Journal of Engineering Science and Technology Review 10 (4) (2017) Research Article. Mădălina Dumitriu

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1 Jestr Journal of Engineering Science and Technology Revie 10 (4 ( Research Article Analysis on the Applicability Domain of the Linear Models During Study of the Lateral Dynamic Behaviour of the Railay ehicles Mădălina Dumitriu Department of Railay ehicles, University Politehnica of Bucharest, 313 Splaiul Independenţei, , Bucharest, Romania Received 15 March 2017; Accepted 10 September 2017 Abstract JOURNAL OF Engineering Science and Technology Revie.jestr.org The use of the linear models for the study in the dynamic behaviour of the railay vehicles provides quality information regarding a series of basic phenomena of the vehicle dynamics. This paper intends to set forth the running conditions for hich the linear models can be utilized during the simulations regarding the lateral dynamic behaviour of the railay vehicles. This action relies on analyses that compare the results of the numerical simulations developed on the linear model of the vehiclith the ones corresponding to the non-linear model, hich considers various characteristics of the track in correlation ith the velocity regime. A good analogy is emphasized beteen the compared units, namely the lateral accelerations calculated in three reference points at the carbody centre and bogies. Keyords: railay vehicles, lateral dynamic behaviour, linear model, non-linear model, track lateral irregularities, lateral accelerations 1. Introduction The study of the dynamic behaviour of the railay vehicles involves the use as an investigation tool of the numerical simulation codes developed on theoretical models of the vehicle and of the vehicle/track interaction [1, 2]. The numerical simulations are useful tools to estimate the dynamic behaviour of the railay, in terms of the running stability [3-6], safety, ride quality and ride comfort, track fatigue [7, 8], and to optimize its dynamic performance [9, 10]. It is evident that the models associated ith such studies call for a high degree of reliability and trustorthiness, as a series of important factors affecting the dynamic behaviour of the railay vehicle should be taken into account [11]. Moreover, in the present context, hen the issue of virtual homologation arises, its model should be an accurate representation of all the aspects having an impact upon the dynamic behaviour of the real vehicle [11-13]. The complexity degree of the models used in the numerical simulations is generally determined as a function of the precision required from the results. The more complex the model, the closer to the reality the results, but it ill be more difficult to derive some general conclusions regarding the basic phenomena of the vehicle dynamics. Quality and even quantity information can be obtained from less complex models. The precision of the models for the study of the lateral dynamic behaviour of the railay vehicle depends a great deal on the modelling of the vehicle-track interaction conditions [14], hich proves difficult due to the nonlinearity of thheel-rail contact of a geometric, elastic (the normal issue of the contact and tribological (the tangential issue of the contact nature. * address: madalinadumitriu@yahoo.com ISSN: Eastern Macedonia and Thrace Institute of Technology. All rights reserved. doi: /jestr The paper deals ith the non-linearity of a tribological nature of thheel-rail contact and, related to it, the issue of the model that needs to be adopted for the calculation of the heel-rail creep forces. Reaching numerical results of a high accuracy regarding the stability or the dynamic performance of the vehicle implies the selection of non-linear models [15-17] that often require long simulation times. For this reason, hen selecting the model to calculate the creep forces, a reasonable compromise is turned to, beteen the accuracy of the results and the duration of the numerical simulations. An effective solution is Shen s non-linear model [18] or Polach s [19]. Despite of the above, there are still issues that can be studied by a linear model of thheel-rail contact, based on hich quality conclusions can be made in terms of the dynamic behaviour of the vehicle under various aspects [9, 20-22]. For instance, a first quality evaluation of the vehicle stability can be conducted via the linear model. Thus, the stability of the vehicle balance position can be defined, based on the analysis of natural values in the matrix of the system of movement linear equations [20]. A series of basic properties of the stable regime of the forced lateral vibrations is still based on the linear model. Moreover, it is possible to have a set of analyses related to the influence of the velocity or the suspension parameters upon the level of vibrations in the vehicle [21]. The interest for using the linear model of thheel-rail contact is linked to the advantages thus derived. It is about simplifications in designing the applications for the numerical simulation of the dynamic behaviour of the vehicle and significantly loer calculation times, compared to the ones in the applications based on a non-linear model of thheel-rail contact. On the other hand, the results coming from the linear model are a reference for the verification of the non-linear model. As a result from the above facts, is it interesting to determine the applicability limits of the linear model in the study of the lateral dynamic behaviour of the railay

