Anselmi Chiara 1041369 – email@example.com
Bizzo Stefano 1034840 – firstname.lastname@example.org
A vehicle without suspensions cannot properly face the irregularities of the road, causing many difficulties to the rider.
Suspensions help maintaining the wheel grip on the road plane under different operating conditions, supporting the weight of the motorcycle, and ensuring a certain degree of comfort to the rider. The level of performance depends on the stiffness, the damping coefficient and other parameters of the spring-damper unit.
Progressive suspensions are more flexible than classic devices since their variable stiffness can be adapted to the strength of disturbances on the road. Thus it is possible to obtain a trim specific to different road conditions. However, a potential problem is that a disturbance at a specific frequency may produce unpleasant resonance conditions, which cause the loss of stability of the system.
The following paragraphs describe the model used and present the simulations and analysis made to show its behavior. We will start by studying the motorcycle in a steady state and then we will analyze the effect of disturbances and velocity on its motion. Finally we will examine the vibration modes of the system and the effect of resonance.
The purpose of this project is to simulate and analyze the behavior of a motorcycles with progressive front and rear suspensions, in in-plane dynamics. During the testing, many situations are considered, to show how the motorcycle reacts to different external stresses, such as bumps.
The dimensions of the components were chosen considering the proportions of a real motorbike. The weights were set so that the front and the rear wheel experience the same load, in steady state.
The front suspension is inspired to the BMW Telelever suspension. This particular kind of suspension is composed by a swinging arm, connected to the chassis by a revolute joint and to the fork by a spherical joint. The upper part of the fork is connected both to the lower part of the fork and to the chassis by a prismatic and a spherical joint respectively.
Since we analyzed the behavior of the motorcycle in in-plane dynamics, the spherical joint between the swinging arm and the fork was replaced, in the model, by a revolute joint, so that it is impossible for the motorbike to turn. It has been reestablished for the study of the wobble. Moreover we placed a planar joint, which makes the xz plane of the chassis coplanar with the xz plane of the ground.
It is important to underline that this particular suspension configuration causes modifications of the caster angle, during the motion. For instance, if the motorbike is breaking, the angle will increase with the increasing of the dive: in this way the directional stability is improved and the system has a better disturbance rejection. In the picture we can see the graph describing the trend of the angle: in abscissa there is the vertical displacement of the wheel, while in ordinate there is the angular motion of the fork.
The parameters describing the spring-damper unit, both of the front and rear suspension, were set to be realistic and valid if compared with real suspension. They were tuned through consecutive simulation and they are:
front suspension free length spring : 239 mm
front suspension spring constant: 35000 N/m
front suspension damping coefficient: 1500 kg/s
rear suspension free length spring: 300 mm
rear suspension spring constant :95000 N/m
rear suspension damping coefficient: 1500 kg/s
Both the front and rear suspensions are progressive suspensions, as can be seen in the following graph, showing in abscissa the vertical displacement of the wheel and in ordinate the normal force felt by the wheel.
Particular attention was devoted to project simulations of different road conditions, in order to analyze the motorcycle motion under variable circumstances. Road irregularities were simulated through road-data, which differ in the wavelength and the amplitude of the bumps.
The behavior of a motorbike in in-plane dynamics is affected by the simultaneous presence of many elementary motions, namely bounce, pitch, rear and front hop and straight mode, which can be excited by the road irregularities, causing discomfort for the rider and loss of wheel adherence.
In the following paragraph we will analyze these motions, to see how it affects the system.
The bounce causes the vertical displacement of the entire system. Thus, the study of this motion is particularly important in order to ensure the rider comfort.
When a motorbike is running on a road which is not plane the irregularities of the road induce some oscillation in the wheels, and consequently in the chassis. A good suspension should damp the chassis oscillations down, in order to prevent damages and inconveniences.
This is actually the case for the following simulation, which models a motorbike running at a speed of 40 km/h in a road with 25cm high bumps.
From the following graphs, showing the vertical displacement of wheels and chassis and the relative accelerations, it is clear that the displacement of the two wheels is grater, in absolute value, than that of the chassis. Similary the acceleration experienced by the frame is lower than that of the wheels. We note that, as the disturbance effect stops, the system settles in the equilibrium status in about 3 seconds.
