1.3 Models for Simulating Ride Comfort on Subsystem
1.3.2 Concept Parameters on Subsystem Level in Ride
In the following, the application of the referred parameters is carried out under the conditions mentioned in the previous section. As already mentioned in Sect.1.3, a derivation of the detailed structure of the subsystems is beyond the scope of this paper, but a summary, illustrating the basic principles, is given for the individual systems.
When examining ride comfort characteristics in the given research, the fre- quency range from zero till 30 Hz is observed. Therefore the vibration of vehicle body, engine, tire-sprung masses and subframe as rigid bodies are of particular interest. Furthermore, natural frequencies of the body structure can occur, but shall be neglected in the investigation. Based on these conditions, in the following the subsystem behavior of tire, suspension, and the mountings of subframe and powertrain need to be modelled. In the current research, the analysis is predomi- nantly performed with focus on the suspension. For the remaining subsystems, conditions for developing an appropriate subsystem approach are given.
The tire, being subsystem and component at the same time, is usually represented by a single-point contact model for long wavelengths occurring for instance at natural frequencies of the body. At higher frequencies shorter obstacles are enclosed (cf. Fig.1.4) requiring a more complex modelling approach. Therefore at low frequencies the predominant tire property on subsystem level is the overall vertical tire stiffness while with rising frequency respectively shorter wavelengths, longitudinal stiffness and geometrical aspects of the tire are getting more important.
With respect to the defined frequency range, a tire model needs to be used, which allows to reproduce the enveloping properties of the tire and which can be param- etrized on a tire test rig in a short time frame, like MF-SWIFT (Pacejka 2006, pp. 412–510).
The vehicle suspension serves as interface between tyre-sprung mass and body respectively the subframe, if latter is mounted elastically on the body. The transfer behavior of the subsystem can then be defined by static and dynamic stiffness in all six directions in space, forming a 6-by-6 matrix with variable coefficients for reproducing dynamic properties. However, as already described in Sect. 1.3.1, instead of using cross-terms of the matrix, a more sophisticated method of abstrac- tion is applied by dividing the transfer behavior into a diagonal stiffness matrix and separate kinematic properties. In this context the stiffness matrix incorporates
Fig. 1.4 Filtering of unevenness of a tire as depicted in Zegelaar (1998, p. 58)
elasto-kinematic properties (for example in longitudinal and lateral direction) as also a stiffness in vertical direction, which is usually attributed to kinematics. On the other hand, the separate kinematics avoids the application of cross-terms in the stiffness matrix by using geometrical relations.
This is done for every connection of tire-sprung mass and body, but also for alternate movement of tire-sprung masses between left and right wheel, if necessary for the respective direction. Additionally every element of the stiffness matrix is wheel-based, meaning the relation is defined between force and displacement at the same location on the wheel, which maintains the independence of axle geometry respectively lever ratios.
Under the described conditions for dividing the subsystem model of the suspen- sion into a diagonal stiffness matrix and kinematics, different concept parameters can be identified.
The overall vertical stiffness of the suspension affects body accelerations over a wide frequency range, being involved in quite all maneuvers relevant for ride comfort, beginning at the natural frequencies of the body. The parameter combines the stiffness of main spring, torsional stiffness of bushings6, and the bump stop (Bindauf et al.2014, p. 78).
Therefore also the vertical damping of the suspension can be defined as impor- tant parameter on subsystem level, affecting the reaction of the axle due to dynamic excitation. In this case, the components contributing to the summarized damping force can be identified as the same as for vertical stiffness. Still, it can be assumed, that the influence of the damper dominates the force generation over a wide frequency range, so that in most cases damping due to torsional deformation of bushings7can be neglected. With rising frequency also the damper top mount has to be considered (Bindauf et al.2014, p. 80).
When the wheels respectively the tire-sprung masses of an axle are unequally deflected in vertical direction, an alternate vertical stiffness comes into effect. The influence can be modelled by defining a wheel-based stiffness as coefficient of vertical force and differential deflection acting between the tire-sprung masses and the respective body connection. Predominantly, the properties of the anti-roll bar are responsible for this effect.
In a similar manner an overall stiffness in lateral direction of the axle can be defined. Therein predominantly the stiffness of bushings is included. A parameter combining several individual damping properties of the bushings can be defined as well. While in driving dynamics a high lateral stiffness is important for maintaining the wheel position when lateral forces are applied (Heiòing et al.2011, p. 456), the influence of both mentioned parameters on ride comfort is mostly unknown.
