# Active Vibration Control on a Tire-Wheel Assembly Using Piezoelectric Spatial Modal Filter

Antoine Carvalho, Kevin Billon, Jonathan Rodriguez, Simon Chesne, Francois Lohr

► To cite this version:

Antoine Carvalho, Kevin Billon, Jonathan Rodriguez, Simon Chesne, Francois Lohr. Active Vibration Control on a Tire-Wheel Assembly Using Piezoelectric Spatial Modal Filter. Journal of Vibration and Acoustics, 2024, 146 (2), pp.021002. 10.1115/1.4065574. hal-04732663

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Submitted on 14 Jan 2026

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# Active Vibration Control on a Tire-Wheel Assembly Using Piezoelectric Spatial Modal Filter

Antoine Carvalho

Université de Lyon, CNRS INSA Lyon, LaMCoS UMR5259, F-69621 Villeurbanne, France

Kevin Billon

Université de Lyon, École Centrale de Lyon, LTDS UMR5513, F-69134 Ecully, France

Jonathan Rodriguez

Université de Lyon, CNRS INSA Lyon, LaMCoS UMR5259, F-69621 Villeurbanne, France

Simon Chesne

Université de Lyon, CNRS INSA Lyon, LaMCoS UMR5259, F-69621 Villeurbanne, France

François Lohr

Manufacture Française des Pneumatiques, Michelin, Cébazat F-63118, France

Vibrations due to tire-road contact in wheeled vehicles induce acoustic discomfort especially beyond 35 km/h. This paper proposes an active control method to reduce the vibration transmission from the tire-road contact to the vehicle through piezoelectric transducers located directly on the wheel spokes. Our approach relies on a double spatial modal filter to physically focus the control energy on the wheel pumping mode while avoiding any spillover phenomena. In addition, a bandpass controller ensures maximum damping on the targeted mode. The proposed control strategy is applied first to the clamped wheel in order to validate the static performance of the spatial controller. Then the wheel is excited through the tire and the efficiency of the controller is evaluated through the measured force passing by the wheel hub. Finally, the tire-wheel assembly (TWA) is placed on an experimental setup recreating the vehicle operating conditions with wheel rotation at different velocities and road excitations. The experimental results confirm the efficiency of the proposed control method and its robustness to the dynamics evolution of the structure in function of the TWA angular velocity.

Keywords: active vibration and noise control, sensors and actuators, smart materials and structures, structural dynamics and control, vibration control

## 1 Introduction

One of the major problems facing transportation vehicles is the noise pollution they produce. This is the case for individual means of locomotion such as cars. These noises are caused by the engine, the exhaust, the airflow, and the contact between the road and the rubber of the tire, more commonly called "road noise." Indeed, vibrations are transmitted from the tires to the passenger compartment of the vehicles. Then, by transmission from the solid to the air domain, they produce sound discomfort which can affect the health and comfort of motorists [1].

In order to offer more optimal driving conditions, the road tire vibrations can be reduced by integrating foam torus inside vehicle tires [2,3] or changing the gas inside the cavity [4,5]. But these solutions only bring improvements on tire cavity resonance. To damp structural modes, it is possible to directly work on the car suspension, by utilizing passive [6] or active solutions [7 10]. It's also interesting to move toward the concept of "smart structures," which offer a wider spectrum of applications and solutions. They

represent an approach aiming at integrating three components into a mechanical assembly: (i) sensors, which allow them to perceive the environment in which they are; (ii) actuators in order to modify their behavior in real time; and (iii) a control system to define the reaction force to apply based on sensors' information. Thanks to it, smart structures can adapt themselves to some of the external constraints and disturbances caused by their environments.

These systems can be greatly optimized in terms of reducing the energy to performance ratio or improving their robustness. This can be achieved by using a modal shaped control system combined with a bandpass filter designed around the modes to be controlled [11] or by designing a modal spatial filter [12,13]. The first category of filters allows to processing sensor data to selectively extract the dynamic components associated with the targeted states to control. This selective processing aids in the precise definition of a reactive force intended to act on the desired dynamic behavior. And the second one is directly based on an optimization of sensor placement or sensor shape, to predefine the states of the system that our sensors can perceive [14,15].

