Modelling performance and retarder chart of off-highway trucks by cubic splines for cycle time estimation

Abstract

Cycle time is a crucial parameter for determining the number of trucks in the transportation system of an open pit mine. During the development of surface mining projects, the haulage roads are planned and the road conditions are predicted. Accordingly, the time that the trucks may spend on that haulage road is estimated. The performance and retarder charts installed within the cabs of the trucks are widely used to calculate the speeds and the time spent for particular road conditions. In this study, a computer-aided system is developed to estimate the truck speed, cycle time and truck number. Speed data for different resistances are read from truck performance and retarder charts and used to build a database. The database with total resistance and speed pairs is modelled by the cubic spline interpolation method. After modelling, the developed system could estimate the speed for any given total resistance. As the road distance is known, the time that any truck would spend travelling on this haul road may be calculated. The approach is coded in C++ and applied to a real overburden stripping mine panel. The results obtained show that the system is convenient for chart modelling and cycle time calculation. Also, the developed system can be used as a part of truck dispatching and simulation system.

Full text

MODELLING PERFORMANCE AND RETARDER CHART OF OFF-HIGHWAY
TRUCKS BY CUBIC SPLINES FOR CYCLE TIME ESTIMATION
KAAN ERARSLAN
Dumlupinar University, Department of Mining Engineering, Kutahya, Turkey
Cycle time is a crucial parameter for determining the number of trucks in transportation system of an
open pit mine. During development of mine projects, road paths are planned and road conditions are
predicted. Accordingly, time that the trucks may spend on that path is estimated. The performance
and retarder charts of the trucks are widely used to calculate the speeds and dependently, time spent
for particular road conditions. In this study, a computer system is developed to estimate the truck
speed, cycle time and truck number. Speed data for different resistances are read from truck
performance charts and used to build a database. The database with total resistance and speed pairs
is modelled by cubic spline interpolation method. The developed system estimates the speed for any
total resistance. As the road distance is known, time that truck will spend is calculated. The approach
is coded in C++ and applied to a real overburden stripping mine panel. The reasonable results reveal
that the system is convenient for chart modelling and cycle time calculation. Also, the developed
system can be used as a part of truck dispatching and simulation system.
1.Introduction
Material transportation by off-highway trucks is one of the most widely applied haulage
method in open pit mines due to their high mobility and flexibility in relatively short
distances. In a large-scale mine, more than hundred trucks may be employed in a truck fleet.
Definitely, these fleets require huge capitals. So, calculation of number of trucks is crucial
from economical point of view. One of the most important factors affecting number of trucks
is total cycle time, which is the time span required by a truck for being loaded, transporting,
dumping, and manoeuvring processes. Cycle time is affected by road conditions such as
distance, grade, various resistances, etc. As a matter of fact, number of trucks in a fleet is
closely related with the cycle time and thus, speed of trucks for particular road conditions.
In mine planning stage, road profiles are predetermined and characteristics of road base,
grade, etc. are estimated (Figure 1).
Figure 1. A mine road profile between load and dump areas.
Before truck fleet sizing, a simulation of transportation system helps for the decision about
truck cycle time and hence, their number. For this purpose, performance and retarder charts of
Load area
Dump area

