Volume 4, Number 3, Pages 394{409 - ualberta.caThe retarder braking torque is sustained even when...

INTERNATIONAL JOURNAL OF c© 2008 Institute for ScientificINFORMATION AND SYSTEMS SCIENCES Computing and InformationVolume 4, Number 3, Pages 394–409

MODELLING THE VEHICLE WITH HYDRODYNAMIC BRAKESON DOWNHILL ROUTES AND DEFINING THE BRAKE

PARAMETERS WITH THE PNEUMATIC VALVE

BRANIMIR D. KIKOVIC1, NEMANJA S. VISNJIC2, AND DRAGUTIN LJ. DEBELJKOVIC2

(Communicated by DRAGUTIN LJ. DEBELJKOVIC)

Abstract. A mathematical model of the road vehicle with hydrodynamic

brakes on road slopes, containing disturbances such as the variable load and

the angle of the slope, is presented. The input is the oil filling and the output

is the vehicle velocity. A mathematical model of the filling sump controlled by

the proportional spool valve is developed. Using different geometry of the valve

output port, the optional geometry is defined so that the pressure response in

the chamber are sufficiently linear and quick. The effects of the nonlinear flow

through the valve and the air compressibility in the chamber were carefully

considered. Finally, a model connecting the control signal in the control unit of

the hydrodynamic brake with the pressure in the sump, i.e. with the oil filling

in the working space of the brake, is developed, thus completing the modelling

and simulation process. The braking torque depends on the oil filling and the

number of revolution of the brake rotor. The mathematical models of both

the vehicle and the brake are nonlinear, but the analysis is carried out using

linear equations valid in corresponding areas limited with nominal values of the

model parameters.

Key Words. Vehicle, Downhill, Mathematical model, Hydrodynamic, Pneu-

matic proportional spool valves, Simulation.

1. Introduction

The hydrodynamic brake is a powerful unit for continuous braking of compactdesign for buses and commercial vehicles located between the gearbox and the rearaxle of the vehicle, Fig.1.2 . The retarder braking torque is sustained even whengear changes are made but it can not secure parked vehicles.

During braking the kinetic energy is transformed into heat. To dissipate theheat, part of the circulating oil is pumped through the heat exchanger.

A retarder stage switch with maximum of 5 switching stages is used to controlthe braking system. When the retarder operates under the function ”Constantspeed ” the present speed is memorized and the retarder holds the vehicle to thisspeed downhill within the limit of the maximum braking torque.

Received by the editors July 1, 2007 and, in revised form, March 22, 2008.

2000 Mathematics Subject Classification. 93A.This research was supported by the Ministry of Science and Environment Protection of the

Republic of Serbia for financing the national research project: The development of the new gen-eration of buses.

394

MODELLING THE VEHICLE WITH HYDRODYNAMIC BRAKES 395

The retarder braking stages 1 to 5 (except ”Constant speed”) generate differentoil filling of the working chamber. The maximum braking torque is obtained whenthe retarder stage switch is in position 5 (for maximum filling).

The control medium is compressed air. When the retarder stage switch is acti-vated, the control unit is fed by an input signal which provides a control currentto the proportional valve. Depending on the magnitude of the current, the pro-portional valve provides a constant pressure on the oil sump and forces a certainquantity of oil into the working chamber depending on the operating conditions(prop shaft speed).

If the cooling water in the return pipe heats up to a predefined temperature, theconstant air pressure decreases linearly over the range of approximately 10oC atthe highest braking stage.

Fig.1.1 Functional scheme of the Fig.1.2 Schematic picture of thehydrodynamic brake hydrodynamic brake

The electro pneumatic systems offer a better alternative to electrical or hy-draulic systems for certain types of applications. Pneumatic actuators provide thepreviously enumerated qualities at low cost. They are also suitable for clean envi-ronments and safer and easier to work with. However, the position and the forcecontrol of these actuators in applications that require high bandwidth is somehowdifficult. This is mainly due to the compressibility of air and highly nonlinear flowthrough pneumatic system components.

