z. 101 Transport 2014
Krzysztof Zboi!ski, Piotr Wo"nica
Warsaw University of Technology, Faculty of TransportFORMATION OF POLYNOMIAL RAILWAY
TRANSITION CURVES OF EVEN DEGREES
The manuscript delivered: May 2013
Abstract: This paper presents new results obtained by its authors while searching for the proper shape of polynomial railway transition curves (TCs). The search for the proper shape means the evaluation of the curve properties based on chosen dynamical quantities and generation of such shape with use of mathematically understood optimisation methods. The studies presented now and in the past always had got a character of the numerical tests. For needs of this work advanced vehicle model, dynamical track-vehicle and vehicle-passenger interactions as well as optimisation methods were exploited. In the model and its software complete rail vehicle model of 2-axle freight car, the track discrete model, and non-linear description of wheel-rail contact are applied. That part of the software, being vehicle simulation software, is combined with library optimisation procedures into the final computer programme.
The main difference between this and previous papers by the authors are the degrees of examinated polynomials. Previously authors tested polynomial curves of odd degrees, now they focus on TCs of 6th, 8th and 10th degrees without curvature and superelevation ramp tangence in the TC’s terminal points. The fundamental demands that refer to values of curvature and superelevation in terminal points of TC are preserved.
The aim of present article is to find the polynomial TCs’ optimum shapes which are determined by the possible polynomial configurations. Two dynamical quantities being the results of simulation of railway vehicle advanced model are exploited in the determination of quality functions (QFs). These are: minimum of integral of vehicle body lateral acceleration (QF14) and minimum of value of vehicle body maximum lateral displacement (QF15).
Keywords: polynomial railway transition curves, computer simulation, optimisation
1. INTRODUCTION
In many recent works [15], [16], [17], [19] authors of this article showed that for the polynomial curves of odd degrees (5th, 7th, 9th and 11th) the best dynamical properties (the smallest values of QF14) have curves with the biggest possible number of their terms. For curves of 5th degree the number of terms was 3, for curves of 7th degree – 5, for curves of 9th degree – 7, and corresponding for curves of 11th degree – 9. In this context, serious difference between the curves of lower and higher degrees was revealed. The curves of 5th and 7th degree had worse dynamical properties than 3rd degree parabola, whereas the curves
of 9th and 11th degree possess such properties better than 3rd degree parabola. It was also shown that use of polynomial TCs in railway conditions could be an advantage. This can only be achieved, however as mentioned, for the curves of high degree with the maximum number of the terms. The best dynamical properties of such TCs were also confirmed through the simulation results representing vehicle body lateral displacements and accelerations. Maximum numbers of the terms correspond to the quite fundamental geometrical demand – curvature and superelevation equal to 0 at the TC’s initial points and 1/R and H, respectively, at the end point. This conclusion was true for all polynomial degrees, from 5th to 11th ones.It was also manifested univocally that the greater degree of the polynomial and number of its terms, the greater flexibility of TCs in terms of their shape. It was shown explicitly, that use of polynomial TCs can be an advantage in the railway conditions. So motivation for the current studies arose from earlier results by the authors and the wish to study polynomial TCs of even degrees (6th, 8th and 10th) with maximum numbers of the terms.
An increase in number of the publications that deal with transition curves, both the railway and the road ones, can indeed be observed [1]-[3], [5]-[14]. Also some qualitative change in content of these works can be noticed. It consists in attempts to diverge from the standard and to look for new, more modern methods of evaluating properties of TCs. Despite these, some earlier visible limitations still exist in those works, in present authors opinion. Namely, the analysis is rather rare which takes account of advanced dynamics of whole vehicle-track system. Present authors do not know method applied in practice (approved as a design tool), which uses complete dynamical model of vehicle in formation of railway TCs. Many methods in use represent traditional approach. They are based on the traditional criteria and often on very simple vehicle model. The authors failed to find publications that exploit directly mathematically understood optimisation methods in formation of TCs, basing on objective functions calculated as a result of numerical simulations. There are some works, where selected quantities of interest, rather than the shape of the curve itself, are optimised instead, e.g. [9].
