Effects of wheel passing frequency on wear-type corrugations

Wear 265 (2008) 1202–1211

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Effects of wheel passing frequency on wear-type corrugations P.A. Meehan ∗ , W.J.T. Daniel CRC for Railway Engineering and Technologies, Mechanical Engineering, University of Queensland, Queensland, Australia

a r t i c l e

i n f o

Article history: Accepted 3 March 2008 Available online 4 June 2008 Keywords: Corrugations Time delay Stability analysis Rail wear

a b s t r a c t The growth behaviour of the vibrational wear phenomenon known as corrugation is investigated using predictive models. A feedback model for wear-type rail corrugation that includes a wheel pass time delay (inverse of wheel passing frequency) is investigated with an aim to determine what effects the value of frequency of successive wheel passages has on the growth of the amplitude of corrugations. A simplified feedback model [P.A. Meehan, W.J.T. Daniel, T. Campey, Wear-type rail corrugation prediction and prevention, in: Proceedings of the Sixth International Conference on Contact Mechanics and Wear of Rail/Wheel Systems (CM2003) Gothenburg, Sweden, June 10–13, 2003, pp. 445–454] that encapsulates the most critical interactions occurring between the wheel–track structural dynamics, rolling contact mechanics and rail wear is investigated further to determine the growth of wear-type rail corrugations over multiple wheelset passages as a function of the passage time delay magnitude. Based on these results, numerical and analytical investigations are performed to identify conditions under which the wheel pass frequency has a significant effect on the growth of corrugations. A useful analytical prediction of maximum reduction in corrugation growth via speed or wheel base variation is developed, providing new insight into the prediction and possible control of wear-type corrugations. © 2008 P.A. Meehan. Published by Elsevier B.V. All rights reserved.

1. Introduction A phenomenon of increasing significance that has persisted across a range of industries for more than a century is weartype corrugation in rolling contact systems. It is characterised by the development of undesirable, undulating wear patterns on rolling contact surfaces that induce growing vibrations. In the transportation industry the phenomena occurs due to repeated vehicle passages over surfaces and is known as rail [1], road [2] or wheel “corrugation”. Similarly, in the material processing industries, production mills utilise rolling contact to process material at continuously higher loads and speeds. In these processes, the rolling contact wear instability known as “chatter” or “barring” occurs frequently on rolled material and or the rolls that supply the rolling contact forces required to achieve desirable homogenised properties in processed metal or paper products [3,4]. The present research focuses primarily on wear-type corrugations in railways, but the methods and insight gained may be intrinsically applicable to other forms of wear-type corrugation. In the railway industry, rail corrugations induce vibrations as vehicles pass over them, causing excessive noise, restricting running speeds and in some cases causing serious track defects. The

∗ Corresponding author. E-mail address: [email protected] (P.A. Meehan).

phenomenon has remained persistent and grown in prevalency, worldwide, in its multiple forms over many decades [5]. Wear-type rail corrugations include those classified as “rutting” and “roaring rails”[5], which can be characterised by both long (100–400 mm) and short pitch wavelengths (25–80 mm). The resultant railway noise due to short pitch wavelengths is in the range of 200–1500 Hz and is particularly undesirable in populated areas. Several techniques [6], such as rail hardfacing and wheel/rail profile control have had some reported success in reducing the growth of corrugations but are not reliable cures for all conditions. At present the only reliable cure for wear-type rail corrugation is removal by grinding, which costs the railway industry substantially in maintenance expenditure per annum [7]. These costs appear to be increasing in line with the significant increase in usage, development and speed of railways throughout the world. Much research has been focused on prediction and prevention of rail corrugation [1]. Research in Germany [8], Sweden [9] and Japan [10] amongst others has resulted in the development of integrated simulation programs incorporating complex finite element models for the dynamics of the track and discrete element models for the rolling contact mechanics. Wu and Thompson [11] have numerically investigated the effect of multiple wheel–track passes using a frequency domain model. To provide fundamental insight, a number of efforts have also been directed towards obtaining analytical predictions of wear-type rail corrugation [7,12,13]. Frederick [14] was one of the first to propose a feedback loop to model rail

0043-1648/$ – see front matter © 2008 P.A. Meehan. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.03.024

P.A. Meehan, W.J.T. Daniel / Wear 265 (2008) 1202–1211

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Nomenclature C␰ Gri Kb

sensitivity of creep to contact force variations modal growth rate parameter sensitivity of the steady-state response of wheel/rail relative displacement to a step change in input longitudinal profile kc contact stiffness k0 wear coefficient Kci modal sensitivity of the steady-state response of wheel/rail relative displacement to a step change in input longitudinal profile L track length per pass (or wheel base) Ł Laplace transform operator mi ,ωi , i modal mass, natural frequency, damping, respectively n closed loop pole number N wheelset pass number element of the modal matrix pi P0 nominal contact force S nondimensionalised Laplace transform complex variable t,  dimensional, nondimensional time t wheel pass time delay (=V/L) V vehicle speed x distance along track X pole imaginary position variable yi , Yi time, Laplace domain modal displacement of vertical wheelset rail dynamics yr vertical displacement of rail vertical displacement of the wheelset yw zin , Zin /zout , Zout time, Laplace domain rail longitudinal profile variation, from steady-state wear, entering/exiting the rolling contact region nominal steady-state change in profile per wheelset z0 pass Greek symbols ˛, ˇi system parameters  optimum grinding interval for corrugations ˚ growth function ωd nondimensional (wrt ωi ) damped oscillation frequency ωd ratio of the natural oscillation period of the dominant mode to the wheel pass time delay Subscripts i modal parameter (mode i) min, max minimum, maximum

