Introduction to the Bicycle Model

The bicycle model is a heavily simplified vehicle model, which is useful in understanding the concept of understeering/oversteering vehicle.

Assumptions of the bicycle model

No lateral load transfer
No longitudinal load transfer
No body rolling and pitching
Tyres are in the linear range
Constant forward velocity (specified by the user)
Steering position is controlled
No aerodynamic effects
Rigid chassis and no suspension compliance effects

Tyres are in the linear region when the lateral acceleration \(a_y\) is less than about 0.4g. [1]

The bicycle model

\(\delta\)	steer angle [rad]
\(u\)	forward velocity [m/s]
\(v\)	lateral velocity [m/s]
\(V\)	velocity vector [m/s]
\(\beta\)	vehicle slip angle [rad], the angle between the vehicle centreline and the velocity vector at c.g.
\(r\)	yaw velocity [rad/s]
\(R\)	radius of turn [m], the distance between the turning centre to c.g.

The 2 DOF of the model are the lateral velocity \(v\) and yaw velocity \(r\). The steer angle \(\delta\) is the input, and the forward velocity \(u\) is specified.

The velocity vector \(V\) is perpendicular to the turn radius \(R\).

The 3 phases of the idealised cornering

Transient turn entry – Initiated with the steering input \(\delta\), the yaw velocity \(r\) and the lateral velocity \(v\) build up from zero to the values in the steady-state cornering
Steady-state cornering – \(r\) and \(v\) as well as tyre slip angles and vehicle slip angle \(\beta\) are constant. The vehicle is driving on the curved path of the constant radius \(R\)
Transient turn exit – the steering input \(\delta\) ramped down to zero. \(r\) and \(v\) reduce to zero

Steady-state low-speed cornering

Low speed
\(\Rightarrow\) the external force due to the lateral acceleration is negligible. No tyre slip angles, the tyres rotate in their directions of the heading
\(\Rightarrow\) real vehicles can be represented by the bicycle model

The Ackermann steer angle \(\delta_{Acker} = \dfrac{l}{R}\) is the geometric steer angle required for a vehicle to follow a turn radius of \(R\) at low speed with the front and rear tyres heading perpendicular to the turn centre.

‘Parallel Steer’ is common for passenger cars. Racing cars have frequently used ‘Reverse Ackermann’ geometry [1].

(1) Steady-State Cornering of the Neutral Steer Car

With an increase in cornering speed, the lateral acceleration is present. The tyres produce the inward centripetal force \[\text{(total lateral force)} = Y_F + Y_R = M \dfrac{u^2}{R}\] where \(M\) is the total mass of the vehicle

Even when lateral acceleration is present, the neutral steer car does not ‘oversteer’ or ‘understeer’ the intended curved path defined by the Ackermann steer angle.

\(Y_F, Y_R\)	tyre side force [N]
\(C_F, C_R\)	tyre cornering stiffness [N/rad]
\(\alpha_F, \alpha_R\)	tyre slip angle [rad]

For the neutral steer car, assume the c.g. is at the mid-wheelbase (\(a=b= l/2 \)).
Also, assume that the tyre cornering stiffness are the same between the front and the rear \[C_F = C_R\]

In steady-state, the extermal moments are balanced.
\[ \begin{align} Y_F a &= Y_R b \\ C_F\alpha_F a &= C_R\alpha_R b \end{align} \] Since \( a = b \) and \(C_F = C_R\) \[\alpha_F = \alpha_R\]

Remarks on the static front/rear axles loads and the required lateral tyre forces

In steady-state, the distribution of the required lateral tyre forces can be related to the static front/rear axles loads. \[ \dfrac{b}{a} = \dfrac{W_f}{W_r} = \dfrac{Y_F}{F_R} \] and \[ \left\{ \begin{array}{l} Y_F = \dfrac{W_f}{g} a_y \\ Y_R = \dfrac{W_r}{g} a_y \end{array} \right. \] where \(W_f\), \(W_r\) are the front and rear axle static loads, respectively.

How to draw the figure

The velocity vectors at the tyres are perpendicular to the radii lines towards the turning centre
The tyres are drawn with the angles \( \alpha_F\) and \(\alpha_R\) to the velocity vectors
The tyre side forces are drawn perpendicular to the tyre headings
Since the rear steer angle is zero, the vehicle centreline coincides with the direction of the rear tyre
Finally, annotate the Ackermann steer angle \(\delta_{Acker}\) (between the vehicle centreline and the front tyre heading) and the vehicle slip angle \(\beta\)

Remarks on the understeer gradient = 0 for the neutral steer car

From the figure, for a given radius \(R\), the required steer angle \(\delta\) is purely geometrical and does not depend on the speed. \(\delta = l/R\)
Perform a constant radius test (= gradually increase the forward velocity, and the driver adjusts the steer angle to maintain a constant radius if necessary). For the NS car, the rate of increase of the front and rear slip angles with the lateral acceleration are the same. \[ \frac{\Delta \alpha_F}{\Delta A_Y} = \frac{\Delta \alpha_R}{\Delta A_Y} \] or \[ \frac{\Delta \alpha_F}{\Delta A_Y} – \frac{\Delta \alpha_R}{\Delta A_Y} = \frac{\Delta ( \alpha_F – \alpha_R )}{\Delta A_Y} = 0 \]

(2) Steady-State Cornering of the Understeer Car

Passenger cars are all designed to understeer, the degree of understeer ranges from modestly to heavily. [1]

If the lateral acceleration is present, the understeer car turns less than the intended curved path defined by the Ackermann steer angle.

the c.g. is at the \(1/3\) of the wheelbase from the front track (i.e. \(a = l/3, \quad b= 2l/3) \).
Then, the front and rear static loads are \(2/3W\) and \(1/3W\), respectively
the tyre cornering stiffness are the same between the front and the rear \(C_F = C_R\).
Also the same value as the one in NS car above

By keeping the wheelbase \( l \), the Ackermann steer angle \(\delta_{Acker} \) for a given radius \(R\) remains the same.

