Consider a car of mass m taking a turn of radius r along a level road. As seen from an inertial frame of reference, the forces acting on the car are :
1. the lateral limiting force of static friction \(\vec{f_s}\) on the wheels-acting along the axis of the wheels and towards the centre of the circular path- which provides the necessary centripetal force,
2. the weight \(m\vec{g}\)acting vertically downwards at the centre of gravity (C.G.)
3. the normal reaction \(\vec{N}\)of the road on the wheels, acting vertically upwards effectively at the C.G. Since maximum centripetal force = limiting force of static friction,
mar = \(\cfrac{mv^2}{r} = f_s....(1)\)
In a simplified rigid-body vehicle model, we consider only two parameters-the height h of the C.G. above the ground and the average distance b between the left and right wheels called the track width.
The friction force \(\vec{f_s}\) on the wheels produces a torque \(\tau_t\) that tends to overturn/rollover the car about the outer wheel. Rotation about the front-to-back axis is called roll.
\(\tau_t\) = fs.h = \(\left(\cfrac{mv^2}{r}\right)h\) ......(2)
When the inner wheel just gets lifted above the ground, the normal reaction \(\vec{N}\) of the road acts on the outer wheels but the weight continues to act at the C.G. Then, the couple formed by the normal reaction and the weight produces a opposite torque \(\tau_r\) which tends to restore the car back on all four wheels
\(\tau_r\) = mg.b/2 .....(3)
The car does not topple as long as the restoring torque \(\tau_r\) counterbalances the toppling torque \(\tau_t\) . Thus, to avoid the risk of rollover, the maximum speed that the car can have is given by
Thus, vehicle tends to roll when the radial acceleration reaches a point where inner wheels of the four-wheeler are lifted off of the ground and the vehicle is rotated outward. A rollover occurs when the gravitational force passes through the pivot point of the outer wheels, i.e., the C.G. is above the line of contact of the outer wheels. Equation (3) shows that this maximum speed is high for a car with larger track width and lower centre of gravity.
There will be rollover (before skidding) if \(\tau_t\)≥ \(\tau_r\), that is if
The vehicle parameter ratio, b/2h , is called the static stability factor (SSF). Thus, the risk of a rollover is low if SSF ≤ µs . A vehicle will most likely skid out rather than roll if µs is too low, as on a wet or icy road.
