Repository - Scientific Journals of the Maritime University of Szczecin - Modeling the air brake system...

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Maritime University of Szczecin

Akademia Morska w Szczecinie

2010, 24(96) pp. 5–11 2010, 24(96) s. 5–11

Modeling the air brake system equipped with the brake

and relay valves

Modelowanie powietrznego układu hamulcowego

wyposażonego w hamulec i zawory przekaźnikowe

Zbigniew Kulesza

1

_{, Franciszek Siemieniako}

2

Bialystok University of Technology, Faculty of Mechanical Engineering Department of Automatics and Robotics

Politechnika Białostocka, Wydział Mechaniczny, Katedra Automatyki i Robotyki, 15-351 Białystok, ul. Wiejska 45a, e-mail: 1z.kulesza@pb.edu.pl, 2frank@pb.edu.pl

Key words: brake system, mathematical model Abstract

The article presents the mathematical model of the air brake system with two commonly used valves: the dual circuit foot brake valve and the relay valve. The system can be seen as a part of a typical brake system of heavy trucks. The model has been developed based on the previously elaborated and experimentally verified models of the components of the air brake systems. The model can be used at a design stage to predict the behavior of the system during the normal operation and during the sudden braking or malfunction. Differential equations of pressure changes in the chambers and motion equations of the pistons of membrane operated cylinders as well as algebraic equilibrium equations describing the motion of the moving parts of the brake and relay valves are presented. The values of all model parameters, such as, spring stiffnesses and pistons’ diameters, etc. have been determined by dismantling the valve or referenced from the literature. The experimental verification of the model is planned in the future.

Słowa kluczowe: układ hamulcowy, model matematyczny Abstrakt

W artykule przedstawiono model matematyczny powietrznego układu hamulcowego z dwoma powszechnie stosowanymi zaworami: dwuobwodowym, nożnym zaworem hamulcowym oraz zaworem przekaźnikowym. Układ może być traktowany jako część rzeczywistego układu hamulcowego pojazdów ciężarowych. Model opracowano na podstawie wcześniej uzyskanych i eksperymentalnie zweryfikowanych modeli elementów powietrznego układu hamulcowego. Opracowany model może być wykorzystywany na etapie projektowania do sprawdzenia zachowania się układu w czasie normalnej pracy oraz w trakcie nagłego hamowania lub uszkodzenia układu. Wyprowadzono równania różniczkowe opisujące zmiany ciśnienia sprężonego powie-trza w poszczególnych komorach układu, różniczkowe równania ruchu trzpieni siłowników membranowych oraz algebraiczne równania równowagi opisujące przemieszczenia elementów ruchomych zaworu hamulco-wego i przekaźnikohamulco-wego. Wartości wszystkich parametrów modelu takich, jak: sztywności sprężyn, średnice tłoków itd. wyznaczono poprzez rozmontowanie zaworu i siłowników oraz na podstawie literatury. Następ-nym etapem badania opracowanego modelu będzie jego weryfikacja eksperymentalna.

Introduction

Air brake systems are used in trucks, buses, tractors, etc. The brake system is one of the main systems ensuring safety of the whole vehicle. Cal-culations of the transient processes in the complex, multi-circuit air brake systems are serious problems still requiring thorough studies and investigations.

However, these problems are not very popular and seldom mentioned in the literature. A few books [1, 2] actually reference only static calculations. The ability to calculate the dynamic characteristics is extremely important during the design process of the new brake system, as it allows one to predict its behavior during the normal operation and during the sudden braking or malfunction. Dynamic

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characteristics can be determined without the need to conduct the expensive laboratory experiments in so far as the verified mathematical and simulation models of the components and circuits are elabo-rated. Consequently the problem of modeling the complex air brake systems is well-justified.

The present paper discusses the mathematical model of the air brake system equipped with the foot, dual circuit brake valve and the relay valve. Most contemporary brake systems of heavy trucks are dual circuit ones, due to their better safety pro-perties, e.g. reliability. That is why the paper presents the model of the air brake system with the brake valve controling separate circuits of the front and rear axles braking cylinders. Usually, the rear axle circuit consists of a relay valve. The not so very correctly called ‘relay valve’ is in fact the pneumatic proportional accelerating valve used to ensure the proportional dependency between the control and output pressure during the normal ope-ration as well as to shorten the response time during the sudden braking. The model of the whole system has been developed on the basis of the previously elaborated models of the valves, pressure contai-ners, connecting pipes and membrane operated cylinders [3, 4, 5, 6, 7, 8, 9].

