 Open Access
 Total Downloads : 385
 Authors : Crinela Pislaru, Arthur Anyakwo
 Paper ID : IJERTV3IS031727
 Volume & Issue : Volume 03, Issue 03 (March 2014)
 Published (First Online): 31032014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
on QuasiNewton Method Applied to 2D WheelRail Contact Models
Crinela Pislaru, Arthur Anyakwo
University of Huddersfield, Institute of Railway Research Huddersfield, United Kingdom
AbstractReliable and proficient numerical methods are required to determine the contact points between wheel and rail. This paper presents the use of QuasiNewton method for determining the solution of a reduced number of nonlinear wheelrail contact geometry equations that arise as a result of the interaction of wheel and rail on the track.
A novel two dimensional (2D) wheelrail contact model is developed by using the wheelrail contact coordinates to calculate the wheelrail normal contact forces without approximating the contact angle. The simulated results are stored in a lookup table and accessed during the simulation of the bogie dynamic behaviour thus reducing the computational time. The reduced number of nonlinear wheelrail contact geometry equations and employment of QuasiNewton method enable the proposed 2D wheelrail contact model to be used for fast and real time simulations of complex and nonlinear wheelrail contact mechanics.
Keywordswheelrail interface; lateral displacement; yaw angle; wheelrail contact model; railway vehicle dynamics; normal forces

INTRODUCTION
Numerical iterative methods have been widely used for solving systems of multidimensional nonlinear equations. Newtons methods [1] are widely used in most engineering problems (especially where accurate details of the system are known) due to the fast speed of convergence. However graphical methods should be employed to understand the system model and offer good guess for both single and multi dimensional equation version. NewtonRaphson methods [1] exhibit a rather fast speed of quadratic convergence once the solution has been found. The major setbacks are the expensive computation of the Jacobian matrix for the solution of non linear equations and the inability to make an initial guess.
QuasiNewton methods [2] eliminate the need for the computation of the Jacobian at every time step. It is often preferable to store an approximation to the Jacobian rather than anapproximation to the inverse Jacobian for solving large systems of nonlinear equations. The updating procedure can be mademore efficient for the approximate Jacobian than for the approximate inverse Jacobianwhen the Jacobian issparse and the locations of the zeroes are known.This approximate Jacobian matrix is used to determine the solutions of the non linear multidimensional equations which are so complex that their differentiation might not be practical. The computational time is reduced so it is possible to run the iteration in conjunction with other iterations and QuasiNewton methods
are used in the present paper to solve wheelrail contact dynamics problems.
Wickens [3] solves the wheelrail contact geometry equations using Newton Raphsons method. The wheelrail contact co ordinates are determined by taking into account the lateral displacement and the roll angle and then used for wheelrail track simulations. But it is difficult to make an initial guess for solving the equations.Sugiyama and Suda [4] apply Newton Raphsons method to solve wheelrail contact equations and determine the contact points by using the online contact search, offline contact search and hybrid contact search methods. They perform multibody railroad vehicle dynamics simulations using the elastic contact method and the hybrid method which combines the online and offline methods for the determination of the wheelrail contact points. However experience of the actual wheelrail contact geometry has to be used to choose the starting guess for the simulation process.
Anyakwo et al [5] develop a novel method for determining the contact positions of wheelrail contact using the analytical noniterative approach. The wheel and rail profiles are divided into various regions of contact and the equations relating the wheel movement and the rail are derived with the lateral displacement as input. QuasiNewtons method is used to determine the wheelrail contact geometry parameters which are saved in a lookup table. This technique does not require the contact position to be determined by adjusting the roll angle repeatedly until minimum difference between the wheel and the rail profile is achieved. The iterative approach is eliminated because the rolling radius difference change is negligible for very small changes in the lateral contact position. But the wheel and rail profile regions are switched repeatedly to determine the contact positions depending on the contact point regions thus making it very tedious to use especially for nonconical wheel profiles where the tread region is nonlinear in shape. Also fourteen nonlinear differential equations have to be solved synchronously which requires increased computational power.
