*1.2.1 Kinematic chain of the suspension system*

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 six joints (*j=6*).

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 revolute joints *R* [16].

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.

**Figure 8.** *Solid axle with leaf spring suspension. Source: Adapted from Rill et al. [15].*

**Figure 9.** *(a) Body motion. (b) Suspension system.*

#### **Figure 10.**

*(a) Movement of suspension system. (b) Kinematic chain of suspension system. (c) Suspension system including actuators.*

The system is composed of six links identified by letters D (vehicle axle), E and F (spring 1), G and H (spring 2), and I (the vehicle body), and the six joints identified by the following numbers: four revolute joints *R* (5, 7, 8, and 10) and two prismatic joints *P* that represent the leaf springs of the system (6 and 9), as shown in **Figure 10b**.

The mechanism of **Figure 10b** has 3-DoF, and it requires three actuators to control its movements, applying the technique developed in Section 1.1, the kinematic chain has a passive actuator in the prismatic joints 6 and 9 (axial deformation of the leaf spring), and a passive actuator in the joints 6 and 9 (torsion spring shear deformation); but the mechanism with four passive actuators is overconstrained, in this case only one equivalent passive actuator is used in the joint 5 or 8, as shown in **Figure 10c**.

#### *1.2.2 Kinematics of the suspension system*

The movement of the suspension is orientated first by the movement of the tyres system, and second by forces acting on the mechanism (vehicle weight (*W*) and inertial force (*may*)). These forces affect the passive actuators of the mechanism, as shown in **Figure 11**.

Eqs. (6)–(12) define the kinematics of the suspension system:

$$
\theta\_{\mathfrak{n}} = \mathbf{r}\_{\mathfrak{m}} \check{\mathsf{z}}\_{\mathfrak{k}\_{\mathfrak{n}}} \tag{6}
$$

*<sup>θ</sup>n*þ<sup>1</sup> <sup>¼</sup> *<sup>β</sup><sup>n</sup>* <sup>þ</sup> *arcsin <sup>b</sup>*

*Stability Analysis of Long Combination Vehicles Using Davies Method*

*r*

*<sup>θ</sup>n*þ<sup>2</sup> <sup>¼</sup> *<sup>θ</sup><sup>n</sup>* <sup>þ</sup> *arcsin <sup>b</sup>*

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

**1.3 The fifth wheel system**

the trailer to one side [18].

**Figure 12.**

**137**

consists of the kingpin and the bolster plate.

and more flexibility of the chassis, as shown in **Figures 12–14**.

(**Figure 15**), it is located over the front suspension mechanism.

*Movement of the fifth wheel—Starting uphill. Source: Adapted from Saf-Holland [18].*

*r*

where *Txn* is the moment around the *x*-axis on the joint *n*, *kts* is the spring's torsion coefficient, *δLS* is the leaf spring deformation [17], Δ*F* is the algebraic change in the initial load, *l* is the length of the leaf spring, *N* is the number of leaves, *B* is the width of the leaf,*T* is the thickness of the leaf, *Es* is the modulus of elasticity of a multiple leaf, *ln* is the instantaneous height of the leaf spring *n*, *FLSn* is the spring normal force *n*, *ls* is the initial suspension height, *kLs* is the equivalent stiffness of the

suspension, *ln* is the instantaneous height of the leaf spring *n*, *b* is the lateral separation between the springs, and *θ<sup>n</sup>* is the rotation angle of the revolute joint *n*.

This system is a coupling device between the tractive unit and the trailer; but in the case of a multiple trailer train, a fifth wheel also can be located on a lead trailer. The fifth wheel allows articulation between the tractive and the towed units. This system consists of a wheel-shaped deck plate usually designed to tilt or oscillate on mounting pins. The assembly is bolted to the frame of the tractive unit. A sector is cut away in the fifth wheel plate (sometimes called a throat), allowing a trailer kingpin to engage with locking jaws in the centre of the fifth wheel [18]. The trailer kingpin is mounted in the trailer upper coupler assembly. The upper coupler

When the vehicle makes different manoeuvres (starting to go uphill or downhill, and during cornering) [18], the fifth wheel allows the free movement of the trailer

Rotation about the longitudinal axis of up to 3° of movement between the tractor and trailer is permitted. On a standard fifth wheel, this occurs as a result of clearance between the fifth wheel to bracket fit, compression of the rubber bushes, and also the vertical movement between the kingpin and locks may allow some lift of

Consider the third movement of the trailer, the mechanism that represents the

fifth wheel has similar design and movements to the suspension mechanism

*sin* ð Þ 90 þ *θ<sup>n</sup>* 

*<sup>θ</sup>n*þ<sup>3</sup> <sup>¼</sup> <sup>90</sup> � *<sup>β</sup><sup>n</sup>* � *arcsin <sup>b</sup>*

*sin* ð Þ 90 þ *θ<sup>n</sup>* 

*ln*þ<sup>1</sup>

� *arcsin <sup>b</sup>*

*ln*þ<sup>1</sup>

� 90 (10)

*sin <sup>β</sup><sup>n</sup>* ð Þ (11)

*sin <sup>β</sup><sup>n</sup>* ð Þ (12)

$$l\_n = \delta\_{LS} + l\_s = \frac{3\Delta F l^3}{8E\_s NBT^3} + l\_s \approx \frac{-F\_{LSn} + F\_{xi}^{start}}{k\_{Ls}} + l\_s \tag{7}$$

$$r = \sqrt{l\_n^2 + b^2 - 2l\_nb\cos\left(90 + \theta\_6\right)}\tag{8}$$

$$\beta\_n = \arccos\left(\left(b^2 + r^2 - l\_4^2\right) / (2br)\right) \tag{9}$$

**Figure 11.** *Movement of suspension system.*

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

$$
\theta\_{n+1} = \beta\_n + \arcsin\left(\frac{b}{r}\sin\left(\Re \mathbf{0} + \theta\_n\right)\right) - \mathbf{90} \tag{10}
$$

$$
\theta\_{n+2} = \theta\_n + \arcsin\left(\frac{b}{r}\sin\left(90 + \theta\_n\right)\right) - \arcsin\left(\frac{b}{l\_{n+1}}\sin\left(\beta\_n\right)\right) \tag{11}
$$

$$
\theta\_{n+3} = \theta 0 - \beta\_n - \arcsin\left(\frac{b}{l\_{n+1}} \sin\left(\beta\_n\right)\right) \tag{12}
$$

where *Txn* is the moment around the *x*-axis on the joint *n*, *kts* is the spring's torsion coefficient, *δLS* is the leaf spring deformation [17], Δ*F* is the algebraic change in the initial load, *l* is the length of the leaf spring, *N* is the number of leaves, *B* is the width of the leaf,*T* is the thickness of the leaf, *Es* is the modulus of elasticity of a multiple leaf, *ln* is the instantaneous height of the leaf spring *n*, *FLSn* is the spring normal force *n*, *ls* is the initial suspension height, *kLs* is the equivalent stiffness of the suspension, *ln* is the instantaneous height of the leaf spring *n*, *b* is the lateral separation between the springs, and *θ<sup>n</sup>* is the rotation angle of the revolute joint *n*.

#### **1.3 The fifth wheel system**

The system is composed of six links identified by letters D (vehicle axle), E and

*(a) Movement of suspension system. (b) Kinematic chain of suspension system. (c) Suspension system including*

The mechanism of **Figure 10b** has 3-DoF, and it requires three actuators to control its movements, applying the technique developed in Section 1.1, the kinematic chain has a passive actuator in the prismatic joints 6 and 9 (axial deformation of the leaf spring), and a passive actuator in the joints 6 and 9 (torsion spring shear deformation); but the mechanism with four passive actuators is over-

constrained, in this case only one equivalent passive actuator is used in the joint 5 or

The movement of the suspension is orientated first by the movement of the tyres system, and second by forces acting on the mechanism (vehicle weight (*W*) and inertial force (*may*)). These forces affect the passive actuators of the mechanism, as

> *θ<sup>n</sup>* ¼ *Txn=*

<sup>8</sup>*EsNBT*<sup>3</sup> <sup>þ</sup> *ls* <sup>≈</sup> �*FLSn* <sup>þ</sup> *<sup>F</sup>start*

<sup>2</sup> � *<sup>l</sup>* 2 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>n</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> � <sup>2</sup>*lnb cos*ð Þ <sup>90</sup> <sup>þ</sup> *<sup>θ</sup>*<sup>6</sup>

*kts* (6)

þ *ls* (7)

(8)

*zi*

� �*=*ð Þ <sup>2</sup>*br* � � (9)

*kLs*

Eqs. (6)–(12) define the kinematics of the suspension system:

*ln* <sup>¼</sup> *<sup>δ</sup>LS* <sup>þ</sup> *ls* <sup>¼</sup> <sup>3</sup>*ΔFl*<sup>3</sup>

*l* 2

*<sup>β</sup><sup>n</sup>* <sup>¼</sup> *arccos b*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*

q

*r* ¼

F (spring 1), G and H (spring 2), and I (the vehicle body), and the six joints identified by the following numbers: four revolute joints *R* (5, 7, 8, and 10) and two prismatic joints *P* that represent the leaf springs of the system (6 and 9), as shown

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

in **Figure 10b**.

**Figure 10.**

*actuators.*

8, as shown in **Figure 10c**.

shown in **Figure 11**.

