interaction optimization in multibody dynamic system2017).pdf · interaction optimization in...

Interaction Optimization in Multibody Dynamic System

VLADIMIR POLIAKOV

Institute of Track, Construction and Structures

Moscow State University of Railway Engineering (MIIT)

9b9 Obrazcova Street, Moscow

RUSSIA

[email protected]

Abstract: - The paper describes the optimization facility of interaction within the “bridge-track-car” system that

concerns high-speed railway traffic on the bridges zones. The singularity of the approach to be discussed lies in

the attempt of integration of the system of elements that work simultaneously and together. The model that takes

into account vertical oscillation of the car body, bogies, wheels, rails, and superstructure of the bridge. Several

criteria allow the estimation of various parameters of dynamic interaction and reach the optimal dynamic

parameters dealing with wheel-rail contact and derailment, comfort of passengers and ballast wearing. Therefore,

we can obtain the system with predetermined dynamic behavior to decrease or to increase the interaction forces

in defined places. It allows decreasing stress of ballast or increasing wheel-rail contact force to prevent

derailment.

Key-Words: High-speed railway, bridge dynamics, safety of motion, track, ballast wearing

1 Introduction The research of the dynamic processes in the

complicated “Bridge-track-car” system (BTCS) -

consists of mechanisms and structures. It requires a

big amount of information analysis dealing with

behavior and trends of the BTCS changes. The trend

analysis is necessary to determine the direction and

resource of optimizing the system. That is why

scientific methods of decision-making are needed.

System analysis could be very useful in this case.

One of the tools of the system analysis is the

mathematical theory of optimal control that

concerned optimal control of a spaceship. This theory

assumes minimizing a criterion simultaneously with

the execution of constraints in the form of equations

and inequations. Minimizing the criteria and

performing of constraints are achieved by means of

controlling functions that depend on time in the

mathematical theory. Though in construction

engineering the characteristics of a structure must not

(and cannot) be dependent on a short period of time.

It is necessary to vary the characteristics on space

coordinate. Thus, it is necessary to imply stationary

controlling functions.

2 Applied Theory of Optimal Control There is a carrying system in methasystem “railroad”.

The aim of the carrying system is taking any load

during its functioning and railing, security of the

strength, stiffness, etc. of the structures. In our

theory, the BTCS is one of the units of the carrying

system. This unit has input and output impacts. Input

impact is the boundary and starting conditions.

Output impact is a state of the BTCS at the moment

when the train has left the BTCS completely. In

accordance with the Eigen frequency, the BTCS

could be decomposed into elements: sprung mass

(car body and bogies), unsprung mass (wheelsets)

and track – intersystem “wheel-rail”, and bridge

superstructure.

It is very important to point out, that

decomposition in our theory does not mean dividing

the system into isolated elements as it usually

happens, but dividing the system into subsystems,

that continuously interact between each other while

oscillations run on.

Thus, the BTCS consists of:

• discrete-continuous structure of the BTCS,

containing continuous elements – track and bridge

superstructure; the BTCS includes the bridge and

transition zones, and discrete subsystem includes

cars;

Vladimir PoliakovInternational Journal of Theoretical and Applied Mechanics

http://www.iaras.org/iaras/journals/ijtam

ISSN: 2367-8992 43 Volume 2, 2017

• mathematical model Z of the BTCS behavior

(including start and boundary conditions z and

parameter t);

• description of the stationary characteristics O(x)

of the BTCS;

• constraints F on BTCS behavior parameters W, W

= Z(x, t, z, O);

• stationary controlling functions u = u(x) within

multitude O, that influence the achievement of the

aim of the BTCS;

• functional D describes the quality of the BTCS;

• quality parameters d of the BTSC, that have to be

optimized, d=D(u,W).

The BTCS could be defined as “purposeful

system”. The target of the BTCS is the train reaching

the bound of the BTCS, while all the constraints

(including limits of comfort, safety, loads, etc.) are

satisfied when the train is inside the BTCS. For an

optimal system, there is an additional requirement of

minimizing some parameters. In this case,

optimization is performed under absolute satisfaction

of all the constraints.

