This tool evaluates a vehicle's response to step-steer inputs by applying a derivative-based analysis using lateral-force and moment-equilibrium equations to model its dynamic behavior. It compares performance between Car A and Car B using a set of default parameters. Users may modify any input values to calculate and contrast the responses of different vehicles.
To determine the vehicle's time-based response, the lateral force and moment equilibrium equations for a cornering maneuver are utilized. The nonlinear dynamics are described by:
Since these nonlinear equations are difficult to solve analytically, the model is linearized by taking the partial derivatives with respect to the state variables, β and r, and the steering input, δ. This yields a locally linear approximation of the system:
By substituting for the sideslip angle (β), these equations are combined into a single second-order ordinary differential equation that describes the yaw rate response:
The solution to this equation for a step steer input is dependent on the system's damping characteristics. The response is categorized into the following three cases.
The yaw rate response is a decaying sinusoid:
where the constants X and ϕ are determined by the initial conditions:
The system returns to equilibrium as quickly as possible without oscillation:
The response is the sum of two decaying exponential terms and approaches the steady-state value without oscillation:
In comparing the two vehicles, Car A exhibits significant understeer, characterized by a high positive understeer gradient and a substantial positive stability factor, resulting in a characteristic speed of 55 mph. Conversely, Car B demonstrates neutral handling, indicated by an understeer gradient of zero and a minimal stability factor, resulting in a characteristic speed exceeding 5000 mph. Additionally, Car B's polar moment of inertia to mass ratio is closer to ideal, facilitating neutral steering with critically damped steering response. Car B's undamped natural frequency is substantially higher at 74 rad/s compared to Car A's 8 rad/s, resulting in Car B settling quickly to its final yaw rate, as clearly depicted in the line graph. Lastly, Car B to is able to achieve double the yaw rate and half the turning radius of Car A in steady-state conditions.
A neutral-steering vehicle requires an understeer gradient of zero. The present linearized model accounts only for the centre-of-gravity effect and assumes a linear tyre lateral-force vs. slip-angle relationship. However, this assumption breaks down when tires are at their limit, as real tire behavior becomes nonlinear. Additional contributors to the understeer gradient include roll steer, aerodynamic centre migration, camber gain and differential action.
Minimizing the stability factor raises the characteristic speed (for understeer) or critical speed (for oversteer). While a zero stability factor is unattainable in practice, ensuring that the characteristic/critical speed exceeds the vehicle's normal operating range yields a car with neutral steering.
The polar-moment-to-mass ratio strongly governs cornering response: an optimal ratio delivers neutral steering and critically damped response. In practice, mass reduction limits and packaging constraints restrict polar-moment adjustments. Around the optimum, steering damping ratio is relatively insensitive to speed, maintaining critical damping across its operational range.