Simulation Tool

Development of a Mathematical Model to Design the Control Strategy of a Full Scale Roller-Rig

17.3 Simulation Tool

In order to design and test the control logic of the roller-rig, a simulation tool was developed based on co-simulation technique.

Figure17.2shows the scheme of the roller-rig and the flow of information between its components. A locomotive/coach is placed on the rollers, which are driven by independent electric motors. Roller-rig and locomotive/coach are coupled by means of contact forces between wheels and rollers. The braking system and/or the traction system (only present in the case of a locomotive) provides the vehicle with the drive/brake torques (um,dem) demanded by the simulated test, taking into account anti-slip and anti-skid devices intervention.

Resistant forces acting on the vehicle during the considered test are reproduced by applying proper torques to the rollers (udem). Coherence between vehicle drive torques, roller resistant torques and roller angular speeds is given by the roller-rig control system. It consists of the regulators controlling speeds/torques of the motors driving the rollers and of a Real Time (RT) simulator of the vehicle longitudinal dynamics. The basic idea behind the control strategy for the roller-rig is to avoid macro-slippage between wheels and rollers in any working condition, even when low adherence tests are simulated [10].

This can be achieved by properly controlling the motors driving the rollers in order to reproduce, in the contact area, the same tangential forces existing between the wheel and the rail during track testing, rather than the same relative velocity.

Thus, torques applied to the rollers and their angular speed must be determined by the RT vehicle model according to the simulated test.

The developed simulation tool mimics the real test bench, based on co-simulation technique. The roller-rig including the railway vehicle (comprising the brake/traction system) and the rollersets (comprising rollers, motors and their regulators) has been modelled in MatLab/Simulink environment and it has been interfaced with the control logic actually present in the test bench (developed in Labview environment), so to test its effectiveness.

In the following the elements of the simulation tool are described, making reference to case in which the tested vehicle is a locomotive.

17.3.1 Roller-Rig Mathematical Model (MatLab/Simulink)

The mathematical model [11] includes a multi-body representation of the locomotive (Fig.17.3) and a lumped parameters schematization of the driveline and of the roller-rig (Fig.17.4). Concerning the locomotive, carbody and bogies are modelled as rigid bodies, while wheelsets are represented as deformable elastic bodies, using a modal superposition approach in order to take into account their torsional deformability [12]. Carbody, bogies and wheelsets are linked one to the other by means of elastic and damping elements reproducing primary and secondary suspensions. The simulation code accounts for the motion of the locomotive in vertical, lateral and longitudinal direction. The d.o.f.s of the system are shown in Fig.17.3. The vehicle

Test bench control logic

Rollersets Rollers Motors

Motor regulators

Railway vehicle

Wheel-roller contact forces (Traction) / Braking

system u_m,dem

u_dem

u_m

RT vehicle model Control logic

Roller rig (MatLab/Simulink)

Control logic (LabView)

Fig. 17.2 Block diagram of the roller-rig

V

s

₁

s

_a

s

₂

s

_c

s

₃

s

_p

s

₄

b

_c

b

_p

b

_a

z₁

b

₁

z₂ z_a

x_a

b

₂

b

₃

b

4

z_c x_c

x_p

z₁ z_a z₂

y_c z_c y_a

y₂ y₁

y_p

y₃ y₄

z₃ z_p z₄ z_p

z₃ z₄

x_c

r

_c

y_c

y_a

r

_a

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y₁

Fig. 17.3 Rail vehicle model

(a) (b)

z_r

yr

x_r

ml

, u

_{l mr}

, u

_r

k

_m

k

_c

k

_m

l r

Motor Rollers Motor

j

j x

x

y d

x

Fig. 17.4 (a) Roller. (b) Roller-motor connection

equations of motion are written with respect to a moving reference system, traveling along the ideal track centreline, which is a straight line in tangent track.

On the other hand, a lumped parameters schematization has been adopted for the transmission system of the locomotive and for each roller and motor constituting the test bench. In particular, for what concerns the driveline, the torsional flexibility of the connection between the driving electric motors and the wheelsets has been considered and backlash has been included accounting for the clearance between the teeth of the transmission [13].

For each pair of rollers and motors (Fig.17.4), 14 d.o.f.s have been considered: 6 d.o.f.s for each roller (displacementsxr, yr,zrand rotationsı,, ) and the rotation®mof each motor. As it can be seen from Fig.17.4, compliance of roller supports

Fig. 17.5 Kinematics of lozenging. (a) Top view, (b) lateral view

Fig. 17.6 Longitudinal friction coefficient vs. longitudinal and lateral creepage

and torsional flexibility of motor shaft and rollers’ connecting shaft have been introduced in order to take into account the influence of the roller-rig dynamics on the tested vehicle. Supports compliance was experimentally identified.

Separate subsets of equations can be respectively written for the locomotive, the driveline and the roller-rig. Wheel-roller contact forces are then introduced as coupling terms between the sets of equation referring to the vehicle and the roller-rig.

