Experimental study of the use of a transfer function to find rail corrugation from axle-box accelerations

Experimental study of the use of a transfer function to find rail corrugation from axle-box accelerations Xinxin Yu a,d,* , Sergio Mu˜ noz b , Pedro Urda a , Javier F. Aceituno c , Miguel Rodríguez G´ omez a , Jos´ e L. Escalona a a Dept. of Mechanical and Manufacturing Engineering, University of Seville, Spain b Dept. of Materials and Transportation Engineering, University of Seville, Spain c Dept. of Mechanical and Mining Engineering, University of Ja´ en, Spain d Automation Technology and Mechanical Engineering, Faculty of Engineering and Natural Sciences, Tampere University, Finland ARTICLE INFO Keywords: Vehicle vibration Normalized acceleration Signal processing Track irregularity ABSTRACT This investigation uses a scale vehicle-track experimental facility to study the calculation of rail corrugation using vertical accelerations measured in the axle-box of rail vehicles and a transfer function (TF). The rail corrugated profile is machined in the rail heads of the scale track following a periodic function with four harmonics. Experiments are performed with a scale bogie-like vehicle at different forward velocities in the range inspection velocities. Two simple analytical forms of the TF are studied: the kinematic TF, that assumes that the axle box follows the rail profile, and the TF of a 2-dof model of the vehicle-track system. For the vehicle response analysis, this work proposes to normalize the measured acceleration with the square of the forward velocity of the vehicle, that is assumed to be approximately constant. This normalized acceleration reduces the effect of the forward velocity on the TF. Experimental results show that the kinematic TF can be used to measure the track corrugation for moderate forward velocities providing reasonable but not accurate results. The limitation of the kinematic TF is mainly due to free flights and wheel rail curvature incompatibility. The measured axle-box accelerations may include frequency peaks that are not excitation frequencies and can distort the rail profile measurement. Results show that linear elastic models like the assumed 2-dof model do not explain the appearance of these non-excitation peaks. 1. Introduction Rail corrugation is a wave-type wear along the rail with a range of wavelengths between 10 and 1000 mm [1]. It often appears in metro lines, urban railways, and high-speed railways, resulting in remarkable vibration and noise that affect the operating performance of rail vehicles. Corrugation formation is a complex process that is affected by the vehicle-track dynamic interactions [2]. Li et al. [3] proposed that longitudinal compression modes and corresponding longitudinal track dynamics are responsible for corrugation initiation, and this study further is validated by using the 1/5 scaled V-Track test rig [4]. Due to the short wavelengths and amplitudes, its measurement has been traditionally done using walking-speed trolleys, at around 1 m/s, pushed by human operators [5]. This technique is commonly known as a direct measuring method. Besides their low speed of operation, the direct measuring method are robust with high accuracy (about the order of microns) and can be used not only to demonstrate the severity of the corrugation but also to quantify the smoothness of the re-profiling process. However, some of their main disadvantages are the fact that they are required to stop the line traffic due to their limitations on speed and the fact that only one rail per passing is measured. That is why manufacturers, and the research community have deeply worked through the last decades in the development of techniques that can be used onboard at commercial velocities. The techniques, which are commonly known as indirect measuring methods, make use of measurements of noise [5], imaging processing algorithms [6] and axle-box acceleration (ABA) measurements. In [7,8] it was proposed that the corrugation of the track can be obtained from the measurement of environmental noise, making it a convenient way for the indirect measurement of rail corrugation. In the work of Liu et al. * Corresponding author at: Dept. of Mechanical and Manufacturing Engineering, University of Seville, Spain. E-mail addresses: [email protected], [email protected] (X. Yu), [email protected] (S. Mu˜ noz), [email protected] (P. Urda), [email protected] (J.F. Aceituno), [email protected] (M.R. G´ omez), [email protected] (J.L. Escalona). Contents lists available at ScienceDirect Measurement journal homepage: www.elsevier.com/locate/measurement https://doi.org/10.1016/j.measurement.2025.117058 Received 11 October 2024; Received in revised form 30 January 2025; Accepted 18 February 2025 Measurement 249 (2025) 117058 Available online 20 February 2025 0263-2241/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). [7], an indirect method