^{1}

^{*}

^{2}

^{3}

Errant vehicles occur as a result of the driver losing control of the vehicle. This may be due to sudden illness, dozing off or skidding while attempting a manoeuvre. In containing such an errant vehicle on a highway, the priority is to avoid collision with other vehicles. A sloped highway median provides a run-off area for such vehicles where the vehicle can be slowed down and stopped without the danger of being re-directed into the path of other vehicles as may occur with edge barriers. Here, the effect of a containment barrier at the bottom of the sloped median is studied with a view to prevent the vehicle from being redirected outside the median after colliding with the barrier. The focus of this work is on the change of kinematic states due to the collision, so a momentum-based vehicle collision analysis is developed, with the collision energy loss related to the vehicle stiffness being considered by coefficient of restitution. The average maximum lateral displacements post-collision are read from the diagram of vehicle x-y trajectories. In this way, the most suitable median slope 1:6 is selected.

A sloped highway median provides a run-off area for errant vehicles where they can be slowed down and stopped without the danger of being re-directed into the path of other vehicles as may occur with edge barriers. According to Gabler et al. [

The research on Highway Median Safety in recent years, with wide application of simulation software, has mainly focused on two themes: one is investigation of double-sloped median terrain’s influence on vehicle behaviour [^{TM} [^{TM} in such research was validated by Uzunsoy et al. [

The results of these researches indicate that in the study of errant vehicle behaviour in a double sloped median, the profile and geometric dimension of the median play decisive roles, and using simulation software of either FEA or Multi-body Dynamics is effective. The focus of this paper is to determine the most suitable median slope to prevent the vehicle from being re-directed outside the median after collision with the containment barrier. A time-saving method is proposed to simulate the kinematic change of the errant vehicle during collision with the containment barrier while CarSim^{TM} is used to simulate the vehicle’s trajectory before and after the collision.

CarSim^{TM} provides a reliable simulation environment for the dynamics of an errant vehicle as its non-linear Multi-Body Dynamics vehicle model has been shown to reflect the vehicle dynamics property of real vehicles running on the sloped terrain surface of a road median [

Various types of mathematical vehicle collision model have been developed by researchers for two purposes, vehicle collision accident reconstruction, and vehicle post-collision active safety control strategy development [^{nd} laws, has been employed in the software PC-crash which specialises in vehicle collision simulation or reconstruction [^{nd} law [

In this study, the kinematic change of the vehicle during the collision with the rigid containment barrier is approximated by using the collision model based on planar impact mechanics. Thus a hybrid simulation platform is used consisting of a multi-body dynamics simulation for the pre- and post-collision phases (using CarSim^{TM}), and planar impact mechanics (using MATLAB^{TM}). The general structure of the simulation, shown in

The collision model based on planar impact mechanics developed by R.M. Brach [^{TM} is incapable of simulating the collision of the vehicle with rigid barrier. Such momentum-based collision model has already been applied in the commercial software PC-crash [

and vehicle-vehicle collision [^{nd} law [

The following assumptions have been established for the application of the collision model:

1) The plane on which the vehicle is running (motion plane) just before the collision and the motion plane right after the collision are assumed to be the same.

2) The common contact surface (

3) The level of force acting over the common contact surface is significantly higher than other forces, such as aerodynamic force, tire-roadway friction, etc. So impulses of all forces other than contact forces are neglected [

4) The duration of the contact force impulse is very short, which implies that during contact, accelerations are high to the extent that velocities change suddenly and displacements (changes in position and orientation) are negligible [

5) Though the deformation during the collision is inevitable, the value of mass and moments of inertia about the Z coordinate, and the position of the sprung mass centre relative to the vehicle are assumed to be unchanged.

6) Coefficient of restitution e is assumed to be in the range of 0.1 - 0.3 [

7) Larger absolute values of pre-collision yaw-angle (collision angle), produce more severe inelastic deformation of the vehicle, which leads to a smaller value of e [

8) Impulse ratio μ can be approximated by an equivalent coefficient of friction. Since it cannot be determined experimentally or estimated by means such as analytical modelling of the mechanism of tangential force generation, it is taken as the coefficient of dynamic friction between the common materials of the barrier and car body for approximation [

Based on Assumption 1, planar impact mechanics including six pre-collision vehicle kinematic states ( v 1 x , v 1 y , ω 1 , v 2 x , v 2 y , ω 2 ) is applied to the collision model. These represent the pre-collision linear and rotational velocities of the vehicle and barrier in the earth-fixed coordinates.

