how to perform a quasi-static curving analysis with ... · how to perform a quasi-static curving...

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How to Perform a Quasi-Static Curving Analysis with SIMPACK Wheel/Rail Version 2005-05-11

Table of Contents 1 Introduction 2

1.1 Methods 2 1.1.1 Solving a nonlinear equation system 2 1.1.2 Time integration 2

1.2 Single calculation vs. parameter variation 2 2 Single calculation 3

2.1 Setting up the model 3 2.1.1 Track settings 3 2.1.2 Vehicle speed 3

2.2 Calculation parameters 4 2.3 Result outputs 4

2.3.1 Common results 4 2.3.2 Wear index 5 2.3.3 Roll coefficient 6 2.3.4 Unbalanced lateral acceleration 7

3 Parameter variation 8 3.1 Setting up the model 8

3.1.1 Track settings 8 3.1.2 Vehicle velocity 8

3.2 Setting up the parameter variation 8 3.2.1 Defining the parameters in the GUI 9 3.2.2 Using parameter variation steering files 10 3.2.3 Ensuring equal travelled distances with different speeds 12

3.3 Calculation parameters 13 3.4 Result outputs 13 3.5 Plotting the results 13

4 Check list for set up 13 5 Example model 15


1 Introduction

Quasi-static curving analysis is a common task in the design process of railway vehicles. The aim is to know how the vehicle behaves in curves, to know the wheel-rail forces and an index of the wheel wear to be expected. This documentation describes how to perform this analysis with SIMPACK. It is strongly recommended that you have attended a SIMPACK Basics training as well as a SIMPACK Wheel/Rail training before using this documentation. The parameter variation requires a “SIMPACK Virtual Testing Lab” license.

1.1 Methods The basic method to assess the curving behaviour is to find the vehicle’s equilibrium state in the curve. For an equilibrium to be found, the track must not have disturbances or irregularities, and the curve radius or the superelevation must not change within the curve. Since the vehicle is not standing but travelling with constant speed, we are not talking about a real equilibrium but a “moving” equilibrium where the speed is con-stant but not zero. Thus it is called a quasi-static rather than a static equilibrium. In general, there are two basic methods for finding an equilibrium.

1.1.1 Solving a nonlinear equation system

This method describes the multi-body system with a non-linear algebraic equation sys-tem and finds the equilibrium by searching iteratively for a root of this equation system:

0),,,,( ≡tpzzz λ&&&

(see SIMPACK documentation to static equilibrium module). This method is very fast, but the equation system may have several different equilibrium states, and it depends on the initial configuration which equilibrium state is found; thus it is not recommended for curving analysis.

1.1.2 Time integration

This method uses a time integration, letting the vehicle travel from a straight track into the curve. It is therefore very close to the reality and ensures that the correct equilib-rium state is found in the curve, even considering bumpers and other strongly non-linear kinematics, e.g. in tilting mechanisms, as well as active elements. In this document we concentrate therefore on the time integration method.

1.2 Single calculation vs. parameter variation In SIMPACK there are two possibilities to perform a quasi-static curving analysis. The first method is the manual start of a single calculation. The second method uses the pa-rameter variation and allows several calculations with different parameters to be exe-cuted automatically, along with an automatic filtering of the output values in up to three stages. This document describes both methods.


2 Single calculation

2.1 Setting up the model The calculation can be performed with any vehicle model and any wheel-rail contact model. The one-point contact, however, has the best performance and is suitable for main-line profiles and not too narrow curve radii. The only necessary settings are

• curve radius R,

• superelevation u,

• vehicle velocity v. From these parameters results the unbalanced lateral acceleration aq as follows:

ϕϕ sincos2

gRvaq −= with

=

0

arcsineuϕ

where 0e is the rail base (in SIMPACK: reference length for superelevation).

2.1.1 Track settings

• Use the standard track type “curve entry”. It consists of a straight track, a transi-tion curve/superelevation ramp and a curve with constant radius and supereleva-tion. Choose preferredly “s-shaped ramp” to avoid the setting of the smoothing length. (The main reason for using a standard track rather than a cartographic track is that the standard tracks can be modified in a parameter variation.)