corrugation and analyse the response based on linear phase relationships. Muller [7] and Nielsen [13] have investigated a nonlinear contact mechanics filter to explain reports of the independence of wavelength with speed for shorter pitch corrugations. However, the investigations neglected the effect of wheel/rail structural dynamic components on growth. Bhaskar et al. [12] and Muller [7] investigated the stability of the interaction between the structural dynamics and contact mechanics over one wheelset passage. Recently, Meehan et al. [15,16] extended this research, providing an analytical prediction of the growth of wear type rail corrugation over multiple wheelset passes independent of initial profile assumptions. The closed form solution provides an extremely efficient means of estimating corrugation growth and optimum

Fig. 1. Feedback model for wear-type rail corrugation.

grinding intervals as compared to sophisticated, but computationally expensive, numerical models. However, in this previous paper, comparison and validation of the analytical results with such numerical models (in which the passage time delay is effectively infinite) precluded an investigation of the effect of dynamic interactions between multiple wheel passages due to a finite passage time delay. Physically, the passage time delay is simply defined by the ratio of the wheel-base (or track length between successive wheels) and the vehicle speed as t = L/V. However, it introduces significant nonlinearity to modelling which is difficult to account or solve for using existing frequency or time domain models, respectively. Modelling of the passage time delay may be important for corrugation growth estimation if each wheelset passage occurs before the vibrational effects on the vehicle/track system of the previous wheelset passage have become negligible. In the present analysis, the growth behaviour of wear-type rail corrugation is investigated to determine specifically what effects the time delay between successive wheel passages (or wheel passing frequency) has on the growth of the amplitude of corrugations. The feedback model developed in [15] is utilised which encapsulates the most critical interactions occurring between the wheel–track structural dynamics, rolling contact mechanics and rail wear. Using this model, numerical and analytical investigations are performed to identify conditions under which the passage time delay has a significant effect on the growth of corrugations. In particular, a stability analysis on the complete system is extended to determine the component of corrugation growth over multiple wheelset passages due to wheel passing frequency interaction. This is investigated further using numerical models. It is noted that, the present analysis does not rely on assumptions of linear, steady state, sinusoidal solutions that many previous frequency domain analyses are based upon. 2. Analysis of rail corrugation over multiple wheel passages In the present section, the analytical results of Meehan et al. [15,16] are primarily reviewed and discussed in order to provide the basis for subsequent new analysis focussed on the effects of passage time delay on corrugation growth. The system diagram shown in Fig. 1 describes the wear-type rail corrugation development feedback mechanism. This feedback model, based on much previous research, describes the basic mechanism of corrugation growth based on four

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main components. Firstly, the initial rail profile excites the dynamic components of the wheelset–track dynamic system (I) on the first wheelset passage. This leads to a variation in rolling contact forces, which in turn affects the contact mechanics (II). The variable contact mechanics leads to a variation in the frictional power dissipated in the slip region of the contact patch and hence the amount of surface wear (III) associated with this. The initial vibration is hence recorded in the output worn rail profile which is then fed back into the next cycle of the system but only after a time delay t = L/V before the next passage (IV). The surface profile therefore evolves over successive multiple wheelset passages recurring at a wheel passing frequency, 1/t. A decoupled modal description for the wheelset–track vibrational dynamics, I, may be utilised based on experimental and theoretical evidence of an approximately constant corrugation pitch associated with a dominant mode of system vibration. The equations for any given mode i may be described as y¨ i + 2i ωi y˙ i + ωi 2 yi =

kc zin (pi − 1) mi

(1)

for i = 1 to M, where M is the number of modes and each yi is a component of the modal displacement of the wheel/rail system, Kc is the contact stiffness and  i , ωi , mi represent the modal damping, natural frequency, and mass for a particular mode i. These relate to the actual wheel and rail displacement by yw =

M 

pi yi ,

yr =

i=1

M 

yi

(2)

i=1

where pi is the modal contribution factor. The numerator of the right hand side of (1) represents the modal excitation force arising from the incoming variation in wear, zin . In this case, a vertical vibration mode is considered, however, the modal Eq. (1) may be used to represent other vibrational modes. The modal parameters may be determined analytically or experimentally using a variety of techniques as described in [16]. The contact mechanics model, II, is based on quasistatic microslip with consideration of small linear variations about nominal nonlinear operating conditions. In this case, variations in longitudinal components of traction, slip and wear are considered as the dominant contributions to corrugation growth (although the linear parameters could be representative of other components if desired). The wear process, III, is assumed to be proportional to the frictional power dissipated. The equations governing the contact mechanics, II, and the frictional wear process, III, (see [15,16] for details) can be combined and solved for each mode to give zouti − zini z0



