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Friction has long been an important issue in multibody dynamics. Static friction models apply appropriate regularization techniques to convert the stick inequality and the non-smooth stick–slip transition of Coulomb’s approach into a continuous and smooth function of the sliding velocity. However, a regularized friction force is not able to maintain long-term stick. That is why dynamic friction models were developed in recent decades. The friction force depends herein not only on the sliding velocity but also on internal states. The probably best-known representative, the LuGre friction model, is based on a fictitious bristle but realizes a too-simple approximation. The recently published second-order dynamic friction model describes the dynamics of a fictitious bristle more accurately. It is based on a regularized friction force characteristic, which is continuous and smooth but can maintain long-term stick due to an appropriate shift in the regularization. Its performance is compared here to stick–slip friction models, developed and launched not long ago by commercial multibody software packages. The results obtained by a virtual friction test-bench and by a more practical festoon cable system are very promising. Thus, the second-order dynamic friction model may serve not only as an alternative to the LuGre model but also to commercial stick–slip models.
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.
Friction has long been an important issue in multibody dynamics. Static friction models apply appropriate regularization techniques to convert the stick inequality and the non-smooth stick-slip transition of Coulomb’s approach into a continuous and smooth function of the sliding velocity. However, a regularized friction force is not able to maintain long-term stick. That is why, dynamic friction models were developed in the last decades. The friction force depends herein not
only on the sliding velocity but also on internal states. The probably best known representative, the LuGre friction model, is based on a fictitious bristle but realizes a too simple approximation. The recently published second order dynamic friction model describes the dynamics of a fictitious bristle more accurately. Its performance is compared here to stick-slip friction models, developed and launched not long ago by commercial multibody software packages.
LuGre or not LuGre
(2023)
The LuGre model is widely used in the analysis and control of systems with friction. Recently, it has even been made available in the commercial multibody dynamics simulation software system Adams. However, the LuGre model exhibits well-known drawbacks like too low and force rate-dependent break-away forces, drift problems during sticking periods, and significant differences in non-stationary situations between the pre-defined friction law and the one produced by the LuGre model. In the present literature, these problems are supposed to come from the model dynamics or its nonlinear nature. However, most of these drawbacks are not simple side effects of a dynamic friction model but are caused in the LuGre approach, as shown here, by a too simple and inconsistent model of the bristle dynamics. Standard examples and a more practical application demonstrate that the LuGre model is not a “what you see is what you get” approach. A dynamic friction model with accurate bristle dynamics and consistent friction force is set up here. It provides insight into the physical basis of the LuGre model dynamics. However, it results in a nonlinear and implicit differential equation, whose solution will not be easy because of the ambiguity of the friction characteristics. The standard workaround, a static model based on simple regularized characteristics, produces reliable and generally satisfactory results but definitely cannot maintain a stick. The paper presents a second-order dynamic friction model, which may serve as an alternative. It can maintain a stick and produces realistic and reliable results.
Dieses Lehr- und Übungsbuch vermittelt auf anschauliche Weise die Methoden der Mehrkörpersimulation und verdeutlicht deren Vor- und Nachteile bei der praktischen Anwendung anhand konkreter Beispiele. Die einzelnen Methoden werden durch Matlab-Skripte und -Funktionen verdeutlicht, wobei die Modellbildung, die mathematische Beschreibung und die numerische Simulation von Systemen starrer Körper die Schwerpunkte bilden. Die vorliegende Auflage wurde unter anderem um Matlab-Live-Skripte erweitert, welche kleine Animationen zur Veranschaulichung der Dynamik der Probleme enthalten. Die Lösungen zu den Übungsbeispielen und die integrierten Matlab-Skripte sowie weitere Beispiele und Anwendungen stehen über QR-Codes zum Download zur Verfügung und ermöglichen dadurch auch ein effizientes Selbststudium.
Nach dem Aufbau eines Mehrkörper-Simulationsmodells muss dieses auf Richtigkeit, Funktionalität und Wirtschaftlichkeit getestet werden. Die Ermittlung der Gleichgewichtslage stellt dabei eine erste Plausibilitäts-Kontrolle dar. Eine Linearisierung mit anschließender Analyse der Eigendynamik liefert Aussagen über die Frequenzen und das Dämpfungsverhalten des Modells. Einfache Erregersignale ermöglichen einen ersten Einblick in das nichtlineare dynamische Verhalten des Modells. Modell-Parameter, die nicht genau bekannt sind, können durch gezielte Variationen plausibel geschätzt oder über eine Optimierung sogar mit optimalen Werten belegt werden. Nach all diesen Tests steht das Mehrkörper- Simulationsmodell dann für praktischeUntersuchungen zurVerfügung, die neben reinen Zeitsimulationen auch Methoden der Inversen Kinematik und der Inversen Dynamik mit einschließen.