In many works, also the recent ones, approach to the track-vehicle interactions is traditional, e.g. [4]. It is limited to discussing the vehicles jointly and studying the selected effects (quantities) in the car body. In such works the traditional criteria of 3-dimensional TCs' formation are in use. They demand from the physical quantities that characterise effects on a passenger and eventually on a cargo not to exceed values that are acknowledged as acceptable [4]. The corresponding relations refer to: unbalanced lateral acceleration a alim, velocity of the a change lim, and velocity of wheel vertical rise
along the superelevation ramp f flim. Some up-to-date works extend these criteria with
additional quantities and search for their courses. Such a quantity is the second time derivative of a. In case of the courses (of the a first and second derivatives most often), the continuity (no abrupt change in values), differentiability (no bends) and so on are demanded. Despite that extension, such criteria do not take account of the dynamical properties of particular vehicle, including track-vehicle interactions in particular conditions, or effects on vehicle bogie. These properties are quite different than those assumed in the traditional criteria, where the track has infinite stiffness and no geometrical irregularities, whereas vehicle is represented by a single rigid body or a particle.
2. METHOD OF THE ANALYSIS USED FOR NEEDS
OF CURRENT STUDIES
In order to demonstrate the method used in the research three elements will be discussed. The first element is railway vehicle, its model, and the simulation software. The second is the software in general. The last one touches the optimisation method, quality functions (QFs) implemented in the program, and applied initial shapes of the TCs.
2.1. THE OBJECT, ITS MODEL, AND THE CORRESPONDING
MODEL
In order to make analysis easier relatively simple object and its model were utilised. This model represents 2-axle HSFV1 freight car of the average values of parameters. It is the same model of the system as used in the earlier studies by present authors [15]-[19]. Its structure is shown in Fig. 1c. It is supplemented with discrete models of vertically and laterally flexible track shown in Fig 1a and 1b, respectively. Linearity of the vehicle suspension was assumed. So, linear stiffness and damping elements in vehicle suspension were applied. The same concerns the track models. Here also linear stiffness and damping elements were applied. One can find all parameters of the used models in [15] and [17].
Vehicle model is equipped with a pair of wheel/rail profiles that corresponds to the real ones. That is a pair of the nominal (i.e. unworn) S1002/UIC60 profiles that are used all over the Europe. Non-linear geometry of this pair is introduced into the model in a form of table with the contact parameters. In order to calculate non-linear tangential contact forces between wheel and rail well known FASTSIM program by J.J. Kalker was applied. Normal forces in the contact are not constant but influenced by both the geometry and the dynamical effects that make value of a wheelset vertical load variable.
Generalised approach to the modelling was used, as explained in [18]. Basically, dynamics of relative motion is used in that approach. This means that description of motion (dynamics) is relative to track-based moving reference frames. Dynamical equations of motion are equations of relative motion with terms depending on motion of the reference frames explicitly recorded. None of such terms is omitted in the equations. According to this method, the kinematic type non-linearities arising from rotational motions of bodies within our multi-body system (MBS) model are taken into account. The term generalised refers to the generalised conditions of motion. So, the same generalised vehicle model describes vehicle dynamics in any conditions, i.e. in straight track ST, circular curve CC, and TC sections. The routes composed of such sections are analysed.
The route (section) of interest is characterised in the method by shape of the track centre line which is the general space (dimensional) curve. In railway systems such 3-dimensional objects are TCs with their superelevation ramps. A necessary condition to apply the method is description of the curves (sections) by parametric equations, with the curve's current length l as the parameter. The cases of CC and ST are treated in the method
as the special cases of 2- and 1-dimensional geometrical objects, respectively. Such an approach was described in [17] and [18].
Fig. 1. System's nominal model: (a) track vertically, (b) track laterally, (c) vehicle
An important element in the method is description of kinematics of the track-based moving reference frames. Their motion comes out directly from the track centre line shape. The applied method of determination of the kinematical quantities on the basis of the parametric equations is presented most recently in [17] and [18].
