Finite Element modeling of force amplification at the spindle due to a tire’s cavity mode: experimental verification Won Hong Choi 1 , J. Stuart Bolton 2 , Kyosung Choo 3 Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University 177 S. Russell Street West Lafayette IN 47907-2099 Matthew Black 4 Ford Motor Company Vehicle Engineering, 20500 Oakwood Boulevard, Dearborn, MI 48121

ABSTRACT Reduction of tire-road noise is an important issue when developing luxury cars and electric vehicles. In this context, the air-cavity mode is an important source of spindle forces transmitted to the sus- pension that then increase interior noise levels. When a tire rotates, the cavity mode near 200 Hz splits into two adjacent modes due to a Doppler effect and tire deformation. That split can lead to increased levels of both longitudinal and vertical spindle forces at the spindle since the two acoustic modes each contribute to both forces when the tire rotates. Thus, it is important to develop tools to identify the contributions of the split air-cavity modes to the spindle force. A FE simulation of the spindle force for a steady-state rolling tire has been verified by a comparison with laboratory test results obtained by using a wheel-force transducer mounted on Purdue’s Tire Pavement Test Appa- ratus. It was observed that the frequency split expands as the rotation speed increases and that the vertical spindle force increases when aligned with an odd-numbered circumferential structural mode.

1. INTRODUCTION

Tire/road noise has gained more attention than ever as EV vehicles have emerged, since, in that case the powertrain noise is eliminated. It is also known that tire noise comprises 35% of NVH prob- lems for EV vehicles (Parmar, 2020, [1]). Tire/road noise can be categorized into both structure-borne noise and airborne noise depending on the transfer paths. In particular, structure-borne noise is crucial since it is easy to transmit the force from a tire into the vehicle chassis via a suspension system in the low-frequency range between 80 Hz and 250 Hz (Michelin, 2020, [2]). Moreover, the air-cavity mode is a source that amplifies the transmitted force in a suspension system, thus contributing to an increase in the level of cabin noise. In a previous study (Cao, 2018, [3-4]), it was proposed, in particular, that

1 choi124@purdue.edu

2 bolton@purdue.edu

3 choo7@purdue.edu

4 mblack4@ford.com

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the force level is amplified when one of a tire’s structural circumferential modes couples with the air- cavity mode near 200 Hz. Furthermore, when a tire is deformed by contact with the road, the funda- mental cavity mode can split into two adjacent modes due to the break of geometrical symmetry (Thomson, 1995, [5]). The fore-aft mode in the air cavity is formed at the lower of the two frequencies owing to the increased particle velocity in the contact patch region that results from the reduction in cross-sectional area due to loading. In contrast, the vertical mode shifts to a higher frequency. For the deformed tire, it would be problematic, for example, if a tire’s odd-numbered structural mode coin- cided with a vertical acoustic mode at the upper natural frequency, thus generating a large net force induced by the acoustic pressure within the tire in the vertical direction. In this regard, it is very important to identify the mechanism controlling the force amplification at the hub with a focus on the interaction between structural modes and the split in the cavity resonance. In addition, the split in the fundamental cavity mode is enlarged when a tire is rolling due to the Doppler effect in which the natural frequency changes depending on whether the wave propagation in the cavity matches the direction of rotation or is in the opposite direction. Structural resonances can also be affected by the rolling condition since the phase speed of traveling structural waves along a tire’s circumference is different depending on whether the positive and negative-going directions are parallel to the direction of rotation. Hence, it becomes more complicated to investigate the char- acteristics of force amplification for a rolling tire than for a static tire. In the current work, TPTA (Tire Pavement Test Apparatus) measurements of the dynamic force and acceleration at the wheel hub are presented, along with the verification of a FE simulation that is intended to help clarify the mechanism of force amplification by the split cavity mode. 2. THE FORCE AMPLIFICATION AT THE HUB DUE TO A TIRE’S CAVITY MODE

The basic mechanism of force amplification due to the air-cavity mode is described in this section. When a tire is deformed, the 1 st order cavity mode is split due to the break of geometrical symmetry since the tire cross-section is distorted near the contact patch as shown in Figure [1]. A fore-aft mode is observed to form at a lower frequency than the fundamental cavity mode. In contrast, a vertical mode is produced at a slightly higher frequency. Formulae to predict the split natural frequencies that result from the static deformation (Thompson, 1995, [5]) are:

𝑐

𝐿 𝑐 +(1−𝒎)𝒍 𝒄𝒑 , (1)

𝑓 𝐻 =

𝑐

𝐿 𝑐 −(1−𝒎)𝒍 𝒄𝒑 . (2)

𝑓 𝑉 =

where 𝑓 𝐻 is the frequency of the horizontal, fore-aft mode and 𝑓 𝑉 is the frequency of the vertical mode. These equations are written in terms of sound speed ( 𝑐 ); the ratio of the tire cross-sectional area in the contact patch region between the loaded and unloaded configurations ( 𝑚 ), the mean cir- cumferential length ( 𝐿 𝑐 ), and the length of the contact patch ( 𝑙 𝑐𝑝 ).