The equations of motion can thus be expressed as follows:

8ˆ ˆ<

ˆˆ :

ŒMt xR_tCŒCt xP_tCŒKt x_tDQ_tr

xP_t;x_t;xP_r;x_r;u_dem;t CQ_tm

xP_t;x_t;xP_m;x_m;u_m;t ŒMr Rx_rCŒCr xP_rCŒKr x_rDQ_r

xP_t;x_t;xP_r;x_r;u_dem;t ŒMm xR_mCŒCm xP_mCŒKm x_mDQ_m

P

x_t;x_t;xP_m;x_m;u_m;dem;t (17.1) where xt,xm andxr are the vectors containing the coordinates of the locomotive, of the driveline and of the roller-rig respectively,Qtr andQr represent the Lagrangian components of wheel-rollers contact forces (including components due to wheelset decrowning and lozenging, see Figs.17.5and17.7) with respect of vehicle and rollers d.o.f.s andQtmandQm

include the Lagrangian components of the drive torques (input to the model,um,dem) provided by the locomotive traction system with respect of vehicle and driveline d.o.f.s. Finally, vectorudem collects the torques computed by the test bench control logic.

The contact model is based on a geometrical analysis of the measured wheel and roller profiles. Contact parameters such as rolling radius, contact angle and decrowning angle (angle˛, Fig.17.5) are reported in table form, as functions of wheel- rail lateral and longitudinal displacements. It must be in fact noted that when the wheelset yaws on a rollerset, the wheels move off the crown of the rollers (Fig.17.5and17.7). As a consequence, the mass of the centre of gravity of the wheelset moves downward and the wheelset lozenges, as shown in Fig.17.5. Then the point of contact at each wheel on the roller moves away from the top of the rollers and the wheelset is said to decrown, as depicted in Fig.17.7.

Fig. 17.7 Forces due to decrowning

Wheel-roller contact force (Fc,i) is assumed to be a non-linear function of the vertical load (FN,i) and of the longitudinal and lateral creepages ("L,i,"T,i) according to (Fig.17.6):

Fc;iD ."tot;i/FN;iI FL;iDFc;i"L;i

"tot;iI FT;iDFc;i"T;i

"tot;i

."tot;i/D _s ^"tot;i

1C_"tot

;i max

₂I "tot;iDq

"²L;iC"²T;i

(17.2)

Lateral and longitudinal creepages are then computed as:

8<

:

"LiD .^VL;i PˇiRi.^Yrel;i//^CPiRd j^PiRdj

"Ti D j^VPiTR^;dⁱj

(17.3)

whereVL,iandVT,iare respectively the longitudinal and lateral velocity of thei-th wheel centre, whileRiandRd.are the i-th wheel and roller radii. As it can be seen, both the locomotive and the roller-rig dynamics are included in the definition of contact forces.

Motors driving the rollers have been introduced into the model through their characteristic curve torque vs. angular speed, while their dynamics has been modelled as a first order time lag, so to account for motors’ bandwidth, nonlinearity and torque saturation.

17.3.2 Control Logic (Labview)

The control strategy implemented in the test bench is based on the algorithm proposed in [10]. As already mentioned, the basic idea of the control logic is to regulate the driving torque of the rollers so to reproduce on the rig the same angular speeds the wheels would have on the rail, but avoiding avoid macro-slippage between wheels and rollers. On the purpose, a RT model simulating the longitudinal dynamics of the locomotive is needed.

According to the proposed algorithm, torque demanded by the control logic to thei-th roller is given by:

udem;iD JyrR JysRd

bCi JyrR²

JysR²_d C1

Rdf_l^._;i^s/Ch kp

ˇPi Pˇ^._i^s/ Cki

ˇiˇ_i^.s/i

(17.4) In Eq. (17.4),RandRdare respectively wheel and roller radii,JyrandJysare respectively the roller and the wheel moments of inertia,ˇPiis the angular speed of thei-th wheel, whileˇPi^.s/andf_l^._;i^s/are respectively the angular speed and the longitudinal force of thei-th wheel predicted by the RT model.

The estimate of the driving/braking torque applied by the locomotive to thei-th wheel is given by:

bCiD R Rd

umeas;iC

Jyr

R² R²_d CJys

ˇRi (17.5)

beingˇRithe angular acceleration of thei-th wheel (which is estimated based on the measurements of the angular speed of the roller and of the axle) andumeas,ibeing the torque measured by the torquemeter.

The longitudinal force of thei-th wheelf_l^._;^s_i^/is calculated by the RT vehicle model simulating the longitudinal dynamics of the locomotive during the considered maneuver:

8<

:

MVP^.s/DP₈

iD1f_l^._;^s_i^/CP.t/

JysˇRi^.s/DbCif_l^._;^s_i^/R

f_l^._;^s_i^/D

"^.i^s/

F^._N^s_;^/_iI "^.i^s/D V^.s/ Pˇ^._i^s/R

V^.s/ (17.6)

whereM is global mass of the locomotive,V^(s)is the longitudinal speed of the RT vehicle model cog,P(t) includes the resistant force applied to the vehicle (aerodynamic resistance, rail slope, effect of other coaches attached to the locomotive, etc.),ˇRi^.s/and"^.i^s/are respectively the angular acceleration and the creepage of thei-th wheel of the RT simulator. Note that with the apex^(s)the quantities calculated by the RT model are indicated.

It is to point out that, in Eq. (17.4), the torque demanded to thei-th roller includes a PI regulator term based on the error between the angular speed of the wheel of the locomotive and of the RT simulator of the control logic, so to avoid differences between the simulated and the performed maneuver.