z. 71 Transport 2006
Edward Rydygier
Municipal Office of the Capital City of Warsaw
Kazimir Pulaski Technical University of Radom, Faculty of Transport and Electrical Engineering
Zygmunt Strzyakowski
Kazimir Pulaski Technical University of Radom, Faculty of Transport and Electrical Engineering, Institute of Automatics and Telematics
APPLICATION OF NUMERICAL METHODS OF
FIELD SOURCES IDENTIFICATION TO SELECTED
ISSUES IN RAIL VEHICLE-TRACK SYSTEM
Manuscript received, October 2007
Abstract: In this paper a numerical method of identification of physical field sources is presented and
its applications to solve selected issues in rail vehicle-track system are indicated. Within mathematical modelling, identification belongs to the group of inverse problems. For this reason the established method for identification was completed by a regularization procedure. Examples presented in this paper illustrate that this method is effective to solve such problems of rail vehicle-track modelling like: identification of tension field sources for torsion of a rail and identification of thermal sources in a heat path on rail surface caused by a wheel-rail rolling contact. Also it is shown in present work that the established approach of identification can be used to verify mechanical features of a wheel-rail rolling contact by applying the heat model of a contact.
Keywords: field source identification, modelling 2-D systems, wheel-rail rolling contact
1. INTRODUCTION
Recently investigations of inverse problems described by the 2-D and 3-D models have been intensively developed. These investigations are carried on in following directions: development of theory of inverse problems and improvement of numerical methods for their solutions and betterment of measurement technology [2, 6, 12, 13]. Inverse problems arise in the course of determining the internal structure of physical systems from system’s measured behaviour, or the unknown input that gives a rise to a measured output signal. Modelling of 2-D and 3-D systems resolves itself into the solution of suitable differential equations set. In this instance calculation is relatively uncomplicated in the dimensionally finite system but becomes very difficult for various partial differential equations such
as wave, heat, electromagnetic ones. When it is impossible, or difficult to obtain an exact solution of partial equations governing a continuous system, the equations set is reduced to a discrete form.
Inverse problems encountered in many branches of natural sciences and engineering such as mathematics (theory and methods), statistics, geophysics, seismology, astrometry, astrophysics, optics and image restoration, high temperature plasma diagnostics, electrodynamics, electrotechnique, scattering in nuclear physics, medicine (medical imaging, impedance tomography, electrocardiogram interpretation), mechanics, material engineering, transport, and railway engineering as well as in biology, economics and even in social sciences. Inverse problems may be divided into following groups: boundary inverse problems, initial inverse problems, shape system problems, and problems of parameter identification and sources identification. In this paper the problem of identification of field sources are investigated. The field sources which are identified mean the sources of physical field appearing in a concrete physical or technical system.
2. THE IDENTIFICATION OF FIELD SOURCES
2.1. DESCRIPTION OF A MODEL
The problem of field sources identification is of importance in the practice. Often, in nature as well as in technological and industry processes, it is necessary to identify internal sources of system. For the reason of different limits that consist more often in troubles with making accurate measurements, the correct determination of position and intensity of sources that generate the certain physical field can not be done. For the correct identification a special mathematical tool must be used to support knowledge based on the possible measure data. To considerations were taken fields that can be described by the Poisson equation represented by second order partial differential equation valid in two-dimensional space (2-D), completed by boundary conditions [6, 12]
, ), , ( ) , ( ) , ( 2 2 2 2 * f x y u y y x u x y x u w w w w (1)
where u(x, y) is a potential function, f(x, y) is a sources’ function, x and y mean coordinate respectively on the OX and OY axis of system of coordinates, * means a boundary of an investigated domain.
With the aim of solving such described problem one of authors of this paper elaborated a special numerical method with the use of computer simulations named the Simulation Method [7, 8]. To make this method simple and effective the special computational tools were constructed on the basis of concepts and objects from a modern combinatorial analysis [1]. In particular, there are used the monic polynomials with coefficients generated by modified numerical triangles. There were applied two kinds of modified triangles.
Elements of the first modified numerical triangle are computed in accordance with special recurrence formula
an,k = 2an -1,k + an -1,k -1 - an -2,k, a0,0 = 1, a1,0 = 1, n = 0, 1, 2, ... , 0dkdn (2)
Such introduced elements generate the non-zero monic polynomials of the first kind in
qR defined by the following recurrence formula
Tn+2(q) = (2 + q)Tn+1(q) - Tn(q), n = 0, 1, 2, ... T0(q) = 1, T1(x) = 1 + q, (3) in such a way Tn(q) =
¦
n k k k n q a 0 , , n = 0, 1, 2, ... . (4)From the above recurrence formula or on the basis of the first modified numerical triangle the Tn(q) polynomials can be calculated
T0(q) = 1 T1(q) = 1 + q T2(q) = 1 + 3q + q2 T3(q) = 1 + 6q + 5q2 + q3 T4(q) = 1 + 10q + 15q2 + 7q3 + q4 T5(q) = 1 + 15q + 35q2 + 28q3 + 9q4 + q5. ... ... ... ... ... ... ….