The goal of this approach is to provide an accurate model of a system controlledby a proportional spool valve. This model is targeted to develop force controllersfor the filling and the emptying of the working circuit of a hydrodynamic brake.The filling and emptying of the circuit must take place quickly due to the requestedshort response times; thus in the case of a hydrodynamic brake for road vehiclesthe response times may not exceed 1 second.

Fig.1.1 shows a schematic representation of a hydrodynamic brake with the pro-portional air valve activated by coil, the air reservoir, the oil reservoir and the oilheat exchanger.

2. Modelling the vehicle with hydrodynamic brakes on a road slope

The longitudinal vehicle model has been created and divided into submodels:the engine , the gearbox, the retarder, the propshaft, the wheels and the chasses

396 B.KIKOVIC, N.VISNJIC, AND D.DEBELJKOVIC

[10],[11],[13]. The torque expression for the retarder component:

(1) Tprop = Tg out − Tretardera

where Tprop is the propeller shaft torque(propeller shaft), Tg out the output shafttorque of the gearbox, Tretarder the applied braking retarder torque, [5]:

(2) Tretarder = λρoilω2retD

5ret

where λ is a non-dimensional power coefficient,ρoil - the density of the operatingmedium (oil), ωret - the angular velocity of the rotor (ωret = ωpropeller, Dret - theprofile diameter of the pump impeller.

Fig.2.1. Forces and torques actingon a wheel

Fig.2.2. Different speed signals from thedriveline

Fig.2.2 shows: ωe which is the engine speed, ωt the turbine speed and ωp beingthe propeller shaft speed. The motion of the wheels in the longitudinal directioncan be described by an ordinary rotating equation:

(3) Fx · r = Tω − Iωωω

The rolling resistance force Frr is acting in an opposite direction to the tire force, Fx.Therefore, the traction force delivered to the chasing component can be describedby:

(4) Ftraction = Fx − Frr

with Frr being the rolling resistance force and Ftr the traction force:

(5) Frr =Cr

100FZ sign(ωω)

with Cr- being the coefficient of the rolling resistance and FZ-the normal forceacting on the wheel.

The differential equation based on the Second Newtons Law for a vehicle in adownhill motion is given in the following way:

(6) mV′

=∑

Fdriving tot − Fslop − Far

with m-being the mass of the vehicle, V′

the acceleration of the vehicle,Fdriving tot

the driving force generated by all wheels,α = arctg(slope/100) the slope of theroad, Fslop = mgsinα the force of the road slope and Far the air resistance force:

(7) Far = CDAρa

2∆V 2

where CD is an aerodynamic constant, A the frontal area of the vehicle, ρa thedensity of the air, ∆V the difference between the vehicle and wind velocities.From (6), Fdriving tot can be found. Assuming that ωp = ωw , Tw = Tprop and


Fdriving tot = Ftraction, we get from (3) and (4) the torque Tprop = Tw in terms ofthe longitudinal vehicle movement parameters such as mass, resistance, slope, airresistance etc.

¿From (1), it is obvious that the influence of applied braking retarder torqueTretarder on vehicle dynamics is as follows:

(8) (mr2 + Iω)

.

ω′

p = Tg out − Tret −Cr

1000rFZsign(ωp) +mgr sinα− CD

2ρArV 2

Fig.2.3. Forces acting on the chassis

The normal forces in Fig.2.3 are determined by the torque balance equations onthe front (Ffa) and on the rear wheels (Fra):

(9) Ffa =(lfr − lfm)mg cosα+ hgmg sinα− haFad

lfr

(10) Fra =lfmmg cosα− hgmg sinα+ Fadha

lfr

Since we consider the rear wheels drive, the normal force FZ on the driving wheelsis the reaction force on the rear wheels Fra.

