6.6 Three-Dimensional Model of a Five-Link Rear
6.6.3 Simulation Results of the Three Dimensional Quarter Vehicle ModelVehicle Model
Every type of wheel suspension has its, dependent on the respective design, characteristic behavior in the deflecting and hopping of the wheel. The most important kinematic parameters of the wheel motion and hence the position of the wheel towards the road are the camber angle and toe angle changes of the resilient axle. Since the rear axle determines the stability of the vehicle, too extreme alterations of the toe angle result in an uncomfortable tracking effect of the axle.
Therefore, the toe angle should stay as neutral as possible. The camber angle should adjust itself in a way in which there is always a possibly beneficial contact between the wheel and the road. With the aid of the earlier mentioned kinematic equations the characteristic quantities of the wheel suspension can now be cal- culated exemplary for a spatial quarter vehicle with a five link wheel suspension.
Trajectory of the wheel centerFor the five-link wheel suspension analyzed in the previous sections, the recording of the trajectory of the tire center is helpful. In Fig.6.37, the wheel trajectory in the three levelszx;zy. andxyis depicted in the vehicle-fixed coordinate system.
Camber- and toe angle graphsThe camber and toe angles (see Sect.6.3) are calculated from the alignment of the wheel normal with respect to the vehicle- fixed coordinate system as depicted in Fig.6.38.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
0.71 0.72
0.72 0.73 -0.1 -0.05 0 0.05 0.1 0.15
0 -0.1 -0.05 0 0.05 0.1 0.15
Fig. 6.37 Trajectory of the tire center of the five-link wheel suspension W124 (Mercedes)
The camber and toe angles are therefore calculated as such:
camber angle: cẳ arctan z ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2þy2 p
!
; ð6:56ị
toe angle: dẳ arctan x
y : ð6:57ị
In Fig.6.39, the calculated camber and toe angle graphs of a five-link wheel suspension in correlation with the compression travel are plotted. Additionally, there are also reference points from a corresponding, actually measured camber curve depicted. The deviations between calculation and measurement are caused by the elasto-kinematic, which was neglected in the modeling presented here. The camber- and toe variation at the wheel suspension can be visualized with suitable animation software, as shown in Fig.6.40.
Fig. 6.38 Camber and toe angle definition and calculation
-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 -80
-60 -40 -20 0 20 40 60 80
camber and toe angle / o
Vertical deflection / mm
toe angle camber angle toe reference camber reference
Fig. 6.39 Camber and toe angle graph of a five-link wheel suspension
Simulation of the dynamics of an axle test benchFinally, the dynamic simu- lation of an axle test bench as presented in Fig.6.41 is shown for exemplary reasons. The wheel is subject to an imposed vibration of the wheel suspension generated by a plate. The amplitude of the wheel movement is larger than the stimulus amplitude, a phase shift between the movement of the ground and the wheel is virtually not identifiable (see Fig.6.42). The occurring areas during the transient period in which the ground force is not observable, the wheel has lifted off and there is no contact to the ground (compare Figs.6.43and6.44).
Fig. 6.40 Wheel position at±80 mm deflection/hopping: (leftthe almost not visible change in toe angle,rightthe change in camber angle)
Fig. 6.41 The quarter vehicle model on the axle test bench
0 1 2 3 4 5 6 7 -0.4
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1
vertical deflection / m
time / s
clearance wheel center
Fig. 6.42 Vertical deflection of the base plate and the wheel carrier
0 1 2 3 4 5 6 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fn/Fn,max
normierte Normalkraft
time / s
rel. wheel load
Fig. 6.43 Standardized vertical wheel load (normal force)
References
Cronin DL (1981) McPherson Strut Kinematics.Mechanism and Machine Theory16. S 631–44 DIN (1993) DIN_70020-1: Strassenfahrzeuge—Kraftfahrzeugbau—Begriffe von Abmessungen.
