3.5 Kinematics of a Complete Multibody System
3.5.5 Example: Double Wishbone Suspension
On the example of the double wishbone suspension, which plays an important role in automotive engineering, as already discussed in Sect.3.4.3 under the title
‘‘topological methods’’, the kinematic analysis of multibody systems with kine- matic loops will be carried out further (Figs.3.29 and3.30).
The topological structure of the system consists of two independent kinematic loops L1and L2, which are coupled over the revolute joint R1(angle of rotationb1) and the spherical joint S1 (spherical angles b2, b3, b4) of the upper wishbone, respectively. One recognizes that the loop L1, which also embodies the wheel carrier, corresponds to the spatial four-link mechanism, introduced in Sects.3.5.2 and 3.5.3, where the wheel carrier is connected with, both upper and lower, wishbones over the corresponding spherical joints. This is required, because later, the steering motionswhich is independent from the spring deflectionb1;can be introduced into the system. Having a separate glance on the spatial four-link
relative kinematics Fig. 3.26 Module for
relative kinematics of a complex multibody system
absolute kinematics Fig. 3.27 Module for
absolute kinematics of a complex multibody system
mechanism, one can recognize that this loop possesses two degrees of freedom, one for the spring deflectionb1of the wheel mount and a second independent DoF, which corresponds to an isolated rotation of the wheel carrier about the connecting straight line between upper and lower spherical links of the wishbone, which is
global kinematics
relative kinematics
absolute kinematics
Fig. 3.28 The global kinematics of complex multibody systems
S
S
S S R
R
P
L
L
upper wishbone wheel carrier lower wishbone tie rod rack and pinion R - revolute joint ( ) P - prismatic joint ( ) S - spherical joint ( ) isolated DoF
Fig. 3.29 Double wishbone suspension—kinematic structure
L
L
,
Fig. 3.30 Relative kinematics of the double wishbone suspension—
topology
required for the guidance of the steering motion over the steering mechanism referred to in Fig.3.29.
In a similar manner, the second kinematic loop L2of the double wishbone axle can be prepared and successively interrelated with the loop L1 over the linear equations in the variablesb2;b3;b4 of the spherical joint S1 (Figs.3.29and3.30), corresponding to the coupling of the loops via these joints, to the already shown block diagram for the calculation of the relative kinematics (Fig.3.31). An important advantage of this approach, which can be applied in its basic ideas to a large number of further technically interesting kinematic structures, is the mini- mized kinematic calculation amount: Only those kinematic variables must be calculated, which are necessary for the further continuation of the calculation. For example, in most practical cases the angles of the spherical joints are not required.
It should still be emphasized once more that in the preceding example the cal- culation for the complete system was carried out in explicit-recursive form (Hiller et al.1986–1988), (Woernle1988) and (Schnelle 1990).
Note: In the later applications (especially in Chaps.6 and 12), the simplified symbolism already demonstrated in Fig.3.23is used for the block diagrams, in order to emphasize the nonlinear transmission behavior inside each kinematic loop. In the case of the double wishbone suspension (Fig.3.31) will be replaced by (Fig.3.32).
L L
Fig. 3.31 Block diagram of the relative kinematics of the double wishbone suspension
1 2
L L
2 3 4
Fig. 3.32 Relative kinematics of the double wishbone suspension
References
Hiller M (1981) Analytisch-numerische Verfahren zur Behandlung rọumlicher ĩbertragungs- mechanismen [habil.]. Habilitation, Universitọt Stuttgart, Dỹsseldorf: VDI-Verlag
Hiller M (1995) Multiloop Kinematic Chains, and Dynamics of Multiloop Systems in Kinematics and Dynamics of Multi-body Systems
Hiller M and Kecskeméthy A (1989) Equations of motion of complex multibody systems using kinematical differentials. InTransactions of the Canadian Society of Mechanical Engineers, Vol. 13, S 113–121
Hiller M, Kecskeméthy A and Woernle C (1986–1988) Computergestützte Kinematik und Dynamik für Fahrzeuge, Roboter und Mechanismen. InCarl-Cranz-Kurs, Vol. 1.16, Carl- Cranz-Gesellschaft, Oberpfaffenhofen
Hiller M, Kecskeméthy A and Woernle C (1986) A loop-based kinematical analysis of complex mechanisms. InPresented at the Design Engineering Technical Conference, Columbus, Ohio Kecskemethy A (1993) Objektorientierte Modellierung der Dynamik von Mehrkửrpersystemen mit Hilfe von ĩbertragungselementen [Dr.-Ing.]. Dissertation, Universitọt Duisburg, Dỹssel- dorf: VDI-Verlag
Li H (1990) Ein Verfahren zur vollstọndigen Lửsung der Rỹckwọrtstransformation fỹr Industrieroboter mit allgemeiner Geometrie Dissertation, Universitọt – Gesamthochschule – Duisburg
Mửller M (1992) Verfahren zur automatischen Analyse der Kinematik mehrschleifiger rọumlicher Mechanismen [Dr.-Ing.]. Dissertation, Universitọt Stuttgart
Reuleaux F (1875) Theoretische Kinematik: Grundzüge einer Theorie des Maschinenwesens.
Vol. 1, F. Vieweg und Sohn - ISBN 978-3-8364-0352-8
Schnelle K-P (1990) Simulationsmodelle für die Fahrdynamik von Personenwagen unter Berỹcksichtigung der nichtlinearen Fahrwerkskinematik Dissertation, Universitọt Stuttgart, Stuttgart
Woernle C (1988) Ein systematisches Verfahren zur Aufstellung der geometrischen Schlieòbe- dingungen in kinematischen Schleifen mit Anwendung bei der Rỹckwọrtstransformation fỹr Industrieroboter [Dr.-Ing.]. Dissertation, Universitọt Stuttgart, Dỹsseldorf: VDI-Verlag
Equations of Motion of Complex Multibody Systems
In this chapter several methods from classical mechanics to state equations of motion of mechanical systems are briefly explored (Sect.4.1–4.4). Principally, all these methods are also suited for the modeling of the vehicles regarded in this book, which can be considered as complex multibody systems. Concerning the practical application of a method, this decision is mainly dependent on the asso- ciated modeling and calculation effort, according to the complexity of the system.
Consequently, the equations of motion for this kind of large scale systems with many bodies and many kinematic loops (and therefore many constraints) will be developed numerically or symbolically with the help of the computer. The com- putational cost is dependent on the number of describing coordinates, including its geometrical significance; further being defined by the formulation of the constraint equations, i.e. through the modeling of the kinematics of the system, which is reflected directly by the number of required mathematical operations.
A particular approach is further detailed in Sect.4.5. Based on d’ALEMBERT’s principle for rigid bodies, a method which is suitable for the development of the equations of motion for mechanisms and complex multibody systems is derived.
This method is based on the kinematic structure of the system. With an adequate formulation of the constraint equations (see Chap.3), compared to other methods, the number of the necessary mathematical operations can be significantly reduced.
The implementation of this method, as well as its practical application by means of a computer is further illustrated in Sect.4.6of this chapter, see also (Hiller1983, 1995; Hiller et al.1986–1988,1986; Kecskemethy1993).