The another recurrence formula introduces elements of the second modified numerical triangle
bn, r = 2bn -1, r + bn - 1, r - 1 - bn - 2, r , b0,0 = 0 , b1,0 = 1, n = 0, 1, 2, ... , 0drdn, (5)
that generate monic polynomials of the second kind defined by the following recurrence formula Pn + 2(q) = (2 + q)Pn + 1(q) - Pn(q) , n = 0, 1, 2, ... , P0(q) = 0, P(q) = 1, (6) in such a way Pn(q) =
¦
n r r r n q b 0 , , n = 0, 1, 2, ... . (7)From the above recurrence formula or on the basis of the second modified numerical triangle the Pn(q) polynomials can be calculated
P0(q) = 0 P1(q) = 1 P2(q) = 2 + q P3(q) = 3 + 4q + q2 P4(q) = 4 + 10q + 6q2 + q3 P5(q) = 5 + 20q + 21q2 + 8q3 + q4 . ... ... ... ... ... ...
It should be noted that the sum of all numbers in a row of the first or second modified numerical triangle equals f2n , n = 0, 1, 2, ... , or f2n + 1, n = 0, 1, 2, ... , respectively, i.e. they are equal to successive elements with even or odd indices of the Fibonacci sequence defined by the following recurrence formula
fn + 2 = fn + 1 + fn, n = 0, 1, 2, ... , (8)
with f0 = 1 and f1= 1 as initial values.
2.2. THE IDEA OF THE ELABORATED METHOD
Identification of field sources are done with the use of the Simulation Method which was previously elaborated to solve inverse problem described by the Poisson equation with boundary conditions (1). This method was named the Simulation Method because computer simulations were used to test the algorithms as well as to reckon the regularization coefficient [8, 9]. To simplify notation without loss of generality the zero boundary conditions are assumed. The discretization of the continuous equation (1) are done by the use of the finite differences method. For the discrete variables (m, n), x = mh,
m = 0, 1, 2, ... , M, y = nh, n = 0, 1, 2, ... , N, where M = l1/h , N = l2/h, h is a step of discretization of domain l1×l2, the difference Poisson equation can be formulated using a five points difference scheme
¦
Pmn p
mp
a um,n = fm,n , (9)
where: um,n = u(m, n), fm, n = f(m, n), and amn means a five points difference operator
amp = °¯ ° ® z m n if h m n if h 2 2 4 , .
The source function is approximated by discrete 2-D Fourier series
fm,n = ¦ 1 1 sin ) ( 2N k m N n k k F S , fm,0 = fm,N = 0 , m = 0, 1, 2, ... , M , (10)
where n is taken as a parameter, F(k) mean amplitudes for k = 1, 2, … N-1. Similarly the potential function is developed into discrete Fourier series
um,n = ¦ 1 1 sin ) ( 2 N k m N n k k U S , u0,n = uM,n = 0 , n = 0, 1, ... , N. (11)
2 1 h ( ( ) 2 ( ) ( )) (4sin 2 ) ( ) ( ) 2 1 1 U k F k N k k U k U k Um m m m m »¼ º «¬ ª S , m = 1, 2, ... , M-1 (12)
with the boundary condition U0(k) = 0 and values UM(k) defined by uM,n = 0, n = 1, ... , N.
Using the monic polynomials P(qk) (6) for the parameter
N k qk 2 sin 4 2 S the solution of
direct problem takes a form
¦ 1 1 2 1( ) ( ) ( ) ) ( ) ( m l m l k l k m m k P q U k P q h F k U , m = 2, 3, ... , M – 1 . (13)
The values U1(k) in the equation (13) are calculated from the boundary conditions (11). When N = M , the system of M-1 equations can be obtained to calculate set of coefficients
UM(k), k = 1, 2, ... , M-1. Then after a substitution of these coefficients to the equation (13)
for m = M, a set of coefficients U1(k) for k = 1, 2, ... , M-1 can be found.