The brake system

Figure 1 presents the schematic diagram of the dual circuit brake system, consisting of: pressure containers 3, 4, 17, foot brake valve 7, relay valve

19, membrane operated cylinders 11, 13, 22, 23, and connecting pipes 5, 6, 8, 9, 10, 14, 18, 20, and 21. The system can be treated as a part of the real dual circuit air brake system of a heavy truck [1].

The compressed air is stored in two containers 3 and 4 (Z1, Z2) supplying separate circuits I and II.

The air from containers 3, 4 through pipes 5 and 6 is directed to supply ports 11 and 21 of brake valve 7. If the vehicle is released, front axle braking cy-linders 11, 13 through port 22 of brake valve 7 as well as the control chamber of relay valve 19 through port 12 of valve 7 are vented. The thrust force of the driver foot on the treadle of brake valve 7 causes the opening of the valve and compressed air flows in circuit I from port 21 to 22 and next to front axle cylinders 11, 13, while in circuit II from port 11 to 12, and next to port 41 of relay valve 19 controling the rear axle cylinders 22, 23. The in-crease of the pressure in circuits I and II follows proportionally to the thrust force and the rotation angle of the treadle. The retention of the treadle in the position of partial breaking causes the stoppage of increase of pressure in the braking cylinders of the front and rear axles at the demanded level. Further thrust of the treadle with maximum force causes full opening of brake valve 7 and equali-zation of pressure in both circuits. The decrease of the thrust force results in the release of the brakes. The chamber of relay valve 19 through port 12, and front axle cylinders 11, 13 through port 22 of brake valve 7 are gradually vented.

Relay valve 19 is supplied from the separate container 17 located in its near neighbourhood. Due to the fact, that the lenght of pipe 18 connecting container 17 with relay valve 19 is much more smaller than the length of control pipe 8 connecting brake valve 7 with relay valve 19, the time of flow-ing the air from container 17 to brakflow-ing cylinders 22, 23 can be considerably shortened. This way the acceleration in the operation of the rear axle cylin-ders in relation to the front axle cylincylin-ders can be obtained, what results in better stability of the vehicle trajectory during the braking.

Physical model

The physical model of the brake system is shown in figure 2. As the length of pipes 5, 6, 8, and 9 in real brake systems can be considerable (from 5 to 10 meters), the pipes are modeled as systems of five constant volume chambers and four throttles of identical cross-sections [5], i.e. as lumped systems. Much more shorter pipes 10, 14, 18, 20, 21 are modeled only as throttles skipping their internal volume.

Fig. 1. Schematic diagram of the air brake system: 1, 15 – compressed air source, 2, 16 – shut-off valves, 3 – circuit I supply container, 4 – circuit II supply container, 5, 6, 8, 9, 10, 14, 18, 20, 21 – connecting pipes, 7 – dual circuit brake valve, 11, 13, 22, 23 – membrane operated cylinders, 12 – junction (tee), 17 – relay valve supply container, 19 – relay valve Rys. 1. Schemat powietrznego układu hamulcowego: 1, 15 – źródło sprzężonego powietrza; 2, 16 – zawory odcinające; 3 – zbiornik zasilania obwodu I; 4 – zbiornik zasilania obwodu II; 5, 6, 8, 9, 10, 14, 18, 20, 21 – rury łączące; 7 – podwójny ob-wodowy zawór hamulcowy; 11, 13, 22, 23 – membrana działa-jących cylindrów, 12 – złącze; 17 – zbiornik przekaźnika za-woru; 19 – zawór przekaźnikowy

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Dual circuit brake valve 7 is presented as the system of five constant volume chambers V'11, V'12,

V13, V'21, V'22, four throttles of varrying cross-

-section d e

f 1, feo1, fed2, feo2, and a throttle of

con-stant cross-section fe13 [3, 6, 8]. Throttles fed1, fed2

correspond to these channels of the valve, through which the air flows from chambers V'11 and V'21

(supply chambers) to chamber V2 (output chamber),

and throttles o e

f ₁, o e

f ₂ – to the channels through which output chambers V'12, V'22 are vented.

Relay valve 19 is modeled as the system of three constant volume chambers V1, V2, V4, and two

throttles of varrying cross-sections fedi, feoi [4, 7].

Throttle fedi corresponds to the channel of the valve

through which the air flows from chamber V1

(supply chamber) to chamber V2 (output chamber),

and throttle feoi – to the channel through which

out-put chamber V2 is vented.

While modeling the braking cylinders, the volumes Vk1, Vk2, Vk3, Vk4 are not constant [5].