trucks published by truck manufacturers could be used for the calculation of speed and total
cycle time. Performance chart shows the velocity that truck could have when it is operating
against positive total resistance. Retarder chart is employed when truck is operating with
negative total resistance. Total resistance consists of resistance due to road conditions and
grade1.
TR = RR + GR (1.1)
where,
TR = total resistance applied to truck (%)
RR = rolling resistance (%)
GR = grade resistance (%)
Rolling resistance is summation of resistances caused by mainly tire penetration, internal
friction and tire flexing2. Rolling resistance values for different road conditions are given in
Table 1.
Table 1. Rolling resistance values for different road conditions3.
Road condition
Rolling resistance (%)
Concrete, asphalt, no tire penetration
2
Stabilized road, very slight tire penetration
2
Compressed, tight, slight penetration
3
Tire penetration about 5 cm
5
Tire penetration about 10 cm
8
Loose (sand, pebble, gravel)
10
After determining total resistance for a specific part of road, as an example, performance chart
shown in Figure 2 can be used.
The vertical axis on the right hand side includes crosswise lines of total resistance, the upper
horizontal axis shows weight of truck and the lower horizontal axis represents speed. For a
known total resistance corresponding speed is read such that, according to truck is loaded or
empty, vertical dashed line is intersected with total resistance and closest gear curve is
coincided. Intersection point is projected down to lower horizontal axis and speed is read.
After reading speed, time spent can be calculated simply by below familiar equation;
T = D / S * 3600 (1.2)
where,
T = time (sec)
D = distance (km)
S = speed (km/h)

Figure 2. Off-highway performance chart9.
Number of trucks required for that road profile can be calculated by using the cycle duration;
where,
n = number of trucks needed for an excavator (or loader)
ti = time spent in ith part of road
P = number of parts in road profile
L = loading and spot time (= time span for manoeuvring into position for loading).
In this study, a computerized modelling and estimation system has been developed to model
truck performance and retarder charts and estimate truck cycle time for particular road
conditions. The system utilizes cubic spline method for modelling the database consisting of
total resistance-speed data pairs. So first, performance and retarder charts are read for empty
and loaded truck cases for different total resistances. Then, total resistance-speed data pairs
are introduced to the modelling utility of the system. The system can estimate speed of truck
and time spent for specific road profile. The estimation approach is based on cubic spline
interpolation, which is one of the most reliable and effective interpolation methods of
numerical analysis. The system has been applied to a real case for validation.
L
t
n
P
ii


1
(1.3)

2. Cubic Spline Interpolation
The developed system utilizes cubic spline interpolation for modelling and estimation.
Physical spline is a long and narrow strip of plastic used to fit curve through specified points
and shaped by lead weights called as ducks4. In mathematics, spline is a high-quality curve
fitting technique. Lagrange interpolation polynomials, Newton Divided Difference methods
are other frequently used curve and model fitting methods. However, cubic spline is superior
because it provides continuity at data points at their first and second derivatives4,5,6. The
algorithm is summarized below.
Cubic spline interpolation is based on the use of a cubic polynomial in each interval
between two consecutive data points5,6. Considering an interval ti

t

ti+1 having span length
of

li = ti+1 - ti and a local co-ordinate x= t - ti, a cubic polynomial for the interval can be
written such that6 ;
p(x) = c1 + c2 x + c3 x2 + c4 x3 (2.1)
In Fig. 3, interpolation range is presented.
Figure 3. Range of cubic spline interpolation.
Initially, p(x) is equalized to known value of f(x) function at x=0 and x=

li,
fi = c1 (2.2)
fi+1 = c1 + c2

li + c3

li 2 + c4

li 3 (2.3)
Here, fi and fi+1 are known values at x=0 and x=

li, respectively6. Additionally, p' and p'' are
continuous at i and i+1. First and second derivatives of function p can be represented as pi'
and pi'' at point i. Second derivative of function given in Eq.2.1 is,
p''(x) = 2c3 +6c4x (2.4)
At i and i+1, p'' becomes,
pi'' = 2c3 (2.5)
pi+1'' = 2c3 + 6c4

li (2.6)

li-1 = ti - ti-1
fi-1

li = ti+1 - ti
fi
fi+1

Coefficients c3 and c4 can be derived in terms of pi'' and pi+1'' such that,
2
3i
p
c

(2.7)
i
ii l
pp
c




6
1
4
(2.8)
Coefficient c1 has been given in Eq.2.2 as fi. The remaining one c3 can be found in Eq.2.3 by
using Eq.2.2, Eq.2.7and Eq.2.8,
i
ii
i
ii l
pp
l
ff
c