• Choose a curve radius and a superelevation.

• Set the length of the straight section so that it is a little longer than the vehicle it-self. Usually a length suffices which is five meters longer than the vehicle.

• Set an appropriate length of the transition curve. For high speeds, high superele-vations and/or small curve radii, use longer transition curves. The following table gives some (generously estimated) examples, assuming a velocity yielding aq ≈ 1 m/s². If in doubt, tend to longer transition curves.

Curve radiusCurve radiusCurve radiusCurve radius Transition length for u=0 mTransition length for u=0 mTransition length for u=0 mTransition length for u=0 m for u = 0.1 mfor u = 0.1 mfor u = 0.1 mfor u = 0.1 m for u = 0.15 mfor u = 0.15 mfor u = 0.15 mfor u = 0.15 m

50 m 10 m 75 m 120 m

100 m 15 m 75 m 120 m

200 m 25 m 75 m 120 m

300 m 50 m 75 m 120 m

500 m 75 m 75 m 120 m

1000 m 100 m 100 m 120 m

> 1000 m 100 - 250 m 100 - 250 m 120 - 250 m

• Set the total track length. Consider that the vehicle will need some time, usually about 10 to 15 seconds, to reach the equilibrium in the curve.

2.1.2 Vehicle speed

Set a vehicle speed that yields the intended unbalanced lateral acceleration. Since the vehicle will lose speed due to frictional energy loss in the curve, it is recommended to change the joint type of the car body from 07 to 09 (v=const.). This resembles that the


vehicle is pulled through the curve by a locomotive. Keep in mind that then the first joint state (s) gets lost and the others (y, z, ϕ, ψ, γ) are shifted one position up.

2.2 Calculation parameters The curving analysis should be executed with the SIMPACK module “Static Equilibrium” rather than with the normal time integration. The Static Equilibrium module allows a global (inertia) damping in roll and lateral direction (ϕ and y) to be applied. These direc-tions follow the track even in the curve. The damping prevents the vehicle from going into a limit cycle in the curve, which is a small periodic lateral and yaw movement, com-parable to an “instability” in straight track. In the curve it is not generally harmful but corrupts the analysis results. Usually a rather small inertia damping is sufficient. Recommended values are about 100 to 1000 Ns/m lateral and 10 to 100 Nms/rad in roll direction, respectively.

2.3 Result outputs

2.3.1 Common results

The following table explains how to get the most important results. Since the equilib-rium is the intended end state of the calculation, only the end values of the outputs are important as results.

DescriptionDescriptionDescriptionDescription From whereFrom whereFrom whereFrom where RemarksRemarksRemarksRemarks

QQQQ Vertical wheel-rail force Output value 26 of wheel-rail force elements ($F_RW_Friction_of_…)

Sum of all contact points of the wheel; in track system, acting towards the rail

YYYY Lateral wheel-rail force Output value 25 of wheel-rail force elements ($F_RW_Friction_of_…)

Sum of all contact points of the wheel; in track system, acting towards the rail

TTTTxxxx Longitudinal creep force Output value 5 of wheel-rail force elements ($F_RW_Friction_of_…)

Attention: only for the current contact point. In contact patch system, acting towards the wheel

Y/QY/QY/QY/Q Derailment coefficient Output value 28 of wheel-rail force elements ($F_RW_Friction_of_…)

Sum of all contact points of the wheel

WWWWssss Wear index in [Nm/m] not directly available May be calculated by an expres-sion, see below

ΣΣΣΣYYYY Lateral wheelset force Output value 29 of wheel-rail force elements ($F_RW_Friction_of_…)

In track system, acting towards the rails

ψψψψ Yaw angle (≈ angle of attack)

Joint state 5 of wheelset joint

yyyy Lateral wheelset position in track

Joint state 2 of wheelset joint

ssssRRRR Roll coefficient not directly available May be calculated by an expres-sion, see below


