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Offtracking Calculation

Offtracking Calculation
By:  Craig Luker 
January 26, 2012

Definitions

Offtracking
- the difference in the paths followed by the frontmost and rearmost axles of a vehicle.  i.e. how far the rearmost axle tracks inside or outside the frontmost axle around a curve.
- OT is positive if it tracks inside and negative if it tracks outside.

Tractrix - the path followed by a trailing point on a body (i.e. a vehicle) when another point follows a curve and the first point is not allowed to slip sideways.

Tire Cornering Stiffness - a parameter defining how much sideways force is developed for a tire for every degree of sideslip (ex. Lbs/deg).
- Affected  by the tire’s construction and materials (large differences between truck tire makes and models) and by the load carried by the tire.

Sideslip or slip angle - difference between the direction a tire is moving and the direction that it is facing.  Unless the vehicle is going straight, this is never zero, but it is often small.

Physics

The physics of a turning truck (whether or not it is articulated) is very complex.  The exact path it follows is affected by these math parameter:

M,
friction between tire and road.
Cx, 
cornering stiffness
x, 
slip angle.
Geometry of truck, wheel base, pivot locations, spacing of axles within
the axle group, etc.
v,  
speed.
Roll Angle 
trailers roll slightly during a turn and this affects
the path they travel and the path of the tractor.
e,
superelevation of road.


Methods

A.  Kinematic Solution - kinematic means that we do not consider the mass of the vehicle, nor the forces acting on it.  Only the geometry is considered.  This approach is also called a "zero-spread" model.

  • Tractrix curves are the shape of the rearmost axle’s path found kinematically when the front axle follows a specified path.  This is the method used by FARO.  It has been used for many years and is accurate when the speed is low enough and the turn is sweeping enough that the trailer axles do not slip significantly sideways.

In practice this method has been shown to be accurate for situations such as:

    • Loaded logging trucks on gravel surfaces turning 50 -> 125 ft. radius turns at low speed.
    • Tractor-trailers (singles) on paved road turning 80 and 100 ft. radius curves at low speed.
    • Lane changes with lateral accelerations below about 0.25g (i.e. Not aggressive lane changes).
This method essentially reduces the vehicle to one wheel on each axle.  Despite the simplifications, it has been shown to be accurate in many typical situations.
  • The kinematic solution can be expanded to consider the effect of large spacing between axles within an axle group, different track widths on axles, etc.
    • The need to do this normally only arises when trailer axles are widely spaced and turn sharply.


B.  Kinetic Solution - Kinetic means that the mass and speed and forces are accounted for.  This is much more complex and requires detailed knowledge of the tires, etc.

Simplified kinetic solutions need tire stiffness, friction, and load on each tire.

More complex kinetic solutions need full details of the vehicle’s suspension.  Information that is not normally available in an accident reconstruction.

The more complex models do account for speed and can even predict rearward amplification and outbound offtracking.

Effect of Speed

Due to geometry, the trailer will track inside of the steering axle.  However, speed tends to make the trailer swing outward and, therefore, outside the steering axle path (due to centrifugal force).

Critical Speed - speed at which the trailer tracks right on the steering axle path (and thus offtrack is zero).

- For typical configurations on pavement this speed is between 32 km/h and 76 km/h.


Guidelines for Using FARO HD Articulated Vehicle Calculation

Feel confident in the simulation result when:

- Speeds are low, such as normal driving around corners in the city.

- Driving in a parking lot or loading bay unless the maneuver is so tight that the tractor steering is very sharp and/or any tires would be expected to slide, scrub, or slip.

- Calculating gentle lane changes.

- The actual offtracking toward the inside of the curve when a truck goes around a highway curve will always be less than the offtracking shown by the software.  As such, the simulation can be used to give the maximum inward offtracking in these situations.
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