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MODEL FOR CALCULATING THE EFFECT OF

LONGITUDINAL PROFILES ON THE SPEED

OF HEAVY VEHICLES

by

G. Carlsson

(2)

MODEL FOR CALCULATING THE EFFECT OF

LONGITUDINAL PROFILES ON THE SPEED

OF HEAVY VEHICLES

by

G. Carlsson

(3)

C O N T E N T S

page

1. I N T R O D U C T I O N 1 2. D E S C R I P T I O N O F T H E M O D E L 2 3. T E S T OF T H E M O D E L 5 4. A P P L I C A T I O N O F T H E M O D E L 7

(4)

I N T R O D U C T I O N

T h e k n o w l e d g e of the effe c t s w h i c h the v e r t i c a l a l i g n m e n t of a r o a d p r o d u c e s on traffic is n e c e s s a r y for p r e p a r a t i o n of a p p r o p r i a t e s t a n d a r d s and

r e c o m m e n d a t i o n s r e l a t i n g to g r a d i e n t s a nd to v e h i c l e - c l i m b i n g lanes* Fu r t h e r m o r e th is k n o w l e d g e c o n s t i t u t e s an i n d i s p e n s a b l e e lement of the

i n f o r m a t i o n w h i c h is r e q u i r e d for e c o n o m i c c a l c u l a t i o n s in tr a f f i c e n g i n eering. In fact, v e r y e x t e n s i v e i n f o r m a t i o n is n e e d e d in order that t he a bove-

m e n t i o n e d ef f e c t s m a y b e d e t e r m i n e d in a s a t i s f a c t o r y man ner. A m o n g othe r things, it is n e c e s s a r y to k n o w t he g e o m e t r i c r o a d d e s i g n features, th e traffic vo lume, a nd the c l a s s i f i c a t i o n of t r a f f i c a c c o r d i n g to ty p e of vehicl e. M o r e o v e r , it is r e q u i r e d to k n o w the ef f e c t s of t h e v e r t i c a l

alignm ent on f r e e - m o v i n g * ^ v e h i c l e s of v a r i o u s types, a n d the i n f l u e n c e of sight c o n d i t i o n s and o p p o s i t e - d i r e c t i o n t r a f f i c on t h e f r e q u e n c y of o v e r ­ t a king and passing. A c o m p l e t e t r e a t m e n t of this p r o b l e m w i l l b e c a r r i e d out w i t h t h e h e l p of s i m u l a t i o n by m e a n s of a u t o m a t i c da ta p r o c e s s i n g equipment. A m o d e l for s i m u l a t i o n of t r a f f i c on t w o - l a n e r o a d s is at presen t in p r e p a r a t i o n at the N a t i o n a l S w e d i s h R o a d R e s e a r c h Institu te.

As has b e e n p o i n t e d out in the above, th e n e c e s s a r y b a s i s for the s i m u l a ­ tion co mprises, a m o n g other things, the k n o w l e d g e of t h e e f fects p r o d u c e d on f r e e - m o v i n g v e h i c l e s by t he v e r t i c a l a l i g n m e n t of t h e road. Stud ies of the v a r i a t i o n s in the sp eeds of f r e e - m o v i n g v e h i c l e s w i t h the d i s t a n c e on u p g r a d e s h a v e t h e r e f o r e b e e n m a d e for th is p u r p o s e a n d t h e r e s u l t s are p r e s e n t e d in t h i s repor t.

1) T h e t e r m fff r e e - m o v i n g ” is u s e d to d e s i g n a t e the m o t o r v e h i c l e s w h o s e drivers can b e c o n s i d e r e d to b e u n h a m p e r e d by oth er traf fic, an d are free to c h o o s e t heir own w a y of dri ving .

(5)

2.

DESCRIPTION OF THE MODEL

In order to describe the speed-distance relation on upgrades, use was made

of an assumed model which was based on physical considerations*

This

model had to comply with two requirements.

First, it should be simple

to use.

Second, in should adequate!}*- represent the variation in the

speed with the distance for each individual vehicle.

In other words,

this model should make it possible to describe the whole vehicle popula­

tion, and not only a single typal vehicle, as has been usual in previous

studies of this problem.