(ed.), Deutsches Institut für Normung e.V., Berlin
Kecskemethy A (1993) Objektorientierte Modellierung der Dynamik von Mehrkửrpersystemen mit Hilfe von ĩbertragungselementen [Dr.-Ing.]. Dissertation, Universitọt Duisburg, Dỹssel- dorf: VDI-Verlag
Mửdinger W, Bublitz HG, K.-J., Schulz W, Minning M and Braun R (1997) Das Fahrwerk des neuen A-Klasse von Daimler-Benz.Sonderausgabe ATZ und MTZDie neue A-Klasse von Daimler-Benz. S 102–9
Schmidt A and Wolz U (1987) Nichtlineare rọumliche Kinematik von Radaufhọngungen—
kinematische und dynamische Untersuchungen mit dem Programmsystem.MESA VERDE - Automobilindustrie 19876. S 639–44
Schnelle K-P (1990) Simulationsmodelle für die Fahrdynamik von Personenwagen unter Berỹcksichtigung der nichtlinearen Fahrwerkskinematik Dissertation, Universitọt Stuttgart, Stuttgart
Schuster H, Balk A and Oehlschlaeger H (1995) Der Sharan—Die Groòraumlimousine von Volkswagen.ATZ Automobiltechnische Zeitschrift97. S 400–15
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
0 1 2 3 4 5 6 7
-1 0 1 2 3 4 5 6 7 8
time / s
tire deformation %
tire deformation %
Fig. 6.44 Elastic vertical deformation of the wheel
Modeling of the Road-Tire-Contact
A sufficiently accurate knowledge of the interaction between the tires and the road surface is of vital importance to describe and evaluate the dynamics of vehicles.
The contact road-tire is, apart from the influence of aerodynamic factors, the only possibility to actively influence the motion of the vehicle. Here all the forces and the torques will be transmitted over a post card sized patch called the tire contact patch.
The wheel is defined as the rotating components, spinning about the wheel spin axis, of the vehicle undercarriage. These include the wheel carrier, which is assumed to be rigid, the rotating parts of the brake, parts of the drive shaft and the drive train as well as the tires. The wheel is fitted to the suspension using a wheel bearing. The wheels have three fundamental properties that, depending on the application, have to be represented in the modeling of the wheel forces:
• the absorption of the wheel loads and the protection of the other vehicle components and the passengers from impact loads,
• the transmission of acceleration and braking forces and
• the lateral forces during cornering.
Physically this means the transfer of forces and torques in all three spatial directions. In the process the wheel is assigned to two functionally separate sub- systems of the vehicle. On the one hand, it is part of the suspension and on the other, a part of the drive train. In both these areas, it is the last element of the causal chain and as such is the immediate interface to the road surface.
Modern tires are, from the production point of view, elaborately designed viscoelastic forms and represent, from the modeling point of view in vehicles, force components with complex nonlinear and dynamic characteristics. The degree of complexity of the tire models, for applications in vehicle models, ranges thus from the simple linear to the very complex nonlinear models, depending on the situation and the application. This chapter deals with the suitable mathematical models required to describe the transfer characteristics of the tires, without delving too much into the physical characteristics of the tire. For this purpose it shall be referred to the comprehensive literature available in this topic. In Ammon (2013) the fundamental mathematical characteristics of the tires are dealt with.
D. Schramm et al.,Vehicle Dynamics, DOI: 10.1007/978-3-540-36045-2_7, Springer-Verlag Berlin Heidelberg 2014
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In Pacejka (2006) one can find a comprehensive introduction to the mathematical tire models as well as an overview over the methods to measure the required data.
The construction, history and the fundamental characteristics of tires and wheels is described in Reimpell and Sponagel (1988) or in Leister (2009). Comprehensive information along with the description of known tire models can be found in Gipser (2010).
While in the case of vehicle dynamics models, the characteristic curves of the tires are often measured first on the test rig and then replicated, as exact as possible, through a model (the empirical method), the physical modeling is based on the knowledge of the exact mechanism, by which the forces are generated.
The physical approach requires a lot more computational effort when compared to the models used in vehicle dynamics. Another difference lies in the fact that the vehicle dynamics models are suitable to recreate the stationary and non-stationary tire characteristics in the frequency range up to 20 Hz of vehicle dynamics (modeling of low frequency forces and deformations). Compared to this the comfort models can represent high dynamic situations up to 80 Hz and more (for example: vibrations on uneven surfaces). Through this it is possible to predict the outcome of working points that cannot be measured realistically. Depending on the job description one is to choose the tire model, which offers the best compromise between computational time and performance.