On the sound radiation of a rolling tyre

Abstract

The sound radiation from rolling tyres is still not very well understood. Although details such as horn effect or directivity during rolling have been investigated, it is not clear which vibrational modes of the tyre structure are responsible for the radiated sound power. In this work an advanced tyre model based on Wave Guide Finite Elements is used in connection with a contact model validated in previous work. With these tools the tyre vibrations during rolling on an ISO surface are simulated. Starting from the calculated contact forces in time the amplitudes of the modes excited during rolling are determined as function of frequency. A boundary element model also validated in previous work is applied to predict the sound pressure level on a reference surface around a tyre placed on rigid ground as function of the modal composition of the tyre vibrations. Taking into account different modes when calculating the vibrational field as input into the boundary element calculations, it is possible to identify individual modes or groups of modes of special relevance for the radiated sound power. The results show that mainly low-order modes with relative low amplitudes but high radiation efficiency in the frequency range around 1 kHz are responsible for the radiated sound power at these frequencies, while those modes which are most strongly excited in that frequency range during rolling are irrelevant for the radiated sound power. This fact is very essential when focusing on the design of quieter tyres.

Introduction

This paper investigates the contribution of different vibration modes of a tyre to the sound radiation from a rolling tyre by means of a theoretical model. Tyre/road noise is the dominant source for passenger cars in ordinary road traffic at driving speeds above 30–50 km/h. While the reduction of tyre/road noise is essential in order to reduce the environmental impact of road traffic the progress in designing quieter tyres is rather moderate. One reason might be the conflict between different performance criteria for tyres, but a more severe problem certainly is the lack of understanding of the relevant generation and radiation mechanisms. This paper focus on one of the key question: which parts of the tyre vibration (expressed as modes and/or waves) are responsible for the sound radiation during rolling? From simple analysis of wave speed on tyre-like structures such as rings or plates, it was rather soon understood that the radiation efficiency of the free waves is too low to explain the radiated sound power from rolling tyres (see e.g. [1]).

In the literature one finds two ideas explaining which waves and/or modes on a rolling tyre are responsible for the radiation of sound. Kim and Bolton [2] suggested that fast waves, identified as in-plane waves, are potentially significant radiators at higher frequencies. They especially pointed out the wave involving a breathing motion at the cross-section plane as driving the radiation. However, a proof of their theory is missing, since for the radiation of sound from a rolling tyre, forced vibrations are relevant rather than the free vibrations on which their reasoning was based. Kropp et al. [3] investigated the radiation efficiency of modes on a cylinder and suggested that low-order modes are the main contributors to radiated tyre/road noise. In the case of a rolling tyre, the contact conditions lead to a distributed force and therefore to an excitation of low-order modes even at higher frequencies. Although their amplitudes might be much smaller than the amplitudes of free waves corresponding to frequencies where maximum sound power is radiated, the radiation efficiency of these modes is considerably higher at these frequencies.

Wullens and Kropp [4] used a three-dimensional contact model to calculate the contact forces for a rolling tyre and studied the resulting velocity field along the tread centre-line in Eulerian coordinates. The contribution of each rotating mode to the total radiated sound power was analyzed using a two-dimensional radiation model based on the equivalent source method. By considering the cumulative contribution of low-order modes, it was found that a limited number are enough to accurately predict the total radiated power. This would simply mean that it is the variation of the contact patch geometry over time, which leads to the radiated sound. One should bear in mind, however, that the results obtained in [4] were based on a radiation model, as well as tyre model, of simplified character. This especially refers to the tyre model, based on the assumption of a plate geometry. The model could hardly reproduce correct wave speed and the variety of modes observed on the tyre. Phenomena such as breathing modes are neglected.

There are a number of experimental attempts to correlate tyre properties such as driving point mobilities or material properties with measured sound pressure levels from rolling tyres. However, most of these studies suffer from the limited number of cases investigated. To get a reliable picture, a variety of tyres and road surfaces has to be considered, allowing for a statistical evaluation. Up to now only a few data sets exist which provide all necessary information required for such studies. One of these sets comes from a Dutch project [5]. These data were used to build a hybrid model for the prediction of Close-Proximity (CPX) levels [6]. In a recent study Beckenbauer and Kropp [7] also made a correlation analysis calculating partial correlation coefficients with the speed as control variable between point mobilities, damping and tread stiffness measured on 14 tyres from the data set and their CPX values measured on six different road surfaces at around 50 km/h. The point mobilities were measured on a freely suspended tyre in radial direction in the middle of the tread. Damping was measured from the half power band width for the first resonances in the measured point mobilities where individual modes could be identified (below 300–400 Hz).

While damping did not show significant correlation and tread stiffness showed correlation especially where the pitch frequency was located, the driving point mobility showed the somewhat surprising result displayed in Table 1. The point mobilities were represented by the third octave band levels SREY according to

$$\mathrm{SREY}=10{\log }_{10}\left({\int }_{f_1}^{f_2}\mathrm{Re}\left\{\frac{Y(f)}{Y_{\mathrm{ref}}}\right\}\mathrm{d}f\right)\mathrm{dB}.$$

Point mobility measurements at 250–400 Hz correlate with CPX levels at 800–1250 Hz. This means that changes observed in the point mobilities of the tyres at lower frequencies are linked to the radiated sound at higher frequencies. This fact was even more pronounced for rough surfaces like a surface dressing with 16 mm maximum aggregate size on asphalt as shown in the lower correlation matrix in Table 2. An interesting example is the third octave band 125 Hz where SREY correlates with the CPX levels at 315, 630, 1000, 1250 and 2000 Hz. Does this mean that modes in the third octave band of 125 influence the radiated sound pressure almost in the whole frequency range? An alternative, of course, could be that tyre parameters which determine the response in the 125 Hz band are also influencing the higher frequency range in a similar way.

This question can only be answered when one understands the modal composition of the vibrational field on a rolling tyre. Therefore the goal of this paper is to contributing to this understanding by utilising a theoretical model for the simulation of tyre/road noise. An appropriate first step was to gain a clear understanding of the types of waves that propagate on a typical tyre construction (not loaded and not rolling), along with their phase speeds. In a paper by Sabiniarz and Kropp [8] an in-house implementation of the model, which was originally presented in [9] (based on the Waveguide Finite Element Method), to study free wave propagation on a tyre that is stationary and not in contact with the ground. The results from this study are briefly summarized in Section 4. The main idea of the theoretical model was to use this very exact tyre model and combine it with a contact model. This model is mainly based on the work by Wullens [10]. The contact model delivers contact forces during rolling which then can be applied to the tyre model. In this work the forces are used to calculate the modal amplitudes on the tyre. Results from these calculations are also shown in Section 4. This allows us to synthesize the vibrational field on the tyre by selecting modes or groups of modes in order to investigate their influence on the sound radiation. The vibrational field is then transformed into the coordinate system fixed to the centre of the contact area. Based on the vibrational field in this coordinate system, the radiated sound can be calculated by a Boundary Element (BE) model as presented in [11]. With the model the complex pressure amplitudes on a reference surface around the tyre are calculated. This allows for superposition of different calculation results with different modal contents in the vibrational field. This methodology is used for a parameter study for a tyre rolling at 80 km/h on an ISO replica. The results from this study are presented and discussed in Section 5. In Section 2 the theory and models behind this procedure are presented. The analysis is carried out for one specific case which is briefly described in Section 3, for which also a validation of the simulations against measurements is presented.