Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 1037-1048, Warsaw 2012

50th Anniversary of JTAM

FUZZY MODEL FOR BRAKING FORCE MAXIMIZATION

Marko Djukić

Railways of Serbia, Belgrade, Serbia; e-mail: marko.djukic@srbrail.rs

Srdjan Rusov

University of Belgrade, Faculty of Transport and Traffic Engineering, Belgrade, Serbia; e-mail: s.rusov@sf.bg.ac.rs

Zoran Mitrović, Aleksandar Obradović

University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Serbia; e-mail: zmitrovic@mas.bg.ac.rs;

aobradovic@mas.bg.ac.rs

Slavǐsa Šalinić

University of Kragujevac, Faculty of Mechanical Engineering, Kraljevo, Serbia; e-mail: salinic.s@ptt.rs;

salinic.s@mfkv.kg.ac.rs

This paper shows the process of braking force realization by air brakes with brake shoes
accompanied by a suitable mechanical model. The complexity of adhesion nature as a phy-
sical phenomenon as well as the limited factors on which the braking force value depends
are pointed out. According to this, themodel of braking force realization based on the fuzzy
set theory is explained. The procedure of fuzzy controller projectingwith a task to regulate
the value of kidding and by that the value of braking torque through the air pressure in the
braking cylinder by maximizing the braking force that can be realized according to adhe-
sion conditions is described. The testing of the optimizationmodel under concrete adhesion
conditions of the wheels on the rails is done at the end of the paper.

Key words: railway vehicle, braking force, wheel, rail, adhesion

1. Introduction

The realization of braking force high values appears as one of the basic demands in railway
exploitations based onwhich the brakingdistance is determined (Bureika andMikaliūnas, 2008).
Besides that, the maximum deceleration value depends on the braking force which is very im-
portant for city and suburban traffic vehicles from the aspect of travelling time, because short
distances between stops do not allow much participation of speed limit travelling.
The braking forcemaximization is a difficult problemdue to highly complex, non-linear and

time-varying nature of adhesion characteristics between the wheel and rail (Arias-Cuevas and
Li, 2008; Bureika, 2008; Lata, 2008). A study on the optimal brake control using the adhesion
coefficient is given by Lee et al. (2007).
The maximum value of the braking force that can be realized in accordance with adhesion

conditions is reached for a certain skidding value. But, because of the significant stochastic
character of the adhesion nature, it is very difficult to model the dependence between these
variables using classic mathematical methods. That is why the use of fuzzy logic (Zadeh, 1965)
is a useful tool for solving the problemof braking forcemaximization, having inmind that fuzzy
controllers are designed using linguistic rules without the existence of the suitablemathematical
model (Zadeh, 1973; Sugeno and Tanaka, 1991).
Many successful industrial applications of the fuzzy set theory are shown by Sivinandam et

al. (2007), and one of them is a fuzzy anti-lock system. An antiskid fuzzy controller with the
emergency brake of trains is successfully applied in Japanese company Mitsubishi (Cheok and



1038 M. Djukić et al.

Shiomi, 1998, 2000). FGPA realization of a fuzzy based subway train braking system is shown
by Reaz and Rahman (2002).
Thepaper describes the process of braking force realizationwith air brakeswith brake shoes.

Theadhesionphenomenonasaphysical phenomenon is analysedbrieflybasedonwhich the fuzzy
model for braking force maximization is formed.

2. Dynamical relations in the braking regime of trains

In a general sense, the task of braking is to stop a moving train of mass mt within a time t
on a braking distance lb (Dinić, 1986). Note that Barney et al. (2001) analysed the relevant
parameters and shownthemethod for calculating the trainbrakingdistance.Thebraking force is
a force that causes thedecrease of the train speedor the stoppingof the train. In aphysical sense,
the decrease of the train speed or stoppingmeans the decrease of the train kinetic energy Ek to
the value that is appropriate for the final speed. Thework-energy theorem (Ginsberg, 2008) for
the braking regime of the train reads

dEk
dt
=−(Fb+W)v (2.1)

where Fb is themagnitude of the braking force, W is themagnitude of the resistance force, and
v represents the train speed at a moment t of the braking process. Since the kinetic energyEk
is determined by

Ek =
mtv

2

2
+
n∑

j=1

Ijω
2
j

2
(2.2)

where Ij and ωj are, respectively, the moment of inertia and the angular velocity of the jth
rotation body, then relation (2.1) takes the following form

Fb+W =−mt(1+̺)
dv

dt
(2.3)

where ̺ is the rotation mass coefficient.