=

C kc yi (1 − pi ) + zini P0



(3)

where P0 is the steady-state contact pressure, z0 is the steadystate wear per pass and C␰ is the nondimensional sensitivity of creep variations to contact force variations. An analytic expression for C␰ can be derived from Vermeulan and Johnson creep theory as in [13] or any other contact mechanics model, based on linearization about a nominal nonlinear condition on the creep curve. z0 can be derived by considering the frictional work hypothesis and in [13] is shown to be given by z0 =

˙ frict −k0 W 2bV

(4)

˙ frict is the steady where k0 is the wear coefficient,  is the density, W frictional power, b is the half width of the contact patch and V is the velocity. For multiple wheel passages, the rail profile variation entering the rolling contact region of the wheelset, zin (x), is the same rail profile variation exiting from the previous wheelset

pass, zout (x). Therefore, assuming a time interval between wheelset passes, t, the profile wear of successive passes of wheelsets is represented by the time delay relationship zin (t) = zout (t − t)

(5)

Using the Laplace transform denoted as



Ł f (ωi t)



= F(S)

(6)

Eqs. (1) and (3) may be solved to obtain Zouti Zini



= 1 + Kb



1 − Kci (S 2 + 2i S + 1)

(7)

where Kb =

C kc z0 P0

,

Kci =

kc (1 − pi )2 mi ωi 2

(8)

Eq. (7) represents the dynamic behaviour of the system over one wheelset passage. Kb represents the sensitivity of wear variations to wheel/rail contact deflection variations. Similarly, Kci may be shown to represent the modal sensitivity of the wheel/rail relative displacement to a change in input longitudinal profile. For realistic railway parameters, Kb , Kci and  i are always positive valued. Under these assumptions, it may be easily shown, using renowned stability analysis techniques, that the second-order system (7), is always stable, in line with the results of [7,12]. To investigate the dynamic behaviour over multiple wheelset passages, the passage time delay Eq. (5) is also transformed into the Laplace form Zini

Zouti

= e−Sωi t

(9)

Eqs. (7) and (9) describe a single input–single output feedback system that may also be represented by a block diagram equivalent to Fig. 1 [15]. The stability behaviour of the system may be determined analytically from the characteristic equation for the complete system. The characteristic equation may be obtained by solving Eq. (7) as



1 − (1 + Kb ) 1 −

ˇi S 2 +2S+1



e−Sωi t = 0

(10)

where ˇi =

Kb Kci

1 + Kb

(11)

The stability behaviour of the system is determined by the dominant real part of the system closed loop poles which are the roots to the characteristic Eq. (10), described by the nondimensional expression S =  + ωd j

(12)

where j denotes the imaginary component and  and ωd relate to the corrugation growth rate and frequency, respectively. The characteristic Eq. (10) may be considered to define both magnitude and phase conditions due to the complex nature of the roots, S. The non-trivial solutions to these conditions are developed under the realistic assumptions [15]: (a) (b) (c)

0 < Kb << i << 1, 0 < ˇi << i << 1, || << i << 1

and are summarised in the following.

P.A. Meehan, W.J.T. Daniel / Wear 265 (2008) 1202–1211

2.1. Phase condition

Table 1 Railway parameters for simulation

The phase condition of (10) provides the solution for the imaginary component of the closed loop poles as

Train speed (m/s) Wheel mass (kg) Wheel radii-long (m) Trans. (m) Wheel load (N) Young’s modulus (N/m2 ) Poisson’s ratio Shear modulus (Pa) Rail disc mass (kg) Track length/pass (m) Rail density (kg/m) Rail radii-long (m) Trans. (m) Coefficient of friction Primary rail damping ratio Bump length (m) Bump height (m) Contact damping ratio

ωd = ± ω2 n, t i

n = 0, 1, 2, . . .

(13)

Eq. (13) defines an infinite number of closed loop poles at equally spaced discrete intervals along the imaginary axis of the root locus. Each solution to (13) represents a frequency (or corrugation wavelength) that will be present in the response of the system. The infinite number of roots (or order) of the system arises due to the nonlinear passage time delay term. The effect of the magnitude of the wheel pass time delay t on the system behaviour via Eq. (13) is investigated and discussed subsequently. 2.2. Magnitude condition

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34.7 49.73 0.085 0.04 400 2.1 × 1011 0.3 7.7 × 1010 32.06 30 7800 0.213 0.05 0.4 0.01 0.012 10−7 0.0021

Solution for the magnitude condition of (10) yields an expression for the real part of the system poles,



