Operational Modal Analysis of a rotating tyre subject to cleat excitation

Peter Kindt(1), Arnaldo delli Carri(2), Bart Peeters(2), Herman Van der Auweraer(2), Paul Sas(1), Wim Desmet(1)

(1)Department of Mechanical Engineering, K.U.Leuven, Belgium

(2)LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium, bart.peeters@lms.be

ABSTRACT Structure-borne tyre/road noise is an important component of the perceived noise annoyance of passenger cars. More in particular, it was observed that crossing road surface discontinuities (e.g. concrete road surface joints, railroad crossing, potholes, …) causes a significant increase in instantaneous exterior noise level. In addition, it has an adverse effect on the interior vehicle NVH in the sense that the passengers experience high-amplitude transient noise and vibrations. Therefore, an extensive research programme was established at the Department of Mechanical Engineering, K.U.Leuven, to study structure-borne tyre/road noise due to road surface discontinuities. As part of the research activities, an original test setup for impact tyre/road noise was developed so that rolling tyre vibrations, radiated noise and dynamic spindle forces could be measured at different rolling speeds. The test setup is based on the tyre-on-tyre principle and a cleat is used to reproduce a road surface discontinuity. This paper concentrates on the data processing techniques used to experimentally obtain the modes of a rolling tyre. Since the forces introduced by the cleat cannot me measured, Operational Modal Analysis was selected as processing technique. A major challenge is the requirement to obtain spatial information on the tire from a single-point measurement device. Therefore, a dedicated triggering and time-domain averaging procedure was elaborated. The purpose of averaging is obviously to reduce random noise whereas triggering is required to be able to correlate different tyre locations that have not been measured at the same time (a single-point Laser Doppler Vibrometer was used).

1 INTRODUCTION The increase of the road traffic density over the past decades resulted in a growing noise burden for most inhabitants of urban areas [1][2][3]. Nowadays, there is a high awareness among policymakers of the problems that traffic noise causes to the society. Therefore, road traffic is subjected to ever tightening noise limits. The three main sources of vehicle noise are: power unit noise, aerodynamic noise and tyre/road noise. Tyre/road noise refers to the noise that is generated by the interaction between the rolling tyre and the road surface. For modern vehicles, the tyre/road noise becomes more important than the power unit noise for driving speeds above approximately 40 km/h. The aerodynamic noise is small for normal driving speeds. Thus, tyre/road noise has become the dominant vehicle noise source for most driving conditions. Although tyre/road noise has been extensively studied for decades, still some of the noise generating phenomena are not yet fully understood and the generation of tyre/road noise for certain tyre-road configurations has never been studied in detail. For instance, the noise caused by passing a road surface discontinuity, such as joints in concrete road surface, railroad crossings, bridge joints, cobbled roads, etc. , has hardly been studied. The interaction between the tyre and a road discontinuity causes a transient noise that reaches significant peak levels and that is perceived as highly annoying. This noise causes serious discomfort, particularly in cities where a large number of these discontinuities are found. Reduction of tyre/road noise requires an integrated approach which comprises both low noise road surfaces and low noise tyres. Moreover, all design aspects – such as durability, wet grip performance, rolling resistance – have to be considered in the development of both new tyres and road surfaces. Therefore, a full understanding of all noise generating phenomena is essential.

