*Numerical and Experimental Studies on Combustion Engines and Vehicles*

#### **1.1 Tyres system**

The tyres system (tyres and rigid suspension) maintains contact with the ground and filters the disturbances imposed by road imperfections [3]. This system allows two motions of the vehicle: displacement in the *z*-direction and a roll rotation around the *x*-axis [4], as shown in **Figure 2**.

### *1.1.1 Kinematic chain for tyres system*

Mechanical systems can be represented by kinematic chains composed of links and joints, which facilitate their modelling and analysis [5–7].

• While tyres 2 and 3 have a lateral deformation and may slide laterally,

contact was modelled as a prismatic joint *P* in the *y*-direction.

*(a) Kinematic chain of the tyres system. (b) Tyres system including actuators.*

by prismatic joints *P*, [11, 12].

*Movement constraints in Tyre-road contact.*

*Stability Analysis of Long Combination Vehicles Using Davies Method*

*DOI: http://dx.doi.org/10.5772/intechopen.92874*

time.

**Figure 5.**

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**Figure 3.**

**Figure 4.**

*Vehicle on a curved path.*

model of the tyres system.

producing a track width change of their respective axles. As a consequence, tyres 2 and 3 have only a constraint on the *z*-direction. Therefore, tyre-ground

• Tyres are assumed as flexible mechanical components and can be represented

• In vehicles with rigid suspension, tyres remain perpendicular to the axle all the

Applying these constraints, **Figure 5a** shows the proposed kinematic chain

The kinematic chain is composed of five links identified by letters A (road), B (outer tyre in the turn), C and D (inner tyre in the turn), and E (vehicle axle); and the five joints are identified by numbers as follows: two revolute joints *R* (tyre-road

The kinematic chain of the tyres system in **Figure 2** has 2-DoF (M *= 2*), the workspace is planar (λ = 3), and the number of independent loops is one (*ν* = 1). Based on the mobility equation, the kinematic chain of tyres system should be composed of five links (*n=5*) and five joints (*j=5*) [7].

To model this system, the following considerations were taken into account:


**Figure 2.** *Tyres system.* *Stability Analysis of Long Combination Vehicles Using Davies Method DOI: http://dx.doi.org/10.5772/intechopen.92874*

#### **Figure 3.** *Movement constraints in Tyre-road contact.*

#### **Figure 4.**

**1.1 Tyres system**

*Simplified trailer model.*

**Figure 1.**

**Figure 2.** *Tyres system.*

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around the *x*-axis [4], as shown in **Figure 2**.

and joints, which facilitate their modelling and analysis [5–7].

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

composed of five links (*n=5*) and five joints (*j=5*) [7].

*1.1.1 Kinematic chain for tyres system*

The tyres system (tyres and rigid suspension) maintains contact with the ground and filters the disturbances imposed by road imperfections [3]. This system allows two motions of the vehicle: displacement in the *z*-direction and a roll rotation

Mechanical systems can be represented by kinematic chains composed of links

The kinematic chain of the tyres system in **Figure 2** has 2-DoF (M *= 2*), the workspace is planar (λ = 3), and the number of independent loops is one (*ν* = 1). Based on the mobility equation, the kinematic chain of tyres system should be

To model this system, the following considerations were taken into account:

or brake force, *Fyi* is the lateral force, and *Fzi* is the normal force;

ground contact was modelled as a pure revolute joint *R* along *x*-axis.

• There are up to three different components of forces acting on the tyre-ground contact *i* of the vehicle [8–10], as shown in **Figure 3**, where *Fxi* is the traction

• However, at rollover threshold, tyres 1 and 4 (outer tyres in the turn, **Figure 4**) receive greater normal force than tyres 2 and 3 (inner tyre in the turn, **Figure 4**), and thus tyres 1 and 4 are not prone to slide laterally. We consider that tyres 1 and 4 only allow vehicle rotation along the *x*-axis. Therefore, tyre*Vehicle on a curved path.*


Applying these constraints, **Figure 5a** shows the proposed kinematic chain model of the tyres system.

The kinematic chain is composed of five links identified by letters A (road), B (outer tyre in the turn), C and D (inner tyre in the turn), and E (vehicle axle); and the five joints are identified by numbers as follows: two revolute joints *R* (tyre-road

**Figure 5.** *(a) Kinematic chain of the tyres system. (b) Tyres system including actuators.*

contact of joints 1 and 4) and three prismatic joints *P*, two that represent tyres of the system (2 and 5), and one the lateral slide of tyre 2 (3).

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

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**.

#### *1.1.2 Kinematics of tyre system*

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 passive actuators of the mechanism, as shown in **Figure 7**.

Eqs. (1)–(5) define the kinematics of the tyres system.

$$l\_i = \delta\_T + l\_r = \frac{\Im \Delta F}{k\_t + a\_c} + l\_r \approx \frac{-F\_{Ti} + F\_{xi}^{start}}{k\_T} + l\_r \tag{1}$$

$$\beta\_i = \Re 0 - \arcsin\left(\frac{l\_{i+1}}{\sqrt{t\_{i+1}^2 + l\_{i+1}^2}}\right) \tag{2}$$

*ti+1* is the axle width, and *θi;j* are the rotation angles of the revolute joints *i* and *j*

*Stability Analysis of Long Combination Vehicles Using Davies Method*

*DOI: http://dx.doi.org/10.5772/intechopen.92874*

assumed that the vehicle has this suspension on the front and rear axles.

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**.

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

The rigid suspension is a mechanism that allows the following movements of the

The system of **Figure 10a** has 3-DoF (M *= 3*), the workspace is planar (*λ = 3*) and the number of independent loops is one (*ν* = 1). From the mobility equation, the kinematic chain of suspension system should be composed of six links (*n=6*) and

To model this system the following consideration is considered: leaf springs are assumed as flexible mechanical components with an axial deformation and a small shear deformation, and can be represented by prismatic joints *P* supported in

To allow the rotation of the body in the z-axis, the link between the body and the leaf spring is made with revolute joint. Applying these concepts to the system, a

model with the configuration shown in **Figure 10b** is proposed.

*Solid axle with leaf spring suspension. Source: Adapted from Rill et al. [15].*

respectively.

six joints (*j=6*).

**Figure 8.**

**Figure 9.**

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*(a) Body motion. (b) Suspension system.*

revolute joints *R* [16].

**1.2 Suspension system**

*1.2.1 Kinematic chain of the suspension system*

$$t\_i = \sqrt{t\_{i+1}^2 + l\_{i+1}^2 + l\_i^2 - 2\left(\sqrt{t\_{i+1}^2 + l\_{i+1}^2}\right)} l\_i \cos\left(\beta\_i\right) \tag{3}$$

$$\delta\_{\dot{i}} = \arcsin\left(l\_{\dot{i}}\sin\left(\beta\_{\dot{i}}\right)/\mathfrak{t}\_{\dot{i}}\right) \tag{4}$$

$$
\theta\_i = \theta\_j = \mathbf{9} \mathbf{0} - \delta\_i - \beta\_i \tag{5}
$$

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 rolling radius of the tyre *i*, *Fstart zi* is the initial normal load *i*, *kT* is the equivalent tyre vertical stiffness, *lr* is the initial dynamic rolling radius of tyre *i*, *ti* is the track width,

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

**Figure 7.** *Movement of tyres system.*

*ti+1* is the axle width, and *θi;j* are the rotation angles of the revolute joints *i* and *j* respectively.