Zheng and Wu [6] determine the solutions of the normal contact problem: normal contact forces, size, shape and orientation of the wheelrail contact patch and normal pressure distribution along the contact patch area. Analytical techniques are used to determine the normal contact forces on the wheel tread region for the left and right wheelrail contact assuming that for nonconformal contact condition exists and Hertz contact model is applied. This method is valid for the computation of the normal forces provided that the effect of the contact angles and the roll angles are small.
Iwnicki [7] presents an approximate analytical method for determining the wheelrail contact normal forcesconsidering the effect of the contact angles: roll angle and yaw angle. The
( , )
= ( , )
(5)
study shows that the computed normal contact forces at flange contact depend on friction coefficient, contact angle and yaw angle of the wheelset and the axle load. This method gives accurate predictions of the normal contact forces occurring at the flange contact region thus enabling yaw angle of the wheelset to be included in dynamic simulations.
This paper presents the use of QuasiNewton method for determining the solution of a reduced number of nonlinear
where is equal to 1 then equation (5) gives = 1 and we have the generic Newtons method. If we start with an initial estimate C1we can write:
+1 = + (6)
The matrix can be determined and expressed as
differential equations of the wheelrail contact geometry thus
reducing the time required for dynamic simulations of the
= (
(7)
, ) ( , )
bogie on the railway track. Also this paper describes the
development of a novel 2D wheelrail contact model and the normal contact problem, tangential contact problem and wheel rail dynamic simulations are implemented to investigate the dynamic behaviour of a bogie on the track. Also the wheelrail normal contact forces are calculated without approximating the contact angle.This noniterative 2D wheel rail contact model is useful for studying wheelset derailment, prediction of
The vectors , and are not yet determined.
expressed as follows because + 1 < :
1
= +1 +
=+1
can be
(8)
wheel climb, wear predictions and lateral stability of the bogie on the track.

QUASI NEWTON METHOD
Computing the Jacobian matrix is very expensive especially if much of the work carried out is used in evaluating the function
. The Jacoian is therefore difficult to evaluate since it is computed using finite differences. Quasi Newton methods replace the true ( ) in the Newton Raphson equation in equation (1) by estimates which can be modelled from value function over the sequence of iterations.
In QuasiNewton iteration method, the sequence of approximations is expressed by the equation similar to Newton Raphson method [2].
If the vectors , and are chosen such that they can be
orthogonal to the subspace spanned by ( < ):
, = , = 0 (9)
The convergence of the iteration expressed in equation (7), (8) and (9) can be used to solve a system of equations since the space of the residuals cannot exceed n because of the linearly independent vectors in the space. The matrix +1 can be structured assuming that , ( ). Thus since the vectors need to be orthogonal, a great deal of flexibility exists for different choices of and . One possible simplification of the quasiNewton method is to express as the follows:
+1 = []1{} (1) Where J is the jacobian matrix, f is the nonlinear equation function and xn is the variable.