2 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( vehicles. This issue comes naturally, as a consequence of the fact that the applicability domain of the linear model has to comply ith meeting certain basic requirements associated ith the presumption that the vehicle has a stable running. What this represents is that thheelsets perform small oscillations ith respect to the track axis, ithout the clearance consumption on the track and the velocity is smaller than the critical speed. On the other hand, the linear model in this paper to calculate thheel-rail contact forces is based on Kalker theory [23], hich means to verify the condition that the creepage in thheel-rail contact points be sufficiently lo ( This paper intends to establish the limits of the applicability domain of the linear models in the study of the lateral dynamic behaviour of the railay vehicles, carried out on the basis of the numerical simulations. The vehicle is represented by a complex model, ith 21 degrees of freedom, hich allos the study of the vehicle dynamics hile running on a track ith lateral irregularities. To calculate thheel-rail creep forces, Polach s non-linear model is applied first [19] that is herein extended by introducing the influence of the load transfer beteen the heels of the samheelset upon the creep coefficients. What is obtained is a non-linear model of the vehicle, ith non-linearity sources in thheel-rail creep forces, the lateral reaction of the rail acting on thheelset hen its clearance on the track is consumed and the load transfer beteen thheels of each heelset. Afterards, thheelrail creep forces are calculated according to Kalker s linear theory [23] and the vehicle movement equations become linear, thus obtaining a linear model of the vehicle. More analyses based on these to models ill be carried out, hile considering various running conditions running on a track ith lateral irregularities of a harmonic, random shape or some that correspond to defects of an isolated type, in correlation ith the speed regime. In this context, the applicability limits of the linear model ill be determined, folloed by comparison of the accelerations in three carbody reference points using the to models. 2. The Model of the ehicle To study the dynamic behaviour of the railay vehicle during running on a track ith lateral irregularities, the model in Fig. 1 and Fig. 2 [24, 25] ill be adopted, hose parameters are described in Annex 1. The vehicle carbody is represented ith a rigid body ith the folloing motions: lateral y c, roll j c and ya a c. The bogie chassis is modelled as a three-degree freedom rigid body: lateral y bi, roll j bi and ya a bi, ith i 1 or 2. The heelsets can perform the folloing independent movements: a lateral translation y j,(j+1 and a motion of rotation around the vertical heelset ya, a j,j+1, here j 2i 1, ith i 1 or 2; the bogie i has thheelsets j and j + 1. Also, thheelset has a rotation around its on axis at the angular speed W j,(j+1 /r + j,j+1, her j,j+1 is the angular sliding speed of thheelset compared to /r, and is the vehicle velocity. Likeise, thheelset makes to more motions, namely rolling and bouncing, hich are not independent but they affect thheelset lateral motion on the track. Fig. 1. ehicle model - front vie. 3. The Non-Linear Movement Equations of the ehicle The equations of the motions of the vehicle carbody are: m c!!y c + c yc 2(!y c + h c!φ c (!y b1 +!y b2 h b2 (!φ b1 +!φ b2 + ( ( y b1 + y b2 h b2 (φ b1 +φ b2 +2k yc 2 y c + h c φ c J!! 2 xc φ c + 2c zc b c 2!φ c (!φ b1 +!φ b2 + +c yc h c 2(!y c + h c!φ c (!y b1 +!y b2 h b2 (!φ b1 +!φ b2 + +(k φc + 2k zc b 2 c 2φ c (φ b1 +φ b2 + +2k yc h c 2( y c + h c φ c ( y b1 + y b2 h b2 (φ b1 +φ b2 m c gh c φ c 0 ; J zc!! α c + 2c xc b 2 c [2!α c (!α b1 +!α b2 ]+ +c yc a c 2a c!α c (!y b1!y b2 h b2 (!φ b1!φ b2 + +2k xc b c 2 [2α c (α b1 + α b2 ]+ +2k yc a c 2a c α c ( y b1 y b2 h b2 (φ b1 φ b ; (1 The equations of motion for the bogie i, for i 1, 2 and j 2i 1, are: m b!!y bi + c yc (!y bi h b2!!y c h c!φ c a c!α c +2c yb [2(!y bi + h b1! (!y j +!y ( j+1 ]+ +2k yc ( y bi h b2 y c h c φ c a c α c +2k yb [2( y bi + h b1 ( y j + y ( j+1 ] 0 ; J xb!! + 2c zc b c 2 (!!φ c +c yc h b2 (h b2!!y bi +!y c + h c!φ c ± a c!α c +2c yb h b1 [2(h b1! +!y bi (!y j +!y ( j+1 ]+ +4c zb b b 2! + (k φc + 2k zc b c 2 ( φ c 2k yc h b2 (h b2 y bi + y c + h c φ c ± a c α c +2k yb h b1 [2(h b1 + y bi ( y j + y ( j+1 ]+ +[4k zb b 2 b g( h 12 m c / 2 + h b1 m b ] 0, ith h 12 h b1 + h b2 ; (2 (3 (4 (5 155

3 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( J zb!! α bi + 2c xc b 2 c (!α bi!α c +2c xb b b 2 [2!α bi (!α j +!α ( j+1 ]+ +2c yb a b [2a b!α bi (!y j!y ( j+1 ]+ +2k xc b c 2 (α bi α c +2k xb b b 2 [2α bi (α j + α ( j+1 ]+ +2k yb a b [2a b α bi ( y j y ( j+1 ] 0. For thheelsets j, respectively j + 1, for j 2i 1 and i 1, 2, the equations of lateral displacement and ya motions and the equation of the rotation motion around the heelset axis are: m!!y j, j+1 + 2c yb [!y j, j+1!y bi h b1! a b!α bi ]+ +2k yb [y j, j+1 y bi h b1 a b α bi ]+ Y j,( j+11 + Y j,( j+12 Y σ j, j+1 (7 (6 J z!! α j, j+1 + 2c xb b 2 b (!α j, j+1!α bi 2k xb b 2 b (α j, j+1 α bi +J y r!φ j, j+1 [X j,( j+11 X j,( j+12 ]. J y!ω j, j+1 r [X j,( j+11 + X j,( j+12 ], (9 here, r the coordinates of thheel-rail contact points hen thheelset is in a median position on the track; the term J y ( / r!φ j, j+1 corresponds to the gyroscopic moment due to the combined effect of the heelset rotation motion around its on axis ith the roll motion ϕ j,j+1 [26]; X j,(j+11,2 are the longitudinal forces and Y j,(j+11,2 represent the guidance forces acting upon the heelsets j and j + 1 respectively, in thheel/rail contact points 1 or 2. (8 Fig. 2. ehicle model - side vie. The motion equations (1 9 can be solved by Runge- Kutta method. The modelling of the situation here thheelsets consume the clearance on the track s is achieved by the introduction of the lateral reaction ith a non-linear characteristic Y σj,(j+11,2, hich acts upon the front heel [27]. The equation of such force is as such Y σ j, j+1 H y j, j+1, j+1 σ 2 sign( y ζ j, j+1 j, j+1 *k yr y j, j+1, j+1 σ, (10 2 The bouncing and roll equations of thheelsets, motions that are dependent on the lateral displacement of the heelset on the track. When considering that the bounce coming from the lateral displacement of thheelset on the track is very lo [26], the inertia effect of thheelset mass in the vertical direction can be thus neglected. Hence, the balance equation of the vertical forces can britten as Q j,( j+11 + Q j,( j+12 2Q, (11 here Q j,(j+11,2 are the vertical loads in thheel/rail contact points and Q is the static load on thheel. The equation of thheelset roll motion is here H(. is the Heaviside s unit step function, and function ζ j,j+1 describes the track irregularities against each heelset. 156