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

Similar to the noise exterior to the vehicle, the tyre/road interaction also contributes to the noise inside the passenger compartment [4]. In addition, the driver of a vehicle experiences vibrations at the seat and steering wheel. Depending on the amplitude and frequency content, these vibrations can reduce the driver’s comfort significantly. For most road surfaces, the interaction between the tyre and the road surface is a major source of vibrations that are transmitted through the suspension towards the vehicle body. The harshness of a vehicle expresses the subjective perception of transient vibrations and noise. Crossing road surface discontinuities causes transient vehicle interior noise and vibrations that can reach significant peak levels. Passengers perceive this kind of excitation as annoying, which results in a considerable reduction of the comfort for vehicle occupants. Improving the Noise, Vibration and Harshness (NVH) characteristics of a vehicle requires a thorough understanding of the different noise sources, vibration sources and transmission paths of structural and acoustic energy in the vehicle. Over the last two decades, the development times for vehicles have decreased significantly due to the introduction of advanced numerical and experimental methods. This evolution, combined with the increasing comfort requirements for new cars, has lead to a demand for more accurate tyre models for vehicle NVH simulations. Therefore, an extensive research programme was established at the Department of Mechanical Engineering, K.U.Leuven, to study structure-borne tyre/road noise due to road surface discontinuities [3]. As part of the research activities, an original test setup for impact tyre/road noise was developed so that rolling tyre vibrations, radiated noise and dynamic spindle forces could be measured at different rolling speeds. This paper concentrates on the data processing techniques used to experimentally obtain the modes of a rolling tyre.

2 TEST SETUP The rotation is known to have an influence on the dynamic behaviour of a tyre [5]. However, it is practically infeasible to measure the excitation forces on the rolling tyre caused by the surface texture. Thus, a classical modal analysis is not applicable to characterize the dynamic behaviour of a rolling tyre. Therefore, an operational modal analysis (OMA) will be used to identify the dynamic behaviour of the rolling tyre out of measured responses only. A novel test setup, which is based on the tyre-on-tyre contact, was developed in order to simulate a tyre rolling on a flat road surface [6][7]. The studied tyre is of size 205/55R16 without tread pattern. Two identical tyres that are statically loaded against each other both deform as if they are loaded against a flat road surface. Figure 1 illustrates the tyre-on-tyre contact between two identical tyres.

Figure 1: Static deformation of two identical tyres loaded against each other. Dotted line represents the tangent line to both deformed tyres in the middle of the contact patch.

The tangent line to the tyre in the middle of the contact patch is equal for both tyres and is perpendicular to the line that connects the two tyre spindles. A road surface discontinuity is simulated by guiding an aluminium cleat through the contact area of the two tyres. The cleat is mounted on the driven tyre (Figure 2). The cleat is attached to the steel wheel by means of four pre-tensioned rubber springs. The pre-tensioned flexible fixture keeps the

cleat also connected to the tyre surface when the cleat approaches the contact area where the belt is deflected radially. A cleat which is rigidly connected to the wheel will not follow the intended trajectory through the contact area. As the cleat passes through the contact area of the two tyres, the cleat is indenting both tyres (Figure 3). This deformation is similar to the one of a tyre rolling over a cleat with half the size of the cleat used in the test setup. The test tyre is mounted on a multiaxial wheel hub dynamometer with built-in encoder (Figure 4). The piezoelectric dynamometer measures the three spindle forces and the three spindle moments. The x, y and z direction correspond to the longitudinal, lateral and vertical tyre direction, respectively.

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Figure 2: (a) Test setup with two tyres mounted; (b) Cleat fixation.

Figure 3: Circular cleat passing through the contact patch between the two identical tyres.

Figure 4: Multiaxial wheel hub dynamometer with built-in encoder.

Figure 5 (a) shows the setup to measure the radial component of the rolling tyre treadband vibrations. A Laser Doppler Vibrometer (LDV) is used for this purpose. A plane mirror is used to reflect the laser beam of the LDV such that the laser hits the tyre tread surface perpendicularly. The mirror is mounted on a support which can be considered rigid in the frequency range of interest. The mirror support can be positioned at different locations around the tyre, allowing a circumferential measurement resolution of 10 degrees. Since the mirror can be fixed at different positions and angles on the support, measurements can be performed over the entire width of the treadband.

Figure 5: (a) Rolling tyre vibration measurement by means of a LDV (dotted line shows the path of the laser beam); (b) Measurement points on the tyre cross-section; (c) Tyre measurement grid.