(11) λ = 0.025 + 0.28p

When the turbine speed becomes higher than the engine speed, a situation that canappear on a downhill slope, the engine is no longer driving the transmission. Theengine is, then, driven by the transmission, and the output torque of the enginecan therefore be considered as negative. In that case, a friction torque is called ashuffle torque.

The expression for the shuffle torque is calculated from a number of measurementpoints, Fig.2.4.

(12) Tg out = (20 + 1.52 ωp)Gratio, for ωp(rad/ sec)

with Gratio being the adjusted gear ratio (3.51, 1.9, 1.43, 1, 0.74, 0.64). Substituting(2), (11) and (12) into (8), we get:(13)(mr2 + Iω)lfr

.ωp−lfrGratio(20 + 1.52ωp)− 0.025lfrρoilD

5pω

2p − 0.28plfrρoilD

5pω

2p+

0.5CDr3ρvA(Crha − lfr)ω2

p = Crrlfmmg cosα− (Crhg + lfr)mgr sinα

The above equation is given in terms of the product of the oil filling coefficient p(the input variable) and the propeller shaft speed ωp (the output variable) and that


Fig.2.4. Shuffle torque Fig.2.5. Coefficients of rolling re-sistance Cr

fact has to be taken into account in the following process of the linearization andnormalization of (13):

(14)∂Tretarer

∂ωp= λ lfrρoilD

52ωpωpN

ωpN= 2 TretarderN

ωpN∂Tretarder

∂p = 0.28lfrρoilDpω2p

pN

pN= 0.28TretarderN

pN

The normalized equation (13) is given as follows:

(15) I.ωp

+(−K1 +K2 +K3 +K4)ωp = −KUp−Kcos cosα+Ksin sinα

and the corresponding constants as follows:

(16)I = (mr2 + Iω)lfr , K1 = [1.52Gratiolfr] , K2 =

[0.05lfrρoilD

2pωp

]N,

K3 =[0.56plfrρoilD

5pωp

]N,K4 =

[(Crha + lfr)CDr

3ρaAωp

]N,

KU =[0.28lfrρoilD

5pω

2p

]N, Kcos = Crrlfmmg , Ksin = (Crhg + lfr)mg.

2.1. Parameter Identification and Simulation. The equation of the vehiclemotion on the slope (15) depends on a large number of parameters calculated for avehicle with certain characteristics: m = 18.000+10275 = 25.650kg, (constant mass18.000 kg), r = 0.54m, Iω = 40kgm2 wheel, tire and other rotation part inertia,ρoil = 850kg/m3 the oil density,ρa = 1.25kg/m3 the air density, Cr = 0.007 orCr = 0.006+0.58∗10−8ω2

p the rolling resistance coefficient, (Fig.2.6), CD = 0.5−1.1the aerodynamic constant.

Geometric characteristics are h = 2.56m, b = 2.5m,hg = 0.854m, lfr = 6m,ha =1.28m, lfm = 3m,A = hb = 6.4m2, (Fig.2.4). Coefficients Ki, i = 1, 2, 3, 4 given in(15) are defined for two different hydrodynamic brakes Dp : 0.320 and 0.250m andbraking retarder torques 4000 and 2000Nm for nominal values of ωN and pN .

The Simulink model was first introduced for the open loop system, i.e. whenthe driver himself chooses the stage of oil filling at the control system input, whilethe output value is that of the angular velocity or the vehicle velocity. The vehiclevelocity depends, above all, on the oil filling and the road slope representing thedisturbance or the second system input.

The input of the system without a feedback is the oil filling and the systemis presented by four subsystems depending on the nominal value of the oil filling.Eight linear models have been formed and boundaries are set within which theyare valid for two types of hydrodynamic brake. Based on the Simulink model, itis possible to investigate the influence of the oil filling, the slope and the type ofbrake, as well as the current gear, on the output angular velocity.

Fig.2.6-2.7 show the simulation results for variable and constant filling. Theangular velocity follows the change of the filling- to a lesser degree, and of the


Fig.2.6. Variable filling and constantslope

Fig.2.7. Constant filling and variableslope

slope- to a greater degree. It also varies in accordance with the change of vehicleweight ( the lighter the vehicle, the bigger the velocity loss).