= ( ) (
where = ( )
(10)
, )
The QuasiNewton method can be expressed as:
+1 = ( ) (2)
where is a complex number and matrix represents the nth approximation of 1(). Introducing a metric into the
This value of the vectors , and can be used to obtain other quasi Newton methods for solving nonlinear differential equations. For instance Powell algorithm [8] is one method that is used for calculating the unconstrained minimum (or maximum) of the quadratic function given by:
residual space it yields:
+1 = +1, +1
=
+1
+1
(3)
=
( )
( , )
(11)
where
is the transpose of and the metric matrix N is the
Equation (11) defines the correction matrix proposed by
Hermitian which is independent of and is positive or negative definite. The complex number, can be calculated to minimize the function +1. This leads to the nonlinear equation of the form:
+1, +1 ( ) = 0 (4)
For the special case if is a nonsingular matrix, equation (4) is linear in and it gives:
DavidonFletcherPowell [8]. This algorithm poses difficulties since problems occur in finding the constrained extremum. The modification suggested was found to be computationally unsatisfactory due to the round off errors obtained during the simulations. These errors are very large and thus lead to various problems. The most stable quasi Newton method that is stable to any system of equations that could be linear and nonlinear is the method proposed by Barnes presented by Rosen [18].Barnes algorithm defines as follows:
= ( ) ( , ) (12)
where is obtained from using Schmidt orthogonalization procedure presented in [18]. The first method assumes that is equal to . is then expressed as follows:
= ( ) ( , ) (13)
The disadvantage of using this algorithm is that it cannot be guaranteed that will be orthogonal thus we cant be very sure of convergence. The second method proposed by Broyden [9] chooses = to realize
= ( ) ( , ) (14)
The inputs for the proposed 2D wheelrail contact geometry are the lateral displacement (uy), the roll angle () and the piecewise cubic interpolation of the wheel and the rail profiles. The standard new P8 wheel profile [13] and BS 113A rail profile [14] are used to develop the wheelrail contact model.
The fixed frame of reference shown in Figure 2 defines the contact point location with respect to the wheelset frame centre of mass. This means that the coordinates at the wheelset centre is AfOfBf (0,0). The coordinate at the right wheel contact with respect to the fixed reference frame is (Ywr,
W(Ywr)) while the coordinates at the left wheel contact is (Ywl, W(Yrl)).
Table 1.1 shows the coordinates of the right/left wheelrail profiles at central position
Table 1.1. Wheelrail contact coordinates at central position
Right wheelrail profile
Left wheelrail profile
Ywr = Yrr
W(Ywr) =
R(Yrr)
Ywl = Yrl
W(Ywl) =
R(Yrl)
742.203 mm
460.625 mm
742.203 mm
460.625 mm
This suffers from problems that will not be orthogonal is most cases. The problem can be solved if does not lie near the solution [9].

WHEELRAIL CONTACT POINT DETERMINATION
DavidonFletcherPowellderived above is used to solve the reduced set of nonlinear differential wheelrail contact equations in MATLAB. The wheelrail contact geometry equations are derived from the wheelrail contact geometry model that would be discussed shortly. The wheelrail contact model used is the 2D wheelrail contact model that considers the movement of the wheelset in two dimensions, the lateral displacement (uy) and roll angle () thus forming two degrees of freedom. The four wheelrail contact coordinates of interest for each wheelrail contact are the lateral wheel contact coordinate (Ywr), lateral rail contact coordinate (Yrr), vertical wheel contact coordinate W(Ywr) and the vertical rail contact coordinate R(Yrr) as shown in the figure below:
Figure 1. Right wheelrail contact at central position
Similarly the contact positions at the left wheelrail contact are the same but the lateral coordinates are negative since it is on the negative axis of the wheelrail contact.
Figure 2. Fixed frame of reference AfOfBf
The kinematic equation describing the contact point location on the wheel and rail interface is:
= + (15) where is a generic point on the rail profile,
defines the movement of the wheelset in the vertical and lateral direction, is the rotation matrix, and is a generic point on the wheelset with respect to the Fixed reference frame and j represents the left (l) or right (r). The rotation matrix is a function of the roll angle and is defined as the rotation of the wheelset about the longitudinal direction of motion. It can be expressed as follows:
= (16)
While coordinates of the wheelset centre of mass is:
=
(17)
+
(28)
+
= 0
The position of a point on the right rail profile is:
Similarly for the left wheelrail contact the tangent to the
=
( )
(18)
wheel and rail profile plane can be found by differentiating Equation (27):
While the position of a point on the right wheel profile is:
+
=
(19)
+ = 0 (29)
( )
Substituting Equation (16) to (19) into Equation (15) and
= + (20)
( ) = + + (21)
Equation (26) to (29) can then be solved using Quasi Newtons to determine the four unknowns , , , for given inputs and. The error difference between the
vertical displacements of the right and left wheel must be less than 1×106mm to ensure that is approximately equal to
.
expanding the expression we have:
Similarly, the equations for the left wheelrail conact geometry can be represented as thus; the position of a point on the left rail profile is defined as:
Figure 3 contains the block diagram showing the steps of the novel algorithm proposed to solve the wheelrail contact geometry equations (26 29).
