MODEL FOR CALCULATING THE EFFECT OF
LONGITUDINAL PROFILES ON THE SPEED
OF HEAVY VEHICLES
by
G. Carlsson
MODEL FOR CALCULATING THE EFFECT OF
LONGITUDINAL PROFILES ON THE SPEED
OF HEAVY VEHICLES
by
G. Carlsson
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 7I 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 .
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
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.
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 • vn
a2
- Cr.
• 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 na
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
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.
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
ä25-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
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 o18 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
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'
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.
Sp eed
Tkm/h]
D i s t a n c e fm]