Figure 5 (b) shows the measurement points on the tyre cross-section. Vibrations are measured in two points on the treadband and in one point on the sidewall. The points are chosen such that the different cross-sectional modes can be identified. The laser vibrometer is equipped with an indicator that shows the intensity of the scattered light received by the vibrometer. When the laser beam is aligned perpendicularly to the tyre surface, a maximum amount of light is scattered back in the direction of the laser beam. The direction of the laser beam is therefore adjusted such that the intensity of the received scattered light is maximized. This assures that the laser

beam is aligned perpendicularly to the tyre surface. This approach is still valid when the laser beam is deflected by a mirror since the light reflected from the tyre surface is also deflected by the mirror. Vibration measurements in the contact area of the tyre are impossible since the laser beam has no access to this region of the tyre. In this setup, measurements on the tread can be performed at circumferential angles between 30 degrees and 330 degrees (Figure 5 (c)). The sidewall rolling tyre vibrations are measured directly without deflecting the laser beam. All the vibration measurements are performed with respect to a fixed reference frame in this setup.

3 REFERENCE ANALYSIS RESULTS This section will review the analysis results as presented in [3] and obtained through rather non-classical data (pre-) processing. These results will be considered as reference results throughout this paper and will be compared to other, sometimes more classical, Operational Modal Analysis (OMA) processing results in next sections. Since it is difficult to measure all responses simultaneously, a sequential measurement will be performed in which the responses are measured separately. Sequential measurements can only yield information about the complete vibration pattern if the phase relation between the different responses is maintained. This can be achieved by measuring simultaneously the response point and reference point vibration. This requires at least two laser Doppler vibrometers. However, if the excitation is perfectly repetitive it is possible to use a time reference instead. The acquisition of the individual responses has to start at the same time instant relative to the excitation. Here, the vertical component of the spindle force zF (Figure 4) is used to synchronize the different response measurements since the spindle force and tyre surface vibrations are always measured simultaneously in this setup. Therefore, the time reference is obtained from the force reference. In Section 4, the force reference will be used directly to obtain correct phase relationships between acquisition runs. Figure 6 shows an example of the vertical spindle force due to four subsequent cleat passages. Here, the time instant at which the spindle force is maximum (indicated by an arrow) is used as the time reference. The time reference is then used to synchronize the reference response and all the other responses such that the auto- and cross-power spectral density functions between these signals can be calculated. A sufficiently high sampling rate is required to determine accurately the time instant of the maximum spindle force. The same time reference is used to calculate the time averaged responses as described below. Alternatively, the auto-correlation function could have been used to identify the time intervals between each rotation.

Figure 6: Maxima of vertical spindle forces used as time reference between different cleat impacts.

The synchronized, time-averaged responses are further processed in the classical OMA way as implemented in LMS Test.Lab [8], as if they were measured simultaneously in a single run. In the presented analysis, two responses on the tread are used as reference (Figure 5). However, the references are not considered

simultaneously during the modal parameter estimation. Two different modal parameter estimations will be performed in which a single reference is considered. Certain resonances will be identified in both analyses while others modes are only identified in one analysis, dependent on the reference location. As is shown in [3], the two considered references provide a complete identification of all the excited rolling tyre modes.

4 THROUGHPUT-DATA OPERATIONAL MODAL ANALYSIS In this section a method will be investigated to conduct an Operational Modal Analysis using throughput-data only. Throughput-data is raw-collected data as good as it comes from a measurement system, without averages in a single run or synchronization between different runs. In the presented application, the basic idea is to compute cross-spectra between the spindle forces and the tyre vibration responses for each measurement run. It should be noted that only spindle (reaction) forces are available and not the contact forces between the obstacle and the tyre. By computing these cross-spectra, no triggering or synchronization is needed between the different runs, since the relative phase between two response quantities is independent from the start time of the acquisition:

)()()()()( * ffGfRfFfG FRxyFR ϕ∠=⋅= (1)

4.1 Time-Data Figure 7 shows the time signal of the vertical spindle force and the vibration response of point (a, 250) on the tyre surface for the entire measurement duration and a zoom around 1 particular cleat impact. The total acquisition time is 13 s, at a sampling frequency of 3200 Hz. Every measurement run counts 32 passages of the cleat in the contact-zone and the driving wheel has a rotation speed of 150 rpm.