Furthermore several other assumptions are taken, simplifying the model of the whole system. The main are as follows: neglecting heat transfers through the walls of pneumatic components, perfect tightness of junctions, constant temperature of the air, flow rate function in the form proposed by Myatlyuk and Avtushko, etc. [10].

Fig. 2. Physical model of the brake system Rys. 2. Fizyczny model systemu hamulcowego

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It is assumed that during the simulations all the containers are filled with the compressed air at 800 kPa and shut-off valves 2 and 16 are closed. Con-sequently the dynamics of such components as pressure sources 1, 15 and shut-off valves 2, 16 can be omited.

Mathematical model

Based on [3, 4, 5, 6, 7, 8], the mathematical model of the system can be presented as the follow-ing:

1. Equations of pressure changes in the subsequent chambers:

• containers 3 and 4 (Z1 and Z2):



1 11



1 1 1 _, d d l z z l z _g _p _p V t p _ ₍₁₎



2 21



2 2 2 _, d d l z z l z _g _p _p V t p _ (2) • internal chambers of pipe 5:



 





11 1 11 12



1 1 11 _, _, d d l l z l l l l _g _p _p _g _p _p V t p _ _ ₍₃₎



 





12 11 12 13



1 1 12 _, _, d d l l l l l l l _g _p _p _g _p _p V t p _ _ (4)



 





13 12 13 11



1 1 13 _, _, d d p p g p p g V t p l l l l l l _ _ ₍₅₎

• internal chambers of pipe 6:



 





21 2 21 22



2 2 21 _, _, d d l l z l l l l _g _p _p _g _p _p V t p _ _ (6)



 





22 21 22 23





 





23 22 23 21



2 2 23 _, _, d d p p g p p g V t p l l l l l l _ _ ₍₈₎

• chambers of brake valve 7:







11 12



11 1 13 11 11 1 11 _, _, d d p p g V p p g V t p d l l     (9)















12 41



12 4 13 12 12 13 12 12 1 11 12 12 1 12 , , , , d d l l a o d p p g V p p g V p p g V p p g V t p          (10)



13 12



13 13 13 _, d d p p g V t p _ (11)







21 22



21 2 23 21 21 2 21 _, _, d d p p g V p p g V t p d l l   _  (12)











22 31



22 3 22 22 2 21 22 22 2 22 , , , d d l l a o d p p g V p p g V p p g V t p        (13)

• internal chambers of pipe 9:



 





31 22 31 32



3 3 31 _, _, d d l l l l l l _g _p _p _g _p _p V t p   (14)



 





32 31 32 33





 





l l l w



l l l _g _p _p _g _p _p V t p , , d d 33 32 33 3 3 33 _ _ ₍₁₆₎

• the chamber of tee 12:











₂



3 6 1 3 5 33 3 3 , , , d d k w l l k w l l l w l l w p p g V p p g V p p g V t p        (17)

• cylinders 11 and 13 (S1 and S2):





01 1 1 1 1 1 1 5 1 , d d V x F p F p p g t p e k e w k l k      (18)





02 2 2 2 2 2 2 6 2 , d d V x F p F p p g t p e k e w k l k      (19)

• internal chamber of pipe 8:



 





41 12 41 42



4 4 41 _, _, d d l l l l l l _g _p _p _g _p _p V t p _ _ (20)



 





42 41 42 43





 





43 42 43 4



4 4 43 _, _, d d p p g p p g V t p l l l l l l _ _ ₍₂₂₎

• chambers of relay valve 19:



4 43



4 4 4 _, d d l l _g _p _p V t p _ (23)







1 2



1 1 3 1 1 9 1 _, _, d d p p g f V p p g V t p edi i z l   _  (24)













o eoi



a



edi i k l k l p p g f V p p g f V p p g V p p g V t p , , , , d d 2 2 2 1 2 2 2 2 2 2 8 1 2 2 7 2          (25)

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• container 17 (Z3):



3 1



9 3 _, d d p p g t p z l z __ ₍₂₆₎

• cylinders 22 and 23 (S3 and S4):





03 3 3 3 3 3 2 3 7 3 , d d V x F p F p p g t p e k e k l k      (27)





04 4 4 4 4 4 2 4 8 4 , d d V x F p F p p g t p e k e k l k      (28)

2. Motion equations of the rods of cylinders S1, S2,

S3, S4: j j t x   d d (29)

