In case when the vehicle velocity is kept constant [3], the closed-loop system isformed, Fig.2.8, the subsystem of which is the open-loop system. The input is thegiven angular velocity and the output is the real output velocity and the brakingtorque.

Fig.2.8. Closed-loop system in case of the constant velocity

Fig.2.9 shows the variation of the angular velocity and torque for a PID controlunit and a stable system response, the slope is constant.

Fig.2.10 shows the velocity change in the case of variable slope for predefinedregulation parameters. The responses are influenced by the periodical variationof the slope only to a lesser degree in contrast to the open-loop system. Also, thevariation of the regulation parameters has smaller influence on the system responsesin comparison to the open-loop system.


So far, the input value has been that of the oil filling. Now, we turn to the analysisof the HD brake itself where the input variable is the voltage or the current whichin turn gives rise to the oil filling and the previously given analysis then follows.

Fig.2.9. Velocity and torque for theconstant slope in the closed-loop

Fig.2.10. Velocity and torque for thevariable slope in the closed-loop system

3. Mathematical model of the pneumatic system

3.1. The model of the chamber. The most general model for describing thebehavior of a gas volume consists of three equations: a state equation (ideal gas), theconservation of mass (or continuity) equation and the energy equation. Assumingthat : (i) the gas is ideal, (ii) the pressure and the temperature within the chamberare homogeneous, and (iii) the kinetic and the potential energy is negligible, theequations can be written for the chamber. Considering the control volume V , withthe air density ρ, the mass m, the pressure P , and the temperature T and the idealgas constant R, the energy equation can be written as follows, [2],[4]:

(17)k − 1k

(qin − qout) +1ρ

(m′

in −m′

out)− V =V

kPP

′

where qin and qout are the heat transfer terms, k is the specific heat ratio, m′

in andm

′

out are the mass flows entering and leaving the chamber.Analyzing the heat transfer terms, further simplifications can be made. If the

process is considered to be adiabatic (qin−qout = 0) and isothermal (T = constant)then the only difference in the pressure change rate will be the specific heat ratioterm k. Thus, both equations can be written as:

(18) P′

=RT

V(αinm

′

in − αoutm′

out)− αP

VV

′

with α, αin and αout taking values between 1 and k, depending on the actual natureof the heat transfer during the process. In (18) one does not have to know the exactheat transfer characteristics, but merely estimate the coefficients α, αin and αout.The fact that the uncertainty of the estimation is banded by k to 1. For the chargingprocess, the value of αin close to k is recommended, while for the discharging ofthe chamber αout should be chosen close to 1.

3.2. The Valve Model. The pneumatic valve is a critical component. It is acommand element, and should be able to provide fast and precisely controlled airflows through the chambers. There are many available designs for pneumatic valves,which differ in geometry of the active orifice, type of the flow regulating element,number of paths and ports.


Fig.3.1. Valve spool dynamic Fig.3.2. The area of radial holes in thevalve sleeve

We have restricted our study to the proportional spool valves, actuated by coils.Two types of the valve geometry will be analyzed: (i) the circular output shapeand (ii) the rectangular output shape. The design presents several advantages, suchas quasi-linear flow characteristic, small time constant, very low hysteresis and lowinternal friction [1],[7].

The spool is balanced with respect to the pressure and positioned at the equi-librium (closed) position using two coil springs. Modelling of the valve involvestwo aspects: the dynamics of the valve spool, and the mass flow through the valvevariable orifices. Analyzing Fig. 3.1, the equation of motion for the valve spool canbe written as:

(19) Msx,,s = −csx

′

s − Ff + ks(xs0 − xs)− ks(xs0 + xs)− Fc

where xs is the spool displacement, xs0 is the spring compression at the equilib-rium position, Ms is the spool and the coil assembly mass, cs is the viscous frictioncoefficient, Ff is the Coulomb friction force, ks is the spool spring constant, andFc = Kfcic is the force produced by the coil (with Kfc being the coil force coeffi-cient, and ic the coil current).