The required slip angles for the understeer car

In steady-state, the sum of external forces and the moments are zero.
Force equilibrium: \[ \begin{align} \text{(centrifugal force)} &= Y_F + Y_R \\ &= C_F\alpha_F + C_R\alpha_R \end{align} \] Moment equilibrium about c.g.: \[ \begin{align} C_F\alpha_Fa &= C_R\alpha_Rb \\ C_F\alpha_F \dfrac{1}{3}l &= C_R\alpha_R \dfrac{2}{3}l \end{align} \] With the assumption of \(C_F = C_R\), the slip angles have to be \[\alpha_F = 2\alpha_R\] The slip angle at the front must be twice that of the rear.
Similar to the static load split, \(2/3\) and \(1/3\) of the total lateral force are produced at the front and rear tracks, respectively.

For the NS car, \[ (\text{total lateral force}) = 2C\alpha_1 \] where \( C = C_F = C_R\) is the cornering stiffness, and \( \alpha_1\) is the slip angle (the same front and rear).

For the understeer car, \[ (\text{total lateral force}) = C(\alpha_F + \alpha_R) = C(2\alpha_R + \alpha_R) \] Therefore, \[\alpha_F = \dfrac{4}{3}\alpha_1, \; \alpha_R = \dfrac{2}{3}\alpha_1 \]

In comparison to the NS car, the rear slip angle \( \alpha_R \) is reduced by \( 1/3\alpha_1 \), and consequently, the vehicle attitude angle \(\beta\) is smaller. Therefore, if the NS car steer angle is kept, the front slip angle is reduced.

In the understeer car, the steer angle has to be increased by \( 2/3\alpha_1 \) from the NS car to counteract this reduced vehicle rotation, as well as the increased front slip angle \(\alpha_F = 4/3\alpha_1 \) to maintain the same radius \(R\).
The value of \(2/3\alpha_1\) comes from \( \alpha_F – \alpha_R = (4/3)\alpha_1 – (2/3)\alpha_1 = 2/3\alpha_1\).

The total steer angle for the understeer car is \[\delta = \delta_{Acker} + (-\alpha_F + \alpha_R) = \dfrac{l}{R} + (-\alpha_F + \alpha_R)\] The 2nd term is the additional steer angle required for the understeer car to maintain the same radiurs of turn \(R\).
(N.B. following the sign convention in use \(\alpha_1 <0\), \(\alpha_F <0\) and \(\alpha_R <0 \) in the illustrated scenario.)
If the steering angle is kept at the Ackermann angle, the understeer car will follow a larger radius.
When the lateral acceleration is not negligible, the car ‘understeers’ the path defined by the Ackermann steer angle.

The understeer gradient – a measure of the degree of understeer

Performing a constant radius test in an understeering car, the driver increases the forward speed gradually and the steer angle has to be increased accordingly to maintain the same radius of turn \(R\).
The understeer gradient is \[ \text{usg} = \dfrac{\Delta(-\alpha_F + \alpha_R)}{\Delta a_y} \] i.e. the slope of the steer angle \(\delta\) against the lateral acceleration \( a_y \) plot.

Summary

	Understeer	Neutral steer	Oversteer
Front slip angle \(\alpha_F \)	\(\alpha_F > \alpha_1\)	\(\alpha_1\)	\(\alpha_F < \alpha_1\)
Rear slip angle \(\alpha_R \)	\(\alpha_R < \alpha_1\)	\(\alpha_1\)	\(\alpha_R > \alpha_1\)
Vehicle slip angle \(\beta\)	\(\beta < \beta_1\)	\(\beta_1\)	\(\beta > \beta_1\)
To maintain the same turn radius \( R \) with increase in forward speed \( u\)	increase \( \delta \)	maintain \( \delta \)	Reduce \( \delta \)
Rate of change of \(\alpha\) with \( a_y\)	\( \dfrac{\Delta\alpha_F}{\Delta a_y} > \dfrac{\Delta\alpha_R}{\Delta a_y}\)	\( \dfrac{\Delta\alpha_F}{\Delta a_y} = \dfrac{\Delta\alpha_R}{\Delta a_y}\)	\( \dfrac{\Delta\alpha_F}{\Delta a_y} < \dfrac{\Delta\alpha_R}{\Delta a_y}\)

The above quantities are in absolute values rather than SAE convention (in SAE, \( \alpha \) is negative for a right-hand turn [1] ).

Following the SAE convention,

\[ \begin{align} \delta &= \delta_{Acker} &+ (-\alpha_F + \alpha_R)
\\ &= \dfrac{l}{R} &+ (-\alpha_F + \alpha_R) \end{align} \]

Reference

[1] W. Milliken and D. Milliken, Race Car Vehicle Dynamics.