


ARTICLE 

Year : 2015  Volume
: 5
 Issue : 1  Page : 813 

Step Response and Estimation of Lateral and Yaw Motion Disturbance of Rail Wheel set
Zulfiqar Ali Soomro
Department of Mechanical, Mehran University of Engineering and Technology, Jamshoro, Pakistan
Date of Web Publication  16Jan2015 
Correspondence Address: Zulfiqar Ali Soomro Department of Mechanical, Mehran University of Engineering and Technology, Jamshoro Pakistan
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.149473
Abstract   
The rail wheelset dynamics has skeleton importance to avoid the slip and slide due to insufficient adhesion level depending on creep forces. Proper study and control upon lateral and yaw motion owing to creep forces may reduce the problem at some extent. In this paper, this dynamics of wheelset with its possible degree of freedom has been discussed and limited to lateral and yaw analysis. These analyses have been computed and simulated by simulink to observe their behavior. The disturbances of these dynamical parameters are estimated using Kalman filter. The correlation of actual and estimated curves has been assessed to invent novel idea. Keywords: Adhesion, creepage, degree of freedom, Kalman estimator, lateral and yaw motion, lateral motion, longitudinal creep, spin moment
How to cite this article: Soomro ZA. Step Response and Estimation of Lateral and Yaw Motion Disturbance of Rail Wheel set. J Eng Technol 2015;5:813 
1. Introduction   
Hence, many of ideas have been invented to determine the running condition of the wheelrail interface ^{[1],[2]} . A model based scheme for condition monitoring at the wheel rail interface is proposed in Charles et al., 2008 ^{[2]} . The adhesion level is identified by measuring the dynamic response of the vehicle to lateral track irregularities under normal running. This work is further forwarded in (Ward) with the basic purpose to estimate the creep forces created at the contact of wheelrail. An inverse modeling for the estimation of the creep coefficients of car body acceleration using measurement was aimed in Xia et al., 2007 ^{[3]} .
The wheelset is connected to the vehicle body or bogie via suspension elements in lateral and longitudinal directions. The longitudinal analysis of the wheelset and the bogie were considered to be solid as the stiffness is usually very high and concerned dynamics are not of the significant correlation to study ^{[4]} .
However, longitudinal springs provide necessary yaw stiffness to stabilize the kinematic motion of the wheelset. All the forces governing the dynamic behavior of the rail wheels are procured at the railwheel contact patch is very small (about 1 cm to 2 cm) ^{[5]} .
It is very necessary to study of rail wheelset dynamics to innovate and utilize a comprehensive model including all concerned motions of the wheelset correlated to the contact forces due to the strong interactions between different motions of the wheelset through the creep forces at the wheelrail contact acting upon both longitudinal and lateral dimensions. The railway wheelset has many degrees of freedom (DOF). The lateral displacement, yaw angle, longitudinal motion and the rotational motion are considered for this study, whereas the roll and bounce motion are constrained by the track and not included in the wheelset model. The dynamic equations are given as follows ^{[6],[7]} . For modeling rail and wheel correlation it is necessary to see the papers of Esveld, Knothe and Jenkins et al. for Hertzian spring ^{[8],[9],[10]} . Furthermore, Yalcin and et al., have analyzed rail vehicle vibrations for different types of track and operating speeds for the dynamic response of vehicletrack interaction ^{[11]} .
In this paper, modeling of wheels is discussed by its dynamics and DOF consisting lateral and yaw motion. Mathematics of model is shaped by simulink to get results and then these results are to be estimated and compared.
2. Modeling of Rail Vehicle Dynamics   
A vehicle is a very complex system, so many of parameters are usually required if a multibody dynamics method is adopted to model it in detail. To minimize the complexity and difficulty of developing a vehicle dynamics control system, it is common to build a relatively simple vehicle model and thus validate the feasibility in the concept design stage.
2.1 Possible rail wheelset motions (degree of freedom)
The possible motion (degree of freedom) for the railway wheelset during static and dynamic conditions are shown as under.
For the test of the dynamic interactions that occur at the interface between the track and the vehicle, for a rail vehicle analysis the electric locomotive is shown in [Figure 1]a below, was chosen. Despite the awareness that it is "the aging" locomotive model, it was chosen because of easy access to technical documentation to allow replay in the virtual space of the essential features of the vehicle ^{[12]} .