This last kind of filters was first introduced as a substitute for state observers in order to mitigate spillover effects in modal control [16 19]. They can be made by designing a spatially continuous distributed sensor. It can be done by designing a piezoelectric transducer in polyvinylidene defined as a modal sensor. This is particularly the case for one dimensional structures [20 22]. However, the limit
of this type of solution lies in its manufacturing process. Final shapes are often complex and difficult to realize, and for industrial applications, common structures have more than one dimension. They can also be made by using a network of multiple parallel sensors. To achieve this, a system of electromechanical transducers made from lead zirconate titanate (PZT) can be implemented to control the mechanical vibration. These components are widely deployed in the field of vibration and noise control, whether it is with passive applications, even on massive structures [23,24], or also for active applications by bringing an external electrical power source to the controlled system [25,26].

The present study is based on assemblies composed of a rim and a Michelin tire. The complex configuration of the rim tire assembly, particularly due to the tire design, led to the focus of the study on the most energetic wheel vibration mode, the pumping mode. This mode is characterized by a homogeneous displacement on the periphery of the wheel and a zero displacement in the embedding. It is similar to a membrane mode where the fixed reference frame is located at the hub.

In this paper, we propose to actively control the dynamics of a tire wheel assembly (TWA) using a spatial modal filter with a discrete array of PZT transducers. With regard to the contribution provided by this work; first, while vehicle suspensions and their tires isolate within the frequency range of [0, 250] Hz, there is no solution for the vibration modes of mounted assemblies, which typically fall between [250, 500] Hz. The proposed solution addresses this frequency band. Additionally, spatial modal filters in the literature are often applied to simple academic structures, whereas the controlled structure here is complex and subjected to real operating conditions with loading and rotation. Section 2 details the controlled structure. The control law and the modal spatial filter design are developed in Sec. 3. Finally, the experimental results are presented and discussed in Sec. 4.

## 2 The Tire-Wheel Assembly Dynamics

### 2.1 Context and Experimental Setup.
The vibrations induced by the contact between the wheel and the road are transmitted from the tire to the passenger compartment through the wheel hubs. Thus, increasing the mechanical damping of the TWA thanks to piezoelectric sensors and actuators, is one solution to reduce vibrations and potentially acoustic discomfort.

The considered structure is a TWA composed of a Michelin tire 235 45 R18 and an 18 × 8 J8 profile rim of the car manufacturer Mercedes illustrated in Fig. 1. The structure is excited by an electrodynamic shaker. The acceleration response A of the structure is measured in the y direction by an accelerometer located at the top of the rim. The force response F is also measured in the y direction. The structure is attached to a seismic foundation using five bolts. All the experimental equipment used in this study are summarized in Table 1.

### 2.2 Modal Analysis.
For the identification process, a white noise signal at a sampling frequency of 10 kHz and a maximum amplitude 5 V is applied as a control voltage to the shaker amplifier. The frequency response function (FRF) between the force F and acceleration A is presented in Fig. 2.

<figure><img src="image_2.png" /><figcaption>Fig. 1 Definition of the static test bench of the 1:1 model used for the laser vibrometer study</figcaption></figure>

We observe a significant modal diversity within the targeted control bandwidth: 200 500 Hz. This frequency range corresponds to the main TWA modes which are not necessarily well filtered by the car body. The TWA weight applies a bending moment on the support structure at the center of the hub on the x axis, generating the two first modes 1 and 2. Then, due to the deformation of the tire, an acoustic mode 3 is present, corresponding to the propagation of a sound wave inside the TWA cavity at the medium radius of the tire. Finally, ovalization mode 4, pitch mode 5, and pumping mode 6 are the characteristic modes of the TWA linked to the wheel behavior. In this paper, the main control objective is to dampen the pumping mode 6. Figure 3 and Table 2 summarize all the measured characteristics of the TWA modes.

