Stability of Composite Braking Produced by Retarder and Braking System

ZHAO Guo-zhu(趙國(guó)柱)，WEI Min-xiang(魏民祥)

(1.College of Engineering，Nanjing Agricultural University，Nanjing 210031，Jiangsu，China;2.College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，Jiangsu，China)

Introduction

As a standard part，the retarder is equipped on various vehicles of non-M1 and non-N1 categories.Currently，the evaluation methods for the matching quality between a retarder and a service braking system can be the distance method between the optimal distribution curve of braking force on front and rear axles，namely I curve，and the real brake force distribution curve，namely β curve，the braking efficiency method and the experiment method［1－2］.The first two methods do not have clear borders and are not suitable for quantitative evaluation，and the third method requires longer time and higher costs.

Consider following two reasons.Firstly，no matter what the matching quality between the retarder and the service braking system is，the performance of the composite braking must meet the related standards.Secondly，in regulation R13 drafted by the Economic Commission for Europe(ECE)，the wheel-locking sequence of the vehicle's front and rear axels and their brake efficiency in braking are defined clearly，and its correctness and feasibility have been proved for decades.This paper establishes a mathematical model of the adhesion coefficient for the composite braking system.Its braking stability is evaluated by using this model and regulation R13 to provide a guideline for matching a retarder to the service braking system.

1 ECE Regulation R13

The required braking performance in ECE regulation R13 for wheeled vehicles other than those of category M1 and N1 is shown in Fig.1.

1)For all load states of all vehicles，the adhesion utilization curve of the rear axle shall not be situated above that for the front axle.When the theoretical coefficient of adhesion φ is between 0.2 and 0.8，the adhesion utilization curve for all vehicles shall comply with the relation，φ≤(z+0.07)/0.85，where z is the braking rate of vehicle.

2)When z=0.15～0.3，for all vehicles of non-M1 and non-N1 categories with a maximum mass more than 3.5 tons，the adhesion utilization curves for each axle are situated between two lines parallel to the line of ideal adhesion utilization given by the equation φ=z+0.08 and φ =z－0.08，as shown in Fig.1，and the adhesion utilization curve for the rear axle in z＞0.3 shall comply with the relation φ≤(z－0.018 8)/0.7.

2 Mathematical Model of Adhesion Utilization for Composite Braking

The adhesion utilization of axle i can be defined as

where Fbiis the longitudinal tire braking force in N on axle i corresponding to z，F(xiàn)niis the normal reaction of road surface on axle i in N corresponding to z［3］.

For a two-axle vehicle，when braking on level road，the normal reaction of road surface on front and rear axle can be written as

where Fgis the vehicle's gravity in N，L is the vehicle wheelbase in m，a and b are from the front and rear axle to vehicle's center of gravity in m，respectively，hgis the height above ground of the centre of gravity in m.

The relationship between the braking forces acting on the front and rear axles is

where Fsfis the braking force acting on the front axle in N，F(xiàn)sris the braking force acting on the rear axle in N，β is a distribution coefficient of brake force.

When composite braking，the total braking forces on rear axle can be expressed as

where Fcris the total braking force on rear axle in N，F(xiàn)rris the braking force of the retarder in N.

The sum of the longitudinal tire braking forces on the front and rear axles is related to the vehicle deceleration.

When the rear axle is locking or both rear and front axles is locking simultaneously in braking，the relationship between the tire braking forces and the braking forces acting on the front and rear axles can be expressed as

Substituting Eq.(5)，(7)and(8)into(6)，we have

Substituting Eq.(2)and(9)into(1)，we have w

here φf(shuō)is the adhesion utilization of the front axle.

Similarly，the adhesion utilization φrof the rear axle can be expressed as

Thus，we can analyze the stability of the composite braking conducted by the retarder and the service brake using Eq.(10)，(11)and ECE R13 quantitatively.As the braking stability is closely related to the initial speed of the vehicle，as long as it is not too high，even though the rear axle locks in a certain time earlier than the front axle，a slight sideslip will occur only;on the contrary，if the initial speed exceeds a certain value，and the rear axle locks earlier than the front axle，a serious sideslip will occur.Therefore，we shall put emphasis on evaluation of the composite braking stability in the medium or high speed.

3 Case Analyses

Table 1 shows the parameters of a medium bus［4］.The retarder's maximal and minimal braking forces Frrmaxand Frrminare 6 188 N and 3 962 N，respectively.

Table 1 Parameters of a medium bus

When there are not any retarder and braking force adjustor，such as proportional valve，load sensing proportional valve and ABS，in the rear axle，the bus's adhesion utilization is shown in Fig.2.It shows the adhesion utilization of the bus meets the requirements of ECE R13 in braking when z is smaller than 0.635.But，the distance between the adhesion utilization curve and the line of ideal adhesion utilization increases with its load.That means the braking efficiency decreases with the increase of load.

Fig.2 Relationship between φ and z when service-braking

3.1 Stability of Composite Braking Under Fullload Condition

The relationship between the bus's adhesion utilization and braking rate under full-load condition is shown in Fig.3.This vehicle's adhesion utilization can satisfy the requirements of the regulation on the fullload condition only when the retarder's braking force is minimal and brake rate is in the range from 0.18 to 0.46.And，with the increase of the retarder's braking force，φrcurve ascends gradually，while φf(shuō)curve descends gradually.The adhesion utilization curve of the rear axle has been thoroughly situated above that for the front axle when Frris up to 4 910 N.Hereafter，the rear axle locks earlier than the front axle in any braking rate.However，F(xiàn)ig.2 also shows that φrand φf(shuō)curves are both situated among three lines given by the equations φ =z+0.08，φ =z － 0.08，and φ =(z －0.018 8)/0.74 while the retarder's braking force increasing from the minimum to maximum.That means the bus's braking efficiency meets the regulation.

