5.5 Exercises for This Chapter Two-Axle Vehicle on Cosine Track Two-Axle Vehicle on Generally Periodic Track...

Size: px

Start display at page:

Download "5.5 Exercises for This Chapter Two-Axle Vehicle on Cosine Track Two-Axle Vehicle on Generally Periodic Track..."

Franklin Marshall
5 years ago
Views:

1 Contents 1 Introduction The Basic Function of the Wheel/rail System Significance of Dynamics on the Operation of Rail Vehicles On the History of Research in the Field of Railway Technology Since A New Start After 1945: Japan and France Research and Development Aimed at Overcoming the Boundaries of the Wheel/rail System Railway Industry in Europe Overview of This Book Classification Vertical and Lateral Vibrations Curving Behavior Frequency- and Time-Domain Calculations References Modeling of Vehicle, Track, and Excitation Prior Considerations and Coordinate Systems Vehicle Modeling Bogie Frame Design, Primary Suspension, and Bogie Guidance Mechanical Model of the Vehicle. Connecting Elements Elastic Carbodies Modeling of Track and Excitation Track Modeling Modeling of the Excitation References ix

2 x Contents 3 Modeling of Wheel/Rail Contact Profile Geometry Contact Kinematics Between Wheel and Rail Contact Kinematics with Conical and Circular Profiles Contact Kinematics for Arbitrary Profiles On the Determination of Equivalent Contact Parameters by Means of Quasilinearization Conversion into Equivalent Circular Profiles Linearized Contact Kinematics with Track Irregularities Creepage Calculation Normal Contact Mechanics Overview of the Calculation of Contact Point Stresses Assumptions of the Normal Contact Problem Nonelliptic Contact Patches Hertzian Treatment of the Normal Contact Problem Spherical and Point Contact Ellipsoidal Contact Contact of Rollers, Line Contact Linearized Replacement Model Tangential Contact Mechanics Introduction to the Tangential Contact Problem Analytical Solution for Rolling Contact (Line Contact) Kalker s Theory of Rolling Contact for Ellipsoidal Contact Approximate Solutions According to Vermeulen Johnson and Shen Hedrick Elkins Simplified Theory of Rolling Contact [54] Adaptation of the Theory to Reality References Vertical Dynamics, Equations of Motion, and Free Vibrations Notation and Assumptions Equations of Motion with the Principle of Linear and Angular Momentum Displacement Degrees of Freedom for a Two-Axle Vehicle Constraints Forces in the Spring and Damper Elements Free-Body Diagrams of the Masses... 87

3 Contents xi Principle of Linear and Angular Momentum for the Formulation of the Equation System Elimination of the Constrained Forces. Final System of Equations Principle of Virtual Displacements for Rigid-Body Systems Initial Remark Formulation of the Principle of Virtual Displacements Inclusion of Geometric Constraints in the Principle of Virtual Displacement Exemplified on a Two-Axle Vehicle Formalized Derivation of the Equations of Motion and the Principle of Virtual Displacements Displacement Vector with the Degrees of Freedom of the Free System Relation Between Spring Elongation and System Displacement Spring Forces and the Formulation of the Virtual Strain Energy Mass Matrix and Formulation of the Virtual Work of the Inertia Forces External and Constraint Forces Equations of Motion for the Free System. Introduction of Constraints Equations of Motion for Elastic Car Bodies Solution for Free Vibrations Exercises for This Chapter Constraint Forces when Constraints are Met Validity of the Rolling Condition References Forced Vertical Vibrations for Excitation with Harmonic and Periodic Track Irregularities (Frequency Domain Solution) Complex Syntax Vertical Vibrations on a Track with Cosine-Shaped Track Irregularity (Cosine Track) Track Irregularities and Base-Point Excitations Solution for the Vertical Motion Interpretation of the Solution for Different Suspensions Between Wheelset and Carbody Vehicle on a General Periodic Track Solution for a Vehicle with Elastic Carbody Modes

4 xii Contents 5.5 Exercises for This Chapter Two-Axle Vehicle on Cosine Track Two-Axle Vehicle on Generally Periodic Track References Random Vibrations due to Stochastic Track Irregularities Characterization of an Irregular Track with Help of Root Mean Square Value and Power Spectrum Determination of Vehicle Response for Stochastic Track Excitation Power Spectra of Track Irregularities How to Derive Power Spectra of Track Irregularities Power Spectra for the DB Network Supplementary Comments Regarding the Relationship Between Measured Spectra of Spatial Angular Frequencies and Spectra of Angular Frequencies as Function of Time Interpretation of Response Spectrum References Human Perception of Vibrations - Ride Comfort Wertungsziffer According to Sperling Periodic Vibrations Random Vibrations ISO CEN Standard EN Simplified Criterion for Mean Comfort: N MV Comfort on Curve Transitions: P CT Comfort on Discrete Events: P DE Final Remarks Measure or Simulate Ride Comfort as System Property Duration of Exposure of Vibration Exercises for This Chapter Calculation of Wz-Values According to Sperling References Introduction to Lateral Dynamics of Railway Vehicles Preliminary Remark Hunting and Klingel Equation Assumptions for Derivation of the Klingel Equation Determination of Equivalent Conicity with Eq. (8.13) Final Remark to This Chapter References

