Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2014 August 17-20, 2014, Buffalo, New York, USA
DETC2014-34159
A DYNAMIC MODEL OF FRICTION DRAFT GEAR Qing Wu*, Maksym Spiryagin, Colin Cole Centre for Railway Engineering, Central Queensland University Rockhampton, Queensland, QLD4701, Australia
ABSTRACT Friction draft gears are the most widely used draft gears. Modeling and prediction of their dynamic behavior are of significant assistance in addressing various concerns. Longer, heavier and faster heavy haul trains mean larger in-train forces and more complicated force patterns, which require further improvements of dynamic modeling of friction draft gears to assess the longitudinal train dynamics. In this paper a forcedisplacement characteristics model named “base model” was described. The base model was simulated after the analyses of a set of field-test data. Approaches to improve the base model to a full advanced draft gear model were discussed; preliminary simulation results of an advanced draft gear model were also presented. INTRODUCTION A coupler system as shown in Figure 1 is a mechanism for connecting cars and locomotives in a train. The auto-coupler with friction draft gear scheme is commonly used. In some modern heavy haul trains, some auto-couplers are replaced with draw bars which can minimize the coupling slack so as to lower the dynamic in-train forces. In both schemes the draft gear unit is an integral part. A friction draft gear is an assembly of springs, wedges and plates, which are enclosed by a housing (see Figure 1). Friction draft gears in train operations play two roles: as a part of the car-to-car connections to transmit the longitudinal forces; and acting as friction dampers to damp longitudinal in-train forces. Modeling and prediction of the dynamic behavior of friction draft gears are of significant assistance in addressing various concerns. During shunting operations, draft gears are probably the most important devices to protect rolling stock systems. Inappropriate draft gear behavior or inadequate
dynamic performance could result in component failure or/and goods damage [1]. During normal main line train operations, the dynamic behavior of friction draft gears not only has implications for longitudinal train dynamics [2], but also for rolling stock fatigue [3]. Under heavy haul conditions, coupler systems could be working close to the limits of the available technology. Longer and heavier trains mean larger in-train forces and more complicated force patterns. Practical experience indicates that the development of fatigue failure of coupler systems in long heavy trains may differ from conventional understanding [4]. Therefore, further or even new understanding of draft gear behavior and the implications for fatigue damage of rolling stock as well as for longitudinal train dynamics are required. A program aiming at improvements of dynamic modeling of friction draft gear is in progress at Centre for Railway Engineering (CRE), Australia. As a starting point, a comprehensive review about dynamic modeling of friction draft gear has been carried out by the authors [5]. This paper presents some work from the second step of the CRE research program, which includes redevelopment of a potential forcedisplacement characteristics model; analyzing field-test data; identification of potential improvements by comparing simulated results and field-tests data. Approaches to improve the previous model are discussed; preliminary simulation results of an advanced model are also presented. CONSIDERATIONS FOR DRAFT GEAR MODELING It has been reported [6,7] that friction draft gears have friction dependent stiffness; and ultimately velocity dependent stiffness. The nature of friction damping gives draft gears nonlinear hysteresis and results in discontinuities between loading and unloading curves (Figure 2). For most cases, smoothing approximations or some transitional characteristics are needed to solve the discontinuity issue mathematically.
* Corresponding author. Email:
[email protected], Phone: (+61)0749309589
1
Copyright © 2014 by ASME
car B
car A yoke
car body coupler
draft gear
Draft gear
or
car body draw bar Draft gear
Coupler system
Coupler system
Figure 1. Schematic of coupler system For the purpose of longitudinal train dynamics simulations, coupler systems are usually simplified into single-element models, so every two draft gears are modeled in series as a single unit. A coupler system unit model has to incorporate characteristics which can simulate the whole coupler system working under both draft and buff conditions. The final model must also consider coupler slack as well as the limiting stiffness which appears after springs are fully compressed. When installed, draft gears are usually pre-loaded, which should also be incorporated. To sum up, a desirable draft gear dynamics model should include the following elements: velocity dependent friction, slack, limiting stiffness, pre-load and transitional characteristics. The first four elements are usually expressed as force-displacement characteristics, so modeling of coupler systems has two general aspects, force-displacement characteristics and transitional characteristics.