The ratio between the vertical displacement of the chassis and the displacement due to the road profile is called transmissibility. Its value is lower than 1 when the ratio between the frequency imposed by the road oscillations and the frequency of the suspension is grater than 1.41. In this case, the suspension is able to reduce the chassis oscillation and hence ensure a certain degree of comfort to the rider.
Unfortunately, suspensions do not always damp down the chassis oscillations. For instance, for a velocity of 20km/h, the amplitude of the frame and wheels oscillations is nearly identical. The following graphs show the vertacal displacement of the chassis ancd the wheels, and the trend of accelerations respectively.
This phenomenon happens when the frequency ratio is equal to the critical value of 1.41.
If the motorbike is travelling at a lower speed (15km/h) the oscillation amplitude is grater than that of the previous cases. That is, at a frequency ratio lower than 1.41, despite the presence of the suspension, the vertical displacement of the chassis is grater than the wheels one. Clearly this is the worse possible behavior for a suspension, since it produces an effect exactly the opposite of the desired one.
The pitch motion concerns mostly the wheel adherence. Using the LMS eigenvalues display, it is possible to find the natural frequency related to this motion.
For a motorbike running on the same road profile used for the bounce analysis, that is a road with wavelength of 4m, the natural frequency vn of the pitch is in the range of 4-5 Hz, This corresponds to a forward velocity of:
V = Lwave vn = 18m/s
After 3 seconds, the system becomes instable, showing the characteristic pitching behavior.
The system instability is underlined by the eigenvalues study, which gives a positive real part for the pitch mode.
By varying the velocity, the road irregularities stop exciting the pitch mode.
The following video shows the simulation of the pitching motion.
The following graphs show the pitch when the motorbike speed is 65 km/h and 68km/h. In the second case the frequency related to the speed is not the critical one and, even though the wheels loose the ground contact, it is possible to maintain the control of the vehicle. On the contrary in the first case the speed excites a frequency which brings to the complete loss of control.
Rear and front hop
Similary to the pitch motion, the rear and front hop affects the wheel adherence.
Even if these two motions are uncoupled, it is easier to study their mutual effect on the motorcycle behavior. To do that, we can consider a different road-data, with a wavelength of 1m, and bumps of 4cm height.
From the analysis of the eigenvalues of the model in this particular situation, it appears that the frequencies interested by this motion are in the range of 10-11 Hz , corresponding to a forward speed of about 10.5m/s.
During the simulation, both the front and rear wheel loose the adherence with the ground, exactly as expected, since the mode frequency is excited by the motorbike speed and the roads irregularities.
In the following video the hop motion can be seen.
The last in-plane motion is the straight mode. This is usually responsible of stable motions even with increasing speed.
Using the same road model than the one used for the hop analysis, we were able to determine that the imaginary part of the mode associated with this motion is always zero in every condition.
As a conclusion to this dynamic analysis, it is interesting to consider at least one out of plane motion, i.e. the wobble. Unlike the motions considered so far, which are mostly related to the rider comfort and wheel adherence, wobble is essentially tied to the stability of the vehicle.
The following video describes the wobble motion.
To study this particular motion, we reestablished the spherical joint between the swinging arm and the fork; in this way the motorbike can steer and eventually show the wobble effect.
For this analysis, we used a plane road model, where the motorbike travels at 76 km/h.
Using the eigenvalues display, it is possible to determine that the mode corresponding to the wobble has a frequency of about 5.9 Hz.
Increasing the speed to 104km/h, we can see that the frequency corresponding to the wobble mode increases, assuming the value of 10-11 Hz. As for the mode stability, the real part of the eigenvalue is either positve or negative varying the speed, the acceleration and the linearisation point.
It is easier to be in resonance condition having high value of speed and acceleration. However it is possible to find istable eigenvalues even in more suitable conditions.
The model behavior, obtained with the simulations described in the previous paragraphs, satisfies to theoretical expectations. In fact, the presence of suspensions guarantees better performances in all situations except when the resonance condition is attained. In this case, the motorbike shows the effect of the elementary motion, typical of the in-plane dynamics. We underline that this peculiar motions disappear, or at least decrease in intensity, when the system is not in the critical range of frequencies, which characterizes the elementary modes.
As for the out of plane dymanics, the results may be altered by the presence of the planar joint, which limits the out of plane motion of the motorbike.