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are theoretically described. This theoretical knowledge is related to measurements of dynamic torsional moments caused by self-excitement and damping of the system in these driving axles. 2. GEAR MECHANISM AT JZ LOCOMOTIVES ACTING AS TORSIONAL OSCILLATING CHAIN; NATURAL TORSIONAL VIBRATION FREQUENCIES Fig. 1 shows the scheme of this mechanism (without a rotor of driving motor): a) complete torsional chain; b) reduced torsional chain on the wheelset. Physical characteristics of the system comprise: moments of mass DURATION OF HIGH SLIDING RATE inertia, shaft stiffness, dimensions of systems and gear RESULTING IN CRACK OCCURRENCE ratio of mechanism (ir = 1 : 3.65) . IN WHEELSET AXLES OF Frequencies of natural free torsional damping vibrations are calculated and they amount to (for Fig. 1-b scheme) ELECTRIC LOCOMOTIVES [1], [5]: Relja JOVANOVIĆ Aleksandar RADOSAVLJEVIĆ Abstract: Wheelset calculation of electric (and other) locomotives includes: concept of power transmission from electric motor to reducing gears, choice of material for wheelset body, and shaft dimensions in accordance with vmax and vmin speeds. Dynamic impacts are essential at vmax and maximum torsion limited by locomotive adhesion is essential at vmin. However, if in operation a locomotive Fig. 1. Gear mechanism at JZ locomotives acting as hauls a heavy train, particularly on gradients, long and torsional oscillating chain high wheel sliding occurs (stick–slip effect). Then the shaft body is exposed to self-exciting and damping ′ ′ ′ ′ I 1 , I 2 , I 3 , I1 , I 2 , I 3 , I 4 - moments of inertia wheel till torsional oscillations. Frequent running of the kind cause torsional cracks in shaft and fractures as well. A solution gear box,bigger gear, second wheel, rotor of driving calls for a limitation in wheel sliding by adequate motor, coupling and smaller gear; c1 and c 2 - shaft protection and reduced wheel sliding time. The subject matter of this paper is time allowed for sliding and how it ′ ′ stiffness; c1 - torsional shaft stiffness; c 2 - coupling can be calculated because of the frequency of this phenomenon on, former JZ, 441 and 461 series ′ * stiffness; c 3 - shaft stiffness of smaller gear; I 2 - locomotives. reduced moment of inertia both gear on axis of shaft - for new wheels in driving axle: ω sI = 316s −1 ; ( f I' = 50.33Hz ), ω sII = 1917 s −1 ; ( f II' = 305Hz ), Kee words: railway, wheelset, axle, crack , locomotive ' ' - for extremely worn wheels (according to the JZ/UIC 1. INTRODUCTION regulations): Wheelsets of series 441 and 461 electric locomotives, '' ( ) ( ω sI = 388s −1 ; f I'' = 61.81Hz , ω sII = 1975s −1 ; f II' = 317Hz '' ' ) licensed by ASEA Sweden, on the Yugoslav Railways (JZ) with single-stage axle reduction gear are subjected to These values negligibly differ from those that would be the effect of torsional oscillations in operation: both from obtained by neglecting the outer (adhesive) and inner driving moment pulses that cause partial