Effect of mass distribution on curving performance for a loaded wagon [original]

ORIGINAL PAPER

Effect of mass distribution on curving performance

for a loaded wagon

Duo Zhang .Yinying Tang .Qiyuan Peng .Chunjiao Dong .Yunguang Ye

Received: 24 February 2020 / Accepted: 17 March 2021 / Published online: 25 March 2021

ÓThe Author(s) 2021

Abstract The location of wagon gravity center for a

loaded wagon is underestimated in a vehicle–track

coupled system. The asymmetric wheel load distribu-

tion due to loading offset significantly affects the

wheel-rail contact state and seriously deteriorates the

curving performance in conjunction with the height of

gravity center and cant deficiency. Optimizing the

location of gravity center and cruising velocity,

therefore, is of interest to prevent the derailment and

promote the transport capacity of railway wagons.

This study aims to reveal the three-dimensional

influencing mechanism of mass distribution on vehicle

curving performance under different velocities. The

wheel unloading ratio is regarded as the evaluation

index. A simplified quasi-static model is established

considering essential assumptions to highlight the

influence of lateral and vertical offset on curving

performance. For a more accurate description, the

MBS models with various locations of wagon gravity

center are built and then negotiate curves in different

simulation cases. The simulation results reveal that the

distribution of wheel unloading ratio determined by

loading offset is like contour lines of ‘basin’. Based on

the conclusions of quasi-static analysis and dynamics

simulations, the regression equation is proposed and

the fitting parameters are calculated for each simula-

tion case. This paper demonstrates the necessity of

optimizing the location of wagon gravity center

according to the running condition and offers a novel

strategy to load and transport the cargo by railway

wagons.

Keywords Railway wagon Mass distribution 

Curving performance Quasi-static analysis 

Dynamics simulation Regression equation

1 Introduction

Symmetric distribution is the basic criterion for the

loading of cargo on wagons. It has been a consensus

that the optimal location of cargo gravity center is at

the center of the vehicle laterally and longitudinally.

D. Zhang Y. Tang Q. Peng

School of Transportation and Logistics, Southwest

Jiaotong University, Chengdu 610031, China

D. Zhang Y. Tang Q. Peng

National United Engineering Laboratory of Integrated and

Intelligent Transportation, Southwest Jiaotong University,

Chengdu 610031, China

C. Dong

Key Laboratory of Transport Industry of Big Data

Application Technologies for Comprehensive Transport,

Ministry of Transport, Beijing Jiaotong University,

Beijing 100044, China

Y. Ye (&)

Institute of Land and Sea Transport Systems, Technical

University of Berlin, 10587 Berlin, Germany

e-mail: [email protected]

123

Nonlinear Dyn (2021) 104:2259–2273

https://doi.org/10.1007/s11071-021-06386-3(0123456789().,-volV)(0123456789().,-volV)

Thus, the general wagon-rail models assume that the

wagon gravity center (WGC) is the same as its

geometry center in the horizontal plane [1–5]. Since

the uneven mass distribution could result in an obvious

unbalance of wheel load and deteriorate the curving

performance seriously, symmetric loading is the basic

prerequisite in the studies with respect to guaranteeing

wagon running safety [6–11].

However, according to practical experience, skew

loading cannot be avoided completely. For the sake of

vehicle running safety, loading guidelines of several

rail organizations are promulgated to specify the

allowed offset values:

(1) International Union of Railways (UIC) [12]

The ratio of masses per bogie should be less than

3:1 and the ratio of load between the wheels

(left/right) of a given axle should be less than

1.25:1. Moreover, the mass per axle should not

exceed the maximum axle load.

(2) The Association of American Railroads (AAR)

[13]

Longitudinally, the center of load weight should

have a certain distance from either truck center,

which depends on the ratio between load weight

and load limit. Laterally, the load must be

located to equalize the weight.

(3) Chinese Railways (CR) [14]

The transversal distance between the cargo’s grav-

ity center and the carbody’s geometrical center should

be within 100 mm. The difference between the masses

per bogie should be no more than 10 ton, and the mass

of cargo on either bogie should not exceed half of the

loading capacity of the wagon.