The target of the optimal system can be achieved

by means of controlling functions that determine

behavior of the system because the starting and

boundary conditions effect is minimized by the

choice of the model parameters. The above-

mentioned formulation of the problem allows using

mathematical theory of optimal controlling. The

difference of our applied theory from the classical

one is application of stationary controlling functions,

i.e. the functions that do not depend on time. The

second difference is minimization of starting and

boundary conditions effect while the behavior of the

BTCS is determined only by the interaction of its

subsystems.

The controlling functions are rigidity 𝐸𝑏𝐽𝑜𝛼4(𝑥𝑘)

and mass 𝛼2(𝑥𝑘)𝜌𝐹0 of the bridge superstructures,

rail bed stiffness and mass of rail and sleepers. The

track parameters can be changed by means of

implementation of different pads stiffness γ(х) and

sleeper’s spacing δ(х). All mentioned functions

depend on space, but not on time. Thus, u(x): {α(x),

γ(x), δ(х)}. For the computer program these functions

are just many dimensional vectors.

The vector criterion D has been worked out for

estimating the quality of the BTCS. D1 is the bridge

superstructure (beam) criterion. D1 will be discussed

in another paper. The D2 criterion consists of three

criteria that allow minimizing rail bed loading (d21),

energy dissipation in ballast (d22), and volatility of

rail bed loading (d23) about predetermined meaning

Q.

𝐷2 = 𝑚𝑖𝑛

|

|

∫ ∫ (𝛾(𝑥)𝛿(𝑥)𝑈′(𝑦р − 𝑦𝑏 ))2

𝑑𝑥𝑑𝑡𝑇𝐿

∫ ∫ (ср (𝜕𝑦р

𝜕𝑡

− 𝜕𝑦𝑏

𝜕𝑡 ))

2


∫ ∫(𝛾(𝑥)𝛿(𝑥)𝑈′(𝑦р − 𝑦𝑏 ) − 𝑄)2


|

|

(𝑑21)⋮⋮

(𝑑22)⋮⋮

(𝑑23)

where U(x,t) = γ(х) δ(х)U'(yr - yb) is rail bed loading,

cp – dissipation coefficient, L – the length of the

BTCS, T – the time interval while the train is inside

the BTCS, y – deflection of the rail or the beam.

The D2 criterion includes physically different

parameters and therefore application of a goal

function that summarizes these parameters is wrong.

Moreover, some criteria may conflict with each

other, i.e. they demand opposite steps of

optimization. Thus, it is necessary to find the

multitude of compromise decisions. Figure 1 shows

the optimization process.

3 The Carrier System Model The model we have introduced above is sufficient

enough to consider the required parameters and, on

the other hand, it is not too complicated to apply the

theory of the optimal controlling. Of cource, multy-

body train should be considered.

The model takes into account the following

parameters shown in Figure 1:

• Mk, mti, mi, m(x), M(x) means mass of the car body,

bogie, wheel, track and superstructure

correspondingly. These parameters may depend on

the x-coordinate if it is pointed out. For the track, it

means dependence of the track mass on sleeper

spacing δ(x). M(x) can be varied by α(x) function.

Jk, jti, J(x) means the moment of inertia of the car

body, bogie and superstructure correspondingly. The

parameters may depend on the x-coordinate if it is

pointed out. U(x) means vertical stiffness of the rail

bed, including rail-sleeper fastening, ballast and

embankment (if it is), that can be varied by γ(x)

function. U(x) depends on δ(x) function as well. It is

important to point out that the non-linear function U

depends on the direction of vertical movement of the

rail. If the vertical movement of the rail is positive

(upwards) respectively the rail bed, the resistance to

the movement is equal only to frictional force. This

feature reflects reality and it is significant for high-

speed traffic. J(x) can be varied by α(x) function.

