Real-Time capable Multibody Model of dual Truck Front Axles

Abstract

Dual front steering axles are quite common in multi-axled heavy duty trucks. In standard layouts of such axle combinations, the steer motions of the wheels depend not only on the rotation of the steering wheel but also on the movements of the axles. As a consequence, the model complexity of the steering system should match with the complexity of the suspension model. The development of new technologies like advanced driver assistance systems or autonomous driving can only be accomplished efficiently using extensive simulation methods. Such kind of applications demand for computationally efficient vehicle models. This paper presents a steering system model for dual front axles of heavy duty trucks which supplements the suspension model of the axles. The model takes the torsional compliance of the steering column as well as the stiffness of the tie rods and the coupling rod into account. A quasi-static solution provides a straight forward computation including the partial derivatives required for an efficient implicit solver. The steering system model matches perfectly with comparatively lean, but sufficiently accurate multibody suspension models.

Full text

Real-Time capable Multibody Model of dual Truck Front Axles
Georg Rill
OTH Regensburg, Galgenbergstr. 30, 93053 Regensburg, Germany
Abstract: Dual front steering axles are quite common in multi-axled heavy duty trucks. In standard layouts of such
axle combinations, the steer motions of the wheels depend not only on the rotation of the steering wheel but also on the
movements of the axles. As a consequence, the model complexity of the steering system should match with the complexity
of the suspension model. The development of new technologies like advanced driver assistance systems or autonomous
driving can only be accomplished efficiently using extensive simulation methods. Such kind of applications demand for
computationally efficient vehicle models. This paper presents a steering system model for dual front axles of heavy duty
trucks which supplements the suspension model of the axles. The model takes the torsional compliance of the steering
column as well as the stiffness of the tie rods and the coupling rod into account. A quasi-static solution provides a straight
forward computation including the partial derivatives required for an efficient implicit solver. The steering system model
matches perfectly with comparatively lean, but sufficiently accurate multibody suspension models.
Keywords: Steering System, Dual Axles, Multibody Model, Vehicle Dynamics, Real-Time
INTRODUCTION
The multibody system approach to vehicle dynamics has become a standard (Bruni et al., 2020). A multibody approach,
which takes the specific features of trucks into account, is illustrated in Rill et al. (2021). Complex models are usually
separated into several subsystems including the tires, the steering system, the suspension, and the vehicle frame work
(Rill, 2006a). Handling tire models, like TMeasy as described in Rill (2013), provide a useful compromise between
accuracy and computation effort. A sophisticated modeling technique combined with an partial implicit solver, tailored
to the structure of general vehicle models, makes real-time applications possible even on small computers (Rill 1997).
Leaf spring suspended front axles are still very popular in particular at heavy duty trucks. A lumped link model, where
in a quasi-static approach the leaf springs act as generalized force elements, suspending and guiding the axle carrier, is
described and studied in Rill et al. (2022). Dual front steering axles suspended by leaf springs are quite common in
multi-axled heavy duty trucks. The modeling of the suspension and the steering system is quite a challenging task. In
particular, if one is interested in lean but sufficient accurate multibody models, which grant low computation costs and
make real-time applications possible even on small computers. The works of Qin et al. (2012) and Topaç et al. (2019)
apply the commercial software package ADAMS or ADAMS/CAR, respectively, for the analysis and the optimization of
the dual axle steering mechanism.
This paper presents a steering system model which fits perfectly to comparatively lean, but sufficiently accurate
multibody suspension models. It takes, in a quasi-static approach, the torsional compliance of the steering column as well
as the stiffness of the tie rods and the coupling rod into account.
THE MULTIBODY SYSTEM APPROACH TO VEHICLE DYNAMICS
General Layout of Dual Front Axles
The model of a virtual test truck (VTT) presented in Rill et al. (2021) encompasses just one front axle. However, the
modular structure makes it possible to supplement the VTT model with a generic subsystem which optionally includes a
second front axle and a more complex steering system acting on both axles.
In general, the axle carriers of the two single tired front axles are guided and suspended by leaf springs, Fig. 1. Each
axle is composed of the axle carrier, two knuckles and two wheels. The displacements 𝜉𝐴𝑖,𝜂𝐴𝑖 ,𝜁𝐴𝑖, and the rotation angles
𝛼𝐴𝑖,𝛽𝐴𝑖 ,𝛾𝐴𝑖 describe the momentary position and orientation of the axle carriers 𝑖=1,2relative to the vehicle-fixed
frame V. The angles 𝛿𝑖 𝑗 describe the king pin rotations of the left ( 𝑗=1) and right ( 𝑗=2) knuckles relative to the axle
carriers 𝑖=1,2. Handling tire models like TMeasy (Rill, 2013) consider the tire as a massless force element. Then, each
wheel consisting of the rim and the tire forms one rigid body, which performs the rotation 𝜑𝑖 𝑗 about an axis fixed to the
knuckle.
The steering system consists of the steering column, the steering box, the pitman arm, the tie rods 1 and 2, the coupling
rod, the coupling lever, and the track rods 1 and 2. The steering box transfers the steer input 𝛿𝑆to the rotation of the pitman
arm. The steering linkage is located here on the left side of the vehicle. As a consequence, the rotation of the pitman arm
is transferred via the coupling rod to the coupling lever and via the tie rod 1 to the rotation 𝛿11 of the left knuckle relative
to the axle carrier 1. The tie rod 2 transmits the rotation of the coupling lever to the rotation 𝛿21 of the left knuckle relative
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Multibody Model of dual Truck Front Axles
Figure 1 – General layout of dual truck front axles with leaf springs suspension and steering linkage.
to the axle carrier 2. The track rods 1and 2couple the knuckle rotations on the left and right side of each axle. The
corresponding constraint equations can be solved analytically and provide at each axle (𝑖=1,2) the knuckle rotations on
the right side 𝛿𝑖2=𝛿𝑖2(𝛿𝑖1)as functions of the knuckle rotations on the left side. Thus, leaving just one degree of freedom
for the steer motion (left knuckle rotations relative to the axle carrier) at each axle.
Each of the subsystems front axle 1and front axle 2has 6+1=7degrees of freedom then. In case a simplified
kinematical suspension model, each axle carrier just performs two unconstrained motions, usually characterized by hub
and roll. Then, each subsystem front axle has just 2+1=3degrees of freedom.
Vehicle Equations of Motion and Numerical Solution
The equations of motion for vehicle models, like the virtual test truck or the the virtual test car, described in Rill et al.
