DURATION OF HIGH SLIDING RATE RESULTING IN CRACK OCCURRENCE IN by sanmelody

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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




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