## 3 Control Strategy

In the following section, the mode shapes of the TWA structure are investigated to support the implementation of a double spatial modal filter with piezoelectric transducers. Then, the transfer

<table><caption>Table 1 Detailed presentation of the experimental materials used</caption><thead><tr><td>Experimental equipment</td><td>Reference</td><td>Manufacturer</td><td>Specification</td></tr></thead><tbody><tr><td>Electrodynamic shaker</td><td>TV 51110</td><td>TIRA</td><td>Rated force: 100 N</td></tr><tr><td>Data acquisition system</td><td>RTI 1202</td><td>Dspace</td><td>Sampling frequency: 10 kHz</td></tr><tr><td>Accelerometer</td><td>4371</td><td>Brüel & Kjær</td><td>Sensitivity: 9869 pC/g</td></tr><tr><td>Force sensor</td><td>8230 001</td><td>Brüel & Kjær</td><td>Sensitivity: 4123 pC/N</td></tr><tr><td>PZT transducer</td><td>P 876.A15</td><td>Physik instrument</td><td>Minimum bending radius: 70 mm</td></tr><tr><td>Glue</td><td>EA 3423</td><td>Loctite</td><td>Shear strength wheel/transducer: 3 N/mm</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Fig. 2 Bode diagram of the real system and modal identification fitting</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3 Experimental modal deformation of (a) cavity 208 Hz, (b) ovalization 253 Hz, (c) pitch 281.9 Hz, and (d) pumping mode 436 Hz of the TWA</figcaption></figure>

<table><caption>Table 2 Experimental eigenfrequencies and loss factor</caption><thead><tr><th>Mode</th><th>Description</th><th>Frequency (Hz)</th><th>Loss factor η (%)</th></tr></thead><tbody><tr><td>1</td><td>Support mode 1</td><td>159.5</td><td>1.24</td></tr><tr><td>2</td><td>Support mode 1</td><td>173.7</td><td>1.75</td></tr><tr><td>3</td><td>Cavity mode</td><td>208.0</td><td>0.58</td></tr><tr><td>4</td><td>Ovalization mode</td><td>253.0</td><td>1.80</td></tr><tr><td>5</td><td>Pitch mode</td><td>281.9</td><td>1.69</td></tr><tr><td>6</td><td>Pumping mode</td><td>436.0</td><td>1.06</td></tr></tbody></table>

function between the actuation ring and the sensing ring is identified. Finally, the controller is presented.

**3.1 Spatial Modal Filter Design.** In order to optimize control lability of the targeted mode, both PZT sensors and actuators need electric transducers can

be utilized both as sensors, known as the direct effect, wherein deformation of the element produces a voltage, and as actuators, known as the indirect effect, wherein applying an electric field to the element causes deformation.

Direct effect equation

$$ \left\{ \begin{matrix} D_1 \\ D_2 \\ D_3 \end{matrix} \right\} = \underbrace{ \begin{matrix} 0 & 0 & 0 & 0 & d_{15} & 0 \\ 0 & 0 & 0 & d_{24} & 0 & 0 \\ d_{31} & d_{32} & d_{33} & 0 & 0 & 0 \end{matrix} }_{\text{coupling}} \begin{matrix} T_{11} \\ T_{22} \\ T_{33} \\ T_{23} \\ T_{31} \\ T_{12} \end{matrix} + \underbrace{ \begin{bmatrix} \varepsilon_{11} & 0 & 0 \\ 0 & \varepsilon_{22} & 0 \\ 0 & 0 & \varepsilon_{33} \end{bmatrix} }_{\text{permittivity}} \begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix} \quad (1) $$

Indirect effect equation

$$ \left\{ \begin{matrix} S_{11} \\ S_{22} \\ S_{33} \\ 2S_{23} \\ 2S_{31} \\ 2S_{12} \end{matrix} \right\} = \underbrace{ \begin{bmatrix} S_{11} & S_{12} & S_{13} & 0 & 0 & 0 \\ S_{12} & S_{22} & S_{23} & 0 & 0 & 0 \\ S_{13} & S_{23} & S_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{66} \end{bmatrix} }_{\text{strain}} \begin{bmatrix} T_{11} \\ T_{22} \\ T_{33} \\ T_{23} \\ T_{31} \\ T_{12} \end{bmatrix} + \underbrace{ \begin{bmatrix} 0 & 0 & d_{31} \\ 0 & 0 & d_{32} \\ 0 & 0 & d_{33} \\ 0 & d_{24} & 0 \\ d_{15} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} }_{\text{coupling}} \begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix} \quad (2) $$

where *D* is the electric displacement (charge per unit area, expressed in C m²), *E* the electric field (V/m), *T* the stress (N m²), and *S* the strain. *ε* is the dielectric constant (permittivity) under constant stress, *S* is the compliance when the electric field is constant (inverse of the Young’s modulus), and *d* is the piezoelectric constant (m/V or C/N).