• y(x,t), y(0,t), y(L,t), yb(x,t) means vertical

displacement of the rail that depends on x-coordinate

and time, starting/boundary conditions and vertical

displacement of the bridge superstructure.

• Gk, Ck, Gt, Ct, Pti, Ri means the forces in the

suspensions of the car and in the rail-wheel contact.



ISSN: 2367-8992 44 Volume 2, 2017

The α(x), δ(x) and γ(x) functions are controlling

function. They are not assigned, but have to be found

using by criterion of optimization. It is important to

underline that the ranges of their changing are limited

by the technologic possibilities.

In addition, the safety condition should be

observed. The stability of a wheel motion on a rail

depends on the ratio between vertical and lateral

forces in wheel-rail contact.

The ratio has been determined for different cases.

Obviously, the single-track superstructure itself does

not cause horizontal oscillation of the car; it causes

only pitching and bouncing oscillation of the car if

the vertical interaction is considered. We do not take

into account the wind load, but take into account the

maximum lateral force that the track can bear.

Therefore, we may assume that horizontal oscillation

of a car is random. In the worst case, the horizontal

force may be equal (but not exceed) to the force that

may shunt the rail in the horizontal lateral direction.

Therefore, we can estimate the minimal permissible

value of the vertical force Rmin that prevents

derailment. According to our conclusion Rmin >

23,814 N [3].

The theory of the optimal controlling supposes

equality and inequality constraints bridge, rail and car

oscillations. Equality constraint F looks like a partial

differential equation of rail oscillation:

(𝑘𝑏𝛿(𝑥)𝜌𝑝𝐹𝑝)𝜕2𝑦𝑝

𝜕𝑡 2 + ср (

𝜕𝑦р

𝜕𝑡 −

𝜕𝑦𝑏

𝜕𝑡 ) + (1)

+𝐸𝑝𝐽𝑝 𝜕4𝑦𝑝

𝜕𝑥4 + 𝛾(𝑥)𝛿(𝑥)𝑈′(𝑦р − 𝑦𝑏 ) = 𝑃(𝑥, 𝑡)

Similarly, superstructures oscillation equations were

used [4]. Oscillations of the cars may be presented by

ordinary differential equations [4].

The inequality constraints F are needed to limit

minimum vertical force Rmin in wheel-rail contact

that prevents derailment and vertical acceleration of

car bodies to obtain acceptable comfort:

Rmin > Rlim = 23,814 N (2)

Wmax < Wlim = 0.35 m/s2 (3)

4 Optimisation Let us consider the optimisation of dynamic

interaction of cars and track inside the bridge zone.

We will use the criterion of minimal irregularity of

the load on the rail bed in reference to preset meaning

of the average load Q:

𝐷 = ∫ ∫(𝛾(𝑥)𝛿(𝑥)𝑈′(𝑦р − 𝑦𝑏 ) − 𝑄)2

𝑑𝑥𝑑𝑡 (4)𝑇𝐿

Simultaneously we will try to maximize Rmin while a

train is passing a bridge zone L including transition

zones for a period of time T, where L is overall length

of the bridge zone including transition zones and T is

the total time of a train motion on the bridge zone.

4.1 Wheel-rail contact forces Figure 3 shows an example of vertical wheel-rail

contact force of a single wheel of a car during its

motion through the superstructure before and after

optimization. Note, the possibility of wheel lift-off is

common knowledge and was discussed for instance

in [5]. Figure 3 demonstrates the wheel lift-off before

optimization on the first iteration. Then, the wheel-

rail contact force becomes more stable due to

optimization and on the sixth iteration we get

acceptable meaning of the force, that is more than

Rlim = 23,814 N.

Figure 4 shows vertical forces of four wheels of

the second car of a train during its motion through the

superstructure at the train speed 400 km/h. We take

into account the second car because oscillation of the

bridge superstructure gets stabilization after the first

car passing and superstructure oscillation amplitude

does not grow while the train is passing the bridge

superstructure. We can see dangerous decreasing of

the forces behind the bridge and the wheel lift-off.