=
( )
(22)
Rail profile
Wheel profile
While the position of a point on the left wheel profile is:
=
( )
(23)
Substituting Equation (22) to (23) into Equation (15) and
= + (24)
( ) = + + (25) expanding the expression we have:
Equations (20) and (21) contain three unknown variables,Yrr(lateral contact point on the right rail profile),Ywr(Lateral contact position of the right wheel profile) and (vertical displacement at the right wheel) while equations (24) and (25) contain (lateral contact point on the left rail profile), (lateral contact point on the right wheel profile) and (vertical displacement at the right wheel).
is the left wheel rolling radius while is the right wheel rolling radius, is the rolling radius and is the
roll angle of the wheelset.
The three unknown variables included in the equations for the right and left wheel can be reduced to two unknown variables by substituting Equation (20) into Equation (21) and Equation
(24) into Equation (25) as follows:
Lateral displacemen t
2D
Wheel rail geometry Equations
Solve equations using
QuasiNewton
e = uzr – uzl
Increase Roll angle
e = 0 No
+ =
+ +
+
(26)
Lateral wheelrail contact positions ( , , , )
Right/left wheel rolling radius
(( ),( ))
Yes
= + + (27)
The wheel and rail profileshave to touch each other only in one contact point location thus fulfilling the nonconformal condition with no interpenetration[1]. For this to be satisfied the tangents to the wheel and rail profile planes must be determined by differentiating Equation (26) with respect to and Equation (27) with respect to . This yields:
Figure 3. Block diagram algorithm for the wheelrail contact
geometry
The outputs of the block diagram which include the lateral wheelrail contact positions, right/left wheel rolling radius and the wheelrail contact angle are used in wheelrail contact model to investigate the dynamic behaviour of the bogie on the railway track. The contact point locations on BS 113A and P8 wheel profiles are determined using the block diagram and can be shown in Figure 4and Figure 5 below;
Figure 4. Right wheelrail contact positions (Positive/Negative lateral displacement)
Figure 4 shows that a significant jump is noticed from 6 mm to 8 mm because the right wheel has reached the flange region. Also for negative lateral displacement, the contact jump is observed from 2 mm to 4 mm due to the rail and wheelgeometry and the rail cant angle.
Left BS 113A rail profile Left P8 wheel profile
10 8 6 4
2 0
440
Vertical axis (mm)
450
460
470
480
490
820 800 780 760 740 720 700 680
Lateral axis (mm)
Figure 5. Left wheelrail contact positions (Positive/Negative lateral displacement)
It can be observed from Figure 4 that there are significant contact point jump from 2 mm to 4 mm due to the wheel profile design and the rail geometry.
3.1. Wheelrail contact geometry results
Figure 5 shows the results obtained for the right and left wheelrail contact positions
(a)
(b)
Figure 6: Right (a) and Left (b) lateral wheel contact positions
There is a contact point jump in the right wheel lateral contact position between points A and B because the right wheel has reached the wheel flange region. For the left wheel, two contact jumps are observed in the lateral left wheel contact position from C to D and from D to E. This is as a result of the cant angle of the rail profile and the geometry of the wheel profile at those regions.
(a)
(b)
Figure 7. Right (a) and Left (b) lateral rail contact positions
Figure 7 shows the contact point location of the right and left rail contact position as a function of the lateral displacement. Results show that for lateral displacement range of 6.5 mm to about 6.8 mm the contact point position jumps from E to F. This indicates that the wheelset has reached the rail gauge region of the right rail as a result of flange contact. Similarly the rail contact point location jumped from G to H for 2 mm to 4 mm range as a result of the cant angle of the rail profile at that region.