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Figure 1: The frequency split in a fundamental cavity mode due to a static deformation.

In addition, the net force at the hub can be different depending on whether the net deformation in a tire’s circumference due to a structural mode matches the modal shape of the acoustic mode (Cao, 2018, [3-4]). For instance, if an even-numbered structural mode coincides with the fore-aft mode, then the net force becomes larger in the horizontal direction. Also, an odd-numbered tire mode that couples with the vertical mode can generate a large net force at the hub in the vertical direction since the net displacement is directed to that direction since the other modal peaks cancel each other out, as illustrated in Figure [2].

Figure 2: The force amplification due to vertical acoustic mode at 205 Hz, 235/50R18.

Meanwhile, the effect of rolling makes the frequency split much wider than in the static tire case owing to the Doppler effect in the air cavity. The first acoustic natural frequency, referred to as 𝑓 1 , originating from the fore-aft mode, moves down to a lower frequency since the negative-going wave traveling around the cavity is directed opposite to the rolling direction (currently, counterclockwise). On the other hand, the second acoustic resonance frequency, defined as 𝑓 2 , derived from the vertical mode, shifts to a higher frequency if the positive-going wave propagation is parallel to the direction of rotation. The lower and upper frequencies can be expressed in terms of rotation speed ( 𝑣 ) as:

𝑐−𝑣

𝑐+𝑣

2𝜋∙𝐿 𝑐 , 𝑓 2 =

2𝜋∙𝐿 𝑐 . (3)

𝑓 1 =

It has been observed that the effect of rolling is normally the dominant factor in expanding the frequency split, rather than the static deformation. Also, the two natural modes are no longer either purely horizontal or vertical. Instead, the modes take on an elliptical shape at a skewed angle when both effects are combined. However, it is still possible for there to be force amplification when an elliptical shape at either frequency aligns with the modal shape of a tire’s structural modes, as de- picted in Figure [3].

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Figure 3: The frequency split for elliptical shape in acoustic pressure at 48 km/h, 235/50R18. In summary, it is important to be able to predict the frequency split as it is enlarged by the effects of rolling at different rotation speeds to avoid the force amplification caused by a match with the natural modes of the tire structure. In other words, it is necessary to decouple the split in cavity mode with any major structural modes to minimize the resultant force at the hub, which thus provides a practical guideline for developing a low-noise tire from a vehicle design perspective.

i t

3. FORCE MEASUREMENT IN A LABORATORY TEST FOR ROLLING TIRE

In this section, the test procedures used to measure dynamic force and acceleration at the hub when a tire is rolling are described. Also, data are presented in terms of frequency response and 2D surface plots to highlight force amplification related to the split in the cavity mode during a rolling test.

3.1. TPTA machine

In the present study, the TPTA (Tire Pavement Test Apparatus) has been used to operate tires at different rotational speeds, as shown in Figure [4]. This equipment was originally designed to evalu- ate road-noise levels over various concrete pavements. It has the capability of running tires at the maximum speed of 48 km/h (30 mph) with a controlled static load that is achieved by adjusting the rotating arms that hold the tires. The details are summarized in Table 1.

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Figure 4: TPTA machine in a semi-anechoic chamber.

Table 1: Specification of performance of TPTA machine.

Max. Speed Max. Load Mean diameter Pavements

48 km/h 4,415 N 4.4 m Six (rough, fine)

The test rig has five connecting arms that are connected to the wheel hub, and which are also connected to the rotating beam (yellow). In order to isolate the cavity mode from a rig resonance near 160 Hz, TMD’s (Tuned Mass Dampers) were applied at the centers of the individual arms. In that way, significant attenuationin of the vibration was achieved: for details see the paper (Choo, 2022, [6]).

A tire under test is set to run over six different concrete pavements once per revolution to reflect a variety of possible road conditions. The TPTA is housed in a semi-anechoic chamber.