is proposed that employs wavelet packet decomposition (WPD) to analyze the energy levels of wheel-rail noise, which is captured by a microphone mounted on the bogie. Wei et al. [8] developed an inspection algorithm based on interior noise to identify the position, characteristic wavelength, and severity of rail corrugation. The method utilizes the measurement data from a microphone installed on the carriage floor of high-speed railway (HSR) trains and ABA. Computer vision technique is one type of noncontact technologies that can be used for the track quality inspections. Chen et al. [9] combined the traditional chord with the laser sensor and cameras to obtain a precise measurement of rail corrugation. However, their system can only run at the maximum speed of 6 km/h. A computer vision-based identification approach is provided in [6] to assess the severity of rail corrugation. Gazafrudi [10] developed a high-speed rail corrugation measurement system based on laser triangulation principle and image processing techniques. But the measurement system in [10] has only been tested on the manufactured corrugated rails by CNC with a constant wavelength of 50 mm and amplitudes from 0.01 mm to 0.1 mm. Furthermore, Lee et al. [11] estimated the corrugation wavelength on rail heads using the computer vision technique and a feature descriptor approach. ABA measurement has the advantage of low cost, easy maintenance, and commercial speeds. In this technology, the acceleration of the axle box is assumed to be equal to the acceleration of a rigid wheelset. Bocciolone et al. [12] studied the rail status based on the correlation between the root mean square (RMS) of the ABA measurements and the corrugation. Similar research was conducted by Tanaka et al. [13] where the detection accuracy between the leading and trailing ABA measurements is compared using a corrugation-ABA measurement correlation. In addition, Salvador et al. [14] suggested the optimal sampling and filtering frequencies and the location of accelerometers on the axle-box for the detection of corrugation and vibration modes. Hassanieh et al [15] developed a machine learning (ML) model to create the moving RMS of the corrugation using the ABA signals measured from a commercial train. The results show that the estimation based on the ML model [15] is more accurate when comparing the one based on the model-based TF [16]. A time-domain approach is developed in [17] to estimate the corrugation using a wheel-rail interaction model with the input of ABA signals. A one-dimensional convolution neural [18] and a data-driven method [18,19] using a vehicle-track coupling simulation model and optimization approaches, are employed to estimate the corrugation wavelength and depth from the ABA signals. Still, the research in [17–19] is carried out under ideal conditions in a simulated environment. If the wavelength of a track defect is relatively long, ranging from 3 to 200 m, it is typically classified as track irregularity. Track irregularities can be measured through both contact and non-contact methods. Contact methods include approaches such as walking-speed trolleys, while non-contact methods involve the use of inspection trains equipped with lasers and cameras. Detailed reviews of track irregularity measurement techniques are provided in [20]. The following paragraph introduces recent developments in the estimation of track irregularities. The Kalman filter (KF) algorithm is used in [21] to identify track irregularities on railway bridges using vehicle acceleration as inputs. The results shows that the developed KF can produce accurate results even in the presence of measurement noise, non-stationary vehicle operating conditions, parameter uncertainties, and boundary spring effects. Strano et al. [22] estimated the lateral wheel-rail contact forces and track irregularities using a Central Difference Kalman Filter with the velocity and acceleration measurements of the bogie frame as inputs. Similarly, an unknown input observer is constructed to estimate the track irregularities [23] using a vehicle suspension model with multi-sensor acceleration measurements as input. However, the model-based approach developed in [23-5 are only demonstrated under the simulation environment. The dynamic interaction between the vehicle and track is complex, making the relationship between track irregularities and vehicle response equally intricate. Lei et al. [24] examines the effect of the spatial coherence of track irregularities on railway vehicle dynamics using a multibody simulation model, revealing that the coherence effect decreases as the irregularity’s wavelength shortens, thereby justifying the assumption of incoherent excitations at high frequencies. Furthermore, Karis et al [25] studied the correlation between vehicle response and track irregularities using simulations and measurements from a passenger car. However, a strong correlation