≈

1 ln (1 + Kb ) ωi t



1+

ˇi (ω2 −1) 2

d

(1−ω2 ) +(2i ωd )2



(14)

d

Eqs. (12)–(14) define the analytical solution for all the closed loop system poles (wavelength spectrum). The stability of the system and hence the growth of corrugations may be determined from Eq. (14). In particular, the growth rate of instability for mode i, defined by parameter Gri can be expressed as the magnitude of the transfer function (see [15]):



Zouti ωi t

Zin = 1 + Gri = e

(15)

i

Note that instability here refers to profile amplitude growth over multiple passes (i.e., exponential growth of the closed loop system) as opposed to the vibrations of the vehicle/track system on any given pass (open loop system) which will be stable. The dominant pole (or mode) magnitude is determined by finding the critical value for the imaginary component, ωd , for which the maximum value for  occurs. This is given by ωd2 = 1 + 2i

(16)

This maximum value for growth is typically not realised exactly as the system phase condition described by (13) must be satisfied. If the parameter ωi t is typically very large, the discretization of ωd , defined by (13), is small enough such that there will always be a pole that approximately satisfies (16) to sufficient accuracy. In this case, each wheelset passage occurs after the dynamic effects of the previous wheelset on the vehicle/track system have settled down to a negligible level. This assumption has been made in previous research, i.e., [8,9,12,13] and may be used to simplify the growth rate to the form



Gri ≈ Kb

1 + Kci

4i (1 + i )



(17)

assuming a small k0 . It is noted, that no assumptions of a steadystate sinusoidal solution has been made to determine the growth rate and a spectrum of growth may be evaluated by superposing a range of modes i. In particular, the present analysis is valid for any initial surface profile and transient conditions such as a rail joint or other imperfection from which corrugations are typically found to be initiated from in practice. This closed form analytical prediction not only can be used to immediately determine the sensitivity of corrugation growth to different railway parameters [16], but is also convenient for estimating optimum grinding intervals for corrugation removal. In particular, according to [6] an optimum

grinding interval for corrugations , may be determined based on minimization of rail mass removal and (hence approximate costs). With reference to [6] it may be shown that the relationship between the optimum grinding interval and the growth parameter is =

t Gr i

(18)

The details of this derivation are provided in the Appendix A and examples of it’s use on field data in [17]. However, for the consideration of adjacent wheels on a vehicle travelling at considerably high speeds, it is expected that the approximation (17), will not be as accurate as for low speeds. Greater accuracy will be obtained if the value for ωd that satisfies (13) and is closest to satisfying (16) is used. In this case, it is interesting to explore whether, the magnitude of the passage time delay could have a substantial effect on the growth rate of corrugations due to the phase of vibrations of the vehicle/track system between the previous and present pass dynamics. This is investigated quantitatively in the subsequent two sections using simplified and finite element, time domain models. 3. Two mode model wear predictions The analytical solutions for rail profile wear (13)–(17) were compared initially with that obtained via numerical integration based on a discrete, two mode, time domain model detailed in [16]. The vibrational dynamics is represented by two vertical vibration modes and incorporated with Hertzian rolling contact mechanics and frictional wear models for rail longitudinal wear. Analytical and numerical simulations of wear resulting from an initial bump on a flat profile over multiple wheel passages were obtained for infinite and short passage time delays. For comparison, both the analytical and numerical results use the parameters of Table 1 with K0 = 10−9 kg/N m unless otherwise stated. These parameters are the scaled railway conditions of [15] for a two-disk test-rig, where the contact stress is equivalent. A numerical step size of 10−4 m was chosen to achieve adequate convergence of solutions. It is noted that Johnson and Gray [18] have previously experimentally and numerically studied the development of corrugations on a twin-disk rig, however, these corrugations were due to plastic deformation as opposed to wear-type and the effect of small changes in rolling speed (i.e., passage delay) was not investigated. An example of the results is shown in Fig. 2. In particular, the rail wear, zouti N , over 50,000 passages for short (0.865 s) and infinite wheel pass delays are plotted versus rail track position variable, x, assuming constant vehicle velocity V = xt.

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Fig. 2. Corrugation growth over 50,000 passes.

The wear is expressed as a profile ratio, which is defined as zouti N /zini 1 for the lower frequency mode of wear. Table 2 summarizes the results for growth rate for both the dominant modes. Figs. 2 and 3 predict that the passage time delay has an effect on the growth of corrugations for the conditions chosen but results are fairly close to the analytical prediction for infinite time delay (17). It was of interest to investigate the effect of small changes in time delay (or speed) on these results. As such, the growth rate of corrugations was determined for the same conditions of Table 1 for a range of vehicle speeds, as shown in Fig. 4, for the low frequency mode.

Fig. 3. Corrugation amplitude growth of 223 Hz mode.