Force
Limiting stiffness
Loading Transitional characteristics
Slack & Preload
approach. Wedge-spring models, further, differentiated by details considered during modeling; simplified models proposed by [7] simplify friction mechanism into a single pair of friction wedge; detailed models [8] consider all components in the draft gear. The detailed wedge-spring approach was originally published by Nikolskii [8] in 1964 in Russian. More recently, this approach was adapted by Ma [9] to model the type MT-2 (Chinese version of Mark 50) friction draft gear. The draft gear studied in this paper has the same structure (see Figure 3) as that in [9]; the force-displacement characteristics model described in this section is a redevelopment. A full cycle in which the draft gear is fully compressed and then released is presented as an example. The working process of this type of draft gear can be divided into four stages. Loading stage 1: The loading process has started but the follower has not touched the movable plates, as shown in Figure 3 (a). During the first stage the components which are directly associated with the draft gear forces are the central wedge, wedge shoes, the spring seat and springs. Examining the force equilibrium of the central wedge (Figure 4 (a)), wedge shoes (Figure 4 (b)) and the spring seat (Figure 4 (c)), the relationships of Equations (1)~(4) can be reached. P2 = (Fe − Fsr )⁄2 P3 = (Fsm − Fsr )⁄2 P1 = P2 − P3 Q1 = Q 2 + Q 3
Unloading
Deflection Figure 2. Integrated draft gear model A FORCE-DISPLACEMENT CHARACTERISTICS MODEL Generally, the force-displacement characteristics of friction draft gears can be modeled by using five approaches [5]: lookup table approach; polynomial fitting approach; spring-friction approach; intelligent system approach and wedge-spring
(1) (2) (3) (4)
where P1 , P2 and P3 are the vertical force components on different surfaces; Q1 , Q 2 and Q 3 are the corresponding horizontal force components on those respective surfaces; Fe is the external force applied on the central wedge; Fsr is the release spring force; Fsm is the main spring force.
2
Copyright © 2014 by ASME
Figure 3. Working stages of friction draft gears: a) loading stage 1, (b) loading stage 2, (c) unloading stage 1, (d) unloading stage 2
Figure 4. Force analyses of movable componentes By using the concept of Friction Angle, the horizontal force components and vertical force components are governed by Equations (5)~(7). (5) Q1 = P1 ⁄tan�γ + arctan(μ1 )� Q 2 = P2 tan�α + arctan(μ2 )� (6) Q 3 = P3 tan�β + arctan(μ3 )� (7)
where α, β and γ are wedge shoe angles as shown in Figure 3 (c); μ1 , μ2 and μ3 are the corresponding coefficients of friction as shown in Figure 3(d). Given this, the draft gear force of the first stage, F1 , can be expressed as: Fe = F1 = Ψ1 Fsm − (Ψ1 − 1)Fsr
Ψ1 =
1+tan �β+arctan (μ3 )�tan �γ+arctan (μ1 )�
1−tan �α+arctan (μ2 )�tan �γ+arctan (μ1 )�
(8) (9)
When installed, draft gears are usually preloaded. Therefore, in Equation (8), the main spring force and the release spring force have to consider the preload:
Fsm = k m (xm0 + xf ) Fsr = k r xr0
(10) (11)
where k m and k r are the stiffness for the main spring and xr0 are the prerelease spring respectively; xm0 and deflection of the main spring and the release spring respectively; xf is the draft gear deflection. Loading stage 2: During the loading process, the follower has touched the movable plates. Both the central wedge and movable plates are pushed by the follower, as shown in Figure 3 (b). During this stage the draft gear force is composed of the force applied on the central wedge and the forces applied on the movable plates. The calculation of the force applied on the central wedge follows the same procedure as for stage 1. Therefore the main task for modeling of the second stage is to determine the forces applied on the movable plates. Examining a stationary plate (Figure 4(d)) can lead to the following relationship: FN4 = FN1 �cos(γ) − μ1 sin(γ)� 3