2.2. THE OPTIMISATION METHOD AND OBJECTIVE FUNCTIONS
The optimisation problem which is solved in the current studies is to find the Ai
polynomial coefficients that define TC’s shape. Type of a TC chosen for optimisation is the polynomial TC of any degree n!4. It is defined by Eqs. (1)-(4) that are related to space curve parametric equations:
" " # $ % % & ' ( ( ( ( ( ( ) * * * * * * * * * * 1 0 3 3 2 0 4 4 5 n 0 3 n 3 n 4 n 0 2 n 2 n 3 n 0 1 n 1 n 2 n 0 n n l l A l l A ... l l A l l A l l A l l A R 1 y , (1)
+
,
+
,+
,
-. / 0 0 1 2 3 ( ( * * ( * ) ) * * * * * 1 0 1 3 3 n 0 3 n 1 n 2 n 0 2 n n 2 2 l l A 2 3 ... l l A 2 n 1 n l l A 1 n n R 1 dl y d k , (2)+
,
+
,+
,
-. / 0 0 1 2 3 ( 3 ( ( * * ( * ) * * * * * 1 0 1 3 2 0 2 4 3 n 0 3 n 1 n 2 n 0 2 n n l l A 2 3 l l A 3 4 ... l l A 2 n 1 n l l A 1 n n H h , (3)+
,+
,
+
,+
,+
,
-. / 3 3 ( ( 0 0 1 2 * * * ( * * ) ) * * * * * 1 0 0 3 3 n 0 4 n 1 n 2 n 0 3 n n l l A 1 2 3 ... l l A 3 n 2 n 1 n l l A 2 n 1 n n H dl dh i , (4)where y, k, h, and i define curve lateral co-ordinate, curvature, superelevation, and inclination of superelevation ramp, respectively. The R, H, l0, and l define curve minimum
radius (at its end), maximum susperelevation (at the curve end), total curve length, and curve current length, respectively. The Ai are polynomial coefficients (i = n, n-1,…., 4, 3)
while n is polynomial degree. Here, n=6, 8 and 10. Number of the polynomial terms (terms in Eqs. (1)-(4)) must not be smaller than 2. On the other hand the smallest degree nmin of
the last term in Eq. (1) must be nmin ! 3. Such definition of the curves gives possibility of
proper k and h values at TCs terminal points. They should equal to 0 at the initial points and to 1/R and H at the end points. Note, that values for both always equal to 0 for l=0. In order to ensure 1/R and H values for l=L, normalisation of the coefficients is necessary, such as in [15]. Finally, coefficients A4i areobtained which satisfy constraints imposed on their values. The problem just formulated is a classical formulation of the static constrained optimisation. It is solved with the library procedure that utilises moving penalty function algorithm combined with Powell's method of conjugate directions.
For needs of current paper authors utilised two quality functions (QFs) marked as in previous works the numbers 14 and 15. They concern minimisation of:
- integral of vehicle body lateral acceleration (QF14) - main criterion
dl | y | L QF C L 0 b 1 C 14
5
* ) , (5)- value of vehicle body maximum lateral displacement (QF15)
| y | max
QF15 ) b , (6)
where and y b y – lateral acceleration and displacement of vehicle body, respectively, and b LC – length of whole TC and the adjacent 100 m of CC.
The difficulty of the problem solution consists in quite complex form and way to determine the objective function (quality function). This function is calculated as a result of the numerical simulation of motion of the dynamical mechanical system as described in Subsection 2.1. The main steps during calculation of the objective function are: generation of the new shape of TC, calculation of the kinematical quantities (velocities and accelerations) that depend on this new shape, and solution of the corresponding 2nd order ordinary differential equations (ODEs) set. Note that here, this system of equations describes dynamical system of 18 degrees of freedom.
2.3. GENERAL LOOK AT THE SOFTWARE
Scheme of the software used in optimisation TCs shape is shown in Figure 2. The major objects within this scheme are two iteration loops visible there. The first loop is the integration loop. This loop is stopped when distance llim, being the length of route (usually
compound route ST, TC and CC or CC, TC and ST), is reached by the model. The second one is the optimisation process loop. It is stopped when number of iterations reaches limit value ilim. This value means that ilim simulations of vehicle motion have to be performed in
order to stop optimisation process. In the calculations done so far ilim=100 was used as
standard value. If the optimum solution is reached earlier, i.e. for i<ilim, then the
optimisation process stops automatically and the corresponding results are recorded. When no optimum solution is reached for ilim=100, then this value has to be increased manually,
while the process has to be repeated.