An experimental study of the TWA modal deformations is performed using a Polytec PSV500 laser vibrometer. In this case, the excitation is also applied through the shaker as in Fig. 1. The emission axis of the laser vibrometer is positioned perpendicular to the radiated surface of the rim at a distance of 2.5 m on the y axis. The results of this analysis are shown in Fig. 3, where we can observe the experimental modal deformations of (a) cavity, (b) ovalization, (c) pitch, and (d) pumping mode.

The TWA pumping mode (mode 6) exhibits low damping with a loss factor *η* of 1.06% and its modal shape reveals that all spokes deform in phase with equal strain. Hence, the target mode is the only one with axisymmetric strain compared to modes 3, 4, and 5. Using this information, the location and wiring of the PZT actuators and sensors can be optimized.

Figure 4 provides a cross sectional view of the TWA depicting the concept of spatial modal filter. When a piezoelectric transducer experiments strain, it generates a proportional voltage whose signs depend on the curvature geometry. Examining the top section of Fig. 4 describes the pumping mode where all transducers deform in phase. As a result, the voltage *V* produced by the two transducers is of the same sign and amplitude. Figure 4 (bottom) section displays the modal shape of a non axisymmetric mode. Both voltages
<figure><img src="image_5.png" /><figcaption>Fig. 4 Schematic representation of spatial modal filtering</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5 Wiring of the ten PZT transducers in two rings</figcaption></figure>

V generated by the transducers are of opposite sign while maintaining the same magnitude as it occurs for the pitching mode for example.

Considering this phenomenon of voltage summation or cancelation due to modal strain, we propose to implement the actuation and sensing functions in rings of transducers wired in parallel. Thus, the PZT wiring network is designed according to Fig. 5 where each ring is composed of five transducers, one per spoke as follows:

* inner ring (actuation): 105 mm average radius

* outer ring (sensing): 185 mm average radius

As a consequence, both inner and outer rings theoretically only control or sense the pumping mode. To verify this hypothesis, the transfer functions between the shaker disturbance F and the voltage output of the inner and outer rings, $U_A$ and $U_S$ respectively are measured and displayed in Fig. 6. We clearly observe that the dominant mode located at 436 Hz is the pumping mode 6 while the others are drastically reduced. In terms of amplitude, the inner ring shows a higher response at the modal frequency. Consequently, the inner ring is used in

<figure><img src="image_7.png" /><figcaption>Fig. 6 Experimental FRFs from shaker disturbance F to inner and outer rings voltages ($U_A$ and $U_S$)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 Definition of the static test bench of the 1:1 model</figcaption></figure>

the following section for the actuation function and the outer ring for the sensing function.

## 3.2 Control System Identification.
The next step is to measure and identify the following transfer functions $H_0$, $H'_0$, $H_1$, and $H_2$ defined by

$$H_0(s) = \frac{A(s)}{F(s)} \qquad (3)$$

$$H'_0(s) = \frac{U_S(s)}{F(s)} \qquad (4)$$

$$H_1(s) = \frac{U_S(s)}{U_A(s)} \qquad (5)$$

$$H_2(s) = \frac{A(s)}{U_A(s)} \qquad (6)$$

where s is the Laplace variable.
<figure><img src="image_9.png" /><figcaption>Fig. 8 Frequency response function, shaker white noise disturbance/inner piezoelectric ring, and shaker white noise disturbance/inner piezoelectric ring</figcaption></figure>