DescriptionDescriptionDescriptionDescription From whereFrom whereFrom whereFrom where RemarksRemarksRemarksRemarks

aaaaqqqq Unbalanced lateral accel-eration

Control element 167 (“AQ-Acceleration sensor”)

At wheelset marker, see below

2.3.2 Wear index

The wear index, which relates the frictional energy dissipated in the contact patch to the travelled distance, can be calculated by dividing the frictional power P by the vehi-cle speed v:

vPWs1

, ⋅= WheelWheel ,

either in the General Plot, or in an expression within the model:

FORCEOV($F_RW_Friction_…, 27) / JOINTST($J_Wheelset, 1, 1)

Here the frictional power is obtained from the appropriate output value of the wheel-rail tangential force element. The vehicle speed comes directly from the wheelset joint.

The best way to visualise an expression result in the General Plot is to define a dummy force element of type 50 (force by expression) and plot the output value of this force element:


2.3.3 Roll coefficient

The roll coefficient is in kinematic or dynamic terms, respectively:

Ground Track,

Track Carbody,

ϕϕ

=Rs or Track

Carbody

yy

sR &&

&&=

It should be calculated by an expression as well. The easiest form is the kinematic one:

IF( ABS(AX($M_Isys_Track_Frame_of_…,$M_Isys_Track_Camera_of_…))-1.e-2: 0, 0, AX($M_Body,$M_Isys_Track_Frame_of_…) / AX($M_Isys_Track_Frame_of_…,$M_Isys_Track_Camera_of_…) )

It uses the fact that the Track Frame marker is canted with the track whilst the Track Camera marker is not. (The IF avoids a division by zero when the track itself is not canted.) Since the Track Frame and Track Camera marker are not automatically created for a car body or a bogie frame, you should either use the markers of the nearest wheelset or define new markers of this kind, see the following pictures. Here $J_Body is the carbody joint.


2.3.4 Unbalanced lateral acceleration

If the unbalanced lateral acceleration aq itself is desired as output value, it has to be measured using the control element 167 (“AQ-Acceleration sensor”), which takes the gravity into account. This sensor should be defined in a way that the measuring marker is located on a wheelset, see the picture below. (Although the unbalanced lateral accel-eration is originally defined on track level, the Track Frame marker must not be used here because it does not experience an acceleration.) Note that the sensor outputs the acceleration seen by the wheelset, the sign of which is contrary to that of the acceleration experienced by the passengers.


3 Parameter variation

3.1 Setting up the model In a parameter variation the settings for curve radius R, superelevation u and vehicle velocity v can be varied automatically, leading to different configurations in the calcula-tion runs. Whilst in a manual calculation the parameters had to be calculated manually for the desired residual lateral acceleration, the parameter variation allows these pa-rameters to be determined automatically. Generally there are four possible scenarios:

Given (fix or varied) paramGiven (fix or varied) paramGiven (fix or varied) paramGiven (fix or varied) parameeeetersterstersters Resulting parameterResulting parameterResulting parameterResulting parameter

1 v, R, u Unbalanced lateral acceleration aq

2 aq, R, u Vehicle speed v

3 aq, v, u Track radius R

4 aq, R, v Superelevation u

The first scenario uses only given parameters that are explicitly defined in the model. The other scenarios have the implicit parameter aq as given parameter. In the parameter variation module it is possible to handle the scenarios 2 and 3 automatically, i.e. to have vehicle speed or track radius set by SIMPACK, yielding the desired aq. For the scenario 4 this function is not yet available. The necessary settings will be described in the the following.

3.1.1 Track settings

• The track settings are the same as for the single calculation, see section 2.1.1. If values will be varied in the parameter variation, set arbitrary dummy values for them.

• Since the length of the transition curve cannot be varied in the parameter varia-tion, set it to the longest curve resulting from the parameter combinations that will be used.

• Ensure that the track is long enough even for the highest vehicle speed used in the simulation. Otherwise the calculation will fail with an error message.

