Turning an articulated truck on a spreadsheet
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The challenge that is addressed by this paper came
about in a very straightforward way. A technical colleague from
the Irish Department of Transport asked the author how much clearance
to the front and to the side would be required for an articulated
truck to leave a testing lane and do a U-turn. This kind of information
would typically be required where a new articulated-truck-testing
lane was to be set up. A more general example of the same type of
problem would involve checking whether a given articulated truck
could negotiate a particular turn or roundabout. Articulated trucks
come in a wide range of sizes and configurations, and it soon became
apparent that even the identification of the worst-case scenario
would require detailed consideration of a range of different cases.
On the face of it, at least some aspects of the problem seemed suitable
for solving by the use of a standard and ubiquitous spreadsheet
application. The functionality of the spreadsheet that was developed
is illustrated by a short video clip.
Microsoft Excel spreadsheet workbook was developed that contained
two worksheets: ‘Dimensions’ and ‘Paths’.
The former allowed the dimensions of the articulated truck to be
specified and displayed to scale in plan view. The latter allowed
steering information to be inputted, and the loci of principal points
on the tractor and trailer to be plotted to scale on a diagram.
A simple animation of the movement of the articulated truck was
also included on the Paths sheet.
The defining dimensions for an articulated truck are shown in Table
1. For the most part these are self-explanatory, although the
term ‘axle group’ needs a little explanation.Tractors
commonly have either one or two rear axles. Multiple axles can carry
a greater load. If the number of rear axles is more than one, there
will be some tyre dragging whenever a tractor turns. Because of
this, some tractor units have the facility to raise one of the rear
axles when the tractor is lightly loaded. In terms of turning behaviour,
a group of rear axles can be represented by an equivalent single
axle halfway between the first and the last of the active axles
in a rear axle group. The turning characteristics of a tractor depend
on the distance between the front steering axle and the centre of
the rear axle group; for example, raising the frontmost of two tractor
rear axles will lengthen the effective wheelbase for turning purposes.
Semi-trailers can have a single rear axle or a rear axle group consisting
of two, three or occasionally four rear axles. One or more of these
could possibly be raised at light load to reduce tyre wear. As in
the case of the tractor unit, for turning analysis purposes the
rear axle group of a semi-trailer can be represented by an equivalent
single rear axle that would be in the middle of the group of rear
axles that are in use.
The ‘maximum steering angle’ is the maximum angle between
the inside steering axle wheel and the centre-line of the tractor.
The steering axle wheel on the inside of a turn always makes a greater
angle with the centre line of the tractor than the steering axle
wheel on the outside of the turn, as can be seen in Figure
1. The greater the maximum steering angle, the shorter will
be the centre line turning radius of the tractor. Another significant
dimensional parameter is the ‘Steering axle width between
contact centres’. This is the distance between the points
where the king pin pivot axes of the steering axle intersect the
road. It is approximately equal to the distance between the centres
of the front tyre contact patches on the road. Table
2 lists truck dimensions that can be calculated from those listed
1. These data were calculated automatically within the Dimensions
sheet of the workbook by the use of the appropriate formulae.
The rear axle group of the tractor rotates about a turning centre
as illustrated in Figure
1. The ‘minimum centre-line turning radius’ of the
tractor is calculated from the tractor dimensions, including the
maximum steering angle: it is the distance from the centre of the
rear axle group to the current turning centre. The ‘minimum
cab front corner turning radius’ is the distance from the
outside front corner of the cab to the current turning centre.
The ‘trailer angle’ describes the angular relationship
between the tractor and the semi-trailer: it is zero when the trailer
is straight in line with the tractor. Figure
2 and Figure
3 illustrate different angular positions of the trailer. These
diagrams were in fact XY scatter plots of data generated from the
input dimensions and the trailer angle. The diagrams updated automatically
when any of the input data changed. If any changes were made to
the X- or Y-scales it would have been necessary to use the drag
handles of this graph to ensure that the grid boxes remained square.
The main reference locations of the articulated truck were also
identified in these figures. The origin of the articulated truck
was taken to be the centre of the rear axle group of the tractor.