**1.2 Suspension system**

contact of joints 1 and 4) and three prismatic joints *P*, two that represent tyres of the

The mechanism of **Figure 5a** has 2-DoF, and it requires two actuators to control its movement. The mechanism has a passive actuator in each prismatic joint of tyres (2 and 5—axial deformation); these actuators control the movement along the *x-*

In this model, the revolute joint (3) and the prismatic joint (4) can be changed by a spherical slider joint (*Sd*), with constraint in the *z*-axis, as shown in **Figure 6**.

The movement of this system is orientated by the forces acting on the mechanism (trailer weight (*W*) and the inertial force (*may*)) [13]. These forces affect the

<sup>þ</sup> *lr* <sup>≈</sup> �*FTi* <sup>þ</sup> *<sup>F</sup>start*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *t* 2 *<sup>i</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>l</sup>* 2 *i*þ1

� � q

*t* 2 *<sup>i</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>l</sup>* 2 *i*þ1

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0

B@

where *δ<sup>T</sup>* is the normal deformation of the tyre [14], Δ*F* is the algebraic change in the initial load, *kt* is the vertical stiffness of the tyre, *ac* is the regression coefficient, *FTi* is the instantaneous tyre normal load, *li* is the instantaneous dynamic

vertical stiffness, *lr* is the initial dynamic rolling radius of tyre *i*, *ti* is the track width,

*zi kT*

1

*δ<sup>i</sup>* ¼ *arcsin li sin β<sup>i</sup>* ð Þ ð Þ*=ti* (4) *θ<sup>i</sup>* ¼ *θ <sup>j</sup>* ¼ 90 � *δ<sup>i</sup>* � *β<sup>i</sup>* (5)

*zi* is the initial normal load *i*, *kT* is the equivalent tyre

*li cos β<sup>i</sup>* ð Þ

þ *lr* (1)

(3)

CA (2)

system (2 and 5), and one the lateral slide of tyre 2 (3).

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

passive actuators of the mechanism, as shown in **Figure 7**. Eqs. (1)–(5) define the kinematics of the tyres system.

*li* <sup>¼</sup> *<sup>δ</sup><sup>T</sup>* <sup>þ</sup> *lr* <sup>¼</sup> <sup>3</sup>Δ*<sup>F</sup>*

*kt* þ *ac*

*<sup>β</sup><sup>i</sup>* <sup>¼</sup> <sup>90</sup> � *arcsin li*þ<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

and *z*-axes, as shown in **Figure 5b**.

*ti* ¼

rolling radius of the tyre *i*, *Fstart*

**Figure 6.** *Tyres system model.*

**Figure 7.**

**134**

*Movement of tyres system.*

*t* 2 *<sup>i</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>l</sup>* 2 *<sup>i</sup>*þ<sup>1</sup> þ *l* 2 *<sup>i</sup>* � 2

s

*1.1.2 Kinematics of tyre system*

This system comprises the linkage between the sprung and unsprung masses of a vehicle, which reduces the movement of the sprung mass, allowing tyres to maintain contact with the ground, and filtering disturbances imposed by the ground [3]. In heavy vehicles, the suspension system most used is the leaf spring suspension or rigid suspension [15], as shown in **Figure 8**. For developing this model (trailer), it is assumed that the vehicle has this suspension on the front and rear axles.

The rigid suspension is a mechanism that allows the following movements of the vehicle's body under the action of lateral forces: displacement in the *z-* and *y*direction and a roll rotation about the *x*-axis [1, 8], as shown in **Figure 9a** and **b**.
