The aim of the vehicle suspension system is to ensure the road-wheel contact and to set a good compromise between handling and comfort.
Forces between vehicle chassis and the ground can only be exchanged through the wheels, thus it is clear the importance to mantain the contact between them in all the conditions and to have a proper weight distribution. If it doesn’t happen the handling of the vehicle can be negatively affected. Consequently vehicle behaviour in cornering eather clearing a hurdle or accelerating/braking phasis depends on suspension system settings: this is due to the fact that vehicle behaviour depends on how forces are trasmitted between wheels and ground.
In the following paragraphs is described the generation of the model in order to analyse the behaviour of a car equipped with suspension system, including antiroll bars. Then are exposed the results of the simulations for different road characteristic, varying the stiffness of the suspensions and antiroll bars.
A car suspension system is composed by suspensions and antiroll bars: the purpose of this project is to analyse vehicle behaviour in different situations varying stiffness of those components. The analisys deal with vehicle behaviour in a corner, on a complete bump (involving wheels of both sides) and on a lateral bump (involving only one side wheels of the car). Results are collected in terms of forces exchanged between wheels and ground as well as cinematic parameters of wheels and car body.
The modelling problem
All models are created using LMS Virtual Lab software.
Suspensions and antiroll bars attachment points are obtained by manual measurment on a BMW 3 series.
- Tire model
In order to make the contact forces between tires and ground more realistic, the model is equipped with “Complex Tire” forces instead of “Simple Tire” ones. In this way it is possible to use a lateral stiffness surface spline instead of a constant cornering stiffness. The lateral force of a tire, called Flat, is considered as a function of slip angle, called λ, and normal force applied on the tire, called Fn. Using a lateral stiffness table it is possible to define a surface spline setting slip angle on the x axis and normal forces on y axis and, consequently, the lateral force on the z axis as a surface. It is possible to find in literature the following formula, that we used to calculate the lateral force spline surface:
Flat = 10 * λ * Fn
Considering that in Burckhardt lateral force model for dry asphalt the maximum of lateral force is reached around 8 and 10 degrees of slip angle, the surface can be adjusted in order to obtain a rational shape. The final result of lateral stiffness surface spline is reported on the following image.
- Front suspension: MacPherson strut
In the following pitcures it is possible to compare a real MacPherson strut and the model used in simulations.
The MacPherson suspension, used as front suspension on the car model, includes the lower arm, the shock absorber (body and rod), the steering arm and a chassis to fix properly all the components.
The lower arm is coupled to the chassis using a rotational joint and to the shock absorber body using a spherical joint. The shock absorber body is then coupled to the shock absorber rod by a cylindrical joint, and finally the shock absorber rod is coupled to the chassis using a spherical joint. The steering arm is coupled to the shock absorber body and to the steering rack (in the complete model) using two spherical joints. In the complete car model it is possible to see the antiroll bars, decribed later on the car model section.
A TSDA force is applied between the chassis attachment of the shock absorber body and the upper point of the shock absorber body (blue spring in the previous picture).
- Rear suspension: Multilink strut
In the following pitchures it’s possible to compare a real Multilink strut and the model used in simulations.
The Multilink suspension, used as rear suspension on the car model, includes five arms, the hub and the chassis. All arms are coupled with the hub and the chassis by spherical joints. In the complete car model is possible to see the antiroll bars, decribed later on the car model section. A TSDA force is applied between the chassis and the lower/rear arm (blue spring in the picture).
- Car body with suspensions, antiroll bars and steering system
In the complete model, front and rear suspensions are joined to the chassis body. Antiroll bars and steering rack are added. In the following pictures it is possible to see all details.
Antiroll bars are splitted into two pieces (not visible in the picture). Each end of the antiroll bar is coupled to a link rod using a spherical joint, the link rod is coupled to the hub (rear axle) or to the shock absorber body (front axle) using another spherical joint. Then, one half antiroll bar is coupled to the chassis body using a revolute joint, and to the other half antiroll bar using another revolute joint. Here, in this last revolute joint (between the two parts of the antiroll bar) is applied a RSDA force, to simulate the antiroll bar torsional stiffness.