Depending on these three representative examples,

we can recognize that there are no universal require-

ments on the mass distribution for a loaded wagon.

Moreover, the allowed offset values stated in the

loading guidelines are sketchy and empirical to some

extent. For a further supplement to loading guidelines,

the method of dynamics simulation has been used to

search for the safe range of wagon’s gravity center.

Shatunov and Shvets [15] proposed that, as for a kind

of 4-axle flat wagon, the maximum lateral offset could

reach 150 mm and longitudinal offset could be

expanded too. Bao et al. [16] focused on a common

open-top wagon in China and demonstrated that CR

criteria were conservative.

The loading guidelines and limited former studies

are based on assumptions that the best location of the

cargo is at the center of a wagon and the height of

WGC should be as low as possible. There are two

confusions brought by the aforementioned documents

and their prerequisites:

(1) The location of the combined center of gravity is

a three-dimensional variable. The decision on

skew loading should be made combined with the

height of gravity center, which is neglected

when defining the allowed offset. Matsumoto

et al. [17] and Bekele [18] pointed out that

lowering the height of gravity center was an

obvious advantage for running safety. Zhang

et al. [19] attempted to figure out the three-

dimensional constraint for the combined center

of gravity of a loaded wagon but failed to draw a

quantitative conclusion.

(2) The assumption of optimal choice is not

convincible. Suda et al. [20,21] proposed that

the asymmetric truck may be better for curving

performance. Keropyan et al. [22] demonstrated

that a longitudinal offset was necessary for the

locomotive to promote its traction ability. Since

the advantage of asymmetric loading had been

proved in other areas, we can infer that

symmetric distribution might not be a perfect

plan for all the loading cases.

Since loading guidelines and former studies have

obvious limitations, this paper analyzes the stereo-

scopic influence mechanism of mass distribution for a

loaded wagon on its curving performance. It has been

demonstrated that the safety indices would be severe

when the wagon negotiates the transition curve (TC)

[19]. Optimizing the location of WGC on TC is

necessary to improve the curving performance. How-

ever, because of the geometry and changing elevation

of TC, it is difficult to reveal the quantitative

relationship between the location of WGC and the

curving performance on TC. Thus, the process of

negotiating the circular curve is the objective of this

study.

In Sect. 2, a simplified quasi-static analysis of an

uneven loaded wagon is conducted considering pro-

fuse assumptions. Section 3establishes the MBS

model and carries out simulations to discuss the

qualitative relationship between the location of WGC

and the selected safety criterion. Based on the

123

2260 D. Zhang et al.

conclusions in Sect. 3and Sect. 4proposes the

regression equation and calculates the fitting param-

eters for each simulation case.

2 Analysis of the vehicle curving performance

with a simplified quasi-static model

2.1 Quasi-static model

This paper focuses on the two-axle wagon with three-

piece bogies. There is no doubt that the suspension

system plays a key role in the vehicle running

performance by means of weakening the impact

between the carbody and wheelset. Nevertheless, as

for the unevenly loaded wagon, the suspension device

has little effect on inhibiting the unbalanced wheel

load which is resulted from the asymmetric mass

distribution and closely related to the vehicle running

safety. Thus, the quasi-static model is established to

highlight the influence of mass distribution on the

wheel load. The wagon in uniform circular motion is

deemed as suffering equilibrium force including the

centrifugal force so that it can be regarded as a whole.

In order to simplify the model, essential assumptions

are put forward as below:

(1) All the parts of the model are rigid bodies.

(2) The track is smooth and has no rail cant.

(3) The wheelset is symmetric with the center line

of the track.

(4) The load of a truck is distributed evenly to each

wheelset.

(5) The carbody is at its original position.

The mass distribution is directly related to the

vertical contact force. Consequently, the wheel

unloading ratio is more sensitive than the derailment

coefficient for this issue in terms of definitions. This

conjecture was demonstrated by Zhang et al. [19].