3. Braking force realization

Basically, in railway vehicles, there are two types of brakingmechanisms (Hasegawa andUchida,
1999). One type is based on using adhesion brakes (mechanical or dynamical) in which adhesion
is a necessary condition for the realization of braking forces, and the second one based on using
non-adhesion brakes (rail brakes or air resistance braking) in which the process of realization
of braking forces is done outside the wheel-rail contact zone, that is, adhesion is not involved
in the generation of braking forces. According to this, the dynamic brakes are used in the zone
of higher speed (Liudvinavičius and Lingaitis, 2007) while in the zone of smaller velocity their
effect weakens when mechanical brakes are used, that is, air brakes with brake shoes or discs.
Having in mind that braking by air brakes is until stopping, these brakes represent the basic
type of brakes on railway vehicles. Thenon-adhesion brakes aremostly used as additional brakes
in high speed trains.
In what follows, the procedure of braking force realization by air brakes with brake shoes is

described. Under the influence of pressure force from the brake cylinder, which is transmitted
to the brake shoes through the brake rigging, there comes to the leaning of brake lining (the
changeable part of brake shoe) on the rolling surface of the wheel. The braking process is



Fuzzy model for braking force maximization 1039

realized by friction forces between the brake linings and wheel, which leads to creating the
braking torque Tb (Fig. 1) that acts in the direction opposite to the wheel rotation causinga
decrease of the speed of motion.

Fig. 1. The realization of braking forces on railway vehicle wheel

The magnitude of friction forces, ~F lµ = −
~Fdµ, between the brake linings and the wheel is

determined by

Fµ =µNbs (3.1)

where µ is the coefficient of friction between the brake linings and the wheel and Nbs is the
magnitude of the brake shoe forces, ~Nlbs =−

~Ndbs, that act on the wheel. The couple of friction
forces creates the braking torque

Tb =2rwFµ (3.2)

which can be further replaced by an equivalent couple of forces ~Fb =−~F
′

b having the torque

Tb =Fbrw (3.3)

wherefrom, taking into account (3.2)

Fb =
Tb
rw
=
2rwFµ
rw

=2Fµ (3.4)

The force ~Fb represents thebraking force thatacts at thewheel center O, and ~F
′

b is the tangential
force that acts at the wheel-rail contact point K. In addition, the following inequality holds

F ′b ¬F
l
a (3.5)

where F la represents the limit value of the adhesion force determined by

F la =GbΨ (3.6)

where Gb is the weight per braked wheel and Ψ is the coefficient of adhesion. The force ~F
′

b is

balanced by force ~Fa, i.e., ~F
′

b =−
~Fa. This force represents the horizontal reaction of the rail.

As a consequence, the forces ~F ′b and
~Fa vanish and the remaining force ~Fb causes the decrease

of the vehicle speed. If the instantaneous center of zero velocity Pv coincides with the point K,
then the wheel performs rolling without slipping with the speed of wheel center

v0 = rwω (3.7)

where ω is the angular velocity of the wheel.



1040 M. Djukić et al.

In the case of local deterioration of adhesion conditions, that is, when the force ~F ′b has the
magnitude F ′b >F

l
a, there comes to skidding of wheels. The friction of rolling changes then into

the friction of skidding so that the coefficient of adhesion is decreasing to the value Ψ ′(Ψ ′ <Ψ)
and it drops with the increase of the vehicle speed. In that way, the limit value of the adhesion
force is reduced, and thus the braking force magnitude which can be realized. In accordance
to this, the braking couple of torque Tb can be decomposed into two couples. One, forming by
friction forces of the magnitude equal to the decreased value of the adhesion force GbΨ

′, and
the second one, causing the decrease of the angular velocity of the wheel. The torque of this
second couple is called the skidding torque Ts representing the part of the torque Tb that was
not realized in the sense of braking force realization because the base did not accept it. Now,
the instantaneous center of zero velocity Pv lies on the direction O-K below the point K so
that the speed of the wheel center is

v0 =OPvω (3.8)