is only observed in the simulation environment, indicating that the real-world track stiffness affects the ABA dynamics. The study of [26] introduces a simple and efficient prediction scheme for train-induced ground and building vibrations. The vibrations are predicted in the frequency domain by considering three key stages: ’emission’ (excitation from railway traffic), ’transmission’ (wave propagation through the soil), and ’immission’ (transfer to buildings). Xu and Zhai [27] examined the impact of spatial variability in the geometry, physical, and mechanical properties of railway tracks on train-track interaction. Simulation results show that the spatial variation of track parameters significantly affects wheel-rail interaction and track vibrations, highlighting the importance of considering these uncertainties in train-track dynamic analysis. A novel drive-by system is proposed by [28] for high-speed railways to detect resonant bridges using the difference of the measured track irregularities between the first and last vehicles of a train. Numerical and experimental results demonstrate that this method can accurately detect resonant bridges with spans ranging from 20 to 60 m. Xu and Liu [29] propose a coupled sleeper-vehicle-track model to analyze the impact of track geometry and irregularities on wheel-rail interaction. The simulation results indicate that the variation in rail geometry at the crossing section has a more significant effect on wheel-rail interaction than rail irregularities. This phenomenon is further validated using the multibody dynamics programs VI-Rail and Simpack by [30]. The TF or frequency response function (FRF) is commonly used in structural dynamics to characterize the dynamic response of a physical system in the frequency domain [31]. It defines the relationship between input (such as loads or excitations) and output (such as dynamic responses in terms of displacement, velocity, or acceleration). The relationship between the ABA and the rail corrugation can be considered as a single input–single output system and the TF is assumed to have the capability to detect corrugation. Based on the ABA spectrum and the TF of the wheel-rail system, Liu et al. [32] developed a methodology to determine the maintenance limit for rail corrugation. However, the proposed estimation of the maintenance limit is restricted to the frequency band of interest and the track type. The research question of this paper is the use the TF for the derivation of rail corrugation using the ABA acceleration measurements as the inputs. The objective of the paper is not to find an accurate TF associated with the vehicle and track used in the experiments. As it will be shown in the content of the paper, the calculation of the specific TF of the problem at hand is not needed to check if the concept of the TF, that is associated with linear response at the operation conditions, works reasonably well when measuring corrugation. Two simple approaches, the kinematic TF and the TF of a 2-dof model, are compared. In the experiments, a 1:10 scaled experimental track and vehicle are used. The corrugation is machined on the rail head and measured with laser to work with a wellknown excitation. Results obtained under laboratory conditions cannot be directly extended to real scale vehicles and track, but they can be helpful for the development of a detection procedure for rail corrugation based on ABA signals. This paper is organized as follows. Section 2 shows the foundation of the two simple TFs considered in this work. Section 3 provides the details of the experimental setup: the track, the vehicle and the machined corrugation profile. Section 4 shows the experimental results and includes a discussion about the use of the TFs for the measurement of corrugation. Summary and conclusions are provided in Section 5. X. Yu et al. Measurement 249 (2025) 117058 2 2. Fundamentals of the acceleration to path-profile transfer function 2.1. Kinematic transfer function The simplest model that can be used to find the TF considers a rigid wheel moving with a constant forward velocity V on an irregular profile, as shown in Fig. 1. This model is called here the kinematic model. The vertical position z w (t) of the rigid wheel is given by: zw(t) = u(s)|s=Vt +R,(1) where s is the arc-length coordinate along the track, u(s)is the rail profile and R is the radius of the wheel. Equation (1) means that the trajectory of the center of the wheel is a translated copy of the rail profile. For this assumption to be true, the fulfilment of the rigid body assumption is not enough. Two other conditions must apply: There are no free flights of the wheel. The wheel stays in contact with the rail. In other words, the contact force is always compressive. There is curvature compatibility between the wheel and the rail profile. In other words, the curvature of the wheel is larger than the curvature of the rail