In this section, in a first step it will be discussed how the interrelation of the kinematic topologies of a tree-type structure and a closed-loop structure can be used for the analysis and the statement of the modeling equations of wheel sus- pensions (see Chap.3). This will be illustrated with the example of a multi-link rear wheel suspension.
5.2.1 Incorporation of the Wheel Suspension Kinematics
Regarding an individual analysis of wheel suspensions, position, velocity, and acceleration of arbitrarily chosen wheel suspension points in a vehicle at rest have to be determined. This means that velocities and accelerations with respect to the
vehicle are determined to be designated with the left lower index ‘‘V’’. Figure 5.3 illustrates the corresponding scheme of interdependencies of the vectorVrT from the vehicle reference system to a selected point T of the wheel carrier, using the body-joint-representation, as introduced in Sect.3.4.
The objective of the individual analysis of the car body kinematics—discussed in the previous section—has been the calculation of position, velocity, and acceleration of arbitrary chassis points with respect to the inertial system. The corresponding general scheme, which represents a tree-type structure, is shown in Fig.5.4. The open or tree-type kinematic chain on the left side of this figure represents the free motion (with 6 DoF) of the chassis with respect to inertia. For the analysis of the complete system, the subsystems containing the kinematic loops (i.e. wheel suspensions, etc.) will be attached to the tree-type structure, as shown in Fig.5.4. One gets position, velocity, and acceleration of the wheel suspension points with respect to the inertial system by applying the well-known equations for the relative motion of two bodies.
Guiding motion, Relative motion,
3
wheel
L6 − 10
3
rear wheel carrier T
Vehicle body
= ( 3)
= ( 3, 3)
= ( 3, 3, 3)
= ( 3, 3)
= ( 3, 3, 3) loop analysis loop
topology
Fig. 5.3 Individual analysis of a multilink rear wheel suspension
= ( … )
= , , , , ,
= , , , , , , , ,
= ( … , … )
= ( … , .. , … )
= ( , , )
(
( )
)
Fig. 5.4 Scheme of the chassis kinematics, inertial system E, chassis system V, calculation of the tree-structure
rẳrV ỵVr; ð5:7ị r_ẳr_V ỵxV Vr ỵVr;_ ð5:8ị
€rẳ€rV ỵx_V VrỵxV ðxVVrị ỵ2xVVr_ỵV€r; ð5:9ị
xẳxV ỵVx; ð5:10ị
x_ ẳx_V ỵxV Vx ỵVx:_ ð5:11ị Equations (5.7)–(5.11) show that the kinematics expressed in the inertial system—which is needed for the equations of motion of the complete vehicle—
can be obtained in a very simple way from the already known loop-kinematics, together with the very simple kinematics of the tree-type kinematic chain ‘‘chas- sis’’. In general, one can say that in systems which contain a combination of tree- type and loop topologies, it is recommended to analyze first trees and loops sep- arately, and to fit them together later-on (Fig.5.5).
For the equations of motion, in addition to the ‘‘real’’ velocities also the so-called pseudo velocities, introduced in Chap.4, are needed. The idea here is to calculate unknown partial derivatives, which are needed later-on, in an intelligent kinematic way (Sect.4.6.1). In Eqs. (5.8) and (5.9), one can see that the calcu- lation of the pseudo velocities can be simplified, depending if the actually regarded degree of freedom (DoF) in this procedure belongs to the tree-type or the loop-type structure. In case of the tree-type motion, the last term on the right hand side of
3
wheel 3
L6 − 10
Fig. 5.5 Combination of chassis and wheel suspension kinematics
Eqs. (5.8) and (5.9) becomes ‘‘0’’; in case of the loop-type motion, left of the last one vanishes, because the corresponding DoF have the velocity ‘‘0’’.
5.2.2 Equations of Motion
For the dynamics of a multibody system of nB rigid bodies, in Sect. 4.5 d’ALEMBERT’s principle has been applied:
XnB
iẳ1
mi€rSiFi
ð ị drSiỵðHSix_iỵxiHSixiTiị dui
ẵ ẳ0; ð5:12ị
with
mi;HSi mass and tensor of inertia,
€rSi acceleration of the center of mass, xi;x_i angular velocity and angular acceleration, Fi;Ti external forces and torques,
drSi;dui virtual displacements for each body ‘‘i’’.
The tensors of inertia and the external torques have to be expressed with respect to the body centers of mass. For the equations of motion after Eq. (5.12), the dependent virtual displacementsdrSi;dui, as well as the accelerations€rSi;x_i;have to be calculated as a function of the virtual displacements and the accelerations of the independent generalized coordinates. Regarding that for the virtual displace- ments the same transformation rules are valid as for the velocities, one gets:
driẳXf
jẳ1
~_rðjịS
idqj; €riẳXf
jẳ1
~_rðjịS
i€qjỵ€~rSi; ð5:13ị
duiẳXf
jẳ1
x~ðjịi dqj; x_iẳXf
jẳ1
x~ðjịi €qjỵx~_i: ð5:14ị
After inserting Eqs. (5.13) and (5.14) into Eq. (5.12), and considering the independent virtual displacementsdqj, the equations of motion in minimal form are:
M€qỵbẳQ: ð5:15ị The coefficients of the generalized ẵff-mass matrix M;as well as of the
ẵf 1-vector of the generalized gyroscopic and centrifugal forces b; and the
ẵf 1-vector of the generalized external forcesQare the following:
Mj;kẳXnB
iẳ1
ẵmi~_rðjịS
i~_rðkịS
i ỵx~ðjịi ðHSix~ð ịikị bjẳXnB
iẳ1
mi~_rðjịS
i
€~
rSiỵx~ðjịi HSix~_iỵxiHSixi
h i
;
QjẳXnB
iẳ1
ẵ~_rðjịS
iFiỵx~ðjịi Ti:
ð5:16ị
In case of the already solved global kinematics of the system, the Eq. (5.16) can be evaluated, following the procedure of Sect.4.6.
References
DIN (1994) Fahrzeugdynamik und Fahrverhalten—Begriffe. in FAKRA(ed.), Vol. Deutsche Norm DIN 70000 ISO 8855, Deutsches Institut für Normung e.V.: Normenausschuss Kraftfahrzeuge im DIN
Hiller M (1983) Mechanische Systeme: Eine Einführung in die analytische Mechanik und Systemdynamik. Springer, Berlin u.a.—ISBN 978-3-540-12521-1
Modeling and Analysis of Wheel Suspensions
The wheel guidance or wheel suspension has made a remarkable development during its centennial history. By the implementation of more sophisticated kinematic structures, the generation of accurate and reproducible wheel movements became possible. As the knowledge about the vibration behavior of vehicles and their driving dynamics increased, the subtleties of the spatial kinematics came to the force.
Accordingly, today a variety of wheel guidance designs and suspension geometries with different objectives and characteristics is available. Due to the use of computer- aided processes, today the desired design of the chassis is possible. As a result, safe driving behavior and best driving comfort can be achieved at the same time.