resonance in the damping (in shaft material); all the afore mentioned whole transmission mechanism [1] and from self-exciting however would not intervene with the conclusions of this torsional oscillations when the system is exposed to the paper. high wheel sliding. The results of voluminous theoretical research of torsional oscillations of locomotive driving 3. DRIVING WHEELSET AT HIGH WHEEL system and records of dynamic moments in practice show SLIDING that torsional cracks on their wheelset bodies cause the system self-excitement due to high sliding [1],[2],[3],[4]. 3.1. Natural torsional vibration frequencies and There is a theoretical support for the procedure to find the "darting factor" high sliding time of driving axles until permissible stress limit is reached [τ] and application to the system and size The dynamic scheme of these wheelsets is given in Fig. 2. at the named former JZ, now Serbian Railways, vehicles If the perimeter speed of a point on wheel perimeter 187 vt = R ⋅ δ& , and v is speed of vehicle on the line, it will be & I 1δ&1 + c1 (δ 1 − δ 2 ) = − M t1 sliding κ as follows [5]: * && I 2 δ 2 + c1 (δ 2 − δ 1 ) + c 2 (δ 2 − δ 3 ) = 0 (3) & I 3δ&3 + c 2 (δ 3 − δ 2 ) = − M t 2 ) ⋅ 100 [%] v κ = (1 − & (1) R ⋅δ According to [5], at κ > 6 ÷ 8% , the change of κ = f ( R, v, δ&) and µ = f (κ ) can be expressed for physical sizes of these shafts at any v=const. and for any wheel, with an error of 2 to 3% as follows: 47 R & κ= ⋅ δ − 44 (4) v ⎛ µ1 − µ 2 ⎞ µ − µ2 µ= ⎜ ⎜ ⋅κ 2 + µ 2 ⎟ − 1 ⎟ κ −κ ⋅ κ (5) ⎝ κ 2 − κ1 ⎠ 2 1 By changing of (eqn 4) into (eqn 5) it will be: Fig. 2. The dynamic shaft load at high wheel sliding µ = a − bδ& (6) where: Depending on sliding, friction coefficient changes in all wheel-rail contact points. This phenomenon is known as µ1 − µ 2 µ − µ 2 47 R "stick-slip" effect [1], [4] (Fig. 3). Fig. 3 shows that at a= (κ 2 + 44) + µ 2 > 0; b = 1 ⋅ > 0. κ 2 − κ1 κ 2 − κ1 v least small sliding is required for rail wheel movement. The highest friction (adhesion) coefficient of 3÷6% is By changing of (eqn 6) into (eqn 2) it will be: found at wheels sliding on dry rails and over oily wheel- rail contact surfaces at κ=2÷3%. Then µ drops to its ( ) M t 1, 2 = Z1, 2 ⋅ R ⋅ µ (κ ) = Z ⋅ R ⋅ a − bδ = A − Bδ & & (7) minimum value µ2 (µ2‘). where: A = Z ⋅ R ⋅ a > 0 ; B = Z ⋅ R ⋅ b > 0 ; s = − B / I , k1, 2 = c1, 2 / I , k1', 2 = c1, 2 / I * . When κ is high, it will be easy to obtain the state of & v << R ⋅ δ , for low running speeds. As adhesion coefficient µ is for dry rails µ = µ ′ , for wet rails it will be µ = µ ′′ ; the adhesion moments are M t′,1, 2 and M t′′,1, 2 , respectively. When (eqn 7) for any rail moisture is included into (eqns 3), the characteristic "λ - polynomial" will be obtained Fig. 3. Changes of adhesion coefficient µ depending on and its coefficients are negative at odd degrees [1], [5]: both wheel-rail relation (dry-1, wet-2) and ′ ′ λ 6 + 2 sλ5 + ( s 2 + k1 + k 2 + k1 + k 2 )λ 4 + s( k1 + k 2 + wheel sliding κ ("stick-slip" effect) ′ ′ [′ ′ ′ ′ ] 2k1 + 2k 2 )λ3 + s 2 (k1 + k 2 ) + k1 k 2 + k1 k 2 + k1 k 2 ⋅ λ 2 (8) The area of self-exciting torsional oscillations varies within 6%< κ <100% [1], [5]. ′ ′ + s ( k1 k 2 + k1 k 2 )λ = 0 Dissipation moments of friction at high wheel sliding Here, the factor s of system damping are given in (eqn 7). (and at vehicle speed below 20÷30km/h; though this also According to (eqn 7), s < 0 , all factors are negative in applying to higher speeds with κ > 6 ÷ 8% ) are (for some (eqn 8) where the s is an odd degree. This means that the wheels) [1], [5]: oscillating system, at κ > 6 ÷ 8% , is dynamically M t1, 2 = FT 1, 2 ⋅ R = Z1, 2 ⋅ R⋅ µ (κ ) [kNm] (2) unstable, that is, as a function of the solution λ = + h ± ωj , h > 0 and e ht > 0 [6]. where: Z 1, 2 = Z 1 = Z 2 ≈ 100kN - wheel pressure In the literature [1], [5] relating to physical states of gear (approximate static pressure); R = 0.59 ÷ 0.625m - radius mechanism and adhesion and the most unconvenient "λ of worn to new wheel, along the circle of rotation; µ (κ ) - polynomial" the following values for λ are obtained by adhesion coefficient as a function of sliding. computer support: For dynamic state (Fig. 2), under the conditions of high λ1 = 0 ; λ 2 = 3.82 + 0 ⋅ j ; λ 3,4 = 12.436 ± 390.8 ⋅ j ; wheel sliding, the equations for free damping torsional (9) oscillations of locomotive shaft will be: λ 5,6 = 9.3 ± 1994.8 ⋅ j. 188 This indicates that hmax = 12.436 at natural frequency B where [5]: s = − ; ( s < 0 , system damping against −1 ω = 391s , i.e., "darting factor" of system is very high in I c c A this case. adhesion at κ >6%); k1 = 1 ; k 2 = 2 ; p = − ; I1 I3 I3 3.2. Gear system at forced torsional oscillations c1 c2 1 ′ k1 = ′ ; k2 = ; I2 = I2 + * I 4 = 33.345 Nms 2 ; I 3 = I1 These gears are driven by electric motors operating at * I2 * I2 ir2 rectified single-phase current 25kV, 50Hz. The shaft itself (Fig. 2) is driven by the moment M z = M tz ; its function c A c c may be expressed by moment M m in small gear (of ′ * ′ * k2 = 2 ; p=− ; k1 = 1 ; k2 = 2 ; I3 I3 I2 I2 motor) as [7], [8]: 1 I2 = I2 + 2 I4 = 33345Nms; I3 = I1 * . 2 M z = M (t ) = ir ⋅ M m (10) ir and the driven M m is as follows: I = I 1 = I 2 = 155.12 Nms 2 , (R = 0.625 m ); ; ⎡ π ⎤ M m = M om ⎢1 + a1 sin Ωt + a 2 sin (2Ωt + )⎥ (11) I = I1min= I2min=10104Nms, (R=0.59m) . 2 ⎣ 2 ⎦ where: ir = 3.65 - transmission gear ratio, a1 = 16 / 33 , c1 = 94.29 ⋅ 10 6 Nm rad −1 ; c 2 = 9.31 ⋅ 10 6 Nm rad −1 . a 2 = −1 / 33 ; M om - mean value of motor moment; Damping s and factors p are analyzed for four main −1 conditions [1], [5] at v = 2.35 m / s , Z = 9.81 ⋅10 4 N : Ω = 2 ⋅ 2πf = 200π s - line frequency of rectified current. a) dry rails and new wheels Now, the moment in driven gear, at Ω p = Ω / i r : s = −17 s −1 ; p = −229.3 s −2 ⎡ π ⎤ M z = M z (t ) = ir ⋅ M om ⎢1 + a1 sin Ω p t + a 2 sin (2 Ω p t + )⎥ b) dry rails and extremely worn wheels ⎣ 2 ⎦ ⎡ π ⎤ s = −24 s −1 ; p = −323.08 s −2 M z (t ) = M o ⎢1 + a1 sin Ω p t + a 2 sin (2 Ω p t + )⎥ (12) ⎣ 2 ⎦ c) wet