The model assumed for the present study was based on the forces which act­

on a motor vehicle on an upgrade, see fig. 1.

pi ^ 1 ,

Vehicle on an upgrade

F

~ Tractive force, in N

F

= Air resistance, in N

F 1

~ Rolling resistance, in

N

m

== Mass of the vehicle,

in kg

2

g ~ Acceleration of gravity, in m per sec

x = Angle of slope

(6)

T h e e q u a t i o n of forc es give s

F - F- - F - m g • sin i = m • (1)

1 r dt

w h e r e

v = the s peed of t h e ve h i c l e , in m per sec t = the time, in sec

T h e t r a c t i v e force, F, w h i c h is d e v e l o p e d at the w h e e l s b y t h e engi n e can b e w r i t t e n

F = — v

w h e r e

P = the p ower output, in W, d e v e l o p e d by the t r a c t i v e f orce

T h e r e s i s t a n c e s a re depe n d e n t on s e veral factors, as m a y b e seen fro m the f o l l o w i n g equ atio ns . T h e air r e s i s t a n c e , F^, is F = C • A • v 2 1 1 r w h e r e 3 C- = a shape f a c t o r o f t ne ve h i c l e , in k g per m ,

2

A = t h e front su r f a c e area of t he veh i c l e , in m

v^ = t h e r e l a t i v e v e l o c i t y of t h e v e h i c l e w i t h r e f e r e n c e to t h e s u r r o u n d i n g air, in m per 6ec, w h i c h is a p p r o x i m a t e l y equal to t h e s p e e d of th e v ehicle, v, in m per sec

T h e r o l l i n g r e s i s t a n c e , F^, is

F = C • m • cosi & G • m

r r r

w n e r e

= a const a n t w h i c h is d e p e n d e n t on the r o a d s u r f a c i n g a n d on the ty p e of t yre

T h i s e q u a t i o n d i s r e g a r d s that c o m p o n e n t of the r o l l i n g r e s i s t a n c e , F^, w h i c h is d e p e ndent on the speed.

(7)

If the e x p r e s s i o n s for F, F^, and F are s u b s t i t u t e d in Eq (1), th e n w e obtain

p

Ck • A • v

n

a

2

- C

r.

• m - m g s m i = m-r-

dv

v 1 r ° dt Thi s e q u a t i o n can be w r i t t e n

a

i

Ci .

A

0

dv _ P 1 1 . ,0 v jT - — • — - --- * v - C - g • sxn i ... (2) dt m v m r Eq (2) is the e q u a t i o n of m o t i o n of a ve hicle.

T h e first h y p o t h e s i s m a d e for the p resent i n v e s t i g a t i o n w as that t h e po wer output, P, in Eq (2) m i g h t b e r e g a r d e d as a p p r o x i m a t e l y constant. F u r t h e r ­ more, it w a s s u p p o s e d that an a p p r o p r i a t e c l a s s i f i c a t i o n of v e h i c l e s in

groups w o u l d m a k e it p o s s i b l e to ass u m e a c o n s t a n t v a l u e of the ” c o e f f i c i e n t C 1 ' A

of air f r i c t i o n ” --- ~ — , for eac h group of v e h i c l e s . T h e c l a s s i f i c a t i o n of v e h i c l e s a d o p t e d for t h i s p u r p o s e is g iven in w h a t follows.

Gro up 1. P r i v a t e ca rs Gr oup 2. T w o - a x l e lo rries

Gro u p 3. T h r e e - a x l e and f o u r - a x l e lorries Gr oup 4. F i v e - a x l e or m u l t i - a x l e lorries

T h e m o d e l d e s c r i b e d in t h e a b o v e c o m p l i e d w i t h t h e r e q u i r e m e n t that it s ho u l d ena b l e each i n d i v i d u a l v e h i c l e to b e c h a r a c t e r i s e d in a s u f f i c i e n t l y

P

simple manner, nam ely , by a v a l u e of the r a t i o — an d by the n u m b e r of h e a v y

1)

m

axles of the vehicle, w h i c h giv e the v a l u e of t h e co e f f i c i e n t of air C 1 • A

f r i c t i o n ” -— — — . In addition, it is n e c e s s a r y to k n o w the spee d of the v e h i c l e at the e n t r a n c e i n t o t h e upgrade.

1) Ax l e s c a r r y i n g a xle load s w h i c h are equal to, or grea t e r than, 2 m e t r i c tons

(8)

TEST OF THE MODEL

In order to test this model, the variation in the speed with the distance

was studied on 18 upgrades, which differed in slope and in length.

All

these sites of observation are located on Europe Roads (E Roads), and the

total data covered about 1000 passenger cars and 2500 heavy vehicles.

The measurements were made in the spring and in the summer of 1968.

The

variation in the speed with the distance was recorded by means of a

special travel time measuring equipment.

This equipment comprised a

number of time recorders, which were connected to detectors placed, on the

carriageway.

The detectors consisted of thin iron wires, which were

scarcely perceptible to road users.

Fig. 2 shows an example of the

location of the detectors on a vertical profile.

9

Fig. 2 .

