Multimodal dispersive waves in a free rail: Numerical modeling and experimental investigation
Pan Zhang, Shaoguang Li, Alfredo Núñez, Zili Li
Mechanical systems and signal processing
In this paper, we present a solution method based on finite element (FE) modeling to predict multimodal dispersive waves in a free rail. As well as the modal behaviors and wavenumber-frequency dispersion relations, the phase and group velocities of six types of propagative waves are also derived and discussed in detail in the frequency range of 0-5 kHz. To experimentally distinguish different types of wave modes, the operating deflection shape (ODS) measurement approach is employed in the
... tory. ODS is measured from the spatial distribution of imaginary parts of the FRFs. We also propose a synchronized multiple-acceleration wavelet (SMAW) approach to experimentally study the propagation and dispersion characteristics of waves in a free rail. The group velocities in the vertical, longitudinal and lateral directions are estimated from the wavelet power spectra (WPSs). The good agreement between the simulation and measurement in terms of mode shapes and ODSs, wavenumber-frequency dispersion curves, and group velocities indicates that the ODS and SMAW approaches are capable of distinguishing different wave modes and measuring wave propagation and dispersion characteristics. In situ experimental results further demonstrate the effectiveness of the ODS measurement for coupled modal identification and the SMAW approach for wave dispersion analysis of the rail in a field track. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l oc a t e / y m s s p models assume a constant cross-section of the rail, which is only valid at low frequencies  . The 2.5D FE model and 3D FE model can both consider rail cross-section deformation, which is important to accurately predict the dispersion relations. Although the computation efficiency of the 2.5D FE model is higher because it treats the longitudinal direction analytically, the 3D FE model has the advantages that it is capable of modeling nonlinear wheel-rail dynamic contact, which is important when studying wheel-rail rolling noise and short pitch corrugation. Many papers [16,     have reported on the dispersion curves and phase and group velocities of rail waves in the ultrasound frequency range (i.e., 100 kHz) for long-range rail inspection. In the frequency range of rolling noise and short pitch corrugation (below 5 kHz), Thompson  and Gavric  predicted the dispersion curves, while the phase and group velocities were not presented. In this paper, we applied a 3D FE model to predict the dispersive waves in a free rail at 0-5 kHz and derived the phase and group velocities. Although many models have predicted the multiple wave modes in a free rail [9, 12, 16] , few experimental methods have been reported in the literature to distinguish different wave modes. One possible reason is that most researchers have focused only on the vertical rail vibration because the wheel-rail excitation mainly acts in this direction [5, 10, 13] . There is only one type of wave mode called the vertical bending mode in this direction; thus, the influence of modal coupling is insignificant. However, when studying three-dimensional rail vibrations, modal coupling of different directions may occur, which makes one natural frequency correspond to multiple rail modes. In this case, the widely used experimental method for the railway structure, frequency response function (FRF) measurement    , has difficulty identifying the coupled modes because it can only obtain the natural frequencies. Another experimental method named operating deflection shape (ODS) measurement can derive both the natural frequencies and the corresponding mode shapes. In the literature     , this method has been successfully applied in specific engineering structures such as beams and plates. The modes were uncoupled in these studies because the beams and plates have relatively simple cross-sections, and the research interest only focused on the vertical direction at frequencies lower than 1600 Hz. In this work, the rail has a more complicated arbitrary cross-section geometry, and our research interest is three-dimensional rail vibrations up to 5 kHz. Considerably more vibration modes (hundreds) need to be dealt with, and they may couple with each other. Coupled modes have dissimilar wavelengths that hinder the ability of sensors to capture them simultaneously. In this paper, the ODS measurement will be used to distinguish coupled wave modes in a free rail. In addition to the multiple modes, dispersion is another feature of the propagative waves in a rail that makes the experimental identification of waves more difficult [9, 12, 16] . For the experimental analysis of the propagation and dispersion of elastic waves, the wavelet transform (WT) has been proven to be an effective tool [18,      . Lanza  applied this timefrequency analysis method in railway tracks and obtained group velocity dispersion curves for the vertical, longitudinal, and lateral rail vibration modes at 1-7 kHz. Only qualitative agreement was achieved with Gavric's numerical simulation results  . The deviations between the measurement and simulation may have been caused by the different boundary conditions of the rail. In Gavric's simulation, the rail had a free-free boundary, while in Lanza's experiment, the rail was supported by steel pads on wooden sleepers, which introduced additional stiffness and damping to the rail. In addition, the accuracy of the single-acceleration wavelet approach used in Lanza's experiment might have been reduced by the wave reflection at the rail ends. To improve the experimental accuracy and achieve better agreement with the simulation, a synchronized multipleacceleration wavelet (SMAW) approach will be developed in this paper. The test boundary condition will also be improved by the experimental setup to better match the free-free boundary of the FE simulation. The purpose of this work is to gain a better understanding of free rail vibrations and provide experimental methods to distinguish different wave modes and measure wave propagation and dispersion characteristics. The paper is organized as follows. In Section 2, a 3D FE model is built to predict the multimodal dispersive waves in a free rail. The modal behavior, wavenumber-frequency dispersion relations, and phase and group velocities of these waves are derived. In Section 3, two methods-ODS measurement and SMAW measurement-are introduced for the experimental investigation of the multimodal dispersive waves. In Section 4, the experimental results, including ODSs, wavenumber-frequency dispersion curves, WPSs and group velocities, are obtained and compared with the simulations. Section 5 discusses the advantages and disadvantages of ODS measurements and SMAW measurements and the effectiveness of these two methods in field tracks. The main conclusions are drawn in Section 6.