Figure 5 demonstrates the result of optimization –

the wheel-rail vertical contact forces remain more

than Rlim = 23,814 N while the train is passing and

vertical force Rmin = 39.3 kN. One of the targets of

optimization has been achieved.

4.2 The main goal of optimization

The main goal of optimization is formulated in (𝑑23).

Let us discuss the interrelation between (4) and

demand (2).

In Figure 6 an example of evolution of D criterion

(4) and the meaning of Rmin during the train passing

the bridge zone is shown. At the first iteration, we can

see the Rmin= 0 and demand (2) is violated.

At the second iteration, the meaning of the

criterion D is nosedivng from 7.5 to 2.18 and

simultaneously Rmin is increasing up to 14.2 kN.

Further decreasing of the D criterion and

simultaneous increasing of Rmin is impossible and the

software is pressed to increase Rmin because the safety

demand (2) is very strong and must be satisfied in any

case. At further iterations, D criterion is almost

constant. At the ninth iteration, the demand (2) is

already hold.

Nevertheless, the consequence of the optimization

may be sufficient. Figure 7 shows the meaning of the

rail bed load direct under the moving wheels of the

second car during motion through the bridge zone

before optimization and Figure 8 – after optimization.



ISSN: 2367-8992 45 Volume 2, 2017

The average meaning of the load is decreasing from

30.8 to 26.2 kN and standard deviation is decreasing

from 2.79 to 1.41kN. Another important result of

optimization is that the maximum load has decreased

to the meaning of 30.69 kN, less than critical meaning

that leads to plastic deformation in ballast.

We supposed that optimization for heavier high-

speed train leads to optimal decision for lighter trains.

Figure 8 demonstrates success of the optimization for

the heavier perspective train and Figure 9 shows that

optimal structure for the perspective train remains

optimal for lighter CHR380 train. The average

meaning and standard deviation for CHR380 train are

lower than for the perspective train.

4.2 The optimal controlling functions The above mentioned results were achieved with the

single γ(x) controlling function that changes the rail

bed stiffness along the track. Our research showed

that the required changes within allowed range of the

rail bed stiffness can the obtained by grading under-

sleeper pads of mass production. The other types of

the controlling function in the cases mentioned are

not required which makes the track structure simpler,

in spite of obligatory usage of flange rails.

Sometimes the application of sleeper spacing

function δ(x) can be required. At last, α(x) controlling

function is used for bridge superstructure

optimization and it will be discussed in detail in

another paper.

Because of integral character of the criterion (4),

the optimal result can be achieved with several

different controlling functions. Managing of

optimization process allows obtaining technically

reasonable controlling function. Figure 10 shows the

unreasonable version of optimal γ(x) function. The

aim of optimization consists in getting some

reasonable version of optimal controlling function

(Figure 11). Slight deviations of the function result

from discrete character of the optimization process.

These deviations are negligible and can be presented

by a line.

If the single controlling function application is

deficient several controlling functions can be used

simultaneously (Figure 12). Figure 11 and Figure 12

show the significance of the integral carrier system

(BTCS) research instead of separate consideration of

different parts of the BTCS because optimal structure

of the track depends on dynamic properties of the

bridge superstructure.

4 Conclusion The developed concept of integrated carrying system

including bridge superstructures, track and cars takes

into account the interaction inside the system.

Controlling of dynamic interaction within

“bridge-track-car” system at the design stage allows

estimation and assurance of safety level. For ensuring

safety and acceptable ballast wearing at the high

speed, it is necessary to explore the whole system.

The research showed the importance of the integral

system dynamic analysis without application of

multiple hypothesis. Thus, the applied theory of

optimal controlling was developed. The applied

theory allows design the system of predicted

behavior.

Planning of motion within many-dimensional

behavior and estimation spaces by means of vector-

specified controlling functions may be considered as

a step forward to development of artificial

intelligence in a certain science. The computer

program can make decisions concerning technical

parameters of the BTCS on the base of analysis of

great amount of information and the designer makes

a final decision about the acceptability of the

computer-aided design.

