Summary from the thesis of Fabio Cagnin, graduate in Mechanical Engineering in the University of Padua

fabio.cagnin@hotmail.itThis paper deal with the multibody analysis of the MG0712, vehicle representative the Race Up Team of the University of Padua in the formula SAE competitions. In this work has been used the software LMS Virtual Lab which provides specifics devices to simulate tires contact forces and spring damper actuators.

PURPOSE

The goal of this analysis is to assembly a simplified model of the MG0712 composed by the bodies related to the suspension system and then valuate the cornering behavior of the car, considering the steering performance and roadholding. The results will be compared for two different tire manufacturer and will be determinate the antiroll bar effects.

### ASSEMBLY OF COMPONENTS

#### Chassis:

All the bodies are rigid elements. Except for the chassis, the mass propertys of each body, calculated in relation of the material, are fixed in the center of gravity of body shape. For the chassis the center of gravity is fixed in a specific point which determinate the weight distribution 53% rear and 47% front. The chassis count the mass property of all the non model components like engine, cooling system, safety devices, and driver too, for a total mass of 256 kg.

#### Suspension subsystem:

The suspension parts involved in the submechanism system are Tire, Rim, Hub, Upright, Top Triangle and the Rocker connected to the Top Triangle by a distance constraint that replace the Pullrod. Then the submechanism systems are connected to the chassis trought two spherical joints in the Top Triangle and a revolute joint for the rocker, while the Low Triangle and the steer link are replaced by distance constraints. The overall weight of each wheel group is around 9,5 kg. A device with Stiffness-Damper property is then used to simulate the spring damper force between rocker and chassis.

#### Transmission

Transmission body is modeled by a Rigid body connected to the chassis by revolute joint. There’s not the differential system. The rotation speed is defined trough a spline engine that plot rpm-time. The axles shaft are connected from the transmission body to the hubs. They transmit the same rotational speed to the wheels and they let free the vertical degree of freedom of the suspension.

#### Steering system and Cover

A simplified mechanism plays the kinematic pinion – rack of the original steer.

A bracket joint in the center of gravity of chassis is used to connect the coachwork. Sensor axis system are used to control if the base of the coachwork collide with the ground in the simulations.

### DEVICES TO SIMULATE FORCES

#### Tsda: Translitional Spring Damper Actuator

The TSDA is the device used to simulate spring-damping force. The movement between rocker and chassis is related to the force calculated trought Stiffness and Damping coefficients in the TSDA.

K=61,3 N/mm

C=4000 kg/s

#### Tire Force: complex tire model

The software provides many models of tire to calculate the contact force with ground. In this case is used the Complex Tire which need some input:

- The values of Vertical Stiffness and Vertical Damping to calculate Normal Force.
- The Curve.Friction.Coefficent spline, that defines the value of longitudinal friction coefficient as a function of rotational slip, for the calculation of Longitudinal Force.
- The Table Lateral Stiffness, a table where for some couples of side slip angle and normal force is related a value of lateral force. This table is used by the software to build a spline surface to evaluate the Lateral Force for each couple of slip angle and normal force during the simulation.

The characteristics of the tires of the most used brands are provided to all the team participating in the formula SAE events. In this analysis, in the complex tire model were inserted values related to tire pressure 0,8 bar and camber angle (IA) zero. The results will be compared for Hoosier and Avon tires.

The characteristic curves of the longitudinal force are inserted by the Curve Friction Coefficent which is an average of the longitudinal friction coefficent obtained by the plots of longitudinal force as function of slip ratio and normal force:

The lateral force is calculate using the spline surface defined trought the characteristic curves of lateral force as function of normal force and sideslip angle:

#### Antiroll Bar:

The antiroll bars are modelled in two half-parts each one fixed in the chassis with a revolute joint. The rotational movement of the bar is related to the rotational movement of the adjacent rocker. A device between the half bar is insert to provide the torsional stiffness of 100 Nm/rad to the bar. Therefore the antiroll bar work only for non symmetric stress. In the next simulations the behavior of the car will be compared in 3 cases: without any antiroll bar, with front antiroll, and with rear antiroll.

