Stability Analysis of Long Combination Vehicles Using Davies Method

The cargo transportation in the world is mostly dominated by road transport, using long combination vehicles ( LCV ’ s). These vehicles offer more load capacity, which reduces transport costs and thus increases the efficiency and competitiveness of companies and the country. But the tradeoff of LCV ’ s is their low lateral stability and propensity to roll over, which has been the focus of many studies. Most vehicle stability models do not consider the longitudinal aspects of the vehicle and the road, such as the stiffness of the chassis, the gravity center location, and the longitudinal slope angle of the road. But, the use of three-dimensional models of vehicles allows a more rigorous analysis of vehicle stability. In this context, this study aims to develop a three-dimensional mechanism model representing the last trailer unit of an LCV under an increasing lateral load until it reaches the rollover threshold. The proposed model considers the gravity center movement of the trailer, which is affected by the suspension, tires, fifth-wheel, and the chassis. Davies method has proved to be an important tool in the kinetostatic analysis of mechanisms, and therefore it is employed for the kinetostatic analysis of the three-dimensional mechanism of the trailer.


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.

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][6][7].
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: • There are up to three different components of forces acting on the tyre-ground contact i of the vehicle [8][9][10], as shown in Figure 3, where F xi is the traction or brake force, F yi is the lateral force, and F zi is the normal force; • 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, tyreground contact was modelled as a pure revolute joint R along x-axis.  • While tyres 2 and 3 have a lateral deformation and may slide laterally, 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 contact was modelled as a prismatic joint P in the y-direction.
• Tyres are assumed as flexible mechanical components and can be represented by prismatic joints P, [11,12].
• In vehicles with rigid suspension, tyres remain perpendicular to the axle all the time.
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   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 xand 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 (S d ), with constraint in the z-axis, as shown in Figure 6.

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 (ma y )) [13]. These forces affect the passive actuators of the mechanism, as shown in Figure 7.
where δ T is the normal deformation of the tyre [14], ΔF is the algebraic change in the initial load, k t is the vertical stiffness of the tyre, a c is the regression coefficient, F Ti is the instantaneous tyre normal load, l i is the instantaneous dynamic rolling radius of the tyre i, F start zi is the initial normal load i, k T is the equivalent tyre vertical stiffness, l r is the initial dynamic rolling radius of tyre i, t i is the track width,  t i+1 is the axle width, and θ i;j are the rotation angles of the revolute joints i and j respectively.

Suspension 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 ydirection and a roll rotation about the x-axis [1,8], as shown in Figure 9a and b.

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.  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 springshear 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.

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 (ma y )). These forces affect the passive actuators of the mechanism, as shown in Figure 11.
Eqs. (6)-(12) define the kinematics of the suspension system: where T xn is the moment around the x-axis on the joint n, k ts is the spring's torsion coefficient, δ LS is the leaf spring deformation [17], ΔF 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, E s is the modulus of elasticity of a multiple leaf, l n is the instantaneous height of the leaf spring n, F LSn is the spring normal force n, l s is the initial suspension height, k Ls is the equivalent stiffness of the suspension, l n is the instantaneous height of the leaf spring n, b is the lateral separation between the springs, and θ n is the rotation angle of the revolute joint n.

The fifth wheel 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. Here l fw is the fifth wheel system's instantaneous height, F FWn is the fifth wheel normal load, and l fi is the fifth wheel system's initial height, b 1 is the fifth wheel width.

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   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 T CG is the torque applied by the forces acting on the CG, T f (T 28 ) is the torque applied on the vehicle front axle,T r (T 27 ) 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: where J f and J r 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 (J r ) is greater than the equivalent polar moment on the front (J f ). These can be expressed as J r = x J f (where x is the constant that allows controlling the torque distribution of the chassis); replacing and simplifying Eq. (13): 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).
where T fx (T x28 ) is the torque applied in the x-axis (this torque acts on the lateral load transfer on front axle),T fy (T y28 ) is the torque applied in the y-axis, and ψ is the steering angle of trailer front axle.

Three-dimensional trailer model
Considering the systems developed, the model of the trailer (Figure 18) is composed of the following mechanisms: • the front mechanism of the trailer is composed for the tyres, the suspension, and the fifth wheel, • the rear mechanism is composed for the tyres, and the suspension, and • the last mechanism is the chassis and links the front and rear mechanism of the model.

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][23][24][25][26][27][28][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.

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. 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, F f and F r 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, d 1 denotes the lateral CG displacement, d 2 denotes the longitudinal CG displacement, and d 3 the vertical CG displacement. Furthermore, Figure 22a and b show that only the weight (W) and the lateral inertial force (ma y ) act on the trailer CG, but, when the model takes into account   the longitudinal slope angle (φ) and the bank angle (ϕ) of the road, these forces have three components, as represented in Eqs. (17)- (19).
where P x is the force acting on the x-axis, P y is the force acting on the y-axis, and P z 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).
where h 2 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.

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.