Since OPv >OK, it is obvious that v0 >rwω. When the distance OPv tends to infinity, then
the angular velocity ω tends to zero causing the complete locking of wheels. The wheels will
then move translationally, that is, they will skid along rails. The skidding value (Ohishi et al.,
2000; Allota et al., 2001) can be expressed through the skidding velocity vs as

vs = v0−rwω (3.9)

or a skidding ratio that, for the process of braking, reads

s=
vs
v0
=
v0−rwω

v0
=1−

rwω

v0
(3.10)

and it goes from 0 for theoretical rolling of the wheel on the rail without skidding (v0 = rwω)
to 1 when the locking of the wheel (v0 ≫ rwω) appears.
The motion of the train with completely locked wheels is an extremely unfavourable case

because there comes to intensive wearing out of wheels followed by the appearance of flattened
places that create hits during each spinning of thewheel, which reflects negatively in the suspen-
sion of vehicle and superstructure of railway track. Besides that, the controllability is lost, that
is, the stability of motion of a vehicle on a rail which, as a consequence, may have derailment
with the least of additional outside disturbances (for example, crossing the switches or any other
interruption in the rail track). In order to stop the locking of wheels, it is necessary to decrease
the value of braking torque, that is, the force of brake shoe that acts on the wheel or improve
the conditions of adhesion of the wheels on the rail (by sanding) which means that for creating
the adhesion it is necessary to satisfy condition (3.5), that is

2µNbs ¬GbΨ (3.11)

The realization of the braking force by brakes with brake discs is done in a similar way as
by brakes with brake shoes. The only difference is that by brakes with discs, the force of brake
lining acts normally to the discs that are firmly tied to the axle creating the friction forces that
forms the braking couple. The magnitude of the braking forces produced by the brakes with
discs is given by

Fb =2
rbd
rw
Fµ (3.12)

where rbd is the radius of the brake disc. The force of brake shoe in a general sense depends
on the parameters of the braking system, that is, the radius of the brake cylinder d, the air
pressure in the brake cylinder pc, and the total brake rigging ratio i as follows

Nbs =
d2π

4
pciη (3.13)



Fuzzy model for braking force maximization 1041

where η is the brake rigging efficiency.With one chosen brake cylinder, the variables are the air
pressure in the brake cylinder and the total brake rigging ratio, whichmeans that the regulation
of these two variables is necessary for managing the braking force.

Theadhesion conditions vary a lot along the track due to: thewheel and rail geometry (wheel
diameter and profile, rail head profile), wheel and rail material type, line configuration (curves,
gradients, tunnels), rail unevenness, rail conditions in terms of presence of various pollutants
(lubricants, dust, leaves, etc.), atmosphere conditions (rain, snow, dew, etc.), temperature, ve-
locity, etc. The value of adhesion coefficient goes between 0.3-0.4 for dry and clean rails to
0.1-0.15 for moist and dirty rails (Vasic et al., 2003). Themost important parameters on which
the value of the coefficient of friction between the brake lining and the wheel depends are the
characteristics of wheel material (steel) and brake lining (chilled iron, composition and sintered
materials, etc.), condition of their contact surfaces regarding the cleanliness, velocity etc. Chilled
iron brake linings have the coefficient of friction that greatly depends on speed, so it takes the
value of 0.05-0.1 in zones of high speed to 0.25-0.3 in zones of small speed instantaneously before
stopping. As a consequence, there is a relatively low value of the braking force in zones of high
speedwhen its value should be the highest in order to decrease the effect of braking because the
length of brakingdistance is increasingwhile the locking ofwheels is possiblewith a small speed.
The use of special composition materials significantly increases the value of friction coefficient
and decreases its dependence on speed. The dynamic friction coefficient of compositionmaterial
brake linings goes from 0.27 to 0.33 depending on the temperature and speed. Brake linings
made of compositionmaterials are a lot lighter than the chilled and they last several times lon-
ger. The basic limitation with composition brake linings is bad thermal conductivity, and that
is why there comes to overheating of wheels that causes creating of micro cracks, that is, small
surface cracks on the rolling wheel surface. Thus, their use in exploitation is greatly limited. In
contrast to brake shoes, linings for disc brakes aremade exclusively of high friction composition
materials with the dynamic coefficient of friction around 0.35 (with high speed railway vehicle
brakes up to 0.42) and theypractically donotdependon speed.Thepresence of grease decreases
the value of friction coefficient while the presence of moisture decreases its value with smaller
speed and it increases the value with higher speed. Brake linings made of high friction sintered
materials (even up to 0.5) and that donot dependon speed are developed recently. The research
with speeds more than 200km/h have shown that brake linings made of sintered materials are
a lot more high temperature andmoisture resistant than composition material brake linings, so
that they enable more efficiency of the brakes. Besides that, the consumption of the mentioned
materials is smaller. The limitation of sintered brake linings is that they are around four times
more expensive than the usual ones (Šubara, 2006). The regulation of force, by which the bra-
ke shoes work on the wheel, can be either manual or automatic. Manual regulation is done by
changing the total brake rigging ratio that is appliedwith freight cars (empty-loaded changeover
device) adjusting the value of the braking force according to vehicle gross mass. The automatic
regulation is done by changing the air pressure in the brake cylinder.

