Ballotta Giacomo (giacomoballotta@gmail.com)

## INTRODUCTION

The vehicle presented in this project is the FSAEprototype of the University of Padua, which competed in the 2013 season of the series.

Fsae is a students engineering competition where student teams from around the world design, build, test, and race a small-scale formula style racing car. The cars are judged, by industry specialists, on a number of criteria in different typologies of events, both statics and dynamics.

One of the dynamic events is a so called “skid pad”, where the cars have to go round a specified radius circular corner, turning left for 2 laps and then right for other 2 laps.

This event is held to evaluate the steady state cornering capabilities of the cars, thus it has been modeled to find out a basic setup and to evaluate under and oversteering capabilities of the model.

## OBJECTIVES

The first part of the analysis aims to benchmark stiffness and damping of the various spring and damper configurations, starting from the equivalent stiffness given by the kinematics of the suspension, which is a double A-arm with pull rod actuator, then the response of the suspension at different inputs has been investigated with a simulation of a k&c machine:natural frequencies, forces and displacements have been taken into account.

In the second part of the analysis the car runs some skid pad laps and the steady state behavior is benchmarked; in this kind of situations the weight and roll stiffness distributions are the two main factors that influences under and oversteer.

## MODELING

The model is composed by the 4 wheels (rim and tire) fixed on the hubs which is connected with the uprights by a rotational joint, the uprights are then connected by two couples of distance constraints (representing the a-arms) and a single distance constraint representing the toe rod to the chassis .On the rear the toe rods are connected with the chassis, at the front they are connected with a steering rack which is connected to the frame with a translational joint.

A TSDA ( spring and damper ) connects the upper A-Arms to the chassis, and simulates the rocker + spring and damper kinematics (stiffness and movements). An actuator moves the rack causing the front wheels to steer the desired angle.

It has to be taken into account that the model is only considering the roll behavior for the skidpad, consequently the roll stiffness is represented only by springs and the damping is suited for the roll behavior of the car; in the real world you have anti roll bars to get different stiffness in bump and roll, and usually the damping is the same in roll and bump, then it has to be a compromise to suit two different stiffness ( ie roll and bump).

The tire forces are modeled with a simple tire model with the following parameters:

Radius: 255 mm

Damping constant: 3000 kg/s

Rolling resistance: 0.3

Friction coefficient: 1.3 – 1.4

Cornering stiffness: 100000 Nm/ rad

Vertical stiffness: 300000 N/m

NOTE: the friction coefficient is different from front to rear to take into account the different size of the treads front and rear, which are 7” on the rear and 6” on the front.

The spring stiffness and damping coefficient tested with the step input are the following (note that the values are not the virtual springs / dampers @ wheel)

A1: front 750000 N/mm, 23400 kg/s rear : 710000 N/mm, 20900 kg/s

A2: front 900000 N/mm, 25500 kg/s rear : 825000 N/mm, 23000 kg/s

A3: front 1050000 N/mm, 27500 kg/s rear : 994000 N/mm, 25000 kg/s

B1: front 750000 N/mm, 54600 kg/s rear : 710000 N/mm, 48800 kg/s

B2: front 900000 N/mm, 59600 kg/s rear : 825000 N/mm, 53600 kg/s

B3: front 1050000 N/mm, 64300 kg/s rear : 994000 N/mm, 58400 kg/s

The chassis is fixed to the ground for the stiffness test by a bracket joint, which is removed for the other tests.

The weight of the suspended mass (including the driver), is assigned entirely to the chassis, and it is 222 kg while the weight of the non suspended masses is assigned to the single uprights, and it is 10.6 kg for the front and 14.2 kg for the rear; the higher weight on the rear is justified by the drive axle.

The dynamic response of other inputs will be presented only for the two better configurations on the step input.

INPUTS:

Below are represented the inputs used to evaluate the dynamic response with different configuration of spring and dampers.

###### Step:

The step input has been used to find out the two best configurations amongst the possibilities and to present only the most relevant and useful results for the other inputs.

A kerb or a cone hit at speed is a good approximation of a step input

###### Ramp:

The ramp is used to benchmark wheel movement which happens at slower speed than the ramp (ie a change of inclination on the tarmac) or even the initial response to a load transfer.

###### Sine function:

The sine is used to represent road perturbations, depending on the frequency and the amplitude it could be roughness of the asphalt or a more severe pattern like circuit kerbs, as in our case.

Note that the input in this case is cyclic.