The pressure drop across the valve orifice is usually large, and the flow has tobe treated as compressible and turbulent. If the upstream to downstream pressureratio is larger than the critical value Pcr, the flow will attain sonic velocity (chokedflow) and will depend linearly on upstream pressure. If this ratio is smaller thanPcs, the mass flow depends nonlinearly on both pressures. The standard equationfor the mass flow trough the orifice of the area Av is:

(20) m′

v =

{Cf AvC1

Pu√T, if Pd

Pu≤ Pcr

C fAvC2Pu√

T( Pd

Pu)1/k

√1− ( Pd

Pu)(k−1)/k, if Pd

Pu≥ Pcr

where m′

v is the mass flow through the valve orifice, Cf is non dimensional dischargecoefficient, Pu is the upstream pressure, Pd is downstream pressure and:

(21) C1 =

√k

R(

2k + 1

)k+1/k−1 ; C2 =

√2k

R(k − 1); Pcr = (

2k + 1

)k/k−1

The area of the valve is given by the spool position relative to the radial holes inthe valve sleeve, as shown in Fig. 3.2 for circular and rectangular figuration.


The valve effective areas for circular (Avr) figuration is given by the followingexpression, [6],[8]:(22)

Avr =

0, if xs ≤ pw −Rh ,

nh

(2R2

h arctan(√

Rh−pw+xs

Rh+pw−xs

)− (pw − xs)

√R2

h − (pw − xs)2)

πnhR2h, if xs ≥ pw +Rh .

,

3.3. Parameter Identification and Simulation. The mathematical model in-cludes a number of geometric and functional characteristics or parameters. Someparameters, such as valve dimensions and mass are provided by the manufacturer orthey can be easily measured. Other parameters, such as the critical pressure ratiofor choked flow, can be calculated using formulas and values for physical constantsinvolved.

References [10] and [9] show that the parameters, such as the valve dischargecoefficient or the spool viscous friction coefficient, cannot be directly measured orcalculated. Instead, they have to be estimated using experiments.

The value for the coil force coefficient is provided in the coil user manual (Kfc =2.78N/m2). The values for the valve mass and the valve assembly parts such asspool diameter and width, hole radius in the valve sleeve, spool spring constantare measured from the prototype design made by the Mihajlo Pupin Institute fromBelgrade (Ms = 0.02kg,Dk = 0.01m, 2pw = 0.005m,Rh = 0.002m,nh = 4, ks =100kg/m).

The two Simulink models, for circular and rectangular figurations of the holesin the valve sleeve, contain all equations, which are present in the mathematicalmodel. Models differ in respect to the two types of the valve areas (Avr and Avb).

Fig.3.3 represents simulation responses pd and Avb for the rectangular figurationof the radial holes in the valve sleeve.

Fig.3.3a. Rectangular figuration re-sponse

Fig.3.3b. Rectangular figuration re-sponse

For two step inputs (the current on the coil), the pressures are proportionalwith the inputs. The response times are approximately 5 s. The valve opening isalso proportional to the input signal and lasts as long as the transient process forpressure.

For the oscillatory input value, the pressure pd and the area Avb also have oscil-latory nature in the transient process that lasts for about 50 s.

Fig.3.4 shows, for reasons of comparison, the responses for Avr and pd for thecircular shape of the radial openings.


All input responses are proportional, but faster and they last 1.5 s. The changeof the valve opening is more uniform.

Based on simulation models, it is possible to examine both the influence of thegeometry and the other elements in the valve construction.