(a)
Figure 8 shows the rolling radius of the left right and left wheel. The rolling radius of the right wheel increase slowly with increasing lateral displacement in the wheel tread region and then increases sharply after 6.5 mm until it gets to flange contact at 6.695942 mm. Also the left wheel rolling radius also shows significant decrease in the rolling radius for lateral displacement range 2 mm to 4 mm as a result of the rail profile geometry.
(a)
(b)
Figure 9. Right (a) and left (b) wheel contact angle
Figure 9 represents the right wheel and left wheel contact angle for lateral displacement range of 0 mm to 10 mm. A sharp increase in right contact angle occurs at lateral displacement near the flange region as a result of right flange/rail gauge contact. The maximum contact angle at flange contact is 68.05 degrees. The contact angle can be mathematicallyexpressed as by the following expression:
( )
=
(
,
)
(30)
=
(b)
Figure 8. Right (a) and left (b) wheel rolling radius
Figure 9 displays the rolling radius difference function obtained by subtracting the rolling radius of the left wheel from the right wheel.
= + , = ,
(33)
Figure10. Rolling radius difference function
The rolling radius difference in Figure 10 was obtained by
where is the lateral creep force developed at the right wheelrail, is the lateral creep force developed at the left wheel wheelrail contact, Fsusp is the lateral suspension force.
In the vertical direction for the right wheelrail contact vertical forces can be expressed as follows;
sin( ) + cos = 0 (34)
Similarly, for the left wheelrail contact the vertical forces can be resolved as follows;
sin( ) + cos = 0 (35)
Where and are the right and left vertical forces applied on the wheelrail contact respectively. The can be expressed as follows [6]:
( )
finding the difference between the right wheel and the left wheel rail contact. The Rolling Radius Difference (RRD) is
= 2 + 2
+
(36)
( + )
expressed as follows
+ 2 +
= ( ) ( ) (31)
The wheelrail contact coordinates Ywr, Yrr,W(Ywr), R(Yrr)
( )
= 2 2 +
(37)
have been determined using Quasinewton method. The are saved in a lookup table and used for dynamic simulations. This offers advantages over the iterative methods whereby the wheelrail contact points are determined for each ateral displacement and used for dynamic simulations.

NORMAL CONTACT PROBLEM
The normal contact problem resolves the vertical and normal contact forces acting on the wheelrail contact. The wheelrail contact forces are derived by analyzed the creep forces developed on the wheelrail contact. Figure 11 shows the wheelrail contact forces acting on the wheelset.
( )
+ 2 +
Where W is the wheelset axle weight, is the half length of the longitudinal spring, and are the distances from the right and left nominal contact positions, m is the mass of the wheelset, and g is the acceleration due to gravity.
Let i = L, R, then applying Kalkers linear theory, the longitudinal ( ), lateral ( )and spin moment ( ) creep forces developed at the wheelrail contact can be expressed as follows:
= 11 , = 22 23 ,
= 23 33 (38)
The longitudinal ( ), lateral ( ) and spin ( ) creepages developed in the contact patch as a result of traction and braking can be defined as follows:
= 1
( ) Â± 0 ,
= cos( ),
0
= Â± sin ( )
(39)
Figure 11. Wheelrail interaction forces on the wheelset
The equation of motion of the wheelset in the lateral direction is expressed as follows:
where is the yaw angle, and Â± indicates the signs for the calculating the creepages. A positive signifies calculation of creepages for the left wheelrail contact while the negative sign indicates calculation of creepages for the right wheelrail contact. The longitudinal (11 ), lateral (22 ), lateral/spin (23 ) and spin ( 33 ) linear creep coefficients as proposed by Kalker by be defined as:
11 = 11 , 22 = 22 , 23 = ( )1.523 ,
cos
+ cos( ) sin( )
= 0 (32)
33 = ( )233 (40)
where G is the modulus of rigidity of the wheel and rail
where
+ sin( ) +
materials given as:
=
(41)
2(1+)
E is the Young modulus of steel equal (207GPa) and is Poissons ratio equal to 0.33 [12], [13]. C11i,C22i,C23iand C33i are the longitudinal, lateral, lateral/spin creep and spin coefficients respectively. They depend on the Poissons ratio,
and the ratio of the semiaxes of the contact patch ai, bi which represent the longitudinal and lateral semiaxes of the contact patch ellipse. They can be expressed as follows:
applied to solve all the equations acting on the vehicle. Details of the equations of motion of the bogie on the railway track can be found in [16]. Details of the calculations of the primary suspensions of wheelset 1 (Fsusp1) and wheelset 2 (Fsusp2) and the suspension moments of wheelset 1(Msusp1) and wheelset 2 (Msusp2) can be found in [16].