(2021) and Rill (2022), result in a set of nonlinear first order differential differential equations
¤𝑦=𝐾(𝑦)𝑧(1)
𝑀(𝑦) ¤𝑧=𝑞(𝑦, 𝑧, 𝑠, 𝑢)(2)
¤𝑠=𝑓(𝑦, 𝑧, 𝑠, 𝑢)(3)
The vector 𝑦collects the generalized coordinates of the vehicle and the kinematic matrix 𝐾defines non-trivial generalized
speeds which are arranged in the vector 𝑧. The principle of virtual power or Jourdain’s principle, respectively, delivers the
elements of the mass matrix 𝑀and the components of the vector 𝑞which summarizes the generalized forces and torques
applied to the bodies of the multibody system. The forces and torques may depend on external control inputs collected
in the vector 𝑢and on additional states 𝑠that are required to describe the dynamics of sophisticated force elements, like
coupled air springs or tires. The partly implicit Euler-Step
𝑦𝑘+1−𝑦𝑘=ℎ 𝐾 (𝑦𝑘)𝑧𝑘+1(4)
𝑀(𝑦𝑘)𝑧𝑘+1−𝑧𝑘=ℎ 𝑞 𝑦𝑘+1, 𝑧𝑘+1, 𝑠𝑘+1, 𝑢𝑘+1(5)
𝑠𝑘+1−𝑠𝑘=ℎ 𝑓 𝑦𝑘, 𝑧 𝑘, 𝑠𝑘+1, 𝑢 𝑘+1(6)
is tailored to vehicle dynamics, because it assumes that the dynamics of the force elements, described by the states 𝑠, is
considerably faster than the dynamics of the vehicle represented by the vectors of generalizes coordinates 𝑦and generalized
speeds 𝑧. The superscripts 𝑘and 𝑘+1abbreviate the states at time 𝑡and 𝑡+ℎwhere ℎdenotes the integration step size.
The control inputs depend on the time 𝑡and deliver 𝑢𝑘+1=𝑢(𝑡+ℎ)straightforwardly. The truncated Taylor expansion
𝑓𝑘+1≈𝑓𝑘+ (𝜕 𝑓 /𝜕𝑠) (𝑠𝑘+1−𝑠𝑘)approximates the function 𝑓as defined in (6) and results into
𝐼−ℎ𝜕 𝑓
𝜕𝑠 𝑠𝑘+1−𝑠𝑘=ℎ 𝑓 𝑦𝑘, 𝑧𝑘, 𝑠𝑘, 𝑢 (𝑡+ℎ)(7)
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G. Rill
where 𝐼denotes the matrix of identity with the same dimension as the matrix of partial derivatives 𝜕 𝑓 /𝜕 𝑠. The vector of
generalized forces and torques required in (5) is approximated by
𝑞𝑦𝑘+1, 𝑧𝑘+1, 𝑠 𝑘+1, 𝑢𝑘+1≈𝑞𝑦𝑘+ℎ𝐾 (𝑦𝑘)𝑧𝑘, 𝑧𝑘, 𝑠 𝑘+1, 𝑢𝑘+1+𝜕𝑞
𝜕𝑦 𝑦𝑘+1−𝑦𝑘+ℎ 𝐾 (𝑦𝑘)𝑧𝑘+𝜕𝑞
𝜕𝑧 𝑧𝑘+1−𝑧𝑘(8)
where 𝑦𝑘+ℎ𝐾 (𝑦𝑘)𝑧𝑘represents an explicit Euler step applied to (1) which enhances the accuracy of the approximation (8)
in particular for strongly nonlinear force elements. Then, the Euler step (5) results in
𝑀(𝑦𝑘) − ℎ𝜕𝑞
𝜕𝑧 +ℎ𝜕 𝑞
𝜕𝑦 𝐾(𝑦𝑘)𝑧𝑘+1−𝑧𝑘=ℎ 𝑞 𝑦𝑘+ℎ 𝐾 (𝑦𝑘)𝑧𝑘, 𝑧𝑘, 𝑠𝑘+1, 𝑢𝑘+1(9)
where (4) was used in addition. As demonstrated in Rill (2006b) the contribution of standard force elements, like springs
and dampers, to the partial derivatives 𝜕𝑞/𝜕𝑦 and 𝜕 𝑞/𝜕 𝑧 is a straight forward task and requires only a small amount of
additional computation effort.
The cascade of partly implicit Euler steps, defined by (7), (5), and (4), provides an extremely fast and robust solver
tailored to vehicle dynamics. When operated with the standard step size of ℎ=1 ms it grants simulation results of sufficient
accuracy and when coded in a higher programming language like ANSI C90 it achieves real-time performance for rather
complex vehicle models even on small personal computers (Rill et al. 2021).
STEERING SYSTEM MODEL
Model Structure
The steering system model shown in Fig. 2 takes the rotational compliance of the steering column 𝑐𝑆, the stiffness of the
tie rods 𝑐1,𝑐2, and the stiffness of the coupling rod 𝑐𝐶into account. The angles 𝛿𝑊and 𝛿𝑆describe the rotations of the
steering wheel and the steering box input. The rotation angles of the pitman arm and the coupling lever are limited by
stops to the ranges 𝛿𝑚𝑖𝑛
𝑃≤𝛿𝑃≤𝛿𝑚𝑎 𝑥
𝑃and 𝛿𝑚𝑖𝑛
𝐿≤𝛿𝐿≤𝛿𝑚𝑎 𝑥
𝐿, respectively. The unit vectors 𝑒𝑃and 𝑒𝐿define the rotation
axes of the pitman arm and the coupling lever. The former is fixed at point H1to the steering box and the latter at point H2
to the truck frame. The tie rod 1 is attached at joint I1to the pitman arm and at joint J1to the left knuckle of axle 1. The
ends of the coupling rod C1and C2are connected to the pitman arm and to the coupling lever. The tie rod 2 is attached at
joint I2to the coupling lever and at joint J2to the left knuckle of axle 2.
Figure 2 – Steering system for dual steered front axles with modeled compliances.
The design position of the attachment points I1, J1, I2, J2, and C1, C2deliver the lengths ℓ10,ℓ20, and ℓ𝐶0of the
undeformed rods. The actual lengths are given by
ℓ1=𝑟𝑇
𝐼1𝐽1,𝑉 𝑟𝐼1𝐽1,𝑉 ;ℓ𝐶=𝑟𝑇
𝐶1𝐶2,𝑉 𝑟𝐶1𝐶2,𝑉 ;ℓ2=𝑟𝑇
𝐼2𝐽2,𝑉 𝑟𝐼2𝐽2,𝑉 (10)
The comma separated index 𝑉indicates that the vectors
𝑟𝐼1𝐽1,𝑉 =𝑟𝑉 𝐽 1,𝑉 −𝑟𝑉 𝐼1, 𝑉 ;𝑟𝐶1𝐶2,𝑉 =𝑟𝑉𝐶 2,𝑉 −𝑟𝑉 𝐶1,𝑉 ;𝑟𝐼2𝐽2, 𝑉 =𝑟𝑉 𝐽2,𝑉 −𝑟𝑉𝐼 2,𝑉 (11)
pointing from one end of each rod to the other are expressed in the vehicle-fixed reference frame V. The compliance of the
rods are modeled by simple linear springs. Then, the forces acting in the tie rod 1, the coupling rod, and in tie rod 2 are
provided by
𝐹1=𝑐1(ℓ1−ℓ10)=𝑐1𝑢1;𝐹𝐶=𝑐𝐶(ℓ𝐶−ℓ𝐶0)=𝑐𝐶𝑢𝐶;𝐹2=𝑐2(ℓ2−ℓ20)=𝑐2𝑢2(12)
where 𝑐1,𝑐𝐶,𝑐2denote the stiffness properties and 𝑢1,𝑢𝐶,𝑢2define the deformations of the corresponding rods.