The inverse problem solution calculated with the use of monic polynomials approach can be expressed by the formula
2 1 1 2 1 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( h q P k F h q P k U q P k U k F k l l i l i k i k l l l ¦ . (14)
A field sources’ function in a form (10) is the solution of the field sources identification problem. Calculations are to be done with the use of special regularization procedure. Inverse problems are the ill-posed problems according to Hadamard’s definition of correctly posed problems [13]. The ill-posed nature of inverse problems causes that various methods solving direct problems are inapplicable for wide range of inverse problems. To solve inverse problems the special numerical procedures must be employed to stabilize the results of calculations. For that reason the Simulation Method was completed with the special numerical approximation procedure elaborated on the basis of an inverse distance method of smoothing the scattered data for 2-D systems [7].
2.3. RESULTS OF INVESTIGATIONS
For applications related to exploitation of rail vehicles possible domains of research contain such phenomena like: tension in a rail due its torsion and various heat problems [7, 8, 9, 10]. Very interesting studies on inverse heat problems can be led within investigations of the rolling path of a wheel and rail [11].
Results of applying of the simulation method to studies on tension in a rail described by the Timoshenko model are presented on fig 1 to fig. 4 . The input data (fig. 1 and fig 2) refer to the auxiliary function \ (x, y) connected with a torsion angle on a cross-section plane [7]. This auxiliary function is defined using the following tension matrix elements [6]
Vxx = Vyy = Vxy = Vzz = 0 , Vxz
=
y w w\ , V yz= -
x w w\ . (15)Moment M acting on a edge z = l is connected with a function \ by the formula ³³ D dxdy y x M 2 \( , ) . (16)
The auxiliary function \ fulfills the Poisson equation (1) with the sources’ function F = GT as a constant where G means the transversal elasticity modulus and T is a torsion angle on a length unit of a rail [6]:
F y x 2 2 2 2 w \ w w \ w . (17)
Fig. 1. Input data for a torsion of a rail Fig. 2. Contour graph
The calculated solution (fig. 3 and fig. 4) shows that a field sources’ function has a constant values in accordance with an analytical solution of inverse problem described by the Poisson equation (17).
The results of identification of heat sources in the heat path generated by a rolling contact of wheel over a rail are shown in fig. 5 to fig. 8. The model of heat transfer for a rail and the temperature data are taken from investigations carried out by the Prof. Piotrowski group from the Institute of Vehicles of Warsaw University of Technology [3].
Fig.5. Temperature distribution in a heat path Fig. 6. Contour graph
Fig.7. Heat sources’ density distribution Fig. 8. Contour graph
2.4. DYNAMICAL ASPECTS OF A WHEEL-RAIL CONTACT
The solution of a problem of a wheel-rail rolling contact is an important part of investigations of dynamical phenomena that appear in a mechanical system vehicle-track. The tangent force T and the torque Mz connected with the rolling of a wheel can be
obtained from following formulas [4]
T = ¸¸¹ · ¨¨© § y x T T =
³³
¸¸¹ · ¨¨© § C y x p p dxdy, Mz=³³
C x y yp dxdy xp . (18)
Tangent forces acting between a rail and a rolling wheel in presence of the microslip are calculated numerically with special computer program called a tangent forces generator based on the theory of a rolling contact between a wheel and rail [5].
Owing to the current possibilities of measurements of contact loads, wheel slips, and a contact area, the parameter identification for the real vehicle is impossible to realize. Therefore there are created mathematical models, based on the wheel-rail contact theory, useful in computer simulations. Because wheels and rails were produced with materials having the distortion properties, then under the vertical loads a geometrical contact point becomes a certain little contact area. Shape and size of a contact area and also a contact pressure distribution have influence on generation of tangent forces in the instance of a wheel that rolls with a slip in presence of friction. Calculation of a contact area is the normal contact problem. For assumptions of Hertz theory a contact area becomes an ellipse with lengths of semi-axis a and b dependent on a vertical force. If the whole assumptions
of the Hertz theory are not accepted, such problem is called nonhertzian.
The tangent contact problem is difficult to solve because the contact area can not be determined due to unknown normal force and relative slips. Solution of the tangent contact problem as well as the normal contact problem may be obtained using iterative calculations at every iterative cycle for every wheel, but it needs a large number of iterative cycles. In practice due to the high costs these calculations are not performed. Instead and of necessity, the reduced problem is solved that consists in excluding a normal contact problem from iterative calculations. This exclusion of a normal contact problem consists in the replacement of contact area with ellipse and the parameterization nonlinear theory of rolling wheel with the use of Kalker parameters for relative slips [5]. Then the normalized tangent forces fx and fy become functions of four variables, i.e. three Kalker parameters ],
K, F and g=a/b fx = N Tx P , fy = N Ty P , (19) ] = c x P UQ , K = c y P UQ , F = c P UI , (20)
where c, U, a, b describe an elliptic contact area under the normal load N, μ means a
friction coefficient, whereas Qx , Qy , I mean the slips connected with the rolling of a wheel.