Fig.3.4a. Circular figuration response Fig.3.4b. Circular figuration response

4. The static characteristics of the hydrodynamic brake

Analyzing the torque characteristics, Fig.4.1, for the hydrodynamic brake withthe maximum braking torque of 2000 Nm in case of the constant power, [12], aswell as the Bernoulli equations for the chosen cross-sections, Fig1.1, the pressureand the oil flow velocity characteristics have been obtained, Fig.4.2-4.4, where:- py-the sump pressure in terms of the oil filling,−px = ∆pa + py−∆pa

H1−h (H1 −H) -the pressure at elevation H, expressed in termsof the oil filling assuming that it varies in direct proportion to the oil level in thesump,-pdin-the dynamic pressure at the brake rotor exit which depends on the oil fillingas well as on the number of revolution of the rotor n,-pdk-the oil pressure in the oil cooler,-pa- the pressure of the oil flow at the brake entry,-Vi; i = 1, 2, 3 and 4-the oil flow velocity at certain cross-sections,-H = 0.1m-the elevation of the non return valve and of the point at which oil re-turns to the oil sump,-h-oil level in the oil sump,-H1 = 0.31m-the elevation at which the oil flows into/out of the brake rotor.

Some other geometry characteristics of the brake will be listed here as well:-Aul = 740mm2-the sum of the entry openings of the brake,-Aiz = 290mm2-the sum of the exit openings of the brake,-Vsump = 2.5l-the volume of the sump filled with oil,-l = 0.3m-the length of the sump basis,-b = 0.1m- the width of the sump basis.


Fig.4.1. The braking torque in terms ofthe angular velocity for various fillings

Fig.4.2. The sump pressures in termsof the oil filling

Fig.4.3. The dynamic and cooler pres-sure in terms of the oil filling

Fig.4.4. The oil flow velocity at the en-try to the rotor in terms of the oil filling

5. The dynamic behavior of the hydrodynamic brake

In contrast to the §3 in which the vehicle motion on the slope was modelled withoil filling as the input variable, we now focus to the modelling of the brake itselfwith the input variable being the current signal at the valve entry. The outputvariable is, in this case, the filling p, i.e. the oil level h in the sump space of thebrake.

The simulation scheme of the hydrodynamic brake given in the Simulink toolboxof the Matlab programme language is considered, Fig.5.1. It enables us to determinethe system responses for various input signals and for different structural brakedesigns as well as the types of alteration of the physical parameters in time domain,i.e. it enables us to simulate the system performance in time domain.

The responses of the following variables are presented: xs[m]- spool displacementas it was previously defined, Av[m2] - the valve orifice area, mv[kg/s] - air massflow, py[Pa] - air pressure in the sump (or the output valve pressure), h[m]- oil levelin the sump.

5.1. The influence of various input signals to the system parameter re-sponses. First, let us present the system responses, Fig5.2, to the Sine Wave inputraised by 0.5 in the positive direction of the y-axis. The amplitude of these oscil-lations is 0.5, the frequency is 10rad/s and the phase is 0. If the same type of theinput signal is applied, but with the ten times smaller amplitude, 0.05, raised inthe positive direction of the y- axis by 0.05, the response as in Fig.5.3 appears.


Fig.5.1. The simulation scheme of the hydrodynamic brake

It is obvious that due to the small input signal, the spool displacement doesn’teven reach the saturation limit and the levelled part of the spool displacement andof the valve orifice area response is gone. On the other hand, the two types of themass flow alteration are present in both cases implying that the pressure exceedsthe critical value separating those two types of alteration.

The next signal, the influence of which is investigated, is of the same type as allthe previous ones, with the amplitude of 0.05 and the frequency of 1 rad/s and thesimulation result is shown in Fig.5.4.

One can easily notice that the frequency of the output signals closely follows thatof the input ones and the mass flow suddenly changes as the pressure py reachesand exceeds the critical value.

Fig.5.5 shows the system responses to the same input signal with a differentfrequency of only 0.1 rad/s.

5.2. The influence of the various structural designs of the HD brake.

5.2.1. The change of the spool spring constant. Instead of the spool springconstant value of ks = 100kg/m which has been used so far, the new value will beintroduced, i.e. ks = 500kg/m.