The tie rods 1 and 2 are attached at J1and J2to the left wheel bodies of the two axles. Their actual positions with
respect to the vehicle fixed reference frame V depend on the generalized coordinates 𝑦𝐴1and 𝑦𝐴2describing the axle
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Multibody Model of dual Truck Front Axles
motions and the rotations of the left wheel bodies about the king pins
𝑟𝑉 𝐽1, 𝑉 =𝑟𝑉 𝐽1,𝑉 (𝑦𝐴1)and 𝑟𝑉 𝐽 2,𝑉 =𝑟𝑉 𝐽 2,𝑉 (𝑦𝐴2)(13)
The momentary positions where the tie rod 1 and the coupling rod are attached to the pitman arm are defined by
𝑟𝑉 𝐼1, 𝑉 =𝑟𝑉𝐻 1,𝐷 +𝐴𝑃(𝛿𝑃)𝑟𝐻1𝐼1,𝐷
| {z }
𝑟𝐻1𝐼1,𝑉
and 𝑟𝑉𝐶 1,𝑉 =𝑟𝑉 𝐻 1,𝐷 +𝐴𝑃(𝛿𝑃)𝑟𝐻1𝐶1,𝐷
| {z }
𝑟𝐻1𝐶1,𝑉
(14)
The matrix 𝐴𝑃=𝐴𝑃(𝛿𝑃)describes the rotation of the pitman arm about an axis defined by the unit vector 𝑒𝑃, 𝑉 . and the
vectors 𝑟𝑉𝐻 1,𝐷 ,𝑟𝐻1𝐼1, 𝐷 , and 𝑟𝐻1𝐶1, 𝐷 characterize the design position of H1and the positions of I1and C1relative to H1.
The momentary positions where the coupling rod and the tie rod 2 are attached to the coupling lever are provided by
𝑟𝑉𝐶 2,𝑉 =𝑟𝑉 𝐻 2,𝐷 +𝐴𝐿(𝛿𝐿)𝑟𝐻2𝐶2,𝐷
| {z }
𝑟𝐻2𝐶2,𝑉
and 𝑟𝑉 𝐼2, 𝑉 =𝑟𝑉𝐻 2,𝐷 +𝐴𝐿(𝛿𝐿)𝑟𝐻2𝐼2,𝐷
| {z }
𝑟𝐻2𝐼2,𝑉
(15)
The matrix 𝐴𝐿=𝐴𝐿(𝛿𝐿)describes the rotation of the coupling lever about an axis defined by the unit vector 𝑒𝐿,𝑉 and the
vectors 𝑟𝑉𝐻 2,𝐷 ,𝑟𝐻2𝐶2, 𝐷 , and 𝑟𝐻2𝐼2, 𝐷 characterize the design position of H2and the positions of C2and I2relative to H2.
Quasi-static Approach
The rotation of the steering box input shaft 𝛿𝑆and the rotation angle of the coupling lever 𝛿𝐿are computed in a quasi-static
approach. The torsional compliance of the steering column generates the steering torque acting at the steer box input as
𝑇𝑆=𝑐𝑆(𝛿𝑊−𝛿𝑆)=𝑐𝑆𝜑𝑆(16)
where 𝑐𝑆denotes the torsional stiffness of the steering column, 𝛿𝑊=𝛿𝑊(𝑡)describes the rotation of the steering wheel
and 𝜑𝑆the torsion angle of the steering column. The principle of virtual work for the steering box and the coupling lever
results in
𝛿 𝜑𝑆𝑇𝑆(𝜑𝑆) + 𝛿 𝑢1𝐹1(𝑢1) + 𝛿 𝑢 𝐶𝐹𝐶(𝑢𝐶) + 𝛿 𝑢2𝐹2(𝑢2) + 𝑇𝑃𝐿 (𝛿𝑃)𝛿 𝛿 𝑃+𝑇𝐿𝐿 (𝛿𝐿)𝛿 𝛿 𝐿=0(17)
The torques 𝑇𝑃 𝐿 =𝑇𝑃𝐿 (𝛿𝑃)and 𝑇𝐿 𝐿 =𝑇𝐿𝐿 (𝛿𝐿)model the impact of stops which limit the rotations of the pitman arm and
the coupling lever. The transmission of the steering box delivers the momentary and the virtual rotations of the pitman
arm as
𝛿𝑃=𝛿𝑃(𝛿𝑆)and 𝛿 𝛿𝑃==
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝛿 𝜑𝑆=
𝜕𝛿 𝑃
𝜕𝛿𝑆
𝜕𝛿𝑆
𝜕𝜑𝑆
𝛿 𝜑𝑆=
𝜕𝛿 𝑃
𝜕𝛿𝑆
(−1)𝛿 𝜑𝑆=−𝑖𝑆𝛿 𝜑𝑆(18)
The definition of the torsion angle 𝜑𝑆=𝛿𝑊−𝛿𝑆as specified in (16) simply results in 𝜕𝛿𝑆/𝜕𝜑𝑆=−1and 𝑖𝑆=𝜕𝛿𝑃/𝜕𝛿𝑆
defines the ratio of the steering box which might depend nonlinearly on the rotation of steering box input shaft. Inspecting
Equations (12) to (15) the virtual deformations of the rods result in
𝛿𝑢1=
𝜕ℓ1