The first three variables are known from the pre-collision simulation result (see Stage 1 in _{t}) and normal impulse (P_{n}).

Newton’s laws in the form of impulse and momentum are applied to the bodies as shown in

m 1 ( V 1 x − v 1 x ) = P t (1)

m 2 ( V 2 x − v 2 x ) = − P t (2)

m 1 ( V 1 y − v 1 y ) = P n (3)

m 2 ( V 2 y − v 2 y ) = − P n (4)

According to Newton’s second law, changes in angular momentum are equal to the moments of the impulses on each body. By taking moments of momentum about vehicle mass centre, two more equations about vehicle yaw rates before and after the collision are obtained:

I z 1 ⋅ ( Ω 1 − ω 1 ) = − d 1 cos ( φ 1 + θ 1 ) P n + d 1 sin ( φ 1 + θ 1 ) P t (5)

I z 2 ⋅ ( Ω 2 − ω 2 ) = − d 2 cos ( φ 2 + θ 2 ) P n + d 2 sin ( φ 2 + θ 2 ) P t (6)

Coefficient of restitution e is defined by Equation (7) [

e = − RelativeReboundVelocityNormaltotheContactPlane RelativeApproachVelocityNormaltotheContactPlane (7)

Equation (8) is obtained by applying Equation (7) to the vehicle and barrier in

e = − V C r n v c r n (8)

V C r n = Ω 1 d 1 cos ( φ 1 + θ 1 ) + V 1 y + Ω 2 d 2 cos ( φ 2 + θ 2 ) − V 2 y (9)

v c r n = ω 1 d 1 cos ( φ 1 + θ 1 ) + v 1 y + ω 2 d 2 cos ( φ 2 + θ 2 ) − v 2 y (10)

Collecting and manipulating Equations (1)-(10), and transforming them into matrix form leads to Equation (11). The values of known variables in Equation (11) are obtained from Vehicle Assembly Screen in CarSim^{TM}: From Assumption 2, m 1 equals to the summation of sprung mass and unsprung mass; I z 1 could be read directly; d 1 and φ 1 are calculated by Pythagorean theorem given the front track width and the horizontal distance between the wheel centre and mass centre; m 2 (kg) and I z 2 (kg∙m^{2} ) are assigned values of 10^{7}; d 2 shares the same value as d 1 , so do the φ 2 and φ 1 , which treats the barrier as a vehicle of the same size as, but far heavier than, the bullet vehicle [

Impulse ratio μ defined by Equation (12) measures how much tangential impulse is generated during the collision [

μ = P t P n (12)

An improper value of μ would produce unrealistic results of post-collision vehicle motion variables contrary to physical laws. Therefore, two criteria are imposed to validate the computation result.

The first criterion is to calculate the kinetic energy of the vehicle before and after the collision by using Equations (13) and (14). If the kinetic energy after the collision becomes higher than that before the collision, then the result of computation by using Equation (6) will be invalid.

T 1 pre-collision = 1 2 × m 1 ( v 1 x 2 + v 2 x 2 ) + 1 2 × I z 1 ω 1 2 (13)

T 1 post-collision = 1 2 × m 1 ( V 1 x 2 + V 2 x 2 ) + 1 2 × I z 1 Ω 1 2 (14)

The second criterion is to check that the impulse ratio assumed is less than the critical impulse ratio μ 0 when the vehicle and barrier reach a common tangential velocity―that is Equation (15). The vehicle is assumed to continue to slide over the contact surface throughout the contact duration, so the ratio of μ to the critical impulse μ 0 must be lower than 1 [

V 1 x − V 2 x − d 1 sin ( φ 1 + θ 1 ) Ω 1 − d 2 sin ( φ 2 + θ 2 ) Ω 2 = 0 (15)

Assembling Equations (1)-(13) into matrix form as shown in Equation (16) and using the definition of impulse ratio, it is possible to calculate μ 0 . If the ratio of μ to μ 0 is larger than 1, then the post-collision motion vector computed by Equation (11) will be invalid.