4 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( J!! x φ j, j+1 J y!α r j, j+1 2b 2 b c zb! 2b 2 b k zb (12 r [Y j,( j+11 + Y j,( j+12 ]+ [Q j,( j+11 Q j,( j+12 ]+ r Y σ j, j+1 by the spin and a j1,2 and b j1,2 stand for the semiaxes of the contact ellipse. The creepage in thheel/rail contact points are calculated ith the equations: here the term J y ( / r!α j, j+1 is given by the gyroscopic moment due to the combined effect of the ya motion ith the rotation motion of thheelset around its on axis [26]. Thheel/rail contact forces are: the longitudinal forces, the guidance forces and the vertical loads. These are expressed as a function on the longitudinal components T xj,(j+11,2 and lateral T yj,(j+11,2 of the creep forces and the normal reactions N j,(j+11,2 in thheel/rail contact points. For example, the equations for thheelsets j (see Fig. 3 can be thereforritten X j1,2 T xj1,2 ; (13 Y j1,2 T yj1,2 cosγ rj1,2 N j1,2 sinγ rj1,2 ; (14 Q j1,2 ±T yj1,2 sinγ rj1,2 + N j1,2 cosγ rj1,2, (15 here γ rj1,2 stand for thheel/rail contact angles against the track reference system and these is calculated ith equation belo 1 e γ rj1,2 γ ± ( y j1,2 + ρ γ, (16 ρ ρ r r γ here γ is thheel/rail contact angle for the median position of thheelset on the track and ρ and ρ r represent the curvature radii of thheel-rail rolling profiles. ν xj1,2 y j γ r e ω r j e "α j ; (19 ν yj1,2 1 (ϕ!y λ! r ζ j j α j, ith λ γ r γ and ϕ 1+ λ ; (20 ν cj1,2 2 2 ν xj1,2 +ν ycj1,2, (21 here γ e is the equivalent conicity and ν ycj1,2 represents the lateral component of the creepage corrected by the spin. In order to calculate the equivalent conicity and lateral component of the creepage corrected by the spin the relations belo are applied: γ e ρ r γ ρ ρ r + ρ r γ r γ (22 ν ycj1,2 ν yj1,2 + a j1,2, f for ν yj1,2 + a j1,2 > ν yj1,2, (23 ν ycj1,2 ν yj1,2, for ν yj1,2 + a j1,2 ν yj1,2, (24 here is the spin, defined as!α j 1 r + ω j γ j1,2, (25 Fig. 3. Explanatory for thheel/rail contact forces. According to Polach s friction nonlinear model [19], the components of the creep forces have the forms T xj1,2 2µN j1,2 π κ j1,2 1+κ j1,2 + arctgκ 2 j1,2 ν xj1,2, (17 ν cj1,2 2 κ j1,2 + arctgκ 2 π 1+κ j1,2 ν yj1,2 + j1,2 ν cj1,2 T yj1,2 µn j1, a K 1+ 6,3 1 a/b j1,2 Mj1,2 ( e( j1,2 ν cj1,2, (18 here m is the friction coefficient, ν xj1,2 and ν yj1,2 is the longitudinal and the lateral components of the creepage in thheel/rail contact points, ν cj1,2 the creepage corrected here γ j1,2 represent thheel/rail contact angles compared to thheelset reference system 1 e γ 1,2 γ ± ( y j1,2 + ρ r γ. (26 ρ ρ r r γ To determine the semiaxes of thheel/rail contact ellipse, the Hertz equations ill be used, as a function on the normal force and the curves of the rolling profiles [29]. The coefficient κ j1,2 (see equations (17 and (18, it can be calculated as belo κ j1,2 1 4 Gπ a j1,2 b j1,2 C ii µn j1,2 ν cj1,2, (27 here G is the transversal elasticity module and C ii is a constant hich depends on the coefficients C 11 and C 22 defined by Kalker [23]: 157

