Theoretical and experimental research on the phenomenon of stick-slip at traction railway vehicles

The stick-slip phenomenon may occur when the limit adhesion force is exceeded at one of the motor axles of the vehicle. In the present paper is analysed the physical phenomenon stick-slip, the conditions for its occurrence and the mechanical model recommended for the study of the phenomenon. Regarding the experimental research, it is presented the stand built in the laboratory of the Rolling Stock Department at Politehnica University of Bucharest.


General considerations
The motion of the drive axle may be accompanied by some intermittents or jerks, a phenomenon known in the speciality literature as stick-slip. As a consequence of the stick -slip, there are important dynamic overloads both in the axle and in its drive system, as well as variations in traction forces at the periphery of the wheels, which cause disturbance of the ride and thus reduce the traction performances of the vehicle.
Issues related to the stick-slip phenomenon will be analyzed for the case of the axle drive system with traction electric motor, at which the motor moment is transmitted to the traction gear by means of a torsion elastic shaft.
The stick -slip vibrations, the primary cause of which lies in the form of the wheel -rail friction and traction force characteristics, occur at low slide speeds, generally at the start of the vehicle, when it is possible the axle slip due to exceeding by the traction force of the limit force given by adhesion. The time variation of the sliding speed of the wheel when the stick -slip occurs, experimentally determined by Schröter and Schönenberger [1], can be seen in fig.1.
Generally, vibrations that occur under the influence of dry friction on the wheel -rail contact are strongly depending on the sliding speed. Thus, at low slip speeds (Fig.2 a), the motion has an adherent phase (stick) and a sliding phase (slip), i.e. an intermittent sliding occurs when the friction force varies between a maximum level T max limited by adhesion and a minimum level T min corresponding to the friction coefficient at the maximum slip speed. At high slip speeds (Fig.2, b) autovibrations are produced, the amplitude of which is much lower than in the case of stick-slip. The occurrence and development of the stick -slip is dependent on the variation law of the wheel -rail friction coefficient versus the slip speed.
In the case of axle sliding there is intense mechanical action between the particles of the wheel-rail contact surfaces, with a significant heat energy generation. The important modifications of the contact surfaces make the friction coefficient vary with the slide speed.

Factors that influence the stick-slip phenomenon
As is was highlighted before, the occurrence and time development of stick-slip depend on the variation of wheel-rail friction coefficient as a function of sliding speed. The friction coefficient  depends on the rail-wheel creep , which is given by the ratio between the slip velocity w the wheelset forward speed (the vehicle speed, practically) , as demonstrated by F. Carter. In Levi's law, the characteristic of the coefficient of friction has an ascending branch up to the creep value  p , corresponding to the maximum coefficient of friction (adhesion coefficient). As shown by Frederick F. [2], this characteristic has a descending branch for > p (see Fig. 1).
The stick-slip oscillations occur when the drive force F and the friction force T=Q, (Q being the wheel load), are decreasing characteristics function of slip velocity w, for a given value of speed see

Mechanical model for the study of stick-slip phenomenon
In Fig. 6 is shown an equivalent model of the driving system of motor wheelset. The occurrence of the stick-slip phenomenon can be explained as follows: in the slip velocity range (0, w p ), the torque of the traction motor causes torsion of its elastic shaft and of the wheelset axle. Due to the fact that at the slip velocity w p the traction force F p >T a , T a being the adhesion force, the wheels start to slip with the acceleration of the drive system. At the same time, there is a tension release of the elastic elements from drive system and a reduction of the forces acting on the wheel circumference up to the point I where F=T. Further, the motion slows down, the adhesion is restored, and in the same time the engine torque increases until it exceeds again the adhesion limit and the motion repeats itself.  Figure 6. Mechanical model for the study of stick-slip.
The system includes: the engine shaft on which acts the drive moment M m , a torque shaft of the motor with a stiffness k c , the assembly consisting of two-wheel gear with the transmission ratio u, transmitting the motion from the engine shaft to the wheelset axle having a stiffness k 1 and the two wheels of the vehicle on which act the moments M t1 , M t2 of friction forces T 1 , T 2 . I r -is the moment of inertia of the engine shaft; I -is the moment of inertia of vehicle wheel; I'=I+u 2 I r -is the moment of inertia of the wheel, including reduced moment of inertia of the two-gear assembly drive axle, axle stiffness portion of it and the axle of the vehicle considering the infinite value; u is the gear transmission ratio;  r -is the angular displacement of electric engine shaft;  1 ,  2 -angular displacements of the vehicle wheels.

Experimental model for the study of stick-slip phenomenon
In the laboratory of the Rolling Stock Department at Politehnica University of Bucharest has been developed a functional model of a motor vehicle bogie on which can be to modelled and experimented the wheel-rail interaction phenomena that occur when a fully equipped bogie runs on the rail ( fig.  7,8,9).  The experimental stand for dynamic performance determination consists of a bogie model fitted with an axle and two wheels driven by an electric motor by means of an elastic shaft. In the lower part of the wheels there are provided two rails operated in the vertical direction by means of a rectangular metal frame provided with a helical screw (Fig. 9) in order to tighten the rails on the wheels. Each frame arm is equipped with four tensiometric marks, the force transducer thus realized having the possibility of measuring the traction-compression force from the bar driving the rails.
The electric motor is of the asynchronous type and has the possibility of modifying the speed, depending on the traction force we want to achieve. Depending on the braking force, the speed of the electric motor will change, even reaching the wheel lock. A tachometric probe with an infrared emitter -receiver system is used to measure the instantaneous speed.
There were determined the accelerations in the vertical and horizontal -longitudinal directions at the left and right journals of the axle by means of piezoelectric accelerometers. The measuring transducer chain is connected to an acquisition system that performs simultaneous 12-channel acquisition.

Conclusion
The stick-slip phenomenon, which occurs generally at the start of traction vehicles, has a negative impact on both the mechanical resistance of the elements of the axle drive system and the traction performance of the vehicle.
The mechanical model proposed for studying the stick-slip phenomenon was validated by the experimental results obtained on the stand in the laboratory.
It is worth mentioning that experimental determinations have also been made on a locomotive in starting regime. The measurements were consistent with those obtained in the laboratory, thus confirming the correctness of the mechanical model.