4. Adhesion in the braking process

The existence of adhesion is caused by elastic deformities of the wheel and rail due to high
pressure in the wheel-rail contact zone. The wheel-rail contact surface (see Fig. 2) consists of
the rolling zone and the skidding zone (Dukkipati, 2000).

Under the influence of braking torque, the differences between the elements of the wheel
center path and the point on the edge of wheel appear.With the increase in the braking torque,
that is, in the air pressure in the brake cylinder, the braking force increases and by that the size
of the skidding and rolling zones decreases while the skidding zone expands. The braking force



1042 M. Djukić et al.

Fig. 2.Wheel-rail contact surface Fig. 3. Adhesion curve and ∆Fb−∆s analysis

shows further increase to the point of reaching themaximum value (adhesion limit) and then it
starts to drop when it comes to the stopping of adhesion. This is the most important moment
in the use of adhesion because themaximumbraking force can be realized then (Fbmax =F

l
a =

=GbΨmax) and the skidding ratio reaches its optimal value sopt (seeFig. 3). This corresponds to
the point of maximum brake performance. The excess of the values sopt leads to the activation
of anti-skid protection and the insufficient exploiting of adhesion questions the required brake
power and braking distance in the rapid braking.
The results of researches (Allota et al., 2001; Yamazaki et al., 2004) have shown that the

adhesion limit in the process of braking is achieved for s=0.05-0.25 depending on the vehicle
speed and the conditions at thewheel-rail contact surface. In accordancewith that,manywheel-
skid prevention systems allow the maximum value of skidding ratio up to 0.2 (Department for
Transport, 2007).

5. Optimization model

Let ∆Fb and ∆s be, respectively, the change of the braking force and the skidding ratio in the
interval ∆t (see Fig. 3) determined by

∆Fb =Fb(t+1)−Fb(t) ∆s= s(t+1)−s(t) (5.1)

where Fb(t+1) and Fb(t) are the braking force values, and s(t+1) and s(t) are the values of
skidding ratio at the moments t+1 and t, respectively. By analysing the values ∆Fb and ∆s,
the distance between the immediate and optimal values of the skidding ratio can be determined.
If signs of ∆Fb and ∆s are identical, then the immediate value of the skidding ratio is in a
stable zone of the curve Fb = f(s). In that case, the optimal value of the skidding ratio is higher
than the immediate one. If signs of ∆Fb and ∆s are different, then the immediate value of the
skidding ratio is in an unstable zone of the curve Fb = f(s). In that case, the optimal value of
the skidding ratio is lower than the immediate one.When ∆Fb is small and ∆s is large, then the
immediate value of the skidding ratio is close to the optimal one. In accordance with that, the
regulation of the braking torque, that is, the braking force can be done. If the immediate value
of the skidding ratio is in the unstable zone of the curve Fb = f(s) it is necessary to decrease
the value of the braking torque quickly. Otherwise, if the immediate value of the skidding ratio
is in the stable zone of the curve Fb = f(s) there is a possibility to increase the value of the
braking torque. Based on that, an optimization algorithm is shown in Fig. 4. The algorithm has
an iterative nature. Therefore, the optimal (maximum) value of the braking force is determined
by regulating the air pressure in the brake cylinder (with the fixed value of total brake rigging
ratio) separately for each previously defined time interval according to the conditions of contacts
between the brake linings and wheels, that is, the adhesion of rail that are of stochastic nature
and subject to specific changes (random variables) having in mind the vehicle gross mass.