Thus, the quasi-static analysis emphatically reveals

the vertical contact force for each wheelset and regards

the wheel unloading ratio as the evaluation index.

In the X–Z plane of the coordinate system of track

centerline, the wagon can be regarded as a simply

supported beam as is shown in Fig. 1.

where P

(i= 1,2,3,4) is the vertical contact force of

wheelset; lis half of the length between bogie pivot

centers; G

and L

are the vertical component force of

wagon’s gravity and centrifugal force, respectively;

(q= 1,2) is distributed G

on each truck; L

(q= 1,2) is distributed L

on each truck; M and N are

the geometrical center of the carbody and the gravity

center of the wagon, respectively; ais the longitudinal

offset; cis the vertical offset.

Since the wagon suffers vertical equilibrium force,

the sum of vertical contact force can be given by:

i¼1

Pi¼GzþLz:ð1Þ

Based on the torque balance, the sum of torques

around the rear truck center can be presented as

follows:

ðP1þP2Þ2l¼ðGzþLzÞðlþaÞ:ð2Þ

According to the assumptions, P

Therefore, the vertical contact force can be calculated

based on Eqs. (1) and (2):

P1¼P2¼ðGzþLzÞðlþaÞ

4lð3Þ

P3¼P4¼ðGzþLzÞðlaÞ

4l:ð4Þ

Furthermore, as a simply supported beam, the

gravity force and centrifugal force are distributed on

the front and rear center plate as below:

Gz1¼GzðlþaÞ

Gz2¼GzðlaÞ

Lz1¼LzðlþaÞ

Lz2¼LzðlaÞ

:ð5Þ

The mass of wagon is distributed to each truck

longitudinally as already stated. Then, lateral offset

Fig. 1 Longitudinal mass distribution of a loaded wagon

123

Effect of mass distribution on curving performance for a loaded wagon 2261

assigns the load of each wheelset to the wheels as

Fig. 2illustrates.

where G

and L

are the gravity and centrifugal

force of the mass loaded on the truck whose order is q,

respectively; subscript qdenotes the front truck when

its value is 1 and the rear truck when its value is 2; Pis

the vertical contact force; Qis the lateral contact force;

subscript idenotes the order of wheelset as is shown in

Fig. 1; subscript land rdenote the left and right wheel

of the wheelset, respectively; arepresents the angle

resulted from superelevation; bis the lateral offset; cis

the vertical offset from N to M; his the vertical

distance from M to the top of rail; dis the half of tape

circle distance. In this paper, we take the right-hand

curve as an example. It is easy to point out that:

i¼1;2q¼1

i¼3;4q¼2

:ð6Þ

As for the truck whose order is q(q= 1, 2), its

distributed vertical load is borne by the corresponding

wheelsets:

XPil þXPir ¼Gzq þLzq:ð7Þ

Moreover, the resultant moment of the truck can be

calculated as:

XPir 2dþðLyq GyqÞðhþcÞ

¼ðLzq þGzqÞðdþbÞ:ð8Þ

Based on Eqs. (6) and (7), the vertical contact force

can be presented as:

Pil ¼ðLzq þGzqÞðdbÞþðLyq GyqÞðhþcÞ

4dð9Þ

Pir ¼ðLzq þGzqÞðdþbÞðLyq GyqÞðhþcÞ

4d:

ð10Þ

For any wheelset, the wheel unloading ratio (UN)

can be derived as:

Pil Pir

Pil þPir

¼Lyq Gyq

Lzq þGzq

hþc

db

d:ð11Þ

Combined with the geometric relationship,

Eq. (11) can be derived as:

Pil Pir

Pil þPir

¼Lzq Gzq tan2a

tan aðLzq þGzqÞhþc

db

d:ð12Þ

Since we have assumed that the truck would

allocate its load to inclusive wheelsets evenly, wheel-

sets of the same truck have identical UN. According to

Eq. (10), it seems that the mass loaded on truck has

effect on the UN. For revealing the effect of the mass

distribution significantly, Eq. (5) is substituted to

Eq. (12) to obtain:

Pil Pir

Pil þPir

¼LzGztan2a

tan aðLzþGzÞhþc

db

¼v2gR tan a

v2tan aþgR hþc

db

dð13Þ

where vdenotes the vehicle running velocity; Rde-

notes the radii of curve; gdenotes the acceleration of

gravity.