**Figure 11.**

**136**

*Movement of suspension system.*

*1.2.2 Kinematics of the suspension system*

This system is a coupling device between the tractive unit and the trailer; but in the case of a multiple trailer train, a fifth wheel also can be located on a lead trailer. The fifth wheel allows articulation between the tractive and the towed units.

This system consists of a wheel-shaped deck plate usually designed to tilt or oscillate on mounting pins. The assembly is bolted to the frame of the tractive unit. A sector is cut away in the fifth wheel plate (sometimes called a throat), allowing a trailer kingpin to engage with locking jaws in the centre of the fifth wheel [18]. The trailer kingpin is mounted in the trailer upper coupler assembly. The upper coupler consists of the kingpin and the bolster plate.

When the vehicle makes different manoeuvres (starting to go uphill or downhill, and during cornering) [18], the fifth wheel allows the free movement of the trailer and more flexibility of the chassis, as shown in **Figures 12–14**.

Rotation about the longitudinal axis of up to 3° of movement between the tractor and trailer is permitted. On a standard fifth wheel, this occurs as a result of clearance between the fifth wheel to bracket fit, compression of the rubber bushes, and also the vertical movement between the kingpin and locks may allow some lift of the trailer to one side [18].

Consider the third movement of the trailer, the mechanism that represents the fifth wheel has similar design and movements to the suspension mechanism (**Figure 15**), it is located over the front suspension mechanism.

**Figure 12.** *Movement of the fifth wheel—Starting uphill. Source: Adapted from Saf-Holland [18].*

because the lateral load transfer is different on the axles of the vehicle. Then, applying the torsion theory, the vehicle frame has similar behaviour with a statically

*T <sup>f</sup> a*

*T <sup>f</sup>* þ *Tr*

Here *TCG* is the torque applied by the forces acting on the *CG*,*Tf* (*T28*) is the torque applied on the vehicle front axle,*Tr* (*T27*) is the torque applied on the vehicle rear axle, *a* is the distance from the front axle to the centre of gravity, and *L* is the wheelbase of the trailer. Applying torsion theory to the statically indeterminate

*<sup>J</sup> fG* <sup>¼</sup> *Tr*ð Þ *<sup>L</sup>* � *<sup>a</sup>*

where *Jf* and *Jr* are the equivalent polar moments on front and rear sections of the vehicle frame respectively, and *G* is the modulus of rigidity (or shear modulus). According to Kamnik et al. [21] when a trailer model makes a spiral manoeuvre, the *LLT* on the rear axle is greater than the *LLT* on the front axle; therefore the equivalent polar moment on the rear (*Jr*) is greater than the equivalent polar moment on the front (*Jf*). These can be expressed as *Jr* = *x Jf* (where *x* is the constant that allows controlling the torque distribution of the chassis); replacing and simplifying Eq. (13):

*a* � *L*

However, when the trailer model makes a turn, the torque applied on the vehicle

front axle has two components, as shown in **Figure 17** and Eqs. (15) and (16).

*JrG* (13)

*ax* <sup>¼</sup> <sup>0</sup> (14)

*Tfx* ¼ *T <sup>f</sup>* cos *ψ* (15)

indeterminate torsional shaft, as shown in **Figure 16**.

*Stability Analysis of Long Combination Vehicles Using Davies Method*

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

shaft, the next equation is defined:

**Figure 16.**

**Figure 17.** *Torque components.*

**139**

*Kinematic chain of the chassis.*

#### **Figure 13.**

*Movement of the fifth wheel—Starting downhill. Source: Adapted from Saf-Holland [18].*

#### **Figure 14.** *Movement of the fifth wheel—Rotation* x*-axis. Source: Adapted from Saf-Holland [18].*

Here *lfw* is the fifth wheel system's instantaneous height, *FFWn* is the fifth wheel normal load, and *lfi* is the fifth wheel system's initial height, *b1* is the fifth wheel width.

#### **1.4 The chassis**

The chassis is the backbone of the trailer, and it integrates the main truck component systems such as the axles, suspension, power train, and cab. The chassis is also an important part that contributes to the dynamic performance of the whole vehicle. One of the truck's important dynamic properties is the torsional stiffness, which causes different lateral load transfers (LLT) on the axles of the vehicle [19].

According to Winkler [20] and Rill [4], the chassis has significant torsional compliance, which would allow its front and rear parts to roll independently; this is *Stability Analysis of Long Combination Vehicles Using Davies Method DOI: http://dx.doi.org/10.5772/intechopen.92874*

**Figure 16.** *Kinematic chain of the chassis.*

because the lateral load transfer is different on the axles of the vehicle. Then, applying the torsion theory, the vehicle frame has similar behaviour with a statically indeterminate torsional shaft, as shown in **Figure 16**.

Here *TCG* is the torque applied by the forces acting on the *CG*,*Tf* (*T28*) is the torque applied on the vehicle front axle,*Tr* (*T27*) is the torque applied on the vehicle rear axle, *a* is the distance from the front axle to the centre of gravity, and *L* is the wheelbase of the trailer. Applying torsion theory to the statically indeterminate shaft, the next equation is defined:

$$\frac{T\_f a}{J\_f G} = \frac{T\_r (L - a)}{J\_r G} \tag{13}$$

where *Jf* and *Jr* are the equivalent polar moments on front and rear sections of the vehicle frame respectively, and *G* is the modulus of rigidity (or shear modulus).

According to Kamnik et al. [21] when a trailer model makes a spiral manoeuvre, the *LLT* on the rear axle is greater than the *LLT* on the front axle; therefore the equivalent polar moment on the rear (*Jr*) is greater than the equivalent polar moment on the front (*Jf*). These can be expressed as *Jr* = *x Jf* (where *x* is the constant that allows controlling the torque distribution of the chassis); replacing and simplifying Eq. (13):

$$T\_f + T\_r \left(\frac{a - L}{a\kappa}\right) = 0\tag{14}$$

However, when the trailer model makes a turn, the torque applied on the vehicle front axle has two components, as shown in **Figure 17** and Eqs. (15) and (16).

$$T\_{f\mathbf{x}} = T\_f \cos \psi \tag{15}$$

**Figure 17.** *Torque components.*

Here *lfw* is the fifth wheel system's instantaneous height, *FFWn* is the fifth wheel normal load, and *lfi* is the fifth wheel system's initial height, *b1* is the fifth wheel

*Movement of the fifth wheel—Rotation* x*-axis. Source: Adapted from Saf-Holland [18].*

*Movement of the fifth wheel—Starting downhill. Source: Adapted from Saf-Holland [18].*

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

The chassis is the backbone of the trailer, and it integrates the main truck component systems such as the axles, suspension, power train, and cab. The chassis is also an important part that contributes to the dynamic performance of the whole vehicle. One of the truck's important dynamic properties is the torsional stiffness, which causes different lateral load transfers (LLT) on the axles of the vehicle [19]. According to Winkler [20] and Rill [4], the chassis has significant torsional compliance, which would allow its front and rear parts to roll independently; this is

width.

**138**

**Figure 15.**

**Figure 14.**

**Figure 13.**

**1.4 The chassis**

*Kinematic chain of fifth wheel system.*

**Figure 18.** *Trailer model.*

$$T\_{\hat{f}\hat{\mathbb{Y}}} = T\_f \cos \psi \tag{16}$$

Usually, the national regulation boards establish the maximum load capacity of the axles of *LCVs*; this is based on the design load capacity of the pavement and bridges, so each country has its regulations. In this scope, the designers develop their products considering that the trailer is loaded uniformly, causing the axle's load distribution to be in accordance with the laws. **Figure 19** shows the example of

However, some loading does not properly distribute the load, which ultimately changes the centre of gravity of the trailer forwards or backwards, as shown in

In **Figures 19** and **20**, *Ff* and *Fr* are the forces acting on the front and rear axles

Generally, the *CG* position is dependent on the type of cargo, and the load distribution on the trailer and it varies in three directions: longitudinal (*x*-axis),

Here, *d1* denotes the lateral *CG* displacement, *d2* denotes the longitudinal *CG*

Furthermore, **Figure 22a** and **b** show that only the weight (*W*) and the lateral inertial force (*may*) act on the trailer *CG*, but, when the model takes into account

lateral (*y*-axis), and vertical (*z*-axis), as shown **Figure 21**.

*Stability Analysis of Long Combination Vehicles Using Davies Method*

displacement, and *d3* the vertical *CG* displacement.

the normal load distribution.

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

**Figure 20** respectively.

respectively.

**Figure 19.**

**Figure 20.**

**Figure 21.** CG *displacements.*

**141**

*Longitudinal* CG *movement.*

*Normal load distribution.*

where *Tfx* (*Tx28*) is the torque applied in the *x*-axis (this torque acts on the lateral load transfer on front axle),*Tfy* (*Ty28*) is the torque applied in the *y*-axis, and *ψ* is the steering angle of trailer front axle.

#### **1.5 Three-dimensional trailer model**

Considering the systems developed, the model of the trailer (**Figure 18**) is composed of the following mechanisms:


The kinematic chain of the trailer model (**Figure 18**) is composed of 28 joints (*j* = 28; 14 revolute joints 'R', 10 prismatic joints 'P', 2 spherical joints 'S', 2 spherical slider joints 'Sd'), and 23 links (*n* = 23).

