COMPLEXITY OF INDEX SETS OF DESCRIPTIVE SET-THEORETIC NOTIONS

A Modular Race Tire Model Concerning Thermal and Transient Behavior using a Simple Contact Patch Description

Tire Science and Technology ◽

10.2346/tire.13.410402 ◽

2013 ◽

Vol 41 (4) ◽

pp. 232-246

Author(s):

Timo Völkl ◽

Robert Lukesch ◽

Martin Mühlmeier ◽

Michael Graf ◽

Hermann Winner

Keyword(s):

Load Distribution ◽

Thermal Condition ◽

Transient Behavior ◽

Wheel Load ◽

Contact Patch ◽

Fundamental Parameters ◽

Dry Conditions ◽

Simple Contact ◽

Tire Forces ◽

Descriptive Set

ABSTRACT The potential of a race tire strongly depends on its thermal condition, the load distribution in its contact patch, and the variation of wheel load. The approach described in this paper uses a modular structure consisting of elementary blocks for thermodynamics, transient excitation, and load distribution in the contact patch. The model provides conclusive tire characteristics by adopting the fundamental parameters of a simple mathematical force description. This then allows an isolated parameterization and examination of each block in order to subsequently analyze particular influences on the full model. For the characterization of the load distribution in the contact patch depending on inflation pressure, camber, and the present force state, a mathematical description of measured pressure distribution is used. This affects the tire's grip as well as the heat input to its surface and its casing. In order to determine the thermal condition, one-dimensional partial differential equations at discrete rings over the tire width solve the balance of energy. The resulting surface and rubber temperatures are used to determine the friction coefficient and stiffness of the rubber. The tire's transient behavior is modeled by a state selective filtering, which distinguishes between the dynamics of wheel load and slip. Simulation results for the range of occurring states at dry conditions show a sufficient correlation between the tire model's output and measured tire forces while requiring only a simplified and descriptive set of parameters.

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DESCRIPTIVE SET THEORETIC PHENOMENA IN ANALYSIS AND TOPOLOGY

Real Analysis Exchange ◽

10.2307/44153655 ◽

1990 ◽

Vol 16 (1) ◽

pp. 15

Author(s):

Becker

Keyword(s):

And Topology ◽

Descriptive Set

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Two Applications of Inner Model Theory to the Study of Sets

Bulletin of Symbolic Logic ◽

10.2307/421049 ◽

1996 ◽

Vol 2 (1) ◽

pp. 94-107 ◽

Cited By ~ 3

Author(s):

Greg Hjorth

Keyword(s):

Los Angeles ◽

Set Theory ◽

Model Theory ◽

Large Cardinals ◽

Descriptive Set Theory ◽

Inner Model Theory ◽

Regular Cardinals ◽

Inner Model ◽

Definable Sets ◽

Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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