 6
2
11
3
(2.9)
Hence, general formula of cubic polynomial is 6,
3
1
2
11 )(
6
)(
2
)(
6
2
)( x
l
pp
x
p
xl
pp
l
ff
fxp
i
iii
i
ii
i
ii
i


















 
(2.10)
Harrington7, Rogers and Adams4, Keryszig5 state the same procedures with different
notations.
Next, first derivative of polynomial function p can be written at x=0 and x=

l,
   
ii
i
ii
i
iff
l
pp
l
p







 11 1
2
6
(2.11)
   
ii
i
ii
i
iff
l
pp
l
p







 111 1
2
6
(2.12)
Here,

l = ti+1 - ti. In between ti-1 < t < ti, pi' becomes,
   
1
1
1
11
2
6










ii
i
ii
i
iff
l
pp
l
p
(2.13)
Here,

li-1 = ti - ti-1. For the continuity of first derivative, an equity using Eq.2.11 can be
written,

































 1
1
1
1
1111 1111
6)22( i
i
i
ii
i
i
iiiiiii f
l
f
ll
f
l
plpllpl
(2.14)

Above equation can be applied to internal points except the two ends. For the end points, an
extrapolation should be performed such that, a set of equations for i=0,...,N, can be written 6,
 
002
1
1
10
0
0
20110 1111
622 plf
l
f
ll
f
l
plpll 





























 
111
1
1
1
1111 1111
622 



 






























iii
i
i
ii
i
i
i
iiiiii plf
l
f
ll
f
l
plpllpl
(2.15)
 
NNN
N
N
NN
N
N
NMNNN plf
l
f
ll
f
l
pllpl 



































1
1
12
2
2
11221 1111
622
Solution of the equation set provides pi''. However, boundary conditions should be specified
either by prescribing or extrapolation. In this study, extrapolation is preferred for
practicability. The p0'' can be extrapolated by 6,
p0'' = 2p1'' - p2'' (2.16)
At the boundary,
pN'' = 2pN-1'' - pN-2'' (2.17)
Using above equations, the set of equations becomes,
 
210
2
12
6
6fff
l
p



(2.18)
 
321
2
321 2
6
4fff
l
ppp 







(2.19)
 
432
2
32
6
6fff
l
p



(2.20)
Here, p1'' and p3'' can be calculated quickly from Eq.2.18 and 2.20. p2'' is determined by
Eq.2.19.
The developed program has been employed to model given database (total resistance-speed)
and estimate speed for any given total resistance.

3. Modelling the Charts by Cubic Spline Method
The developed system is utilized for chart modelling and estimation of truck speed, time
needed for particular road parts and number of truck estimation. The flowchart of the system
is given in the Figure 4.
Figure 4. The flowchart of the system.
Initially, the system generates a cubic spline interpolation function for total resistance (%)
and corresponding speed values (km/h) of several truck models. Besides, it estimates the
speed of truck for any given grade and rolling resistance values in terms of degrees, %. The
system also calculates time spent for a particular road part and total cycle time as well. After
determining the total cycle time, number of truck is simply found.
The developed system has been applied to 9 models of Caterpillar off-highway trucks
having capacity of 36.8 to 218 tons. Table 2 summarizes some properties of trucks.
input: total
resistance &
speed data
Model fitting by
cubic spline
interpolation
Grade (%) &
Rolling
resistance (%)
Truck
chart
database
Speed estimation
from truck chart
database
Time calculation
for road parts
and total cycling
Calculation of
number of trucks