Fig.3 Relationship between φ and z in full-load when composite braking

3.2 Stability of Composite Braking Under Noload Condition

The relationship between the vehicle's adhesion utilization and braking rate under the no-load condition is shown in Fig.4.The sequence of locking the front and rear axles can not meet the regulation in any retarder's braking force.However，the braking efficiency meets the regulation.

Fig.4 Relationship between φ and z in no-load when composite braking

Based on the above analysis，for a rear-wheel drive vehicle，even if the retarder does not take part in braking，the service braking system with a constant distribution for front and rear brake forces can not fully meet ECE R13.Because the line β is a straight line，it can not be completely consistent with the curve I of the ideal front and rear braking force distribution of the service braking system.After the retarder joins into braking，the range of brake rate in which the rear axle locks in advance enlarges with the increase of the retarder's braking force without the change of β.Namely，the adhesion utilization on the front and rear wheels can not meet the regulation with the increase of the retarder's braking force more and more.Therefore，the value of β must be readjusted to improve the stability of the composite braking after a retarder is added.

4 Adjustment Methods

4.1 Generalized Distribution Coefficient of Front and Rear Braking Forces

Define a generalized distribution coefficient βgof the front and rear braking forces as

where Fsis the total braking force of the service braking system in N.Then，the distribution ratio of the braking forces for the vehicles of non-M1 and non-N1 categories is limited by Eq.(12)to meet the regulation constraint［5］.

where

4.2 Adjustment of β

The value of β can be adjusted according to the position relationship between line β and line βg.The line βgis plotted according to(13).

Based on Eq.(12)，Eq.(13)shall meet Eq.(14).

The line β can be plotted according to Eq.(15)in the composite braking.

Eq.(15)should meet Eq.(16)

Moreover，a generalized I curve needs also be defined to easily explain the theoretical basis for adjustment of β.It is the ideal distribution curve for the brake force on the front axle and the composite braking force on the rear axle when the front and rear axles are locked simultaneously［2］.The composite braking force on the rear axle is the sum of the rear axle braking force produced by the service braking system and the braking force produced by the retarder.The generalized I curve is the I curve of the service braking system when the braking force produced by the retarder is zero.The synchronous adhesion coefficient φ0and generalized synchronous adhesion coefficient φ0can be obtained from the relationship among line β，line βgand generalized I curve，as shown in Fig.5.

The retarder's maximum and minimum braking forces during emergency braking in the medium or high speed are represented by point C and D in Fig.5，respectively.It can be seen from the figure that，when the retarder's braking force seesaws between the minimum and maximum values，for the service braking system with a constant β，the synchronous adhesion coefficient φ0can only lie in the range from point A to B in the generalized I curve to make the composite braking stability meet the regulation.

Fig.5 Relationship among βg，β and generalized I curve

Considered the load moves forward when braking，the value of β should be as big as possible on the premise of meeting the regulation，therefore，the line BC in Fig.5 can be taken as the braking force distribution line of the service braking system in composite braking.The generalized distribution coefficient βgcorresponding to point B in Fig.5 is

where FcrBis the sum of the rear axle braking force from the service braking system and the retarder's braking force，corresponding to point B in Fig.5，F(xiàn)sfBis the braking force on front axle corresponding to point B in Fig.5.

4.3 Value of β and βg

For the case analyzed above，the value of β or βgis bounded by(17).

Based on the adjustmentmethod mentioned above，β and βgare 0.716 and 0.67 under the no-load condition;0.674 and 0.622 5 under the full-load condition，respectively.Thus，the relationship between the bus's adhesion utilization and braking rate under the no-load and full-load conditions are shown in Fig.6 and Fig.7，respectively，in composite braking.

Fig.6 Relationship between φ and z in full-load when composite braking after adjusting β

Fig.7 Relationship between φ and z in no-load when composite braking after adjusting β

Figure 6 shows that both the locking sequence of the wheel axles and the adhesion utilization of the bus in full-load meet the regulation well.But，the distance between the adhesion utilization curve and the line of ideal adhesion utilization increases with the retarder's brake force，and it means the braking efficiency decreases slightly.It can be seen from Fig.7 that both the locking sequence of the wheel axles and the adhesion utilization of the bus in no-load can basically meet the regulation when the retarder's brake force is minimum.However，the interval of small braking rate that the locking sequence of the wheel axles does not satisfy the regulation will be extended with the retarder's braking force growing.It will be extended to 0＜z＜0.27 when the retarder's braking force is maximum.But，this kind of braking rate does not mean the emergency brake，and normally it will not affect the vehicle stability greatly.In addition，the bus meets the regulation in braking efficiency when composite braking.Thus，the value of β of the bus should be adjusted to 0.716 in no-load and 0.674 in full-load when emergent braking in medium or high speed.Also，the value of β is still 0.561 when the retarder fails.

5 Conclusions

1)A vehicle's braking stability is generally going down with the retarder braking force's growing when emergent braking in medium or high speed，if β does not change after installing a retarder.

2)The stability of the composite braking produced by a retarder and a service braking system in medium or high speed can be improved significantly after β is appropriately adjusted based on the position relationship among the line β of the service braking system，the line βgand the generalized I curve，and the constraints in ECE R13.

3)The braking efficiency of the composite braking in medium or high speed can meet the regulation if the retarder is selected properly，the value of β is readjusted appropriately and some regulators，such as proportional valve，load sensing proportional valve and ABS，for braking force between the front and rear axles are installed.

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