3.2. Test setup with wheel force transducer

A single tri-axial accelerometer was attached at the hub, and a WFT (Wheel Force Transducer), as commonly used by vehicle manufacturers, was used to obtain the dynamic force: see Figures [5]. Since the tire is rotating, it is impossible to record the signals using conventional wired connections. Instead, a wireless transfer via a router was used by strapping the data acquisition devices to the top of the machine, which enables monitoring and recording of the signals in a control room. In total, eleven channels were used including acceleration, force, moment, and trigger recording. A 205/30/R18 tire was selected for the tests described here.

Table 2: List of DAQ and sensors.

Type Brand Model Remark

DAQ B&K 3560-B/C-130 Eleven Ch.

Accelerometer PCB 356B18 Triaxial

WFT Michigan Scientific LW12.8-50 Passenger car

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Figure 5: The layout of measurement devices.

3.3. Signal processing

During the data acquisition, the trigger pulse was used to identify the starting point of each revo- lution, at which point the tire under test was positioned at the beginning of the smooth pavement section. One second of data acquisition prior to the trigger signal was recorded during each rotation of the rig, and the total recording spanned for four minutes. The Fourier transforms of each one second record were then averaged to give the spectral estimates shown here. The sampling parameters are given in Table 3, and sample data is shown in Figure [6].

Table 3: The parameters in a signal processing.

Sampling freq. Time window Timespan Averaging

4000 Hz Hanning 1s >100 times, Frequency spectrum

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Figure 6: Time window along with trigger signal.

3.4. Test result

The averaged frequency spectra of both the force at the hub and the acceleration in both directions, X (traveling direction) and Z (vertical to the road surface), respectively are presented here. Figure 7 shows the frequency spectra in the low-frequency range between 50 and 300 Hz, for the force re- sponse at the hub at three different speeds: 16 km/h (10 mph), 32 km/h (20 mph), 48 km/h (30 mph). First, it can be observed that the force amplification due to the cavity mode takes place near 208 Hz, which is the higher cavity resonance normally associated with the vertical mode for a loaded tire. Compared to that peak, the force amplification at the lower cavity resonance, originating from the fore-aft mode, is relatively less clear. It is thought that the reason for this is that an odd-numbered structural mode, the 7 th mode in the circumferential direction, interacts with an elliptical acoustic mode at the higher frequency. Moreover, the magnified amplitude can be seen, especially at the sec- ond natural mode, 𝑓 2 , in the two acceleration graphs in Figure 8. The amplification can be seen in both horizontal and vertical directions.

Figure 9 shows the 2D surface plots in which the magnitude of the force is plotted against fre- quency at different rotational speeds between 6 km/h and 48 km/h, in steps of 3 km/h. It can be seen that there are two lines diverging from the initial cavity mode frequencies near 200 Hz. As expected from theory, the frequency split clearly increases with increasing speed. Also, both horizontal and vertical forces are amplified significantly for the loaded, rolling tire.

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Figure 7: Frequency response in force measurement.

Figure 8: Frequency response in acceleration measurement.

Figure 9: The 2D surface plot in force measurement.

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4. FE SIMULATION

A finite element model of the tire structure and air cavity in contact with the road has been estab- lished. The effect of rolling was modeled by using SST (Steady-State Transport analysis) in the com- mercial software, Abaqus 2021. In previous papers, the frequency split due to static deformation was investigated in simulation, including information about the material properties (Choi, 2020, [7]). Also, the frequency split owing to the rolling effect was numerically analyzed by introducing steady-state transport analysis (Choi, 2021, [8]). In addition, the frequency split and associated characteristics have previously been studied in simulation and experiment for a static tire (Choi, 2021, [9]).

4.1. FE modeling

As a first step in the modeling, a full-scale model was constructed as shown in Figure 10 based on the quarter-symmetric cross-sectional area consisting of the tire structure (treandaband, sidewall) on the rim and air-cavity by revolving it into the symmetric plane referred to as SMG (Symmetric Model Generation). Next, a static analysis of the deformation at the rated inflation pressure was performed, during which the tire is loaded by a contact patch that is modeled as a rigid body. After that, a har- monic analysis was performed in the frequency domain by imposing an oscillatory input at the contact patch for the deformed tire. Simultaneously, the effect of rotation that is assumed to be continuous over the tire’s structure and air-cavity was applied by assigning a designated rotation speed in SST analysis supported in Abaqus software (ver. 2021).

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Figure 10: The FE model.