Table 2. Truck property 9.
Model
Weight (empty)
(kg)
Max. Carrying Capacity
(kg)
Max. Speed
(km/h)
Cat 769-D
31.250
36.800
75.0
Cat 771-D
33.975
40.000
56.3
Cat 773-D (24R)*
40.188
52.300
66.0
Cat 773-D (21R)*
40.188
52.300
66.0
Cat 775-D
43.953
60.000
66.0
Cat 776-D
64.359
90.000
60.0
Cat 784-B
96.353
136.000
56.0
Cat 789-B
121.922
177.000
54.0
Cat 793-C
146.937
218.000
55.0
*These are the same models with different tire size
Speeds from performance and retarder charts of off-highway trucks have been read once
for total resistances of 2 % to 10 % and a database has been formed. Table 3 represents the
readings from the performance charts for empty truck case. Similarly, readings for loaded and
continuous grade retarding also take place in the database.
Table 3. Speed of Caterpillar off-highway trucks from performance charts (empty).
(km/h)
Model
Resist., %
Cat-
769D
Cat-
771D
Cat-
773D
24R
Cat-
773D
21R
Cat-
775D
Cat-
776D
Cat-
784B
Cat-
789B
Cat-
793C
2
75.0
56.3
66.0
66.0
66.0
60.0
56.0
54.0
53.8
4
71.8
54.3
65.0
60.1
64.3
62.1
51.9
53.7
52.4
6
53.2
50.7
57.1
55.4
55.0
51.1
46.2
48.2
50.0
8
41.8
37.5
41.2
41.2
42.3
42.1
33.9
35.6
37.1
10
33.9
30.0
35.0
35.0
32.3
32.9
27.3
27.4
31.4
The system models input resistance-speed data. Figure 5 represents the fitted curve for CAT-
769D empty truck after modelling the data by cubic spline interpolation.
Figure 5. Model fitted for CAT-769D.

It is obviously seen that as the total resistance increases the speed decreases. After
modelling the given data, the system can also estimate corresponding speed for given grade
and rolling resistance. Besides speed, time that is spent at the road part can be calculated as
well.
After forming and modelling database, the system estimates truck speed for any given total
resistance by cubic spline modelling and enable calculation of truck cycle time and number of
truck that should be operated with an excavator or loader. The only what to give to the system
is road profile, grade and resistance conditions of road parts.
4. Case Study
The developed system and database has been applied to the Garp Lignite Enterprise,
Tuncbilek, Civilicam C-1 stripping panel in Kutahya, Turkey. Road profile consists of mainly
three parts (Figure 6).
Figure 6. Profile of panel.
Regarding Table 1, the road has slight penetration and the rolling resistance is found to be 3
(RR=3). The system calculates corresponding speeds and dependently, duration of haulage for
each part of road profile, which are presented in Table 4 and 5 for empty and loaded cases,
relatively.
Table 4. Speed and duration of travel for loaded truck (haulage).
Model
Cat-
769D
Cat-
771D
Cat-
773D
24R
Cat-
773D
21R
Cat-
775D
Cat-
776D
Cat-
784B
Cat-
789B
Cat-
793C
Part
1
Speed
(km/h)
25.70
20.71
23.21
22.30
20.71
22.14
18.46
18.46
19.29
Time (sec)
35.72
44.33
39.55
41.17
44.33
41.46
49.73
49.73
47.59
Part
Speed
(km/h)
41.00
37.14
40.71
40.05
33.57
33.21
30.77
32.31
35.00
2
Time (sec)
43.90
48.46
44.22
44.94
53.62
54.20
58.50
55.71
51.43
Part
Speed
(km/h)
20.71
19.28
18.93
19.23
17.50
17.88
14.62
14.62
16.43
3
Time (sec)
43.46
46.68
47.54
46.80
51.43
50.34
61.56
61.66
54.78
Load Area
Dump Area
255 m
500 m
250 m
Grade= 3.9 %
Grade= 1.0 %
Grade= 5.0 %

