Journal of Civil Engineering and Architecture 9 (2015) 368-372 doi: 10.17265/1934-7359/2015.03.014

Mathematical Modelling of Work of Modern

Friction-Polymer Shock Absorbers and Determining the

Dynamical Force during the Impact

Hristijan Mickoski, Ivan Mickoski and Petar Simonovski

Faculty of Mechanical Engineering, University of Skopje, Skopje 1000, Macedonia

Abstract: Shock absorbers are main elements into construction of train wagons that secure protection from longitudinal forces which appear during transitional regimes of movement. Besides, development of new constructive solutions for shock absorbers is quite popular development of their working mathematical models. This paper presents modern shock absorber with elastic block made from polymer elements that increase quantity of absorbed energy. This is achieved by increasing the stiffness characteristic of polymer elastic block. The construction is relatively simple and technology used to create the construction is with more or less low price. If there is not enough elastic stiffness of the polymer block, there is a possibility for not meeting the UIC (International Union of Railways) norms for absorbed energy. Therefore, according to the mentioned characteristic, shock absorbers are divided into three groups. The mathematical model presented in this paper allows calculating the necessary elastic characteristic of the polymer block for a short time. Differential equation of movement of the shock absorber elements is presented in this paper. Force change of polymer block for various impact velocities participates in the differential equation of movement where initial velocity V0 and the current meaning of the velocity

x are taken into consideration. The presented equation is solved by using program language MATLAB/Simulink by developing a

simulation model. Key words: Shock absorber, mathematical modelling, MATLAB/Simulink, simulation, dynamical force.

1. Introduction

Rail freight transport occupies an important role in

the transport system of each country in the world.

Increasing the transport of goods in recent years

respectively led to increased weight of train wagons

and speed collision during their manoeuvrability and

during forming the compositions of arranged stations.

All this significantly increased the level of activity of

the longitudinal forces acting on the train wagons as a

result of load occurrence of massive repairs which

appreciably reduce revenue from transportation of

goods.

The main element for reducing the level of

longitudinal forces in the operation of railway vehicles,

especially freight vehicles, is shock absorbers that are

Corresponding author: Ivan Mickoski, Ph.D., professor,

research fields: mechanics/mechatronics and railway vehicles. E-mail: ivan.mickoski@mf.edu.mk.

embedded on the front of each train wagon. Therefore,

for such a role in the recent years, new constructive

solutions have been developed, where absorption

capacity-efficiency of longitudinal forces takes a

package made by polymeric materials which are

embedded in the shock absorber.

The influence of the speed impact on shock absorber

is a factor studied for a long time by many authors, but

it remains an unexplored issue regarding the influence

of the speed impact on the characteristics of modern

polymer elements.

According to the experimental data, a mathematical

model was created for changing the force F acting on

the retaining polymer package which takes into

account the impact of initial speed V0 as well as the

change of the current speed x . The error of this

resulting force does not exceed 6% in terms of

maximum 1% compared to the maximum of Ref. [1].

D DAVID PUBLISHING

Strict UI

requirement

freight wago

than 2.0 M

capacity not

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capacity if

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polymer blo

This pape

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Movement o

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polymer blo

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and 2. A com

2. Mathem

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requirement

Fig. 1 Princ

Mathemat

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mplete shock

matical Mod

to develop

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ot achieve the

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ressing forc

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ortant and res

he calculation

ymer block

elements of s

ial equation w

e the stiffness

satisfy the abo

rmined. Prin

tion device (

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delling

mathematica

s due to the a

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f elastic-friction

ng of Work ofmining the Dy

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ire forces sm

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given in Fig. 3

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tem swept pa

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fs. [4-6]:

.101a

Force F expre

d up to now w

ss of the mov

ocity and sys

m known esti

ment particip

r Shock Absoe Impact

fness charact

and reduce the

ation of moti

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x g

ement of cone

of cone pressu

tion of cone p

ient of transm

force of the sh

ce of the emb = 5 is tak

2, 3]. Follow

vaF ( 0

66.302. 3 x

b 00.0

essed in kN ha

was estimated

ing system. F

stem position

imation in wh

pate and fric

orbers and

teristics of th

eir cost for ob

on of the ele

as the follow

F

M

e pressure par

ure part;

pressure part;

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hock absorber

mbedded polymken. Calculat

ing equations

bx )0

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as nonlinear c

d experimenta

Force P depen

n, which can b

hich angles o

ction coeffici

369

he embedded

btaining. The

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ing form:

(1)

rt;

;

wing for how

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mer block F.ion of is

s are given in

(2)

062.0

characteristic

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nds on system

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of cone metal

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9

d

e

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)

w

s

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c

e

m

d

l

n

370

Fig. 2 Prindevice (shock

Fig. 3 Comp

Mathemat

nciple scheme k absorber) wit

plete shock abs

tical ModellinDeterm

of elastic-fricth package of p

sorber

ng of Work ofmining the Dy

ctional absorppolymer eleme

f Modern Fricynamical Forc

ption ents.

them

Nev

exp

F

with

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diff

syst

3. M

F

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ction-Polymerce during the

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vertheless, up

perimentally.