#### **2. Static analysis of the mechanism**

Several methodologies allow us to obtain a complete static analysis of the mechanism. For this purpose, the Davies method was used to analyse the mechanisms statically [11, 22–29]. This method was selected because it offers a straightforward way to obtain a static model of the mechanism, and this model can be easily adaptable using this approach.

#### **2.1 External forces and load distribution**

In the majority of *LCVs*, the load on the trailers is fixed and nominally centred; for this reason, the initial position lateral of the centre of gravity is centred and symmetric. *Stability Analysis of Long Combination Vehicles Using Davies Method DOI: http://dx.doi.org/10.5772/intechopen.92874*

Usually, the national regulation boards establish the maximum load capacity of the axles of *LCVs*; this is based on the design load capacity of the pavement and bridges, so each country has its regulations. In this scope, the designers develop their products considering that the trailer is loaded uniformly, causing the axle's load distribution to be in accordance with the laws. **Figure 19** shows the example of the normal load distribution.

However, some loading does not properly distribute the load, which ultimately changes the centre of gravity of the trailer forwards or backwards, as shown in **Figure 20** respectively.

In **Figures 19** and **20**, *Ff* and *Fr* are the forces acting on the front and rear axles respectively.

Generally, the *CG* position is dependent on the type of cargo, and the load distribution on the trailer and it varies in three directions: longitudinal (*x*-axis), lateral (*y*-axis), and vertical (*z*-axis), as shown **Figure 21**.

Here, *d1* denotes the lateral *CG* displacement, *d2* denotes the longitudinal *CG* displacement, and *d3* the vertical *CG* displacement.

Furthermore, **Figure 22a** and **b** show that only the weight (*W*) and the lateral inertial force (*may*) act on the trailer *CG*, but, when the model takes into account

**Figure 19.** *Normal load distribution.*

*Tfy* ¼ *T <sup>f</sup>* cos *ψ* (16)

where *Tfx* (*Tx28*) is the torque applied in the *x*-axis (this torque acts on the lateral load transfer on front axle),*Tfy* (*Ty28*) is the torque applied in the *y*-axis, and *ψ* is the

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

Considering the systems developed, the model of the trailer (**Figure 18**) is

• the front mechanism of the trailer is composed for the tyres, the suspension,

• the last mechanism is the chassis and links the front and rear mechanism of the

The kinematic chain of the trailer model (**Figure 18**) is composed of 28 joints (*j* = 28; 14 revolute joints 'R', 10 prismatic joints 'P', 2 spherical joints 'S', 2 spherical

Several methodologies allow us to obtain a complete static analysis of the mechanism. For this purpose, the Davies method was used to analyse the mechanisms statically [11, 22–29]. This method was selected because it offers a straightforward way to obtain a static model of the mechanism, and this model can be easily

In the majority of *LCVs*, the load on the trailers is fixed and nominally centred; for this reason, the initial position lateral of the centre of gravity is centred and symmetric.

• the rear mechanism is composed for the tyres, and the suspension, and

steering angle of trailer front axle.

and the fifth wheel,

model.

**Figure 18.** *Trailer model.*

**1.5 Three-dimensional trailer model**

composed of the following mechanisms:

slider joints 'Sd'), and 23 links (*n* = 23).

**2. Static analysis of the mechanism**

**2.1 External forces and load distribution**

adaptable using this approach.

**140**

**Figure 20.** *Longitudinal* CG *movement.*

**Figure 21.** CG *displacements.*

**Figure 22.** *(a) Longitudinal slope of the road. (b) Banked road.*

the longitudinal slope angle (*φ*) and the bank angle (*ϕ*) of the road, these forces have three components, as represented in Eqs. (17)–(19).

$$P\_{\pi} = \mathcal{W}\sin\varphi\tag{17}$$

$$P\_{\mathcal{Y}} = -W\sin\phi\cos\varphi + ma\_{\mathcal{Y}}\cos\phi\tag{18}$$

$$P\_x = W \cos\phi \cos\varphi + ma\_\mathcal{y} \sin\phi \tag{19}$$

where *Px* is the force acting on the *x*-axis, *Py* is the force acting on the *y*-axis, and *Pz* is the force acting on the *z*-axis.

Finally, the load distribution of the trailer on a road with a slope angle is given by the **Figure 23** and Eq. (20).

$$P\_x h\_2 - P\_x (a \pm d\_2) + F\_r L = 0\tag{20}$$

**Figure 26.**

**143**

**Figure 24.**

**Figure 25.**

*Variables of the mechanism position (model of the front of the trailer).*

*Stability Analysis of Long Combination Vehicles Using Davies Method*

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

*Variables of the mechanism position (model of the rear of the trailer).*

*Vector along the direction of the screws axis (model of the front and rear of the trailer).*

where *h2* is the instantaneous *CG* height, *L* is the wheelbase of the trailer, and *a* is the distance from the front axle to the centre of gravity.

#### **2.2 Screw theory of the mechanism**

Screw theory enables the representation of the mechanism's instantaneous position in a coordinate system (successive screw displacement method) and the representation of the forces and moments (wrench), replacing the traditional vector representation. All these fundamentals applied to the mechanism are briefly presented below.

### *2.2.1 Method of successive screw displacements of the mechanism*

In the kinematic model for a mechanism, the successive screws displacement method is used. **Figures 24**–**28** and **Table 1** present the screw parameters of the mechanism.

**Figure 23.** *Load distribution of a trailer on a road with slope angle.*

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

**Figure 24.**

the longitudinal slope angle (*φ*) and the bank angle (*ϕ*) of the road, these forces

where *Px* is the force acting on the *x*-axis, *Py* is the force acting on the *y*-axis, and

Finally, the load distribution of the trailer on a road with a slope angle is given by

where *h2* is the instantaneous *CG* height, *L* is the wheelbase of the trailer, and *a*

Screw theory enables the representation of the mechanism's instantaneous position in a coordinate system (successive screw displacement method) and the representation of the forces and moments (wrench), replacing the traditional vector representation. All these fundamentals applied to the mechanism are briefly presented below.

In the kinematic model for a mechanism, the successive screws displacement method is used. **Figures 24**–**28** and **Table 1** present the screw parameters of the mechanism.

*Px* ¼ *W sin φ* (17)

*Py* ¼ �*W sin ϕ cos φ* þ *may cos ϕ* (18) *Pz* ¼ *W cos ϕ cos φ* þ *may sin ϕ* (19)

*Pxh*<sup>2</sup> � *Pz*ð Þþ *a* � *d*<sup>2</sup> *FrL* ¼ 0 (20)

have three components, as represented in Eqs. (17)–(19).

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

is the distance from the front axle to the centre of gravity.

*2.2.1 Method of successive screw displacements of the mechanism*

*Pz* is the force acting on the *z*-axis.

*(a) Longitudinal slope of the road. (b) Banked road.*

**2.2 Screw theory of the mechanism**

*Load distribution of a trailer on a road with slope angle.*

the **Figure 23** and Eq. (20).

**Figure 22.**

**Figure 23.**

**142**

*Variables of the mechanism position (model of the front of the trailer).*

**Figure 25.** *Variables of the mechanism position (model of the rear of the trailer).*

**Figure 26.** *Vector along the direction of the screws axis (model of the front and rear of the trailer).*

**Figure 27.** *Variables of the mechanism position (side view of the trailer).*

**Figure 28.** *Variables of the mechanism position (three-dimensional model).*

In **Figures 24–28** and **Table 1**, *l13* is the distance between the fifth wheel and the front axle, *l1;2;7;8* are the dynamic rolling radii of tyres, *t1;3* are the front and rear track widths of the trailer respectively, *t2;4* are the front and rear axle widths respectively, *b* is the lateral separation between the springs, *b1* is the fifth wheel width, *θ<sup>i</sup>* is the revolution joint angle rotation *i*, *l3;4;9;10* are the instantaneous heights of the leaf spring, *l12* is the height of *CG* above the chassis, and *ψ* is the trailer/trailer angle.

This method enables the determination of the displacement of the mechanism and the instantaneous position vector *s0i* of the joints, and the centre of gravity (The vector *s0i* (**Table 2**) is obtained from the first three terms of the last column of equations shown in **Table 3**).

vector *\$*

**145**

**Table 1.**

moments.