The steering rack is coupled to the chassis body by a cylindrical joint and to each steering arm using a spherical joint. Steering arms are connected to the absorber bodies using a spherical joint. A joint position driver is applied to control rack traslation and, consequently, steer angle.
In the following pictures it is possible to appreciate details of front and rear suspensions.
The overall car mass is settled on 1610 kg. The chassis mass is settled on 1400 kg, applied on the middle point between the two axles in order to achieve an equal weight distribution on the four wheels. This point (center of gravity) is about 520 mm high from the ground.
Wheelbase is: 2760 mm.
Front track with is: 1500 mm.
Rear track with is: 1513 mm.
In the following pictures it is possible to appreciate the addition of a body shell to improve the graphical aspect of the model.
Simulations and analysis of results
– Suspensions characterization
The first analysis step is to characterize separately front and rear suspensions. The aim of those analysis is to identify the spring stiffness that makes front and rear suspensions equivalent stiffness very similar. It is reported in a diagram the normal force acting between wheel and ground (abscissa) and the wheel center of gravity displacement in vertical direction (ordinate): the stiffness is represented by the diagram curve slope. The suspension chassis is fixed to the ground and the wheel runs over a bump 190 mm in height. In order to make the suspensions equivalent stiffness diagram similar, it has to be used a progressive spring for the rear suspension; if a costant value for the spring stifness is assumed, the Multilink equivalent stiffness rate decreases. Conversely, using a costant value of the spring stiffness in the MacPherson strut, equivalent stiffness is slightly increasing. The spring constant values and the spring free lenghts that make similar the front and rear suspensions equivalent stiffness are:
MacPherson: spring constant: 30000 N/m; free lenght: 330 mm.
Multilink: progressive spring (see the spline below, reporting forces on the abscissa and lenght variation of the spring on the ordinate); free lenght: 300 mm.
The following diagrams show front and rear suspension equivalent stiffness.
Furthermore, it is possible to observe the wheel characteristic angles in both suspensions, i.e. camber angle and steer angle. It is important to have a small steer angle variation to avoid bump steer problems: if the steer angle vary widely with the suspension stroke, the vehicle handling results negatively affected. On the other hand, it is preferable to have an increasing negative camber during suspension stroke in order to contrast tire deformation in cornering and, consequently, have a proper contact area between tire and ground, thus it’s useful to have a decrease of camber angle while the suspension is being loaded. This normally increases the lateral force on the tire.
As it is possible to see in the following diagrams, suspensions partially meet both the aspects of decreasing camber angle and limited variation of steer angle. MacPherson camber angle decreases but in the final part of the stroke camber angle increases.
Here there are two videos showing the path used for suspensions characterization:
– Control system
For the entire model simulation, it is possible to build up a Matlab Simulink control, integrated with LMS. In the following picture it is shown the block diagram used in the simulation:
The forward speed of the vehicle from LMS, measured on the rear axle, is the control input. The forward speed is derived to obtain vehicle acceleration, and is multiplied for the car mass in order to obtain the force. This force is then multiplied for the radius of the wheel and divided by two, to obtain the torque for each rear wheel: this is the output of the control (and the input for the LMS model plantout). An additional term of force is calculated multiplying the square of target force for a coefficient calculated multiplying air density, char area of the vehicle and drag coefficient. To compensate the error between target speed and real forward speed, there is a P.I.D. controller (gains: Kp=2500, Kd=100, Ki=50). A ramp limiter and a saturation for the torque signal is set up in order to avoid loss of grip due to an excessive torque to the wheels. Target speed for all the simulations is generated by a ramp (slope: 2 m/s) and a saturation at 30 m/s (that is 108 km/h) for cornering simulation and 15 m/s (that is 54 km/h) for car over a bump.
