Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

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Display screen or portion thereof with graphical user interface

Statistical collision model for Monte Carlo simulation of polyatomic gas mixture

Traffic flow is a kind of many-body system of strongly interacting vehicles. Traffic jams are a typical signature of the complex behaviour of vehicular traffic. Various models are presented to understand the rich variety of physical phenomena exhibited by traffic. Analytical and numerical techniques are applied to study these models. Particularly, we present detailed results obtained mainly from the microscopic car-following models. A typical phenomenon is the dynamical jamming transition from the free traffic (FT) at low density to the congested traffic at high density. The jamming transition exhibits the phase diagram similar to a conventional gas-liquid phase transition: the FT and congested traffic correspond to the gas and liquid phases, respectively. The dynamical transition is described by the time-dependent Ginzburg-Landau equation for the phase transition. The jamming transition curve is given by the spinodal line. The metastability exists in the region between the spinodal and phase separation lines. The jams in the congested traffic reveal various density waves. Some of these density waves show typical nonlinear waves such as soliton, triangular shock and kink. The density waves are described by the nonlinear wave equations: the Korteweg-de-Vries (KdV) equation, the Burgers equation and the Modified KdV equation. Subjects like the traffic flow such as bus-route system and pedestrian flow are touched as well. The bus-route system with many buses exhibits the bunching transition where buses bunch together with proceeding ahead. Such dynamic models as the car-following model are proposed to investigate the bunching transition and bus delay. A recurrent bus exhibits the dynamical transition between the delay and schedule-time phases. The delay transition is described in terms of the nonlinear map. The pedestrian flow also reveals the jamming transition from the free flow at low density to the clogging at high density. Some models are presented to study the pedestrian flow. When the clogging occurs, the pedestrian flow shows the scaling behaviour.

http://iopscience.iop.org/article/10.1088/0034-4885/65/9/203/pdf

The physics of traffic jams

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

Preface. Acknowledgments. 1. Overview of Aerodynamics. 1.1. Introduction and Notation. 1.2. Fluid Statics and the Atmosphere. 1.3. The Boundary Layer Concept. 1.4. Inviscid Aerodynamics. 1.5. Review of Elementary Potential Flows. 1.6. Incompressible Flow over Airfoils. 1.7. Trailing-Edge Flaps and Section Flap Effectiveness. 1.8. Incompressible Flow over Finite Wings. 1.9. Flow over Multiple Lifting Surfaces. 1.10. Wing Stall and Maximum Lift Coefficient. 1.11. Wing Aerodynamic Center and Pitching Moment. 1.12. Inviscid Compressible Aerodynamics. 1.13. Compressible Subsonic Flow. 1.14. Supersonic Flow. 1.15. Problems. 2. Overview of Propulsion. 2.1. Introduction. 2.2. The Propeller. 2.3. Propeller Blade Theory. 2.4. Propeller Momentum Theory. 2.5. Off-Axis Forces and Moments Developed by a Propeller. 2.6. Turbojet Engines: The Thrust Equation. 2.7. Turbojet Engines: Cycle Analysis. 2.8. The Turbojet Engine with Afterburner. 2.9. Turbofan Engines. 2.10. Concluding Remarks. 2.11. Problems. 3. Aircraft Performance. 3.1. Introduction. 3.2. Thrust Required. 3.3. Power Required. 3.4. Rate of Climb and Power Available. 3.5. Fuel Consumption and Endurance. 3.6. Fuel Consumption and Range. 3.7. Power Failure and Gliding Flight. 3.8. Airspeed, Wing Loading, and Stall. 3.9. The Steady Coordinated Turn. 3.10. Takeoff and Landing Performance. 3.11. Accelerating Climb and Balanced Field Length. 3.12. Problems. 4. Longitudinal Static Stability and Trim. 4.1. Fundamentals of Static Equilibrium and Stability. 4.2. Pitch Stability of a Cambered Wing. 4.3. Simplified Pitch Stability Analysis for a Wing-Tail Combination. 4.4. Stick-Fixed Neutral Point and Static Margin. 4.5. Estimating the Downwash Angle on an Aft Tail. 4.6. Simplified Pitch Stability Analysis for a Wing-Canard Combination. 4.7. Effects of Drag and Vertical Offset. 4.8. Effects of Nonlinearities on the Aerodynamic Center. 4.9. Effect of the Fuselage, Nacelles, and External Stores. 4.10. Contribution of Running Propellers. 4.11. Contribution of Jet Engines. 4.12. Problems. 5. Lateral Static Stability and Trim. 5.1. Introduction. 5.2. Yaw Stability and Trim. 5.3. Estimating the Sidewash Gradient on a Vertical Tail. 5.4. Estimating the Lift Slope for a Vertical Tail. 5.5. Effects of Tail Dihedral on Yaw Stability. 5.6. Roll Stability and Dihedral Effect. 5.7. Roll Control and Trim Requirements. 5.8. The Generalized Small-Angle Lateral Trim Requirements. 5.9. Steady-Heading Sideslip. 5.10. Engine Failure and Minimum-Control Airspeed. 5.11. Longitudinal-Lateral Coupling. 5.12. Control Surface Sign Conventions. 5.13. Problems. 6. Aircraft Controls and Maneuverability. 6.1. Longitudinal Control and Maneuverability. 6.2. Effects of Structural Flexibility. 6.3. Control Force and Trim Tabs. 6.4. Stick-Free Neutral and Maneuver Points. 6.5. Ground Effect, Elevator Sizing, and CG Limits. 6.6. Stall Recovery. 6.7. Lateral Control and Maneuverability. 6.8. Aileron Reversal. 6.9. Other Control Surface Configurations. 6.10. Airplane Spin. 6.11. Problems. 7. Aircraft Equations of Motion. 7.1. Introduction. 7.2. Newton's Second Law for Rigid-Body Dynamics. 7.3. Position and Orientation: The Euler Angle Formulation. 7.4. Rigid-Body 6-DOF Equations of Motion. 7.5. Linearized Equations of Motion. 7.6. Force and Moment Derivatives. 7.7. Nondimensional Linearized Equations of Motion. 7.8. Transformation of Stability Axes. 7.9. Inertial and Gyroscopic Coupling. 7.10. Problems. 8. Linearized Longitudinal Dynamics. 8.1. Fundamentals of Dynamics: Eigenproblems. 8.2. Longitudinal Motion: The Linearized Coupled Equations. 8.3. Short-Period Approximation. 8.4. Long-Period Approximation. 8.5. Pure Pitching Motion. 8.6. Summary. 8.7. Problems. 9. Linearized Lateral Dynamics. 9.1. Introduction. 9.2. Lateral Motion: The Linearized Coupled Equations. 9.3. Roll Approximation. 9.4. Spiral Approximation. 9.5. Dutch Roll Approximation. 9.6. Pure Rolling Motion. 9.7. Pure Yawing Motion. 9.8. Longitudinal-Lateral Coupling. 9.9. Nonlinear Effects. 9.10. Summary. 9.11. Problems. 10. Aircraft Handling Qualities and Control Response. 10.1. Introduction. 10.2. Pilot Opinion. 10.3. Dynamic Handling Quality Prediction. 10.4. Response to Control Inputs. 10.5. Nonlinear Effects and Longitudinal-Lateral Coupling. 10.6. Problems. 11. Aircraft Flight Simulation. 11.1. Introduction. 11.2. Euler Angle Formulations. 11.3. Direction-Cosine Formulation. 11.4. Euler Axis Formulation. 11.5. The Euler-Rodrigues Quaternion Formulation. 11.6. Quaternion Algebra. 11.7. Relations between the Quaternion and Other Attitude Descriptors. 11.8. Applying Rotational Constraints to the Quaternion Formulation. 11.9. Closed-Form Quaternion Solution for Constant Rotation. 11.10. Numerical Integration of the Quaternion Formulation. 11.11. Summary of the Flat-Earth Quaternion Formulation. 11.12. Aircraft Position in Geographic Coordinates. 11.13. Problems. Bibliography. Appendixes. A Standard Atmosphere, SI Units. B Standard Atmosphere, English Units. C Aircraft Moments of Inertia. Nomenclature. Index.