In longitudinal direction also an overall stiffness and damping can be defined.
Thereby the individual locations, orientations and properties of the involved
6The wheel-based stiffness due to bushings is usually called secondary spring rate, probably being mainly dependent on the torsional stiffness of bushings.
7Analogue to the secondary spring rate this effect will be called secondary damping rate.
bushings are abstracted and an approach independent of the suspension concept is generated. The overall longitudinal elasticity of the suspension comes into effect when the tire generates longitudinal forces as a result of the road profile. It can be assumed, that this predominantly occurs with rising frequency of the excitation.
Therefore, the influence on ride comfort can directly be deduced when analyzing associated maneuvers with the help of an appropriate concept model. For example, in Fig.1.5the influence of longitudinal stiffness on seat rail acceleration, when a cleat is passed, is depicted.
As can be seen in the figure, a higher longitudinal stiffness results in a higher longitudinal peak acceleration, a higher vibration frequency and a longer decay process. This occurs due to a higher resistance of the axle in longitudinal direction, when the tire is passing the obstacle and an associated decreased effectivity of the damping.
In longitudinal and lateral direction also an alternate stiffness can be defined.
Though, the relevance of these effects depends on the usage of coupling elements in the suspension, like a subframe. For instance, without subframe the alternate longitudinal stiffness of an axle can be neglected during static maneuvers, when certain conditions are met (Bindauf et al.2014, p. 79).
Beside the response of a suspension due to longitudinal and lateral excitation in the same direction, a coupling between these directions and the vertical direction is generated by axle kinematics. At subsystem level, this behavior can be described by a support angle or an instantaneous center of rotation (Matschinsky 2007, pp. 23 and 48). Considering their dependence of vertical, longitudinal and lateral wheel deflection, they can serve as subsystem parameters. On the one hand, these kinematic properties characterize the amount of vertical force generated by longi- tudinal and lateral forces on the wheel, on the other hand they define the kinematic movement of the tire-sprung mass with respect to the body (Matschinsky 2007, p. 41). Therefore an additional vertical force occurs when longitudinal forces are generated in specific maneuvers, as described before when addressing longitudinal Fig. 1.5 Influence of longitudinal stiffness on seat rail acceleration in longitudinal direction when passing a cleat
stiffness. The behavior in lateral direction can be described similar, but corresponding maneuvers are different. As currently cornering is mostly not con- sidered in maneuvers defining ride comfort, lateral forces predominantly are gen- erated during compression and rebound of the suspension, as a consequence of changes in track and toe or when obstacles are asymmetrically enveloped by the tire with respect to the wheel center plane.
Another effect which influences suspension response, but can generally not be attributed to stiffness or damping properties is static axle hysteresis or axle friction.
The effect has been researched to some extent in literature (Yabuta et al.1981;
Gillespie 1992, pp. 166–168; Nakahara et al. 2001), but first investigations concerning finding an integral approach for modelling, designing and parametriza- tion of axle friction in the development process are given by Angrick et al. (2015, pp. 377–403).
When the suspension includes an elastically mounted subframe, additionally it becomes necessary to consider the connection between subframe and body. For this purpose, the stiffnesses and dampings of the individual components are summa- rized in generalized stiffnesses and dampings for the whole mounting. As the properties in longitudinal and vertical direction are of main importance for ride comfort, focus lies on the associated parameters.
The powertrain mounting is abstracted with the same method, but as the cou- pling of the different degrees of freedom is generally more complex (due to asymmetric stiffness and hydraulic properties of the bushings) than in the subframe mounting, the described method is only partially applied by disregarding less important parameters in the stiffness matrix and maintaining the hydraulic proper- ties on component level.
Individual bodies, which are connected by mentioned different mountings, are represented on subsystem level by summing up mass and inertia properties of related components. In this case, properties, which are irrelevant for ride comfort, can be neglected, like yaw inertial torque of the vehicle body or of tire-sprung masses. As a consequence, the aggregated mass and inertia properties of the body correspond to subsystem parameters. This enables the possibility for parametrizing whole subsystems in the development process, when component parameters are still unknown. In particular, this concerns tire-sprung masses (separately for each side of front and rear axle), powertrain, vehicle body as well as subframe and differential, if necessary.