The article deals with the conclusions of analytical modeling cargoes displacement relative wagon and tension in fastening on exposure to spatial system of forces

Elena TIMUKHINA

Khabibulla TURANOV

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TRANSPORT PROBLEMS                                                                                                               2008 PROBLEMY TRANSPORTU                                                                                          Tom 3 Zeszyt 3     Khabibulla TURANOV, Elena TIMUKHINA* Ural State University of Railway Transport (USURT) Kolmogorov st., 66, Ekaterinburg 620034, Russia *Corresponding author. E-mail: ETimuhina@user.usurt.ru      ANALYTICAL MODELLING CARGOES DISPLACEMENT IN WAGON AND TENSION IN FASTENING  Summary. The article deals with the conclusions of analytical modeling cargoes displacement relative wagon and tension in fastening on exposure to spatial system of forces.    АНАЛИТИЧЕСКОЕ МОДЕЛИРОВАНИЕ СДВИГА ГРУЗА В ВАГОНЕ И НАТЯЖЕНИЙ В КРЕПЛЕНИЯХ  Аннотация.  В  статье  изложены  выводы  аналитических  формул  сдвига  груза относительно  вагона  и  натяжений  в  креплениях  при  воздействии  пространственной системы сил.   1. FORMULATION OF A PROBLEM  Up to date there has not been developed a theory of cargo fastening on the open rolling stock un-der the impact of spatial force systems on the system „wagon-cargo-fastening”.The regulations do not include assessment of safety operation on the criterion of cargo shear relative to the wagon because in all availlable regulations there isn’t any notion of cargo shear in a wagon, to say nothing of calculating its value. Due to this fact there has not been done any assessment of operation safety of a wagon with asymmetrically allocation cargo both lengthwise and crosswise the wagon as compared with the ope-ration of wagons with symmetrical allocation. It would be reasonable to assess operation safety using the criterion of cargo shear relative to the wagon by constructing summarized dynamic and mathema-tical models of fastenings of asymetrically allocation cargo both lengthwise and crosswise the wagon as compared to symmetrical allocation. In connection with this let us assume as a basis of a mathema-tical model that the action of spatial system of forces just as in reality is received by the main (the wagon) and additional constraints (flexible elastic and persistent wooden fastening means). Let us consider the general case when cargo weighing G , is allocation asymmetrically (or symmetrically) relative to the longitudinal and transversal symmetry axes of the wagon the physical model of which is shown in Fig. 1,а,b.   70                                                                                                                        K.Turanov, E.Timukhina    Fig. 1a. Physical modeling allocation cargoes in wagon, running on the curve section of railway descent (side view) Рис. 1a. Физическая модель размещения груза в вагоне, движущегося по кривому участку пути на спуск (вид сбоку)     Fig. 1b. Physical modeling allocation cargoes in wagon, running on the curve section of railway descent (top view) Рис. 1b. Физическая модель размещения груза в вагоне, движущегося по кривому участку пути на спуск (вид сверху)  Analitical modelling cargoes displacement....                                                                                        71  In Fig. 1 the following symbols are accepted  jand  i are indexes showing the numbers of rack brackets in wagon and elastic fastening elements ( р n i , 1 =  - a number of flexible elastic fastening ele-ments); 2L , 2B  и 2 H  - are cargo length, width and height accordingly;  i a  и  ai a  - are the projec-tions of flexible elastic fastening elements of one direction on the longitudinal wagon axis x ( pi a  и api a  -are also of another direction); i b  и  ai b  - are projections of flexible elastic fastening elements of one direction on the transversal axis of wagon  y  ( pi b  и  api b  - are also of the other direction);  i h , ai h ,  pi h  и  api h  - are projections of flexible elastic fastening elements on the vertical axis of wagon z ;  i l  и  ai l - is the length of flexible elastic fastening elements of one direction ( pi l  и  api l  - are also of the other direction);  wi l  и  wai l - are projections of the length of flexible elastic fastening elements of one direction on the transversal axis of wagon  y ( wpi l  и  wapi l  - are also of the other direction); i a  и  ai a -a re the angles which are formed by the fastening elements with the flat surface of the floor of wagon of one direction ( pi a  и  api a  - are also of the other direction);  i b  и  ai b  - are the angles that are formed by the projections of fastening elements ( iH l ,  aiH l ,  piH l ,  apiH l ) on the plane of the floor of the wagon of one direction with axis  x ( pi b  и  api b  - are also of the other direction ( i 0 b ,  ai 0 b ,  pi 0 b  и  api 0 b  - are the same angles only they are acute);  h D  - is superelevation; 2S  -is the distance between wheel rolling circles of wagon wheelset of gauge 1520 mm (2S  = 1580 мм); q - is the angle characterizing superelevetion; z – is the angle taking into account tilting of the frame of the wagon with cargo being displaced on the transversal axis of wagon  y  by the value  yM ± .   