rails and new wheels Therefore, M z (t ) is beating moment with mean value s = −7 s −1 ; p = −102.54 s −2 M 0 . For all wheelsets and supply of current, f = 50 Hz ; d) wet rails and extremely worn wheels it is circular frequency in the shaft gear, for the main pulsing tone of transformed current of motor: s = −10 s −1 ; p = −148.6 s −2 Ω 200π Ωp = = = 172s−1; 2Ωp = 344 s−1, Therefore, p1≈ p2≈ p; s1≈ s2≈ s. ir 3,65 Elasticity factors ( k1, 2 , k1', 2 ) are: fΩp =100 Hz; f2Ωp = 200 Hz. a) for new wheels: k1= 0.607845715⋅106; k2= 0.06002⋅106 For further analysis (eqn 12) could be written as: k1’= 2.82771⋅106; k2’= 0.2792023⋅106 rad -1 -1 s M z (t ) = M o + M ′(t ) + M ′′(t ) (13) b) for extremely worn wheels: k1= 0.9331947⋅106 ; where: k2= 0.0921417⋅106 rad -1 -1 s M 0 = const. ; M ′(t ) = M o ⋅ a1 sin Ω p t = M ′ ⋅ sin Ω p t ; ′ ′ where: k1 = ω11 ; k 2 = ω 23 ; k1 = ω12 ; k 2 = ω 22 . 2 2 2 2 π π At forced damping oscillations of wheelset (Fig. 2), M ′′(t ) = M o ⋅ a 2 sin (Ω p t + ) = M ′′ sin (2 Ω p t + ) including the change (eqn 12), that is, (eqn 13) in (eqn 2 2 14), it will be: The stated equation system (eqns 3), with the help of function (eqn 6) and (eqn 7) at natural damped torsional δ&1 + sδ&1 + ω11 (δ 1 − δ z ) = p & vibrations can be expressed by the following system of δ&z + ω12 (δ z − δ 1 ) + ω 22 (δ z − δ 2 ) = M o + M ′(t ) + M ′′(t ) (15) & δ&2 + sδ&2 + ω 23 (δ 2 − δ z ) = p equations & & δ&1 + sδ&1 + k1 (δ 2 − δ 1 ) = p For expected and mentioned damping s at v≈8.4 km/h = 2.35 m/s, for new and extremely worn wheels, for the & δ&2 + k1 (δ 2 − δ 1 ) + k 2 (δ 2 − δ 3 ) = 0 ′ ′ (14) shaft of the said stiffness c1,2 as well as for the factors of pulsing moment in driven shaft gear, according to (eqn & δ&3 + sδ&3 + k 2 (δ 3 − δ 2 ) = p 13): 189 a) for dry rails and new wheels (µ =0.303): δ 3 − δ 2 = e ht [( A3 − A2 ) cos ωt + ( B3 − B 2 ) sin ωt ] + M0 = 37.2 kNm; M′ = 18 kNm; M″ = -1.13 kNm; ( A3 − A2 ) cos Ω1t + ( B3 − B2 ) sin Ω1t + ′ ′ ′ ′ (18) b) for dry rails and extremely worn wheels: ( A3′ − A2′ ) cos 2Ω1t + ( B3 − B2 ) sin 2Ω1t ′ ′ ′ ′ M0 = 35 kNm; M′ = 17 kNm; M″ = -1.06 kNm; If the constants would be adopted: A3-A2=K1;B3-B2=K2; c) for wet rails and new wheels (µ =0.163): ′ ′ ′ ′ ′ ′ ′ A3 − A2 = A3 A3, 2 ; B3 − B2 = B3,2 ; M0 = 20 kNm; M′ = 9.7 kNm; M″ = -0.606 kNm; d) for wet rails and extremely worn wheels: ′ ′ ′ ′ ′ ′ A3′ − A2′ = A3′, 2 ; B3′ − B2′ = B3′,2 , M0 = 18.87 kNm; M′ = 9.15 kNm; M″ = 0.57 kNm, expression (18) gives: the system (eqns 15) is computer calculated. Therefore, δ 3 − δ 2 = e ht (K 1 cos ωt + K 2 sin ωt + A3, 2 cos Ω 1t ) ′ for Ωp1=Ω1 and Ωp2=2Ω1; Ω1=172 s-1 the coefficients Ai’ ′ ′ ′ + B3, 2 sin Ω1t + A3′, 2 cos 2Ω1t + B3′, 2 sin 2Ω 1t = and Bi’ (i=1,2,3) are found; for the following functions of the angles of deformation of the system: e ht K 12 + K 2 sin(ωt + θ ) + 2 (19) ′ δ 1 = e ht ( A1 cos ωt + B1 sin ωt ) + A1 cos Ω 1t + ′ ′ ( A3, 2 ) 2 + ( B3, 2 ) 2 sin (Ω 1t + ϕ1 ) + ′ ′ ′ B1 sin Ω 1t + A1′ cos 2Ω 1t + B1′ sin 2Ω 1t ′ ′ ( A3′, 