5 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( C ii C 11 ν xj1,2 ν j1,2 2 ν + C yj1,2 22 ν j1,2 2, ith The equations (37 are used iteratively for each integration step in the calculation of thheel/rail contact normal forces. ν j1,2 2 2 ν xj1,2 +ν yj1,2. (28 The coefficient K Mj1,2 (see the equation (18 is K Mj1,2 κ sj1,2 here δ j1,2 κ 2 sj1,2 for 2 κ sj1,2 1 3 δ 2 j1, (1 δ 2 3, (29 j1,2 3 δ j1,2 +1, (30 κ sj1,2 8 Gb j1,2 a j1,2 b j1,2 C ν 23 ycj1,2, (31 3 µn j1,2 1+ 2π 1 e (a/b j1,2 here C 23 is the coefficient calculated by Kalker for the spin [23]. To determine the normal reactions N j1,2 in thheel/rail contact points of thheelsets j the equations (11 and (12 arritten as belo q j1 N j1 + q j2 N j2 2Q ; (32 (r p j1 q j1 N j1 + (r p j2 + q j2 N j2 c j, (33 in hich c j J x!! φ j J y ( / r!α j 2b b 2 c zb! 2b b 2 k zb r Y σ j (34 ν q j1,2 1 K yj1,2 ν j1,2 + K sj1,2 γ rj1,2 ; cj1,2 ν cj1,2 p j1,2 K j1,2 ν yj1,2 ν cj1,2 + K sj1,2 ν cj1,2 ± γ rj1,2, (35 4. The Linear Movement Equations of the ehicle To express the linear movement equations of the vehicle, a starting point is the non-linear equations in the previous section, hich ill be linearized in linith the belo procedure applied around the vehicle equilibrium position in an ideal track. The movement equations of the carbody and bogies can be noticed to be linear (Eq. 1-6, hile thheelset movement equations (7-9 here the creep forces and the rail lateral reaction are present are non-linear. Similarly, the algebraic equations (37 ill insert non-linear terms, based on hich the normal forces on the rolling surfaces are calculated. The linear movement equations describe the regime of the small oscillations around the balance position of the vehicle on the track. To linearize the movement equations, the lateral rail reaction that only acts upon the clearance consumption on the track ill consequently be taken out (Y sj,(j+1 0 from the equation of ya movement in the heelsets (Eq. (7. According to Kalker s theory [23], the non-linear expressions of the creep forces for thheelsets j for small creepage ( , are given in the equations T xj1,2 N j1,2 χ x v xj1,2 ; (38 T yj1,2 N j1,2 (χ y v yj1,2 + χ s v sj1,2, (39 here the longitudinal and lateral creepage are found in the equations (19 and (20, hile the (adimensional spin creepage comes from the relation r!α j γ j1,2. (40 The creepage coefficients χ x, χ y şi χ s are defined as follos: here K j1,2 2µ π κ j1,2 1+κ j1,2 + arctgκ 2 j1,2 ; ( K sj1,2 9µ 16 a K 1+ 6,3 1 e (a/b j1,2 j1,2 Mj1,2. (36 The solution of the set of non-linear equations (32 33 can be derived by simple iterations via the belo equations N j1 N j2 2(r p j2 + q j2 Q + q j2 c j r (q j1 p j2 q j2 p j1 2 q j1 q j2 ; q j1 c j 2(r p j1 q j1 Q r (q j1 p j2 q j2 p j1 2 q j1 q j2. (37 χ x Ga j1,2 b j1,2 N j1,2 C 11 ; χ y Ga j1,2 b j1,2 N j1,2 C 22 χ s G(a j1,2 b j1,2 3/2 r N j1,2 C 23, (41 in hich, for linearization reasons, N j1,2 Q, a j1,2 a and b j1,2 b are considered, here a and b are the semiaxes of the contact ellipse corresponding to the load Q. The parameters q j1,2 and p j1,2 (Eq. (35 ill be therefore ritten as: q j1,2 1 (χ y ν yj1,2 + χ s γ rj1,2 ; p j1,2 χ y ν yj1,2 + χ s ± γ rj1,2. (42 After processing, relation (42 becomes: 158

6 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( ϕ "y χ j λ ζ " j y α j + q j1,2 1 * r +χ "α j s γ ε y j e ; (43 y * γ ± ε j r ϕ!y p j1,2 χ j λ ζ! j y α j + r +χ!α j s γ ε y j, (44 ±γ + ε r y j χ y Q j1,2 Q 1 γ χ s ± ϕ "y j χ " ζ j r "α j α j + ± γ ε y j e J "" x φ j J y "α r j 2b 2 b c zb " 2b 2 b k zb 2( r γ (51 The linearized movement equations of thheelsets are: - the ya movement equation m!!y j,( j+1 + 2c yb!y j,( j+1!y bi h b1! a b!α bi + +2k yb y j,( j+1 y bi h b1 a b α bi + for ε r + ρ γ e ; ε ρ ρ r r γ + ρ r γ. (45 ρ ρ r r γ To obtain certain linear shapes for the reactions N j1,2, the second degree terms ill be left out in (43 (44 and the belo approximations ill be considered q j1,2 1; p j1,2 ±γ. (46 The expressions for the reactions ill be derived from (37 and (46 +2ϕ χ y Q J y γ r +2Q ε r 1 χ s ε!y j,( j χ r Q s!α j,( j+1 1 χ s!α r γ j,( j+1 + b 2 2γ (1 χ s b c zb!φ r γ bi y oj,( j+1 2χ y Q α j,( j+1 b 2 2γ (1 χ s b k zb φ r γ bi 2λ χ Q y ζ! j,( j+1 + (52 N j1,2 Q ± c j 2( r γ here c j comes from (34 for Y sj 0 (47 c j J!! x φ j J y!α r j 2b 2 b c zb! 2b 2 b k zb. (48 The linear expressions of the contact forces can be no ritten: X j1,2 χ x Q y j γ r e r ω j "α j ; (49 1 χ +2Q ε s ε r ζ e j,( j+1 only to mention that the inertial term due to thheelset roll γ (1 κ s J x!! φ r γ j has been left out, since it is much smaller than the influence of thheelset mass. - the hunting movement equation J z!! α j,( j+1 + 2c xb b 2 b (!α j,( j+1!α bi 2k xb b 2 b (α j,( j+1 α bi +J y r λ(!y ζ! 2 j,( j+1 j,( j+1 2χ x Q 2χ x Q γ e r ζ j,( j+1. γ e r y j,( j+1 + e 2!α j,( j+1 (53 ϕ!y χ j χ ζ! j r y α j + χ!α j s γ ε Y j1,2 Q y ±γ + ε j r J!! x φ j J y!α r j 2b 2 b c zb! 2b 2 b k zb +γ (1 χ s 2( r γ y j + (50 - thheelset rotation movement around its on axis J y!ω j + 2χ x r 2 Q ω j 0 (54 Folloing the linearization, the rotation movement around its on axis is noticed to become independent. 5. Numerical Analysis In order to establish the applicability domain of the linear models ithin the lateral dynamics of the railay vehicles, 159