References:

[1] Polyakov V., "Dynamic Interaction within a

"Bridge-Track-Car" System on a High-Speed

Railway", in J. Pombo, (Editor), "Proceedings

of the Third International Conference on

Railway Technology: Research, Development

and Maintenance", Civil-Comp Press,

Stirlingshire, UK, Paper 136, 2016.

doi:10.4203/ccp.110.136

[2] Poliakov V.Y., “Optimal bridge superstructures

synthesis for high-speed railways”, Structural

Mechanics and Constructions Designing, 3 35-

42, Moscow, Russia, 2016.

[3] Poliakov V.Y., “Safety of high speed traffic on

bridges”, World of Transport and

Transportation Journal, 6, 182-188, Moscow,

Russia, 2014.

[4] Poliakov V.Y., “Computational modeling of the

vehicle–structure interaction on high-speed

railways”, Structural Mechanics and

Constructions Designing, 2, 54-60, Moscow,

Russia, 2016.

[5] N. Matsumoto & K. Asanuma. Some

experiences on track-bridge interaction in Japan.

Track-Bridge Interaction on High-Speed

Railways, 2009 Taylor & Francis Group,

London, UK, pp. 80-97.



ISSN: 2367-8992 46 Volume 2, 2017

Figure 1: The scheme of solving the BTCS optimization

Figure 2: The model of BTCS

Figure 3: An example of vertical wheel-rail contact force before and after optimization.

0.00E+00

5.00E+04

1.00E+05

1.50E+05

2.00E+05

2.50E+05

2.7

0E-

01

2.9

8E-

01

3.2

6E-

01

3.5

4E-

01

3.8

2E-

01

4.1

0E-

01

4.3

8E-

01

4.6

6E-

01

4.9

4E-

01

5.2

2E-

01

5.5

0E-

01

5.7

8E-

01

6.0

6E-

01

6.3

4E-

01

6.6

2E-

01

6.9

0E-

01

7.1

8E-

01

7.4

5E-

01

7.7

3E-

01

8.0

1E-

01

8.2

9E-

01

8.5

7E-

01

8.8

5E-

01

9.1

3E-

01

9.4

1E-

01

9.6

9E-

01

9.9

7E-

01

1.0

3E+

00

1.0

5E+

00

Wh

ee

l-ra

il co

nta

ct f

orc

e R

, N

Time, sec

R2-2 iteration 1

R2-2 iteration 6



ISSN: 2367-8992 47 Volume 2, 2017

Figure 4: Vertical forces of four wheels of the car of a train during motion through the bridge before

optimization

Figure 5: Vertical forces of four wheels of the car of a train during motion through the bridge after optimization

Figure 6: Vertical minimal wheel-rail contact force Rmin and D criterion

0.00E+00

2.50E+04

5.00E+04

7.50E+04

1.00E+05

1.25E+05

1.50E+05

1.75E+05

2.00E+05

0.3

25

0.3

58

0.3

90

0.4

23

0.4

55

0.4

88

0.5

20

0.5

53

0.5

85

0.6

18

0.6

50

0.6

83

0.7

15

0.7

48

0.7

80

0.8

13

0.8

45

0.8

78

0.9

10

0.9

43

0.9

75

1.0

08

1.0

40

1.0

73

1.1

05

Wh

eel-

rail

con

tact

fo

rce

R, N

Time, sec

R2-1 R2-2

R2-3 R2-4

BRIDGE 34m

Static force 90 kN

motion

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

1.40E+05

1.60E+05

1.80E+05

2.00E+05

0.2

60

0.2

93

0.3

25

0.3

58

0.3

90

0.4

23

0.4

55

0.4

88

0.5

20

0.5

53

0.5

85

0.6

18

0.6

50

0.6

83

0.7

15

0.7

48

0.7

80

0.8

13

0.8

45

0.8

78

0.9

10

0.9

43

0.9

75

Wh

eel-

rail

con

tact

fo

rce

R, N

Time, sec

R2-1

R2-2BRIDGE 34m

motion

Static force 90 kN

1, 7.5

2, 2.18 9, 2.09

2, 14.2

9, 24.6

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9

D

Rmin, кН

iteration number



ISSN: 2367-8992 48 Volume 2, 2017

Figure 7: Rail bed load before optimization (perspective train)

