In this article, we will consider a mathematical formula for the basics of cornering. Cornering, together with accelerating and braking, is one of the characteristic motions of a vehicle. According to Newton’s laws of motion, the motion of an object is expressed as a relational equation between force and acceleration. During cornering, acceleration occurs in the lateral direction of the vehicle. It is therefore the purpose of this article specifically to express the lateral acceleration in mathematical formulae. The key factors are the vehicle yaw rate and the body slip angle.
In the field of vehicle dynamics, the primary focus lies on the act of ‘cornering’ among the basic vehicle operations of accelerating, cornering, and braking.
This focus is due to the unique ability of vehicles to freely change direction based on the driver’s steering, distinguishing them from trains and other rail-bound systems.
Now we consider the vehicle motion when the speed, i.e. the magnitude of the velocity, is constant.To understand vehicle dynamics, let’s first consider the motion of a point mass – an idealized object with mass but no size or shape.
Imagine a point mass moving at a constant speed in a counterclockwise circle.
Here’s what happens:
When we scale up from a point mass to an actual vehicle, the complexity increases. A vehicle’s motion is not just about moving along a path but also involves its orientation. Thus, we consider:
In vehicle dynamics, these two motions are intertwined. Notably, the vehicle’s orientation might not always correspond with the direction of motion, as vividly demonstrated in drifting maneuvers.
Vehicle motion is typically analyzed in a moving coordinate system fixed to the vehicle.
The origin is at the vehicle’s center of gravity, with the x-axis aligned with the vehicle’s longitudinal direction (forward being positive).
Neither the vehicle’s position nor its direction is crucial; instead, motion is considered in terms of velocity and angular velocity (yaw rate, denoted as \(\gamma\)).
The angle of velocity relative to the x-axis (counter-clockwise being positive) is called the body slip angle (denoted as \(\beta\)).
Circular motion is caused by a force transverse to the direction of travel. According to Newton’s laws of motion, a lateral (y-axis) acceleration is produced at that time, so let us consider an equation to express this.
It is the sum of the following two components:
When discussing handling, the body slip angle \(\beta\) can be considered sufficiently small, so the following approximation holds:
$$V_y=V\sin\beta\approx V\beta$$ $$V_x=V\cos\beta\approx V$$
Thus, the lateral acceleration is described as:
$$a_y=\dot+V_x\cdot\gamma\approx V\cdot\dot+V\cdot\gamma=V(\dot+\gamma)$$
This implies that:
Vehicle motion can be described by two variables: the body slip angle \(\beta\) and the yaw rate \(\gamma\) .
Using the formula \(a_y=V(\dot+\gamma)\) , various vehicle motions can be expressed.
In steady-state cornering, the body slip angle stays constant. i.e.
$$\dot=0$$
Therefore,
$$a_y=V\gamma$$
The exploration of vehicle dynamics, particularly the mathematical modeling of ‘turning,’ provides invaluable insights into the intricate dance between a vehicle and the laws of physics. It empowers engineers to push the boundaries of automotive design and safety. As we continue to advance in technology and understanding, the principles laid out in this field will undoubtedly lead to more innovative and efficient vehicles, altering our experience of mobility and transportation. In essence, the mathematics of vehicle movement is not just about numbers and equations; it’s about understanding the harmony between man, machine, and motion.
2024-02-18
2023-12-25