The growth in amplitude of the dominant frequency wear is plotted in Fig. 3 using an FFT analysis and the analytical results of (13)–(17). For the parameters of Table 1, the growth related parameters take the values; Kb = 1.62 × 10−4 , Kc1 = 0.92,  1 = 2 × 10−3 , Kc2 = 0.08,  2 = 0.01 for dominant modes 1 and 2, respectively. Table 2 Growth rate comparisons

t = 0.865 s numerical t = 0.865 s analytical t = ∞ s numerical t = ∞ s analytical

Gr1 of low (223 Hz) mode

Gr2 of high (954 Hz) mode

0.0171 0.0144 0.0156 0.0183

0.0005 0.0005 0.0005 0.0005

Fig. 4. Corrugation length/pass = 30 m.

growth

rate

vs.

time

delay

50,000

passes.

Track

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Fig. 4 shows that the growth rate can be very sensitive to vehicle speed (or passage time delay). In particular, as the vehicle speed increases (or time delay gets smaller) the growth rate becomes more sensitive to its value. This is evident in the numerical results as well. The comparison between analytical and numerical results is good with the small offset likely due to numerical errors associated with the frequency discretization of the FFT method and/or nonlinearities. Importantly there is a good comparison between the variation (or range) of growth rate close to any particular speed. By inspection of Eqs. (13) and (14) it may be seen that this sensitivity of growth rate is an artefact of the phase relationship discussed previously. In particular, the blowup section of Fig. 4 indicates that the growth rate variation is periodic with a period equal to ωi t ≈ 2␲. This is consistent with the discretization of the closed loop poles defined by Eq. (18). The maximum growth rate at any speed occurs when ωi t = n √ 2

1+2i

,

n = 0, 1, 2, . . .

(19)

which is in accordance with Eqs. (13) and (16). This time delay related growth phenomena is highlighted in Figs. 5 and 6 for further clarity. In these figures the root locus poles (or corrugation modes) of Eqs. (13) and (14) are plotted for three different speeds (or time delays). Fig. 5 shows the effect of changing the vehicle speed (or pass delay) by 0.1% at a nominal speed of approximately 96 km/h. At 96 km/h the growth is determined by the dominant pole which is at the maximum growth rate defined by (17) and (19). A decrease in speed (or increase in pass delay) by 0.1% changes the position of the dominant pole in accordance with (13) such that the growth rate has been significantly reduced. It is interesting to note that in this case the growth rate has been reduced by approximately 28% for only a 0.1% change in speed. Physically this has occurred because the passage delay period has been detuned from being an integer multiple of the natural period of oscillation for that mode. Conversely, Fig. 6 shows the case where the speed has been reduced to approximately 38.4 km/h (a decrease of 60%). In this case there is a much smaller change in the growth rate of only 6% even though the pass delay has been detuned from the modal period of oscillation. The reason for the smaller effect is that the discretization of the pole spacing is much finer in accordance with (13) and hence the poles are found to lie closer to the position of maximum possible growth rate. Hence at lower speeds the passage time delay has less effect.

Fig. 5. Root locus plot of dominant poles for a speed change of 0.1% at approximately 96 km/h. Track length/pass = 30 m.

Fig. 6. Root locus plot of dominant poles for a speed change of 60% from 96 km/h. Track length/pass = 30 m.

Corrugation growth was also investigated under conditions of larger variations from nominal conditions. Fig. 7 illustrates such a case for a shorter track length per pass over 1.5 million passes where corrugations are initiated with a 1 ␮m bump. Under these conditions, the short time delay growth greatly exceeds the infinite case, particularly once the wavelength ratio becomes an integer. In particular, the wavelength of corrugations for the short time delay case becomes fixed at an integer value of 9 in accordance with the predictions of (13). In contrast, the infinite time delay case shows a wavelength that is varying with the number of passages most likely due to the nonlinearities associated with the large contact force variations. At the final pass, the variation in contact force is ±80% of the nominal condition indicating highly nonlinear conditions have been reached. Although this case may not be of common practical concern, it would be of interest to determine the exact nature of the nonlinear behaviour involved. It is noted that for linear conditions (small number of passes) the corrugation growth was found to be similar in both cases. For a more detailed numerical investigation, accounting for all the modes of vibration and wave propagation, a previously benchmarked finite element model was used for investigation as described subsequently.

Fig. 7. Modal growth of wear profile for 1,500,000 passes. Track length/pass = 1.34 m.