2 ) 2 + ( B3′, 2 ) sin( 2Ω1t + ϕ 2 ), ′ δ 2 = e ( A2 cos ωt + B 2 sin ωt ) + A2 cos Ω 1t + ht (16) ′ ′ ′ B 2 sin Ω 1t + A2′ cos 2Ω 1t + B 2′ sin 2Ω 1t where: ′ δ 3 = e ht ( A3 cos ωt + B3 sin ωt ) + A3 cos Ω 1t + K2 B − B2 θ = arctg = arctg 3 ; B3 sin Ω 1t + A3′ cos 2Ω 1t + B3′ sin 2Ω1t ′ ′ ′ K1 A3 − A2 K = K 12 + K 2 ; 2 3.3. Time of permissible duration of system self- ′ B3, 2 ′ ′ B3 − B 2 excitement ϕ1 = arctg = arctg ; ′ A3, 2 ′ A3 − A2′ As the ratio of shaft stiffness is c2<<c1 (Fig. 2), it was ′ B3′, 2 ′ ′ B3′ − B 2′ proven in practice that the part of shaft between masses ϕ 2 = arctg = arctg . ′ A3′, 2 ′ ′ A3′ − A2′ * I z and I 2 fails due to torsional cracks and deformation (δ 1 − δ 2 ) but (δ 2 − δ z ) is not important from the aspect For all wheel inertia Ω1 and 2Ω1, of the stiffness c1,2, the corresponding coefficient values ( Ai′ ) and (Bi′ ) , as well of this paper. Also, the pulsing is important in connection as the subsequent angles ϕ1,2 are calculated by the ( with the natural frequency of system ω I = 390 s −1 as ) computer for analyzed state of wheels high sliding. In well as system function at self-excitement from the order to define the corresponding angle θ, and the "darting factor" h=hmax=12.436, for the reason that e ht is constant factor K = K 12 + K 2 , 2 the initial shaft the most dangerous. This is the case when the wheels are movement will be used, as follows (eqn 19): extremely worn out and at high sliding (κ >6÷8%) on dry rails; then the friction between the wheels and rails is "the δ 3 − δ 2 = Ke ht sin (ωt + θ ) + C1 sin(Ω1t + ϕ1 ) + (20) most negative" [1]. C 2 sin (2Ω1t + ϕ 2 ), When applying theoretical analysis in this paper to where the constants are: practice, where locomotives are supplied from 50 Hz network and the system physical characteristics are ′ ′ ′ ′ C1 = ( A3, 2 ) 2 + ( B3, 2 ) 2 ; C 2 = ( A3′, 2 ) 2 + ( B3′,2 ) 2 . identical to those of JZ locomotives, it would be necessary to find permissible time for this process for the Using the differentiate expression (eqn 20) per time, it worst case of high wheel sliding (frequent at JZ); shaft will be obtained: body deformation, c=c2 of stiffness, does not exceed the deformation which causes torsional stresses δ&3 − δ&2 = Ke ht [h sin (ωt + θ ) + ω cos (ωt + θ )] + (21) τmax=τpermited=[τ] in theory of elasticity. Due to torsion, the C1Ω1 cos ϕ1 + 2C 2 Ω1 cos(2Ω1t + ϕ 2 ) deformation limit should be [9]: Adopting the initial conditions that: t=0, M t2 M M0 (δ 3 − δ 2 ) max = 0.25 [o / m ′] = π 720 [rad ⋅ m ] −1 (17) δ&3 − δ&2 = 0; δ 3 − δ 2 = c2 , (where: M t 2 = 2 = 2 adhesion moment in more distant wheel from gear), from An example for this calculation t=tpermited is as follows. the expression (eqn 20) and (eqn 21) will be obtained: Therefore (eqn 16) is [1]: for the first initial condition: 190 K (h sin θ + ω cosθ ) + C1Ω1 cosϕ1 + 2C 2 Ω1 cosϕ 2 = 0 (22) occurrence of torsional cracks and the fracture of these shafts, which will not occur immediately. for the second initial condition: 