Figure 8: Rail bed load after optimization (perspective train)

-50,000

-45,000

-40,000

-35,000

-30,000

-25,000

-20,000

-15,000

-10,000

-5,000

0

3.87E-01

4.08E-01

4.30E-01

4.51E-01

4.73E-01

4.94E-01

5.16E-01

5.37E-01

5.59E-01

5.80E-01

6.01E-01

6.23E-01

6.44E-01

6.66E-01

6.87E-01

7.09E-01

7.30E-01

7.52E-01

7.73E-01

7.95E-01

8.16E-01

8.38E-01

8.59E-01

8.81E-01

9.02E-01

9.24E-01

9.45E-01

Ra

il b

ed

lo

ad

, N

Time, sec

Q2-1 Q2-2

Q2-3 Q2-4

Q average = -3.08Е+04 Std. deviation 2.79+03

motion BRIDGE

-30,690

-50,000

-45,000

-40,000

-35,000

-30,000

-25,000

-20,000

-15,000

-10,000

-5,000

0

3.58E-01

3.80E-01

4.01E-01

4.23E-01

4.44E-01

4.66E-01

4.87E-01

5.08E-01

5.30E-01

5.51E-01

5.73E-01

5.94E-01

6.16E-01

6.37E-01

6.59E-01

6.80E-01

7.02E-01

7.23E-01

7.45E-01

7.66E-01

7.88E-01

8.09E-01

8.31E-01

8.52E-01

8.74E-01

8.95E-01

9.17E-01

9.38E-01

Ra

il b

ed

lo

ad

, N

Time, sec

Q2-1 Q2-2

Q2-3 Q2-4

Q average = -2.62E+04Std. deviation =1.41E+03



ISSN: 2367-8992 49 Volume 2, 2017

Figure 9: Rail bed load under the second car of CHR380 train after track optimization for perspective train

Figure 10: The unreasonable version of optimal γ(x) function

-50,000

-45,000

-40,000

-35,000

-30,000

-25,000

-20,000

-15,000

-10,000

-5,000

0

3.56E-01

3.78E-01

3.99E-01

4.21E-01

4.42E-01

4.63E-01

4.85E-01

5.06E-01

5.28E-01

5.49E-01

5.70E-01

5.92E-01

6.13E-01

6.34E-01

6.56E-01

6.77E-01

6.99E-01

7.20E-01

7.41E-01

7.63E-01

7.84E-01

8.06E-01

8.27E-01

8.48E-01

8.70E-01

8.91E-01

9.12E-01

9.34E-01

Ra

il b

ed

lo

ad

, N

Q2-1Q2-2Q2-3

Q average = -2.03E+04 НStd. deviation = 1.37кН

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1 6

11

16

21

26

31

36

41

46

51

56

61

66

71

76

81

86

91

96

10

1

10

6

11

1

11

6

12

1

12

6

13

1

13

6

14

1

14

6

Sleepers



ISSN: 2367-8992 50 Volume 2, 2017

Figure 11: The reasonable version of optimal γ(x) function

Figure 12: The optimal controlling functions in case of less rigid superstructure

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

1.40E+00

1 6

11

16

21

26

31

36

41

46

51

56

61

66

71

76

81

86

91

96

10

1

10

6

11

1

11

6

12

1

12

6

13

1

13

6

14

1

BRIDGE

ACTIVE γ(x) ZONE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91

𝜸(𝒙)

𝜹(𝒙)



ISSN: 2367-8992 51 Volume 2, 2017

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