Mathematical Modelling of Work of Modern Friction … Fig. 2 Prin device (shock Fig. 3 Comp Mathemat...
Transcript of Mathematical Modelling of Work of Modern Friction … Fig. 2 Prin device (shock Fig. 3 Comp Mathemat...
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: [email protected].
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
3.0 MN to a
Then, when
polymer blo
capacity if
increase o
determinatio
polymer blo
This pape
force acting
Movement o
expressed w
force is taken
polymer blo
can be accu
elastic-fricti
with packag
and 2. A com
2. Mathem
A need
absorption d
requirement
Fig. 1 Princ
Mathemat
IC (Internat
s for towing
ons and locom
MN in order
t less than 70
achieve energy
there is insu
ock, it will no
the rigidity
of the pr
on of stiffnes
ck is an impo
er presents th
g on the poly
of the swept e
with differenti
n, from wher
ock that can s
urately deter
onal absorpt
ge of polymer
mplete shock
matical Mod
to develop
devices arises
s, and it woul
iple scheme of
tical ModellinDeterm
tional Union
g shock abso
motives requi
to achieve
kJ and the fu
y capacity no
fficient stiffn
ot achieve the
is increased
ressing forc
ss characteris
ortant and res
he calculation
ymer block
elements of s
ial equation w
e the stiffness
satisfy the abo
rmined. Prin
tion device (
r elements is
absorber is g
delling
mathematica
s due to the a
ld further be u
f elastic-friction
ng of Work ofmining the Dy
n of Railw
orber devices
ire forces sm
energy abso
ull force less
ot less than 90
ness of embed
e required en
d, there will
ces. Theref
stic of embed
sponsible issu
n of the dyna
during collis
shock absorb
where dynam
s characterist
ove requirem
ciple scheme
(shock absor
s given in Fig
given in Fig. 3
al modelling
above-mentio
used to determ
nal absorption
f Modern Fricynamical Forc
ways)
s for
aller
orber
than
0 kJ.
dded
ergy
l be
fore,
dded
ue.
amic
sion.
er is
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ic of
ments
e of
rber)
gs. 1
3.
g of
oned
mine
the
poly
diff
syst
whe
x
xx
muc
the At
give
Ref
F
and
mas
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from
elem
n device.
ction-Polymerce during the
optimal stiff
ymer block a
ferential equa
tem swept pa
ere:
x is displace
x is velocity o
x is accelerat
is coeffici
ch pressure fo
pressure forcour case, en in Refs. [2
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
arts (Fig. 1) ha
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;
mission, show
hock absorber
mbedded polymken. Calculat
ing equations
bx )0
98.36 2 xx
xe24083
as nonlinear c
d experimenta
Force P depen
n, which can b
hich angles o
ction coeffici
369
he embedded
btaining. The
ements of the
ing form:
(1)
rt;
;
wing for how
r P oversizes
mer block F.ion of is
s are given in
(2)
062.0
characteristic
ally. M is the
nds on system
be calculated
of cone metal
ient between
9
d
e
e
)
w
s
. s
n
)
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
will
W
diff
syst
3. M
F
dev
MA
of i
give
imit
velo
(Eq
ction-Polymerce during the
m, so force P
vertheless, up
perimentally.
Force F can b
h replacing th
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)