<table><caption>Table 3 Identified modal parameters from the accelerometer (A) and outer ring (US) to the force sensor (F) and the inner ring (UA)</caption><thead><tr><th>Mode</th><th>Magnitude (dB)</th><th>Phase (deg)</th><th>a<sub>i</sub><sup>0</sup></th><th>b<sub>i</sub><sup>0</sup></th></tr></thead><tbody><tr><td>H<sub>0</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>5.64</td><td>37.17</td><td>5.37</td><td>1.00 &times; 10<sup>3</sup></td></tr><tr><td>2</td><td>2.24</td><td>79.74</td><td>7.97</td><td>1.09 &times; 10<sup>3</sup></td></tr><tr><td>3</td><td>3.07</td><td>66.46</td><td>3.76</td><td>1.30 &times; 10<sup>3</sup></td></tr><tr><td>4</td><td>10.03</td><td>76.07</td><td>12.45</td><td>1.58 &times; 10<sup>3</sup></td></tr><tr><td>5</td><td>16.70</td><td>85.47</td><td>15.01</td><td>1.76 &times; 10<sup>3</sup></td></tr><tr><td>6</td><td>18.88</td><td>84.66</td><td>14.41</td><td>2.74 &times; 10<sup>3</sup></td></tr><tr><td>Mode</td><td>Magnitude (dB)</td><td>Phase (deg)</td><td>a<sub>i</sub><sup>0</sup></td><td>b<sub>i</sub><sup>0</sup></td></tr><tr><td>H<sub>0</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>2.74</td><td>154.74</td><td>6.41</td><td>1.00 &times; 10<sup>3</sup></td></tr><tr><td>2</td><td>7.23</td><td>124.61</td><td>8.60</td><td>1.09 &times; 10<sup>3</sup></td></tr><tr><td>3</td><td>2.15</td><td>150.55</td><td>3.36</td><td>1.30 &times; 10<sup>3</sup></td></tr><tr><td>5</td><td>5.91</td><td>157.36</td><td>13.30</td><td>1.76 &times; 10<sup>3</sup></td></tr><tr><td>6</td><td>33.25</td><td>77.38</td><td>14.46</td><td>2.74 &times; 10<sup>3</sup></td></tr><tr><td>Mode</td><td>Magnitude (dB)</td><td>Phase (deg)</td><td>a<sub>i</sub><sup>1</sup></td><td>b<sub>i</sub><sup>1</sup></td></tr><tr><td>H<sub>1</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>6</td><td>50.50</td><td>76.30</td><td>17.02</td><td>3.40 &times; 10<sup>3</sup></td></tr><tr><td>Mode</td><td>Magnitude (dB)</td><td>Phase (deg)</td><td>a<sub>i</sub><sup>2</sup></td><td>b<sub>i</sub><sup>2</sup></td></tr><tr><td>H<sub>2</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>6</td><td>66.50</td><td>99.56</td><td>13.20</td><td>2.73</td></tr></tbody></table>

The experimental setup displayed in Fig. 7 is sequentially excited by the shaker and the inner ring with a white noise disturbance to obtain the acceleration response of the structure and voltage response of the outer ring. The consideration of the acceleration is motivated by the necessity of having an out of the loop performance sensor.

The models of these transfer functions are established through the rational fraction polynomial method [27,28], giving us pole zero

<figure><img src="image_10.png" /><figcaption>Fig. 9 Closed-loop system block diagram</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10 Open-loop transfer function $H_1(s)$. $H_{control}(s)$ for $g_1 = 10$</figcaption></figure>

systems. Figure 8 shows the measured and identified transfer functions $H_0(f)$ and $H_1(f)$. The most important feature to notice is the double spatial modal effect between the inner ring and outer ring visible in $H_1(f)$. This characteristic ensures to reduce or avoid any spillover effect in the closed loop and focuses the control energy on the targeted pumping mode. Finally, all the identified parameters are summarized in Table 3.

The complete representation of the control problem is displayed in the block diagram of Fig. 9.

### 3.3 Control Law Design
Based on the open loop transfer function $H_1$ previously identified, we define a controller $H_{control}$ to achieve satisfactory performance and stability margins.

The present transducer configuration represents a non minimum phase system. The controller transfer function presents a $-\frac{\pi}{2}$ phase at 436 Hz corresponding to the pumping mode. Consequently, a simple proportional controller can provide damping in the closed loop at the mode frequency. The controller is there defined as

$$H_{control} = g_1 \times \frac{s}{s + \omega_1} \times \frac{\omega_2}{s + \omega_2} \quad (7)$$
<figure><img src="image_12.png" /><figcaption>Fig. 11 Simulated frequency response functions without control (dotted) and with control (light)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12 Experimental FRFs between acceleration A and disturbance force F</figcaption></figure>

where $g_1$ is the global gain, $\omega_1$ and $\omega_2$ are the cutoff frequencies of a bandpass filter, respectively 100 rad/s and 4000 rad/s.

The stability margins estimated by simulating the complete system are displayed in Fig. 9 and shown in Fig. 10. It can be observed that for a gain $g_1$ of 10, the resulting margins are still substantial, namely $M_G = 17.2$ dB and $M_{\omega_1} = 83.3$ deg. The gain $g_1$ has been tuned to the value of 10 that corresponds to the optimal damping value of the targeted poles.