*<sup>A</sup>* = [*Mx My Mz Fx Fy Fz*]

*Screw parameters of the mechanism.*

*T*

• for the *x*-direction a steady-state model was used in the analysis;

• disturbances imposed by the road and the lateral friction forces (*Fy*) (tyre-ground contact) in the joints 3 and 19 were neglected; and

To simplify the model of **Figure 28**, the following considerations were made:

**Joints and points** *s s0 θ d* Joint 1 1 0 0 �*l13* 0 0 *θ<sup>1</sup>* 0 Joint 2 0 0 1 �*l13* 0 00 *l1* Joint 3a 0 1 0 �*l13* 0 00 *t1* Joint 3b 1 0 0 �*l13* 0 0 *θ<sup>3</sup>* 0 Joint 4 0 0 1 �*l13* 0 00 *l2* Joint 5 1 0 0 �*l13* (*t2* � *b*)*/2* 0 *θ<sup>5</sup>* 0 Joint 6 0 0 1 �*l13* (*t2* � *b*)*/2* 0 0 *l3* Joint 7 1 0 0 �*l13* (*t2* � *b*)*/2* 0 *θ<sup>7</sup>* 0 Joint 8 1 0 0 �*l13* (*t2 + b*)*/2* 0 *θ<sup>8</sup>* 0 Joint 9 0 0 1 �*l13* (*t2 + b*)*/2* 0 0 *l4* Joint 10 1 0 0 �*l13* (*t2 + b*)*/2* 0 *θ<sup>10</sup>* 0 Joint 11 1 0 0 0 (*t2* � *b1*)*/2* 0 *θ<sup>11</sup>* 0 Joint 12 0 0 1 0 (*t2* � *b1*)*/2* 0 0 *l5* Joint 13 1 0 0 0 (*t2* � *b1*)*/2* 0 *θ<sup>13</sup>* 0 Joint 14 1 0 0 0 (*t2 + b1*)*/2* 0 *θ<sup>14</sup>* 0 Joint 15 0 0 1 0 (*t2 + b1*)*/2* 0 0 *l6* Joint 16 1 0 0 0 (*t2 + b1*)*/2* 0 *θ<sup>16</sup>* 0 Joint 17 1 0 0 �*L* 0 0 *θ<sup>17</sup>* 0 Joint 18 0 0 1 �*L* 0 00 *l7* Joint 19a 0 1 0 �*L* 0 00 *t3* Joint 19b 1 0 0 �*L* 0 0 *θ<sup>19</sup>* 0 Joint 20 0 0 1 �*L* 0 00 *l8* Joint 21 1 0 0 �*L* (*t4* � *b*)*/2* 0 *θ<sup>21</sup>* 0 Joint 22 0 0 1 �*L* (*t4* � *b*)*/2* 0 0 *l9* Joint 23 1 0 0 �*L* (*t4* � *b*)*/2* 0 *θ<sup>23</sup>* 0 Joint 24 1 0 0 �*L* (*t4 + b*)*/2* 0 *θ<sup>24</sup>* 0 Joint 25 0 0 1 �*L* (*t4 + b*)*/2* 0 0 *l10* Joint 26 1 0 0 �*L* (*t4 + b*)*/2* 0 *θ<sup>26</sup>* 0 Joint 27 1 0 0 �*L t4/2 0 θ<sup>27</sup>* 0 Joint 28 1 0 0 0 *t2/2* 0 *θ<sup>28</sup>* 0 Point 29 0 0 1 0 *t2/2* 0 *ψ* 0 *CG* (30) 1 0 0 �*a* � *d2 (t4/2)* � *d1 l12* � *d3* 0 0

*Stability Analysis of Long Combination Vehicles Using Davies Method*

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

, where *F* denotes the forces, and *M* denotes the

#### *2.2.2 Wrench—Forces and moments*

In the static analysis, all forces and moments of the mechanism are represented by wrenches (*\$ <sup>A</sup>*) [13]. The wrenches applied can be represented by the


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

#### **Table 1.**

In **Figures 24–28** and **Table 1**, *l13* is the distance between the fifth wheel and the front axle, *l1;2;7;8* are the dynamic rolling radii of tyres, *t1;3* are the front and rear track widths of the trailer respectively, *t2;4* are the front and rear axle widths respectively, *b* is the lateral separation between the springs, *b1* is the fifth wheel width, *θ<sup>i</sup>* is the revolution joint angle rotation *i*, *l3;4;9;10* are the instantaneous heights of the leaf spring, *l12* is the height of *CG*

This method enables the determination of the displacement of the mechanism and the instantaneous position vector *s0i* of the joints, and the centre of gravity (The vector *s0i* (**Table 2**) is obtained from the first three terms of the last column of

*<sup>A</sup>*) [13]. The wrenches applied can be represented by the

In the static analysis, all forces and moments of the mechanism are

above the chassis, and *ψ* is the trailer/trailer angle.

*Variables of the mechanism position (three-dimensional model).*

*Variables of the mechanism position (side view of the trailer).*

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

equations shown in **Table 3**).

**Figure 28.**

**144**

**Figure 27.**

represented by wrenches (*\$*

*2.2.2 Wrench—Forces and moments*

*Screw parameters of the mechanism.*

vector *\$ <sup>A</sup>* = [*Mx My Mz Fx Fy Fz*] *T* , where *F* denotes the forces, and *M* denotes the moments.

To simplify the model of **Figure 28**, the following considerations were made:


## *Numerical and Experimental Studies on Combustion Engines and Vehicles*


**Joints and points Instantaneous position matrix**

Joint 28 *p'<sup>28</sup> = A29 A1 A2 A5 A6 A7 A11 A12 A13 A28 p28 CG* (30) *p'CG = A17 A18 A21 A22 A23 A27 ACG pCG*

> *Fyi* 01 0 *Fzi* 00 1 *Myi* 01 0 *Mzi* 00 1

> *Fzi* 00 1 *Txi* 10 0 *Myi* 01 0 *Mzi* 00 1

*Fni* 0 *cos θi*�*<sup>1</sup> sin θi*�*<sup>1</sup> Mxi* 10 0 *Myi* 01 0 *Mzi* 00 1

*si* **Inst. position**

*Fxi* 1 0 0 Revolute joints

*Fxi* 1 0 0 Spherical slider joints 3 and 19 *Fzi* 00 1

*Fxi* 1 0 0 Revolute joints 5, 11, and 21 *Fyi* 01 0

*Fxi* 1 0 0 Prismatic joints

*FTi* 0 �*sin θi*�*<sup>1</sup> cos θi*�*<sup>1</sup>* Prismatic joints

*FLSi* 0 �*sin θi*�*<sup>1</sup> cos θi*�*<sup>1</sup>* Prismatic joints

*FFWi* 0 �*sin θi*�*<sup>1</sup> cos θi*�*<sup>1</sup>* Prismatic joints

**vector** *s0i*

1, 7, 8, 10, 13, 14, 16, 17, 23, 24, and 26

2, 4, 6, 9, 12, 15, 18, 20, 22, and 25

2, 4, 18, and 20

6, 9, 22, and 25

12 and 15

Joint 20 *p'<sup>20</sup> = A19*<sup>a</sup> *A19b A20 p20* Joint 21 *p'<sup>21</sup> = A17 A18 A21 p21* Joint 22 *p'<sup>22</sup> = A17 A18 A21 A22 p22* Joint 23 *p'<sup>23</sup> = A17 A18 A21 A22 A23 p23* Joint 24 *p'<sup>24</sup> = A17 A18 A24 p24* Joint 25 *p'<sup>25</sup> = A17 A18 A24 A25 p25* Joint 26 *p'<sup>26</sup> = A17 A18 A24 A25 A26 p26* Joint 27 *p'<sup>27</sup> = A17 A18 A21 A22 A23 A27 p27*

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

**Table 3.**

**Joints and reference points**

Revolute joints 1, 7, 8, 10, 13, 14, 16, 17, 23, 24, and 26

Spherical slider joints 3 and 19

Revolute joints 5, 11, and 21

Prismatic joints 2, 4, 6, 9, 12, 15, 18, 20, 22 and

Prismatic joints 2, 4, 18, and 20

Prismatic joints 6, 9, 22, and 25

Prismatic joints 12 and 15

25

**147**

*Instantaneous position matrix.*

**Constraints and forces**

*si* ¼ sin *θici* ¼ cos *θ<sup>i</sup>*

*\*h*<sup>1</sup> ¼ � ð Þ <sup>2</sup>*l*12�2*d*<sup>3</sup> *<sup>s</sup>*27þ23þ21þ17�*d*<sup>1</sup> *<sup>c</sup>*27þ23þ21þ17�*b*<sup>2</sup> *<sup>c</sup>*23þ21þ17þ2*l*<sup>9</sup> *<sup>s</sup>*21þ17þ2*l*<sup>7</sup> *<sup>s</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>c</sup>*<sup>17</sup>

2*: \*h*<sup>2</sup> ¼ � �*d*<sup>1</sup> *<sup>s</sup>*27þ23þ21þ17þ �ð Þ <sup>2</sup>*d*3�2*l*<sup>12</sup> *<sup>c</sup>*27þ23þ21þ17�*b*<sup>2</sup> *<sup>s</sup>*23þ21þ17�2*l*<sup>9</sup> *<sup>c</sup>*21þ17�2*l*<sup>7</sup> *<sup>c</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>s</sup>*<sup>17</sup> 2

#### **Table 2.**

*Instantaneous position vector* s0i*.*


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


#### **Table 3.**

**References** *s0i*

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

<sup>21</sup> �*<sup>L</sup>* � <sup>2</sup>*l*<sup>17</sup> *<sup>s</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>c</sup>*<sup>17</sup>

*\*h*<sup>1</sup> ¼ � ð Þ <sup>2</sup>*l*12�2*d*<sup>3</sup> *<sup>s</sup>*27þ23þ21þ17�*d*<sup>1</sup> *<sup>c</sup>*27þ23þ21þ17�*b*<sup>2</sup> *<sup>c</sup>*23þ21þ17þ2*l*<sup>9</sup> *<sup>s</sup>*21þ17þ2*l*<sup>7</sup> *<sup>s</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>c</sup>*<sup>17</sup> 2*: \*h*<sup>2</sup> ¼ � �*d*<sup>1</sup> *<sup>s</sup>*27þ23þ21þ17þ �ð Þ <sup>2</sup>*d*3�2*l*<sup>12</sup> *<sup>c</sup>*27þ23þ21þ17�*b*<sup>2</sup> *<sup>s</sup>*23þ21þ17�2*l*<sup>9</sup> *<sup>c</sup>*21þ17�2*l*<sup>7</sup> *<sup>c</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>s</sup>*<sup>17</sup> 2