Example of location of vehicle detectors on an

upgrade

Thus, the time recorders furnished data on the travel times over

consecutive road sections.

The entrance and exit speeds, as well as the

total travel time over the upgrade, were used to calculate the mean value

of the power output developed per unit mass of vehicle, and averaged,

over this total travel time.

This mean value was then used as a supposed

constant in the assumed model.

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After that, at the points where the time and the speed had been measured,

their observed values were compared with the corresponding values

calculated with the help of the model.

The agreement between the

respective values was found in all cases to be very close.

An example

of this comparison is shown in fig. 3.

These graphs represent the 25-,

50- and /5-percentiles in tne distributions of the percentual differen­

ces in speed and in time at those points of the upgrade at which the

speed and the time have been measured.

This example refers to the same

location of the detectors on the upgrade that is shown, in fig. 2.

Road

co-ordinate

[m]

600

500

400

300

200

100

t

-4

m

v

4

100

%

Road

co-ordinate

[m]

600

500

400

300

200

100

*—&

j

•T**®'“r—

r “trj

§-t

1

S+ m

1

m

4

100

%

m

m

Fig. 3 ,

Example of graphical comparison of calculated and

observed values of the speed and the travel time at points

defined by different road co-ordinates

A

B

C

f

— )

A

ä

25-percentile

B = 50-percentile

0 ~ 75-percentile

Vm = ^Pe€c^ calculated from the model

v - Observed speed

t s Time calculated from the model

t ~ Observed time

u

Average gradient of the upgrade 54 per mil

Number of vehicles 57

(10)

APPLICATION OF THE MODEL

Some results which can be obtained from the model are exemplified. The

first examples show speed-distance curves computed from the theory on

various grades (see fig 4-7).

These examples are related to the

median vehicle in each group.

In the diagrams the speeds at the entrance into the upgrade are 100 k

m/l

for passenger cars and 75 krn/h for heavy vehicles.

The curves can also

be used for entrance-speeds lower than the values mentioned.

Gradient

0

d / o o

18 o/oo

20 o/oo

30 o/oo

40 o/oo

Speed

[km

/h

j

100

50 o/oo

60 o/oo

70 o/oo

80 o/oo

200

400

600

800

1000

Distance [m]

Fig. 4 .

Speed-distance curves for the median vehicle of

passenger cars

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Speed

[km/h

Gradient

0 o/oo

18 o/oo

20 o/oo

30 o /oo

40 o/oo

50 o/oo

60 o/oo

70 ö/oo

80

o!oo

200

400

600

800

1000

Distance [tn]

Fig.

5 .

Speed-distance curves for the median vehicle of

two-axle lorries

60

40

20

o/oo

o/oo

o/oo

o /oo

o/oo

ö/oo

o/oo

200

400

600

800

1.000

Distance fm1

Fig. 6. Speed-distance curves for the median vehicle of

three-axle and four-axle lorries

Speed

[km/h'

(12)

Speed

[km/h]

60

40

20

Gradient

Ö o/oo -

5 o/oo

20 o/oo

30 o/oo

40 o/oo

50 o/oo

60 o/oo

70 o/oo

80 o/oo

1000

g,

«

7.

Speed~distance curves for median vehicle of five-axle

and multi-axle lorries

Another example relates to a whole population of free-moving heavy lorries

which is typical of traffic on Europe Roads in Sweden.

The upgrade in

question has a gradient of 60 per mil, and is I 200 m in length.

Further­

more, this example is based on the simplified assumption that all vehicles

travel at the same speed, 75 km/h, at the entrance into the upgrade.

On

the basis of these data, fig. 8 represents the 10-, 25-, 50-, 75-, and

90-percentiles in the speed distribution for any arbitrary points on the

upgrade.

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Sp eed

Tkm/h]

D i s t a n c e fm]

Fig, 8 ,

Speed-distance graph for an upgrade.

Population of

heavy lorries.

Breakdown of lorries by number of axles:

Two-axle

39 per cent

Three-axle or four-axle

32 per cent

Five-axle or multi-axle

29 per cent

Gradient of the upgrade 60 per mil.

(14)

Figure

Fig.  2 . Example  of  location  of  vehicle  detectors  on  an  upgrade
Fig.  3 , Example  of  graphical  comparison  of  calculated  and observed values  of  the  speed  and  the  travel  time  at  points defined by  different  road  co-ordinates
Fig.  4 . Speed-distance  curves  for  the median vehicle of  passenger  cars
Fig.  5 . Speed-distance  curves  for  the median vehicle  of  two-axle  lorries 60 40 20 o/ooo/ooo/ooo /ooo/ooö/ooo/oo 200  400  600  800  1.000 Distance  fm1 Fig
+2

References

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