Vibration modes of motorcycles by Vittore Cossalter, Roberto Lot

The motorcycle is assumed to run straight on a flat, level road surface at a certain velocity. The rider is thought to be a rigid body fixed to the rear frame, who can exert no control on the motorcycle.
For simplicity's sake, the motorcycle is considered to have rigid suspension; from a kinematic point of view it is a spatial system whose motion can be described by means of the following four coordinates:

• the steering angle;
• the roll angle;
• the yaw angle;
• the lateral displacement of the mass centre.

The system "motorcycle" may be considered to consist of two parts:

• the rear frame including the rider, the engine, the petrol tank and the seat and rear wheel;
• the front frame including the forks, the handlebars and the front wheel.

The rear frame and the front frame are hinged togheter at the steering axis by means of a revolute pair.
During the motion the tyres are free to sideslip; so they produce lateral forces which are a linear function of the sideslip angles and the camber angle. These forces, from a practical point of view, may be considered to be restoring forces like those produced by springs.

Keeping the above simplification in mind, the steering axis is considered to be constrained so that it can't move laterally. So the motorcycle may be thought of as two decoupled systems each having only one degree of freedom:

• the front frame oscillating around the steering axis on which the lateral front tyre force (with the normal trail as a lever) acts as a restoring force;
• the rear frame oscillating around the steering axis on which the lateral rear tyre force (with a lever proportional to the wheelbase) acts as a restoring force.

The two oscillation modes are:

• an oscillation of the front frame around the steering axis called the wobble mode.
• an oscillation of the rear frame around the steering axis called the weave mode.
  
  

Wobble
The frequency of the wobble mode is:

where:

a_n is the normal trail;
I_f is the moment of inertia of the front frame around the steering axis;
k_{l f} is the front tyre stiffness;
e is the steering head angle.

For example: a_n =110 mm;I_f =0.9 Kgm²; k_{l f} =15 rad^-1; e =27°. load on the front wheel=1000 N, the wobble frequency is equal to 6.43 Hz. you have to remember that the formula don't take into acconut the fork flexibility, the gyroscopic effects, the dependance on the forward velocity......

The wobble mode is essentially a steering oscillation of the front forks; this mode doesn't involve the rear frame in any significant way.
Typical values of the frequencies vary from 4 Hz, in the case of heavy motorcycles, to 9 Hz, in the case of lightweight motorcycles.
The frequency of the wobble mode goes up as the trail increases and as the front inertia decreases: this depends essentially on the sideslip stiffness of the front tire and is also influenced by the lateral flexibility of the front fork.
When the speed of the motorcycle ranges from 10 to 20 m/s (40-80 km/h), the wobble is only a little damped and can therefore become unstable: the adoption of a damping steer increases the damping and consequently the degree of stability.

Weave
The frequency of the weave mode is:

where:

l is the lever of the rear tyre force with respect to the steering axis;
I_r is the moment of inertia of the rear frame around the steering axis;
k_{l r} is the rear tyre slip stiffness.

• For example: l =1.38 m; I_r =30 Kgm²; k_{l r} 10 rad^-1;load on the rear wheel=1500 N, the weave frequency is equal to 3.94 Hz.The weave mode is a side-to-side motion of the entire motorcycle characterised by a coupling of the yaw and roll oscillations with a significant lateral displacement of the steering axis.
  The weave mode has a natural frequency increasing from about 0.2 Hz at a low speed to about 3-4 Hz at a high forward speed;
  Its value depends on numerous factors:
• the position of the mass centre of the rear frame;
• the inertia of the wheels;
• the castor angle;
• the trail.

The mode may be unstable at a low speed; at a high speed it may be difficult to control since its natural frequency may be too high to be controlled by the driver.

In reality, the modes are different in view of the fact that the steer axis is not blocked.

Capsize
Now a non-oscillating mode is consided.
It is well known that motorcycles, without the action of control exercised by the rider, are unstable in roll, this motion is a non-oscillatory motion and consists in the capsize of the two-wheel vehicle.
In fact, the motorcycle in uniform rectilinear motion will eventually fall laterally: the motorcycle is therefore an intrinsically unstable system, above all in the low speeds range when the stabilising action of the gyroscopic effects turns out to be negligible.
The capsize motion is essential for driving the motorcycle; it is used and controlled by the rider by means of the steering.

The time of the vehicle's capsize depends on numerous factors:

• the speed of the vehicle;
• the inertia of the wheels (gyroscopic effect);
• the height of the mass centre;
• the mass of the motorcycle;
• the roll inertia of the vehicle;
• the castor angle;
• the trail.

The next figure shows an example of wobble and weave vibration of a racing motorcycle:

For more information see: V. Cossalter Motorcycle Dynamics.