                             j j j pj j j pj j j j e a kj j j j j j e a kj j j pj j pj j j j e a kj j j s x b s x c P x c F p p s x P x c F p p x b x c P x c F p p m t for , 0 for , 0 for , 1 d d 1 1 1    (30) where j = 1, ..., 4. Effective cross-sections o e f ₁, d e f ₁ of the channels in section I, and f_eo₂, f_ed₂ of the channels in section II of the brake valve depend on displacements y1, y2

of the pistons of the valve in the following manner [3]: 0 max 1 1 31 1 e o e _a f y f  , d e d e _q f y f ₂_max 3 32 1  0 max 2 2 41 2 e o e _a f y f  , d e d e _q f y f ₂_max 4 42 2 (31) where: q3 = q1 – a1, ... q4 = q2 – a2 (32)            0 for 0 for for 0 1 1 1 1 1 1 1 1 31 y a a y y a a y y            1 1 3 1 1 1 1 1 1 1 32 for for for 0 q y q q y a a y a y y (33)            0 for 0 for for 0 2 2 2 2 2 2 2 2 41 y a a y y a a y y (34a)            2 2 4 2 2 2 2 2 2 2 42 for for for 0 q y q q y a a y a y y (34b)

Displacements y1, y2 of the pistons of the brake

valve are calculated using the following depende-cies [3]: • for y1 < a1: 1 1 1 _c _c F y h s   (35) • for y1  a1: 3 1 3 1 1 _c _c _c F F y h s s     (36) • for y1  y2 + a3:  if 0 < y2 < a2, then: 2 3 1 2 3 1 1 _c _c _c _c F F F y h s s s       (37)  if a2  y2 < q2, then: 4 2 3 1 4 2 3 1 1 _c _c _c _c _c F F F F y h s s s s         (38) • for y2 < a2: 2 2 2 _c F y _ s ₍₃₉₎ • for y2  a2: 4 2 4 2 2 _c _c F F y s s    (40)

where forces Fs1, Fs2, Fs3, Fs4 loading the pistons of

the brake valve are as follows [8]:

10 12 12 11 1 p A p A c y F F_s  _a   _h _h (41) 20 22 22 21 13 2 p A p A F Fs    (42) 30 1 3 31 11 32 12 3 p A p A ca F F_s     (43) 40 2 4 41 21 42 22 4 p A p A ca F Fs     (44)

In the above equations g(p, pj) is the modified

Myatlyuk-Avtushko flow rate function [3]:





              p p p p p p p p p p p p p p p p g j j j j j j j j for 13 . 1 for 13 . 1 , (45)

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The switching functions fedi, feoi of the relay

valve are as follows [7]:

        max max max max max for for y y f y y y y f f edi edi edi       4 2 4 2 max for 0 for p p p p f f_eoi eoi (46) where 2



₄ ₂



4 π p p c D y i   is the displacement, D

piston diameter, and ci spring stifness of the relay

valve.

Other parameters are as follows [5]:

3

654 .

0 

lk fek RT [m3/s], where k = 1, ..., 9; R = 287.14 [J/kgK] – air gas constant;

T = 293 [K] – air absolute temperature;

 = 1.4 – air adiabatic index;

fek = k fk [m2] – effective cross-section of the kth

pipe;

k – flow rate coefficient of the kth pipe;

fk [m2] – cross-section of the kth pipe;

Vlk [m3] – volume of the kth internal chamber of the

pipes, where Vlk = d /4, and dk2 k is the internal

diameter of the pipe;



2



2 2 1 2 1 12 π j j j j ej D D D D F    [m2_{] – effective cross-}

-section of the brake cylinder membrane,

j = 1, ..., 4;

D1j [m] – bigger diameter of the cylinder

mem-brane;

D2j [m] – smaller diameter of the cylinder

mem-brane;

V0j [m3] – initial volume (for t = 0) of the cylinder;

cj [N/m] – cylinder spring stiffness;

Pj [N] – constant force loading the rod of the

cy-linder;

cpj [N/m] – stiffness of the covers of the cylinder;

bpj [Ns/m] – damping coefficient of the covers of

the cylinder;

sj [m] – maximum displacement of the rod of the

cylinder;

mj [kg] – reduced mass of the moving parts of the

cylinder.

Differential equations (1)–(30) along with the algebraic equations (31)–(40) are the mathematical model of the presented system. They can be used to calculate displacements x1, x2, x3, x4 of the rods as

well as pressures pk1, pk2, pk3, pk4 inside the

cham-bers of membrane cylinders. The input signal to the model is displacement yh of the piston of the brake

valve resulting from the thrust force of the driver pushing on the treadle of the valve. It can be seen that these are the nonlinear differential equations that can be solved only using proper numerical procedures, e.g. ode34, ode56 from Matlab.