<sup>22</sup>–<sup>23</sup> �*<sup>L</sup>* � <sup>2</sup>*l*<sup>17</sup> *<sup>s</sup>*17þð Þ *<sup>b</sup>*2�*t*<sup>4</sup> *<sup>c</sup>*17þ2*l*<sup>9</sup> *<sup>s</sup>*21þ<sup>17</sup>

Joint 1 *p'<sup>1</sup> = A29 A1 p1* Joint 2 *p'<sup>2</sup> = A29 A1 A2 p2* Joint 3 *p'<sup>3</sup> = A29 A3*<sup>a</sup> *A3b p3* Joint 4 *p'<sup>4</sup> = A29 A3*<sup>a</sup> *A3b A4 p4* Joint 5 *p'<sup>5</sup> = A29 A1 A2 A5 p5* Joint 6 *p'<sup>6</sup> = A29 A1 A2 A5 A6 p6* Joint 7 *p'<sup>7</sup> = A29 A1 A2 A5 A6 A7 p7* Joint 8 *p'<sup>8</sup> = A29 A1 A2 A8 p8* Joint 9 *p'<sup>9</sup> = A29 A1 A2 A8 A9 p9* Joint 10 *p'<sup>10</sup> = A29 A1 A2 A8 A9 A10 p10* Joint 11 *p'<sup>11</sup> = A29 A1 A2 A5 A6 A7 A11 p11* Joint 12 *p'<sup>12</sup> = A29 A1 A2 A5 A6 A7 A11 A12 p12* Joint 13 *p'<sup>13</sup> = A29 A1 A2 A5 A6 A7 A11 A12 A13 p13* Joint 14 *p'<sup>14</sup> = A29 A1 A2 A5 A6 A7 A14 p14* Joint 15 *p'<sup>15</sup> = A29 A1 A2 A5 A6 A7 A14 A15 p15* Joint 16 *p'<sup>16</sup> = A29 A1 A2 A5 A6 A7 A14 A15 A16 p16*

Joint 17 *p'<sup>17</sup> = A17 p17* Joint 18 *p'<sup>18</sup> = A17 A18 p18* Joint 19 *p'<sup>19</sup> = A19*<sup>a</sup> *A19b p19*

<sup>2</sup> � <sup>2</sup>*l*<sup>13</sup> *<sup>s</sup>*29þ*t*<sup>2</sup> *<sup>c</sup>*29�*t*<sup>2</sup>

<sup>2</sup> � ð Þ <sup>2</sup>*l*<sup>1</sup> *<sup>s</sup>*1þ*t*<sup>2</sup> *<sup>c</sup>*29þ2*l*<sup>13</sup> *<sup>s</sup>*29�*t*<sup>2</sup>

<sup>2</sup> � <sup>2</sup>*l*<sup>13</sup> *<sup>s</sup>*29þð Þ *<sup>t</sup>*2�2*t*<sup>1</sup> *<sup>c</sup>*29�*t*<sup>2</sup>

5–16 ⁞⁞ ⁞ 17 �*L* 0 0 18 �*L* �*l*<sup>7</sup> *s*<sup>17</sup> *l*<sup>7</sup> *c*<sup>17</sup> 19 �*L t*<sup>3</sup> 0 20 �*L t*<sup>3</sup> � *l*8*s*<sup>19</sup> *l*8*c*<sup>19</sup>

24–28 ⁞⁞ ⁞ *CG* �*a* � *d*<sup>2</sup> \**h*<sup>1</sup> \**h*<sup>2</sup>

**Joints and points Instantaneous position matrix**

<sup>2</sup> � <sup>2</sup>*l*<sup>13</sup> *<sup>s</sup>*29þð Þ <sup>2</sup>*l*<sup>2</sup> *<sup>s</sup>*3þ*t*2�2*t*<sup>1</sup> *<sup>c</sup>*29�*t*<sup>2</sup>

2

2

<sup>2</sup> <sup>0</sup>

<sup>2</sup> *<sup>l</sup>*1*c*<sup>1</sup>

<sup>2</sup> <sup>0</sup>

<sup>2</sup> *<sup>l</sup>*2*c*<sup>3</sup>

ð Þ *t*4�*b*<sup>2</sup> *s*17þ2*l*<sup>17</sup> *c*<sup>17</sup> 2

ð Þ *t*4�*b*<sup>2</sup> *s*17þ2*l*<sup>17</sup> *c*17þ2*l*<sup>9</sup> *c*21þ<sup>17</sup> 2

1 *<sup>t</sup>*<sup>2</sup> *<sup>s</sup>*29�2*l*<sup>13</sup> *<sup>c</sup>*<sup>29</sup>

2 ð Þ <sup>2</sup>*l*<sup>1</sup> *<sup>s</sup>*1þ*t*<sup>2</sup> *<sup>s</sup>*29�2*l*<sup>13</sup> *<sup>c</sup>*<sup>29</sup>

3 ð Þ *<sup>t</sup>*2�2*t*<sup>1</sup> *<sup>s</sup>*29�2*l*<sup>13</sup> *<sup>c</sup>*<sup>29</sup>

*si* ¼ sin *θici* ¼ cos *θ<sup>i</sup>*

*Instantaneous position vector* s0i*.*

**Table 2.**

**146**

4 ð Þ <sup>2</sup>*l*<sup>2</sup> *<sup>s</sup>*3þ*t*2�2*t*<sup>1</sup> *<sup>s</sup>*29�2*l*<sup>13</sup> *<sup>c</sup>*<sup>29</sup>

*Instantaneous position matrix.*



The mechanism of the **Figure 28** is represented by the direct coupling graph of the **Figure 29**. This graph has 23 vertices (links) and 31 edges (joints and external

The direct coupling graph (**Figure 29**) can be represented by the incidence matrix [*I*]*23X31* [30] (Eq. (25)). The incidence matrix provides the mechanism cutset matrix [*Q*]*22X31* [11, 25–28, 30] (Eq. (26)) for the mechanism, where each line represents a cut graph and the columns represent the mechanism joints. Besides, this matrix is rearranged, allowing 22 branches (edges *1–3, 5–9, 11–15, 17–19, 21–25, and 27*—identity matrix) and 9 chords (edges *4, 10, 16, 20, 26, 28, Px, Py,* and *Pz*) to

*Stability Analysis of Long Combination Vehicles Using Davies Method*

forces (*Px*, *Py*, and *Pz*)).

**Figure 29.**

**Figure 30.**

**149**

*Cut-set action graph of the mechanism.*

*Direct coupling graph of the mechanism.*

be defined, as shown in **Figure 30**.

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

#### **Table 4.**

*Wrench parameters of the mechanism.*

• the components of the trailer weight (*W*) and the inertial force (*may*) are the only external forces acting on the trailer *CG*.

Considering a static analysis in a three-dimensional space [7], the corresponding wrenches of each joint and external forces are defined by the parameters of **Table 4**, where *si* represents the orientation vector of each wrench *i*.

All of the wrenches of the mechanism together comprise the action matrix [*Ad*] given by Eq. (21) (or the amplified matrix of the Eq. (22)).

$$[A\_d]\_{6 \times 148} = \begin{bmatrix} \mathfrak{s}\_{Fx1}^A & \mathfrak{s}\_{Fy1}^A & \mathfrak{s}\_{Fz1}^A & \cdots & \mathfrak{s}\_{Fz}^A & \mathfrak{s}\_{Fy}^A & \mathfrak{s}\_{Fz}^A \end{bmatrix} \tag{21}$$

$$[A\_d] = \begin{bmatrix} \mathbf{0} & \mathbf{0} & -p\_1 F\_{x1} & \cdots & \mathbf{0} & h\_2 P\_y & -h\_1 P\_x \\ \mathbf{0} & \mathbf{0} & -p\_2 F\_{x1} & \cdots & h\_2 P\_x & \mathbf{0} & (-a \pm d\_2) P\_x \\ p\_1 F\_{x1} & p\_2 F\_{y1} & \mathbf{0} & \cdots & -h\_1 P\_x & -(-a \pm d\_2) P\_y & \mathbf{0} \\ F\_{x1} & \mathbf{0} & \mathbf{0} & \cdots & P\_x & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & F\_{y1} & \mathbf{0} & \cdots & \mathbf{0} & -P\_y & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & F\_{x1} & \cdots & \mathbf{0} & \mathbf{0} & -P\_z \end{bmatrix} \tag{22}$$

where *pi* is a system variable.