A full rail vehicle consists of car body, bogie, wheelset, primary and secondary suspension. Due to the number of these components and their moving modes, model DOF are determined. In [Figure 1]b, the rotation around x (Φ), y (λ), z (Ψ) axes are called rolling, pitch and yaw respectively. And also vertical (z), longitudinal (x) and lateral (y) motions are shown in [Figure 1]. Chudzikiewicz presented a wider expression by giving equations for all kinds of motion and also creep in his study ^{[13]} . As wheelset moves to x and y directions hence mainly longitudinal and lateral are considered while z plays the motion with coaxes.
2.2 Rail wheelset dynamics
According to the schematic diagram represented in [Figure 2]a, the car body is connected to the bogie frames by means of suspension elements usually known as the primary suspensions, then the bogie frames are connected to the wheelsets by means of other spring elements defined as the secondary suspensions. The forces applied to the wheelset are transmitted upwards through these elements. The vehicle performance and dynamic behavior are affected by the characteristics of these elements ^{[14]} .
[Figure 2]b shows a plan view of a wheelset with a yaw angle relative to the track known as the angle of attack, contributes to the lateral creepage through a component of the wheelset's rotational velocity. If the wheelset has a lateral velocity in addition to the component of lateral velocity due to its rotation, the net lateral velocity of the wheelset at the contact zone, assuming the angle of attack to be small. It is clear that if the wheelset is moving towars flange contact with a positive angle of attack, the lateral velocity tends to reduce the effective angle of attack. The direction of the lateral creep force depends upon the resultant of the contribution of both the lateral and spin creepages ^{[15]} .
2.3 Mathematical formulation of wheelset
For the practicality purpose, it is necessary that the design of the estimators should be as simple as possible by considering the wheelset dynamics, which are directly related to the contact condition. Previous studies have shown that the lateral and yaw dynamics are sufficient to detect these changes ^{[2],[16],[17],[18]} .
Therefore for the estimator design some simplifications in the wheelset model are introduced. The simplified dynamic equations involving only yaw and lateral dynamics are given below. The Longitudinal creepage of left and right rail wheelset are extracted as under.
and
If the horizontal force F at the wheeltread is perpendicular to the plane of the wheel, then the motion of the wheel in the direction of F, divided by the distance it travels in its own plane, will be called the "lateral creepage (λy)
The yaw angle is the result of the difference in the longitudinal creep forces between the two wheels and the lateral dynamics are determined by the total creep force of the two wheels in the lateral direction. The relationship for the lateral and the yaw motions of the railway wheelset are given in below Equations (6,7).
Where m _{w} is the wheelset mass, F _{c} is the centrifugal component of the force and can be ignored if the wheelset is not running on the curved track, I _{w} is the yaw moment of inertia and kw is the yaw stiffness of a spring used to stabilize the wheelset. F _{yL} , f _{yR} , f _{xL} and f _{xR} are the creep forces of left and right wheels in lateral and longitudinal directions.
3. Estimation of Rail Wheelset Model   
3.1 Linearization of model
The derivative terms represent the slope on the creep curve at the point of linearization which is simplified and represented by g _{11} and g _{12} .
All the creep forces are linearized in the similar way and the linearized small signal model of the lateral and yaw dynamics of the railway wheelset is obtained in Equation (10).
The final small signal model of the simplified wheelset model is given in the following equation in the below state space form.
This shows that not only a change in contact condition has a significant effect on wheelset dynamics, but the dynamics are also varied during the application of traction and braking.
3.2 Proposed KalmanBucy filter
The most commonly used state estimator is the Kalman filter. KalmanBucy filter (which is a continuous time counterpart of the discrete Kaman filter) is chosen for this research because it gives good results due to optimality and robustness in presence of stochastic noises applied by the track. A Kalman filter is based on small signal model of equation (10) can be designed to estimate the states of the rail wheel system at specific point on the creep curve for adhesion.
This is not a trivial application of the Kalman filter because the track irregularities denoted by y_{t} excite the lateral and yaw dynamics of the railway wheelset are not directly measurable ^{[16]} and cannot be neglected either, which makes the design of the Kalman filter more complicated. Hence, the equation (10) is reformulated where the unknown unknown parameter y_{t} is considered as the part of the state vector so that the dynamics of the system are unchanged ^{[17]} .
Equation (11) shows the reformulated model where y_{t} is treated as part of the state vector by assuming following relation.
The wheelset dynamics are excited by the unknown track disturbance. In order to obtain good estimation results the unknown track disturbance is treated as part of the state vector as shown in the Equation (11), where it is assumed that ∆yÿ _{t} = −0.001+ yÿ _{t} .