(12)
Copyright © 2014 by ASME
where FN1 and FN4 are friction normal forces; Ff1 and Ff4 are friction forces. Examining the central wedge, two wedge shoes and the spring seat as a whole can lead to the following relationship: FN1 =
F e −F sm
2�μ1 cos (γ)+sin (γ)�
(13)
Therefore the draft gear force for the second stage can be determined as: F2 = Ψ2 Fsm − (Ψ2 − 1)Fsr
Ψ2 = Ψ1 +
2�1−μ1 tan (γ)�μ4 (Ψ 1 −1) μ1 +tan (γ)
(14) (15)
Unloading stage 1: The unloading process has started, but the spring seat has not touched the movable plates, as shown in Figure 3 (c). Unloading stage 2: During the unloading process, the spring seat has touched the movable plates. Both the spring seat and the movable plates are pushed by the main spring, as shown in Figure 3 (d). By performing similar analyses which have been demonstrated in the modeling of loading stages, the draft gear forces for unloading stages can also be determined: Fi = Ψi Fsm − (Ψi − 1)Fsr , Ψ3 =
Ψ4 =
(i = 1, 4)
1+tan �β−arctan (μ3 )�tan �γ−arctan (μ1 )� 1−tan �α−arctan (μ2 )�tan �γ−arctan (μ1 )� (tan (γ)−μ1 )Ψ 3
tan (γ)(1−2μ1 μ4 +2μ1 μ4 Ψ 3 )+2μ4 Ψ 3 −2μ4 −μ1
To integrate friction models into the force-displacement characteristics model, the relative velocities between different components have to be determined. The velocity analyses can be performed as shown in Figure 6, where vf is the velocity of the follower, v1 is the absolute velocity of the wedge shoe, v2 is the relative velocity of the wedge shoe with respect to the central wedge, v3 is the relative velocity of the wedge shoe with respect to the spring seat and v4 is the absolute velocity of the spring seat.
Figure 6. Velocity analyses Examining the geometrical information in Figure 6, the following relationships can be reached: v1 =
v2 =
(16)
v3 =
(17)
(19)
sin (γ) v cos (α+γ) f
cos (α)sin (γ) v cos (α+γ)cos (β) f
cos (α)cos (γ−β)
v4 = �cos (α+γ)cos (β) vf
(18)
Note that the expression of Equation (16) can be used for all four stages; i = 1, 4 are for loading stage 1, loading stage 2, unloading stage 1 and unloading stage 2 respectively.
cos (α) v cos (α+γ) f
vf
ANALYSIS OF FIELD-TEST DATA
(20) (21) i=4 i=2
(22)
MODELING OF FRICTION
Figure 5. Friction models Various friction models are available for friction draft gear modeling. Three types of models are commonly used: staticdynamic model, Coulomb model and exponential model as shown in Figure 5. The exponential model is selected for this paper.
Figure 7. Field-test data A set of filed-test data (Figure 7) were measured from impact tests; the measured cases were that one loaded car
4
Copyright © 2014 by ASME
Hysteresis and stiffening behavior The hysteresis is caused by the change of friction force directions and characterized by the separation of envelops for loading stages and unloading stages. The stiffening behavior means draft gears become much stiffer whenever they are approaching the end of loading stages.
stiffness of the draft gears are decreasing until the stiffening behavior appears. SIMULATION OF THE BASE MODEL 2000
Coupler force(kN)
weighed 93 tonnes stroke the other identical stationary loaded car at various velocities on a section of tangent track. The data were sampled at the frequency of 1,000 cycles per second and filtered by using a 100Hz low-pass filter. Three sequences of data corresponding to the initial velocities of 5km/h, 6km/h and 7km/h are illustrated in Figure 7 in the form of forcedisplacement characteristics. From the field-test data, several types of friction draft gear dynamic behavior can be identified.