=
3(12 )
1/3
,

NUMERICAL SIMULATION RESULTS
=
3(12 )
1/3
2( + )
1
, + = 0.5 +
1 (42)
For application purposes the saved wheelrail contact geometry coordinates; the rolling radius, contact angle, roll angle and lateral wheelrail coordinates was used to perform
2( + )
( )
dynamic simulations of a single bogie running on a straight
RRiis the principal transverse radii of curvature for the rail profile while Riis the principal radi of curvature for the wheel profile.
Substituting ai and bifrom equation (42) into equation (40), equation (40) into equation (38) and equation (38) into the equation (34) whereby i = L or R yields:
track. The bogie and the wheelset considered here are from the Manchester benchmark bogie used in British Rail Mark IV trains in the U.K. The parameters used for the simulation of the bogie model can be found in [16]. Figure 12 below shows the lateral behaviour of the front wheelset for forward speeds 10, 30, and 50 m/s using Heuristic nonlinear method.
where
= 22 2/3 + 23 (43)
22 =
3(1 2)
2( + )
3(1 2)
2/3
22 (44)
23 = ( )3/2
2( + )
23 (45)
The normal contact forcesdeveloped on the right and left wheel can be solved using Quasi Newton method by substituting Equation (43) into Equation (34) or (35) to obtain the expression:
(22 2/3 + 23 )sin( ) + cos = 0 (46)
It is important to note that an initial guess is required for the normal contact forces of the right and left wheel rail contact for a feasible solution to be found. The starting guess is usually at the initial normal load of the wheel at central position. The simulation time required for computing the normal force is reduced since the wheelrail contact co ordinates have already been saved in a lookup table and hence are easily accessible for the calculation of the normal force and hence for dynamic simulations of the bogie on the track.
The wheelrail contact problem is nonlinear hence using a linear theory to relate the creep forcecreepage leads to errors due to the nonlinear geometric functions and the adhesion limits. The heuristic nonlinear model computes the creep forces at the linear and nonlinear region of the creep force creepage curve. It includes the effect of spin creepage of which is neglected in Johnson and Vermeulen. The theory of heuristic nonlinear creep force model is discussed in [14],
[15] and is used to calculated the tangential creep forces developed at the wheelrail contact.The equations of motion describing the movement of the bogie on the track can be implement by summing all the lateral forces and spin creep moment forces. Newtons law is then
Figure 12. Lateral displacement of the front wheelset of the bogie for forward speeds 10m/s, 30 m/s and 50 m/s.
For forward speeds of 10 m/s and 30 m/s it can be observed that for the initial lateral misalignment of the front wheelset at
0.005 m decays and then returns to its central position on the track at nearly zero lateral displacement. As the velocity of the wheelset is increased from 10 m/s to 30 m/s and then 50 m/s the lateral oscillations increases and finally saturates at 50 m/s leading to hunting. For a velocity of 50 m/s the hunting motion has a null decaying rate hence the lateral behaviour of the wheelset shows harmonic oscillation. Hence the critical speed of the bogie model is 50 m/s or 180 km/hr.