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝛿 𝜑𝑆with 𝜕ℓ1
𝜕𝛿 𝑃
=
𝑟𝑇
𝐼1𝐽1,𝑉
ℓ1
𝜕𝑟𝐼1𝐽1,𝑉
𝜕𝛿 𝑃
=𝑒𝑇
𝐼1𝐽1,𝑉
𝜕𝑟𝐼1𝐽1,𝑉
𝜕𝛿 𝑃
(19)
𝛿𝑢𝐶=
𝜕ℓ𝐶
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝛿 𝜑𝑆+𝜕ℓ𝐶
𝜕𝛿 𝐿
𝛿 𝛿𝐿with 𝜕ℓ𝐶
𝜕𝛿 𝑃, 𝐿
=
𝑟𝑇
𝐶1𝐶2,𝑉
ℓ𝐶
𝜕𝑟𝐶1𝐶2, 𝑉
𝜕𝛿 𝑃, 𝐿
=𝑒𝑇
𝐶1𝐶2,𝑉
𝜕𝑟𝐶1𝐶2, 𝑉
𝜕𝛿 𝑃, 𝐿
(20)
𝛿𝑢2=
𝜕ℓ2
𝜕𝛿 𝐿
𝛿 𝛿𝐿with 𝜕ℓ2
𝜕𝛿 𝐿
=
𝑟𝑇
𝐼2𝐽2,𝑉
ℓ2
𝜕𝑟𝐼2𝐽2,𝑉
𝜕𝛿 𝐿
=𝑒𝑇
𝐼2𝐽2,𝑉
𝜕𝑟𝐼2𝐽2,𝑉
𝜕𝛿 𝐿
(21)
The pitman arm and the coupling lever perform rotations about axes defined by the unit vectors 𝑒𝑃and 𝑒𝐿and change
the momentary positions of the attachment points I1, C1and C2, I2, respectively. The partial derivatives of the position
vectors defined in (11) are then provided by
𝜕𝑟𝐼1𝐽1,𝑉
𝜕𝛿 𝑃
=−𝜕𝑟𝑉 𝐼 1,𝑉
𝜕𝛿 𝑃
=−𝜕𝑟𝐻1𝐼1,𝑉
𝜕𝛿 𝑃
=−𝑒𝑃,𝑉 ×𝑟𝐻1𝐼1, 𝑉 (22)
𝜕𝑟𝐶1𝐶2, 𝑉
𝜕𝛿 𝑃
=−𝜕𝑟𝑉 𝐶1, 𝑉
𝜕𝛿 𝑃
=−𝜕𝑟𝐻1𝐶1,𝑉
𝜕𝛿 𝑃
=−𝑒𝑃,𝑉 ×𝑟𝐻1𝐶1, 𝑉 (23)
𝜕𝑟𝐶1𝐶2, 𝑉
𝜕𝛿 𝐿
=
𝜕𝑟𝑉 𝐶2, 𝑉
𝜕𝛿 𝐿
=
𝜕𝑟𝐻2𝐶2,𝑉
𝜕𝛿 𝐿
=𝑒𝐿,𝑉 ×𝑟𝐻2𝐶2,𝑉 (24)
𝜕𝑟𝐼2𝐽2,𝑉
𝜕𝛿 𝐿
=−𝜕𝑟𝑉 𝐼 2,𝑉
𝜕𝛿 𝐿
=−𝜕𝑟𝐻2𝐼2,𝑉
𝜕𝛿 𝐿
=−𝑒𝐿,𝑉 ×𝑟𝐻2𝐼2,𝑉 (25)
Inserting Equations (18) to (25) into Equation (17) results in
𝛿 𝜑𝑆𝑐𝑆𝜑𝑆−𝑖𝑆𝛿 𝜑𝑆𝑒𝑇
𝐼1𝐽1,𝑉 −𝑒𝑃, 𝑉 ×𝑟𝐻1𝐼1,𝑉 𝐹1+𝑒𝑇
𝐶1𝐶2,𝑉 −𝑒𝑃, 𝑉 ×𝑟𝐻1𝐶1,𝑉 𝐹𝐶+
𝛿 𝛿𝐿𝑒𝑇
𝐶1𝐶2,𝑉 𝑒𝐿, 𝑉 ×𝑟𝐻2𝐶2,𝑉 𝐹𝐶+𝑒𝑇
𝐼2𝐽2,𝑉 −𝑒𝐿,𝑉 ×𝑟𝐻2𝐼2,𝑉 𝐹2=0(26)
private use only

G. Rill
Virtual displacements and rotations, like 𝛿 𝜑𝑆and 𝛿 𝛿 𝐿are infinite small but do not vanish permanently. Hence, the virtual
work (26) delivers two relations
𝑇1=𝑐𝑆𝜑𝑆+𝑖𝑆𝑒𝑇
𝑃,𝑉 𝑟𝐻1𝐼1, 𝑉 ×𝑒𝐼1𝐽1,𝑉 𝐹1+𝑟𝐻1𝐶1,𝑉 ×𝑒𝐶1𝐶2,𝑉 𝐹𝐶+𝑖𝑆𝑇𝑃𝐿 =0(27)
𝑇2=𝑒𝑇
𝐿,𝑉 𝑟𝐻2𝐼2,𝑉 ×𝑒𝐼2𝐽2,𝑉 𝐹2−𝑟𝐻2𝐶2,𝑉 ×𝑒𝐶1𝐶2,𝑉 𝐹𝐶+𝑇𝐿𝐿 =0(28)
where the property 𝑎𝑇(𝑏×𝑐)=𝑏𝑇(𝑐×𝑎)of a scalar triple product was used to rearrange the equations in the form of
torque balances at the steering box and the coupling lever. Introducing the shortcuts
𝑎11 =𝑒𝑇
𝑃,𝑉 𝑟𝐻1𝐼1, 𝑉 ×𝑒𝐼1𝐽1,𝑉 𝑎1𝐶=𝑒𝑇
𝑃,𝑉 𝑟𝐻1𝐶1, 𝑉 ×𝑒𝐶1𝐶2,𝑉 (29)
𝑎22 =𝑒𝑇
𝐿,𝑉 𝑟𝐻2𝐼2,𝑉 ×𝑒𝐼2𝐽2,𝑉 𝑎2𝐶=𝑒𝑇
𝐿,𝑉 𝑟𝐻2𝐶2,𝑉 ×𝑒𝐶1𝐶2,𝑉 (30)
the torque balances simply read as
𝑇1=𝑐𝑆𝜑𝑆+𝑖𝑆(𝑎11 𝐹1+𝑎1𝐶𝐹𝐶+𝑇𝑃𝐿 )=0(31)
𝑇2=𝑎22 𝐹2−𝑎2𝐶𝐹𝐶+𝑇𝐿𝐿 =0(32)
Numerical Solution
The torque balances represent a set of two nonlinear equations which can be solved iteratively by the Newton-Raphson
scheme
𝑇(𝑦𝑖
𝑆) + 𝜕𝑇
𝜕𝑦𝑆𝑦𝑖+1
𝑆) − 𝑦𝑖
𝑆)=0or 𝑦𝑖+1
𝑆)=𝑦𝑖
𝑆−𝜕𝑇
𝜕𝑦𝑆−1
𝑇(𝑦𝑖
𝑆)𝑖=0,1, ... (33)
The torsion angle of the steering column 𝜑𝑆and the rotation of the coupling lever 𝛿𝐿as well as the torques 𝑇1and 𝑇2are
hereby collected in 2×1-vectors
𝑦𝑆="𝜑𝑆
𝛿𝐿#, 𝑇 (𝑦𝑆)="𝑇1(𝜑𝑖
𝑆, 𝛿𝑖
𝐿)
𝑇2(𝜑𝑖
𝑆, 𝛿𝑖
𝐿)#,and 𝜕𝑇
𝜕𝑦𝑆
=
𝜕𝑇1
𝜕𝜑𝑆
𝜕𝑇1
𝜕𝛿 𝐿
𝜕𝑇2
𝜕𝜑𝑆
𝜕𝑇2
𝜕𝛿 𝐿

(34)
which, as a consequence, define the elements of the 2×2-matrix 𝜕𝑇 /𝜕𝑦𝑆of partial torque derivatives. In general, the
rotations of the pitman arm 𝛿𝑃and the coupling lever 𝛿𝐿will not differ significantly. Then
𝜑0
𝑆=0and 𝛿0
𝐿=𝛿𝑃(𝛿𝑊)(35)
provide appropriate values to start the iteration. The derivatives of the torques 𝑇1(𝜑𝑖
𝑆, 𝛿𝑖
𝐿)and 𝑇2(𝜑𝑖
𝑆, 𝛿𝑖
𝐿)as defined in
(27) and (28) result in