The wrench can be represented by a normalised wrench and a magnitude. Therefore, from the Eq. (22) the unit action matrix and the magnitudes action vector are obtained, as represented by Eqs. (23) and (24).

$$\begin{bmatrix} \dot{A}\_{d} \end{bmatrix}\_{6 \times 148} = \begin{bmatrix} 0 & 0 & -p\_{1} & \cdots & 0 & h\_{2} & -h\_{1} \\ 0 & 0 & -p\_{2} & \cdots & h\_{2} & 0 & (-a \pm d\_{2}) \\ p\_{1} & p\_{2} & 0 & \cdots & -h\_{1} & -(-a \pm d\_{2}) & 0 \\ 1 & 0 & 0 & \cdots & 1 & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & -1 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 & -1 \end{bmatrix} \tag{23}$$

$$[\Psi]\_{148 \times 1} = \begin{bmatrix} F\_{x1} & F\_{\mathcal{I}1} & F\_{x1} & \cdots & P\_{x} & P\_{\mathcal{I}} & P\_{x} \end{bmatrix} \tag{24}$$

#### *2.2.3 Graph theory*

Kinematic chains and mechanisms are comprised of links and joints, which can be represented by graphs, where the vertices correspond to the links, and the edges correspond to the joints and external forces [5, 7].

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

The mechanism of the **Figure 28** is represented by the direct coupling graph of the **Figure 29**. This graph has 23 vertices (links) and 31 edges (joints and external forces (*Px*, *Py*, and *Pz*)).

The direct coupling graph (**Figure 29**) can be represented by the incidence matrix [*I*]*23X31* [30] (Eq. (25)). The incidence matrix provides the mechanism cutset matrix [*Q*]*22X31* [11, 25–28, 30] (Eq. (26)) for the mechanism, where each line represents a cut graph and the columns represent the mechanism joints. Besides, this matrix is rearranged, allowing 22 branches (edges *1–3, 5–9, 11–15, 17–19, 21–25, and 27*—identity matrix) and 9 chords (edges *4, 10, 16, 20, 26, 28, Px, Py,* and *Pz*) to be defined, as shown in **Figure 30**.

**Figure 29.** *Direct coupling graph of the mechanism.*

• the components of the trailer weight (*W*) and the inertial force (*may*) are the

*Fzi* 00 1 *Txi* 10 0 *CG* (30) *Px* 10 0 *CG* (30) *Py* 0 �1 0 *Pz* 0 0 �1

Considering a static analysis in a three-dimensional space [7], the corresponding

All of the wrenches of the mechanism together comprise the action matrix [*Ad*]

*Fz*<sup>1</sup> <sup>⋯</sup> \$*<sup>A</sup>*

h i

*Px* \$*<sup>A</sup> Py* \$*<sup>A</sup> Pz*

� � (24)

(21)

(22)

(23)

*si* **Inst. position**

*Fxi* 1 0 0 Spherical joints 27 and 28 *Fyi* 01 0

**vector** *s0i*

wrenches of each joint and external forces are defined by the parameters of **Table 4**, where *si* represents the orientation vector of each wrench *i*.

> *Fx*<sup>1</sup> \$*<sup>A</sup> Fy*<sup>1</sup> \$*<sup>A</sup>*

0 0 �*p*1*Fz*<sup>1</sup> ⋯ 0 *h*2*Py* �*h*1*Pz* 0 0 �*p*2*Fz*<sup>1</sup> ⋯ *h*2*Px* 0 ð Þ �*a* � *d*<sup>2</sup> *Pz p*1*Fx*<sup>1</sup> *p*2*Fy*<sup>1</sup> 0 ⋯ �*h*1*Px* � �ð Þ *a* � *d*<sup>2</sup> *Py* 0 *Fx*<sup>1</sup> 0 0 ⋯ *Px* 0 0 0 *Fy*<sup>1</sup> 0 ⋯ 0 �*Py* 0 0 0 *Fz*<sup>1</sup> ⋯ 0 0 �*Pz*

The wrench can be represented by a normalised wrench and a magnitude. Therefore, from the Eq. (22) the unit action matrix and the magnitudes action

> 0 0 �*p*<sup>1</sup> ⋯ 0 *h*<sup>2</sup> �*h*<sup>1</sup> 0 0 �*p*<sup>2</sup> ⋯ *h*<sup>2</sup> 0 ð Þ �*a* � *d*<sup>2</sup> *p*<sup>1</sup> *p*<sup>2</sup> 0 ⋯ �*h*<sup>1</sup> � �ð Þ *a* � *d*<sup>2</sup> 0 10 0 ⋯ 10 0 01 0 ⋯ 0 �1 0 00 1 ⋯ 0 0 �1

½ � *<sup>Ψ</sup>* <sup>148</sup>�<sup>1</sup> <sup>¼</sup> *Fx*<sup>1</sup> *Fy*<sup>1</sup> *Fz*<sup>1</sup> <sup>⋯</sup> *Px Py Pz*

Kinematic chains and mechanisms are comprised of links and joints, which can be represented by graphs, where the vertices correspond to the links, and the edges

only external forces acting on the trailer *CG*.

**Constraints and forces**

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

given by Eq. (21) (or the amplified matrix of the Eq. (22)).

½ � *Ad* <sup>6</sup>�<sup>148</sup> <sup>¼</sup> \$*<sup>A</sup>*

vector are obtained, as represented by Eqs. (23) and (24).

correspond to the joints and external forces [5, 7].

½ �¼ *Ad*

**Joints and reference points**

**Table 4.**

Spherical joints 27 and 28

> *A*^ *d* h i

*2.2.3 Graph theory*

**148**

<sup>6</sup>�<sup>148</sup> <sup>¼</sup>

*Wrench parameters of the mechanism.*

where *pi* is a system variable.

**Figure 30.** *Cut-set action graph of the mechanism.*

must be equal to zero. Thus, the statics of the mechanism can be defined, as

<sup>132</sup>�148½ � <sup>Ψ</sup> *<sup>T</sup>*

<sup>148</sup>�<sup>1</sup> <sup>¼</sup> ½ � <sup>0</sup> <sup>132</sup>�<sup>1</sup> (28)

*Fx*<sup>1</sup> *Fy*<sup>1</sup> ⋮ *Px Py Pz*

<sup>¼</sup> ½ � <sup>0</sup> <sup>132</sup>�<sup>1</sup> (29)

<sup>3</sup>�<sup>1</sup> <sup>¼</sup> ½ � <sup>0</sup> <sup>132</sup>�<sup>1</sup> (30)

� �*<sup>T</sup>* (31)

*:*

exemplified in Eq. (28) (or the amplified matrix of the Eq. (29)):

*A*^ *n* h i

*Stability Analysis of Long Combination Vehicles Using Davies Method*

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

0 0 ⋯ 00 0 0 0 ⋯ 00 0 *p*<sup>1</sup> *p*<sup>2</sup> ⋯ 00 0 1 0 ⋯ 00 0 0 1 ⋯ 00 0 0 0 ⋯ 00 0 ⋮⋮⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 *h*<sup>2</sup> �*h*<sup>1</sup> 0 0 ⋯ *h*<sup>2</sup> 0 ð Þ �*a* � *d*<sup>2</sup>

0 0 ⋯ �*h*<sup>1</sup> � �ð Þ *a* � *d*<sup>2</sup> 0 0 0 ⋯ 10 0 0 0 ⋯ 0 �1 0 0 0 ⋯ 0 0 �1

and divided in two sets, as shown by Eq. (30).

132�145

In this case, the primary variable vector is:

and the secondary variable vector is:

solution system provides the next equation:

<sup>Ψ</sup>*<sup>s</sup>* ½ �*<sup>T</sup>*

Ψ*<sup>p</sup>* � �

*Fz*<sup>3</sup> þ

� *<sup>h</sup>*<sup>1</sup> <sup>þ</sup> *<sup>P</sup>*<sup>1</sup> *t*<sup>1</sup> *cos ψ*

*P*<sup>1</sup> þ *t*<sup>3</sup> *t*<sup>1</sup> *cos ψ*

*<sup>A</sup>*^ *ns* h i

It is necessary to identify the set of primary variables [Ψ*p*] (known variables), among the variables of Ψ. Once identified, the system of the Eq. (28) is rearranged

132�3

Ψ*<sup>p</sup>* � �*<sup>T</sup>*

<sup>145</sup>�<sup>1</sup> <sup>þ</sup> *<sup>A</sup>*^ *np* h i

ables, and *<sup>A</sup>*^ *ns* h i are the columns corresponding to the secondary variables.

where [Ψ*p*] is the primary variable vector, [Ψ*s*] is the second variable vector (unknown variables), *<sup>A</sup>*^ *np* h i are the columns corresponding to the primary vari-

<sup>3</sup>�<sup>1</sup> <sup>¼</sup> *Px Py Pz*

*<sup>Ψ</sup><sup>s</sup>* ½ �<sup>145</sup>�<sup>1</sup> <sup>¼</sup> *Fx*<sup>1</sup> *Fy*<sup>1</sup> *My*<sup>1</sup> *Mz*<sup>1</sup> <sup>⋯</sup> *Fz*<sup>1</sup> *Fz*<sup>3</sup> *Fz*<sup>17</sup> *Fz*<sup>19</sup> � �*<sup>T</sup>* (32)

Solving the system Eq. (30) using the Gauss-Jordan elimination method, all secondary variables being function of the primary variables, the last row of the

*Fz*<sup>19</sup> þ

*h*2 *t*<sup>1</sup> *cos ψ*

*Pz* þ

*P*1 *t*<sup>1</sup> *cos ψ*

*Fz*<sup>17</sup>

*Py* ¼ 0 (33)

*Cut* 1

⋮

*Cut* 22

**151**

All constraints are represented as edges, which allows the amplification of the cut-set graph and the cut-set matrix. Additionally, the tyre normal load (*FTi*), spring normal load (*FLSi*), fifth wheel normal load (*FFWi*), and the passive torsional moment (*Txi*) are included.