Figure 13. Yaw angle of the front wheelset of the bogie for forward speeds 10m/s, 30 m/s and 50 m/s.
Figure 13 shows the yaw angle of the front wheelset of the bogie for speeds, 10 m/s, 30 m/s and 50 m/s. For low forward speed 10 m/s the yaw angle response decays with time and settles to about zero radians, thus indicating that the front wheelset has returned to its central position. As the speed increases the yaw angle amplitude oscillations increase significantly. At 50 m/s hunting is observed which is similar to the lateral displacement of the wheelset.
Figure 14. Normal contact force developed on the front wheelset for a forward speed of 10 m/s, 30 m/s and 50 m/s.
Figure 14 shows the normal contact force response of the front wheelset at forward speeds 10 m/s, 30 m/s and 50 m/s. The normal contact force decays at low speed with respect to time and settles at the central position at about 64 kN. For high forward speeds hunting occurs which show sustained oscillations of the normal forces. This indicates that critical speed of the bogie has been reached. The simulation time is reduced since the since the saved wheelrail contact co ordinates are stored offline and used to the simulation of dynamic movement of the wheelrail contact in the system. In most iterative procedures for determining the heelrail contact, the wheelrail contact points are determined online and used for dynamic simulations of the bogie on the track. This slows down the computation time since at every time step the wheelrail contact point must be determined.

CONCLUSIONS
This paper presents the use of QuasiNewton method for determining the solution of a reduced number of nonlinear wheelrail contact geometry equations that arise as a result of the interaction of wheel and rail on the track. The simulation time is reduced since the since the saved wheelrail contact co ordinates are stored offline and used to the simulation of dynamic movement of the wheelrail contact in the system. In most iterative procedures for determining the wheelrail contact, the wheelrail contact points are determined online and used for dynamic simulations of the bogie on the track. This slows down the computation time since at every time step the wheelrail contact point must be determined. A novel two
dimensional (2D) wheelrail contact model is developed by using the wheelrail contact coordinates to calculate the wheelrail normal contact forces without approximating the contact angle. The simulated results have been stored in a lookup table and accessed during the simulation of the bogie dynamic behaviour thus reducing the computational time. The reduced number of nonlinear wheelrail contact geometry equations and employment of QuasiNewton method enable the proposed 2D wheelrail contact model to be used for fast and real time simulations of complex and nonlinear wheel rail contact mechanics and advanced condition monitoring systems for railway vehicles.
The wheelrail contact coordinates geometry determined using a reduced number of nonlinear wheelrail geometry equations have been investigated using QuasiNewtons method. The usage of QuasiNewton method is investigated for the solution of the nonlinear wheelrail contact geometry equations that arise as a result of the interaction of the wheel and the rail on the track. The results indicate that the Quasi Newton method provides an efficient solution strategy for the determination of the wheelrail contact coordinates. The wheelrail normal contact forces are then calculated without approximating the contact angle. The results of the simulations in are stored in a lookup table and accessed during the dynamic analysis of bogies thus reducing the computational time. The process of solving the ordinary differential equations representing the bogie dynamic behaviour becomes faster by using the lookup table. The simulation time is reduced since the since the saved wheelrail contact co ordinates are stored offline and used to the simulation of dynamic movement of the wheelrail contact in the system. In most iterative procedures for determining the wheelrail contact, the wheelrail contact points are determined online and used for dynamic simulations of the bogie on the track. This slows down the computation time since at every time step the wheelrail contact point must be determined.
The developed novel 2D wheel rail contact model is useful for studying wheelset derailment, prediction of wheel climb, wear predictions and lateral stability of the bogie on the track. The critical velocity of the bogie model using Heuristic nonlinear model is 50 m/s. The proposed 2D wheelrail contact model could be used for fast and realtime simulations of complex and nonlinear wheelrail contact mechanics.
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