Lengths a and b are chosen in the way that the rate g=a/b equals the rate a length to a
width of contact area. Parameters c, U are the geometrical parameters of elliptical contact area. Parameter c called the supplementary radius of contact area is calculated for the
known contact area A
S
A
c . (21)
The second geometrical parameter of supplementary elliptical contact area is called the characteristic length described by the following formula
2 3 2 ) 1 ( 3 4 ¸¸¹ · ¨¨© § nanb A N E S V U , (22) where E means the Young modulus, means the Poisson coefficient, whereas coefficients
na and nb are tabulated.
For the force N’ that not differs much from the force N the geometrical parameters can be calculated by factorization 3 1 ' ' ¸ ¹ · ¨ © § N N c c , U’=U, g’= g (23) The problem of calculation the tangent force resolves itself into the task consisting in a calculation of the tangent surface torsion p = (px, py) in a contact area which is subject to the Coulomb law of dry friction
p μpz , p = μpz s 1s for s 0, (24) s = x x v y v y x w w » ¼ º « ¬ ª I I u + l w w u, (25)
where: the contact pressure pz (x, y), relative slips Qx , Qy , I and the friction coefficient μ
have given values, u means the difference of tangent displacements for a wheel and a rail in relation to rigid solids of wheel and rail.
It should be pointed out that tangent deformations of rolling planes are generated by friction and the quantities u and p are connected by certain constitutive equation [5]. The new variable l is defined as follows l =
³
t dt t v 0 ' ) '
( . The geometry of the contact problem under consideration is following: a wheel moves with translatory velocity v and turns with angular velocity and rolls slantwise at the angle to the direction of velocity v, the x-axis is located along a rail turned towards a direction of motion, the y-x-axis is lied on a plane perpendicularly to the rail, and the z-axis completes a dextrorotatory coordinates system.
Solving the problem described by equations (24) the tangent force T and the torque Mz connected with the rolling of wheel can be obtained on the basis of the equation (18).
Exemple numerical methods useful to solve the problem of a wheel-rail rolling contact are the methods elaborated by J. J. Kalker implemented as computer programs called DUVOROL and CONTACT [4, 5]. The DUVOROL program is based on the complete theory of steady rolling of a wheel. In this program a rolling is taken as transitional process which tends to the steady state. The CONTACT is the most general program among the programs that are based on the theory of a rolling contact for elastic bodies. This program is useful to solve many detailed problems like a steady and unsteady process of rolling,
calculation of slips coefficients and contact area (a normal contact problem) as well as a reckoning the field of torsions and deformations in elastical semi-space. Solving the steady rolling problem for the given quantities like a direction of rolling, discrete contact area, contact pressure, friction coefficient, and constants of material, this program calculates a tangent surface torsion distribution, a slide velocity distribution, the tangent forces and a friction forces work.
For the numerous set of Kalker parameters values and g = a/b the distribution of values of tangent forces can be calculated in a form of normalized numerical results called the Kalker table. Tangent forces acting between a rail and a rolling wheel in presence of the slip are calculated numerically with the numerical tangent forces generator based on the theory of a rolling contact between a wheel and rail. For the enough small slips the linear theory can be successfully applied and the solutions are obtained in an analytical form. Otherwise the nonlinear theory must be used. In the large number of applications a theory of steady rolling is used. This theory can not be used to study high frequency phenomena.
J.J. Kalker included calculation of derivatives to the CONTACT program. In new version of this program calculations of derivatives replaced calculations of coefficients existing in linear theory for the Hertz size of a contact area. The use of derivatives extends applications of linear theory to nonhertzian surroundings. It makes stability investigations possible for singular kinds of contact where a suitable theory does not exist [4].
Mechanical parameters of a wheel-rail rolling contact can be verified using the results of investigations of a heat path on the rolling plane of rail, carried out with the Simulation Method. The identification of field sources in the heat path on a rail is useful for the dynamical system identification which makes possible a verification of mechanical properties of this contact as well as the obtaining local tensions, slips, and a friction distribution [3, 4]. Using the Simulation Method a mechanical model of a wheel-rail contact can be verified by its thermal model.