Figure 11 displays the simulated performances for gains $g_1$ of 5 and 10 of the controller $H_{control}$ on the closed loop on the out of the loop sensor. A substantial reduction in the amplitude of the pumping mode at 436 Hz is observable, reaching an attenuation of -8 dB and -12 dB respectively.

# 4 Experimental Validation

Now that the controller is designed and tuned, its performance and robustness are assessed on three different experimental test benches. The first one is the same as in Fig. 7, the second one allows the application of a static load on the tire. Finally, the third one replicates real operational conditions with the TWA rotation and road tire contact.

## 4.1 Static Test Bench.
The controller $H_{control}$ is experimented on the test static test bench (Fig. 7) between $U_s$ and $U_A$. The results are displayed in Fig. 12 for gains $g_1$ of 5 and 10. Compared to Fig. 11, the achieved reduction of the pumping mode is coherent (-10 dB and -15 dB). Such performance level confirms the design of the controller. In addition, the double modal spatial filter effectively focuses the control energy on the pumping mode without any spillover effect.

Figure 13 presents the root mean square (RMS) value of the control voltage directly applied on the inner ring and the attenuation of the control gain $g_1$. We

<figure><img src="image_14.png" /><figcaption>Fig. 13 Attenuation of the pumping mode and RMS consumption of the control as a function of gain "g"</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14 Experimental bench equipped with sensors in the hub. Application of a static load vertically (weight of a vehicle) and transverse displacement (rolling effect).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15 Schematic representation of the test bench allowing a loading of the TWA</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 16 (a) Transmissibility and (b) dynamic stiffness measured without and with control</figcaption></figure>

observe a saturation on the attenuation level for gains g₁ from 10 to upper values which validates the chosen control tuning. One can observe that for this level of excitation, the RMS voltage applied to the PZT actuators is relatively low (7 V).

at a specific frequency bandwidth. It is defined as

$$T(s) = \frac{X_{\text{output}}(s)}{X_{\text{ref}}(s)} \qquad (8)$$

**4.2 Pneumatic Cylinder Test Bench.** The principle of this second test bench illustrated in Figs. 14 and 15 is to verify the effect of the controller on the force transmission at the wheel hub level. A vertical static load of 5000 N is applied to the TWA corresponding to a quarter weight of a vehicle. A slider allowing y translation is placed at the interface between the pneumatic actuator and the tire. A shaker is placed to excite the TWA through this slider recreating realistic conditions of system disturbance. The hub is equipped with a force sensor, enabling the observation of the control impact on the forces transmitted by the wheel to the rest of the vehicle.

The same controller H<sub>control</sub> is applied in the closed loop as in Sec. 4.1, and the results are displayed in Fig. 16(a) introduces the transmissibility between the acceleration of the slider X̃₀ and the acceleration of the outer rim X̃₁ where a -7.5 and -11 dB reduction is observable on the pumping mode. Transmissibility refers to the ratio between the amplitude of a response (such as displacement or acceleration) and the amplitude of the input excitation or force. It characterizes how the system transmits or attenuates responses

<figure><img src="image_18.png" /><figcaption>Fig. 17 Instrumented wheel mounted on the ZF-M500 with slip ring</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 18 F_y force measured in the hub of the tire-wheel assembly studied, for Ω₁ and Ω₂ equivalent rotational velocity with control (light) and without (dotted) control</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 19 Evolution of performance as a function of Ω rotational velocity, without control (dotted) with control with g₁ = 10 (light)</figcaption></figure>

where s is the Laplace variable, T is the transmissibility, and X the compared physic component.

Figure 16(b) shows the effect of the controller on the dynamic stiffness on the y direction at the wheel hub level. Dynamic stiffness measures the ability of a structure or system to resist deformation or displacement under dynamic loading conditions, such as vibrations or oscillatory forces. It quantifies the relationship between applied dynamic loads and resulting dynamic displacements or deformations, analogous to how static stiffness relates to static loads and dis placements. It is defined as

$$K_{\text{dyn}}(s) = \frac{F_{\text{dyn}}(s)}{d_{\text{dyn}}(s)} \qquad (9)$$

where s is the Laplace variable, K<sub>dyn</sub> is the transmissibility, F<sub>dyn</sub> the dynamic force, and d<sub>dyn</sub> the resulting dynamic displacements of deformations.