profile, 1/R≥uʹʹ, where uʹʹ is the second spacederivative of the rail profile, or an approximation of the profile curvature. If the model applies, the acceleration of the center of the wheel can be obtained by differentiating Eq. (1) using the chain rule, as follows: ¨ zw=V2uʹʹ,(2) where the ‘dot’ represents time derivative and ‘prima’ the space derivative. Taking the Fourier transform of both sides of Eq. (1) shows that Zw(f) = U(f), where f is time frequency, in cycles/s, Zw(f)is the Fourier transform of zw(t), and U(f)is the Fourier transform of u(Vt). This equality is valid for all frequencies except at f =0, due to the distance R between the parallel trajectories. The Fourier transform of the vertical acceleration of the wheel, ¨ Zw(f),is related to the Fourier transform of the displacement by ¨ Zw(f) = − (2 π f)2Zw(f). Using this relationship, the following TF is obtained: Tkin ABA(f) = ¨ Zw(f) U(f)= − (2 π f)2,(3) where this function is called here kinematic transfer function from acceleration to profile. TFs are usually defined as the transform of the input (profile) divided by the transform of the output (acceleration). However, in Eq. (3) the definition is taken as output divided by input. Without losing generality, this definition is used throughout this paper for convenience, because the resulting expressions are simpler and more familiar in the context of the theory of mechanical vibrations. Because the profile is a space function, it makes more sense to turn it into a function of the space frequency f, in cycles/m. This is very simple because f=Vf. Substituting yields: Tkin ABA(f) = − (2 π Vf)2.(4) Obviously, the TF depends on the forward velocity of the wheel V. If the measured ABA acceleration is normalized by the square of the forward velocity, the kinematic TF becomes velocity independent, as follows: anor = ¨ zw V2=uʹʹ⟹T kin nor(f) = Anor(f) U(f)= − (2 π f)2,(5) where anor is the normalized acceleration, with units of curvature (m −1 ), and Anor its Fourier transform. 2.2. Simple dynamic transfer function The kinematic model is not generally valid because in the wheel-rail systems there are many sources of flexibility: 1. The local elasticity in the wheel-rail interface, sometimes called Hertzian stiffness [33]. 2. The flexibility of the bearing in the axle-box. 3. The structural flexibility of the rail. 4. The structural flexibility of the wheel. The simplest model of the wheel-rail system that accounts for flexibility is the 2-dof dynamic model shown in Fig. 2. The vertical displacement of the center of the wheelset is zw, and mw,cwand kw are its generalized mass, damping and stiffness constants. The vertical displacement of the rail section under the wheelset is zr, and mr,crand kr are its generalized mass, damping and stiffness constants. The equations of motion of the system about the vertical static equilibrium position are given by: [mw0 0mr][¨ zw ¨ zr]+[cw−cw −cwcw+cr][˙ zw ˙ zr]+[kw−kw −kwkw+kr][zw zr] =[kwu+cw ˙ u −kwu−cw ˙ u]⇒M¨ q+C˙ q+Kq =F(t)(6) where m, c, and k are the mass, equivalent viscous damping coefficient and stiffness constant of the system, respectively. Representing the track dynamics using concentrated elements (point mass, dashpot, spring) that run with the vehicle is a common modeling technique in railroad dynamics, as done for example in the model used in the Manchester Benchmark [34]. The research group of the authors of this paper has developed a computational methodology [35], called Moving Modes Method, that can be used to obtain the “running” constants associated with the track flexibility using a detailed finite element model. As it is generally the case, assume that kw≫kr. Let us define the stiffness ratio N=kw/kr≫1. Assume also that the ratio of damping constants fulfils N=cw/cr. The mass ratio is defined as: α =mr/mw. Note that in N the rail constant is in the denominator while in α the wheelset constant in in the denominator. This is done on purpose because kr and mw are easier to measure than kw and mr, respectively. The assumption that N=cw/cr implies stiffness-proportional damping. In this case, the eigenvalue analysis based on the mass and stiffness matrices provide the natural frequencies and modes of vibration, Fig. 1. (a) Kinematic wheel-track model. (b) Curvature incompatibility. Fig. 2. Two-dof vehicle-track model. X. Yu et al. Measurement 249 (2025) 117058 3 yielding: K− ω 2M=0⇒ ω = ω n1, ω n2 [K− ω ni2M]ϕ=0⇒ϕ=ϕ1,ϕ2⇒ Φ = [ϕ1ϕ2],(7) where ω ni and ϕi, i=1,2, are the natural frequencies and modes of vibration and Φ is the modal matrix. These equations can be solved symbolically, yielding: ω n1≅ kr mw(1+ α ) √, ω n2≅ Nkr(1+ α ) α mw √,Φ ≅⎡ ⎢ ⎣1+1 N(1+ α )− α (1−1 N(1+ α )) 1 1 ⎤ ⎥ ⎦,(8) where the assumption N≫1 has been used for the simplification. For the standard modal transformation, the new set of modal coordinates p= [p1p2]T, such that q=Φp, are substituted into the equations of motion, yielding: ΦTMΦ¨ p+ΦTCΦ˙ p+ΦTKΦp=ΦTF(t)⇒m¨ p+c˙ p+kp =f(t),(9) where the modal mass, damping and stiffness matrices and the modal force are given by: m≅[mw(1+ α )0 0 α mw(1+ α )],c≅[cr0 0Ncr(1+ α )2],k ≅[kr0 0Nkr(1+ α )2],f(t) ≅ ⎡ ⎢ ⎣ 1 N(1+ α ) − (1+ α ) ⎤ ⎥ ⎦(kwu+cw ˙ u).