Usually calculation time of the single process on the PC computer with Inter Core 2 Duo 2GB processor lasted from 5 to 60 minutes. No calculation times longer than 80 minutes happened, so far.
3. RESULTS OF THE STUDIES
3.1. INFORMATIONS ON POLYNOMIAL TSC OF EVEN DEGREES
Each polynomial of even degree has two different standard transition curves (INI) that guarantee minimum of the centrifugal force integral. This property differs even degree polynomials from polynomials of odd degrees, where there is just one standard TC. In this case, the functions of inclination of superelevation ramp – formula (3) – are symmetrical about a vertical axis passing through the point l0/2. The method of receiving two standard
TC for even degrees is presented in [15]. The list of standard TC of 6th, 8th and 10th is demonstrated in Tab. 1.
For each polynomial curve (also of odd degree) geometrical demands were imposed, depending if one wants or does not want to take them into account. Possible combinations of coefficients for these demands are shown in Tab. 2. In this paper, as mentioned, authors focused only on curves with maximum number of the terms.
Each TC has minimal length which is calculated in accordance with the method presented in [8]. This minimal length arises from two conditions. They demand that two values are not allowed to be exceeded. First one is the velocity of the unbalanced lateral acceleration change and second one is the velocity of wheel vertical rise along the superelevation ramp f. Minimum lengths for two vehicle velocities v=24.26 m/s and v=30.79 m/s corresponding to lateral acceleration a equal to 0 and 0.6 m/s2, radius of circular arc R=600 m and superelevation H=0.15 m are presented in Tab. 3. Velocity of wheel vertical rise along the superelevation ramp f is equal to 56 mm/s and velocity of the unbalanced lateral acceleration change is equal to1 m/s3. For needs of this work authors always took a greater length calculated for both conditions (lf min and l( min).
Table 1
Standard (initial) TCs of 6th, 8th and 10th degrees Degree of
polynomial Two standard (initial) TCs (INI) 6th "" # $ %% & ' ( * ) 3 0 5 4 0 6 1 l l 5 1 10l l R 1 y "" # $ %% & ' ( * ) 2 0 4 3 0 5 4 0 6 2 l l 2 1 l l 5 2 10l l R 1 y
Table 1- continuation 8th "" # $ %% & ' ( * ( ) 4 6 5 7 6 8 1 l l 2 1 l l 7 4 l l 28 5 R 1 y 0 0 0 "" # $ %% & ' ( * ( * ) 3 5 4 6 5 7 6 8 2 l l l l 2 3 l l 7 6 l l 28 5 R 1 y 0 0 0 0 10th "" # $ %% & ' ( * ( * ) 5 0 7 6 0 8 7 0 9 8 10 1 l l 3 4 l l 2 5 l l 3 5 l l 18 7 R 1 y 0 "" # $ %% & ' ( * ( * ) 4 0 6 5 0 7 6 0 8 7 0 9 8 0 10 2 l l 3 7 l l 3 16 l l 5 l l 9 20 l l 18 7 R 1 y Table 2
Possible polynomial configurations for different geometrical demands
Type of demand 6 ________________
Polynomial degree (terms number in the
initial polynomial) 7
Demand IDZ=1 (proper values of r and h
in TCs' terminal points) Number of terms Demand IDZ=2 (tangence of r and h functions at TCs' terminal points) Number of terms Demand IDZ=3 (tangence of h slope, i.e.