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Figure 7: Throughput-data for the vertical spindle force and the vibration response at point (a, 250). (Left) Entire measurement duration; (Right) zoom around 1 cleat impact.

4.2 Pre-processing The so-called correlogram was used as cross-spectrum estimate. The tyre surface velocity time histories (as measured by the LDV) and the vertical spindle force time histories (as measured by the wheel hub dynamometer) were processed into cross-correlations. The vertical force has been selected as a reference since this component was the most significant from the 6 spindle force/moments. For those computations, 640 time lags and no windowing have been used. Afterwards, so called “half cross-spectra” were obtained by applying a single DFT to the positive time lags of the cross-correlations. This procedure is repeated for each of the measurement runs. Each run contains a different tyre surface response point (Figure 5), but the spindle forces as well. Figure 8 (Left) shows some of the calculated cross-spectra. More information about this particular pre-processing, which is very suited for OMA, can be found in [9][10]. Figure 8 (Right) shows the spindle force power spectra for all measurement runs. A tyre rolling over a discontinuity in the road surface will partially envelope this discontinuity, causing the spindle force to be

significantly smoother than the geometry of the road discontinuity. Despite this enveloping, a road discontinuity causes an unequal distribution of the excitation energy in the frequency domain [11][3]. At distinct frequencies, there is almost no excitation of the tyre. These frequencies are determined by the rolling speed, the length of the contact patch and the geometry of the road discontinuity. These frequencies of low excitation are clearly visible in the spectra of Figure 8 (Right). The different spectra show the same trend, but there are small variations in amplitude which are caused by variations in the tyre temperature between the different measurement runs. This time-invariance can provoke several errors in modal parameters identification.

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Figure 8: (Left) Some of the computed cross-spectra. (Right) Spindle force power-spectra for all runs.

4.3 PolyMAX parameter extraction In order to extract the modal parameters, all runs are analyzed simultaneously using the Operational PolyMAX method [10][12]. Although in the analysis only the cross spectra between tyre surface velocity and spindle force are used, it is assumed in such a global approach that the operational forces are the same for each run. These operational forces introduced by the cleat could not be measured, but at least when considering the spindle reaction forces, it is observed that these are not exactly the same for each run (Figure 8 – Right). The frequency range for the estimation is set from 0 to 445 Hz and the modal order is 45. As can be seen in Figure 9, this method yields quite some global estimates for the poles.

Figure 9: Operational PolyMAX stabilization diagram.

In Figure 10, the mode set identified here is compared to a mode set identified according to the pre-processing discussed in Section 3. Although both mode sets do not match perfectly, a good agreement is found between both processing ways. In general, the mode shapes look very plausible and clean.

Figure 10: Matching of poles and mode shapes.

4.4 Modal model validation The Modal Assurance Criterion (MAC) was used to validate the modal model. The MAC is a mathematical tool to compare two vectors: assuming that {X} and {Y} are two vectors with the same number of elements, MAC is defined as:

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If the modal assurance criterion is unity, then both vectors are perfectly identical within a scale factor. If the modal assurance criterion is zero, no linear relation exists between both vectors and the estimated modal scale factor has no meaning. Here, the modal assurance criterion is used as a tool to compare different sets of estimated mode shapes or to investigate the validity of the estimated modes within one set (also referred to as the AutoMAC). Figure 11 (Left) shows the AutoMAC, which indicates how much a single mode is self-independent. Ideally, the off-diagonal elements should be low, indicating that the sensor number and locations were well selected to distinguish the mode shapes from each other. Figure 11 (Right) shows the MAC between the classical throughput OMA (this section) and the reference analysis using synchronized, time-averaged signals (Section 3). Especially the lower modes agree very well. Some typical mode shapes are represented in Figure 12.