### SIMULATIONS AND RESULTS

#### Skidpad simulation:

The skidpad test is one of the official races of formula SAE events. The vehicle have to ride a track which have the shape of the 8 and the teams receive score for best lap time. In this simulation the car is driven by a closed loop control that compare the position of the chassis with the track to follow. The circular path have a radius of 9,125 m. Was sought the best performance increasing the speed of the car and comparing results with different tires and different layout of the antiroll bar.

The following table show the best laptime obtained in each case:

As shown in the table, with Hoosier tire the best laptime is obtained with a front antiroll bar, while with Avon tire the best is obtained with rear antiroll bar.

Next plots shown the normal and lateral forces of the wheels in the skidpad simulation for no antiroll bar case:

In the skidpad simulation the steerwheel position is continually changed by the closed loop control in relation of the position of the vehicle in the track. To understand how the antiroll bar work we have to try another simulation in which the steer wheel remains hold in the same position.

#### Simulation of circular path:

In this simulation the steer wheel is blocked in a position that fix the wheels with an angle δ and make the car to tread a circular track with the same radius of the skidpad R_{0}=9,125m. In the joint velocity driver of the transmission a spline curve command the rotational speed starting from zero and it will be increased until the car goes into a spin.

The simulation will be repeat with Hoosier and Avon tires, and with different layout of the antiroll bar. The car can follow the ideal track since the speed is low, but the characteristics of the tires influence the behavior of the car in under steer or over steer when the car speed increase.

The next graphs plot the Normal, Lateral and Longitudinal force in the case with no antiroll bar:

In this circular track simulation is used the same rotational speed in the transmission, but the longitudinal force show different plots with Hoosier and Avon Tires. Remember that in this model there’s no differential system, however we can analyze the differences related to the Tire characteristics.

The Under Steer index is the ratio R_{0}/R that tell us if the car is in understeer or oversteer conditions:

WB = car wheelbase = 1600 mm

δ = steer angle of the wheel = 10°

R_{0} = ideal radius of the curve = p/ δ = 9,125 m

R = effective radius of the traveled route = *v*/ψ

where *v * = car velocity

Ψ = yaw velocity

The following plots show how the ratio R_{0}/R change with the increasing speed with the different tires and how the antiroll bar can modify the behavior of the vehicle.

The plot could explain why in the skidpad simulation the hoosier tire can be a little faster than avon: with Avon tire the ratio R_{0}/R is 0,9, it means that the car is treading a curve with an higher value of radius than the ideal track. So in the skidpad simulation with Avon tire we have to decrease the speed of the car to complete the skidpad without go off-road. Next graphs plots the index R_{0}/R with different antiroll bar layout and they could explain why with avon tire it’s possible to obtain a lower laptime if a rear antiroll is used. As shown in the next plots the Front antiroll bar provide an under steer behavior of the car. Instead the rear antiroll bar determinate an over steer behavior of the car. So, in the skidpad simulation, the Avon tires need to install a rear antiroll bar to correct the route to follow. Conversely, the hoosier tire can reach an higher speed with the front antiroll, without that the index R_{0}/R deviates so much from the value of 1.

### CONCLUSIONS:

The simulations explain how antiroll bar generally modifiy the steering behavior of the car. The front antiroll provide a better roadholding and the vehicle reach an higher speed before going off-road, but the front antiroll bar make the car to follow a curve with higher radius than the one obtained with no bar installed. It means the front antiroll bar make the vehicle to be under steering in relation of the no bar case. Conversely, the rear antiroll bar make the car becoming over steering, and the car go off road with lower value of speed. Therefore, the better configuration of the antiroll bar is not only one, but it depends from the tire characteristics, and we can use the right bar to correct the steering behavior.