of i, at TCs' terminal points) Number of terms 6th
(IWI=2, 3) IW=2; IW=3; IW=4 IW=2(single curve); IW=3 - 8th
(IWI=3, 4) IW=2; IW=3; IW=4; IW=5; IW=6
IW=2(single curve); IW=3; IW=4; IW=5;
IW=3(single curve); IW=4 10th
(IWI=4, 5) IW=2; IW=3; IW=4; IW=5; IW=6; IW=7; IW=8
IW=2(single curve); IW=3; IW=4; IW=5; IW=6; IW=7
IW=3(single curve); IW=4; IW=5; IW=6
Table 3
Minimum lengths of TCs
Degree of polyn. lf min [m] - v=24.26 m/s (v=30.79 m/s) l& min [m] - v=24.26 m/s (v=30.79 m/s)
6th
115.47 (146.59) 0 (32.83)
8th
134.75 (171.06) 0 (38.31)
10th 152.71 (193.87) 0 (43.42)
3.2. RESULT OF OPTIMISATION AND DYNAMICAL
SIMULATIONS
Graphical representation of the results refers to polynomial TCs of 6th, 8th and 10th degrees. Only one geometrical demand is taken into account, namely - IDZ=1 (see Table 2). Polynomial configurations are limited to those with the maximum number of terms, i.e. 4, 6, and 8 terms (see Tab. 2). As concerns configuration of test routes they are always composed of ST, TC, and CC. Lengths of ST sections are the same and equal 50 m. Similarly for CC, their lengths are the same and equal 100 m. Besides, single curve radius
R and superelevation H for CC were considered. Their values were R=600 m and H=0.15 m. Two sorts of the TCs' parameters for such CC were considered for each of the TC's degrees. For both sorts maximum velocity of wheel rise along superelevation ramp was f=56 mm/s. Different for this sorts are velocities v and lengths as in Tab. 2. The bigger velocity represents maximum admissible vehicle velocity in curved track. The smaller velocity guarantees ideal balance between transversal components of gravity and centrifugal forces, on the other hand. In addition final length l0 is a function of numerical
coefficient proper for degree of the particular polynomial, eg. [15]. So, differences between the routes exist for TCs only.
The parameters of the four routes selected from all 6 routes tested are as follows: Route 1 (6th degree, v=30.79 m/s, l0=146.59 m, INI=y1, QF14); Route 2 (8th degree,
v=24.26 m/s, L=134.75 m, INI=y2, QF14); Route 3 (10th degree, v=24.26 m/s, l0=152.71 m,
INI=y1, QF14); Route 4 (6th degree, v=30.79 m/s, l0=146.59 m, INI=y1, QF15).
Each of the selected routes is represented by its own group of four figures. Content of the figures in particular groups is analogous. So, the first figure in the group is representing superelevation ramps h corresponding to all TCs' shapes tested in optimisation process. Note, that courses of the curvatures 1/r from the same process are identical in shape with those for h. The only difference is scale of the vertical axis. The skew straight lines in that kind of figures are of no importance. They arise from recording results for all the shapes in a single file. The second figure in the group is representing curvature of the initial and optimised TCs. The third one in the group is representing vehicle body lateral displacements for the both initial and optimised TCs. The forth figure is representing vehicle body lateral accelerations for the initial and optimised TCs. Denotations INI, QF14, and QF15 mean results for the initial and optimised TCs. Subscripts accompanying these denotations characterise particular curves. For example 6d4t means curve of degree 6, possessing 4 terms at the same time. To make the figures better readable different line types were also applied. The solid line represents results after optimisation, while dashed line at the beginning of the optimisation process.