1.1. Man-made assumption  Let us assume that rolling stock is moving along a downgrade at the angle  0 y both in the regime of release and the regime of service braking at a speed of  v  on a curve with curvature radius  r of trajectory in the particular point of the curve. The cargo is fastened to the wagon binding devices by the fastening flexible elastic elements at points  j A ,  aj A ,  pj A  и  apj A ,and to its load gripping hinges – at points  i M ,  ai M ,  pi M and  api M  (Fig. 1). Let us assume that the cargo is placed on the floor of the wa-gon and is kept from shear by flexible elastic and persistent fastening means, the wagon floor being the main constraint and flexible elastic fastening elements and persistent wooden bars (if they are available) the additional constraint. [1, 2]. We assume that external constraints undergo the impact of cargo weight G , longitudinal ( ex I ), lateral ( ey I ), vertical ( ez I ) transient forces of inertia; forces of aerody-namic resistance  в F  and axial force of inertia ( n I ), taking into account the motion of rolling stock on a curve. We’ll take into consideration the fact that the maximum normative values of transversal tran-sient accelerations  ex ex a a =max  are equal to  g aex 3 , 0 =  −on a tangent,  g aex ) 2 , 1 7 , 0 ( - =  −during service braking,  g aex ) 2 2 , 1 ( - =  − during impacting of wagons in the hump yard, and the vertical transient accelerations  ez ez a a =max , due to aberrations in track maintenance −  g aez ) 66 , 0 46 , 0 ( - = . According to this it is possible to assume  G Iex 3 , 0 =  −on the tangent,  G Iex ) 2 , 1 7 , 0 ( - =  −during service  braking  G Iex ) 2 2 , 1 ( - =   −during  impacting  of  wagons  in  hump  yards  and G Iez ) 66 , 0 4 , 0 ( - = . Vertical transient force of inertia  ez I  occurs (i. e.  0 ¹ ez I ) during the movement 72                                                                                                                        K.Turanov, E.Timukhina  of the wagon with cargo on the tangent without braking and in the regime of release along a down-grade and in the regimes of braking and impacting wagons    2. METHODS OF SOLUTION  Let us apply the principle of clear constraints and the law of relative transient motion [1-3]. We introduce the notion “shearing” and “retaining” forces [3], acting lengthwise and crosswise the wagon and received by the fastenings of one (longitudinal or and transverse) direction without taking into account friction forces  t F  and elastic fastening forces  i R , but with account of tension of preliminary twistings of wire fastenings. Definition 1. „Shearing” force is the sum of all forces causing the shift of cargo relative to the flo-or of the wagon. Statement 1.”Shearing ‘ force is a harmful force for cargo threatening traffic safety and hazardous for cargo safekeeping and it is the major force damaging the elements of rolling stock creating poten-tially the situation of near-accident. Definition 2. „Retaining” force is the sum of all reactive forces keeping the cargo from shifting relative to the floor of the wagon. To reactive forces with points of application can be referred:  k N j F i F R y x + + = t t  − is the reac-tion of the main constraint (wagon floor) to unknown application point data  R x  and  R y ,  x тр x F F . = t , y тр y F F . = t  and  N  being the projections of reaction of constraint  R  on coordinate axes  x ,  y  and z ;  k R j R i R R iz iy ix i + + =  −tension in  i-х flexible elastic elements of fastening cargo with given points of application  i x ,  i y  и  i z , ix R ,  iy R ,  iz R  −being the projections of constraints i R  on coordinate axesx ,  y  and  z ;  i R0  −are tensions of preliminary twistings in  i-flexible elastic fastening elements where ix R0 ,  iy R0 ,  iz R0  −are projections of tensions  i R0  on coordinate axes  x ,  y  and  z ;  р n i , 1 =  − the number of flexible elastic fastening elements. Statement 2."Retaining” force is a beneficial force for cargo ensuring safety of movement, cargo safekeeping and preventing the elements of rolling stock from damage.   