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 for prediction of bogie stability Comparison of methods on examples of wheel/rail contact geometry Comparison of resultant critical speeds Conclusions
Motivation Typical task during the suspension design of railway vehicles: What is the maximum speed at which the vehicle will run stable for the specified equivalent conicity? or Which suspension parameters are needed to run stable for the given speed and equivalent conicity? Questions to solve by the specialist: Which method and criteria should be used? and How to model the specified equivalent conicity in the nonlinear simulations?
Bogie and Carbody Stability % of critical damping Critical speed 5% of critical damping stable area Equivalent conicity Carbody hunting Bogie hunting
Definition of Stability Limit Difference: Mechanics Railway Standards Mechanics Railway Standards stable unstable stable unstable y y y y t t t t y t
Classification of Methods for Stability Analysis Vehicle Model Wheel/Rail Contact Calculation Method Excitation Type Linear Linear Eigenvalue analysis Nonlinear Simulation Nonlinear Linear Simulation No excitation Nonlinear Simulation Single excitation Stochastic excitation
Simulation of Run with Decreasing Speed (No Excitation) Wheelset lateral displacement y [mm] 6 2-2 - -6-5 1 15 2 25 Time [s] 32 Speed v [km/h] 3 2 26 2 22 5 1 15 2 25 Time [s]
Excitation by a Single Lateral Irregularity stable unstable
Bifurcation Diagram Amplitude of the limit cycle as function of speed stable cycle Lateral amplitude stable area unstable area unstable saddlecycle v cr Speed
Simulation of Run on Measured Track Irregularity Track Irregularity Simulation Limit values according to standards (UIC 51, pren 1 363): Acceleration, rms value Sum of guiding forces, rms value [m/s 2 ] ÿ 2 + 1 5-5 -1 Signal Analysis Acceleration on bogie frame - time signal 2 6 1 Acceleration on bogie frame - rms value s ÿ 2 + [m/s 2 ] 6 2 2 6 1 s ΣY 2 [kn] 25 2 15 1 5 Sum of guiding forces - rms value 2 6 1
Vehicle Model St A four-car articulated vehicle modelled in Simpack Wheel/rail friction coefficient. (dry)
Examples of Contact Geometry Wheelset/Track Equivalent conicity: Specified for wheelset lateral amplitude of 3 mm Four examples of wheel/rail contact geometry A +5-5 +5-5 y [mm] 6A + - + - y [mm] B +5-5 +5-5 y [mm] 6B +5-5 +5-5 y [mm]
Equivalent Conicity Function Conicity. Conicity.6 1 A 1 6A Conicity [ - ]..6. Rigid contact Quasi-elastic contact Conicity [ - ]..6..2.2 1 2 3 5 6 7 1 2 3 5 6 7 Lateral amplitude [mm] Lateral amplitude [mm] 1 B 1 6B.. Conicity [ - ].6. Conicity [ - ].6..2.2 1 2 3 5 6 7 1 2 3 5 6 7 Lateral amplitude [mm] Lateral amplitude [mm]
Simulations of Run with Decreasing Speed Conicity. Conicity.6 A 6A y [mm] - y [mm] - - 5 1 15 2 25 3 B Time [s] - 5 1 15 2 25 3 6B Time [s] y [mm] - y [mm] - - 5 1 15 2 25 3 3 Time [s] - 5 1 15 2 25 3 3 Time [s] 25 2 15 25 2 15 1 5 1 15 2 25 3 1 5 1 15 2 25 3 Time [s] Time [s]
Method with Single Excitation Conicity. Conicity.6 A 6A 22 23 2 25 26 27 2 29 3 22 Speed [km/h] 23 2 25 26 27 2 29 3 Speed [km/h] B 6B 1 16 1 2 22 2 26 2 3 1 Speed [km/h] 16 1 2 22 2 26 2 3 Speed [km/h]
Bifurcation Diagrams Conicity. Conicity.6 12 A 12 6A Lateral amplitude [mm] 1 6 2 Lateral amplitude [mm] 1 6 2 1 15 2 25 3 35 5 5 1 15 2 25 3 35 5 5 12 B 12 6B Lateral amplitude [mm] 1 6 2 Lateral amplitude [mm] 1 6 2 1 15 2 25 3 35 5 5 1 15 2 25 3 35 5 5
