What is a "linear bicycle model"? Is it a model of a bicycle?
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Roughly speaking, it simplifies a car seen from above into two wheels, a front one and a rear one. It is called that because, like a bicycle, it steers with the front wheel. This simulator calculates two things, the car's "yaw rate" (how fast it rotates) and its "sideslip angle" (the difference between where the car points and where it travels), to reproduce its behaviour. Move the "Steering angle δ" slider above and you will see right away how the car responds.
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Really! Can this model reproduce understeer and oversteer?
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Of course! In practice this model is used to predict those characteristics. For example, if you reduce "Front cornering Cf" on the right and steer, the front wheels lose grip more easily and the "understeer" behaviour, where the car runs wide, gets stronger. Conversely, reducing "Rear cornering Cr" makes the rear slide more easily and the car tends towards "oversteer". Play with the parameters and watch how the graphs change.
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What is the "characteristic speed"? How does it relate to the simulator's results?
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It exists only for understeering cars: it is the speed at which the car's response to the steering wheel (the yaw rate) is most sensitive. Sports cars, for example, are often tuned with this characteristic speed in mind. As you raise "Speed V" in this simulator, there should be a speed at which the yaw-rate response peaks; that is close to the characteristic speed. Change the parameters and see how the peak moves.
Frequently Asked Questions
Look at the transient response graphs of yaw rate and sideslip angle. If the steady-state yaw rate is low for the steering input (the car is reluctant to turn), it is understeer; if it is high (the car turns too much), it is oversteer. The sign and phase lag of the sideslip angle are also indicators.
The model is valid only in the linear range of the tyres (small slip angles). Near large lateral accelerations or the tyres' friction limit, non-linear characteristics appear and the results depart from real vehicle behaviour. Suspension effects and load transfer are also not considered.
In this tool the speed V is set between 5 and 50 m/s and the steering angle δ between 0 and 5° (UI units). Large steering angles or low speeds break the assumptions of the linear model, so interpret the results with care. The time response always starts from zero.
If the yaw-rate response oscillates and diverges without decaying, the oversteer tendency is strong and the vehicle is unstable. If the response converges smoothly to a steady value, it is stable. A sideslip angle that keeps growing over time is also a sign of instability. Compare these waveforms while changing the parameters.
Real-World Applications
Vehicle development and tuning:Widely used in the early design of a new car as basic data for choosing suspension characteristics and weight distribution to achieve the target handling stability (amount of understeer). Changing parameters in the simulator is exactly what happens in development.
Driving simulators:The physics engines of simulators for games and professional driver training use non-linear models developed from this linear model. It is an important starting point for realistic behaviour.
Safety analysis:Applied as the basic theory for predicting vehicle behaviour during sudden steering or emergency avoidance and for designing the control logic of ESC (electronic stability control). The sideslip-angle response is particularly important.
Education and research:The most standard teaching material for understanding handling stability in automotive engineering and dynamics courses. Beyond the equations, a simulator like this lets you learn it intuitively.
Common Misconceptions and Points to Note
There are a few points to keep in mind when you start using this simulator. First, never forget that this is alinearmodel. In other words, it assumes that the tyre's grip force (cornering power) isproportionalto the slip angle. Real tyres, however, reach a ceiling or even lose grip once they slide beyond a certain amount (non-linearity). So if you enter a large steering angle in this tool the yaw rate can become arbitrarily large in the calculation, which is physically impossible. In practice, treat this model asa first step for assessing basic behavioural trends with small steering inputs.
Next, a pitfall in setting parameters. You can see that changing "Front cornering power Cf" changes the understeer tendency. On a real car, this value changes when you alter tyre pressure or the suspension camber angle. Thinking of "reducing Cf" in the simulation as "deliberately worsening the grip of the front wheels" makes the link to real-car tuning easier to understand.
Finally, the importance of the speed V. This model calculates the response atconstant speed. That is different from real cornering with acceleration or braking. In particular, the "characteristic speed" is only a theoretical value derived from this linear steady-state circular-turning model. In real sports-car development the idea is extended to evaluate the balance across a range of speeds.
Enter the vehicle mass (kg), yaw moment of inertia (kg·m²), front axle distance (m) and rear axle distance (m)
Set the front steering angle (degrees) and travel speed (m/s); the results are recalculated automatically (there is no run button)
Watch the yaw-rate response and the time response of the sideslip angle, and judge the vehicle characteristic (understeer / neutral / oversteer) from the stability factor K
Change the parameters to see how the steady-state yaw rate and characteristic speed change, and understand the effect of the axle-distance ratio and mass distribution quantitatively
Worked Example
A typical passenger car (tool defaults): mass 1500 kg, yaw moment of inertia 2500 kg·m², front axle distance lf = 1.1 m, rear axle distance lr = 1.4 m, front and rear axle cornering stiffnesses Cf = 60000 and Cr = 55000 N/rad, speed V = 20 m/s, steering angle δ = 2°. The stability factor is K = m/L²·(lr/Cf − lf/Cr) ≈ 8.0×10⁻⁴ s²/m², so the car is judged to understeer. The steady-state yaw rate is r = Vδ/(L(1+KV²)) = 0.279/1.32 ≈ 0.212 rad/s, the lateral acceleration V·r ≈ 4.2 m/s² (≈ 0.43 G), and the characteristic speed 1/√K ≈ 35.4 m/s (about 127 km/h). The yaw rate in the response graph also settles at about 0.21 rad/s. Lowering Cf (less front grip) increases K and strengthens understeer, while lowering Cr makes K < 0 and turns it into oversteer.
Practical Notes
The sum of the axle distances (L = lf + lr) equals the wheelbase, which is a fixed parameter in vehicle design, so avoid geometric inconsistencies when changing it
At high speeds above the characteristic speed, non-linear effects (saturation of tyre cornering stiffness) can no longer be ignored and are outside the scope of this 2-DOF model
In real development, extending to a model with 3 or more DOF that includes roll angle is essential when changing suspension stiffness
A vehicle on the understeer/oversteer boundary (K = 0) is neutral in theory, but variation in tyre characteristics makes this hard to achieve, so a slight understeer setting is the industry standard for safety
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