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4. Finite element wear predictions Rail wear due to two 350 kg wheels, 2.4 m apart, repeatedly traversing a track has been studied with a conventional finite element rail model. The model has been benchmarked and documented in [16] and the same parameters are used presently, except for the sleeper spacing, which is set at 0.685 m to correspond to Queensland practice. This spacing corresponds to 3.5 sleepers over the wheelbase. The rail consists of five Timoshenko beam elements per sleeper. The sleepers are lumped masses, as are the wheels. Ties and ballast are modelled with discrete springs and dampers. The equations for vertical motion are derived from equilibrium under gravity, in order to remove any excitation due to multiples of sleeper-passing frequency. A contact-smoothing algorithm is used to avoid any artifacts in the spectrum from the wheels crossing from one rail element to the next. The profile of the track is computed from estimated frictional power, filtering the highest frequencies from the rail profile, to allow for the finite size of the wheel-rail contact zone. The two wheels are first moved along the rail in a series of direct time integration runs with the train speed set to 35 m s−1 giving a short pass delay, t, of 0.0686 s. The peak growth of corrugation in response to an initial random profile is found to be at 791.4 Hz corresponding to a corrugation wavelength of 44 mm. The spectrum of corrugation obtained is shown in Fig. 8. Response around each natural frequency is discretized at intervals of wheel-passing frequency (1/t) in accordance with (12). The same discretization of the spectrum at both wheel passing and wheel rotation frequencies was obtained recently from CHARMEC’s more comprehensive 3D train–track, time–domain, multibody, corrugation model for wheel and rail wear [19] which was tuned to field measurements. In contrast to the present results, the wheel rotation frequency was also present due to the additional modelling of wheel wear. In the present analysis, the damped natural frequencies and modes of vibration of the track model with two wheels on it have been found from a complex eigenvalue analysis in the Abaqus package. This reveals that the mode responding near 791.4 Hz corresponds to three semi-wavelengths of bending of the rail between the wheels. This agrees with the observation of Igeland [20] that peak corrugation responses occur in modes with integer multiples of a semi-wavelength between the wheels. The peaks near 1200 Hz

Fig. 8. Spectrum of rail wear at wheel spacing 2.4 m, speed 35 m/s.

Fig. 9. Spectrum of rail wear at wheel spacing 2.74 m, train speed of 31.09 m/s.

correspond to about four semi-wavelengths between the wheels. In order to tune the excitation of the 791.4 Hz mode, the speed of the train is adjusted slightly to 35.1725 m s−1 , making this resonance 54 times wheel-passing frequency. This tuning of the excitation increases the growth of corrugation at this resonant frequency by 33%. If the wheel spacing is changed to 2.74 m, making it match four sleeper spacings, then a peak response at speeds close to 31 m s−1 occurs largely in wheel/rail modes around 1080 Hz. However, very slight changes in train speed greatly influence the response in these modes. The spectrum of corrugation resembles that shown in Fig. 9 and is again discretized at intervals of wheel-passing frequency, 11 Hz. Which of the 4 peaks seen in the spectrum near 1080 Hz grows the most (1064 Hz, 1075 Hz, 1086 Hz or 1097 Hz) changes with small speed changes in line with the previous section results. Fig. 10 shows the changes in the peak growth of corrugation after 15 million simulated passes and the vibration frequency associated with the corrugation in each case. Each data point is from a two-wheel simulation at a different speed in a very small range

Fig. 10. Effect of speed on corrugation growth near 31.1 m/s.

P.A. Meehan, W.J.T. Daniel / Wear 265 (2008) 1202–1211

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about 31.1 m/s. Fig. 10 highlights that while the overall envelope of corrugation growth versus frequency, is predictable, the detail of growth at a particular frequency is critically speed dependent. These results are consistent with analytical and numerical predictions shown in the previous section (e.g., Fig. 4), although the presence of multiple modes has added complexity. 5. Prediction of corrugation growth variation bounds The results of the previous sections highlight the extreme sensitivity of corrugation growth rate on passage delay under different conditions. This sensitivity may be useful in order to predict and or control the growth of wear-type corrugations. In particular, this sensitivity may serve to clarify perplexing field observations of the varying occurrence and growth of corrugation under similar conditions. Alternatively, if precise speed control is achievable then minimisation of corrugation growth rate may be possible. For these reasons it seemed useful to analytically predict the range of growth rate due to pass delay effects for any particular speed (or pass delay). Based on the insight gained from the previous results, Eqs. (13)–(16) may be used to determine the minimum value for growth rate near any particular speed (or pass delay). In particular, referring to Figs. 5 and 6 it is deduced that the minimum growth rate occurs when two poles (as opposed to one) become equally dominant, either side of the point of maximum growth rate. By inspection of Eqs. (14) and (15) this condition may be expressed as ˚min =

ωd 2 −1

1 2 (1−ωd 2 ) +(2i ωd )2 1 1

=

ωd 2 −1 2

(20)

2 (1−ωd 2 ) +(2i ωd )2 2

2

where ˚ is a growth function defined by Gr ≈ Kb (1 + Kci ˚)

(21)

and ωd1 and ωd2 are the imaginary values of the equally dominant closed loop poles, respectively, such that 2 ωi t

ωd = ωd2 − ωd1 =

=

2 V ωi L

(22)

in accordance with (13). Based on the intuition gained in [15] the solution to (20) in terms of the pass delay may be obtained using the simplifying variable substitution ωd1,2 =



1+X

(23)

such that (20) reduces to the form ˚min =

X X 2 +(2i )2 (1+X)

(24)

Eq. (24) may be rearranged to be in the form of a quadratic to which the difference in the two solutions is given by



X = X2 − X1 =



˚min

−1

− (2i )2

2

− 16i 2

(25)