**Figure 30** presents the cut-set action graph and the Eq. (27) presents the expanded cut-set matrix ([*Q*]*22X148*), where each line represents a cut of the graph, and the columns are the constraints of the joints as well as external forces on the mechanism.

ð27Þ

#### **2.3 Equation system solution**

Using the cut-set law [24], the algebraic sum of the normalised wrenches given in Eqs. (23) and (24), that belong to the same cut [*Q*]*22X148* (**Figure 30** and Eq. (27)) must be equal to zero. Thus, the statics of the mechanism can be defined, as exemplified in Eq. (28) (or the amplified matrix of the Eq. (29)):

$$\begin{aligned} \begin{bmatrix} \hat{A}\_{\pi} \end{bmatrix}\_{132\times146} \begin{bmatrix} \mathbf{w} \end{bmatrix}\_{148\times1}^{\mathsf{T}} &= [0]\_{132\times1} \end{aligned} \tag{28}$$

$$\begin{bmatrix} 0 & 0 & \cdots & 0 & 0 & 0\\ 0 & 0 & \cdots & 0 & 0 & 0\\ p\_1 & p\_2 & \cdots & 0 & 0 & 0\\ 1 & 0 & \cdots & 0 & 0 & 0\\ 0 & 1 & \cdots & 0 & 0 & 0\\ 0 & 0 & \cdots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & \cdots & 0 & h\_2 & -h\_1\\ 0 & 0 & \cdots & h\_2 & 0 & (-a \pm d\_2)\\ 0 & 0 & \cdots & -h\_1 & -(-a \pm d\_2) & 0\\ 0 & 0 & \cdots & 1 & 0 & 0\\ 0 & 0 & \cdots & 1 & 0 & 0\\ 0 & 0 & \cdots & 0 & -1 & 0\\ 0 & 0 & \cdots & 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} F\_{x1} \\ F\_{y1} \\ \vdots\\ F\_{y1} \\ \vdots\\ F\_{y1} \\ \end{bmatrix} = [0]\_{132\times1} \quad \text{(29)}$$

It is necessary to identify the set of primary variables [Ψ*p*] (known variables), among the variables of Ψ. Once identified, the system of the Eq. (28) is rearranged and divided in two sets, as shown by Eq. (30).

$$\left[\hat{\mathbf{A}}\_{\mathfrak{m}}\right]\_{\mathbf{1}\mathbf{3}\times\mathbf{1}\mathbf{4}\mathbf{5}}\left[\boldsymbol{\Psi}\_{\mathfrak{s}}\right]\_{\mathbf{1}\mathbf{4}\mathbf{5}\times\mathbf{1}}^{T} + \left[\hat{\mathbf{A}}\_{\mathfrak{m}}\right]\_{\mathbf{1}\mathbf{3}\mathbf{2}\times\mathbf{3}}\left[\boldsymbol{\Psi}\_{\mathfrak{p}}\right]\_{\mathbf{3}\times\mathbf{1}}^{T} = \left[\mathbf{0}\right]\_{\mathbf{1}\mathbf{3}\mathbf{2}\times\mathbf{1}}\tag{30}$$

where [Ψ*p*] is the primary variable vector, [Ψ*s*] is the second variable vector (unknown variables), *<sup>A</sup>*^ *np* h i are the columns corresponding to the primary variables, and *<sup>A</sup>*^ *ns* h i are the columns corresponding to the secondary variables. In this case, the primary variable vector is:

$$\left[\Psi\_{\mathcal{P}}\right]\_{\mathfrak{Z}\times\mathbf{1}} = \left[\begin{array}{cc} P\_{\mathfrak{x}} & P\_{\mathcal{Y}} & P\_{\mathfrak{x}} \end{array}\right]^{T} \tag{31}$$

and the secondary variable vector is:

$$\begin{bmatrix} \mathbf{Y}\_{\mathbf{z}} \end{bmatrix}\_{\mathbf{145} \times \mathbf{1}} = \begin{bmatrix} F\_{\mathbf{x1}} & F\_{\mathbf{y1}} & M\_{\mathbf{y1}} & M\_{\mathbf{z1}} & \cdots & F\_{\mathbf{z}1} & F\_{\mathbf{z}3} & F\_{\mathbf{z}17} & F\_{\mathbf{z}19} \end{bmatrix}^{\mathrm{T}} \tag{32}$$

Solving the system Eq. (30) using the Gauss-Jordan elimination method, all secondary variables being function of the primary variables, the last row of the solution system provides the next equation:

$$F\_{x3} + \frac{P\_1 + t\_3}{t\_1 \cos \psi} F\_{x19} + \frac{P\_1}{t\_1 \cos \psi} F\_{x17}$$

$$-\frac{h\_1 + P\_1}{t\_1 \cos \psi} P\_x + \frac{h\_2}{t\_1 \cos \psi} P\_y = 0\tag{33}$$

ð25Þ

ð26Þ

ð27Þ

All constraints are represented as edges, which allows the amplification of the cut-set graph and the cut-set matrix. Additionally, the tyre normal load (*FTi*), spring normal load (*FLSi*), fifth wheel normal load (*FFWi*), and the passive torsional

**Figure 30** presents the cut-set action graph and the Eq. (27) presents the expanded

Using the cut-set law [24], the algebraic sum of the normalised wrenches given in Eqs. (23) and (24), that belong to the same cut [*Q*]*22X148* (**Figure 30** and Eq. (27))

cut-set matrix ([*Q*]*22X148*), where each line represents a cut of the graph, and the columns are the constraints of the joints as well as external forces on the mechanism.

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

moment (*Txi*) are included.

**2.3 Equation system solution**

**150**

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

replacing *Py* and *Pz*:

$$F\_{x3} + \frac{P\_1 + t\_3}{t\_1 \cos \psi} F\_{x19} + \frac{P\_1}{t\_1 \cos \psi} F\_{x17}$$

$$-\frac{h\_1 + P\_1}{t\_1 \cos \psi} \left( W \cos \phi \cos \phi + ma\_\gamma \sin \phi \right)$$

$$+ \frac{h\_2}{t\_1 \cos \psi} \left( ma\_\gamma \cos \phi - W \sin \phi \cos \phi \right) = 0 \tag{34}$$

The normal forces *Fz3* and *Fz17* depend on the *LLT* coefficient in the front and rear axles respectively [4, 21, 34]. Furthermore, this coefficient depends on the

This information demonstrates that the *SRT*3*Dψϕφ* factor of a vehicle (Eq. (36)) is, in general, inferior to the *SRT* factor for a two-dimensional model vehicle [35], as

*<sup>g</sup>* <sup>¼</sup> *<sup>t</sup>*

<sup>2</sup>*<sup>h</sup>* (37)

*SRT*2*<sup>D</sup>* <sup>¼</sup> *ay*

**Parameters of the trailer Value Units** Trailer weight—*W* 355.22 kN Front and rear track widths (*t1,3*) 1.86 m Front and rear axles widths (*t2,4*) 1.86 m Stiffness of the suspension per axle (*ks*) [37] 1800 kN.m�<sup>1</sup>

Vertical stiffness per tyre (*kT*) ([37]) 840 kN.m�<sup>1</sup> Initial suspension height (*l3,4,9,10*) (*ls*) 0.205 m Initial dynamic rolling radius (*l1,2,7,8*) (*lr*) (Michelin XZA® [36]) 0.499 m Initial height of the fifth wheel (*lfi*) 0.1 m Lateral separation between the springs (*b*) 0.95 m Fifth wheel width (*b1*) 0.6 m *CG* height above the chassis (*l12*) 1.346 m Distance between the fifth wheel and the front axle (*l13*) 0.15 m Wheelbase of the trailer (*L*) 4.26 m Distance from the front axle to the centre of gravity (*a*) 3m Offset of the cargo *d1* 0.1 m Trailer/trailer angle (*ψ*) 0 °

Number of axles at the front (trailer) (four tyres per axle) 2 Number of axles at the rear (trailer) (four tyres per axle) 3

vehicle type, speed, suspension, tyres, etc.

*Dynamic rollover test. Source: Adapted of Cabral [33].*

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

*Stability Analysis of Long Combination Vehicles Using Davies Method*

where *h* is the *CG* height, *t* is the vehicle track.

shown in Eq. (37).

**Figure 32.**

**Table 5.**

**153**

*Parameters of the trailer model.*

where *P1* is a system variable (*P*<sup>1</sup> ¼ ð Þ 2*l*<sup>13</sup> sin *ψ* þ *t*2ð Þ cos *ψ* � 1 *=*2), *h1* is the instantaneous lateral distance between the zero-reference frame and the centre of gravity, and *h2* is the instantaneous *CG* height (**Table 2**). Simplifying the equation, and making tan ð Þ¼ *ϕ e*, where *e* is the tangent of the bank angle, we have:

$$\frac{a\_{\mathcal{Y}}}{\mathcal{g}} = \frac{h\_1 \cos \phi + h\_2 \cos \phi}{h\_2 - (h\_1 + P\_1)e} \times$$

$$\left(1 - \frac{t\_1 F\_{x3} \cos \psi + P\_1 (F\_{x37} - W \cos \phi \cos \phi) + (P\_1 + t\_3)F\_{x19}}{W \cos \phi (h\_1 \cos \phi + h\_2 \cos \phi)}\right) \tag{35}$$

According to the static redundancy problem known as the four-legged table [31, 32], a plane is defined by just three points in space and, consequently, a fourlegged table has support plane multiplicities. This is why when one leg loses contact with the ground, the table is supported by the other three, as shown in **Figure 31**.