(10) The uncoupled equations of motion in terms of the modal coordinates are given by: mw(1+ α )¨ p1+cr ˙ p1+krp1=1 (1+ α )(kru+cr ˙ u), mr ¨ p2+cw(1+ α )˙ p2+kw(1+ α )p2= − (kwu+cw ˙ u).(11) As shown in Fig. 3, the system behaves at a modal level as two suspended vehicles moving on irregular tracks. The mass, suspension properties and level of irregularities that these vehicles “see” can be observed in the figure. Taking the Fourier transform of these equations and reorganizing yields: P1( ω ) = (kr+i ω cr) kr− ω 2mw(1+ α )+i ω cr 1 (1+ α )U( ω ) = 1 (1+ α )T1( ω )U( ω ), P2( ω ) = − (kw+i ω cw) kw(1+ α )− ω 2mr+i ω cw(1+ α )U( ω ) = − 1 (1+ α )T2( ω )U( ω ), (12) where i= −1 √, P1( ω ), P2( ω ), U( ω )are the Fourier transforms of p1(t), p2(t), u(s/V), respectively, and T1( ω )and T2( ω )are modal transmissibility functions. The transmissibility functions can be rewritten using the usual non-dimensional parameters in the theory of vibrations, as follows: T1( ω ) = (1+2iξ1 τ 1) 1− τ 12+2iξ1 τ 1 ,T2( ω ) = (1+2iξ2 τ 2) 1− τ 22+2iξ2 τ 2 ,(13) where ξ1=cr/(2 ω n1mw(1+ α )), ξ2= (1+ α )cw/(2 ω n2mr)are the damping factors associated with each mode, and τ 1= ω / ω n1, τ 2= ω / ω n2 are the non-dimensional track irregularity frequencies. Using the modal transformations from Eq. (8), the Fourier transform of the original coordinates yields: [Zw( ω ) Zr( ω )]=Φ[P1( ω ) P2( ω )] =⎡ ⎢ ⎣(1+1 N(1+ α ))P1( ω )− α (1−1 N(1+ α ))P2( ω ) P1( ω )+P2( ω ) ⎤ ⎥ ⎦ ≅[P1( ω )− α P2( ω ) P1( ω )+P2( ω )].(14) Substituting Eq. (12) into Eq. (14) yields: Zw( ω ) ≅ (1 (1+ α )T1( ω )+ α (1+ α )T2( ω ))U( ω ) = Tw( ω )U( ω ).(15) That means that the transmissibility form track corrugation to axle-box displacement, Tw( ω ), is a linear combination of T1( ω )and T2( ω ). When the vehicle goes very slow, the frequency of the excitation ω tend to zero and both, T1( ω )and T2( ω ), tend to one, therefore: V⟶0⟹ ω ⟶0⟹T1( ω )⟶1,T2( ω )⟶1⟹Tw( ω )⟶1 (1+ α )+ α (1+ α ) =1. (16) If the corrugation to axle-box transmissibility tends to one, the axle-box perfectly follows the rail corrugation for low forward velocity V, as expected. The dynamic transfer function from rail corrugation to axle-box normalized acceleration, that is defined as ¨ zw(t)/V2, is given by Tdyn nor (f) = 1 (1+ α )Tnor,1(f)+ α (1+ α )Tnor,2(f),Tnor,1(f) = − (2 π f)2(1+2iξ1 τ 1) 1− τ 12+2iξ1 τ 1 ,Tnor,2(f) = − (2 π f)2(1+2iξ2 τ 2) 1− τ 22+2iξ2 τ 2 , (17) The dynamic TF from rail corrugation to axle-box acceleration is given by Tdyn ABA(f) = V2Tdyn nor (f).(18) Apart from speed V, the dynamic TF based on a 2-dof system depends on five parameters: the natural frequencies fn1 and fn2, the damping factors Fig. 3. Modal response of two-dof vehicle-track model. X. Yu et al. Measurement 249 (2025) 117058 4 ξ1 and ξ2, and the mass ratio α . Assuming the values: fn1=332 Hz, fn2= 1035 Hz, ξ1=0.0295, ξ2=0.0173, α =0.11, Fig. 4 shows Tdyn ABA(f) (top) and Tdyn nor (f)(bottom) for different forward velocities of the wheelset ranging from 0.5 to 2.5 m/s. Section 3.2 will show that the assumed valued for the five parameters are actually the experimentally identified values of the wheel-rail system used in this paper. The plot on top is the TF based on the ABA acceleration and the plot at the bottom is the TF based on the normalized acceleration. As is can be observed, the use of a normalized acceleration is still convenient in the dynamic case because for low frequencies all the TF’s collapse to the kinematic one, that is a straight line with slope 2 when plotted in logarithmic scale, as follows: log(Tkin nor)=2[log(f)+log(2 π )],(19) The resonance peaks move to smaller space frequencies when the forward velocity increases. It can be interpreted that, the lower the forward velocity V, the wider is the range of space frequencies where the kinematic model is valid. In practice, this means that the lower the inspection velocity the wider is the range of measurable corrugation wavelengths avoiding resonance effects, that clearly introduce uncertainty into the measurement. If the wavelength of the track defect is relatively long (3––200 m), then they are called “irregularity” instead of “corrugation”. Contrary to the problem of resonance effects, measuring irregularities using ABA at low inspection velocity may have a sensitivity problem, because the vertical acceleration induced in the axle-box may be too small. More complex and detailed dynamic model of the wheel-track system can be developed to yield a more accurate TF that in turn depends on a larger set of parameters. Anyway, the concept of the TF works only if the system dynamics is linear. The purpose of this paper is to check the validity of the TF experimentally using a scaled vehicle-track system with corrugation that is described in next section. 