## Introduction

The De Dion tube is a particular type of automobile rear suspension, developed for the first time in 1894 by Charles Trepardoux the co-founder, with Albert De Dion, of the automobile manufacturer De Dion-Bouton. During the years several car manufacturers used this technology like Aston Martin, Lancia, Opel and in recent times even Smart. However probably the car manufacturer that adopted most this technology is Alfa Romeo specially during the ’70s and ’80s. In fact De Dion tube is particularly useful for rear-wheel drive and , in Alfa Romeo’s case, applied with a transaxle system.

The picture represents the De Dion tube of an Alfa Romeo 75. In this case it is composed by a triangular structure of solid tubolar beams used to hold the opposite wheels in parallel, in fact this is a rigid axle suspension, meaning that wheels can’t move independently. There is also a Watt transverse linkage that guides vertical displacements.

The main advantage of this system is the reduction of unsprung weights since the differential, half shafts and brake system (for Alfa Romeo) are connected to the chassis. On the other side the main disadvantages are the decrease of the footprint of the wheel during the rolling phase because of the rigidity and a greater construction complexity.

## Objectives

This projects objectives are the following:

- Kinematic analysis of De Dion tube considering different configurations of constraints and rigid bodies to recreate the real multi-body system composed by flexible bodies.
- Kinematic analysis after a vertical displacement and calculation of the rolling center along longitudinal plane.
- Kinematic analysis after rolling phase and calculation of the rolling center along frontal plane.Using these fundamental formulas:

## The modelling problem

The model is composed by 15 bodies starting from the red “triangle” wich is connected to the two wheels whose vertical movement is controled by two cylinder, while the rotation is locked and controlled by two links that are connected to the differential box. Finally there is a Watt transverse linkage to assure a vertical displacement.

List of bodies and joints:

Model used in this study in XY plane and YZ plane:

A very simplified chassis called “mass” was modelled and fixed in the ground but it was hid for better visual clarity to the other more important bodies.The interaction of these bodies depends on the objective considered.

Knowing that F=fixed, R=revolute, T=tranlsational, S=spherical, C=cylindrical, PI=inplane, H=Hooke joint, the next kinematic chain was developed for objectives 2 and 3, while for objective 1 are used the joints signed with the number (1).

The two translational motion trasl_cyl_dx and sx are used to control the vertical displacement of the wheels; the two translational motion trasl_link_dx and sx are used to lock or release the relative internal translation of link_dx and link_sx. Finally rot_diff and rel_trasl_diff_differential are used to lock the absolute rotation of diff and the relative translation between diff and differential.

Two springs and two dumpers are inserted between mass and triangle with the default values that Adams View gives: k=0,16N/mm ; c=3.1Ns/mm. This is a kinematic analysis, so it isn’t so important to know the real springs and dumpers’ value; but it’s important to consider them from a modelling point of view.

## Simulation and analysis of results

The simulation starts with all joints and motions activated, so there are same warnings. However these warnings aren’t a problem because during simulations, thanks to scripted simulations, same joints and motions will be deactivated.

1a) Simulation of a rigid configuration: triangle connected to mass with a spherical joint and no internal translational displacement along left and right links.

Grubler count: N=-4 because of 0 dof and 4 redundant constraints.

The simulation fails as soon as it starts because the motion is completely locked. In fact, even if the constraints that are used are the real ones, all bodies are rigid while in reality they are flexible and thanks to this flexibility the motion is allowed. In particular the flexion of Watt transverse linkage and the torsion of “triangle” body. So if we want to study the kinematic using rigid bodies we have to release same dof.

1b) Simulation of a less rigid configuration: triangle connected to mass with a spherical joint in the tip and internal translational displacement along left and right links released.

Grubler count: N=2 because of 2 dof and 0 redundant constraints.

Even if the links’ internal displacement are released the joints relative to the watt linkage have to be changed. In fact Watt linkage can’t allows just planar displacement along XY plane because we are imposing a rotation along YZ plane around a spherical joint. So we have to disability “wattdx_mass_R” and “wattsx_mass_R” and ability “wattdx_mass_C” and “wattsx_mass_C”. passing from two revolute joints to two spherical joints. In this way also a longitudinal (z) displacement is allowed, whose maximum value is 6mm, that is completely compliant with the length of link’s Watt linkage.

2) Kinematic analysis after a vertical displacement and calculation of the rolling center along longitudinal plane. The model is handled like in point 1b and this means we are considering the flexibility of the system with the axial displacements of “link_sx” and “link_dx” and the mobility of Watt linkage even along z axis.

Grubler count: N=2 because of 2 dof and 0 redundant constraints.

Vertical motion obtained considering the function: [step(time,0,0,1,-100)+step(time,1,-100,2,0)+100] applied to both cylinder at the same time. Calculation of rolling center respect to Om=marker_5 in “triangle” considering formulas above but along the plane ZY.

Because there’s a spherical joint in the triangle’s tip, a rolling center is imposed by this joint. In fact the rolling center coordinates are constant: Y=0mm and Z=1200mm that are the coordinates of the spherical joint. It’s also important to underline that rolling center’s Y-coordinate has two picks caused by the changing of motion direction that causes an angular velocity equal to zero.

Finally we can analyse rolling center along YZ plane:

3) Kinematic analysis after rolling phase and calculation of the rolling center along frontal plane. Same model used before, but to create a rolling one cylinder is locked and the other one is vertical moved.

Grubler count: N=2 because of 2 dof and 0 redundant constraints.

Calculation of rolling center respect to Om=marker_15 in “triangle” considering formulas .

Zooming the X coordinate of the rolling center:

Because the vertical displacement of the “wheel_dx” is locked and the rigidity of the system causes a rotation of the “wheel_dx”. The rolling center is not exactly in the contact point between “wheel_dx” and “cylinder_dx”. However it is near it.

Finally we can analyse rolling center along XY plane:

## Conclusions

This project had the purpose of modelling the De Dion tube, a particular type of automobile rear solid suspension. The motion of this system is based on the flexibility of its parts but, wanting to model it using rigid bodies, it was necessary first of all to understand how recreate this flexibility releasing same degrees of freedom thanks to particular joints instead. After same tries the best solution was releasing the longitudinal dof of the Watt linkage. In this way the rolling center was calculated: along the longitudinal plane after a vertical displacement and along the frontal plane after a rolling phase.

Thanks to this model it is possible to calculate the rolling center of the system during the rolling phase just using rigid bodies.