5 Contents xiii 9 Derivation of Equations of Motion for Lateral Dynamics Principle of Virtual Displacements for a Suspended Wheelset with All Degrees of Freedom Evaluated System and Acting Forces Formulation of Principle of Virtual Displacements Determination of the Virtual Displacements Equilibrium Conditions in the x-direction and Around the y-axis Equilibrium Conditions in the y-direction and Around the z-axis Exercises for This Chapter Interpretation of Creep Force Terms in Eq. (9.13) Rolling Resistance Due to Spin Creepage Equations of Motions for Forced Lateral Vibrations Rolling Resistance in the Simplified Theory Equations of Motion of a Suspended Wheelset with Numbers Creep Forces Under the Assumption of a Reference State with Creep Forces References Lateral Eigenbehavior and Stability of a Wheelset on Straight Track Determination of Eigenvalues and Eigenvectors Root Loci Curves Approximate Solution for Low Speeds Stability Investigation with Hurwitz Criterion Critical Speed of a Wheelset Interpretation of Stability Criteria of a Single Wheelset Exercises for This Chapter Characteristic Equation Transformation of Wheelset Equations of Motion Graphical Illustration of the Root Loci Curves of a Suspended Wheelset and Determination of the Critical Speed Independently Rotating Wheels Reference Lateral Eigenbehaviour and Stability of Bogies Numerical Calculation of Eigenvalues and Critical Speed Analytic Approximations for Bogies Coordinate Transformation to Introduce Generalized Degrees of Freedom Bogie with Infinitely High Bending and Shear Stiffness

6 xiv Contents Realization of Infinitely High Bending Stiffness or Shear Stiffness Bogies with Infinite Shear Stiffness Bogies with Infinite Bending Stiffness Bogies with Finite Bending and Shear Stiffness Exercises for This Chapter Equations of Motion of a Bogie Equations of Motion of a Freely Rolling Wheelset at Low Speed Equations for Bending and Shear Stiffness References Lateral Eigenbehavior and Stability of Bogie Vehicles Stability of a Train with Two Vehicles General Conclusions Regarding the Stability of a Bogie Vehicle According to Matsudaira [4] Theory and Simulation of Bogie Vehicle Hunting Bogie Hunting Influence of Friction Yaw Damping on Bogie Hunting (Nonlinear Stability Investigation) Suggestions for Further Work for This Chapter Influence of Bending and Shear Stiffness on the Stability of Bogie Vehicles Stability of a Vehicle with Independently Rotating Wheels Friction and Viscous Yaw Damping References Introduction to Non-linear Stability Investigations Preamble Nonlinear Critical Speed Fourier Decomposition of Nonlinear Limit Cycle Motions: The Method of Urabe and Reiter Simplified Investigation of Nonlinear Stability with the Method of Quasilinearization Limits of Fourier Decomposition Nonlinear Stability Investigations in the Time Domain Ideas for Further Study in this Chapter Stability Investigation for the Boedecker Vehicle References Quasistatic Curving Behavior Historical Introduction General Remarks A Single Wheelset in a Curve Free Wheelset in a Curve (Kinematic Curving)

7 Contents xv 14.4 Wheelset Guided in Track Following Frame Curving of Bogies and Entire Vehicles Curving According to Uebelacker and Heumann Curving of Bogies with Suspension Wear Calculation in the Wheel Rail Contact Exercises for This Chapter Directions of Creep Forces for Different Wheelset Positions Deviation of Angle of Attack or Lateral Displacement of Wheelsets Curving of Single Wheelset References Determination of Load Collectives for Vehicle Components Introduction General Procedure Stress Calculation in Components Finite Element Calculation in Each Time Step Stress Calculation with the Help of Transformation Matrices Determination of Load Collectives Determination of Sustainable Stresses Cycle Counting Methods to Determine Load Collectives Conversion from a Two-Parameter to a One Parameter Collective Superposition to Total Load Collective Damage Accumulation: Proof of Strength Damage Accumulation Hypotheses Concepts for Variable-Amplitude Fatigue Strength Proof for Rail Vehicles Exercises for This Chapter Transformation Matrix Between Degrees of Freedom in a Multibody Simulation Model and Stresses in the Bogie Frame Determination of Collective of Spring Forces with Spectral Analysis References Appendix List of Symbols Coordinate Systems

8 xvi Contents 16.3 Fundamentals of Contact Mechanics Hertz Contact Mechanics Contact Equation Basic Equations for the Tangential Contact Problem According to Carter Function Φ for the Vermeulen Johnson Solution Basic Equations of the Simplified Theory of Rolling Contact Stability Criteria of Characteristic Equations with Hurwitz Criterion Critical Speed of Single Wheelset Taking Nondiagonal Elements of Creepage Damping Matrix into Account References Index

Methods for Running Stability Prediction and their Sensitivity to Wheel/Rail Contact Geometry

Methods for Running Stability Prediction and their Sensitivity to Wheel/Rail Contact Geometry Oldrich POLACH and Adrian VETTER Bombardier Transportation Winterthur, Switzerland Contents Motivation Methods