3. CONCLUSIONS
In this paper there are presented possible directions of application of numerical method specialized to identification of field sources in selected issues in rail vehicle – track system. This method named the Simulation Method was elaborated generally with the aim of solving inverse problems described by the Poisson equation valid in two-dimensional space and can be used to identify the field sources in the 2-D systems. In this paper it is shown that this method is effective to identify tension field sources for torsion of a rail and to identify thermal sources in a heat path on rail surface caused by a wheel-rail rolling contact. The examples presented in this paper relate to real problems of railway vehicle – track systems. The elaborated method may be useful to dynamical system identification in the case of investigations of a wheel-rail rolling contact.
References
1. Bryant V.: Aspekty kombinatoryki. WNT, Warszawa 1997.
2. Groetsch Ch.: Inverse Problems in the Mathematical Science., Vieweg, Braunschweig/Wiesbaden 1993. 3. Nagórski Z., Piotrowski J.: Modelowanie pola temperatury w szynie wywoanego toczeniem si koa.
Zeszyty Instytutu Pojazdów Politechniki Warszawskiej 4(39), 2000.
4. Piotrowski J.: Poprzeczne oddziaywanie midzy pojazdem szynowym i torem. Podstawy modelowania numerycznego. Prace Naukowe Politechniki Warszawskiej. Mechanika, Z. 118, 1990.
5. Kalker J.J.: Survey of wheel-rail rolling contact theory. Veh. Sys. Dyn., 5(1979)317-142.
6. Kcki E.: Równania róniczkowe czstkowe w zagadnieniach fizyki i techniki, WNT, Warszawa 1995. 7. Rydygier E.: Studies on tension in a rail with the special numerical method to solve inverse problems
for the real 2-D systems. Archives of Transport 15,1(2003)103-113.
8. Rydygier E.: Symulacje komputerowe zagadnie identyfikacji ukadów 2-D w badaniach szyny kolejowej w zastosowaniu do diagnostyki toru. Prace Naukowe Politechniki Radomskiej, Transport 1(17), 2003, pp. 551-556.
9. Rydygier E., Strzyakowski Z.: Uycie symulacji komputerowych do opracowania metody obliczeniowej identyfikacji róde w inynierskich zastosowaniach transportowych. Prace Naukowe Politechniki Radomskiej. Transport 2(20), 2004, pp. 481-486.
10. Rydygier E., Strzyakowski Z.: Komputerowe wspomaganie rozwizywania inynierskich problemów odwrotnych w zagadnieniach mechaniki i transportu. Prace Naukowe Politechniki Radomskiej, Transport 3(23), 2005, pp. 455-460.
11. Rydygier E., Strzyakowski Z.: Symulacje komputerowe w badaniach kontaktu tocznego koo-szyna. Materiay XIII Warsztatów Naukowych PTSK nt. „Symulacja w badaniach i rozwoju”, Kazimierz Dolny, Poland, August/September 2006, pp. 85-86.
12. Taler J., Duda P.: Rozwizywanie prostych i odwrotnych zagadnie przewodzenia ciepa. WNT, Warszawa 2003.
13. Tikhonov A. N., Goncharsky A. V., Stepanov V. V., Yagola A. G.: Numerical Methods for the Solution of Ill-posed Problem. Kluwer, Dordrecht 1995.
ZASTOSOWANIE METOD NUMERYCZNYCH IDENTYFIKACJI RÓDE POLA DO WYBRANYCH ZAGADNIE W UKADZIE
POJAZD SZYNOWY-TOR
Streszczenie: W pracy przedstawiono metod numeryczn identyfikacji róde pola i jej zastosowania w
rozwizywaniu wybranych zagadnie w ukadzie pojazd szynowy - tor. W modelowaniu matematycznym identyfikacja róde pola naley do klasy problemów odwrotnych. Z tego wzgldu opracowana metoda identyfikacji zostaa uzupeniona przez procedur regularyzacyjn. Zamieszczone w pracy przykady stanowi ilustracj efektywnoci obliczeniowej metody w rozwizywaniu takich problemów modelowania ukadu pojazd szynowy-tor jak: identyfikacja róde pola napre szyny kolejowej wywoanych jej skrcaniem oraz identyfikacja róde ciepa w ladzie cieplnym na powierzchni szyny powstaym w wyniku kontaktu tocznego koo-szyna. W pracy wskazano take na to, e opracowany sposób identyfikacji moe by wykorzystany do weryfikacji cech mechanicznych kontaktu tocznego koo-szyna poprzez uycie cieplnego modelu kontaktu.
Sowa kluczowe: identyfikacja róde pola, modelowanie ukadów 2-D, kontakt toczny koo-szyna