3. Experimental setup 3.1. Vehicle-track system The 1:10 scaled track is located on the roof of the School of Engineering at the University of Seville [36]. It is 90 m long and is formed by straight segments, two curves with 24 m and 6 m radii, and transition segments, as shown in the plan view of Fig. 5 (a). The rails are manufactured using rectangular steel beams, where a scaled version of the UIC-54 rail profile has been machined just in the rail head area. Fig. 5 (b) shows the geometry of the centreline of the track and the location of the corrugated segments. The scaled vehicle, shown in Fig. 6, is a bogie with two rigid wheelsets. The primary suspension includes eight helical springs connecting both wheelsets with the bogie frame. The vehicle instrumentation includes: Two piezoelectric accelerometers (brand is PCB Piezotronics and reference is 352C33 with 50 g-range) installed on the axle-boxes in the front wheelset and a third one in the central part of the bogie frame. All accelerometers measure vertical accelerations. A high precision encoder (brand is Kubler and reference is 05.2400.1122.0360) that registers the rotation of the front wheelset. A data acquisition system (brand is National Instruments and reference is NI myRio) with acquisition rate of 5 KHz. The vehicle is driven with a Maxon electric motor using conical gears in the transmission. 3.2. Experimental modal analysis The experimental modal analysis of the wheelset on the track was performed as observed in Fig. 7. An impact hammer was used in the tests. Two accelerometers were used. One was installed on the axle-box while the second was installed in the foot of the rail. Locations were selected to measure the accelerations associated with the 2-dof model shown in Fig. 2. Fig. 8 shows the measured accelerations. The accelerations are approximately a tw0-harmonic signal, where the highfrequency harmonic, with an approximate frequency of 1 MHz, dies out approximately at 8 ms. Afterwards, only the low frequency harmonic, with an approximate frequency of 300 Hz, remains. The high frequency has little effect on the acceleration of the rail. It can also be observed that for the low frequency vibration the axle-box and the rail vibrate in phase, with higher amplitude on the axle-box. However, for the high frequency harmonic, vibration happens at approximately 180◦ of phase difference and much higher amplitude on the axle-box. All these remarks are consistent with the 2-dof model presented in Section 2.2 and its analytical modal analysis. Fig. 4. Top: dynamic TF based on acceleration. Bottom: dynamic TF based on normalized acceleration. X. Yu et al. Measurement 249 (2025) 117058 5 Fig. 9 shows the receptance of the axle-box obtained with the power spectral densities of the contact force at the impact hammer and the acceleration at the axle-box and their cross-spectral density. The curve is fitted to the results of the modal analysis of the 2-dof model shown in Section 2.2. The modal properties, needed to find the transmissibility functions T1( ω )and T2( ω )given in Eq. (13) are: ω n1=2 π ×332rad/s, ω n2=2 π ×1035rad/s,ξ1=0.0295,ξ2=0.0173 Φ=[1.5787 −2.1438 1 1 ].(20) Numbers given in Eq. (20) coincide with those used to plot the TFs in Fig. 4. Therefore, the TFs shown in Fig. 4 are not just an example, but the actual functions associated with the scale wheel-track system used in this research according to the experimental modal analysis. In railway dynamics, the P2 frequency is the lowest natural frequency of the vertical vibration of a wheelset running on the track. An approximation of this frequency can be obtained assuming a simple mass-spring system in which the wheelset mass and vertical stiffness of the track are used to set the model parameters. The natural frequency ω n1, whose analytical formula is given in Eq. (8) and has an experimentally measured value of 332Hz, can be considered as the P2 frequency of the scaled track. Considering that in real tracks this frequency lies in the interval 30 – 100 Hz, it can be concluded that the scale track is, in relative terms, much stiffer than a real track [37]. 