3.1.2 Vehicle velocity

Set a vehicle velocity that yields the intended unbalanced lateral acceleration. If the velocity is to be varied in the parameter variation, use an arbitrary dummy value (but not zero).

3.2 Setting up the parameter variation There are two different methods to set up the parameter variation. The first method is to define the varied parameters and their variation range in the parameter variation GUI, the second method makes use of parameter variation steering files. The first method is easier to set up, the second one is more flexible.


Although the SIMPACK parameter variation allows nested variation loops, we concen-trate on the case of one loop. Feel free to use the middle loop for variation of other parameters. (The outer loop is not useable here.)

3.2.1 Defining the parameters in the GUI

SIMPACK is able to calculate the vehicle velocity or the track radius automatically ac-cording to a given unbalanced lateral acceleration. The following table shows some common scenarios. Since the definition in the GUI allows only equidistant variation steps, it is not reasonable to vary more than one parameter, e.g. R and u, at once.

Given, constantGiven, constantGiven, constantGiven, constant Given, variedGiven, variedGiven, variedGiven, varied Automatically calcAutomatically calcAutomatically calcAutomatically calcuuuulatedlatedlatedlated

1 R, u v –

2 u, aq R v = f(aq, R, u, e0)

3 R, aq u v = f(aq, R, u, e0)

4 v, u aq R = f(aq, v, u, e0)

All these parameters are of the main type “wheel-rail global”. They have the following sub-types:

• Vehicle velocity v: type 01 in [m/s] or 04 in [km/h].

• Track radius R: type 70.

• Superelevation u: type 71.

• Unbalanced lateral acceleration aq, leading to v: type 72, )sin(cos

ϕϕ

gaRv q −= .

• Unbalanced lateral acceleration aq, leading to R: type 73, ϕ

ϕsin

cos2

gavRq −

= .

ϕ is here the track roll angle, and g is the gravitational acceleration. The resulting val-ues of type 72, 73 and 74 are not rounded and may therefore be quite unaccustomed. If R, u or v are constant, they are set in the model setup rather than in the parameter variation. If aq shall be constant, however, it has to be defined as a “varied” parameter, but with equal start and end value. The automatically calculated parameters have to be defined as varied parameters as well. The following pictures show the scenario 2 from the table above, with seven track radii from 300 to 1000 m and a constant aq of –1 m/s² (the superelevation is taken from the model setup):


3.2.2 Using parameter variation steering files

Parameter variation steering files allow arbitrary variation steps and parameter combi-nations. It is possible to define all parameters to be varied at once in the steering file or to combine the steering file with additional parameters defined in the GUI. Parameter variation steering files have the extension “.pvs” and are stored in the SIM-PACK data base directory “parvar_steering_files”. To keep the data structure clean it is recommended to use the model or project specific data base rather than the user spe-cific data base when working with steering files.

The steering files used here must be of type 10 (“input parameters”). This type allows arbitrary model parameters in .sys file format to be set. The wheel-rail global parame-ters, which have no according .sys file entries, have the following names:

• Vehicle speed v: ‘#Vehicle.v_m/s‘ or ‘#Vehicle.v_km/h‘

• Track radius R: ‘#Vehicle.track.R‘

• Superelevation u: ‘#Vehicle.track.u‘

• Automatically calculated v: ‘#Vehicle.aq:v(aq,R,u)‘


• Automatically calculated R: ‘#Vehicle.aq:R(aq,v,u)‘ This is an example for a parameter variation steering file:

1 ! Example steering file for static equilibrium in curves 10 ! file format = 10: input parameters 4 ! number of variations 3 ! number of input parameters '#Vehicle.v_km/h' '#Vehicle.track.R' '#Vehicle.track.u' ! varied parameters 60 500 0.1 ! R=500 m, u=0.1 m 80 500 0.1 100 500 0.1 60 1000 0.1 ! R=1000 m 80 1000 0.1 100 1000 0.1 60 1000 0.15 ! u=0.15 m 80 1000 0.15 100 1000 0.15 …

Once the steering file is finished, it has to be selected in the parameter variation con-figuration GUI. The number of variations is automatically read from the file and cannot be changed:


3.2.3 Ensuring equal travelled distances with different speeds

Usually, when the vehicle speed is varied, the vehicle travels different distances in each parameter variation calculation. Often this is undesirable. In a parameter variation there are two different methods to ensure that the travelled distance is the same, re-gardless of the vehicle speed.