7 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( this section ill focus on a series of results obtained from numerical simulations developed on the models presented in the previous sections in this paper. The compared parameters are the lateral accelerations calculated in three carbody reference points (at the carbody centre and bogies during the running of the vehicle on a track ith lateral irregularities: a in a harmonic shape; b isolated defect-type; c random shape. The parameters of the numeric model are featured in Table 1 and are representatives for a passenger vehicle. To calculate the geometrical parameters of thheel/rail contact, the S78 profilheel and the UIC 60 rail ith the standard cant at C.F.R. of 1/20 are considered (see Table 2 [30]. Table 1. The parameters of the numerical model. m c kg; 2c xc 50 kns/m m b 3200 kg m 1650 kg J xc kgm 2 J zc kgm 2 J xb 3200 kgm 2 J zb 5000 kgm 2 J x J z kgm 2 J y kgm 2 2a c 19 m 2a b 2.56 m h c 1.3 m h b m; h b2 0.2 m 2b b 2b c 2 m c yc kns/m 2c zc kns/m 2k xc 340 kn/m 2k yc 340 kn/m 2k zc 1.2 MN/m k ϕc 10 knm 4c xb 100 kns/m 4c yb kns/m 4c zb kns/m 4k xb 140 MN/m 4k yb 10 MN/m 4k zb 4.4 MN/m k yr 100 MN/m σ 12 mm m 0.36 Table 2. The geometrical parameters of thheel/rail contact. r m; r r m g m; r m γ e The critical speed of the vehicle of km/h derives from the values in Tables 1 and 2 [20]. The validity limits of the linear model should be firstly established, hich involves the determination of the running conditions corresponding to hich the creepage in the heel-rail contact points are sufficiently lo (< so that the linear theory underlining the model be applicable. To calculate the creepage in thheel-rail contact points, the relation belo applies ν j,( j+11,2 2 2 ν xj,( j+11,2 +ν yj,( j+11,2, for j 2i 1, ith i 1, 2. (55 a Track lateral irregularities in a harmonic shape The lateral irregularities of the track are considered harmonic ith thavelength Λ and amplitude ζ 0, in the form of ζ (x ζ 0 cos 2π Λ x, (56 here the position of the coordinate x t corresponds to the middle of the vehicle. The track irregularities are out of phase in conformity ith the vehiclheelbase against each heelset, ζ 1,2 (x ζ 0 cos 2π Λ (x + a c ± a b ; ζ 3,4 (x ζ 0 cos 2π Λ (x a c ± a b. (57 The functions ζ j,(j+1, ith j 2i 1 for i 1, 2, can be expressed as time harmonic functions ζ 1,2 (x ζ 0 cosω t + a ± a c b ; ζ 3,4 (x ζ 0 cosω t a a c b, (58 in hich 2π/Λ represents the angular frequency induced by the track excitation ( 2πf, here f is the excitation frequency. Thavelength of the track lateral irregularities is adopted so that the excitation frequency coincides ith the frequencies that dominate the poer spectral density of the carbody acceleration, hich become important in terms of the dynamic behaviour of the vehicle. It is about the natural frequencies of the coupled movement of ya-roll (0.46 Hz and 1.21 Hz and the carbody hunting frequency (0.78 Hz [21]. As for the amplitude of the lateral irregularities, values beteen mm ill be used. The Fig. 4 (for 100 km/h and Fig. 5 (for 200 km/h feature the values of the creepage in thheel-rail contact points during running on a track ith lateral irregularities of a harmonic shape, ith the above-mentioned characteristics. The limits of the applicability domain of the vehicle linear model can be noticed to change along ith velocity and excitation frequency. At the speed of 100 km/h (Fig. 4, the linear model can be applied for the excitation frequencies already considered unless the amplitude of the track irregularities exceeding 1 mm. For a velocity of 200 km/h (Fig. 5, at excitation frequencies corresponding to the lo frequency of the coupled movement of ya-roll, the model can be applied for amplitudes of the track irregularities of up to 1 mm. Should the excitation frequency is 0.78 Hz, equal to the carbody hunting frequency, or 1.21 Hz - equal to the high frequency of the coupled movement of ya-roll, the linear model is then valid for amplitudes of the track irregularities loer than 0.5 mm. 160

8 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Fig. 4. Wheelset creepage for 100 km/h: (a 0.46 Hz; (b 0.78 Hz; (c 1.21 Hz., 1 heelset;, 2 heelset;, 3 heelset;, 4 heelset. Fig. 5. Wheelset creepage for 200 km/h: (a 0.46 Hz; (b 0.78 Hz; (c 1.21 Hz., 1 heelset;, 2 heelset;, 3 heelset;, 4 heelset. 161

9 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Once determined the applicability limits of the linear model, it is interesting to see to hat extent the derived results are comparable to the results reached by using the non-linear model of the vehicle. The Tables 3-4 sho the maximum values of the accelerations in three reference points (at the carbody centre and to bogies, coming from the use of the to models. It is obvious that the previous cases have been taken into account and for each the study has been narroed don to the domain for hich the linear model is applicable. A good correspondence beteen the results from the to models is visible. Table 3. The maximum values for the carbody acceleration at 100 km/h velocity Frequency Wavelength Amplitude Linear model Non-linear model excitation (Hz Λ (m ζ 0 (mm Acceleration (m/s 2 Acceleration (m/s at the carbody centre front bogie rear bogie at the carbody centre front bogie rear bogie Table 4. The maximum values of the carbody lateral acceleration at 200 km/h velocity Frequency Wavelength Amplitude Linear model Non-linear model excitation (Hz Λ (m ζ 0 (mm Acceleration (m/s 2 Acceleration (m/s at the carbody centre The Fig. 6-8 include the carbody lateral accelerations hile running at 200 km/h speed, on a track ith the amplitude of the harmonic irregularities of ζ mm and various avelengths: Λ m (corresponding to the excitation frequency of 0.46 Hz; Λ m (corresponding to the excitation frequency of 0.78 Hz; Λ m (corresponding to the excitation frequency of 1.21 Hz. What is obvious is the harmonic shape of the accelerations, hich proves the permanent regime of vibration. At frequency of 0.46 Hz, the accelerations in those three carbody reference points are in phase. For the frequencies of 0.78 Hz and 1.21 Hz, the acceleration at the carbody centre is in phase only ith the acceleration above the front bogie. As for the level of vibrations in the carbody reference points, it changes as a function of the excitation frequency and thavelength of the track lateral irregularities respectively, in the same direction for both models. To front bogie rear bogie at the carbody centre front bogie rear bogie further examine this aspect, the concept of carbody critical point is introduced, as being that reference point here the level of vibrations is the highest (calculated on the basis of lateral acceleration. Should the excitation frequency coincides ith the natural la frequency of the coupled movement of ya-roll (0.46 Hz, then the maximum value of acceleration is found rear bogie, hich thus becomes the carbody critical point; the acceleration ill be at is loest value front bogie (Fig. 6. In case the excitation frequency is equal ith the carbody hunting frequency (0.78 Hz, the highest value of the lateral acceleration occurs front bogie and the loest at the carbody centre (Fig. 7. For the natural high frequency of the coupled movement of ya-roll (1.21 Hz, the critical point remains front bogie, hile the loest acceleration is above the rear bogie (Fig