Simulations of Run on Measured Track Irregularities Speed 27 km/h Speed 2 km/h [m/s 2 ] ÿ 2 + 1 5-5 Acceleration on bogie frame - time signal + ÿ 2 [m/s 2 ] 1 5-5 Acceleration on bogie frame - time signal Lateral acceleration, time signal [m/s 2 ] s ÿ 2 + -1 6 2 2 6 1 Acceleration on bogie frame - rms value [m/s 2 ] s ÿ 2 + -1 6 2 2 6 1 Acceleration on bogie frame - rms value Lateral acceleration, rms value (UIC 51) s ΣY 2 [kn] 25 2 15 1 5 2 6 1 Sum of guiding forces - rms value 2 6 1 s ΣY 2 [kn] 25 2 15 1 5 2 6 1 Sum of guiding forces - rms value 2 6 1 Sum of guiding forces, rms value (UIC 51)
Results of Simulations on Track Irregularities Conicity. Conicity.6 Nondimensional max. [%] 2 175 15 125 1 75 5 25 A Sum of guiding forces (UIC 51) Acceleration, rms value (UIC 51) Acceleration, peak value (UIC 515) Limit value Nondimensional max. [%] 2 175 15 125 1 75 5 25 6A 22 23 2 25 26 27 2 29 3 31 32 1 19 2 21 22 23 2 25 26 27 2 Nondimensional max. [%] 2 175 15 125 1 75 5 25 B Nondimensional max. [%] 2 175 15 125 1 75 5 25 6B 22 23 2 25 26 27 2 29 3 31 32 1 19 2 21 22 23 2 25 26 27 2
Dynamic Behaviour after a Single Excitation [m/s 2 ] ÿ 2 + 15 1 5-5 -1-15 Speed 26 km/h Acceleration on bogie frame - time signal 5 1 15 2 25 3 [m/s 2 ] ÿ 2 + 15 1 5-5 -1-15 Speed 2 km/h Acceleration on bogie frame - time signal 5 1 15 2 25 3 Acceleration on bogie frame - rms value Acceleration on bogie frame - rms value s ÿ 2 + [m/s 2 ] 6 2 s ÿ 2 + [m/s 2 ] 6 2 5 1 15 2 25 3 5 1 15 2 25 3 25 Sum of guiding forces - rms value 25 Sum of guiding forces - rms value 2 2 s ΣY 2 [kn] 15 1 5 s ΣY 2 [kn] 15 1 5 5 1 15 2 25 3 5 1 15 2 25 3
Stability Assessment of Behaviour after a Single Excitation Nondimensional max. [%] 2 175 15 125 1 75 5 25 Sum of guiding forces (UIC 51) Acceleration, rms value (UIC 51) Acceleration, peak value (UIC 515) Limit value A y max [mm] 7 6 5 3 2 1 A Nondimensional max. [%] 2 175 15 125 1 75 5 25 1 16 1 2 22 2 26 2 3 B 1 16 1 2 22 2 26 2 3 B y max [mm] 7 6 5 3 2 1 1 16 1 2 22 2 26 2 3 B 1 16 1 2 22 2 26 2 3
Comparison of Resultant Critical Speeds Conicity. Critical speed [km/h] 3 25 2 15 1 5 A Contact geometry B No excitation, decreasing speed Bifurcation diagram Single excitation, oscillation damped Single excitation, sum of Y-forces, rms value [UIC 51] Single excitation, acceleration, rms value [UIC 51] Single excitation, acceleration, peak value [UIC 515] Track irregularity, sum of Y-forces, rms value [UIC 51] Track irregularity, acceleration, rms value [UIC 51] Track irregularity, acceleration, peak value [UIC 515]
Comparison of Resultant Critical Speeds Conicity.6 Critical speed [km/h] 3 25 2 15 1 5 6A Contact geometry 6B No excitation, decreasing speed Bifurcation diagram Single excitation, oscillation damped Single excitation, sum of Y-forces, rms value [UIC 51] Single excitation, acceleration, rms value [UIC 51] Single excitation, acceleration, peak value [UIC 515] Track irregularity, sum of Y-forces, rms value [UIC 51] Track irregularity, acceleration, rms value [UIC 51] Track irregularity, acceleration, peak value [UIC 515]
Conclusions (1): Nonlinear Method for Stability Analysis Difference in definition of stability between Mechanics and Railway Engineering The methods presented are comparable if no limit cycles with small amplitude occur For a specified conicity, differences between the results occur dependent on the method, limit value and the contact geometry If small limit cycles occur stability limits from the railway standards should be used to judge the results y y t t
Conclusions (2): Specification of Wheel/Rail Contact Specification of the shape of wheel profile, rail profile, rail inclination and gauge Separate specification of the maximum wheelset related equivalent conicity and the maximum track related equivalent conicity Only the maximum equivalent conicity specified: Recommended to use wheel/rail contact geometry with increasing or constant conicity function to avoid small limit cycles Conicity more exact less 3 mm Amplitude