– Car simulation while left cornering
The aim of the simulation is to model the behaviour of the whole car while cornering in order to understand forces on the four wheels and, consequently, weight distribution on them. Car speed is costant and settled on 108 km/h while steering wheels turn very slowly. The rack translates 20 mm in 20 seconds: the wheel load changes have to be smooth. Of course when the car starts the corner, loads on external wheels increase and on internal wheels decrease: that happens till there is a loss of grip, after the maximum of lateral force carried out by the wheel. In this simulation, the maximum external front wheel lateral force is considered as the point of loss of grip, anyway both the external wheels lose grip about at the same time. At this time it is also considered the maximum normal force (on external wheels) and the slip angle of each wheel. On the following pictures it is possible to observe particular diagrams using front antiroll stiffness 4000 Nm/rad and rear antiroll stiffness 330 Nm/rad: these values are the torsional stiffnes calculated for the real antiroll bars supposing steel material (from measurement, the diameter of the bars are 27 mm for the front axle and 15 mm for the rear axle). In the pictures is possible to see respectively lateral force, normal force and slip angle (please note that red: front right wheel, green: front left wheel, blue: rear right wheel, yellow: rear left wheel).
Values of lateral forces, normal forces and slip angles are collected while changing antiroll bars stiffness, first plotting diagrams showing influences of varying antiroll bar stiffness on the rear axle, keeping constant the front axle, then plotting diagrams showing influences of varying antiroll bar stiffness on the front axles, keeping constant the rear axle.
Here follow diagrams showing forces and slip angles varying rear antiroll bar stiffness, keeping constant the front one (4000 Nm/rad front, 0-165-330 Nm/rad rear):
Here follow diagrams showing forces and slip angles varying rear antiroll bar stiffness, keeping constant the front one (2000 Nm/rad front, 0-165-330 Nm/rad rear):
Here follow diagrams showing forces and slip angles varying rear antiroll bar stiffness, with no front antiroll (0 Nm/rad front, 0-165-330 Nm/rad rear):
Here follow diagrams showing forces and slip angles varying front antiroll bar stiffness, keeping constant the rear one (0-2000-4000 Nm/rad front, 330 Nm/rad rear):
Here follow diagrams showing forces and slip angles varying front antiroll bar stiffness, keeping constant the rear one (0-2000-4000 Nm/rad front, 165 Nm/rad rear):
Here follow diagrams showing forces and slip angles varying front antiroll bar stiffness, with no rear antiroll (0-2000-4000 Nm/rad front, 0 Nm/rad rear):
First of all it can be noticed that for each wheel the trend of lateral force and normal force is similar, due to the fact that there is an approximately linear dependency between them. Furthermore, it can be appreciate that front antiroll bar has a greater effect on load distribution on the wheels. Considering the analysis that increases just the rear antiroll bar stiffness, it can be noticed that the difference of load on the front wheels decreases and the difference between the rear wheels increases. Conversely, increasing just the front antiroll bar, the difference of load on the rear wheels decreases and the difference between the front wheels increases.
Increasing the rear antiroll bar stiffness, the slip angle on all wheels is slightly increasing. Conversely, increasing the front antiroll bar stiffness, the slip angle decreases.
To observe the behaviour of the vehicle varying stiffness of the suspensions, another couple of spring stiffness values are adopted. To individuate other similar values of equivalent stiffness, it is repeated the comparison made for the suspensions characterization. Once the MacPherson spring stiffness constant is fixed, it is individuated a spline for the Multilink progressive spring in order to make the equivalent stiffness of the suspensions similar. In this way, two configuration for each suspensions are created: the following table shows the values of stiffness and free lenght of the spring for each configuration and suspension.
SPRING STIFFNESS [N/m]
SPRING FREE LENGHT [mm]
In the following picture results of simulations are reported with four configurations of suspensions:
1 – MacPherson: normal configuration; Multilink: normal configuration;
2 – MacPherson: normal configuration; Multilink: sport configuration;
3 – MacPherson: sport configuration; Multilink: normal configuration;
4 – MacPherson: sport configuration; Multilink: sport configuration;
To appreciate only the effect of suspensions stiffness, antiroll bars were disabled in the model.