In above Equation (12), the v is a vector represents the noise level of sensors. It is considered that there is no any relation between the process and measurement noises by tuning the Kalman filter parameters (R as measurement noise covariance, Q as process noise covariance and N as small constant) and the contact condition parameters (g_{11} and g_{22} ).
4. Simulation Results   
The simulation results for the evaluation of the wheelset are divided into two major two portions.
4.1 Wheelset dynamics simulations
The dynamics of the rail wheelset is simulated by simulink blocks as under through graphic correlation. The various dynamic parameters like lateral and yaw analysis having motion, velocity and acceleration accessed as under.
In [Figure 3], the simulink subsystem denoted by wheelset having input of forward speed of 30 m/s and track inputs. The track input has a gain of 0.005 with random input to both yaw and lateral parameters.
This wheelset system has five outputs divided into yaw movement, yaw rate and three lateral motion, velocity and acceleration.
In [Figure 4], the simulation results of yaw analysis premising the random track input, yaw movement and yaw rate graphics with respect to obtained curves are less dense showing a less noise. Whereas the raw rate has fewer evaluation results are as below from above dynamic wheelset simulink.
Disturbances than time have been displayed from "scope". The curves of random track input are dense showing a lot of disturbance while the yaw movement given random track input and higher noise than yaw movement.
In [Figure 5] the simulated curves of lateral movement, velocity and acceleration are compared to random track input. This random input has less intensity with small peaks than that of yaw track input used. The lateral movement has very thinner disturbance than lateral velocity, while the curves of lateral acceleration resemble to track input with a little bit disturbance but by great peaks.  Figure 5: Simulation results of lateral motion, velocity and acceleration
Click here to view 
4.2 Kalman filter estimation of wheelset parameters
The Kalman filter is used for following four parameters of the wheelset for the small scale of the railway vehicle.
4.2.1 Lateral motion (y)
In [Figure 6], the lateral motion of the wheelset has been estimated. Here lateral distance in "meters" is varied by time factor in "seconds". In above [Figure 1], the green curve shows actual line of the state, while the blue line as estimated value. There is so much visible difference between two curves.
The estimated curve rises from 0 meters up to 2 meters, and then it goes fall upto2 to nearly6 meters down at 2000 seconds. Then this blue curve goes up on 40008000 seconds with some fluctuations till it rises up from actual green curve to go down on 10,000 seconds. While actual curve of lateral movement shows the same behavior like that of scope results.
In [Figure 7], the yaw motion of the wheelset has been estimated. Here rotational distance is denoted in "radians" is varied by time factor in "seconds". The green curve shows actual line of the yaw motion, while the blue line as estimated value curve. There is a little bit visible difference between these two curves as they travel parallel with each other by very small space. The blue curve goes higher than blue after 20004000 seconds while mixed on 8000 seconds on going down.
In the [Figure 8] above, the lateral velocity of the wheelset has been estimated. Here lateral velocity in "m/s" is varied by time factor in "second". In [Figure 8], the green curve shows actual line of the state, while the blue line as estimated value. Lateral in the [Figure 8] above, the lateral velocity of the wheelset has been estimated. Here lateral velocity in "m/s" is varied by time factor in "seconds". In [Figure 8], the green curve shows actual line of the state, while the blue line as estimated value lateral.
There is a little bit visible difference between these two curves, with some coincidence. This [Figure 8] of lateral velocity resembles to that of [Figure 7] of yaw motion with smaller variation. The vertical plane of the yaw motion is displayed in "radians" and that of this velocity in "meter/second" with different scale values. The similarity is only to the behavior of the both curves like that of yaw motion.
In [Figure 9], The Yaw rate of the wheelset has been estimated. Here yaw velocity in "rad/s" is varied by time factor in "second." The green curve shows actual line of the state, while the blue line as estimated value. There is large visible difference between these two curves, with some coincidence at few instants. Both curves start for 0 radian under 2000 seconds, then estimated curve goes downward abruptly up to 4000 seconds. Then this green curve rises below actual curve to fall on 8000 seconds then goes upward the actual line to end on 10000 seconds.
5. Conclusion   
The dynamics of rail wheelset system has been discussed properly along with possible degree of freedom. In this paper, two important degree of freedom (DOF) like lateral and yaw in terms of dynamics were computed. A suitable simulink block was framed depending upon its mathematical formulation of lateral and yaw analysis. A graphical behavior of these dynamical parameters has been simulated, which show the disturbance with different variation. These disturbances are measured and estimated using "Baucy Kalaman filter". The behavioral correlation of actual and estimated curves has been observed. It is been accessed that there is similarity between yaw motion and lateral velocity, while lateral motion and yaw rate very differently.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9]