Figure 11: (Left) AutoMAC for throughput-data Operational Modal Analysis. MAC between classical throughput OMA and the reference analysis using synchronized, time-averaged signals.

Figure 12: Typical rotating tyre mode shapes obtained by applying operational PolyMAX to cleat excitation data.

5 SCALED CROSS-SPECTRA OPERATIONAL MODAL ANALYSIS Figure 8 reveals that the spindle force spectra are not equal for the different runs, which violates the basic assumption of a global analysis. In an attempt to overcome this problem, every response cross-spectra can be weighted relative to its respective force auto-spectrum. Under the assumption of system linearity, doubling the force also doubles the response. Thus, dividing by the force yields:

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Figure 13 shows some of the obtained scaled cross-spectra. The PolyMAX method applied to this set of cross-spectra yields very good estimates of the system poles, which are nearly identical to the poles obtained by the reference Operational Modal Analysis (see Figure 14). Despite this very good estimation of the system poles, the

estimated mode shapes look less clean than in the unscaled case (Section 4), whereas rather the opposite was expected. It is speculated that the vertical spindle force (which is used to scale the responses) may not be a good indication of the real forces that are injected into the system by the cleat.

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Figure 13: some scaled cross-spectra.

Figure 14: PolyMAX method for scaled cross-spectra set.

6 SPINDLE FORCE EXPERIMENTAL MODAL ANALYSIS Another approach has been investigated in which the vertical spindle force is used as an input force for a classical experimental modal analysis. This approach was inspired by the approach of previous section in which the tyre responses are scaled by the spindle forces. This is exactly what happens in a classical Frequency Response Function (FRF) estimation assuming that the spindle forces are representative for the real forces. Unfortunately, again the estimated mode shapes are not very clean, which indicate that the vertical spindle forces may not be a good indication of the real forces that are injected into the system by the cleat. The coherence function between vertical spindle forces and tyre surface velocities are shown in Figure 15. It can be seen that above the 160 Hz, the responses are not so well correlated to the chosen input force.

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Figure 15: Coherence functions between vertical spindle forces and tyre surface velocities.

7 CONCLUSIONS This paper investigated several possibilities to process data from a rotating tyre with the aim to extract the modal parameters in operational conditions. Following data processing methods can be distinguished:

1. Synchronized, time-averaged responses > auto- and cross spectra using response as reference > OMA: Section 3 and [3];

2. Synchronized, time-averaged responses > single-block DFT of each response (without references) > OMA: not discussed here, but also valid approach;

3. Raw time responses > auto- and cross spectra using vertical spindle force measured in each run as reference > OMA: Section 4;

4. Raw time responses > auto- and cross spectra using vertical spindle force measured in each run as reference > scale cross spectra by auto spectra of spindle forces > OMA: Section 5;

5. Raw time response > H1 estimator considering vertical spindle force as reference > EMA: Section 6. Essentially, the last two approaches (4 and 5) are identical, except for the fact that in the OMA processing as presented in [8][10], the correlogram approach is used to calculate auto- and cross-spectra, whereas in the traditional H1 FRF estimate, the periodogram approach is used. The scaling by the reaction force was an attempt to compensate for force differences that may exist between runs and also for reducing the non-system-related dips in the response spectra due to the specifics of the cleat excitation. This phenomenon seems to have some resemblance with the well-known “double impact” in impact testing (Figure 7). Despite the good hypothesis, these scaling approaches did not yield satisfactory results, leading to the speculation that the spindle forces may not be entirely representative for the (unmeasurable) forces injected into the tyre by the passing cleat. The unscaled approaches (1, 3, and also 2 although this method is not discussed in the paper) yielded high-quality operational modal parameters with realistic frequency and damping estimates and very clean mode shapes. The advantage of method 3 is that no special pre-processing is required to yield synchronized, time-averaged responses between different measurement runs.

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