## SIMULATIONS AND RESULTS

For the first part of the simulation, as mentioned above, the equivalent stiffness and the natural frequencies of the non suspended masses are obtained using a bracket joint between chassis and the ground and moving 4 patches of “road” elements, one for each tire, which are connected to the ground via a translational joint that allows only vertical displacements.

A joint driver moves the road patches upward following a low angle ramp for the stiffness evaluation, and following the different inputs listed before for the dynamic response of the system.

The bracket joint during the dynamic response simulation is removed to have the correct behavior of the suspended and non suspended mass system.

Here are presented the force and displacements graphs for the four tires in each of the six configuration mentioned in the modeling section.

##### A1) Soft spring and low damping coefficient:

##### B1) Soft spring and high damping coefficient

##### A2) Medium spring and low damping coefficient:

##### B2) Medium spring and high damping coefficient:

##### C1) Hard spring and low damping coefficient:

##### C2) Hard spring and high damping coefficient:

The configuration that better couples low chassis movement and lower time of vertical load loss during the step is as expected the low stiffness, low damping configuration.

It would be compared with the high stiffness, high damping configuration because of the slope of the vertical forces in this case has virtually no overshoot, and this could be a good thing if you’re searching a stable grip during a corner with an obstacle like a bump or a kerb.

Moreover generally for a driver low chassis movements are better for the feeling.

#### Eigenvectors and eigenfrequencies calculation:

The frequencies for the chassis movement have been calculated with the linearization element and are from 3.2 Hz to 4 Hz depending on the springs, it is to be noticed that the Eigenvectors analysis gives four very high frequencies, (magnitude 1E8) that are related to the tire stiffness and damping, and that couldn’t be suppressed without modifying too much the tire parameters.

Video of the Eigenmode of the chassis at 3.2 Hz

Video of the Eigenmode of the front wheels at 11.1 Hz

#### Response to the ramp:

Soft configuration; normal tire force and displacements of rims and chassis:

Hard configuration; normal tire force and displacements of rims and chassis:

#### Response to the sine function:

Soft configuration; normal tire force and displacements of rims and chassis:

Hard configuration; normal tire force and displacements of rims and chassis:

As expected the so called soft configuration gives lower vertical forces at the tire, wich reflects in a bigger “mechanical grip” and a little more body movement, the stiff configuration gives rim and chassis movement more phased, wich could be good for driver feeling and reaction.

#### Skid pad behaviour:

The second part of the simulation is about the skid pad, so the car now is no more constrained over four independent patches but trimmed to let it gain the neutral position on a single road and then a torque is applied symmetrically at the two rear wheels following a spline in order to avoid slip on the first phases; the speed is then stabilized via the very high rolling resistance given to the tire and the steering rack joint driver positions the rack in order to reach the radius of the typical curve (around 18 m).

Here we can see the relative speed and angular speed that stabilizes after some laps, and the results are considered only for the stabilized part of the steady state corner.

The two previous configurations are tested, the values taken into account are the lateral forces acting on the tires (with the goal to maximize it), and the yaw acceleration of the car (aiming to keep it as near to zero as possible) as this values gives a strong indication on the under/oversteering behavior of the car, because in a steady state corner the angular speed should be constant, consequently its variation should be zero.

Here we can see that the relative speeds ( in red the longitudinal component) of the chassis stabilizes in the second part of the simulation.

Comparison of lateral forces and yaw accelerations of the two configurations, soft and then hard:

##### Lateral forces:

##### Yaw accelerations:

A test has been made even with a different stiffness distribution: here we can see the lateral forces with soft spring on the front and stiff spring on the rear:

## CONCLUSIONS

As expected lowering force transmission equates usually with the use of soft spring and dampers, and this analysis justify the fact that often in racing if you’re searching for mechanical grip you try first of all to reduce stiffness, obviously if you’re not concerned with aerodynamics, this is true in the step, sine and ramp test.

On the other side low chassis movements sometimes are useful to let the driver feel better the behaviour of the car, even if higher stiffness is penalising the grip, moreover the ground clearance required always in racing limits the use of even softer springs.

In the skid pad the difference between the two configurations is very subtle because of the fact that in steady state is the stiffness distribution, which is the same, that gives you some difference.

In fact with softer spring on the front the difference is a little higher, but in every skid pad simulation the grip limits of the car are NOT reached, and consequently, as is evident in the yaw acceleration graphs, the car remains neutral.

It would be interesting to investigate further to see how much is the influence of a different stiffness distribution configuration.

All the results of the simulations are valid only for steady state analysis, low stiifness spring and low damping could be bad in transient situations, not only for the driver feeling.