It is possible to notice that increasing the stiffness of the rear suspension (configuration 2), load on rear right and front left wheels increase, while load on front right and rear left wheels decrease: the trend is to uniform wheel loads of the front axle. On the other hand, increasing only stiffness of front suspension (configuration 3), is it possible to observe the opposite trend: wheel loads are more uniform on the rear axle. With the configuration 4, that is the one with higher stiffness for both suspensions, it is possible to notice that the difference of load of same axle wheels decrease compared to configuration 1. Because of the relationship between normal and lateral force, the trend of lateral forces is the same of normal forces.
Regarding slip angle, increasing stiffness of the rear suspensions (configuration 2), the slip angle of all wheels increase; increasing stiffness of the front suspensions (configuration 3), slip angle of all wheels decrease (more accentuated trend for rear left wheel), except for the rear right wheel (compared to configuration 1).
- Car behaviour while crossing a complete bump and a left side bump
The aim of the simulation is to model the behaviour of the whole car while crossing a complete bump and a left side bump in order to understand how the antiroll bar and the stiffness of spring can affect the comfort (measured as vertical acceleration of the chassis) as well as the handling (measured as chassis roll angle). For this kind of simulations, characterized by transitory dynamic phenomenons of very small duration, it is taken a small integration step (1e-4). With higher values of integration step, it is not possible to observe the behaviour of the car. In these simulations car speed is 54 km/h.
Three configurations have been considered:
- normal spring configuration with no antiroll
- normal spring configuration with 4000 Nm/rad antiroll stiffness on front and 330 Nm/rad antiroll stiffness on rear
- sport spring configuration with no antiroll
In the following video it is shown the car crossing a complete bump, 100 mm on high and 1200 mm on lenght.
In the following pictures it is shown typical trend of the nomal force on all tires and the vertical acceleration of the car center of gravity compared in the different configuration analysed:
It is compared the maximum vertical acceleration and chassis roll of all configurations, in order to compare comfort and handling:
It is possible to appreciate how the antiroll bar does not affect negatively the comfort crossing a complete bump. Conversely, increasing spring stiffness increases the vertical acceleration and consequently the comfort is negatively affected; on the other hand the chassis roll decreases.
Applying the antiroll bar it is possible to see how the chassis roll decreases dramatically without compromising comfort.
At this point it is interesting to simulate the complete car crossing a left side bump, where just the left side wheels are involved. As in the previous simulation it is possible to notice the influence of increasing spring stiffness and applying antiroll bar for the three configurations.
In the following pictures it is shown typical trend of the nomal force on all tires crossing a left side bump and the vertical acceleration of the car center of gravity compared in the different configurations analysed:
It is shown the maximum vertical acceleration and chassis roll in order to compare comfort and handling for all configurations:
It is possible to appreciate how the antiroll bar affect slightly negatively the comfort crossing a left side bump. Conversely, increasing spring stiffness increases the vertical acceleration and consequently the comfort is negatively affected; on the other hand the chassis roll decreases.
Applying the antiroll bar it is possible to see how the chassis roll decreases dramatically with very little losses of comfort.
A properly selected antiroll bar will reduce body roll in corners for improved cornering traction, but will not increase the harshness of the ride, or reduce the effectiveness of the tire to maintain good road surface contact due to overly stiff spring. Considering one axle, the antiroll bar helps increase the mechanical downforce of the outside tire during cornering. This increases the traction of that tire, and that end of the car (front or rear). Conversely, on the opposite axle an uniformity of downforce on tires will occur. The most confortable solution to improve downforce without sacrifing excessively the comfort of the ride is that to use proper antiroll bars, instead of incrementing heavily the stiffness of the suspensions. Many other parameters influenced the behaviour of the car: camber angle, toe in angle, caster angle, king pin angle, pressure of tires and tires blend are only some of them. Furthermore, each parameter controls many sides of the behaviour of a car, like understeer or oversteer, tires consume and temperature and many others in addition to the ones considered in this presentation. It will be interesting to improve this model, making possible to study the influence of all these parameters on the vehicle behaviour.