Fuzzy model for braking force maximization 1043

Fig. 4. An algorithm for determining the optimal (maximum) value of the braking force

The whole procedure begins at the moment t with specified values of Fb(t) and s(t).With
the change of the air pressure in the brake cylinder there comes to a change of the braking force
and by that the skidding ratio so that, at the next moment t+1, their values are Fb(t+1)
and s(t+1), respectively. Based on Eqs. (5.1), the values ∆Fb and ∆s are determined (the
input variables of fuzzy controller) and, depending on them, the skidding ratio values are set at
the next moment t+2

s(t+2)= s(t+1)±|s′| (5.2)

according to which the adjustment of air pressure in the brake cylinder is done, that is, the
decision about its decreasing or increasing is made. The value of the skidding ratio change s′

represents the value of fuzzy controller output variable. The values Fb(t+2) and s(t+2)become
the current values Fbcurr and scurr, that is, the input values for the next iteration (t= t+1).
The procedure is repeated again and it lasts to themoment when the optimal values of the bra-
king force and skidding ratio are reached under the given adhesion conditions. A fuzzy controller
is composed of four major parts: the fuzzifier, the rule base, the inference and the defuzzifier.
Furthermore, the following linguistic values of the quantities ∆Fb and ∆s are chosen: negative
big (NB), negative medium (NM), negative small (NS), neutral (NE), positive small (PS),
positivemedium (PM), and positive big (PB). Also, the following linguistic values of the quan-
tity s′ are chosen: negative big (NB), negative highmedium (NHM), negativemedium (NM),
negative low medium (NLM), negative small (NS), neutral (NE), positive small (PS), posi-
tive lowmedium (PLM), positivemedium (PM), positive highmedium (PHM), and positive
big (PB) (see Fig. 5).
The numerical values x , (xNB,xNM, . . . ,xPB)∈ℜ

1×6, y , (yNB,yNM, . . . ,yPB)∈ℜ
1×6,

and z , (zNB,zNHM, . . . ,zPB)∈ℜ
1×10 of the quantities ∆Fb, ∆s, and s

′, respectively, repre-
sent the values that depend on the actual braking conditions (see Fig. 5). The control strategy
is expressed as a set of linguistic “if-then” type rules. The fuzzy rule base contains 49 rules



1044 M. Djukić et al.

Fig. 5. Linguistic values of the input and output fuzzy variables shownwith suitable fuzzy sets

as shown in Table 1. These rules were defined by a human expert and determined heuristically
(Zadeh, 1973; Jang and Sun, 1995). The process of deduction, that is, the decision-making pro-
cess, is defined by the algorithm of approximate or fuzzy reasoning based on the principle that
each rule represents a fuzzy relation between different categories of fuzzy variables. For example:
∆Fb – negative big (NB) and ∆s – negative big (NB) is a fuzzy phrase P in the Cartesian
space NB×NB with the related function

µP(∆Fb,∆s)=min{µNB(∆Fb),µNB(∆s)} (5.3)

The “If P, then s′ is PLM” rule is also a fuzzy phrase Q in theCartesian space ∆Fb×∆s×s
′

with the related function

µQ(∆Fb,∆s,s
′)=min{µP(∆Fb,∆s),µPLM(s

′)} (5.4)

The rules are associated by means of the term “ELSE” and make up a fuzzy phrase S in the
Cartesian space ∆Fb×∆s×s

′ with the related function

µS(∆Fb,∆s,s
′)=max{µQ1(∆Fb,∆s,s

′),µQ2(∆Fb,∆s,s
′), . . . ,µQn(∆Fb,∆s,s

′)} (5.5)

Table 1. Fuzzy rule base

s′
∆s

NB NM NS NE PS PM PB

NB PLM PM PHM NE NM NHM NB
NM PS PLM PM NE NLM NM NHM
NS PS PS PS NE NS NLM NM

∆Fb NE NE NE NE NE NE NE NE
PS NM NLM NS NE PLM PS NE
PM NHM NM NLM NE PHM PM PS
PB NB NHM NM NE PB PM PLM

The “max-min” composition is used for the inference module, and the Center of Gravity
method is used for the defuzzification module. This way, from the resulting fuzzy set, obtained
via the algorithm of approximate reasoning, a single (representative) numerical value of the
output variable is chosen. The non-linear function of the system associates the values of change



Fuzzy model for braking force maximization 1045

of the braking force ∆Fb and the skidding ratio ∆swith themagnitude of thenext change of the
skidding ratio s′, shown in the formof its increase or decrease. The controller describedhas been
tested for various values of ∆Fb and ∆s, arbitrarily specified in the range of −0.08 to 0.08, and
presented in Table 2 for numerical values of the input and output fuzzy variable. The obtained
values of output variable s′, depending on the values of input variables ∆Fb and ∆s, are shown
in Figs. 6a and 6b.