The values of

Equations (1)-(18) are implemented as MATLAB™ M-files for the computation of the collision model.

CarSim^{TM} is applied to simulate the vehicle motion before collision (pre-collision simulation): the vehicle is running at specified encroachment speed and encroachment angle before passing the median border; when the vehicle just passes the median border, all the control actions exerted on the vehicle are released, resulting in a control-free vehicle whose motion status is only subject to environmental factors such as gravity and terrain; the simulation is terminated instantly the vehicle contacts the rigid barrier. These conditions are implemented in CarSim^{TM} as VS commands. At this point the coordinates of the vehicle in the earth-fixed coordinates (x, y, z) are recovered as well as the vehicle’s dynamic state (velocities, orientation etc.) in the vehicle-fixed coordinates (X, Y, Z). The vehicle-fixed coordinate system is a moving coordinate system which is fixed to the centre of gravity of the vehicle whereas the earth-fixed coordinate system is a coordinate system whose origin is permanently fixed to the earth (see _{1x}, V_{1y}, ω_{1}) and used as input into the collision model to compute the post-collision kinematic state. The resulting post-collision kinematic state (V_{1x}, V_{1y}, Ω_{1}) is then transformed into the vehicle-fixed coordinates (V_{1X}, V_{1Y}, Ω_{1}). These, along with the pre-collision coordinates constitute the initial state of the vehicle for the post-collision simulation in CarSim^{TM}. Again, all controls are released for the post-collision simulation. The post-collision simulation needs to end when the vehicle just runs out of the median or contacts the barrier again. These conditions are again implemented in CarSim^{TM} as VS commands.

In this study, the effect of median slope on the effectiveness of the containment barrier has been investigated. Median slopes of 1:4, 1:5, and 1:6 [

The encroachment angles used were 5˚, 10˚, and 15˚ and the encroachment speed was chosen to be 37.55 mph (about 60 km/h). These values are representative of the values of typical vehicle encroachment parameters. Encroachment angles smaller than 15˚ cover 75% of vehicle encroachment accident cases [

speed interval 35.1 mph to 40 mph, which accounts for the largest proportion of the relevant traffic accidents (35%) among other intervals [

The vehicle encroachment simulation tests were carried out for four classes of vehicles, taken from the CarSim^{TM} database, namely: C-Class Hatchback 2012, D-Class Sedan, E-Class Sedan and Pickup. These represent a wide range of existing vehicle product, such as Audi A3, Ford Focus, Chevrolet Silverado, BMW 3 Series, BMW 5-Series, etc [^{TM} were kept.

From the result of pre-collision simulation, the collision angle

As described in Assumption 8, the value of μ could be related to the coefficient of friction when no experimental value is available. For approximation, the value of 0.3 is assumed in this study based on some sliding friction coefficient data [

The intermediate result refers to the kinematic variables (see

The first two derived variables are metrics for validating the vehicle-barrier collision. The results show that in all cases, the imposed criteria for validating

Class C Hatchback | Median Slope | ||||||||
---|---|---|---|---|---|---|---|---|---|

1:4 | 1:5 | 1:6 | |||||||

Encroachment angle (deg) | 5 | 10 | 15 | 5 | 10 | 15 | 5 | 10 | 15 |

Rate of kinematic energy loss | 0.2755 | 0.3126 | 0.3766 | 0.2976 | 0.3360 | 0.3923 | 0.2800 | 0.3134 | 0.3700 |

Ratio of μ to μ_{0} | 0.2854 | 0.3029 | 0.3376 | 0.2957 | 0.3162 | 0.3478 | 0.2885 | 0.3045 | 0.3354 |

Increase rate of yaw angle magnitude | 2.7925 | 1.0975 | 0.5882 | 3.0280 | 1.1897 | 0.6255 | 2.7946 | 1.0796 | 0.5570 |

Change rate of longitudinal speed | 0.0196 | 0.0230 | 0.0265 | 0.0175 | 0.0223 | 0.0258 | 0.0094 | 0.0155 | 0.0191 |

Change of roll angle (deg) | −0.3013 | −0.4550 | −0.5899 | 0.8173 | 2.0116 | 2.1755 | 8.9402 | 8.8387 | 8.6747 |