𝜕𝑇1
𝜕𝜑𝑆
=𝑐𝑆+𝑖𝑆𝑒𝑇
𝑃,𝑉 𝜕𝑟𝐻1𝐼1, 𝑉
𝜕𝛿 𝑃
×𝑒𝐼1𝐽1,𝑉 𝐹1+𝑟𝐻1𝐼1,𝑉 ×𝑒𝐼1𝐽1,𝑉
𝜕𝐹1
𝛿𝑃
+𝜕𝑟𝐻1𝐶1,𝑉
𝜕𝛿 𝑃
×𝑒𝐶1𝐶2,𝑉 𝐹𝐶+𝑟𝐻1𝐶1,𝑉 ×𝑒𝐶1𝐶2,𝑉
𝜕𝐹𝐶
𝜕𝛿 𝑃(−𝑖𝑆)
+𝑖𝑆
𝜕𝑇𝑃 𝐿
𝜕𝛿 𝑃
𝑖𝑆(−1)
(36)
𝜕𝑇1
𝜕𝛿 𝐿
=𝑖𝑆𝑒𝑇
𝑃,𝑉 𝑟𝐻1𝐶1, 𝑉 ×𝑒𝐶1𝐶2,𝑉
𝜕𝐹𝐶
𝜕𝛿 𝐿
(37)
𝜕𝑇2
𝜕𝜑𝑆
=−𝑒𝑇
𝐿,𝑉 𝑟𝐻2𝐶2,𝑉 ×𝑒𝐶1𝐶2,𝑉
𝜕𝐹𝐶
𝜕𝛿 𝑃
(−𝑖𝑆)(38)
𝜕𝑇2
𝜕𝛿 𝐿
=𝑒𝑇
𝐿,𝑉 𝜕𝑟𝐻2𝐼2,𝑉
𝜕𝛿 𝐿
×𝑒𝐼2𝐽2,𝑉 𝐹2+𝑟𝐻2𝐼2,𝑉 ×𝑒𝐼2𝐽2,𝑉
𝜕𝐹2
𝜕𝛿 𝐿
−𝜕𝑟𝐻2𝐶2,𝑉
𝜕𝛿 𝐿
×𝑒𝐶1𝐶2,𝑉 𝐹𝐶−𝑟𝐻2𝐶2,𝑉 ×𝑒𝐶1𝐶2,𝑉
𝜕𝐹𝐶
𝜕𝛿 𝐿+𝜕𝑇𝐿 𝐿
𝜕𝛿 𝐿
(39)
The partial derivatives of the position vectors are already computed in (22) to (25). The force derivatives result in
𝜕𝐹1
𝜕𝛿 𝑃
=𝑐1
𝜕ℓ1
𝜕𝛿 𝑃
=𝑐1𝑒𝑇
𝐼1𝐽1
𝜕𝑟𝐼1𝐽1
𝜕𝛿 𝑃
=𝑐1𝑒𝑇
𝐼1𝐽1(−𝑒𝑃×𝑟𝐻1𝐼1)=−𝑐1𝑒𝑇
𝑃(𝑟𝐻1𝐼1×𝑒𝐼1𝐽1)=−𝑐1𝑎11
𝜕𝐹𝐶
𝜕𝛿 𝐿
=𝑐𝐶
𝜕ℓ𝐶
𝜕𝛿 𝐿
=𝑐𝐶𝑒𝑇
𝐶1𝐶2
𝜕𝑟𝐶1𝐶2
𝜕𝛿 𝐿
=𝑐𝐶𝑒𝑇
𝐶1𝐶2(𝑒𝐿×𝑟𝐻2𝐶2)=𝑐𝐶𝑒𝑇
𝐿(𝑟𝐻2𝐶2×𝑒𝐶1𝐶2)=𝑐𝐶𝑎2𝐶
𝜕𝐹𝐶
𝜕𝛿 𝑃
=𝑐𝐶
𝜕ℓ𝐶
𝜕𝛿 𝑃
=𝑐𝐶𝑒𝑇
𝐶1𝐶2
𝜕𝑟𝐶1𝐶2
𝜕𝛿 𝑃
=𝑐𝐶𝑒𝑇
𝐶1𝐶2(−𝑒𝑃×𝑟𝐻1𝐶1)=−𝑐𝐶𝑒𝑇
𝑃(𝑟𝐻1𝐶1×𝑒𝐶1𝐶2)=−𝑐𝐶𝑎1𝐶
𝜕𝐹2
𝜕𝛿 𝐿
=𝑐2
𝜕ℓ2
𝜕𝛿 𝐿
=𝑐2𝑒𝑇
𝐼2𝐽2
𝜕𝑟𝐼2𝐽2
𝜕𝛿 𝐿
=𝑐2𝑒𝑇
𝐼2𝐽2(−𝑒𝐿×𝑟𝐻2𝐼2)=−𝑐2𝑒𝑇
𝐿(𝑟𝐻2𝐼2×𝑒𝐼2𝐽2)=𝑐2𝑎22
(40)
private use only

Multibody Model of dual Truck Front Axles
where all comma separated subscripts 𝑉, indicating that the corresponding vector is expressed in the frame-fixed coordinate
system, are omitted for the sake of simplicity. The final results incorporate the partial derivatives 𝜕𝜑 𝑃/𝜕𝜑𝑆,𝜕ℓ1/𝜕 𝜑𝑆,
𝜕ℓ𝐶/𝜕𝜑𝑆,𝜕ℓ𝐶/𝜕𝛿𝐿, and 𝜕 ℓ2/𝜕𝛿 𝐿defined in (18) and (19) to (21) as well as the abbreviations (29) and (30).
The analytical computation of the elements of the matrix of partial torque derivatives 𝜕𝑇/𝜕𝑦𝑆makes the Newton
algorithm (33) converge very fast and robust even with the rather simple starting values as provided by (35). The partial
derivatives provided in this section and in particular the inverse of the 2×2-matrix 𝜕𝑇/𝜕𝑦𝑆also deliver the effective
stiffness properties of the tie rods which are required for an implicit integration step.
Effective Stiffness Properties
The tie rod forces 𝐹1and 𝐹2as defined by (10) to (12) depend on the position vectors 𝑟𝑉 𝐽1,𝑉 and 𝑟𝑉 𝐽 2,𝑉 which describe
the momentary positions of the axle-fixed joints J1and J2. As a consequence, the torsional angle of the steering column
𝜑𝑆and the rotation angle of the coupling lever 𝛿𝐿computed via the torque balances (31) and (32) also depend on 𝑟𝑉𝐽 1,𝑉
and 𝑟𝑉 𝐽2, 𝑉 . Then, the total derivatives of the vectorized torque balance 𝑇with respect to 𝑟𝑉𝐽 1,𝑉 and 𝑟𝑉 𝐽 2,𝑉 result in
𝜕 𝑇
𝜕 𝑟𝑉 𝐽 1,𝑉 𝑡𝑙
=
𝜕𝑇
𝜕𝑟𝑉 𝐽 1,𝑉
+𝜕𝑇
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 1,𝑉
=0and 𝜕 𝑇
𝜕 𝑟𝑉 𝐽 2,𝑉 𝑡𝑙
=
𝜕𝑇
𝜕𝑟𝑉 𝐽 2,𝑉
+𝜕𝑇
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 2,𝑉
=0(41)
and deliver the derivatives of the vector 𝑦𝑆with respect to the position vectors 𝑟𝑉 𝐽 1,𝑉 and 𝑟𝑉 𝐽 2,𝑉 . The partial derivatives
of the torque vector 𝑇with respect to the position vectors 𝑟𝑉 𝐽1, 𝑉 and 𝑟𝑉 𝐽2,𝑉 result in
𝜕𝑇
𝜕𝑟𝑉 𝐽 1,𝑉
=
𝜕𝑇1