10 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Fig. 6. Carbody lateral acceleration during running on a track ith lateral irregularities of a harmonic shape (Λ m, ζ mm: (a linear model; (b non-linear model;, at the carbody centre;, front bogie;, rear bogie. Fig. 7. Carbody lateral acceleration during running on a track ith lateral irregularities of a harmonic shape (Λ m, ζ mm: (a linear model; (b non-linear model;, at the carbody centre;, front bogie;, rear bogie. Fig. 8. Carbody lateral acceleration during running on a track ith lateral irregularities of a harmonic shape (Λ m, ζ mm: (a linear model; (b non-linear model;, at the carbody centre;, front bogie;, rear bogie. b Lateral irregularities of the track, corresponding to isolated defects Against each heelset, the isolated defect can be analytically described in a function such as belo: ζ j,( j+1 (x j,( j+1 ζ 0 sin 2 (π x j,( j+1 / Λ, for 0 x j,(j+1 L ; (59 ζ j,( j+1 (x j,( j+1 0, for x j,(j+1 < 0 or x j,(j+1 > L, here, depending on thheelset position ithin the vehicle, x j,(j+1 is: - for i 1, x 1 x ; x 2 x 2a b ; - for i 2, x 3 x 2a c ; x 4 x 2a b 2a c, 163

11 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( here abscissa x is calculated in dependence on the moment t, x t. Thavelength L 10 m is considered for this case; according to the UIC 518 Leaflet [31], and as for the defect amplitude, unless the velocity exceeds 200 km/h, ζ 0 4 mm can be adopted for a QN1 quality track; for a QN2 track, the value of ζ 0 is 6 mm. When folloing a similar methodology ith the running on a track ith lateral irregularities in a harmonic shape for establishing the applicability domain of the linear model, the creepagill be calculated in thheel-rail contact points and conditions ill be set forth for hich these results do not exceed As seen in Fig. 9 and Fig. 10, these conditions are complied ith for all the situations being examined. Fig. 9. Wheelset creepage for 100 km/h:, 1 heelset; 2, heelset;, 3 heelset;, 4 heelset. Fig. 10. Wheelset creepage for 200 km/h:, 1 heelset and 3 heelset;, 2 heelset and 4 heelset. The Table 5 features the maximum values of the lateral accelerations in the carbody reference points at velocities of 100 km/h and 200 km/h, derived from using the to models, hereas the Fig. 11 and 12 sho a time history of the carbody lateral accelerations upon passing over an isolated defect ith a 6 mm-amplitude. While using the linear model of the vehicle, higher values of acceleration are noticed, as ell as the fact that the differences beteen the results from the to models increase along ith the velocity and the amplitude of the isolated defect. For instance, if referring to the maximum acceleration calculated at the carbody centre, it can be proved that there occur the folloing differences beteen the results derived from the linear model and nonlinear model of the vehicle: for ζ 0 4 mm 5.79%, at 100 km/h; 10.02%, at 200 km/h; for ζ 0 6 mm 9.07%, at 100 km/h; 15.90%, at 200 km/h. Table 5. The maximum values of the carbody lateral acceleration during running on a track ith lateral irregularities corresponding to isolated defects Amplitude of elocity Linear model Non-linear model the isolated (km/h Acceleration (m/s 2 Acceleration (m/s 2 defect at the at the ζ 0 (mm carbody front bogie rear bogie carbody front bogie rear bogie centre centre As for the position of the carbody critical point in terms of the level of vibrations, it can be immediately identified rear bogie at velocity of 100 km/h, both during running on a QN1 quality track QN1 (ζ 0 4 mm or QN2 (ζ 0 6 mm, irrespective of the applied model. At the velocity of 200 km/h, the acceleration to bogies ill have close value, a reason for hich the vehicle carbody has to critical points. 164