4.2. Simulation result in comparison with test First, a dispersion diagram can be plotted to investigate the characteristics of wave propagations in the tire structure based on the mobility data: i.e., the surface velocity on the sidewall normalized by the input force plotted against wavenumber ( 𝑘=

2𝜋

𝜆 ) and the driving frequency ( f ). The dispersion diagram is obtained by performing a DFT (Discrete Fourier Transform) on the mobility data at spatial points along the sidewall. Figure 11 shows the dispersion curves at two different speeds: 0 km/h and 48 km/h. Inside the main slopes, which represent the phase speed of circumferential flexural waves, two spots exists, which represent the split in the 1 st cavity mode. More interestingly, the split is visibly increased at 48 km/h due to the Doppler effect, The upper natural frequency occurs at 208 Hz, which is in a good agreement with the measurement result. Furthermore, the relation between rotation speed

and the frequency split is accurately reproduced, as shown in Figure 12. The split does not change significantly up to 20 km/h, and then it increases in proportion to the rotation speed, a behavior that matches qualitative expectations. Figure 13 shows the comparison between the measurement and simulation for the force at the hub at 48 km/h. Not only is the force amplification visible at the two natural frequencies, 𝑓 1 = 197 Hz and 𝑓 2 = 208 Hz , but also lower frequency features associated with the structural resonances of the tire show a good match, although there is an offset in modal frequency at the lower frequency region around 70 Hz, which needs to be further investigated. Finally, based on the correlated simulation model at 48 km/h, 2D surface plots were calculated in the range between 6.4 km/h and 48.3 km/h increasing by 3 km/h steps in the simulation: see Figures 14-15. The increasing frequency split with increasing speed is clear, and the simulation captures the main features of the measurement. That is, in the simulation, the first natural frequency, originating from the fore-aft mode is more significant in horizontal force direction than it is in the vertical direc- tion.

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Figure 11: The dispersion diagram, 205/30/R18.

Figure 12: The relation between rotation speed and frequency split for a tire, 205/30/R18.

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Figure 13: The comparison of force between test and simulation at 48 km/h, 205/30/R18.

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Figure 14: The comparison of 2D surface plot between test and simulation, Fx, 205/30/R18.

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Figure 15: The comparison of 2D surface plot between test and simulation, Fz, 205/30/R18.

prce, F, Vertical Force, F, Vertical Force, F, Vertical Fe 20u oe : = ie aS ae fect He i | 200 : : 200 200 = : | = cy > a z 2. 3 5 = 08 8 g g = © © = = ce i im im 18 “150 150 150 150 -20 20 20 ne. -25. 25.

5. CONCLUSIONS

A laboratory test environment was established for measuring force and acceleration at the hub of a rolling tire to create a database related to the interaction between the air-cavity mode and the force output at the hub. In the current work, the amplification due to the air-cavity mode is clearly observed from the hub force and acceleration data. Also, the measured data were quantitatively compared with simulation results, which demonstrated the possibility of estimating dynamic force at the hub numer- ically. In future, different sizes of tires will be tested, and these results will be used to verify and enhance the simulation prediction accuracy. Finally, a design optimization based on geometry and material properties will be carried out to see whether it is possible to reduce the amplified force by adjustment of tire properties. 6. ACKNOWLEDGEMENTS

The authors are grateful for the financial support provided by Ford Motor Company, through the Ford/Purdue Alliance. We are also grateful to the Ford project manager, Matt Black, for his technical and logistical support. 7. REFERENCES

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verification. Proc. INTER-NOISE 18 (2018). 4. Rui Cao, J. Stuart Bolton, and Matthew Black. Force transmission characteristics for a loaded

structural-acoustic tire model. SAE Int. J. Passeng. Cars – Mech. Syst , 11(4) (2018). 5. J.K. Thompson. Plane-Wave Resonance in the tire air cavity as an interior vehicle noise source.

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mass dampers to the Tire Pavement Test Apparatus to minimize the impact of rig resonance. Proc. NOISE-CON 22 (2022). 7. Won Hong Choi and J. Stuart Bolton. Prediction of split in fundamental air-cavity mode of loaded

tires based on experimental observations and computational simulations. Proc. NOISE-CON 20 (2020). 8. Won Hong Choi and J. Stuart Bolton. Simulation of the frequency split of the fundamental air

cavity mode of a loaded and rolling tire by using steady-state transport analysis. Proc. INTER- NOISE 21 (2021). 9. Won Hong Choi and J. Stuart Bolton. FE simulation of split in fundamental air-cavity mode of

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