Taking into account the dominant poles are located either side of the location of the maximum growth rate, i.e., ωd2 > ωdmax > ωd1 and that the pass delay is typically much greater than the modal period of oscillation, it may be shown using (16) (22) and (23) that X ≈ 2 ωd



1 + 2i

(26)

Substituting (26) into (25) and solving using (21) the analytical expression for the lower bound for growth rate near any speed (or time delay) may be obtained as



Grmin ≈ Kb

1+



2

Kci (1 + 2i )ωd 2 + 4i 2 + 4i 2



(27)

This analytical solution is plotted in the case of Fig. 4 and is shown to provide a very good approximation of the lower bound

Fig. 11. Effect of speed on corrugation growth near 31.1 m/s.

of growth rate that may be achieved at any particular speed. The variation in growth rate may also be expressed as a nondimensional ratio with respect to the maximum possible growth rate using (27) and (17): Kci Gr ≈ Grmax Kci + 4i (1 + i )

 1−



2



4i (1 + i )

(1 + 2i )ωd 2 + 4i 2 + 4i 2 (28)

where Gr = Grmax − Grmin . This function provides the maximum possible fractional change in growth rate due to speed (or pass delay) variation at any nominal speed (or pass delay) and is only dependent on the dynamic parameters, Kci ,  i and ωd . The function is plotted in Fig. 11. In this figure, the nominal parameters of Table 1 are used in (a) while for cases (b) and (c) Kci and  i have been decreased by 10 times and increased by two times, respectively. The abscissa ωd represents the ratio of the natural oscillation period of the dominant mode to the pass delay (21) and is proportional to vehicle speed. It is interesting to note in (a) the large range of growth rate possible, particularly at higher speeds. For instance, vehicles travelling at greater than 50 km/h with a wheel pass average spacing of 6.2 m will have a mode-pass delay ratio of ωd ≥ 0.01 and hence according to Fig. 11 case (a) will cause corrugations with a growth rate that may vary by more than 60% for small variations in speed (or pass delay). However, for a speed of 10 km/h this variation drops to 10%. This variation may be considered to be the error range with which the growth rate can be predicted without precise knowledge of the mode-pass delay ratio. Alternatively if the mode-pass delay ratio can be controlled precisely then this variation indicates the maximum reduction in growth rate achievable. Case (b) in Fig. 11 highlights that these results are relatively insensitive to the parameter Kci which has been reduced by 10 times in this case. It may be deduced by inspection of (28) that the dependence on Kci is weak for typical conditions under which i << Kci . Since high growth rates, Gr , will be characterised by low damping and high Kci of the dominant mode, this will be true for typical occurrences of corrugation. These results highlight that the predictions of variation in corrugation growth of Fig. 11 due to wheel passing frequency are applicable to a very wide range of railway

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P.A. Meehan, W.J.T. Daniel / Wear 265 (2008) 1202–1211

conditions.1 Conversely case (c) in Fig. 11 highlights that the results are relatively dependent on damping ratio with the increase of 200% causing as much as 25% reduction in growth rate variation. It is noted that cases (b) and (c) had maximum growth rate reductions of approximately 10 and 2 times, respectively in line with the predictions of (17) and detailed in [15]. 6. Conclusions Analytical predictions have been developed for the growth of wear-type rail corrugation showing the effect of the time delay between successive multiple wheel passages. These predictions are based on a simplified feedback model that encapsulates the most critical interactions occurring between the wheel/rail structural dynamics, rolling contact mechanics and rail wear. It is noted that previous frequency domain models cannot properly quantify the effect of passage time delay (wheel passing frequency) on corrugation growth due to required linear, steady-state (sinusoidal) solution assumptions. Numerical and analytical investigations have identified conditions under which the passage time delay has a significant effect on the growth of corrugations. The results indicate that the phase relationship between the wheel–track vibrations and wheel passages is a critical factor determining the magnitude of the effects on growth when the passage time delay is small. In particular the growth is shown to be very sensitive to the time delay when it is small, however, this variation is well predicted by the analytical solution. The model of two wheels on finite element rails with sleepers shows the sensitivity of growth to a short time-delay, as a discretization of the spectrum of corrugation as confirmed by recent results from other multi-wheel, time–domain, numerical models (tuned to field measurements). Due to the close presence of numerous modes, the growth at higher frequencies is found to be very sensitive to this parameter, the peak response switching by one or more multiples of wheel-passing frequency with a very small change of vehicle speed. This effect is similar to corrugation growth amplification of modes at integer multiples of sleeper passing frequency based on a breaking of rail spatial homogeneity at sleeper spacing. The analytical model provides a useful means by which to predict the sensitivity of growth on wheel-passing frequency. This significant sensitivity may be useful in order to predict and or control the growth of wear-type corrugations. As such, an analytical expression for prediction of the range of growth rate for any particular speed (or pass delay) is developed. The results for typical railway conditions highlight that large variations in growth rate may occur due to passage delay effects, particularly at speeds >30 km/h. This growth rate range prediction may be considered to be a confidence with which the growth rate can be predicted without precise knowledge of the mode-pass delay ratio. Alternatively if the mode-pass delay ratio can be controlled precisely (i.e., via speed or wheel-base control) then this closed form algorithm may be used to estimate the maximum reduction in corrugation growth rate achievable. The prediction was found to be insensitive to variations of railway parameters and hence applicable to most conditions. A limitation of the analytical solution is that it is restricted to the initiation of corrugation growth when the amplitude is small such that some linear assumptions are valid. For larger amplitude growth numerical simulation indicates that corrugation growth exceeds the analytical predictions due to nonlinearities. This could