12 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Fig. 11. Carbody lateral acceleration during passing over an isolated defect at velocity of 100 km/h, ith a 6 mm-amplitude: (a linear model; (b nonlinear model, at the carbody centre;, front bogie;, rear bogie. Fig. 12. Carbody lateral acceleration during passing over an isolated defect at velocity of 200 km/h, ith a 6 mm-amplitude: (a linear model; (b nonlinear model, at the carbody centre;, front bogie;, rear bogie. c Track lateral irregularities in a random shape The track lateral irregularities are described by a pseudostochastic function ζ(x, ritten as [32] N ζ (x K ζ f (x U k cos(ω k x +φ k, (60 ith K ζ k0 N ζ adm max f (x U k cos(ω k x +φ k k0, (61 here: K ζ is a scaling coefficient of the amplitudes in the track lateral irregularities, ζ adm is the maximum value of the track lateral irregularities as per UIC 518 Leaflet [31]; f(x is an adjustment function applied on the distance L 0, in the form of f (x 6 x 5 L 15 x 4 0 L +10 x 3 0 L 0, (62 *H(L 0 x H(x L 0 here H(. is the Heaviside s unit step function; U k is the amplitude of the spectral component corresponding to the ave number W k, and j k is the lag of the spectral component k for hich a uniform random distribution is selected. The amplitude of each spectral component is established on the basis of the poer spectral density of the track irregularities described in accordancith ORE B176 [33] and the specifications included in the UIC 518 Leaflet [31] regarding the track geometrical quality described by the quality levels QN1 and QN2. Against each heelset, the track lateral irregularities are described by the function ζ j,,(j+1 (x j,(j+1,. dependent on the distance along the track, as such ζ j, j+1 (x j,( j+1 0, for x j,(j+1 0; ζ j, j+1 (x j, j+1 ζ (x j,( j+1, for x j,(j+1 > 0, (63 ith, x 1 x ; x 2 x 2a b, for i 1; x 3 x 2a c ; x 4 x 2a b 2a c, for i

13 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Fig. 13. Wheelset creepage for QN1:, 1 heelset; 2, heelset;, 3 heelset;, 4 heelset. To determine the applicability limits of the linear model, the velocities of 100 km/h and 200 km/h ill be taken into account, along ith both quality levels for the track, QN1 and QN2 [31], for hich the creepagill be calculated in thheel-rail contact points for all four heelsets of the vehicle (see Fig. 13 and Fig. 14. The prerequisite regarding the value of the creepage (ν < , for hich the vehicle linear model is valid, is only met during running on a QN1 Fig. 14. Wheelset creepage for QN2:, 1 heelset; 2, heelset;, 3 heelset;, 4 heelset. quality track. For this case, the carbody lateral accelerations calculated in the three reference points via the linear model and the non-linear model ill be compared (see Fig. 15 and Fig. 16. The comparison units are represented by the maximum values (the encircled peaks and the root mean square deviation of the lateral acceleration, as seen in Table 6. It has to underline that no flange contact has been signalised in numerical simulation results. Fig. 15. Carbody lateral acceleration during running at 100 km/h velocity on a QN1 quality track: linear model (a at the carbody centre; (b above the front bogie; (c rear bogie; non-linear model (a at the carbody centre; (b front bogie; (c rear bogie. 166

14 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Fig. 16. Carbody lateral acceleration during running at 200 km/h velocity on a QN1 quality track: linear model (a at the carbody centre; (b above the front bogie; (c rear bogie; non-linear model (a at the carbody centre; (b front bogie; (c rear bogie. When examining the results for the velocity of 100 km/h, the differences beteen the maximum accelerations calculated on the basis of the to models can be shon to fall ithin the limits 0.91% %. The root mean square deviation of the acceleration for the linear model is at least 2.07% higher than the same deviation for the acceleration in the non-linear model of the vehicle. The differences increase along ith the velocity; they ill be found in the interval 2.56% % for the maximum acceleration at 200 km/h velocity and 2.56% % for the root mean square deviation of the acceleration. Finally, the carbody critical point ill be rear bogie for both models, at the 100 km/h velocity. Should velocity goes up to 200 km/h, the critical point ill shift to the front bogie. 167

15 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( Table 6. The values of the carbody lateral accelerations during running on a QN1 quality track elocity (km/h Acceleration (m/s 2 Linear model Non-linear model Maximum value Root mean square deviation Maximum value Root mean square deviation at the carbody centre front bogie rear bogie at the carbody centre front bogie rear bogie Conclusions The paper features a set of analyses based on hich there have been determined the limits here the linear models can be used for the study of the lateral dynamic behaviour during running on a track ith lateral irregularities, depending on the velocity regime. It has been demonstrated that these limits do not cover a domain here all the specific situations in the exploitation of the railay vehicles be included; the reference here is made to the least unfavourable conditions running at high velocity on a track ith lateral irregularities, hose amplitude can have the maximum value admitted. Within the limits of the applicability domain of the linear model, the analyses made via the comparison of the accelerations at the carbody level derived from the linear model ith the non-linear ones have proved close results by using the to models. Significant differences, hich can reach up to 16%, ill be visible in conditions of running at a high velocity on a track ith lateral isolated defects. It is orthhile noticing that the differences beteen the results of the to models occur at the applicability limit of the linear model (for close creepage or equal ith the limit value When introducing the concept of critical point as that carbody reference point here the level of vibrations is the highest, it as demonstrated that its position as not affected by the model in use but by the running conditions, namely the excitation frequency or the velocity. Acknoledgments This ork has been funded by University Politehnica of Bucharest, through the Excellence Research Grants Program, UPB GEX Research project title: Research on developing mechanical and numerical models for the virtual evaluation of the dynamic performances in the railay vehicles (in Romanian. Contract number: 48/ This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence References 1. J. Evans, M. Berg, Challenges in simulation of rail vehicle dynamics, ehicle System Dynamics, 47, pp ( G. Schupp, Simulation of railay vehicles: Necessities and applications, Mechanics Based Design of Structures and Machines, 31 (3, pp ( T. Mazilu, M. Dumitriu, C. Tudorache, Instability of an oscillator moving along a Timoshenko beam on viscoelastic foundation, Nonlinear Dynamics, 67 (2, pp ( T. Mazilu, M. Dumitriu, C. Tudorache, On the dynamic effect beteen a moving mass and an infinite one-dimensional elastic structure at the stability limit, Journal of Sound and ibration, 330 (15, pp ( T. Mazilu, Instability of a train of oscillators moving along a beam on a viscoelastic foundation, Journal of Sound and ibration, 332 (19, pp ( Y.C. Cheng, C.T. Hsu, Hunting stability and derailment analysis of a car model of a railay vehicle system, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 226 (2, pp ( B. Suarez, J.M. Mera, M.L. Martinez, J.A. Chover, Assessment of the influence of the elastic properties of rail vehicle suspensions on safety, ride quality and track fatigue, ehicle System Dynamics, 51 (2, pp ( Y.Q. Sun, C. Cole, M. Spiryagin, Study on track dynamic forces due to rail short-avelength dip defects using rail vehicle-track dynamics simulations, Journal of Mechanical Science and Technology, 27 (3, pp ( Y. He, J. McPhee, Optimization of the lateral stability of rail vehicles, ehicle System Dynamics, 38 (5, pp ( L. Mazzola, S. Alfi, S. Bruni, A method to optimize stability and heel ear in railay bogies, International Journal of Railay, 3 (3, pp ( S. Bruni, J. inolas, M. Berg, O. Polach, S. Stichel, Modelling of suspension components in a rail vehicle dynamics context. ehicle System Dynamics, 49 (7, pp ( C. Funfschilling, Y. Bezin, M. Sebès, DynoTRAIN: Introduction of simulation in the certification process of railay vehicles, Transport Research Arena, Paris, ( C. Funfschilling, G. Perrin, S. Kraft S, Propagation of variability in railay dynamic simulations: Application to virtual homologation, ehicle System Dynamics, 50, pp ( N. Burgelman, M. Sh. Sichani, R. Enblom, M. Berg, R. Dollevoet, Influence of heel rail contact modelling on vehicle dynamic simulation, ehicle System Dynamics, 53 (8, pp ( O. Polach, On non-linear methods of bogie stability assessment using computer simulations, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 220, pp ( Y.C. Cheng, S.Y. Leeb, H.H. Chen, Modeling and nonlinear hunting stability analysis of high-speed railay vehicle moving on 168