The measured value goes from 4 × 10<sup>7</sup> N/m to 8.8 × 10<sup>6</sup> N/m for gain g₁ = 10 (–78%). The most important observation here is that the controller does have authority over the effort transmission between the tire and the wheel hub. Again, since the excitation level is drastically lower than the real operational conditions, the voltage applied to the inner ring is maintained at a low level (1.74 V RMS).

### 4.3 Michelin M500 Test Bench.
The final part of the experimental results is achieved on the ZF M500 test bench shown in Fig. 17. In this experimental setup, the wheel is placed on a rotating shaft with the same mechanical assembly as for real vehicle. A cylinder whose metallic external surface is machined using a six axis CNC machine to replicate the surface of different scanned road profiles is driven by a motor. The cylinder is then translated in the x direction until contact with the TWA driving its rotation in addition with a 5000 N static load. Hence, the disturbance on the TWA is directly induced by the contact between the tire and the road cylinder. Finally, the forces transmitted through the wheel hub are measured on the y axis. During the rotation, the control and measurement signals are transmitted through a slip ring as depicted on the upper right side of Fig. 17.

Now, the same controller H<sub>control</sub> is applied to the rotating test bench. Figure 18 shows the power spectral density of the hub force F<sub>y</sub> measured for two angular velocities Ω₁ (Fig. 18(a)) and Ω₂ (Fig. 18(b)) where Ω₂ > Ω₁. The results are compared with and without control for g₁ = 10. A significant reduction of 40% in the amplitude of the force due to the pumping mode is noticeable at 436 Hz for Ω₁ and 30% for Ω₂.

It is important to notice that the proposed control approach remains performant despite real rolling conditions and a significantly higher disturbance level. In addition, the controller appears insensitive to the frequency shift of the pumping mode between Ω₁ and Ω₂ (10 Hz). This robustness can be explained by the control system design where the phase of H₁ at the pumping mode will always be –π/2 regardless of its frequency.

Finally, Fig. 19 shows the different effort spectra with and without control for two higher angular velocities, Ω₃ and Ω₄, where Ω₄ > Ω₃ > Ω₂. Several observations can be made. First, the control remains efficient and localized in frequency around the pumping mode, and no undesirable effects are visible. Finally, the dynamics of the TWA appear to be evolving with the velocity increasing but the controller still reduces the effort amplitude by 20–30% for Ω₃ and Ω₄. The design of the control system provides the desired robustness, despite the evolving dynamics with no adverse effects observed.

Nevertheless, a shift in the TWA response is clearly observable between Ω₁ and Ω₄, both in terms of measured force levels and frequency response. It appears that the applied control law is no longer well suited for elevated rotation velocities and further robust control methods are to be evaluated in the near future to mitigate the target mode for higher velocities.

## 5 Conclusions and Perspectives

This paper proposed an active modal vibration control method based on a spatial modal filter applied to a tire wheel passenger car assembly. The structure was equipped with two rings of PZT transducers, used both as actuators and sensors, and were bonded to the wheel rim. Initially, a modal analysis of the TWA has been performed. Then the placement of the piezoelectric transducers and the wiring was optimized to design a double spatial modal filter. Subsequently, a bandpass control law has been designed to attenuate the targeted pumping mode. Thus, the proposed control approach was successfully applied to the TWA on a test bench rereating real operational conditions including rotation and tire road contact. The experimental results validated the relevance of using a spatial modal filter for different angular velocities of the TWA. The
control strategy achieved an attenuation of -10 dB and -8 dB for $Ω_1$ and $Ω_2$.

Due to the rotation, the damping of the pumping mode seemed to increase with the angular velocity and other dynamic phenomena also appeared in the same bandwidth. Thus, a modification of the control law could be necessary to increase the performance at higher velocities. Finally, the proposed control approach did not produce adverse effects despite the evolving dynamics of the TWA and demonstrated its ability to control the target pumping mode with a suitable controller, and placement of the actuators and sensors.

## Acknowledgment

The authors would like to thank the Association Nationale de la Recherche et de la Technologie (ANRT) for having provided the financial resources which allowed to obtain these results through the CIFRE contract No. 2020/1173.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

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