𝜕𝑟𝑉 𝐽 1,𝑉
𝜕𝑇2
𝜕𝑟𝑉 𝐽 1,𝑉

=
𝑖𝑆𝑎11
𝜕𝐹1
𝜕𝑟𝑉 𝐽 1,𝑉
0
and 𝜕𝑇
𝜕𝑟𝑉 𝐽 2,𝑉
=
𝜕𝑇1
𝜕𝑟𝑉 𝐽 1,𝑉
𝜕𝑇2
𝜕𝑟𝑉 𝐽 1,𝑉

=
0
𝑎22
𝜕𝐹2
𝜕𝑟𝑉 𝐽 2,𝑉 
(42)
where 𝜕𝐹1
𝜕𝑟𝑉 𝐽 1,𝑉
=𝑐1𝑒𝑇
𝐼1𝐽1,𝑉
𝜕𝑟𝐼1𝐽1,𝑉
𝜕𝑟𝑉 𝐽 1,𝑉
=𝑐1𝑒𝑇
𝐼1𝐽1,𝑉
𝜕𝑟𝑉 𝐽 1,𝑉
𝜕𝑟𝑉 𝐽 1,𝑉
=𝑐1𝑒𝑇
𝐼1𝐽1,𝑉 (43)
𝜕𝐹2
𝜕𝑟𝑉 𝐽 2,𝑉
=𝑐2𝑒𝑇
𝐼2𝐽2,𝑉
𝜕𝑟𝐼2𝐽2,𝑉
𝜕𝑟𝑉 𝐽 2,𝑉
=𝑐2𝑒𝑇
𝐼2𝐽2,𝑉
𝜕𝑟𝑉 𝐽 2,𝑉
𝜕𝑟𝑉 𝐽 2,𝑉
=𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 (44)
provides the partial derivatives of the tie rod forces 𝐹1and 𝐹2with respect to the position vectors 𝑟𝑉𝐽 1,𝑉 and 𝑟𝑉 𝐽 2,𝑉 . The
partial derivative of the torque vector 𝑇with respect to the vector 𝑦𝑆is defined in (33) and (36) to (39) provide it element
by element. Then, the total torque derivatives (41) deliver
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 1,𝑉
=−𝜕𝑇
𝜕𝑦𝑆−1𝜕 𝑇
𝜕 𝑟𝑉 𝐽 1,𝑉 𝑡𝑙
=−𝜕𝑇
𝜕𝑦𝑆−1
𝑖𝑆𝑎11
𝜕𝐹1
𝜕𝑟𝑉 𝐽 1,𝑉
0
=−𝜕𝑇
𝜕𝑦𝑆−1𝑖𝑆𝑎11 𝑐1𝑒𝑇
𝐼1𝐽1,𝑉
0(45)
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 2,𝑉
=−𝜕𝑇
𝜕𝑦𝑆−1𝜕 𝑇
𝜕 𝑟𝑉 𝐽 2,𝑉 𝑡𝑙
=−𝜕𝑇
𝜕𝑦𝑆−1
0
𝑎22
𝜕𝐹2
𝜕𝑟𝑉 𝐽 2,𝑉 
=−𝜕𝑇
𝜕𝑦𝑆−10
𝑎22 𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 (46)
where the 2×2-matrix of partial torque derivatives 𝜕𝑇 /𝜕𝑦𝑆is defined in (34) and its inverse is already used in (33). The
total changes of the tie rod forces 𝐹1and 𝐹2with respect to the position vectors 𝑟𝑉 𝐽1,𝑉 and 𝑟𝑉 𝐽 2,𝑉 are defined by
𝜕 𝐹1
𝜕 𝑟𝑉 𝐽 1,𝑉 𝑡𝑙
=
𝜕𝐹1
𝜕𝑟𝑉 𝐽 1,𝑉
+𝜕𝐹1
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 1,𝑉
=𝑐1𝑒𝑇
𝐼1𝐽1,𝑉 +𝜕𝐹1
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝜕𝐹1
𝜕𝛿 𝐿 −𝜕𝑇
𝜕𝑦𝑆−1𝑖𝑆𝑎11 𝑐1𝑒𝑇
𝐼1𝐽1,𝑉
0!
=𝑐1𝑒𝑇
𝐼1𝐽1,𝑉 −−𝑐1𝑎11 (−𝑖𝑆)0𝜕𝑇
𝜕𝑦𝑆−1𝑖𝑆𝑎11 𝑐1
0𝑒𝑇
𝐼1𝐽1,𝑉
=𝑐1 1−1 0 𝜕𝑇
𝜕𝑦𝑆−11
0𝑐1(𝑖𝑆𝑎11)2!𝑒𝑇
𝐼1𝐽1,𝑉 =𝑐𝑞
11 𝑒𝑇
𝐼1𝐽1,𝑉
(47)
𝜕 𝐹2
𝜕 𝑟𝑉 𝐽 1,𝑉 𝑡𝑙
=
𝜕𝐹2
𝜕𝑟𝑉 𝐽 1,𝑉
+𝜕𝐹2
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 1,𝑉
=0+𝜕𝐹2
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝜕𝐹2
𝜕𝛿 𝐿 −𝜕𝑇
𝜕𝑦𝑆−1𝑖𝑆𝑎11 𝑐1𝑒𝑇
𝐼1𝐽1,𝑉
0!
=−𝑖𝑆0𝑐2𝑎22 𝜕𝑇
𝜕𝑦𝑆−1𝑐1𝑎11
0𝑒𝑇
𝐼1𝐽1,𝑉 =𝑐𝑞
21 𝑒𝑇
𝐼1𝐽11,𝑉
(48)
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G. Rill
𝜕 𝐹1
𝜕 𝑟𝑉 𝐽 2,𝑉 𝑡𝑙
=
𝜕𝐹1
𝜕𝑟𝑉 𝐽 2,𝑉
+𝜕𝐹1
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 2,𝑉
=0+𝜕𝐹1
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝜕𝐹1
𝜕𝛿 𝐿 −𝜕𝑇
𝜕𝑦𝑆−10
𝑎22 𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 !
=−𝑖𝑆𝑐1𝑎11 0𝜕𝑇
𝜕𝑦𝑆−10
𝑎22 𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 =𝑐𝑞
12 𝑒𝑇
𝐼2𝐽12,𝑉
(49)
𝜕 𝐹2
𝜕 𝑟𝑉 𝐽 2,𝑉 𝑡𝑙
=
𝜕𝐹2
𝜕𝑟𝑉 𝐽 2,𝑉
+𝜕𝐹2
𝜕𝑦𝑆
𝜕𝑦𝑆
𝜕𝑟𝑉 𝐽 2,𝑉
=𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 +𝜕𝐹2
𝜕𝛿 𝑃
𝜕𝛿 𝑃
𝜕𝜑𝑆
𝜕𝐹2
𝜕𝛿 𝐿 −𝜕𝑇
𝜕𝑦𝑆−10
𝑎22 𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 !
=𝑐2𝑒𝑇
𝐼2𝐽2,𝑉 −0𝑐2𝑎22 𝜕𝑇
𝜕𝑦𝑆−10
𝑎22 𝑐2𝑒𝑇
𝐼2𝐽2,𝑉
=𝑐2 1−0 1 𝜕𝑇
𝜕𝑦𝑆−10
1𝑐2𝑎2
22!𝑒𝑇
𝐼2𝐽12,𝑉 =𝑐𝑞
22 𝑒𝑇
𝐼2𝐽12,𝑉
(50)
Where 𝑐𝑞
11,𝑐𝑞
22, and 𝑐𝑞
12 =𝑐𝑞
21 represent the effective stiffness properties of the tie rods and the corresponding coupling
term.