3.3. Corrugated rails Four rail segments of 1.8 m, which are installed two on the left side and two on the right side, have been machined to create a corrugated railhead profile. The corrugated area covers more than 3.6 m because the starting and end points in the left and right sides do not coincide. Therefore, there are areas at the entrance and exit with corrugation only in one side and a central area with both rails corrugated. The corrugation profile is built by adding four harmonic functions without phase difference, as follows: zr(s) = ∑ 4 i=1 Bisin(2 π λi s),(21) where the amplitudes Bi and the wavelengths λi are given in Tab 1. According to the standard EN-13231–2 [38], and considering the scale, the selected wavelength lies within the ranges 30–100 mm and 100–300 mm. However, the amplitudes Bi are not scaled but exaggerated due to the difficulty to machine a profile with amplitude of just a few microns. Machining was done with a CNC model LAGUN L-650 with a spherical 2-mm end-milling cutter. This machine, shown inf Fig. 10 has a Fig. 5. (a): Plan view of the scaled track: aerial photograph and (b) scheme of the track centre line. Fig. 6. Scaled vehicle: a) left view, b) right view. X. Yu et al. Measurement 249 (2025) 117058 6 Fig. 7. Experimental modal analysis of wheel on track. Fig. 8. Accelerations measured with experimental modal analysis. Fig. 9. Axle-box receptance measured with experimental modal analysis. X. Yu et al. Measurement 249 (2025) 117058 7 1-µm precision in Cartesian displacement and a total operating length of 600 mm. That means that machining the 1.8 rail segments requires 3 phases, with the resulting inaccuracies at the connecting sections. Including the time required for calibration and alignment, a period of 61 h was used in the machining process, being 2 h and 40 min the time required to machine each 600 mm-segment. The installed corrugated rails, shown in Fig. 11, are located at the track distance s =54.9 m in the left side and at the track distance s = 55.6 m in the right side. After machining, the railhead profiles were measured with a laser profilometer. Fig. 12 shows the left and right rail profiles along 1.8 m. It can be observed that: •In the first 0.8 m the left rail is corrugated, but the right one is not. •Corrugation, this is, short wavelength irregularity, appears superimposed over a longer wavelength irregularity, as it happens in real tracks. •The machined corrugation is not perfectly periodic, as expected. Fig. 13 shows the measured vertical profile of the right rail after detrending, that is the one that will be used in the results presented in this paper. Fig. 14 shows the spectra of this vertical profile. The top plot of Fig. 14 shows the RMS spectrum, and the bottom plot shows the PSD. Both spectra are shown because both will be used in the calculation of the TF in later sections. Peaks at the design wavelengths (5, 10, 20, 30 mm) are clearly observed. The peak values of the RMS spectra coincide with the values given in the fourth row of Table 1 divided by  2 √. To analyse the results that will be shown in the next section, it is important to observe that the profile function given in Eq. (21) can be considered as a periodic function whose period is the least common multiple of the wavelengths given in the third row of Table 1, this is, λlcm =60mm. Therefore, the profile function can be written as a Fourier series, as follows: zr(s) = a0+∑ ∞ i=1 aisin(i2 π f0s),(22) being f0=1 λlcm =16,61/m the fundamental period, and a2=B1,a3= B2,a6=B3,a12 =B4,and ai=0 for all other i. 3.4. Free flights and curve compatibility In the experiments made in this investigation the vehicle travelled with approximate forward velocities of 0.5, 1.0, 1.5, 2.0 and 3.0 m/s. If the kinematic wheel-rail model shown in Fig. 1 applies, a simple force balance shows that the wheel rail normal contact force is given by: Fc=m(g+¨ zw).(24) Therefore, the normal contact force will tend to change sign whenever the vertical acceleration of the wheel is negative and equal in norm to the acceleration of gravity. Using the kinematic assumption, and considering the equation of the profile given in Eq. (21), the vertical acceleration of the wheel yields: ¨ zw(t) = ∑ 4 i=1−Bi(2 π λi V)2 sin(2 π λi Vt).(25) Therefore, the amplitude of the vertical acceleration associated with each of the four harmonics is given by Bi(2 π λiV)2 ,i=1,2,3,4. A simple force balance shows that the wheelset separates from the track when the amplitude of the inertia force equals the static force transmitted by the wheelset, this is, its own weight plus half the weight of the rest of the vehicle, as follows: Fig. 10. (a): CNC machine model LAGUN L-650. (b): Machining of a corrugated segment. Fig. 11. (a): Installation of left corrugated rail on the scaled track. (b): Detail of corrugated rail. X. Yu et al. Measurement 249 (2025) 117058 8 munBi(2 π λi V)2 =(mun +msp 2)g⟹Bi(2 π λi V)2 =(1+msp 2mun)g=2.63g, (26) where mun is the unsprung mass of the vehicle (1.964 kg, mass of the wheelset, axle-boxes and bearings) and msp is the sprung mass (6.397 kg, mass of the rest of the vehicle). Fig. 15 shows the value of the amplitudes Fig. 12. Corrugated rail profiles measured with laser profilometer. Fig. 13. Right rail vertical profile after detrending. Fig. 14. Right rail spectra. Top: RMS spectrum. Botton: PSD. Table 1 Design and real properties of periodic corrugated profiles. Wave Number 1 2 3 4 Space frequency 1/λ (cycles/m) 33 50 100 200 Wavelength λ (mm) 30 20 10 5 Amplitude B ( μ m) 44.7 51.6 25.6 24.7 Maximum curvature (1/m) 2.0 5.1 10.1 39.0 X. Yu et al. Measurement 