Calculation of non-linear critical speed

Non-linear critical speed can be defined and calculated in many different ways. The most common way of calculating critical speed is probably according to the instability criterion, defined in in EN14363. However, calculating the the instability criterion according to EN14363 requires many simulations, and is a bit cumbersome.

To quickly calculate the critical speed with only one simulation. The following method can be used:

Copy the master file tang_ideal.tsimf to a new case e.g. crit_ideal.tsimf
cp -ip runf/tang_ideal.tsimf runf/crit_ideal.tsimf
Set the speed of the moving coordinate system with a formula:
```
 func const vkmh_deacc=  5                          # Deacceleration when calculating critical speed
 func operp vkmh= vkmh_init - vkmh_deacc * time     # Vary vkmh linearily
```
The initial speed must be chosen differently depending on type of vehicle.
For high speed passenger trains vkmh_init can be chosen to ~400 km/h.
For a freight waggon the vkmh_init can be chosen to ~200 km/h
Simulation time shall be chosen so the final speed is below the critical speed of the vehicle. E.g. the critical speed is supposed to be in the range vkmh_init and vkmh_final. The simulation time can be set equal to:
time= (vkmh_init - vkmh_final) / vkmh_deacc

Track and sleeper irregularities.
For bogies which only have viscous dampers, the stability calculations can be made without track irregularities:

func const Track_Gauge= 1435.    # Gauge to be used in the intpl_track_irr-commands
func const Xtrac_start=    0.
func const Xtrac_stop= 30000.
func intpl_track_irr2 Xtrac_start Xtrac_stop Ideal_track Track_Gauge

For bogies which have friction dampers some disturbances might be necessary to keep the friction surfaces moving. However, the size of the track irregularities should preferable be regulated with respect on the vehicle speed.

if_then Vo .gt. 80
 func operp YMtrac= 0.25e-3 * ( Vo - 80 ) / (vkmh_init-80) + 1e-3 * ( vkmh_init - Vo ) / (vkmh_init-80)
 func operp ZMtrac= 0.25e-3 * ( Vo - 80 ) / (vkmh_init-80) + 1e-3 * ( vkmh_init - Vo ) / (vkmh_init-80)
else
 func copy  YMtrac= 1e-3
 func copy  ZMtrac= 1e-3
endif

For friction damped bogies, also disturbances from the sleeper can be added. E.g.:

  func operp kzrt_111 =  230e6 + 27e6 * sin( 2*pi/0.65 * axl_111.X )
  func operp kzrt_112 =  230e6 + 27e6 * sin( 2*pi/0.65 * axl_112.X )

Initial values to start the hunting motions and longitudinal retardation forces.
Set initial values on the yaw angle of the bogie(s), and apply a longitudinal force on all masses in the model, so the vehicle will retardate with a rate of vkmh_deacc [km/h/s].

substruct crit_excit           [ # $1= bog_no                                                         
 initval set_var  bog_$1.p = .015                                                                     
 initval set_var  axl_$11.y= .015*aba_$1                                                              
 initval set_var  axl_$12.y=-.015*aba_$1                                                              
 initval set_var  axl_$11.p= .015                                                                     
 initval set_var  axl_$12.p= .015                                                                     
 force rel_lsys1  deacc_bog_$1   bog_$1  0 0 -hbcg_$1  -mb_$1*vkmh_deacc/3.6  0. 0.  0. 0. 0.                     # Deacceleration vkmh_deacc in [km/h/s]
 force rel_lsys1  deacc_axl_$11  axl_$11 0 0 -ro_$11  -(ma_$11+Jka_$11/ro_$11^2)*vkmh_deacc/3.6 0. 0.  0. 0. 0.   # as external forces
 force rel_lsys1  deacc_axl_$12  axl_$12 0 0 -ro_$12  -(ma_$12+Jka_$12/ro_$12^2)*vkmh_deacc/3.6 0. 0.  0. 0. 0.
]
in_substruct crit_excit   [ 11 ]   # $1= bog_no                             
in_substruct crit_excit   [ 12 ]                           

force rel_lsys1  deacc_car_1    car_1   0 0 -hccg_1   -mc_1*vkmh_deacc/3.6   0. 0.  0. 0. 0.      # Deacceleration vkmh_deacc in [km/h/s]

The forces on the masses depends on the weight of the mass, the forces shall be applied in the center of gravity for the masses.

Wheel/rail friction mu_ and Kalker factor mulf_
To find bogie instability the wheel/rail contact shall be in dry conditions, and to be on the safe side, mulf_ shall be set equal to "1".
```
mu_  = 0.6
mulf_= 1.0
```
Use wheel/rail-coupling func wr_coupl_pr3. Coupling func wr_coupl_pr3 has a non-linear creepage - creep force relation all the way down to speed zero. Its non-linear characteristics makes the critical speed higher compared to wheel/rail-couplings that are linear for small creepages.
Evaluate the critical speed with script crit_bobo.sh.
Script crit_bobo.sh, crit_bobo_2.5.sh and print_vkmh_bobo.mplotf can be found in tutorial intro_tutor_3_bobo_pe3.

With the above descibed method the results may change a little depending on the deacceleration rate vkmh_deacc. A large value in vkmh_deacc will normally lead to a lower critical speed. The critical speed for vkmh_deacc= 0. can be calculated in the following way:

Make a critical speed calculation for vkmh_deacc= 2
Make a critical speed calculation for vkmh_deacc= 1
Extrapolate the critical speed down to vkmh_deacc= 0 with the following formula: Vcrit_0= Vcrit_1 - (Vcrit_2-Vcrit_1)/(2-1)