0 100 200 300 distance s; [m] 0 0.04 0.08 0.12 0.16 s u p e re le v a ti o n h ; [m ] a) 0 40 80 120 160 distance s; [m] 0 0.0004 0.0008 0.0012 0.0016 0.002 c u rv a tu re 1 /r ; [1 /m ] QF146d4t INI6d2t b)
Fig. 3. Route 1 - features of TCs: a) superelevation ramps corresponding to all TC shapes tested in optimisation, b) curvatures of the initial and optimised TCs
0 100 200 300 distance s; [m] -0.02 -0.016 -0.012 -0.008 -0.004 0 0.004 v e h ic le b o d y l a t. d is p l. yb ; [m ] INI6d2t QF146d4t a) 0 100 200 300 distance s; [m] -0.15 -0.1 -0.05 0 0.05 0.1 v e h ic le b o d y l a t. a c c e l. y "b ; [m /s 2] INI6d2t QF146d4t b)
Fig. 4. Route 1 - results of simulation for vehicle body for the initial and optimised TCs: a) lateral displacement, b) lateral acceleration
0 100 200 300 distance s; [m] 0 0.04 0.08 0.12 0.16 s u p e re le v a ti o n h ; [m ] a) 0 40 80 120 160 distance s; [m] 0 0.0004 0.0008 0.0012 0.0016 0.002 c u rv a tu re 1 /r ; [1 /m ] b) INI8d4t QF148d6t
Fig. 5. Route 2 - features of TCs: a) superelevation ramps corresponding to all TC shapes tested in optimisation, b) curvatures of the initial and optimised TCs
0 100 200 300 distance s; [m] -0.004 0 0.004 0.008 0.012 v e h ic le b o d y l a t. d is p l. yb ; [m ] a) INI8d4t QF148d6t 0 100 200 300 distance s; [m] -0.03 -0.02 -0.01 0 0.01 0.02 0.03 v e h ic le b o d y l a t. a c c e l. y " b ; [m /s 2] b) INI8d4t QF148d6t
Fig. 6. Route 2 - results of simulation for vehicle body for the initial and optimised TCs: a) lateral displacement, b) lateral acceleration
0 100 200 300 distance s; [m] 0 0.04 0.08 0.12 0.16 s u p e re le v a ti o n h ; [m ] a) 0 40 80 120 160 distance; [m] 0 0.0004 0.0008 0.0012 0.0016 0.002 c u rv a tu re 1 /r ; [1 /m ] b) INI10d4t QF1410d8t
Fig. 7. Route 3 - features of TCs: a) superelevation ramps corresponding to all TC shapes tested in optimisation, b) curvatures of the initial and optimised TCs
0 100 200 300 distance s; [m] -0.004 0 0.004 0.008 0.012 v e h ic le b o d y l a t. d is p l. yb ; [m ] QF1410d8t INI10d4t a) 0 100 200 300 400 distance s; [m] -0.008 -0.004 0 0.004 0.008 0.012 v e h ic le b o d y l a t. d is p l. y "b ; [m /s 2] b) QF1410d8t INI10d4t
Fig. 8. Route 3 - results of simulation for vehicle body for the initial and optimised TCs: a) lateral displacement, b) lateral acceleration
0 100 200 300 distance s; [m] 0 0.04 0.08 0.12 0.16 s u p e re le v a ti o n h ; [m ] a) 0 40 80 120 distance; [m] 0 0.0004 0.0008 0.0012 0.0016 0.002 c u rv a tu re 1 /r ; [1 /m ] b) INI6d2t QF156d4t
Fig. 9. Route 4 - features of TCs: a) superelevation ramps corresponding to all TC shapes tested in optimisation, b) curvatures of the initial and optimised TCs
0 100 200 300 distance s; [m] -0.02 -0.016 -0.012 -0.008 -0.004 0 0.004 v e h ic e l b o d y l a t. d is p l. yb ; [m ] INI6d2t QF156d4t a) 0 100 200 300 distance s; [m] -0.15 -0.1 -0.05 0 0.05 0.1 v e h ic le b o d y l a t. a c c e l. y "b ; [m /s 2] b) INI6d2t QF156d4t
Fig. 10. Route 4 - results of simulation for vehicle body for the initial and optimised TCs: a) lateral displacement, b) lateral acceleration
3.3. DISCUSSION OF THE RESULTS OBTAINED
Results of numerical calculations presented above can be divided into two categories. First is the category for the Route 1, Route 2 and Route 4. Second is the category related to the Route 3. In case of the first category, optimum shapes of the TCs lead to curvatures (and superelevation ramps) quite close to the linear curvatures (superelevations) for the 3rd degree parabolic TC in case of Route 2 and Route 4. In case of Route 1, it is not so obvious. In case of the second category the optimum TC’s shape leads to curvature (and superelevation ramp) being something between the initial curve and the linear shape for 3rd degree parabolic TC.
Concluding, one may state that in case of the higher degrees (the 10th degree) optimum curves differ from the initial curves and the parabolic TC. In case of the lower degrees (the 6th and 8th) optimum TC are close to the parabolic TC of the 3rd degree. This conclusion is similar to that obtained for the odd curves.