3. SOLUTION RESULTS  Let us consider a general case when cargo shear ( s D ) occurs relative to the action of the resultant of spatial forces systems ) (i F D  (Fig.2). Fig. 2 contains the following symbols:  i M  - are points showing mounting hinges and their pro-jections on horizontal and vertical planes;  i l  и  i a ,  i b ,  i h  - are lengths of flexible elastic fastening elements and their projections;  s D  - cargo shear in relation to the action of the resultant of spatial forces systems ) (i F D , which is to be found;  i x  - angle characterising cargo shear in the plane of the wagon floor. In accordance with the method of identification of deformation at minor displacements let us pro-ject a „new” position of cargo fixation point (point  ki M ) on the „original” or „old” direction  i iM A of flexible  elastic  element  [1,  2].  After  simple  mathematical  computation  we  get  the  dependence  of lengthening of flexible elastic fastening elements on cargo shear in the plane of the wagon floor   Analitical modelling cargoes displacement....                                                                                        73         a  b  Fig. 2. General modelling cargoes displacement: a) - axonometric; b) - top view Рис. 2. Обобщённая модель сдвига груза: a) - аксонометрия; b) - вид сверху                                                    l + l D = D) ( ) ( sin cosiii iiii lblas l                                                           (1) where  i a  и  i b  – are the projections of the length of fastening wire  i l  lengthwise and crosswise the wagon; ) (i l  – is the angle characterising the direction of spatial forces systems relative to the lon-gitudinal axis of the wagon. Thus the lengthening in the flexible elastic fastening element under the action of spatial forces system occurs only when there is cargo shear in the plane of wagon floor by the value  s D . Leaving out interim mathematical computation we find the dependence of cargo shear in the plane of the wagon cargo under the action of the resultant spatial forces systems  74                                                                                                                        K.Turanov, E.Timukhina                                    ( ) ( )2 ) ( 2 ) (12) (sin cos 854 . 7iiii ini iiiiD C BlndFsрl + lD= D∑=                                    (2) where  i d ,  i n  и  i l  – is the diameter (mm), the number of threads (pcs.) and the length of fastening wire;  i B ,  i C  и  i D  - are nondimensional variables   l + l =) ( ) ( sin cosiii iiii lblaB ;    + =iiiii lalhf C ;    + =iiiii lblhf D . Taking into account that i h  – is the projection of the length of fastening wire  i l  on vertical axes; f  – is friction coefficient between the surfaces of wagon floor and cargo. In formula 1 let us present the resultant of spatial forces systems ) (i F D  (Fig 1), received by the flexible elastic fastening elements as                                        j F i F Fiyixi ) ( ) ( ) ( D + D = D ,                                              (3) where the projections of constituent forces on longitudinal and transverse axes and on the direction cosine are presented as                                      ; cos) (1 1) ( iniizniixixр рR f R F l + = D ∑ ∑= =                                                               (4)                                  ) (1 1) ( sininiizniiyiyр рR f R F l + = D ∑ ∑= =;     ) () () (0 cos iix iFFDD= l ,                                  (5) It is known that in expressions (4) and (5) the projections of elastic forces ( ix R , iy R  и  iz R ) i-х flexible elastic fastening elementsг are connected by the following correlations                                iii ix laR R = ;iii iy lbR R = ; iii iz lhR R =                                                              (6) In its turn, in (4) и (5) ) (ix F D and ) (iy F D re presented by the following correlations                                   ) ( 0 ) ( cosi e ixix F F F l - D = D t ;  ) ( 0 ) ( sini e iyiy F F F l - D = D t                                 (7) where ) ( 0 ix F D – is the longitudinal force received by the fastenings of one direction as the difference of „shearing” and „retaining” forces                                                    ix уд x сдix F F F . .0 - = D                                                                        (8) ) ( 0 iy F D – s the transverse force received by the fastenings of one direction as the difference of shearing and retaining