Generalized Forces and Derivatives
The differential equation (2) defining the dynamics of the vehicle is driven by the vector 𝑞which collects the generalized
forces and torques applied to the multibody model. According to the principle of virtual power the contributions of the tie
rod forces 𝐹1and 𝐹2are defined as follows
𝑞𝑆1=𝜕𝑣1
𝜕𝑧 𝐴1𝑇
𝐹1and 𝑞𝑆2=𝜕𝑣2
𝜕𝑧 𝐴2𝑇
𝐹2(51)
where 𝜕𝑣1/𝜕𝑧 𝐴1and 𝜕𝑣2/𝜕𝑧 𝐴2denote the partial velocities of the tie rod deformations. Similar to (1) the differential
equations
¤𝑦𝐴1=𝐾𝐴1(𝑦𝐴1)𝑧𝐴1and ¤𝑦𝐴2=𝐾𝐴2(𝑦𝐴2)𝑧𝐴2(52)
provide the generalized speeds of the axle subsystems 𝑧𝐴1,𝑧𝐴2via the kinematic matrices 𝐾𝐴1,𝐾𝐴2. The virtual velocities
𝛿 𝑣𝑖of the tie rod defirmations 𝑖=1,2provide the partial velocities of the tie rod deformations as
𝛿 𝑣𝑖=
𝜕𝑣𝑖
𝜕𝑧 𝐴𝑖
𝛿 𝑧 𝐴𝑖 =
𝜕¤𝑢𝑖
𝜕¤𝑦𝐴𝑖
𝜕¤𝑦𝐴𝑖
𝜕𝑧 𝐴𝑖
𝛿 𝑧 𝐴𝑖 =
𝜕𝑢𝑖
𝜕𝑦 𝐴𝑖
𝐾𝐴𝑖 𝛿 𝑧𝐴𝑖 or 𝜕 𝑣𝑖
𝜕𝑧 𝐴𝑖
=
𝜕𝑢𝑖
𝜕𝑦 𝐴𝑖
𝐾𝐴𝑖 =
𝜕𝑢𝑖
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
𝜕𝑦 𝐴𝑖
𝐾𝐴𝑖 (53)
The subsystems front axle 𝑖=1and front axle 𝑖=2provide the partial derivatives of the position vectors 𝑟𝑉𝐽 1,𝑉 and 𝑟𝑉 𝐽 2,𝑉
with respect to the generalized axle coordinates 𝑦𝐴1and 𝑦𝐴2as well as the kinematic matrices 𝐾𝐴1and 𝐾𝐴2. Similar to
(19) and (21) the partial derivatives of the tie rod deformations with respect to the position vectors are simply given by
𝜕𝑢𝑖
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
=𝑒𝑇
𝐼𝑖 𝐽 𝑖,𝑉
𝜕𝑟𝐼𝑖 𝐽𝑖,𝑉
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
=𝑒𝑇
𝐼𝑖 𝐽 𝑖,𝑉 (−1)=−𝑒𝑇
𝐼𝑖 𝐽 𝑖,𝑉 (54)
where (11) was processed in addition. Then, the contributions of the tie rod forces (51) read as
𝑞𝑆1=−𝑒𝑇
𝐼1𝐽1,𝑉
𝜕𝑟𝑉 𝐽 1,𝑉
𝜕𝑦 𝐴1
𝐾𝐴1𝑇
𝐹1and 𝑞𝑆2=−𝑒𝑇
𝐼2𝐽2,𝑉
𝜕𝑟𝑉 𝐽 2,𝑉
𝜕𝑦 𝐴2
𝐾𝐴2𝑇
𝐹2(55)
The partly implicit Euler step (5) requires the derivatives of the generalized force vector 𝑞with respect to the vector of
generalized coordinates 𝑦in the form of the matrix 𝜕𝑞/𝜕𝑦 𝐾 , where 𝑞and 𝑦are the vectors of generalized forces and
coordinates and 𝐾names the overall kinematic matrix. The steering forces 𝐹1and 𝐹2depend on the vectors 𝑦𝐴1and 𝑦𝐴2
which collect the generalized coordinates describing the subsystems front axle 1 and front axle 2. Then, the generalized
force vectors 𝑞𝑆1and 𝑞𝑆2contribute the submatrix
𝜕𝑞 𝑆1/𝑆2
𝜕𝑦 𝐴1/𝐴2
𝐾𝐴1/𝐴2=
𝜕𝑞 𝑆1
𝜕𝑦 𝐴1
𝐾𝐴1
𝜕𝑞 𝑆1
𝜕𝑦 𝐴2
𝐾𝐴2
𝜕𝑞 𝑆2
𝜕𝑦 𝐴1
𝐾𝐴1
𝜕𝑞 𝑆2
𝜕𝑦 𝐴2
𝐾𝐴2

(56)
to the overall matrix of partial derivatives. The concept of non-perfect multibody vehicle models, as applied in Rill et al.
(2021) and Rill (2022), considers the change of the partial velocities due to the vector of generalized coordinates or speeds
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Multibody Model of dual Truck Front Axles
as comparatively small. Then, the partial derivative of the vector 𝑞𝑆1with respect to the vector 𝑦𝐴1for example can be
approximated by
𝜕𝑞 𝑆1
𝜕𝑦 𝐴1
≈𝜕𝑣1
𝜕𝑧 𝐴1𝑇𝜕 𝐹1
𝜕𝑦 𝐴1
=𝜕𝑣1
𝜕𝑧 𝐴1𝑇𝜕 𝐹1
𝜕𝑟𝑉 𝐽 1,𝑉 𝑡 𝑙
𝜕𝑟𝑉 𝐽 1,𝑉
𝜕𝑦 𝐴1
(57)
The subsystem front axle 1 provides the partial derivative of the position vector 𝑟𝑉𝐽 1,𝑉 with respect to the generalized axle
coordinates 𝑦𝐴1and (47) delivers the total derivative of the tie rod force 𝐹1with respect to 𝑟𝑉𝐽 1,𝑉 . Then, the parts of the
submatrix (56) are defined as follows
𝜕𝑞 𝑆𝑖
𝜕𝑦 𝐴 𝑗
𝐾𝐴𝑗 ≈𝜕𝑣𝑖
𝜕𝑧 𝐴 𝑗 𝑇𝜕 𝐹𝑖
𝜕𝑟𝑉 𝐽 𝑗 ,𝑉 𝑡𝑙
𝜕𝑟𝑉 𝐽 𝑗 ,𝑉
𝜕𝑦 𝐴 𝑗
𝐾𝐴𝑗 =−𝑒𝑇
𝐼𝑖 𝐽 𝑗,𝑉
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
𝜕𝑦 𝐴 𝑗
𝐾𝐴 𝑗 𝑇
𝑐𝑞
𝑖 𝑗 𝑒𝑇
𝐼𝑖 𝐽 𝑗,𝑉
𝜕𝑟𝑉 𝐽 𝑖, 𝑉
𝜕𝑦 𝐴 𝑗
𝐾𝐴𝑗
𝑖=1,2
𝑗=1,2(58)
where (47) to (50) as well as (53) and (54) are taken into account and the kinematic matrices 𝐾𝐴𝑗 for 𝑗=1,2follow from
the definition of non-trivial generalized axle speeds in (52). The parts 𝜕𝑞𝑆1/𝜕𝑦𝐴2and 𝜕𝑞𝑆2/𝜕𝑦 𝐴1represent the effect of
the coupling rod which applies forces to both axle subsystems.
RESULTS
The force based steering system model can easily be integrated into multibody truck models. To avoid overshoots or
high-frequent oscillations in the steering system, the tie rod and coupling rod forces, defined in (12) as simple springs, are
supplemented by viscous parts which are simply adjusted to the stiffness properties of the rods. The main features of the
virtual test truck representing a fully laden 10-wheeler with a total mass of 40 t are provided in Fig. 3.
Figure 3 – Main model properties of a 10-wheeler.