16 Mădălina Dumitriu/Journal of Engineering Science and Technology Revie 10 (4 ( curved track, Journal Sound and ibration, 324, pp ( T. Mazilu, M. Dumitriu, On the instability of the railay vehicles, Mechanical Journal Fiability and Durability, 2 (8, pp. 1-6 ( Z.Y. Shen, J.K. Hedrick, J.A. Elkins, A comparison of alternative creep force models for rail vehicle dynamic analysis, Proceedings of the 8th IASD Symposium, Cambridge, MA, pp ( O. Polach, A fast heel rail forces calculation computer code, ehicle System Dynamics, 33, pp ( M. Dumitriu, Influence of the suspension parameters upon the hunting movement stability of the railay vehicles, Mechanical Journal Fiability and Durability, 1, pp ( M. Dumitriu, Influence of the longitudinal and lateral suspension damping on the vibration behaviour in the railay vehicles, Archive of Mechanical Engineering, 62 (1, pp ( C.E. Bell, D. Horak, J.K. Hedrick, Stability and curving mechanics of rail vehicles, Transaction of the ASME Journal of Dynamic, Systems, Measurement and Control, 103, pp ( J.J. Kalker, On the rolling contact of to elastic bodies in the presence of dry friction, Ph.D. Thesis, Delft ( M. Dumitriu, Numerical analysis of the influence of lateral suspension parameters on the ride quality of railay vehicles, Journal of Theoretical and Applied Mechanics, 54 (4, pp ( M. Dumitriu, C. Crăciun, Evaluation of stability in railay vehicles on basis of bogie lateral acceleration, Applied Mechanics and Materials, , pp ( M. Dumitriu, I. Sebeşan, Ride quality of railay vehicles, (in Romanian, Matrix Rom, Bucharest ( S.Y. Lee, Y.C. Cheng, Hunting stability analysis of high-speed railay vehicle trucks on tangent tracks, Journal of Sound and ibration, 282, pp ( A.H. Wickens, Fundamentals of rail vehicles dynamics: Guidance and stability, Taylor & Francis e-library ( I. Sebeşan, Dynamic of the railay vehicles, (in Romanian Matrix Rom, Bucharest ( M. Dumitriu, Modelling the geometric contact beteen heels and the rails of a track ith horizontal irregularity, Mechanical Journal Fiability and Durability, 1, pp ( UIC 518 Leaflet, Testing and approval of railay vehicles from the point of vie of their dynamic behaviour Safety Track Fatigue Ride Quality ( M. Dumitriu, Numerical synthesis of the track alignment and applications. Part I: The synthesis method, Transport Problems, 11 (1, pp ( ORE B 176, Bogies ith steered or steering heelsets, Report No. 1: Specifications and preliminary studies, ol. 2, Specification for a bogiith improved curving characteristics (1989. Annex 1. The parameters of the vehicle model. m c Carbody mass 2c zc ertical damping of the secondary suspension * m b Mass of suspended bogie c yc Lateral damping of the secondary suspension * m Wheelset mass 2c xc Longitudinal damping of the secondary suspension * J xc J zc The inertia moments of the carbody 2k zc ertical rigidity of the secondary suspension * J xb J zb The inertia moments of the bogie chassis 2k yc Lateral rigidity of the secondary suspension * J x J z J z 2a c The vehiclheelbase k jc Torsional stiffness of the secondary suspension * The inertia moments of thheelset 2k xc Longitudinal rigidity of the secondary suspension * 2a b The bogiheelbase 2c zb ertical damping of the primary suspension ** 2b c The lateral base of the secondary suspension 2c yb Lateral damping of the primary suspension ** 2b b The lateral base of the primary suspension 2c xb Longitudinal damping of the primary suspension ** h c h b1 h b2 r Distance of the carbody mass centre compared to the plan of secondary suspension Distance of the bogie mass centre compared to thheelset axis Distance of the bogie mass centre compared to the plan of secondary suspension The radius of the rolling circlhen heelset occupies the median position on the track * per bogie; ** per axle. 2k zb ertical rigidity of the primary suspension ** 2k yb Lateral rigidity of the primary suspension ** 2k xb Longitudinal rigidity of the primary suspension ** k yr The rail lateral stiffness 169