249 (2025) 117058 9 TF and its application. Machining in the rail head a profile that can be considered as Gaussian white noise would have been more appropriate to study the TF. However, this type of profile would be totally different to the corrugated profiles in real track. The solution adopted in this investigation can be considered as a trade-off solution. The studied TFs are the simplest possible functions. The kinematic TF assumes that the axle box of the wheel, where the accelerometer is installed, follows a trajectory that is a translated copy of the railhead profile. This TF is valid if the wheel-axle box system behaves as a rigid body in the vertical direction, the wheel keeps contact with the rail (no free flights) and there is no curvature incompatibility. If the vehicle behaves as a deformable 2-dof system, the TF takes an analytical form that depends on a few parameters that can be identified experimentally. Experimental modal analysis has been used in this work to find the appropriate parameters of the 2-dof model. The calculation of the TF using the 2-dof model, being very simple, can be used to explain resonance effects in the measurements. A simple rule to avoid resonance effects in the measurements of corrugation using ABA is to keep the inspection velocity V as low as possible. The analytical forms of the mentioned TFs, and the linear transformation of space-frequencies into time-frequencies when the vehicle moves with different forward velocities V, suggests that normalization of the measured accelerations with V can help in the interpretation of the results. This paper suggests processing a normalized acceleration that is obtained dividing the measured acceleration by the square of the forward velocity, that is assumed to be approximately constant. If this normalized acceleration is used, the kinematic TF is forward velocity independent and the 2-dof TF too, but just for frequencies below the resonance peak. This resonance peak moves forward in the frequency axis linearly with the inverse of the forward velocity. The experimental results shown in this paper confirms the benefits of processing the normalized accelerations to reduce the effect of the forward velocity used during the measurements. Experimental results show that the measured axle box acceleration increases with the forward velocity, but the normalized acceleration decreases with the forward velocity. The analysis of the measured accelerations in frequency domain shows that the axle-box vertical motion has the same frequency content than the rail profile, but it also includes peaks at frequencies that are multiples of the fundamental frequency of the rail profile but are not excitation frequencies. This phenomenon has not been explained. One possible explanation, not checked in this investigation, is that this phenomenon is the result of nonlinear dynamics effects due to free flights and curve incompatibility of the wheel rail relative motion. Although the experimental conditions are not favourable, the TF has been attempted to be obtained by signal processing of the input (rail profile) and the output (axle box acceleration). The kinematic TF approximates well the experimentally obtained TF at the excitation frequencies, if frequencies are not too high so free flights and curvature incompatibility do not appear. The H1 approximations of the TF are very noisy, because the selected excitation is not appropriate for this type of estimation. The parabolic form of the kinematic TF can be slightly observed. Using this H1 approximation, it has been shown that the high peaks of the accelerations at non-excitation frequencies are probably not due to resonance of the vehicle system. These results suggest that more complex TFs based on more advanced but linear models of the vehicle would not help to improve the measurements. The rail profile has been reconstructed using the axle-box accelerations and the kinematic TF. Results show that for relatively low and relatively low space frequencies the reconstructed results provide inaccurate but reasonable representation of the real rail profile. CRediT authorship contribution statement Xinxin Yu: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Investigation, Formal analysis. Sergio Mu˜ noz: Investigation, Funding acquisition, Conceptualization. Pedro Urda: Resources, Investigation, Data curation. Javier F. Aceituno: Writing – review & editing, Resources, Project administration, Investigation, Funding acquisition. Miguel Rodríguez G´ omez: Writing – review & editing, Validation, Data curation. Jos´ e L. Escalona: Writing – original draft, Visualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research is supported by the Spanish Department of Economy, Science, Enterprise and University of the Andalusian Regional Government, under the PAIDI 2020 program with project reference P18-RT1772. 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