Pick Up, Full Size | Median Slope | ||||||||
---|---|---|---|---|---|---|---|---|---|

1:4 | 1:5 | 1:6 | |||||||

Encroachment angle (deg) | 5 | 10 | 15 | 5 | 10 | 15 | 5 | 10 | 15 |

Rate of kinematic energy loss | 0.2842 | 0.3149 | 0.3706 | 0.2808 | 0.3095 | 0.3625 | 0.2645 | 0.2950 | 0.3510 |

Ratio of μ to μ_{0} | 0.2439 | 0.2568 | 0.2840 | 0.2425 | 0.2543 | 0.2797 | 0.2348 | 0.2477 | 0.2746 |

Increase rate of yaw angle magnitude | 2.9116 | 1.1294 | 0.5912 | 2.8646 | 1.1015 | 0.5700 | 2.6795 | 1.0188 | 0.5253 |

Change rate of longitudinal speed | 0.0336 | 0.0363 | 0.0379 | 0.0309 | 0.0341 | 0.0355 | 0.0238 | 0.0280 | 0.0305 |

Change of roll angle (deg) | −0.4441 | −0.7045 | −1.0522 | 3.9291 | 4.7617 | 5.4827 | 9.0980 | 9.0561 | 8.7428 |

the simulation results are satisfied i.e. in terms of the loss in kinetic energy and ratio of the assumed impulse ratio to the critical impulse ratio i.e. (

The last three variables compare the vehicle kinematic status post-collision with that of the initial condition for the pre-collision simulation. The important observations from these results are:

• As the encroachment angle increases (collision angle increases too), the rate of kinematic energy loss increases, accounted for by the fact that a larger collision angle results in an increased inelastic deformation.

• As the vehicle drives downward the sloped terrain, the yaw angle can increase considerably (up to 290% in case of the 5˚ encroachment angle). However the sloped terrain has little effect on the longitudinal speed.

• The most notable effect is on the roll angle where there is a change of up to 900% for the 1:6 median slope which is quite substantial, whereas for the flat bottomed 1:4 slope, the change is quite modest. The indication is that the flat bottom enables the suspension to level before the collision, resulting in less roll on impact.

The CarSim^{TM} simulation of the post-collision trajectory of the vehicle allows an assessment to be made of the susceptibility of the vehicle to be re-directed outside the median into the path of other vehicles.

The post-collision trajectories for all four vehicles tested on the three median slopes at the same three encroachment angles described earlier are shown in

Median Type Parameters | 1:4 median | 1:5 median | 1:6 median |
---|---|---|---|

Average maximum lateral displacement (m) | 4.1222 | 2.8520 | 1.7076 |

Standard Deviation of maximum lateral displacement (m) | 1.2634 | 1.2634 | 0.6569 |

This study investigates the use of a sloped road median with a containment barrier to prevent an errant vehicle from being redirected into the path of other vehicles. A time-saving simulation method has been developed for assessing the effectiveness of different median slopes in containing such errant vehicles within the sloped median. A simple collision model based on planar impact mechanics has been implemented to simulate the collision of the vehicle with the containment barrier, greatly reducing the computational cost associated with the collision phase of the errant vehicle’s trajectory in comparison with other methods such as Finite Element analysis. This model is implemented in MATLAB^{TM} and works in conjunction with CarSim^{TM} to simulate the trajectory of the errant vehicle. The total time for simulating one errant vehicle sloped median encroachment including pre- and post-collision simulation in CarSim^{TM} plus collision in Matlab^{TM} is less than 30 seconds.

By comparing vehicle x-y trajectories for median profiles with each other for different encroachment angles, the 1:6 median slope was shown to be the most effective one in containing the errant vehicle. Future study is anticipated to include 4 DOF vehicle collision models, which could predict the post-collision roll motion, in the simulation method, and to validate the simulation method by comparing the simulation results with the result for the same vehicle situation generated by professional vehicular collision software, EDSMAC 4 or PC-crash.

Olatunbosun, O.A., He, R. and Shitta-Bey, O. (2018) Analysis of Highway Sloped Median Performance for Containment of Errant Vehicles. World Journal of Engineering and Technology, 6, 68-80. https://doi.org/10.4236/wjet.2018.61004