Betterment in the system dynamical properties for the optimised TCs' shapes in comparison to the initial curves is confirmed by simulation results of the lateral displacements yb and accelerations y bof vehicle body. This is the case for both these
quantities in Figures 4, 6, 8 and 10. The betterment is obviously, or even first of all, expressed when values of QFs are compared. If one assumes these values for the initial curves equal to 100% then QFs' values in % for the optimum TCs are as follows: Route 1 –
19.66% (optimum coefficients:A46=-0.0208394, A45=0.0521111, A44=0.0178607, 3 A4=0.0614385), Route 2 – 27.42% (A48=-0.0243623, A47=0.115327, A46=-0.20364, 5 A4=0.154465, A44=-0.0543948, A43=1.4778422), Route 3 – 34.06% (A410=-0.108791, 9 A4=0.46701, A48=-0.700877, A47=0.373322, A46=-0.000717601, A45=0.000688863, 4 A4=-0.0105537, A43=0.14507) and Route 4 – 84.37% (A46=-0.0020122, A45=0.0029034, 4
A4=-0.00533217, A43=0.177714). Note that the highest value of QF after the optimisation refers to QF15 that can be treated as some supplement to the basic quality function QF14.
4. CONCLUSIONS
As a result of the discussed studies some original and important conclusions can be drawn by the authors. In [15], [17] it was shown univocally that the polynomial TCs of odd degrees with the biggest possible number of their terms have the smallest values of QF14. Therefore authors of this paper wanted to use such an approach in polynomial TCs of even degrees. It filled the gap in the range of polynomial TCs’ of degrees from 5 to 11. Results of optimisation gave authors the conclusions that without any doubt the standard curves from Tab. 1 do not have a chance to be the optimum solution for railway TCs for their standard lengths. This is because the main aim of research done for needs of this work, i.e. finding TCs’ shapes better than standards TCs’ shapes was achieved by the authors. It is also worth saying that for TCs of even degrees with biggest possible number of their terms, similarly as for the transition curves of odd degrees, still valid is statement: the greater degree of polynomial TC, the bigger flexibility of TC’s shape is observed.
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KSZTA#TOWANIE KOLEJOWYCH KRZYWYCH PRZEJ$CIOWYCH STOPNI PARZYSTYCH
Streszczenie: W artykule przedstawione wyniki optymalizacji kszta>tów wielomianowych krzywych przej@ciowych (KP). Poszukiwanie w>a@ciwego kszta>tu krzywej przej@ciowej oznacza ocenX w>asno@ci krzywej opartej na w>a@ciwo@ciach dynamicznych i wygenerowanie takiego kszta>tu z wykorzystaniem matematycznych metod optymalizacji. Wyniki zaprezentowane teraz i w przesz>o@ci zawsze mia>y charakter zaplanowanych testów numerycznych. Wykonywano je za pomoc\ programu do symulacji ruchu pojazdu szynowego, po>\czonego z biblioteczn\ procedur\ optymalizacyjn\. Tak po>\czone oprogramowanie umo^liwia optymalizacjX kszta>tu krzywych przej@ciowych ze wzglXdu na kryteria dynamiczne. W badaniach nad optymalnym kszta>tem krzywych przej@ciowych wykorzystano model dwuosiowego wagonu towarowego. W pracy wykorzystano nieliniowy model opisu kontaktu ko>o-szyna.
G>ówn\ ró^nic\ pomiXdzy niniejszym a poprzednimi artyku>ami autorów by> stopie" wielomianu. Poprzednio autorzy badali krzywe stopni nieparzystych. Teraz skupili siX na stopniach - 6., 8. i 10 bez styczno@ci krzywizny i rampy przychy>kowej w skrajnych punktach. Fundamentalne wymaganie dotycz\ce krzywizny i rampy przechy>kowej zosta>o zachowane.
Celem niniejszego artyku>u jest znalezienie optymalnych kszta>tów KP ze wzglXdu na przyjXte konfiguracje wielomianu. PrzyjXto dwie funkcje celu bXd\ce wynikiem symulacji ruchu pojazdu. S\ nimi: ca>ka z przyspieszenia poprzecznego @rodka masy nadwozia oraz maksymalne wychylenie poprzeczne @rodka masy nadwozia.
