Reconstruction of transient pressure and acceleration over a tire surface using the inverse time domain boundary element method

. The inverse time domain boundary element method (ITBEM) that is derived from the direct time domain boundary element method by eliminating the retarded time is able to reconstruct the transient pressure and ﬂux on the surface of an arbitrarily shaped source by measuring the pressure on a hologram surface. In the present work, the ITBEM is applied to reconstruct the transient pressure and acceleration over the surface of a tire which is supported away from the ground in a semi-anechoic chamber. The tire is impacted by a rigid sphere to generate a transient sound ﬁeld, and the measurement is controlled by a trigger which is connected to an acceleration sensor stuck on the surface of the tire. The pressure and acceleration on the surface of the tire are reconstructed from the holographic pressure measured by array microphones. By visualizing the pressure and acceleration with respect to the elapsed time, the wave propagation phenomenon of the pressure and acceleration on the surface of the tire is shown clearly. The comparison of the reconstructed surface acceleration to the measured one demonstrates the effectiveness of ITBEM for transient sound ﬁeld reconstruction.


Introduction
Nearfield acoustic holography (NAH) was first documented in the early 1980s [1,2] for the reconstruction of acoustic field based on the measurement of acoustic pressures on a hologram plane. The conventional NAH was carried out via Fourier acoustics and it achieved great success in analysing sources of planar, cylindrical, and spherical shapes [3]. In order to reconstruct the acoustic field radiated by arbitrarily shaped sources, the inverse boundary element method (IBEM) was proposed in the late 1980s [4,5]. Since then, a large number of studies on the IBEM-based NAH have emerged.
Use Boundary element method can not only analyse stationary problems in frequency domain but also can simulate transient problems in time domain, which is usually called time domain boundary element method (TBEM). Many researches on acoustic radiation and scattering problems have been conducted using the TBEM [6][7][8][9][10], and the widely known instability problem of TBEM has also been deeply studied [10][11][12][13]. Recently, an inverse TBEM (ITBEM) was developed for transient sound field reconstruction by Zhang et al. [14]. They combined the surface and field time domain integral equations and shifted the time axis of the field integral equation to rule out the retarded time, and then solved the combined equations implicitly in a marching-on-intime (MOT) way. In the ITBEM, the truncated singular value decomposition (TSVD) method was applied to suppress the measurement noise in order to acquire stable results.
In the present paper, the ITBEM is applied to reconstruct the transient pressure and acceleration over an impacted tire surface from the measured pressure. The reconstruction results are visualized with respect to the elapsed time. The effectiveness of ITBEM is then illustrated through the comparison of the measured and reconstructed accelerations.

Theory of ITBEM
The ITBEM is based on the time domain boundary integral equation which can be expressed as [15]    can be carried out numerically using standard Gaussian quadrature method. Finally, by assembling coefficients into corresponding global matrices, the following linear algebraic equations can be obtained The ITBEM formulations can be derived based on Eq. (2), and the detailed derivation can be found in the work of Zhang et al. [14]. For concision, the ITBEM formulations are directly given here

Experiment of a tire
To validate the proposed method, an experiment of a slick tire was conducted in a semi-anechoic chamber, as shown in Fig.1. The tire was supported away from the ground to get rid of the influence of the reflection. A steel sphere which was hung up as a simple pendulum was used to impact the tire for generating transient sound field. The sphere was controlled by a fixed electromagnet so that it was released at the same position at each measurement. An array of 7 microphones was used to scan the transient sound field, as shown in Fig.1(a). The measurement was triggered by an acceleration sensor which was stuck near the impact point on the surface of the tire so that all the measurement data was time-synchronized. To evaluate the reconstruction results, the acceleration on the surface of the tire was also measured, as shown in Fig.1(b).
The schematic setup of the experiment is shown in Fig.2 The spatial distributions of the measured pressure at variant instants are visualized in Fig.3. The first image in Fig.3 shows the sound pressure at the very beginning of the impact. The impact position indicated by the pressure is consistent with that in Fig.2 (b).  The pressure and acceleration on the surface of the tire are then reconstructed using ITBEM from the measured pressure. The numerical model of the tire is simplified to a cylinder with the same radius and width. A mesh of 1152 quadrilateral elements is adopted in the calculation. The time step size is chosen The spatial distributions of the reconstructed pressure and acceleration are given in Fig.4 and Fig.5, respectively. The reconstructed acceleration presents the vibration status of the tire surface after the impact. From the first image in Fig.5 the impact position can be identified directly. It can also be observed that the energy of the acceleration dissipates more quickly than that of the pressure, which exactly matches the common sense.   To evaluate the reconstructed results more precisely, the reconstructed acceleration at 4 points are compared with the corresponding measured values, as shown in Fig.6. It can be seen that the waveforms of the reconstructed acceleration agree well with the measured values, although the magnitudes of the reconstructed results are overestimated or underestimated at some temporal points. The over-or under-estimation may come from the numerical errors which are caused by the difference between the numerical model and the real tire. The reconstructed results demonstrate that the ITBEM can reconstruct transient sound field effectively.

Conclusions
In this paper, the ITBEM is validated through an experiment of a tire. The tire is impacted by a sphere to generate a transient sound field. By visualizing the measured pressure, the wave propagation phenomenon can be seen clearly. The pressure and acceleration on the surface of the tire are reconstructed through ITBEM. Moreover, the reconstructed acceleration is compared to the measured values. The agreement between them demonstrates that the ITBEM is effective in reconstructing transient sound field.