be investigated more thoroughly. The analytical model, based on modal analysis, also neglects the effects of transient wave motion in the rail, although this is expected to be very small at typical vehicle speeds well below the wave propagation speeds (as confirmed by the present FE results and previous moving load analyses [21]). Recent research has also highlighted the significant effect of randomness in successive vehicle passage speed on corrugation growth predictions [22]. It is unclear whether these effects could be primarily superposed on the present results so future research on the combined influence of speed variation, wheel- and sleeperpassing frequency on corrugation growth would be prudent. Acknowledgements The authors are very grateful for the reviewers insightful comments and support of the Rail CRC, Queensland Rail, Rail Infrastructure Corporation and the Australian Rail Track Corporation and the assistance of Mr. F. Fraysse and Mr. G. Lasserre with simulations. Appendix A. Relationship between grinding interval and corrugation growth rate According to [6] and [15] (amongst many others), corrugation amplitude growth may be approximated as an exponential over time,



zout (t) = zout (0) et/

(29)

where |zout (0)| is the roughness/corrugation amplitude immediately after regrinding. The grinding cost is typically approximately proportional to the mass (or height) of material removed, therefore the grinding cost per unit time per unit length of affected track, C(t), may be expressed as

C(t) =

K zout (t)

(30)

t

where K is the cost proportionality constant typically determined by the cost per grinder pass per kilometre. Eqs. (29) and (30) define a cost curve which has a minimum at an optimum grinding interval. This minimum cost occurs when dC(t) =0 dt

(31)

By solution of Eqs. (29)–(31), it is shown that the minimum cost occurs at the optimum grinding interval of t=

(32)

Finally, as derived in [15], the corrugation growth may also be shown to be defined in terms of wheelset passes N, as



zout (N) = zout (0) eGri N , N = t

t

(33)

Comparison of (29) and (33) reveals that the relationship between the optimum grinding interval and the growth parameter is =

t Gr

(34)

An example of the use of (34) on field data is provided in [17]. It is noted that this cost analysis neglects the costs of mobilization and demobilization of the grinder and the discretization effects of the maximum material removal per grinder pass. References

1 Note parameters K and K encapsulate the contact mechanics and wear conci b ditions as defined by (8).

[1] Y. Sato, A. Matsumoto, K. Knothe, Review on rail corrugation studies, Wear 253 (2002) 130–139.

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[13] J.B. Nielsen, Evolution of rail corrugation predicted with a non-linear wear model, J. Sound Vib. 227 (1999) 915–933. [14] C.O. Frederick, A rail corrugation theory, in: Proceedings of the Second International Conference on Contact Mechanics and Wear of Rail/Wheel Systems in Kingston, Waterloo Press, Rhode Island, July, 1987, pp. 181–211. [15] P.A. Meehan, W.J.T. Daniel, T. Campey, Prediction of growth of wear-type rail corrugation, Wear 258 (2005) 1001–1013. [16] P.A. Meehan, W.J.T. Daniel, T. Campey, Wear-type rail corrugation prediction and prevention, in: Proceedings of the Sixth International Conference on Contact Mechanics and Wear of Rail/Wheel Systems (CM2003), Gothenburg, Sweden, June 10–13, 2003, pp. 445–454. [17] W.J.T. Daniel, R.J. Horwood, P.A. Meehan, N. Wheatley, Wear-type rail corrugation prediction: field study, in: Proceedings of the Ninth Conference on Railway Engineering (CORE2006) in Melbourne, Australia, April 30–May 3, 2006. [18] K.L. Johnson, G.G. Gray, Development of corrugations on surfaces in rolling contact, Proc. Inst. Eng. 189 (1975) 567–580. [19] A. Johansson, Out-of-Round Railway Wheels—Causes and Consequences, Ph.D. Thesis, Chalmers University of Technology, Goteborg, Sweden, 2005, F15–18. [20] Igeland, Railhead corrugation growth explained by dynamic interaction between track and bogie wheelsets, Proc. Inst. Mech. Eng. F: J. Rail Rapid Transit. 210 (1996) 11–20. [21] K.L. Knothe, S.L. Grassie, Modelling of railway track and vehicle/track interaction at high frequencies, Vehicle Syst. Dyn. 22 (1993) 209–262. [22] P.A. Bellette, P.A. Meehan, W.J.T. Daniel, Effects of variable pass speed on weartype corrugation growth, J. Sound Vib. 314 (2008) 616–634.