Force F can b

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l have first or

With replacin

ferential equa

tem swept pa

MATLAB/

For solving E

veloped b

ATLAB/Simu

imitational m

en with Eq. (

tational mod

ocity of mo

q. (1)).

r Shock Absoe Impact

P can be estim

p to now, sa

be expressed w

dP

he expression

rder differentdMx x

P M

ng Eq. (5) in

ation of moti

arts Eq.(1).

Simulink M

Eqs. (1) and (2

by using

ulink. Fig. 4

model for so

(2). Fig. 5 pre

del for estima

oving system

orbers and

mated with Eq

ame force w

with equation

d

d

Mx

t

d

dx

tx

int

tial equation: dx P x

M g F nto Eq. (4), w

on of the ele

Modelling

2), imitationa

program

presents a bl

lving dynam

esents a bloc

ation of displ

m from sho

q. (3) bellow.

was estimated

n:

(3)

to Eq. (3), we

(4)

(5)

we will have

ements of the

al models are

package

lock diagram

mical force F

k diagram of

lacement and

ock absorber

.

d

)

e

)

)

e

e

e

e

m

F

f

d

r

Mathematical Modelling of Work of Modern Friction-Polymer Shock Absorbers and Determining the Dynamical Force during the Impact

371

Fig. 4 Block diagram of imitational model for solving dynamical force F.

Fig. 5 Block diagram of imitational model for estimation of displacement and velocity.

Fig. 6 Changing of dynamical force F, estimated from imitational model from Fig. 4.

See Fig. 5

See Fig. 7

90

80

70

60

50

40

30

20

10

0

Dynamical force vs. displacement

For

ce (

kN)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Displacement (m)

Mathematical Modelling of Work of Modern Friction-Polymer Shock Absorbers and Determining the Dynamical Force during the Impact

372

Fig. 7 Displacement and velocity obtained with simulation of imitational model shown in Fig. 6.

4. Conclusions

Based on the mathematical model for changing the

dynamic force F (Fig. 6) acting on polymer embedded

package that takes into calculation influence of initial

speed of the impact v0, as well as change of the current

speed x , imitational model for force calculation is

developed, from where it can be accurately determined

stiffness characteristic of the polymer block that can

satisfy the requirements of UIC without doing any

further tests, which are very expensive from a

economical point of view.

Using the data obtained for dynamic force and

change of the current speed, we are recommending for

future work optimization of polymeric embedded

package and satisfying the energy capacity of the

absorber in accordance with UIC standards.

An imitational model for calculation of displacement

and velocity of moving system from shock absorber

during period of collision (collision of the train wagons)

is developed, and the results are given in Fig. 7.

Future optimization of the slopes of pressure

elements (Fig. 1) can be very important component for

more precise calculation of the influence of the self

exciting vibrations of the buffer during collision of the

wagons.

References

[1] Nikolskii, L. N. 1986. Railway Vehicles Shock-Absorbers. Moscow: Машиностроение (Machinery Construction). (in Russian)

[2] Mjamlin, S. V., Naumenko, N. E., and Nikitcenko, A. A. 2008. “Designing Mathematical Model for Friction Polymer Shock Absorber.” Vìsnik Dnìpropetrovs’kogo Nacìonal’nogo Unìversitetu Zalìzničnogo Transportu (Bulletin of Dnipropetrovsk National University of Railway Transport) 24: 25-33. (in Russian)

[3] Manashkin, L., Myamlin, S., and Prikhodko, V. 2009. “Oscillation Dampers and Shock Absorbers in Railway Vehicles (Mathematical Models).” Monograph, Ministry of Transport and Communication of Ukraine, Dnepropetrovsk/Ukraine.

[4] Zirov, P. D. 2011. “Modeling of Exploitation Factors with Influence on Effective Work of Modern Shock Absorbers.” In Proceedings of III International Scientific Practice Conference, 24-5. (in Russian)

[5] Zirov, P. D. 2012. “Influence Evaluation of Exploitation Factors for Work Efficiency of the Adsorption Devices at Shock-Absorbers.” Ph.D. thesis, Bryansk State Technical University.

[6] Zirov, P. D. 2012. “Development of Mathematical Model and Characteristics Calculation of Shock-Absorber Adsorption Device with Polymer Elements on Different Temperatures of Working Environment.” Scientific and Technical Journal “Vestnik BSTU” 4: 9-11. (in Russian)

Dis

plac

emen

t (cm

) an

d ve

loci

ty (

m/s

) 5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 0 0.0.2 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s)