Table 5. Speed and duration of travel for empty truck (return).
Model
Cat-
769D
Cat-
771D
Cat-
773D
24R
Cat-
773D
21R
Cat-
775D
Cat-
776D
Cat-
784B
Cat-
789B
Cat-
793C
Part
1
Speed
(km/h)
75.00
56.30
66.00
66.00
66.00
60.00
56.00
54.00
55.00
Time (sec)
12.24
16.31
13.91
13.91
13.91
15.30
16.39
17.00
16.69
Part
Speed
(km/h)
75.00
56.30
66.00
66.00
66.00
60.00
56.00
54.00
55.00
2
Time (sec)
24.00
31.97
27.27
27.27
27.27
30.00
32.14
33.33
32.73
Part
Speed
(km/h)
75.00
56.30
66.00
66.00
66.00
60.00
56.00
54.00
55.00
3
Time (sec)
12.00
15.99
13.64
13.64
13.64
15.00
16.07
16.67
16.36
In order to complete other time parameters of total cycle time such that loading, spot,
dumping time, necessary data are given in Table 6.
Table 6. Time parameters of a cycle excluding haulage and return. (sec)
Model
Parameter
Cat-
769D
Cat-
771D
Cat-
773D
24R
Cat-
773D
21R
Cat-
775D
Cat-
776D
Cat-
784B
Cat-
789B
Cat-
793C
Load
79
82
87
93
99
103
105
111
114
Spot
17
18
18
19
19
22
23
25
26
Dump
62
64
65
67
72
72
73
78
81
Manoeuvre
33
34
34
35
35
35
36
38
39
Total
191
198
204
214
225
232
237
252
260
Here, excavator capacities are proportional with truck capacities. So, in the calculations, the
same loading time could have been used. The system can utilize this information and calculate
total cycle times and number of trucks/excavator (Equation 3) as given in Table 7.
Table 7. Cycle times of off-highway trucks
Model
Cat-
769D
Cat-
771D
Cat-
773D
24R
Cat-
773D
21R
Cat-
775D
Cat-
776D
Cat-
784B
Cat-
789B
Cat-
793C
Total Cycle
Time (sec)
362.32
401.74
390.13
401.73
429.20
438.30
471.39
486.10
481.05
Truck Per
Excavator
3.77 (4)
4.02 (4)
3.72 (4)
3.59(4)
3.64 (4)
3.51 (4)
3.6 (4)
3.57 (4)
3.43 (4)
Decimals are rounded to integer values. Besides, for each excavator, a spare truck can be
added to fleet. According to Table 7, 4 trucks per excavator should be used for all truck

models. Even they look to be identical in number, their excavator capacities are different and
convenient with truck models. If excavator capacities were the same, load time would
increase and number of trucks would decrease. Obtained results are reasonable and validate
the approach. It has been revealed that using the developed system, truck fleet size can be
determined.
5. Conclusion and Discussion
In this research, cubic spline interpolation method is utilized for modelling performance and
retarder charts of off-highway trucks. The aim is to generate a system that can calculate the
speed that truck can perform for specified road conditions and cycle time between load and
dump areas. Using time needed for a cycle of a truck, number of truck per excavator can be
calculated which is crucial from economical point of view. The developed system estimates
speed and dependently, duration. Thereafter, number of truck, needed for an excavator or
loader can also be calculated. In the study, a database including Caterpillar off-highway trucks
are formed, a C++ coded system is generated and applied to a real case. The results reveal that
performance and retarder charts can be modelled by cubic spline polynomials and can be
utilized for determining truck fleet size. The system could also be integrated to a truck
dispatching or transportation system regarding cycle time calculation function.
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3. S., Saltoglu, Open Pit Mines, Istanbul Technical University Publications, No 1472,
Istanbul, 1992, 208 p.
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5. Kreyszig, E., Advanced Engineering Mathematics, John Wiley and Sons, Inc., 7th ed.,
New York, 1993, 1271 p.
6. S. Nakamura, Applied Numerical Analysis in C, Ptr Prentice Hall, 1993, 604 p.
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