300Hz[2]对于例如，在直径为 2m 的钢桶表面进行夹板测试[15]，使其垂直倾斜到轮胎滚动平面。测量轮胎轮毂力并用于调整平面内环模型。类似的测试装置用于 [9]，其中采用激光多普勒测振仪来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。无驱动轮安装在测功轮毂上，可以测量测试过程中产生的力。激发由安装在从动轮上的铝制夹板提供，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 [9,13,16–18]. 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.
许多作者在滚动轮胎测试中观察到的一个现象是分岔效应[16]在静态条件下，输入力会产生渐进和回归波它们以相同的速度沿轮胎结构传播。因此，它们的叠加导致驻波振动模式。在滚动条件下，由于科里奥利加速度，行进波和回归波以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，则存在多普勒频移。科里奥利效应阻止了驻波的实现，产生的模式将很复杂;因此，它们将呈现传播速度[9,13,16–18]分岔效应取决于轮胎滚动速度、所考虑的模态数和相应的振型。基于这些考虑，经典的振型定义不能应用于旋转系统，识别技术应基于波传播模型。

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在低于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。在钢桶上进行夹板测试，并通过扫描轧制胎面的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下收集数据，以研究这两个参数对轮胎动态的影响。其次，从轮胎沿轮胎圆周多个位置的轮胎表面振动出发，提出了一种识别波传播场的策略。相对于以前的出版物，本文使用波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] 或操作模态分析技术

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，每5 个节点从 –33到 +212 间隔（它们的位置在固定参考系统中定义）。出于可视化目的，仅突出显示节点1、10、30和50。

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I.La Paglia等人机械系统与信号处理211（2024）111181

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;在识别过程中不涉及任何分析模型，例如在[11,12]中，由于接触面引起的轮胎径向振动的不连续性被考虑在内启用实周模式数的标识。此外，为了解释结果，将测试数据与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的相关性。

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。在第2 节中，描述了在防滑钉测试期间测量轮胎响应的实验设置。在第3 节中，描述了同步数据并从记录的时程中提取轮胎振动的数据处理技术;然后，给出了波传播算法，并详细介绍了拟合过程。在第4 节中，比较了不同的数据集，以研究速度和充气压力的影响;此外，还将结果与轮胎分析模型的结果进行比较。最终，第5 节得出了这项工作的结论。

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.
为了测量轮胎在滚动条件下的振动，已经设计了一个专门的实验装置，并在现有设施中安装了防滑钉测试装置执行。测试台由电动机驱动的钢桶组成。采用激光三角测量传感器 （MELMicroelectronicsM7L50） 扫描轮胎表面。该传感器在95mm 的焦距周围具有 50mm 的测量范围，通过支撑结构与轮胎保持固定距离，它测量轮胎在锁片碰撞后的响应。振动测量已沿轮胎圆周的 50个位置（节点）进行。采用了 5的测量网格，总覆盖角为245由于接触贴片和相邻区域，可以激光束无法到达，鉴于滚筒的存在，无法进行超过115的测量为了清晰起见，无花果。1为测量网格：出于可视化目的，仅报告了少量节点。接触面的中心位于角度位置270，而第一测量位置与角度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。在这项工作中，使用了横向夹板（垂直于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成坚硬和稳定，以便测量轮胎振动，尽可能减少任何干扰。为此，该结构被安装在通过悬空基础与测试台地下室动态解耦的支架上。无花果。图 2显示了所设计系统的 3DCAD模型，它由两组元件组成：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;一种与所述固定框架相连的移动结构，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，它位于水平梁的中心，与轮胎的中心对齐。这

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
结构的旋转（图 1 中的红色虚线）。2b）由电动机提供。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 时为 30秒。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，在开始下一次采集之前，再考虑30秒以达到静态条件。在测试结束时，将收集 50 个时间历史记录。

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）用于测量滚动轮胎的实验室设置的 3DCAD模型。b） 中的红色虚线表示测量系统在运行过程中所遵循的轨迹。c）侧视图。（有关此图例中对颜色的引用的解释，读者请参阅本文的网络版本。

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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显示了测试台的图片。在图 .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的夹板（高度为 5mm，沿滚动方向长度为 10mm）的细节。

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

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.
在本节中，介绍了为研究滚动轮胎的动态行为而开发的方法。在第3.1 节中，描述了沿轮胎周长收集的时间历史的处理;在第3.2 节中，介绍了波场标识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
第2 节中描述的实验设置允许测量由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
在50和90公里/小时的速度以及2.2和2.7巴的充气压力下收集，以研究这些影响

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。已记录轮胎和鼓旋转的触发信号，以实现数据同步和

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。在下文中，单个时间历史（节点 50,212 个可用节点中的 50,212 个，在 50 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，作为参考案例来呈现实验结果和数据处理。

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在两次后续冲击中采集的位移信号。可以注意到，该信号由两个主要部分组成：第一个，周期为0.15s，与轮胎有关不均匀性;第二个也是最重要的一个与轮胎由于锁片冲击而产生的动态响应有关，它显示0.57s 的周期（由垂直虚线）。

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无响应。因此，需要两个信号处理步骤：一方面，去除轮胎不均匀性贡献;另一方面，强制响应（与Cleat-Tyre交互相关的初始时间时刻）被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（与轮胎换位周期性）用于隔离与每次轮胎通过相关的时间窗口。后来，利用时间平均法生成单次轮胎旋转的非均匀性信号。然后，复制该信号以具有与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，获得仅与防滑钉激励相关的轮胎响应。4a和b.所获得的结果可以在图 4 中观察到。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.
消除轮胎不均匀性后，评估平均防滑钉响应以限制背景噪声的影响。为此，使用鼓触发器来识别与防滑钉撞击相关的时间窗口（图 1 中的垂直虚线）。4c），并将时域平均应用于总共52 个 cleat响应。此外，与鼓旋转的一阶相关的信号被删除，因为该信号分量与夹板响应无关。此过程的结果如图 1 所示。5显示为黑线。

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 次测量值的标准差被量化为约50μm。与背景噪声相关的该值与图 1 中所示的振动水平相比要小得多。5此外，平均运算进一步降低了这种测量不确定性。