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.     In Figures 24-28 and Table 1, l 13 is the distance between the fifth wheel and the front axle, l 1;2;7;8 are the dynamic rolling radii of tyres, t 1;3 are the front and rear track widths of the trailer respectively, t 2;4 are the front and rear axle widths respectively, b is the lateral separation between the springs, b 1 is the fifth wheel width, θ i is the revolution joint angle rotation i, l 3;4;9;10 are the instantaneous heights of the leaf spring, l 12 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 s 0i of the joints, and the centre of gravity (The vector s 0i ( Table 2) is obtained from the first three terms of the last column of equations shown in Table 3).

Wrench-Forces and moments
In the static analysis, all forces and moments of the mechanism are represented by wrenches ($ A ) [13]. The wrenches applied can be represented by the   To simplify the model of Figure 28, the following considerations were made: • for the x-direction a steady-state model was used in the analysis; • disturbances imposed by the road and the lateral friction forces (F y ) (tyre-ground contact) in the joints 3 and 19 were neglected; and    • the components of the trailer weight (W) and the inertial force (ma y ) 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 s i represents the orientation vector of each wrench i.
All of the wrenches of the mechanism together comprise the action matrix [A d ] given by Eq. (21) (or the amplified matrix of the Eq. (22)).
where p i 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). (23)

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].  Table 4.
Wrench parameters of the mechanism.
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 (P x , P y , and P z )).
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][26][27][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, P x , P y, and P z ) to be defined, as shown in Figure 30.   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 (F Ti ), spring normal load (F LSi ), fifth wheel normal load (F FWi ), and the passive torsional moment (T xi ) 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.

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)): In this case, the primary variable vector is: and the secondary variable vector is: 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: replacing P y and P z : where P 1 is a system variable (P 1 ¼ 2l 13 sin ψ þ t 2 cos ψ À 1 ð Þ ð Þ =2), h 1 is the instantaneous lateral distance between the zero-reference frame and the centre of gravity, and h 2 is the instantaneous CG height ( Table 2). Simplifying the equation, and making tan ϕ ð Þ ¼ e, where e is the tangent of the bank angle, we have: 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 (F z19 = 0), and the front inner tyre (F z3 ) 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 (F z19 = 0), and thus: where SRT 3D ψϕφ factor is the three-dimensional static rollover threshold for a trailer model with trailer/trailer angle (ψ), bank angle (e), and slope angle (φ). The normal forces F z3 and F z17 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 3D ψϕφ 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).
where h is the CG height, t is the vehicle track. Dynamic rollover test. Source: Adapted of Cabral [33].

Parameters of the trailer Value Units
Trailer With Eq. (36), it is possible to obtain a better vehicle stability representation and the SRT 3D ψϕφ factor value attainments closer to reality.
To simplify the solution of the system of equations in Eq. (30), the following hypotheses were considered: • in the majority of LCVs, the load on the trailers is uniformly distributed (Eq. (20)); • the lateral load transfer of the trailer model is controlled through the torsional moment of the chassis (spherical joints 27 and 28 (Eqs. (15) and (16))).
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, (P x -force acting on the x-axis, P y -force acting on the y-axis, and P z -force acting on the z-axis).

Case study
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 l i . The proposed model considers Michelin 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.
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 SRT all factor reduces to 0.3364 g. Finally, the proposed model shows how the lateral offset of the cargo (d 1 = 0.1 m) influences the SRT off factor: 2 cm of lateral offset corresponds to a loss of stability of around 0.01 g a reduction similar to that reported by Winkler [20,32].
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 shown in Figure 33b [32].
This model also allows the determination of the lateral (h 1 ) and vertical (h 2 ) CG location (Figure 34).
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 trailer/trailer angles (ψ) [32]. In the worst-case scenario, the trailer model, for a downhill corner with a bank angle of 0%, the longitudinal slope of the road of 8%, and a trailer/trailer angle of 20°c an reduce the SRT factor of the model by 59.6%, using 0.4511 g as a reference [32].  An analysis of Table 6 leads to the following conclusions for the critical conditions of the trailer: • a 1% bank angle corresponds to gain in the stability of around 0.01 g; • when the trailer is in downhill corners, a 1% slope angle corresponds to a loss of stability of around 0.0021 g; • the trailer/trailer angle is inversely proportional to the SRT factor since when the trailer makes a horizontal curve with a small radio, and the trailer/trailer angle and inertial force are large, the SRT factor is lower.

Conclusions
This study demonstrates that the longitudinal characteristics of a trailer model have an essential influence on the SRT factor calculation. In this case, the SRT factor is approximately 38% lower than the previously reported standard value. This value is very close to that reported by Winkler [20] (i.e. 40%), which suggests that the proposed model provides consistent results [32].
This model also shows that the change in the lateral separation between the springs (b) plays an important role, and thus it should be considered in the design and construction of trailers. Greater lateral separation between the springs will increase the trailer model stability [32].
We also found that the parameters of the road, such as the bank angle and the longitudinal slope angle, can affect vehicle stability. This situation is closer to the actual problem: when the road is not planar, the lateral and the longitudinal load transfer play an important role in reducing the stability. On the other hand, this provides a very important warning, because some simplifications carried out when estimating the SRT factor can lead to a considerably higher stability value. This is a point of concern, leading to the perception that our roads are safer than they really are [32].