The input signal is the same as the signal used to give rise to the results inFig.5.2 and its effect on the system responses is given in Fig.5.6 .

The Fig.5.6 shows that due to the increased value of the spool spring constant,the input signal induces smaller intensity responses compared with the previousresults. If the value chosen for the spool spring constant is ks = 50kg/m, Fig.5.7shows the result of that change when the input signal applied to the system differsfrom the previous one in having the amplitude of only 0.05.

The results given in Fig.5.6 and Fig.5.7 are very similar because the increase ofthe rigidity in the former case is compensated by the decrease in the input signalamplitude in the latter, which makes the results pretty much even.

5.2.2. The change of the brake component dimensions. Let us first makean increase in the brake dimensions - those of the sump volume and the elevationof the rotor (stator) axis with respect to the chamber (H1).


Fig.5.2. The system variable responses Fig.5.3. The system variable responses



The new dimensions are now being introduced: b = 0.2m, l = 0.6m,H1 = 0.450mand H = 0.100m.

Fig.5.8 and Fig.5.9 show the system responses for the input signals with theamplitudes 0.5 and 0.05, respectively, and the frequency of 10rad/s.

The response in Fig.5.9 shows the biggest exceptions as a consequence of thesmall input amplitude and increased sump dimensions and, consequently, increasedsystem inertia. Due to this slow change of the input signal, neither the pressure py

nor the change of the oil level in the oil sump h, reach their ultimate values withinthe standard time interval of 10 s.



Now we should show the system responses of the system with decreased dimen-sions for various inputs. The previously mentioned geometry characteristics are asfollows b = 0, 05m, l = 0, 15m,H1 = 0.21m and H = 0.1m.

The following input signals are applied to the system:a) the Sine Wave input raised by 0.5 in the positive direction of the y- axis. The

amplitude of these oscillations is 0.5, the frequency is 10rad/s and the phase is 0,Fig.5.10.


b) the Sine Wave input raised by 0.05 in the positive direction of the y- axis.The amplitude of these oscillations is 0.05, the frequency is 10rad/s and the phaseis 0, Fig.5.11.

The small amplitude of the input, Fig.5.11, causes the small amplitudes of theresponses which don’t reach the saturation limits imposed to the spool displacementxs and to the pressure py . The oil level variations h are such that the level decreasesfrom 0.1m to 0 in the first three seconds and then starts to oscillate around zerowhich is actually the response that is the closest to what is happening in real system.

c) the Sine Wave input raised by 0.05 in the positive direction of the y- axis.The amplitude of these oscillations is 0.05, the frequency is 1rad/s and the phaseis 0, Fig.5.12.

d) the Sine Wave input raised by 0.05 in the positive direction of the y- axis.The amplitude of these oscillations is 0.05, the frequency is 0.1 rad/s and the phaseis 0, Fig.5.13.



Similar trends are noticed in Fig.5.12 and Fig.5.13 as in Fig.5.10 and Fig.5.11,the only difference being the smaller frequency which gives rise to the responseswhich are less ”ruffled”.

6. Conclusion

In §2 a mathematical model of the vehicle with the hydrodynamic brake on aslope has been formed and all the necessary parameters have been identified. Theinput is the oil filling of the working space of the brake having its values from 0 to1, and the output is the angular velocity of the output shaft ωp. The disturbancevariables are the slope and the variable vehicle load. It is possible to analyze theinfluence of certain parameters such as, for example, the dimensions of the brake.The nonlinear model has been linearized through the creation of eight linear modelsvalid around certain nominal points in which the angular velocity and the oil fillinghave certain nominal values. The simulation results have given us the informationabout the relation between the output angular velocity and the inputs- the oilfilling, the slope and the load.

The model was formed in its closed-loop variant (constant velocity). Using theMatlab simulation, the angular velocity and the braking torque responses in cases ofthe constant and variable slope, and for various regulation parameters, are obtained.