The problem of the four-legged table is observed in dynamic rollover tests when the rear inner tyre loses contact with the ground (*Fz19 = 0*), and the front inner tyre (*Fz3*) does not, as shown, for example, in **Figure 32**.

Applying this theory to the vehicle stability, when a vehicle makes a turn, it is subjected to an increasing lateral load until it reaches the rollover threshold [32]. During the turning, the rear inner tyre is usually the one that loses contact with the ground. For this condition (*Fz19 = 0*), and thus:

$$SRT\_{3D\_{\varphi\phi\varphi}} = \frac{h\_1 \cos \varphi + h\_2 \cos \varphi}{h\_2 - (h\_1 + P\_1)e} \times$$

$$\left(1 - \frac{t\_1 F\_{x3} \cos \varphi + P\_1 (F\_{x17} - W \cos \phi \cos \varphi)}{W \cos \phi (h\_1 \cos \varphi + h\_2 \cos \varphi)}\right) \tag{36}$$

where *SRT*3*Dψϕφ* factor is the three-dimensional static rollover threshold for a trailer model with trailer/trailer angle (*ψ*), bank angle (*e*), and slope angle (*φ*).

**Figure 31.** *Redundancy problem of the four-legged table.*

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

**Figure 32.**

replacing *Py* and *Pz*:

*Fz*<sup>3</sup> þ

*Numerical and Experimental Studies on Combustion Engines and Vehicles*

� *<sup>h</sup>*<sup>1</sup> <sup>þ</sup> *<sup>P</sup>*<sup>1</sup> *t*<sup>1</sup> *cos ψ*

*h*2 *t*<sup>1</sup> *cos ψ*

*ay*

(*Fz3*) does not, as shown, for example, in **Figure 32**.

ground. For this condition (*Fz19 = 0*), and thus:

**Figure 31.**

**152**

*Redundancy problem of the four-legged table.*

þ

*P*<sup>1</sup> þ *t*<sup>3</sup> *t*<sup>1</sup> *cos ψ*

*Fz*<sup>19</sup> þ

where *P1* is a system variable (*P*<sup>1</sup> ¼ ð Þ 2*l*<sup>13</sup> sin *ψ* þ *t*2ð Þ cos *ψ* � 1 *=*2), *h1* is the instantaneous lateral distance between the zero-reference frame and the centre of gravity, and *h2* is the instantaneous *CG* height (**Table 2**). Simplifying the equation,

> *<sup>g</sup>* <sup>¼</sup> *<sup>h</sup>*<sup>1</sup> *cos <sup>φ</sup>* <sup>þ</sup> *<sup>h</sup>*2*ecos<sup>φ</sup> h*<sup>2</sup> � ð Þ *h*<sup>1</sup> þ *P*<sup>1</sup> *e*

<sup>1</sup> � *<sup>t</sup>*1*Fz*<sup>3</sup> *cos <sup>ψ</sup>* <sup>þ</sup> *<sup>P</sup>*1ð*Fz*<sup>17</sup> � *Wcosϕcosφ*Þ þ ð Þ *<sup>P</sup>*<sup>1</sup> <sup>þ</sup> *<sup>t</sup>*<sup>3</sup> *Fz*<sup>19</sup> *Wcosϕ*ð Þ *h*<sup>1</sup> *cos φ* þ *h*2*ecosφ* 

According to the static redundancy problem known as the four-legged table [31, 32], a plane is defined by just three points in space and, consequently, a fourlegged table has support plane multiplicities. This is why when one leg loses contact with the ground, the table is supported by the other three, as shown in **Figure 31**. The problem of the four-legged table is observed in dynamic rollover tests when the rear inner tyre loses contact with the ground (*Fz19 = 0*), and the front inner tyre

Applying this theory to the vehicle stability, when a vehicle makes a turn, it is subjected to an increasing lateral load until it reaches the rollover threshold [32]. During the turning, the rear inner tyre is usually the one that loses contact with the

*SRT*3*Dψϕφ* <sup>¼</sup> *<sup>h</sup>*<sup>1</sup> *cos <sup>φ</sup>* <sup>þ</sup> *<sup>h</sup>*2*ecos<sup>φ</sup>*

<sup>1</sup> � *<sup>t</sup>*1*Fz*<sup>3</sup> *cos <sup>ψ</sup>* <sup>þ</sup> *<sup>P</sup>*1ð Þ *Fz*<sup>17</sup> � *Wcosϕcos<sup>φ</sup> Wcosϕ*ð Þ *h*<sup>1</sup> *cos φ* þ *h*2*ecosφ* 

where *SRT*3*Dψϕφ* factor is the three-dimensional static rollover threshold for a trailer model with trailer/trailer angle (*ψ*), bank angle (*e*), and slope angle (*φ*).

*h*<sup>2</sup> � ð Þ *h*<sup>1</sup> þ *P*<sup>1</sup> *e*

�

and making tan ð Þ¼ *ϕ e*, where *e* is the tangent of the bank angle, we have:

*P*1 *t*<sup>1</sup> *cos ψ*

*Wcosϕcos<sup>φ</sup>* <sup>þ</sup> *may sin <sup>ϕ</sup>*

*Fz*<sup>17</sup>

*may cos <sup>ϕ</sup>* � *Wsinϕcos<sup>φ</sup>* <sup>¼</sup> <sup>0</sup> (34)

�

(35)

(36)

*Dynamic rollover test. Source: Adapted of Cabral [33].*

The normal forces *Fz3* and *Fz17* depend on the *LLT* coefficient in the front and rear axles respectively [4, 21, 34]. Furthermore, this coefficient depends on the vehicle type, speed, suspension, tyres, etc.

This information demonstrates that the *SRT*3*Dψϕφ* factor of a vehicle (Eq. (36)) is, in general, inferior to the *SRT* factor for a two-dimensional model vehicle [35], as shown in Eq. (37).

$$\text{SRT}\_{2D} = \frac{a\_\text{y}}{\text{g}} = \frac{t}{2h} \tag{37}$$

where *h* is the *CG* height, *t* is the vehicle track.


**Table 5.** *Parameters of the trailer model.*

### *Numerical and Experimental Studies on Combustion Engines and Vehicles*

With Eq. (36), it is possible to obtain a better vehicle stability representation and the *SRT*3*Dψϕφ* factor value attainments closer to reality.

**3. Case study**

that reported by Winkler [20, 32].

shown in **Figure 33b** [32].

trailer/trailer angles (*ψ*) [32].

location (**Figure 34**).

**Figure 33.**

*(a) Roll angle of the trailer (*θ*). (b) Change in the* SRT *factor.*

In this study, a B-train trailer with two axles on front and three axles on the rear is analysed. This model has a suspension system with a tandem axle, and its parameters depend on the construction materials. Another important parameter of the model is the dynamic rolling radius or loaded radius*li*. The proposedmodel considersMichelin XZA® [36] radial tyres. **Table 5** shows the parameters of the trailer used in this analysis [32, 38]. To calculate the *SRT* factor, the inertial force is increased until the lateral load transfer in the rear axle is complete (the entire load is transferred from the rear inner tyre to the rear outer tyre when the model makes a turn). The reduction in the *SRT* factor (Eq. (36) and the solution of the system of Eq. (38)) results from the combined action of the trailer systems, which allows a body roll angle of the trailer model (**Figure 33**) [32]. In this figure, it can be seen how the stability factor varies according to the influence of some of the parameters of the developed model.

*Stability Analysis of Long Combination Vehicles Using Davies Method*

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

When the model considers all parameters, the *LLT* coefficient on the front axle is approximately 70% of the *LLT* coefficient on the rear axle [21]. Applying this concept, the *SRTall* factor reduces to *0.3364 g*. Finally, the proposed model shows how the lateral offset of the cargo (*d1* = 0.1 m) influences the *SRToff* factor: 2 cm of lateral offset corresponds to a loss of stability of around *0.01 g* a reduction similar to

Additionally, the proposed model shows how a change in the lateral separation between the springs (*b*) influences the *SRT* factor. Some *LCVs* with tanker trailers have a greater lateral separation between the springs, which leads to a decrease in the roll angle and thus an increase in the *SRT* factor: 1 cm of lateral separation between the springs corresponds to a gain or loss of stability of around *0.001 g,* as

This model also allows the determination of the lateral (*h1*) and vertical (*h2*) *CG*

Finally, if we consider the recommended maximum lateral load transfer ratio for the rear axle of 0.6 [39, 40], and also include the recommended bank angle and longitudinal slope of the road [41, 42], we can calculate the *SRT* factor for a trailer model on downhill and uphill corners. **Table 6** shows a trailer model with different

To simplify the solution of the system of equations in Eq. (30), the following hypotheses were considered:


Eq. (38) shows the final static system for the stability analysis, solving this system using the Gauss-Jordan elimination method, all secondary variables are a function of primary variables, (*Px*—force acting on the *x*-axis, *Py*—force acting on the *y*-axis, and *Pz*—force acting on the z-axis).