Based on the draft gear working process description, three change-points of draft gear responses are expected when draft gears are changing working stages, i.e., change from loading stage 1 to loading stage 2; switch between loading and unloading and change from unloading stage 1 to unloading stage 2. Among them, the switch between loading and unloading is actually reflected by the hysteresis. The change from loading stage 1 and loading stage 2 can be identified from Figure 7, which is characterized by the evident stiffness changes when the deflections are around 12mm. However, the third change-point, i.e., change from unloading stage 1 to unloading stage 2 cannot be identified from the field-test data. A theoretical preview of the third change-point can be gained from the simulation results in Figure 8. Theoretically, the unloading stage 1 only exists shortly and has slightly higher forces than that of unloading stage 2. By examining the fieldtest data, it can be seen that, in reality, it is almost the opposite. When the draft gear is switching from loading to unloading, the draft gear force will drop to a level lower than the corresponding force of unloading stage 2 before the unloading stage 2 is commenced. This type of behavior is the “locked unloading” behavior explained in [7], which is also the reason why the third change-point is unidentifiable. Locked unloading behavior The locked unloading behavior refers that the draft gear remains locked during the first stage of the unloading. At the end of the loading stages, the draft gear is locked. When the draft gear is switching from loading to unloading, usually, the draft gear cannot be unlocked immediately, and it can remain locked for a short period of time. Softening behavior The “softening” behavior is characterized by that after draft gears reached the second stage of the loading process, the
1600 1200 800 400 0
Change-of-stage behavior
7km/h 6km/h 5km/h
0
15
30
45
60
75
90
Deflection(mm) Figure 8. Base model A model was redeveloped by the authors according to the descriptions and parameters in [9]. This model will also be used as the base for future improvements; therefore it is named the “base model” in this paper. The base model is expressed by Equations (16); no transitional characteristics are used. The friction model is a conventional exponential model (see Figure 5). Simulation scenarios were set the same to the field-test conditions. The corresponding simulation results are shown in Figure 8. Compared with field-test data, the simulation results give a good presentation of the stiffening behavior and can reproduce the change-of-stage behavior. The base model gives the change point for the switch between the loading stage 1 and loading stage 2, as well as that between unloading stages. Also there is a good match in terms of the maximum forces. However, the simulation results present a big difference in terms of the maximum deflections. The match of the loading characteristics between the base model and the field-test data need to be improved. Specifically, the accuracy for the loading stage 1 need to be improved and the softening behavior is missed. PROPOSED IMPROVEMENTS By analyzing the simulation results against the field-test data, potential improvements have been identified. Based on further examinations of the draft gear structure and authors’ modeling experience, the following improvements are proposed. Improvement for force-displacement characteristics model Previous spring force models, Equations (10~11), assume that the deflection of the main spring equals the deflection of
5
Copyright © 2014 by ASME
the draft gear; and the deflection of the release spring will remain constant during the whole process. By examining Figure 3, it can be found that during the loading and unloading processes, due to the existence of wedge angles, the wedge shoes have lateral movements. Therefore, the release spring will be extended or released during the loading process and be compressed during the unloading process. Meanwhile, the deflection of the main spring will be larger than that of the draft gear. Locked-stiffness Transitional characteristics are necessary to address the issue of discontinuity. A variety of approaches are available from the literature [5]. From the perspective of transitional characteristics, the locked stiffness can act as an intermediate slope and has physical meaning rather than being just a mathematical smoothing strategy [7]. Improvements for analytical friction model Generally analytical friction models (e.g., Figure 5) assume that the coefficient of friction is only