forces                                                 iy уд y сдiy F F F . .0 - = D                                                                           (9) ) (i l  – the direction angle where i  raised to a power means that the angle depends on the number of flexible elastic fastening elements and it has only one value; e Ft  – is friction force                                    + q ¢ + + J + y ¢ ++ - q u + y=∑=tр niiz вy n вxezeR F I FI Gf F10 00 00 sin ) ( ) sin() cos ) cos( (                       (10) where  f  - is sliding friction coefficient ( сц f f 7 , 0 =  taking into account that  сц f  - is cohesion fric-tion coefficient between contacting surfaces of cargo and wagon floor( taken according to reference data.); Analitical modelling cargoes displacement....                                                                                        75  0 y  и  0 J  - is track gradient (the angle of ascend and the angle of descend in relation to the hori-zontal line) and the angle of „yawing” of the wagon with cargo in case of emergency or service brak-ing of the train moving both on a curve and a tangent and also in case of impacting of wagons in the hump yard. In expression (8) „shearing” and „retaining” forces are equal                              ) sin( 0 0 . u + y + = G I F ex x сд ;   ) ψ cos( 0 0 01. J + ¢ + =∑=вхniix ix уд F R Fр                  (11) In expression (9) „shearing” nd „retaining” forces are equal                          q ¢ + + = cos ) ( . вy n ey y сд F I I F  ;   . 0 sin1. ∑=+ q =р niiy iy yд R G F                              (12) On the basis of the value of cargo shear in the plane of the wagon floor ( s D ) we find ten-sion  i R  in i-х flexible elastic fastening elements according to the formula                                          [ ] iniiiii i R Blnd s Rр£ × × D = ∑=12 854 . 7                                                          (13) where [ R D ] - is the admissible value of tension in fastening defined according to specification for cargo placement and fastening in wagons and containers depending on the number of threads  i n  and diameter  i d  of fastening wire. Under the action of spatial forces systems  i R  in i-flexible elastic fastening element on the system „wagon-cargo-fastening” it is necessary to find according to the formula (13) only for those fastening elements the final length of which  ki l  is larger than the original  i l  Statement 3. Cargo shear in relation to the wagon occurs (i.e.  s D  > 0) only when ) (i F D  > 0.  Statement 4. Break of flexible elastic fastening elements doesn’t’ happen only when the condition s D  £ [ s D ] is observed where [ s D ] - is the admissible value of shear cargo defined according to the value (mm) [ R D ].   4. CONCLUSION  Analyzing the received results of mathematical modelling of cargo fastening in a wagon it is nec-essary to note that for the first time there have been derived generalized formulas for defining cargo shear  and  tension  in  flexible  elastic  fastening  elements  under  the  action  on  mechanical  system „wagon-cargo-fastening” of spatial forces systems. The deduced formulas take into account physic-geometrical characteristics of elastic elements (i. e.  l d n E , , , ), the values of external forces (G ,  ex I , ey I ,  ez I ,  в F ), received by the fastenings and cargo and the condition of contacting surfaces of cargo and wagon floor by means of friction coefficient. The derived analytical formulas serve for assessing rolling stock movement safety with asymmetrical and symmetrical allocation of cargo centre- of –mass both lengthwise and crosswise the wagon as a mechanical system „wagon-cargo-fastening”. The results of the research can be looked upon as an important contribution in the theory of cargo al-location and cargo fastening and are practically oriented making it possible to work out a program for calculation cargo fastening in a wagon using computing aids.   Bibliography  1. Туранов Х.Т., Бондаренко А.Н.: Теоретическая механика в задачах погрузки-выгрузки и перевозки грузов в вагонах. - Екатеринбург: УрГУПС, 2006. 76                                                                                                                        K.Turanov, E.Timukhina  2. Туранов Х.Т.: Размещение и крепления грузов в вагонах. - Екатеринбург: УрГУПС, 2007. 3.  Комаров  К.Л.,  Яшин  А.Ф.:  Теоретическая  механика  в  задачах  железнодорожного транспорта. - Новосибирск: Наука, 2004.   Received 21.01.2008; accepted in revised form 25.09.2008  

Analytical modelling cargoes displacement in wagon and tension in fastening

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