1. Set the .num6 file parameters ‘tend‘ (simulation end time) and ‘ntout‘ (number of output time steps) in the parameter variation steering file according to the vehicle speed: 1 ! Number of comment lines ! Comment line 10 ! Steering file format type 4 ! Number of variations 4 ! Number of varied parameters ’#Vehicle.track.R’ ’#Vehicle.v_m/s’ ’tend’ ’ntout’ 300 12.5 60 601 300 25 30 301 600 12.5 60 601 600 25 30 301

2. Use the force element 234 “Stop Integration” (available with SIMPACK 8.616) to

stop the integrator when the vehicle has reached a given position in the track. See the SIMPACK documentation to force element 234 and the following exam-ple, which stops the integration when the first wheelset has reached a position in the track of 500 m. The configuration is shown in the following picture:


3.3 Calculation parameters The curving analysis should be executed with the SIMPACK parameter variation module “Static Equilibrium” rather than with the normal time integration. This parameter varia-tion uses the settings from the manual static equilibrium module, see section 2.2.

3.4 Result outputs Unlike the manual time integration, the parameter variation yields only outputs that have been previously defined in the parameter variation configuration. Otherwise even short calculations could produce huge amounts of output data. The type of outputs is generally the same as it is in a manual calculation – see section 2.3 for their definition. They are configured on the notepage “Results”:

(The output definitions may be re-used in different models by means of the “Tem-plates” and the “Global MBS Elements” feature of SIMPACK.)

3.5 Plotting the results The results can be plotted in the parameter variation plots “Static Equilibrium”. The results may there also be exported to a .csv file (for Excel). Another possibility is to plot the results in the General Plot. Ensure to select the appro-priate parameter variation case for the plot.

4 Check list for set up

Here is a short check list for setting up a quasi-static curving analysis with parameter variation and steering files. Please refer to the above sections for further explanations.

StepStepStepStep DescriptionDescriptionDescriptionDescription RemarksRemarksRemarksRemarks SectionSectionSectionSection


StepStepStepStep DescriptionDescriptionDescriptionDescription RemarksRemarksRemarksRemarks SectionSectionSectionSection

1 Define a track Standard track “Curve entry”. Appropriate straight track and transition curve/superelevation ramp. S-shaped ramp. Remove all excitations.

2.1.1

2 Set constant parameters Parameters (R, u, v) that will not be varied in the parameter variation.

2.1.1, 2.1.2

3 Define additional meas-urements

Track Frame/Camera markers, expressions and dummy force elements, additional sensors etc.

2.3

4 Configure parameter variation

Close Model Setup before because parameter varia-tion configuration is written to .sys file.

Create parameter variation steering file in model specific data base and select it in configuration GUI.

Configure further parameters, if necessary.

3.2.2

5 Configure results Use filters if necessary 2.3, 3.4

6 Configure static equilib-rium calculation

Method time integration. Appropriate simulation time and inertia damping.

2.2

7 Start calculation “ParVariation” – “Perform – Static Equilibrium” – “Calculation + Measurements”.

8 Plot results Parameter variation plots or General Plot.

Define additional meas-urements, if necessary

Sensors, expressions with dummy force elements, markers etc.

Start measurements “ParVariation” – “Perform – Static Equilibrium” – “Measurements only”.

If there are new active elements (providing forces or torques etc.), the calculation must be restarted in-stead.

Plot results


5 Example model

The example model “Example_StaticEquilibriumInCurves” is a standard railway vehicle with pre-configured static equilibrium calculation using the parameter variation. It uses a steering file. Expressions have been defined for the wear indices and the roll coeffi-cient. The model uses the “Stop Integration” element to ensure equal travelled dis-tances.

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