Parameters of the model

In Eqs (1)–(30) and (31)–(40) constituting the mathematical model of the system, several parame-ters, such as dimensions of the pistons, sizes of inlet and outlet channels, spring stiffnesses, chambers volumes, etc. of the brake and relay valve are present. Their values have been determined by dis-mantling the brake valve and direct measurements of its subsequent components. Below are the values obtained.

Volumes of the chambers:

V11 = V21 = 9753.85 mm3, V12 = 35 321.14 mm3,

V13 = 30 243.85 mm3, V22 = 191 349.77 mm3.

Stiffness coefficients of the springs:

ch = 140 208.96 N/m, c1 = 2658.31 N/m,

c2 = 712.48 N/m, c3 = c4 = 5340.61 N/m.

Initial loads of the springs: F10 = 72.31 N,

F20 = 13.47 N, F30 = F40 = 108.95 N.

Diameters of the pistons: D1 = 60.00 mm,

D2 = 110.00 mm, D3 = D4 = 22.36 mm.

Clearances: a1 = a2 = a3 = 1 mm.

Maximum displacements of pistons:

q1 = q2 = q3 = q4 = 2.1 mm.

Diameters of the channels: d₁d= 10.95 mm,

o

d₁ = 12.51 mm, d13 = 3.00 mm, d2d=10.95 mm,

o

d₂= 12.51 mm.

Parameters of the relay valve have been refe-renced from [11]. They are as follows: volumes of the supply, valve, and control chambers:

V1 = V2 = V4 = 108 356 mm3, piston diameter

D = 80 mm, maximum displacement of the piston ymax = 3.5 mm, stiffness coefficient of the spring

c = 1600 N/m.

The system consists also four identical mem-brane cylinders. Their parameters are as follows: bigger D1 j = 145 mm and smaller D2 j = 114 mm

diameters of the membrane, maximum displace-ment of the rod s = 20 mm, initial volume

V0 j = 0.01 dm3, stiffness coefficient of the spring

cj = 21 429.4 N/m, stiffness cp j = 109 N/m and

damping bp j = 105 Ns/m coefficients of te covers,

reduced mass of the moving parts mj = 0.75 kg,

constant force loading the rods Pj = 0 N. These

values have been also determined by dismantling one of the cylinders.

The diameter of all connecting pipes is 12 mm. The lenght of pipes 10, 14, 18, 20, 21 is 1 m, and

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the length of pipes 5, 6, 8, and 9 can be varried from 5 to 10 m.

Conclusion

Modeling the presented air brake system is diffi-cult mainly due to the complexity of the models of subsequent components: foot brake valve, relay valve, braking cylinders, connecting pipes, etc. The presented mathematical model has been simplified only to the extent required to obtain the unambi-guous numerical solution in a finite, short time. The best verification of the presented model will be the experimental tests. The preliminary experimental results of the brake valve [6], relay valve [4], and simple pneumatic systems [5] are very promising and give the hope that the model of the whole sys-tem will be also correct. After the full experimental verification, and after introducing required modifi-cations and extensions, the presented model could be used for the design of new air brake systems.

References

1. ŁOMAKO D.M.,STAŃCZYK T.L.,GRZYB J.: Pneumatic brake

systems of road vehicles. Wyd. Politechniki Świętokrzy-skiej, Kielce 2002.

2. MIATLUK M.,KAMIŃSKI Z.: Brake systems of road vehicles. Calculations. Wyd. Politechniki Białostockiej, Białystok 2005.

3. KULESZA Z.: Modeling the pneumatic brake valve.

Hydrau-lics and Pneumatics / Hydraulika i Pneumatyka 2007, 1, 9–12.

4. KULESZA Z.: Modeling pneumatic brake systems with the relay valve. Pneumatics / Pneumatyka 2004, 4.

5. KULESZA Z.: Modeling the multi-circuit pneumatic brake systems. Ph.d. thesis, Warsaw University of Technology, Warsaw 2003.

6. KULESZA Z.,CZAPKO Ł.,SIEMIENIAKO F.: Influence of the

type of braking on pressure changes in pneumatic brake cylinders. Pneumatics / Pneumatyka 2006, 4, 32–35. 7. KULESZA Z.,MIATLUK M.: Analysis of transient processes

in the pneumatic drive system equipped with the relay valve. Archives of Motorization / Archiwum Motoryzacji 2008, 1, 33–43.

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Recenzent: dr hab. inż. Kazimierz Peszyński, prof. UTP Uniwersytet Technologiczno-Przyrodniczy