associated with velocity, independent of the normal stress on contacting surfaces. Practically, metallic frictional behavior is associated with the shearing behavior of metallic junctions formed at asperities as well as the ploughing of asperities. The forming and ploughing of asperities can be affected by the normal stress, therefore, it is reasonable to assume that the coefficient of friction will also be associated with the normal stress. Simulation of the locked unloading behavior A summary of the measured locked unloading behavior find that unlock usually occurs when the draft gear force is 80kN~130kN lower than the corresponding force of the unloading stage 2. PRELIMINARY RESULTS OF AN ADVANCED MODEL
Coupler force(kN)
2000
7km/h 6km/h 5km/h
1600 1200 800 400 0
0
10
20
30
40
50
Deflection(mm) Figure 9. Advanced model
60
70
An advanced model was developed by implementing the improvements proposed above. The preliminary results of the advanced model are shown as Figure 9. It can be seen that the advanced model has inherited the abilities to simulate the stiffening behavior and the change-of-stage behavior from the base model. Besides, three significant improvements are achieved by using the advanced model: the softening behavior and the locked unloading behavior have been well simulated and the accuracy for the loading stage 1 has been evidently improved. A good overall agreement between the field-test data (Figure 7) and the simulation data (Figure 9) has been reached. In the simulation results of the advanced model, the locked stiffness can also be referred. CONCLUSION A desirable draft gear dynamic model should include velocity dependent friction, slack, limiting stiffness, pre-load and transitional characteristics. The base model gives a good presentation of the stiffening behavior and can reproduce the change-of-stage behavior; also there is a good match in terms of the maximum forces. However, the simulation results present a big difference in terms of the maximum deflections. The match of the loading characteristics between the base model and the field-test data need to be improved. Three significant improvements are achieved by using the advanced model: the softening behavior and the locked unloading behavior have been well simulated and the accuracy for the loading stage 1 has been evidently improved. A good overall agreement between the field-test data and the simulation data has been reached. The locked stiffness has been integrated so as to broaden the behavior spectrum which the model can simulate as well as to meet the requirements of longitudinal train dynamics simulations. ACKNOWLEDGMENTS The support of the Centre for Railway Engineering is acknowledged. The first author is the recipient of an International Postgraduate Research Award (IPRA) and University Postgraduate Research Award (UPRA) from Central Queensland University, Australia. REFERENCES [1] Kasbekar PV, Garg VK, Martin GC. Dynamic simulation of freight car and lading during Impact. Transactions of the ASME J Eng Ind. 1977; 99(4): 859-866. [2] Kaufhold HT, Jarvis DJ, Steffen JJ. The effects of free and controlled slack, car weight, length, articulated joints, and preload draft gear on longitudinal train forces. Rail Transportation: Winter Meeting of the American Society of Mechanical Engineers; 1991 December 1-6; Atlanta (GA): American Society of Mechanical Engineers. 1991. p.11-20. [3] Cole C, Sun YQ. Simulated comparisons of wagon coupler systems in heavy haul trains. Proc Inst Mech F J
6
Copyright © 2014 by ASME
Rail Rapid Transit. 2006; 220(3): 247-256. [4] Boelen R, Curcio P, Cowin A, Donnelly R. Ore-car
[5]
[6]
[7]
[8]
[9]
coupler performance at BHP-Billiton Iron Ore. Eng Fail Anal. 2014; 11(2): 221-234. Wu Q, Cole C, Luo S, Spiryagin M. A review of dynamics modelling of friction draft gear. Veh Syst Dyn. Published online 14 April 2014. DOI: 10.1080/00423114.2014.894199. Hsu T-K, Peters DA. A simple dynamic model for simulating draft-gear behavior in rail-car impacts Transactions of the ASME J. Manuf. Sci. Eng. 1978; 100(4): 492-496. Cole C. Improvements to wagon connection modelling for longitudinal train simulation. Conference on Railway Engineering Proceedings: Engineering Innovation for a Competitive Edge, Rockhampton; (Rockhampton) Australia: Central Queensland University. 1998. p.187194. Nikolskii LN. [Friction shock absorbers: The calculation and design]. Moscow: Mechanical Engineering Publishing House; 1964. Russian. Lei Ma. Performance simulation of MT-2 draft gear [Masters Thesis]. Chengdu: Southwest Jiaotong University; 2012. Chinese.
7
Copyright © 2014 by ASME