At first, the virtual test truck is slowly driven on a flat and horizontal surface. The steering wheel is ramp-like moved
to the right and to the left as defined by Tab. 1. The results related to the steering system of the 10-wheeler during
Table 1 – Lookup table providing the steering wheel angle as a function of time.
𝑡in seconds 0.0 5.0 8.0 12.0 18.0 21.0 24.0 25.0
𝛿𝑊in degree 0.0 0.0 -720.0 -720.0 +720.0 +720.0 0.0 0.0
tight cornering at low driving velocity are plotted in Fig. 4. It can be seen that the pitman arm rotation 𝛿𝑃(𝑡)and the
rotation of the coupling lever 𝛿𝐿(𝑡)follow the pre-defined steering wheel input 𝛿𝑊(𝑡)where the ratio of the steering box
can be estimated roughly as 𝑖𝑆≈𝛿𝑚𝑎𝑥
𝑃/𝛿𝑚𝑎 𝑥
𝑊=50/720 =0.07 by inspecting Tab. 1 and the left plot in Fig. 4. The ramp
like steer input results in impulse like tie rod forces. The knuckle rotations 𝛿11 to 𝛿22 about the king pin axes correspond
approximately to the wheel steering angles. Their time histories trace the Ackermann steering geometry by generating
larger rotation angles at the inner curve wheels and by reducing the steering angles at the second axle, in general. However,
the two twin-tired rigid axles within the tandem rear axle suspension are not steered which makes the Ackermann steering
geometry just a rough guess. That is why, the standard layout of a dual front axle suspension simply consists of two
identical axles. As a consequence, the time histories of the lateral forces show quite different relations at front axle 1 and
2 between the the left (𝐹𝑦11,𝐹𝑦21 ) and right tires (𝐹𝑦12,𝐹𝑦22 ). Even at this extreme steering maneuver, the number of
iterations, required to solve the nonlinear torque balances (31) and (32), is limited to a single digit number.
private use only

G. Rill
Figure 4 – Steering system related results of a 10-wheeler during tight cornering at low speed.
The virtual test truck models each leaf spring by a 5-link model which reproduces the suspension kinematics very
accurately (Rill et al. 2022). In general, the layout of the steering system is adjusted to the kinematics of the front axle
suspension system which makes it less sensitive to the impact of axle motions induced by varying wheel loads.
Figure 5 – Wheel steering angles, wheel loads, and steering forces when crossing an angled bump.
Some results of a bump crossing are plotted in Fig. 5. At the beginning (𝑡=0 s) the truck is driving with a velocity of
𝑣(𝑡=0) ≈ 5 km/h. During the bump crossing it is slowly accelerated to 𝑣(𝑡=6 s) ≈ 25 km/h. The steering wheel is kept
private use only

Multibody Model of dual Truck Front Axles
fixed to 𝛿𝑊=0◦. The suspension travel at the front axles related to the loading condition of the truck results in a small
self-steering effect indicated by the wheel steering angles of 𝛿11 (𝑡=0)=𝛿12 (𝑡=0)=0.16◦and 𝛿21 (𝑡=0)=𝛿22 (𝑡=0)=0.10◦,
respectively. Due to the angled orientation of the bump, the left wheel of the first front axle comes at first into contact
with the bump. The time history of the corresponding wheel load 𝐹𝑧11 (𝑡)in the impact interval of 1.2 s ≤𝑡≤1.8 s shows
a double peak on the wheel load increase and a simple peak on the wheel load decrease. The double peak is caused by
the compliance of the axle suspension and the steering system. The time histories of the tie rod forces 𝐹1(𝑡)and 𝐹2(𝑡)
roughly correspond to the steering movements 𝛿11(𝑡) ≈ 𝛿12 (𝑡)and 𝛿21 (𝑡) ≈ 𝛿22 (𝑡)of the wheels at both front axles. In the
time interval 2.0 s ≤𝑡≤3.5 s the right wheel at the first and the left wheel of the second front axle hit nearly simultaneously
the bump. The impacts on the wheel steering motions counteract itself and cause a distinct reaction in the time history of
the coupling rod force 𝐹𝐶(𝑡).
In this standard layout of a dual front axle suspension system the axle hub and roll movements caused by the bump
crossing induce only slight steer motions. A standard steering system is also less sensitive to the S-shaped bending modes
of the leaf springs occurring at full braking maneuvers (Rill et al. 2022).
SUMMARY
The presented steering system restricts the axle motions just by the tie rod forces and not by kinematical constraints.
That is why, it can easily be integrated in multibody truck models. It forms a perfect supplement to comparatively lean,
but sufficiently accurate multibody suspension models. The quasi-static solution is sufficiently accurate and fast. It also
provides the partial derivatives required for an implicit solver in analytical form. In this particular case, the simulation of
the 10-wheeler truck model, as specified in Fig. 3, executed on a Personal Computer with a 2,7 GHz Quad-Core Intel Core
i7 was 10-times faster than real-time. The virtual test truck, coded in ANSI-C, incorporates the 5-link model of a leaf
spring suspension, the TMeasy tire model, and it solves the set of first order differential equations with a partial implicit
solver at a constant step-size of 1 ms.
REFERENCES
Rill, G., 1997, “Vehicle Modeling for Real Time Applications”, Journal of the Brazilian Society of Mechanical Sciences
- RBCM XIX.2, pp. 192–206.
Rill, G., 2006a, “Vehicle Modeling by Subsystems”, Journal of the Brazilian Society of Mechanical Sciences & Engi-
neering - ABCM XXVIII.4 pp. 431–443.
Rill, G., 2006b, “A modified implicit Euler Algorithm for solving Vehicle Dynamic Equations” Multibody System
Dynamics, Vol. 15, Issue 1, pp. 1–24.
Qin, G., Sun, Y., Yunqing Zhang, Y., and Chen, L., 2012, “Analysis and Optimization of the Double-Axle Steering
Mechanism with Dynamic Loads”, The Open Mechanical Engineering Journal, Vol. 6, (Suppl 1-M2) pp. 26–39.
Rill, G., 2013, “TMeasy – The Handling Tire Model for all Driving Situations”, Proceedings of the XV International
Symposium on Dynamic Problems of Mechanics (DINAME 2013). Ed. Savi, M., Buzios, RJ, Brazil.
Topaç, M. M., Karaca, M., and Kuleli, B., 2019, “A Design Optimization Study for the Multi-axle Steering System of an
8×8 ARFF Vehicle”, Proceedings of the Applied Physics, System Science and Computers III, Ed. Ntalianis, K et
al., pp. 342–347.
Bruni, S., Meijaard, J.P., Rill, G., Schwab A. L., 2020, “State-of-the-art and challenges of railway and road vehicle
dynamics with multibody dynamics approaches”, Multibody System Dynamics, Vol 49, pp. 1–32.
Rill, G., Bauer, F., and Kirchbeck, M., 2021, “VTT – a virtual test truck for modern simulation tasks”, Vehicle System
Dynamics, Vol. 59, No. 4, pp. 635–656.
Rill, G., Bauer, F., and Topcagic, E., 2022, “Performance of leaf spring suspended axles in model approaches of different
complexities”, Vehicle System Dynamics, Vol. 60, No. 8, pp. 2871-2889.
Rill, G., 2022 “A Three-Dimensional and Nonlinear Virtual Test Car”, https://enoc2020.sciencesconf.org/381007/
document
RESPONSIBILITY NOTICE
The author is the only responsible for the printed material included in this paper.
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