Fig. 6. Diagrams s′ = f(∆s) for (a) positive values and (b) negative values of ∆Fb

6. Realization of the optimization model

The optimization model is tested under concrete adhesion conditions between the wheel and
the rail. To compare the resulting numerical values with the real ones, the results of research
by Allota et al. (2001) are used (Fig. 7) and the modelling with the numerical values of fuzzy
variables shown in Table 2 is done.

Table 2.Numeric values of fuzzy variables ∆Fb,∆s, and s
′

NB NHM NM NLM NS PS PLM PM PHM PB

x −0.07 – −0.035 – −0.001 0.001 – 0.035 – 0.07

y −0.08 – −0.04 – −0.001 0.001 – 0.04 – 0.08

z −0.065 −0.05 −0.035 −0.02 −0.002 0.002 0.0125 0.025 0.0375 0.05

Fig. 7. Fb = f(s) curve-experimental data

Note that the numerical values of the fuzzy variable ∆Fb are shown in the form of the
values Fb/Gb. Using the optimization procedure, the braking force and skidding ratio grow to
themoment t=11when they reach the optimal values Fb(11)= 0.1604 and s(11)= 0.2217. In
the case of excessive increasing of the air pressure in the brake cylinder (or sudden deterioration
of adhesion conditions), there comes to dropping of the braking force and sudden increasing of
the skidding ratio. Then the immediate value of the skidding ratio is in the unstable zone of the



1046 M. Djukić et al.

curve Fb = f(s). The regulation of the air pressure in the brake cylinder lasts to the moment
t=6when the braking force and the skidding ratio reach the optimal values Fb(6)= 0.16 and
s(6) = 0.2077. The testing results of optimization model through finding the optimal values of
the braking force and skidding ratio are shown in Fig. 8.

Fig. 8. Optimal values of the braking force and skidding ratio obtained by testing the
optimizationmodel

Comparing the obtained results with themeasured values shown in Fig. 7, a suitable degree
of coincidence is observed. The difference in the obtained optimal values of the braking force
is a consequence of the defined parameter values of suitable fuzzy controller sub-systems and
goes up to 1%. The parameters of suitable fuzzy controller sub-systems can be defined in some
other way, which means that in order to improve it, the designing of the controller is still an
open problem for further research. Besides that, it is possible to widen the model with a larger
number of input variables.

7. Conclusion

Having in mind the stochastic character of adhesion that represents the complex function of
many variables, mostly without the mathematical interpretation, the problem of braking force
maximization is solved by fuzzy logic. The conveniences of fuzzy logic that does not demand the
modelling of the existing suitablemathematical model are used. Based on themechanical model
of the braking force realization process, an optimizationmodel is presentedwithinwhich a fuzzy
controller is formed and whose task is to regulate the value of skidding in order to maximize
the braking force that can be realized in accordance with the conditions of adhesion. The degree
of coincidence between the obtained values and themeasured ones is satisfactory. This signifies
the possibility of the application of the controller described. By applying the described model,
it is possible to reach higher adhesion exploitation that brings to improving the braking effect
in terms of increasing deceleration and shortening the braking distance length. Besides that, the
model can represent the base for designing control systems for preventing the locking of wheels.

Acknowledgement

This research was supported under grant No. TR35006 by the Serbian Ministry of Science. This

support is gratefully acknowledged.

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Maksymalizacja siły hamowania za pomocą modelu z logiką rozmytą

Streszczenie

W artykule opisano proces generowania siły hamującej w pneumatycznych hamulcach szczękowych
za pomocą odpowiedniego modelu mechanicznego. Szczególny nacisk położono na złożoność natury ad-
hezji jako zjawiska fizycznego oraz ograniczoność czynników, od których zależy wartość siły hamującej.
W tym kontekście zbudowanomodel oparty na logice rozmytej jako teoretyczne narzędzie do określania
siły hamowania. Opisano procedurę projektowania sterownika rozmytego, którego zadaniem jest kontro-
la wartości poślizgu poprzez dobór momentu hamującego wynikającego z ciśnienia w zbiorniku układu
pneumatycznego oraz aktualnych warunków adhezji. Na zakończenie przeprowadzono test modelu opty-
malizującego siłę hamowania dla konkretnychwarunkówadhezji pomiędzy szyną, a kołemporuszającego
się po niej pojazdu.

Manuscript received October 26, 2011; accepted for print February 10, 2012