In §3, a detailed mathematical model of an air filling chamber controlled by aproportional coil activated valve is presented. The air pressure is used to pump theoil into the work circuit of the hydrodynamic brake. The pressure response mustbe linear and quick.

The valve dynamics, the nonlinearity of the valve effective area with respect tothe coil current, and the nonlinear turbulent flow through the valve orifice havealso been taken into account. The proposed model is not only accurate, but alsosufficiently simple so that it can be used online in control applications.

The two Simulink models are presented for the circular and the rectangularfiguration of the radial holes in the valve sleeve.

Simulation results show the advantages of the circular over the rectangular fig-uration.

In §4, the HD brake is treated as a separate element. First, the static analysis ofpressures and velocities in certain cross-sections has been performed and the torquecharacteristic has been determined.

The dynamic analysis in §5 deals with the behavior of the brake itself, withthe current at the proportional valve being the input signal and the filling of thebrake working space being the output variable. The influence of the variable input


signal, the spool spring constant and the dimensions (the geometry) of the brakecomponents.

All this gives us the possibility of the complete analysis of the hydrodynamicbraking system of the vehicle.

References

[1] Blackburn, J.,Fluid Power Control, Cambridge, The Massachusetts Institute of Technology,Fourth printing, April 1972.

[2] Kagava, T., Simulatiosmodell fur pneumatische Zylinderantriebe, Olhydraulik und Pneumatik,

Nr.2, (34), 1990., pp115-120.[3] Kikovic,B, Mathematical model parameters of vehicle with the hydrodynamic brake, Proc. 50.

ETRAN (In Serbian),Belgrade 6.-8. June 2006.,

[4] Kikovic B. , Electropneumatic positionering Drawtube of the regulation Hydrodynamic cou-pling,Proc. XXIX HIPNEF2004, (In Serbian) pp.443-448, Vrnjacka Banja, May 2004.

[5] Kikovic B. , Modeling Problematic of Hydrodynamics Regulation Coupling, Proc. 49. ETRAN

(In Serbia) , Vol. I pp 274 - 277, Budva (Yu), 5- 10. June 2005.[6] Kikovic B. ,Defining the optimal Geometry of Proportinal Valve Using Computer Simulation,

Proc. of IEEE EUROCON 2005-The international Conference on Computer as Tool, Belgrade,

pp. 1271-1274, Nov. , 2005.[7] Merritt, H.E, Hydraulic control system, New York: John Wiley& Sons, 1967.

[8] Ohligschlager, O.,Pneumatiche Ventile, Olhydraulik und Pneumatik, Nr.9, (32), 1988., pp614-620.

[9] Richer, E., A High Performance Pneumatic Force Actuator: Part I Nonlinear Mathematical

Model, Transactions of the ASME, Vol. 122, September 2000., pp.416-425.[10] Skoko, D.,Kikovic, B., Debeljkovic, D. Lj.,Modeling the road vehicle with hydrodynamics

brakes, Proc. HIPNEF 2006. (In Serbian), 24.- 26. May 2006.,Vrnjacka Banja, pp 61- 68,

[11] Sonderg, T.,Heavy Truck Modeling For Fuel Consumption Simulations and Measurments,D.S. thesis, dept. Elect. Eng., Linkoping University, Sweden,2001.

[12] VOITH, Overview of Service Documentacion VR 120-2 VR 133-2, 2005.

[13] Zackrrisson, T., Modeling and Simulation of driveline with an automatic gearbox, MastersThesis, December, 2003.

1Mihajlo Pupin Institute, Volgina 15, 11000 Belgrade, Serbia

E-mail : [email protected]

2University of Belgrade, Faculty of Mechanical Eng, Department of Control Engineering,

Kraljice Marije 16, 11000 Belgrade, Serbia

E-mail : [email protected] and [email protected]

Volume 4, Number 3, Pages 394{409 - ualberta.caThe retarder braking torque is sustained even when...

Documents

Transcript of Volume 4, Number 3, Pages 394{409 - ualberta.caThe retarder braking torque is sustained even when...