Mechanics of Flight

The classical solution to Prandtl's well-known lifting-line theory applies only to a single lifting surface with no sweep and no dihedral. However, Prandtl's original model of a finite lifting surface has much broader applicability. A general numerical lifting-line method based on Prandtl's model is presented. Whereas classical lifting-line theory is based on applying the two-dimensional Kutta-Joukowski law to a three-dimensional flow, the present method is based on a fully three-dimensional vortex lifting law. The method can be used for systems of lifting surfaces with arbitrary camber, sweep, and dihedral. The accuracy realized from this method is shown to be comparable to that obtained from numerical panel methods and inviscid computational fluid dynamics solutions, but at a small fraction of the computational cost

Modern Adaptation of Prandtl's Classic Lifting-Line Theory

Adetailed survey ispresentedoftheliteratureonattituderepresentationdatingfromtheearlyworkofEulerand Hamilton to recent publications in e elds such as navigation and control. The scope is limited to the development of the aircraft kinematic transformation equations in terms of four different attitude representations, including the well-known Euler angles, the Euler-axis rotation parameters, the direction cosines, and the Euler ‐Rodrigues quaternion.Theemphasisisdirectedattheapplicationofthequaternionformulationtoaircraftkinematics.Results are presented that reinforce observations that the quaternion formulation, typically implemented to eliminate singularities associated with the Euler angle formulation, is far superior to the other commonly used formulations based on computational efe ciency alone. A development of quaternion constraints necessary to independently constrain roll, pitch, yaw, bank angle, elevation angle, and/or azimuth angle is presented. For verie cation of simulation codes, a general closed-form solution to the quaternion formulation, for the case of constant rotation, is also presented. Additionally, a discussion is provided of numerical integration methods and numerical errors for the quaternion formulation. This discussion is especially important for simulations that may still utilize a common error reduction scheme originally developed for analog computers.

Review of Attitude Representations Used for Aircraft Kinematics

A statistical model for the flow of traffic on an n lane highway is presented. The model is based on the hypothesis that each driver will maintain a minimum space, between himself and the next car, which is proportional to his speed. Both the first and second order continuum formulations, which are derived from the basic statistical formulations, are also presented.

A kinetic model for traffic flow with continuum implications

A more practical form of the analytical solution for the effects of geometric and aerodynamic twist (washout) on the low-Mach-number performance of a finite wing of arbitrary planform is presented. This infinite series solution is based on Prandtl's classical lifting-line theory and the Fourier coefficients are presented in a form that only depends on wing geometry

Warren F. Phillips

Papers

Mechanics of Flight

Modern Adaptation of Prandtl's Classic Lifting-Line Theory

Review of Attitude Representations Used for Aircraft Kinematics

A kinetic model for traffic flow with continuum implications

Lifting-Line Analysis for Twisted Wings and Washout-Optimized Wings