2.2 Analysis of derivative results

Equation (11) illustrates that apart from the lateral and

vertical offset, cant deficiency affects the value of UN

significantly. The values of (h?c), dand Lzq þGzq



are definitely positive. Thus, the positive/negative

signs of Lyq Gyq



and bdetermine the trend of

change in the absolute value of UN as below:

(1) Lyq\Gyq,b[0orLyq [Gyq,b\0

The absolute value of UN is positively corre-

lated with |b| and c.

(2) Lyq\Gyq,b\0orLyq [Gyq,b[0

It is complicated to describe the trend of the

absolute value of UN since it is denoted as the sum of a

positive expression and a negative expression con-

taining variables. Numerical computation is needed to

reveal the distribution rules of UN.

Fig. 2 Lateral mass distribution of loaded wagon

123

2262 D. Zhang et al.

The difference between the lateral component

forces of centrifugal force and gravity of distributed

mass reflects cant deficiency, which is derived to

obtain Eq. (13). Based on Eq. (13), we can demon-

strate the relationship among the loading offset,

velocity and UN clearly. As an example, we assume

that tan aequals 0.1045, the gauge equals 1435 mm,

dequals 0.75 m. For a better description, we use the

variable of zto denote the sum of hand cand set the

scopes of zand b. When the loaded wagon negotiates

curves whose radii are 350 m and 600 m, respectively,

with velocities from 10 m/s to 25 m/s, the absolute

value of UN can be calculated as Fig. 3illustrates.

Figure 3reveals the influence of loading offset and

velocity on UN that can be concluded as below:

(1) For the selected velocity and curve radius, the

distribution of UN is similar as the contour lines

of ‘basin.’ With the increase in velocity, the

location of the basin moves to the right of the

wagon.

(2) For most locations, the absolute value of UN

increases with the increase in vertical offset. But

if there is a large negative offset laterally, the

absolute value of UN will decrease as the

vertical offset increases when the velocity is

low.

(3) The balancing velocity (m/s) can be calculated

as Eq. (14):

V0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

gR tan a

p:ð14Þ

For curves whose radii are 350 m and 600 m,

their balancing velocities are 19 m/s and 25 m/

s, respectively, based on Eq. (14). Figure 3

demonstrates that when the wagon negotiates

the curve at the balancing velocity, UN can be

constrained efficiently with the fluctuations of

lateral and vertical offset.

By means of quasi-static analysis, the roles of

lateral and vertical offset in UN can be revealed. In

order to demonstrate the conclusions of quasi-static

analysis and study the effect of longitudinal offset on

UN, dynamics simulation should be implemented.

3 Dynamics simulation

3.1 Dynamic equations of carbody

In this paper, we adopt C

70H

as the analysis object,

which is one of the commonly used open-top cars in

China. The MBS model of C

70H

is made up of cargo,

carbody and two three-piece bogies. The schematic

diagram is shown in Fig. 4.

where H

(j= 1, 2, 3) denotes the distance between

the gravity centers of different components; rdenotes

the rolling radius of the wheel; idenotes the angle

resulted from the superelevation; adenotes the tilt

angle of carbody; sdenotes half of the tape circle

distance.

Due to the loading offset, the dynamic equations of

carbody play a primary role in building the MBS

model of C

70H

. Because it is deemed that the bolster is

fixed with carbody in each degree of freedom except

roll, the carbody can be regarded as being exerted by

the lateral and vertical forces of the secondary

b (m)

z (m)

v (m/s)

Absolute value of UN

(a) R=350 m

b (m)

z (m)

v (m/s)

Absolute value of UN

(b) R=600 m

Fig. 3 Distribution of the absolute value of UN

123

Effect of mass distribution on curving performance for a loaded wagon 2263

Loading more pages...