3) Attempts to reduce the danger from shaft cracks occurrence using the bodies made of material with M t2 K sin θ + C1 sin ϕ1 + C 2 sin ϕ 2 = , (23) Rm>1200 MPa, were negative, and the suggestion to solve c2 the problem by limiting the time t of sliding was not where: K and θ are unknown constants of initial applied. M t2 4) Analyzed shafts are calculated on standard principles conditions and the angle of shaft body deformation for these assemblies, which are in use at railways c2 worldwide. The paper shows that it should be important to of stiffness c2, due to the static activity of moment Mt2 to recognize the need for more extensive calculation by more distant wheel from the gear. introducing operating problems, calling them "new Solving the system of equations (eqn 22) and (eqn 23) for problems" that occurred with the driving designs that the θ and K, it will be obtained: ASEA Company (Sweden) applied to their electric Ω1 C1cosϕ1 + 2C 2 cosϕ 2 h locomotives, 441 and 461 series, on the Yugoslav ctgθ = ⋅ − (24) Railways (similar to the JZ surrounds). ω M t2 ω C1sinϕ1 + 2C 2 sinϕ 2 − c2 1 ⎡ M t2 ⎤ REFERENCES K= ⋅⎢ − (C1 sin ϕ1 + C 2 sin ϕ 2 )⎥ (25) sin θ ⎣ c 2 ⎦ [1] JOVANOVIĆ, Relja, Naponsko stanje lokomotivskih For afore mentioned: h, ω, Ω1, ϕ1, ϕ2, vratila u eksploatacionim uslovima - Locomotive Axle Stress in Operation (Doctoral Thesis), ′ ′ ′ ′ A3, 2 , A3′, 2 , B3, 2 , B3′, 2 and for the stiffness c2, constants University of Mechanical Engineering, Belgrade, C1, C2 are calculated; from (eqn 24) and (eqn 25) the 1978, (in Serbian) angle θ and the constant K [1] is calculated, too. As the [2] JZ und ÖBB, Schwingungs und Lauftechnische body, c2 of stiffness, is lt≈1000 mm long, the expression untersuchungen an einer E-Lok Reihe 441 der (eqn 20) will be transformed into a form suitable for Jugoslawischen Staatbahner, Wien, 1974 obtaining permissible time for "darting of system": [3] Report UIC ORE B 44/ RP10/ XI 73, Die π Schwingungen des antriebsystems elektrischer ≥ Ke ht sin(ϖt + θ ) + C1sin(Ω1t + ϕ1 ) + C 2 sin(Ω 2 t + ϕ 2 ) (26) Lokomotiven bei freiem schleudern der Räder auf den 720 Schienen, 1973 Members with C1 and C2 can have resonant action so their [4] JUD, W. , La resistance en service des essieux, influence should be examined separately; at those Simposium, Tenu á Poprod, 1974 systems, which are unstable at self-excitement, the factor [5] JOVANOVIĆ, Relja, Odredjivanje frekvencija Keht is important for its influence upon growth of angle of sopstvenih slobodnih prigusenih torzionih oscilacija i deformation (δ3-δ2); that is, it will have an effect upon uslova dinamicke stabilnosti oscilijuceg sistema duration time t for obtaining [τ], when (δ3- lokomotivskih vratila - Determination frequency of δ2)≤ π 720 [rad ⋅ m ]. −1 own free damped torsional oscillations and conditions of dynamical stability oscillating system of locomotive axles, Tehnika, No 1, Belgrade, 1979 (in For hmax=12.436, it follows that tmin=0.12 s which is very Serbian) short. Therefore, K= − 982 ⋅ 10 −6 [1]. However, [6] HARTOG, DEN, Vibracije u masinstvu - Vibration measurements of dynamic values, particularly Mtz=Mtzdin in mechanical engineering, Gradjevinska knjiga, at sliding in the period over 4 and more seconds, it will be Belgrade, 1972 (in Serbian) proved that inner damping in the body (material, steel) [7] JOVANOVIĆ, Relja, Funkcija promjene obrtnog will restrain the extreme of the member (Keht) to momenta vucnog motora elektrolokomotiva -teorijsko M tz max ≈ 15.5M tz [2], [10], [11], [12], [13], [14]. din stat razmatranje - Function torque changing of traction motor of electric locomotives–theoretical discusses, Zeleznice, No. 1, Belgrade, 1979 (in Serbian) 4. CONCLUSIONS [8] RADOJKOVIĆ, Božidar, Električna vuča - Electric traction, Naučna knjiga, Belgrade, 1974 (in Serbian) 1) When high wheel sliding occurs (higher than 6 to 8%, [9] SERENSEN, S.V., Nesuscaja spososbnost i rasceti approximately) on locomotive shafts, these are exposed to detalej masin na procnost, Moscow, 1975 (in self-excitement torsional oscillations of a great factor Russian) (Keht), where hmax>12.436 on dry rails and when wheels [10] JOVANOVIĆ, Relja et al., Calculation of Electric are minimum worn out. Approximate calculation Locomotive Axles and their Reliability in indicates that the shaft, in this situation, will be Exploitation at High Rate of Skidding, 6th Mini torsionally stressed to [τ] already at t=0.12 s in extreme Conference on Vehicle System Dynamics, case and inner damping in material makes it possible to Identification and Anomalies, Budapest, 1998, pp have tmax≈ 1÷ 2 s. 209-217 2) Inner damping of shaft material limits the final value [11] JOVANOVIĆ, Relja et al., Possible Reduction of (Keht) to about 15.5 (δ 3 − δ 2 ) stat , indicating the quick Wheel Flange Wear and Occurrence of Cracks on the 191 Axle-Drives by Improving the Electric-Locomotive CORRESPONDENCE Dynamic Movement, 6th Mini Conference on Vehicle System Dynamics, Identification and Anomalies, Relja JOVANOVIĆ,Full-time Prof.,Ph.D, Budapest, 1998., pp 171-181 Institute of Transportation CIP, [12] JOVANOVIĆ, Relja et al., Uzroci pojave i Mechanical Engineering Department, mogucnosti eliminacije naprslina i lomova pogonskih Nemanjina 6, osovina zeleznickih vozila - Causes of Occurrence 11000 Belgrade, Serbia, and Possibility of Eliminating Cracks and Fractures Faculty of Mechanical Engineering East of Railway Vehicle Motor Axles, Edvard Kardelj Sarajevo Institute, MIN, Niš, 1987 jovanovicr@sicip.co.yu [13] DOLEČEK, Vlatko, Dinamika sa oscilacijama - Dynamic with Oscillations, Faculty of Mechanical Aleksandar RADOSAVLJEVIĆ,Ph.D, Engineering University of Sarajevo, (in Serbian) Research Fellow, 1981 Institute of Transportation CIP, [14] JOVANOVIĆ, Relja, RADOSAVLJEVIĆ, Mechanical Engineering Department, Aleksandar, Impact of High Whell Slipping on the Nemanjina 6, Life Time of Rubber-Metal Elements and Wheel Sets 11000 Belgrade, Serbia, on Electric Locomotives, IHHA, Moscow, 1999, pp radosavljevica@sicip.co.yu 433-438 192